What’s Wrong with “Multiplication Is Repeated Addition”? (2008)(denisegaskins.com)
denisegaskins.com
What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
https://denisegaskins.com/2008/07/28/whats-wrong-with-repeated-addition/
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> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously
I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides to teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct...", and that a different approach is possible. One can argue about the pros and cons of each approach, but I don't think it is helpful to just dismiss one side of the argument as "pedantic falsehoods".
I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides to teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct...", and that a different approach is possible. One can argue about the pros and cons of each approach, but I don't think it is helpful to just dismiss one side of the argument as "pedantic falsehoods".
The general concept is known as https://en.wikipedia.org/wiki/Lie-to-children and I think you'd be hard pressed to teach/learn/apply anything of significance without this "well actually it's more complicated than that" approach and recognizing how far you need to go with it to aptly do some task.
There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.
The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there, even when explaining, say, Kubernetes to adults! It can be a confidence crusher.
There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to the "who decides what you're smart enough to hear about now" question in its own way. Or "oh the teacher doesn't actually know what the difference is, or why this isn't 'true' 100%."
Despite the cutesy name, I don't think omission of detail is the same as lying. It's often impossible to tell 100% the truth. You probably don't even know it yourself!
There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to the "who decides what you're smart enough to hear about now" question in its own way. Or "oh the teacher doesn't actually know what the difference is, or why this isn't 'true' 100%."
Despite the cutesy name, I don't think omission of detail is the same as lying. It's often impossible to tell 100% the truth. You probably don't even know it yourself!
> It can be a confidence crusher.
If confidence gets priority over truth in explanations, then society will churn out people who are confidently wrong. This is a bad idea even if everybody does that and even if it is the traditional approach.
If people were honest that they don't know something then the world at large would be a lot nicer to live in.
If confidence gets priority over truth in explanations, then society will churn out people who are confidently wrong. This is a bad idea even if everybody does that and even if it is the traditional approach.
If people were honest that they don't know something then the world at large would be a lot nicer to live in.
I think this is backwards. The public at large are pretty honest that they don't know math. The world would be a much better place if they were able to do practical algebra, even if the higher truths of abstract mathematics never enter their minds.
I really don't see people being confidently wrong about abstract mathematics as an issue. I certainly don't know the rigorous definitions of an integral, but I can apply the concepts of calculus to everyday life perfectly fine. People who care about rigorous math can do rigorous math, and I'm glad if they teach me an intuitive understanding that lets me live a happy, productive life.
Personally, I find layers of abstraction necessary for learning. Maybe there are people who don't, I suspect they would have to be prodigies though. Tell me how to add fractions practically, then teach me the principles when I need to know them. Framing that as a lie seems wrong to me, I'd call it "bounded knowledge".
I really don't see people being confidently wrong about abstract mathematics as an issue. I certainly don't know the rigorous definitions of an integral, but I can apply the concepts of calculus to everyday life perfectly fine. People who care about rigorous math can do rigorous math, and I'm glad if they teach me an intuitive understanding that lets me live a happy, productive life.
Personally, I find layers of abstraction necessary for learning. Maybe there are people who don't, I suspect they would have to be prodigies though. Tell me how to add fractions practically, then teach me the principles when I need to know them. Framing that as a lie seems wrong to me, I'd call it "bounded knowledge".
The confidence is in the ability to learn a topic, not confidence about the knowledge.
It seems to me that the way to build people's confidence in their ability to learn is to allow them to learn. That means not making oversimplified statements because you don't think they're "ready" for more details. It means giving them the details, and letting them decide when they've had enough for now. (Of course one's time, say in a classroom, will be limited, so at some point one has to say "we don't have time to go into this further in class now"--but that's still letting them know that there are more details, and they can dig into them further on their own.)
That doesn’t work for lots of people (including myself). If you’re spending a lot of time on details that don’t actually matter for the topic at hand, it takes away mental bandwidth that should be spent on the topic being covered.
Refusing to answer curious students questions isn't helpful. Giving them a hint and telling them you'll get back to it works fine. They're smart enough to understand they aren't having to wait because they're incapable of understanding, but because other things need teaching first. I know I did. The good teachers were encouraging of the curiosity while back-burnering something, others responded less patiently with things like "we're not there yet!", which made them seem like bad teachers to me, even back to elementary.
> The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there
The qualifications are there; that's just a fact. Being told about them, or at least about their existence if not every detail of them, up front seems better to me than finding out about them later on when your mental model is solidified around the simplified version that you then find out doesn't always work.
> I don't think omission of detail is the same as lying.
Saying "multiplication is repeated addition", without qualification and without any caveats, is not "omission of detail". It's a false, categorical statement, i.e., lying.
As for where the line is where you stop giving details, obviously that will depend on the circumstances. A teacher who says "we don't have time to talk about that during class today, but yes, there is much more detail here that you can look into on your own" is not lying and is not saying the child is "not smart enough" to take in all the detail now. (Bonus points if the teacher says "see me after class and I'll give you some pointers on where to go for more information".) A parent who says something similar because they have to get dinner ready and the child needs to do the rest of their homework before bed is also not lying and not saying the child is "not smart enough". Limitations of time are a fact of life, and children need to deal with it just like the rest of us.
A teacher who just says "we're not talking about that", or who doesn't even know about the qualifications, or who gets snippy when a child asks a natural question, is obviously not doing the child any good; but that is because of the teacher fixating on a simplified model and treating it as "the Truth", so doing more of that won't fix it.
The qualifications are there; that's just a fact. Being told about them, or at least about their existence if not every detail of them, up front seems better to me than finding out about them later on when your mental model is solidified around the simplified version that you then find out doesn't always work.
> I don't think omission of detail is the same as lying.
Saying "multiplication is repeated addition", without qualification and without any caveats, is not "omission of detail". It's a false, categorical statement, i.e., lying.
As for where the line is where you stop giving details, obviously that will depend on the circumstances. A teacher who says "we don't have time to talk about that during class today, but yes, there is much more detail here that you can look into on your own" is not lying and is not saying the child is "not smart enough" to take in all the detail now. (Bonus points if the teacher says "see me after class and I'll give you some pointers on where to go for more information".) A parent who says something similar because they have to get dinner ready and the child needs to do the rest of their homework before bed is also not lying and not saying the child is "not smart enough". Limitations of time are a fact of life, and children need to deal with it just like the rest of us.
A teacher who just says "we're not talking about that", or who doesn't even know about the qualifications, or who gets snippy when a child asks a natural question, is obviously not doing the child any good; but that is because of the teacher fixating on a simplified model and treating it as "the Truth", so doing more of that won't fix it.
If you don't lie to children you confuse and discourage them. I know because my own internal desire for precision often is to the detriment of my pedagogy.
Let's not take the word "lie" out of its context and then get hung up on it. We're not actually lying to children by simplifying the explanation down to what they can comprehend with the tools they have at that stage.
> Let's not take the word "lie" out of its context
I'm not the one that used that word. If people are going to reference a concept that's been published in a book and uses that word, they should own it and be prepared to justify it. If they can't do that, they should not use the word in the first place.
To be clear, I'm not saying you are one of those people. But the poster I was responding to is.
I'm not the one that used that word. If people are going to reference a concept that's been published in a book and uses that word, they should own it and be prepared to justify it. If they can't do that, they should not use the word in the first place.
To be clear, I'm not saying you are one of those people. But the poster I was responding to is.
I'm not saying you shouldn't use the word. I'm saying you shouldn't ignore its very specific and specialized context. Taking things out of their context often leads to problems, just like in this case.
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But you can define multiplication based on abstracting repeated addition. That how we did it my analysis class, although I forget some of the details. So if you say multiplication is repeated addition, it really isn't a lie.
But did your analysis class define addition on the naturals? Then that's an operator for the naturals. And then for integers, rationals, etc.
Yep and how to construct, the naturals from set theory, the integers from the naturals, the rationals from the integers and the reals from the rational. I've forgotten the details at this point, but I remember the conclusions.
They don't need this information when learning that 3 groups of 5 fruit are 15 fruits. It's not in the natural order of understanding principles.
> They don't need this information when learning that 3 groups of 5 fruit are 15 fruits.
They also don't need the "information" that multiplication is repeated addition, period, full stop. But that's what others in this discussion appear to be trying to argue for.
They also don't need the "information" that multiplication is repeated addition, period, full stop. But that's what others in this discussion appear to be trying to argue for.
I'm really not sure how you could arrive at an understanding of multiplication without first going through the exercise of adding up X groups of Y to yield X * Y. We don't need to declare to children that the precise definition of multiplication is repeated addition. If that's what you're arguing, we agree. If it's that 3*5 shouldn't be explained as 5+5+5, I'm not sure how kids would be expected to learn the subject. We can't transactionally insert comprehensive math knowledge, so it's got to be incomplete along the way.
> I'm really not sure how you could arrive at an understanding of multiplication without first going through the exercise of adding up X groups of Y to yield X Y.*
Arriving at an understanding of one particular special case of multiplication might indeed be a necessary step towards arriving at an understanding of multiplication in general, but that doesn't mean the two are the same.
Arriving at an understanding of one particular special case of multiplication might indeed be a necessary step towards arriving at an understanding of multiplication in general, but that doesn't mean the two are the same.
Well, that would be true if all children were as smart and analytically adept as you.
It looks like splitting hairs, but I also think it makes a difference to state upfront that it’s an oversimplification.
Kids actually understand the point, and they can decide if they care enough to ask more questions or if it’s good enough for them.
We faced that when teaching divisions. Saying upfront we’d explain falsehoods for the sake of simplicity helped set aside the more difficult questions (infinity, etc.) that came right after. We just say it’s the complicated parts and move on.
The best aspect is they are more receptive to have their mental model broken afterwards, instead of clinging to what you explained as fully true.
Kids actually understand the point, and they can decide if they care enough to ask more questions or if it’s good enough for them.
We faced that when teaching divisions. Saying upfront we’d explain falsehoods for the sake of simplicity helped set aside the more difficult questions (infinity, etc.) that came right after. We just say it’s the complicated parts and move on.
The best aspect is they are more receptive to have their mental model broken afterwards, instead of clinging to what you explained as fully true.
I think this viewpoint is pernicious. A child doesn't have to be "smart" in order to deserve being told the truth.
Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)
Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)
An abstraction/simplification/shorthand is not a lie. People don't say "multiplication is repeated addition" because they are trying to hide the truth for some selfish reason. It's a pedagogical strategy to help people learn a new abstraction by analogy to an old one. These kind of crutches are a necessity, you can't introduce all the complexity of the world to someone all at once. This applies to every subject - science, history, writing. Simple notions and shorthands are introduced first, and complexity and nuance added on later.
Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?
In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.
Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?
In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.
> An abstraction/simplification/shorthand is not a lie.
"Multiplication is repeated addition" is not "an abstraction/simplification/shorthand". Doing that for multiplication would be saying something like "multiplication is a distinct primitive operation, but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now." Is that really so hard?
> Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?
If you tell them everything you just said, no, you're not lying. But if you just tell them "always protect your queen", without explaining anything about why and without saying that there are some advanced situations where you might break this rule, yes, you're lying. It only takes a couple of sentences to add that extra information. Again, is that really so hard?
> In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.
Um, what? We're assuming they already know about addition of whole numbers. So it is simple to tell them "this addition thing that you learned, that's an example of an operation", and "those whole number thingies that you learned how to add, those are numbers". Once more, is that really so hard?
"Multiplication is repeated addition" is not "an abstraction/simplification/shorthand". Doing that for multiplication would be saying something like "multiplication is a distinct primitive operation, but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now." Is that really so hard?
> Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?
If you tell them everything you just said, no, you're not lying. But if you just tell them "always protect your queen", without explaining anything about why and without saying that there are some advanced situations where you might break this rule, yes, you're lying. It only takes a couple of sentences to add that extra information. Again, is that really so hard?
> In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.
Um, what? We're assuming they already know about addition of whole numbers. So it is simple to tell them "this addition thing that you learned, that's an example of an operation", and "those whole number thingies that you learned how to add, those are numbers". Once more, is that really so hard?
> but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now
I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary.
People may prefer to consider the extended definition more "real", but I think the argument for going the other way is that usually the original limited form of something is more likely agreed upon by most people, whereas the generalization can be done in multiple ways which may owe something to history and culture, or context.
I feel like math is fundamentally different than physics, where the more advanced theory is objectively closer to correct. With math, it's more of an arbitrary aesthetic or social judgment. Nothing ever stops you from generalizing anything even more than anyone did yet, right?
I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary.
People may prefer to consider the extended definition more "real", but I think the argument for going the other way is that usually the original limited form of something is more likely agreed upon by most people, whereas the generalization can be done in multiple ways which may owe something to history and culture, or context.
I feel like math is fundamentally different than physics, where the more advanced theory is objectively closer to correct. With math, it's more of an arbitrary aesthetic or social judgment. Nothing ever stops you from generalizing anything even more than anyone did yet, right?
> I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary.
Axiomatically, I think it can go either way. But one still has to recognize, as you do, that there are more cases than just the whole number case, and that what works for the whole number case might not work for other cases.
> the generalization can be done in multiple ways
There are certainly cases of this, but I don't think the case under discussion is one of them. There is only one generalization of the whole numbers under discussion here, the one from whole numbers to rationals to reals (and on to complex numbers if you want to take it that far, and still further on to matrices for some people in this discussion). There aren't multiple ways to do that: the rationals, reals, and complex numbers are all unique sets.
Axiomatically, I think it can go either way. But one still has to recognize, as you do, that there are more cases than just the whole number case, and that what works for the whole number case might not work for other cases.
> the generalization can be done in multiple ways
There are certainly cases of this, but I don't think the case under discussion is one of them. There is only one generalization of the whole numbers under discussion here, the one from whole numbers to rationals to reals (and on to complex numbers if you want to take it that far, and still further on to matrices for some people in this discussion). There aren't multiple ways to do that: the rationals, reals, and complex numbers are all unique sets.
Along with what allturtles said, I don't think it's right to call it a lie, which, to me, implies moving someone's model away from the truth (on the basis of them trusting you to convey the correct one). An oversimplified, "wrong" model doesn't do that; it moves them from ignorance toward the correct model (i.e. increases their prediction accuracy).
And yes, even when I'm in the learner's shoes, I prefer that a teacher start with an approximate model, and then refine it as they go further. Starting with the full thing is barely comprehensible.
And yes, even when I'm in the learner's shoes, I prefer that a teacher start with an approximate model, and then refine it as they go further. Starting with the full thing is barely comprehensible.
> I don't think it's right to call it a lie
It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later".
It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later".
Yeah, and that's still too much inferential distance to feed to an elementary schooler.
You appear to have a much more impoverished view of kids and their ability to learn than I do. My experience (not to mention my memory of how I was myself as a kid) is that kids grasp the fact that there can be more to a subject than adults are able to teach them at a particular time and place, so they're ok with adults honestly admitting that. But they do not like adults telling them categorical statements that later turn out to be wrong.
You’re view of children seems to be more based on your memory of high school. That’s a lot different than your behavior as a 6 year old when these simple primitives are being taught.
Little kids don’t care about nuance when they’re still having difficulties with carries in addition. Your ideal world where we first explain children that base10 isn’t the only way to represent numbers and whatever other caveats simply doesn’t exist.
They don’t have the knowledge required yet to even understand the scenarios when “exceptions to the rule” apply.
Little kids don’t care about nuance when they’re still having difficulties with carries in addition. Your ideal world where we first explain children that base10 isn’t the only way to represent numbers and whatever other caveats simply doesn’t exist.
They don’t have the knowledge required yet to even understand the scenarios when “exceptions to the rule” apply.
There's a difference between prefacing a course with "oh hey these are simplifications that you'll improve upon in higher grades" vs loading down literally every claim with that long chain of caveats.
> loading down literally every claim with that long chain of caveats.
I have never proposed doing the latter, so you are attacking a straw man. Once it's understood that you're teaching a simplified, approximate model, you don't have to repeat in every sentence that you're teaching a simplified, approximate model. You just have to not say it's "the Truth", without approximation and without qualification.
I have never proposed doing the latter, so you are attacking a straw man. Once it's understood that you're teaching a simplified, approximate model, you don't have to repeat in every sentence that you're teaching a simplified, approximate model. You just have to not say it's "the Truth", without approximation and without qualification.
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How would you teach physics without lying to children?
Um, by not lying to them?
Do you have a specific example of where you think you need to lie to teach physics?
Do you have a specific example of where you think you need to lie to teach physics?
> Do you have a specific example of where you think you need to lie to teach physics?
If we are defining "lie" to mean "simplifying the model to allow students to understand what's going on" (which is what we're talking about) then basically every field of physics required some simplifications in order to teach.
For instance, it would be "lying" to teach Newtonian dynamics and Newton's theory of gravity at all without explaining relativity -- Newton's theory of gravity is "wrong" after all. It would be "lying" to teach electromagnetism without first explaining quantum mechanics. However it would be impossible to teach anything to a high-school or undergraduate student if you first had to explain everything else in order to teach F=ma or Gauss's law.
I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to or that they're being told they're "not smart enough to understand the full theory". Physics was developed over many thousands of years, it's not really reasonable to expect a student to be able to learn it in one shot. Physics courses are often structured so that you learn concepts in historical order so that you learn how concepts were developed and what motivated future discoveries -- that is also a very useful thing to learn in addition to the actual physics.
I believe the same applies to mathematics education.
Source: I majored in physics. Every year we had an electromagnetism course and we learned that the model we had learned last year was too simplified to work with. Same goes for thermodynamics.
If we are defining "lie" to mean "simplifying the model to allow students to understand what's going on" (which is what we're talking about) then basically every field of physics required some simplifications in order to teach.
For instance, it would be "lying" to teach Newtonian dynamics and Newton's theory of gravity at all without explaining relativity -- Newton's theory of gravity is "wrong" after all. It would be "lying" to teach electromagnetism without first explaining quantum mechanics. However it would be impossible to teach anything to a high-school or undergraduate student if you first had to explain everything else in order to teach F=ma or Gauss's law.
I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to or that they're being told they're "not smart enough to understand the full theory". Physics was developed over many thousands of years, it's not really reasonable to expect a student to be able to learn it in one shot. Physics courses are often structured so that you learn concepts in historical order so that you learn how concepts were developed and what motivated future discoveries -- that is also a very useful thing to learn in addition to the actual physics.
I believe the same applies to mathematics education.
Source: I majored in physics. Every year we had an electromagnetism course and we learned that the model we had learned last year was too simplified to work with. Same goes for thermodynamics.
> If we are defining "lie" to mean "simplifying the model to allow students to understand what's going on" (which is what we're talking about)
No, it isn't what we're talking about. We're talking about making categorical statements that are false. Telling students that you're teaching them a simplified model and there are complications that will be added back later is not lying. But telling them categorically that the simplified model is true and not even talking about the fact that it's a simplified model and there are complications being left out, is lying. And the latter is what the posters I've been responding to are attempting to defend.
> I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to
I have never said that is necessary. Of course it's not. Once you've explained at the outset that you're teaching them a simplified model and there are complications being left out, obviously you're not going to repeat it with every single sentence. Any more than the people who are advocating telling students categorically "multiplication is repeated addition" with no mention of all the complexities lurking underneath are advocating repeating that with every single sentence.
> or that they're being told they're "not smart enough to understand the full theory"
I have never said they should be told that. Teaching the simplified model could be due to nothing more than limited teacher time and knowledge. There's nothing wrong with the teacher honestly telling the students that. Bonus points for encouraging them to investigate further on their own if they're interested.
No, it isn't what we're talking about. We're talking about making categorical statements that are false. Telling students that you're teaching them a simplified model and there are complications that will be added back later is not lying. But telling them categorically that the simplified model is true and not even talking about the fact that it's a simplified model and there are complications being left out, is lying. And the latter is what the posters I've been responding to are attempting to defend.
> I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to
I have never said that is necessary. Of course it's not. Once you've explained at the outset that you're teaching them a simplified model and there are complications being left out, obviously you're not going to repeat it with every single sentence. Any more than the people who are advocating telling students categorically "multiplication is repeated addition" with no mention of all the complexities lurking underneath are advocating repeating that with every single sentence.
> or that they're being told they're "not smart enough to understand the full theory"
I have never said they should be told that. Teaching the simplified model could be due to nothing more than limited teacher time and knowledge. There's nothing wrong with the teacher honestly telling the students that. Bonus points for encouraging them to investigate further on their own if they're interested.
> No, it isn't what we're talking about. We're talking about making categorical statements that are false.
That's not what we're talking about. Describing multiplication as repeated addition as a way of teaching a new concept to elementary school children is simply not lying.
Your issue is that the fact it is a simplified model is not being explicitly told to students, which is being called lying. While it would be false to say that a simplified model is the complete picture, teachers aren't standing up in class and saying "this is all there is to this subject, no need to learn any more!".
My main issue with your suggestion is that young students simply aren't going to remember a side comment at the beginning of their classes that this isn't the full picture -- so you will either have to repeat this regularly (which will be demoralising and confusing) or there really is no strong benefit to doing so.
You say that the "categorical statement" is not being repeated with each sentence -- but the thing is that when students use or discuss the tools they've learned, they're reinforcing their mental model of how the tool works. So in a way, the simplified model is being repeatedly reinforced in their mind. If you want to counteract it, saying once at the beginning of each class that introduces a new concept "this isn't the full picture" won't help most students overcome the issues they hit when they learn their old model doesn't work for more complicated problems.
> I have never said they should be told that.
You misunderstood what I said -- I said that constantly being told that what they're learning is "not the real theory" (or however you want to phrase) is demoralising in of itself -- it needlessly gives the impression to students that they're not smart enough to understand "the real theory". I never claimed that you said that teachers should say that explicitly.
That's not what we're talking about. Describing multiplication as repeated addition as a way of teaching a new concept to elementary school children is simply not lying.
Your issue is that the fact it is a simplified model is not being explicitly told to students, which is being called lying. While it would be false to say that a simplified model is the complete picture, teachers aren't standing up in class and saying "this is all there is to this subject, no need to learn any more!".
My main issue with your suggestion is that young students simply aren't going to remember a side comment at the beginning of their classes that this isn't the full picture -- so you will either have to repeat this regularly (which will be demoralising and confusing) or there really is no strong benefit to doing so.
You say that the "categorical statement" is not being repeated with each sentence -- but the thing is that when students use or discuss the tools they've learned, they're reinforcing their mental model of how the tool works. So in a way, the simplified model is being repeatedly reinforced in their mind. If you want to counteract it, saying once at the beginning of each class that introduces a new concept "this isn't the full picture" won't help most students overcome the issues they hit when they learn their old model doesn't work for more complicated problems.
> I have never said they should be told that.
You misunderstood what I said -- I said that constantly being told that what they're learning is "not the real theory" (or however you want to phrase) is demoralising in of itself -- it needlessly gives the impression to students that they're not smart enough to understand "the real theory". I never claimed that you said that teachers should say that explicitly.
We lie when when we teach calculus by relying on hand-wavy 'limit' proofs, without grounding students in the foundational aspects of point set topology.
It's much more productive pedagogically to get an intuition for slope and area than it is to get an intuition for compactness and the infinite intersection of open sets, but slope and area are A LIE.
It's much more productive pedagogically to get an intuition for slope and area than it is to get an intuition for compactness and the infinite intersection of open sets, but slope and area are A LIE.
Well, I'd be curious what your first ten sentences to a group of, say, 11th graders would be about Newtonian gravitation, and how those would compare to what you'd say about the same subject to 8th graders.
It seems elsewhere in the thread that you don't consider it lying if one gives a general disclaimer that models aren't perfect, but I wonder whether or not that general disclaimer wouldn't inoculate the idea that multiplication is repeated addition? It's an imperfect model that super useful for all the numbers available to students when they learn what multiplication is. And I wonder further whether there's not an implicit disclaimer that what you're learning isn't the final word inherent in the structure of the educational system?
It seems elsewhere in the thread that you don't consider it lying if one gives a general disclaimer that models aren't perfect, but I wonder whether or not that general disclaimer wouldn't inoculate the idea that multiplication is repeated addition? It's an imperfect model that super useful for all the numbers available to students when they learn what multiplication is. And I wonder further whether there's not an implicit disclaimer that what you're learning isn't the final word inherent in the structure of the educational system?
Explaining the structure of an atom without having to explain atomic orbitals and standing waves. Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
> Explaining the structure of an atom without having to explain atomic orbitals and standing waves.
My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out of scope for that class was enough.
> Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
My high school physics teacher had no problem explaining classical mechanics including the caveats. The caveats took only a few minutes early in the semester. What's the problem?
My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out of scope for that class was enough.
> Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
My high school physics teacher had no problem explaining classical mechanics including the caveats. The caveats took only a few minutes early in the semester. What's the problem?
I remember my High School chemistry lesson from the 90's clearly explained we were working with historical models, we talked about nature of the models developments and the experiments that led to them.
We started with the JJ Thomson "plumb pudding model"
Then we learnt about Rutherford and the gold foil experiment which invalidated Thomson's model.
Then we learnt about emission spectrum and the Bohr model which invalidated Rutherford's model.
The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Same thing in physics. We started with debate about nature of light and examined things like lumniferous aether and Michaelson Morley experiment, double slit experiment etc. Then Maxwell and electromagnetism leading to Heinrich Hertz and the photoelectric effect, which led to Planck and Einstein.
I certainly never thought I was being lied to. We were treading along the same path as those who had came before.
We started with the JJ Thomson "plumb pudding model"
Then we learnt about Rutherford and the gold foil experiment which invalidated Thomson's model.
Then we learnt about emission spectrum and the Bohr model which invalidated Rutherford's model.
The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Same thing in physics. We started with debate about nature of light and examined things like lumniferous aether and Michaelson Morley experiment, double slit experiment etc. Then Maxwell and electromagnetism leading to Heinrich Hertz and the photoelectric effect, which led to Planck and Einstein.
I certainly never thought I was being lied to. We were treading along the same path as those who had came before.
> I certainly never thought I was being lied to.
That's because you weren't. You said:
> The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Which is exactly what I think should be told to students when a simplified model is being taught. But the people I've been responding to are advocating for telling students the simplified model as if it were exactly correct and covered all cases. That is what I am saying is lying.
That's because you weren't. You said:
> The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Which is exactly what I think should be told to students when a simplified model is being taught. But the people I've been responding to are advocating for telling students the simplified model as if it were exactly correct and covered all cases. That is what I am saying is lying.
What about earlier than high school? I learned about both of those things much earlier.
Why would you explain classical mechanics at all? It's an inaccurate simplification.
It's much simpler to compute answers with classical mechanics, and the answers are accurate enough for many practical purposes. As the saying goes, all models are wrong but some are useful.
Classical mechanics covers the vast majority of the interactions in day to day life people will have. There is a reason it’s “classical mechanics”.
"When they performed [this experiment] at [experimental accuracy] they observed [that result] which makes them think [atomic orbitals and standing waves]."
Someone who thinks they understand physics without considering experimental accuracy doesn't understand physics.
Someone who thinks they understand physics without considering experimental accuracy doesn't understand physics.
The goal of primary education isn't to understand any particular thing, it's to provide the intellectual training to understand anything. Someone doesn't need to leave middle school or even high school "understanding physics", but with a base for future understanding. The Bohr model is more than sufficient to help understand chemical reactions.
People seem to be interpreting "lie-to-children" as meaning "[you must] lie[ ]to[ ]children". I don't think there's anyone who thinks that children should be actively deceived into not knowing about quantum mechanics, just that it's OK to use simplified explanations as one step in the process of learning.
People seem to be interpreting "lie-to-children" as meaning "[you must] lie[ ]to[ ]children". I don't think there's anyone who thinks that children should be actively deceived into not knowing about quantum mechanics, just that it's OK to use simplified explanations as one step in the process of learning.
It costs nothing to tell someone: "I'm going to teach you one way of thinking about this, but there are other ways you will learn later".
> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..."
My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had to do rote memorization of the tables), but hey they can recite tables now too.
Easing into multiplication via addition doesn't feel wrong IMHO. In fact, at later grades, I'd have to learn about associative/commutative properties anyways.
My daughter in particular has a somewhat peculiar story: she is in kinder and learning multiplication from a game[0] that her older brother started playing, and the way they introduced multiplication is by telling her that to calculate the area of a rectangle, she needs to literally count the number of blocks that compose it.
Is it "wrong"? I guess. Blocks aren't really proper units, but she gets that a rectangle made of 4x5 blocks has an area of 20 blocks, first by stumbling n' counting, then adding rows then eventually just memorizing the multiplication fact, so mission accomplished? Things build up from there, first without the grid, then into area of more complex polygons, spin off into word problems, etc.
IMHO what helps kids learn is just repeatedly seeing where the memorized mechanisms can be applied and where they can't. Eventually multiplication becomes second nature and the addition/subtraction crutches come off.
[0] https://www.prodigygame.com/
My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had to do rote memorization of the tables), but hey they can recite tables now too.
Easing into multiplication via addition doesn't feel wrong IMHO. In fact, at later grades, I'd have to learn about associative/commutative properties anyways.
My daughter in particular has a somewhat peculiar story: she is in kinder and learning multiplication from a game[0] that her older brother started playing, and the way they introduced multiplication is by telling her that to calculate the area of a rectangle, she needs to literally count the number of blocks that compose it.
Is it "wrong"? I guess. Blocks aren't really proper units, but she gets that a rectangle made of 4x5 blocks has an area of 20 blocks, first by stumbling n' counting, then adding rows then eventually just memorizing the multiplication fact, so mission accomplished? Things build up from there, first without the grid, then into area of more complex polygons, spin off into word problems, etc.
IMHO what helps kids learn is just repeatedly seeing where the memorized mechanisms can be applied and where they can't. Eventually multiplication becomes second nature and the addition/subtraction crutches come off.
[0] https://www.prodigygame.com/
> Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`).
This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system.
Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying.
Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together.
18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108
vs.
18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108
Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.
This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system.
Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying.
Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together.
18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108
vs.
18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108
Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.
> I honestly don't feel like they're being lied to, or confused about what they are being taught.
They might not be. If they're just being taught how to multiply in particular cases (whole numbers) using repeated addition, then they're not being lied to. If they're being told straight up that, while multiplication is not identical to repeated addition, they can ease into multiplication by learning repeated addition, even better.
They're only being lied to if they're being told, authoritatively, that multiplication is repeated addition, no ifs, ands, or buts. And yes, I have had teachers like that, and I suspect many others have too.
They might not be. If they're just being taught how to multiply in particular cases (whole numbers) using repeated addition, then they're not being lied to. If they're being told straight up that, while multiplication is not identical to repeated addition, they can ease into multiplication by learning repeated addition, even better.
They're only being lied to if they're being told, authoritatively, that multiplication is repeated addition, no ifs, ands, or buts. And yes, I have had teachers like that, and I suspect many others have too.
> (e.g. `9 x N` is initially taught as `10 x N - N`)
That isn't a crutch, that is how large numbers can be multiplied in one's head!
90 * 21 is (90 * 20) + 90
Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally.
Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14.
Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memorized and I'll add my way from there. I don't remember 7 * 6 but I know its 7 * 7 - 7.
That isn't a crutch, that is how large numbers can be multiplied in one's head!
90 * 21 is (90 * 20) + 90
Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally.
Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14.
Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memorized and I'll add my way from there. I don't remember 7 * 6 but I know its 7 * 7 - 7.
> Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14.
Yeah, that’s interesting how we do things in our head like that.
I would frequently break that down as 9*10 + 9*4 in my head.
But I might also do it as 10*14 - 10 - 4.
I’ll try different ways in my head until I can solve one of them easily.
Yeah, that’s interesting how we do things in our head like that.
I would frequently break that down as 9*10 + 9*4 in my head.
But I might also do it as 10*14 - 10 - 4.
I’ll try different ways in my head until I can solve one of them easily.
Pre-university mathematics is mostly like this.
But there is a way to teach mathematics from axioms that is intuitive: geometry. Euclid's Elements was the key mathematics textbook for about 2000 years.
Of course, you simply can't cover as much ground if you must derive everything.
But I do wonder if the collapse in public discourse is partly because of faith-based mathematics. i.e. taught as: it's true because we tell you it's true, not because you can see for yourself that it's true.
But there is a way to teach mathematics from axioms that is intuitive: geometry. Euclid's Elements was the key mathematics textbook for about 2000 years.
Of course, you simply can't cover as much ground if you must derive everything.
But I do wonder if the collapse in public discourse is partly because of faith-based mathematics. i.e. taught as: it's true because we tell you it's true, not because you can see for yourself that it's true.
I never got over being lied to about electron shells and the 2-8-8 "rule".
To be fair, it once was the best model we had, before we figured out quantum mechanics. That brings up an interesting point, though: the way many of these things are taught closely mirrors the history of how we discovered them in the first place. This is definitely not a coincidence, but I'm also not sure whether it is the most effective way to teach.
How were you lied to about that?
In GCSE Chemistry (taught to 15-16ish year olds), we were taught that the first shell contained two electrons, and subsequent shells contained 8 (up to the final, which may contain fewer).
Then, after GCSEs, during A-levels, we were told (by the same teachers, in the same classrooms) to forget this model, and that the situation was actually more complicated, with s, p, d, f orbitals etc.
I realise that this is was a simplification and not necessarily an outright lie, and can understand why they did it this way. But it was the first time I realised I’d been deliberately taught something that was incomplete or inaccurate.
Then, after GCSEs, during A-levels, we were told (by the same teachers, in the same classrooms) to forget this model, and that the situation was actually more complicated, with s, p, d, f orbitals etc.
I realise that this is was a simplification and not necessarily an outright lie, and can understand why they did it this way. But it was the first time I realised I’d been deliberately taught something that was incomplete or inaccurate.
Wait until you hear about Px, py orbitals and their actual quantum numbers
I still encounter this as a grown adult. I'm working through a course on a programming framework I'm unfamiliar with right now, and several times have gone through a half-hour setup to encounter "Now that I've shown you the awful way to do this, here's a built-in way that's far simpler and less error-prone!" Extremely irritating.
Teaching Newtonian Mechanics to students is also a lie. But I don't believe starting with General Relativity is going to do anything but confuse students.
I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.
I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.
> Teaching Newtonian Mechanics to students is also a lie.
Not if you tell the students that it's an approximate model that works well in the domain they're currently studying, but doesn't work well in a more expanded domain.
Of course if you insist on acting like an authority and telling students that Newtonian Mechanics is "the Truth", then yes, you are lying to them. But you don't have to tell them that to teach them Newtonian Mechanics.
Not if you tell the students that it's an approximate model that works well in the domain they're currently studying, but doesn't work well in a more expanded domain.
Of course if you insist on acting like an authority and telling students that Newtonian Mechanics is "the Truth", then yes, you are lying to them. But you don't have to tell them that to teach them Newtonian Mechanics.
> Of course if you insist on acting like an authority and telling students that Newtonian Mechanics is "the Truth", then yes, you are lying to them. But you don't have to tell them that to teach them Newtonian Mechanics.
Do you think teachers really say "Newtonian Mechanics is the Truth" in class? Maybe it's different in the US but in Australia I've never had an interaction with a teacher like that.
Or is your issue that teachers are omitting that information? If so, that's quite a different thing to discuss.
Do you think teachers really say "Newtonian Mechanics is the Truth" in class? Maybe it's different in the US but in Australia I've never had an interaction with a teacher like that.
Or is your issue that teachers are omitting that information? If so, that's quite a different thing to discuss.
Idk discovering all the lies I believe is one of the most fun parts of learning for me.
> discovering all the lies I believe
I have no problem with discovering lies I believe because I told them to myself (and I think most of the lies anyone believes are lies they've told themselves). What I have a problem with is discovering that someone else deliberately lied to me. But probably not everyone feels the same way I do about such things.
I have no problem with discovering lies I believe because I told them to myself (and I think most of the lies anyone believes are lies they've told themselves). What I have a problem with is discovering that someone else deliberately lied to me. But probably not everyone feels the same way I do about such things.
>I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods
Are you sure it's pedantic? As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again - because it isn't helpful as you incorporate fractions, and negative numbers.
Are you sure it's pedantic? As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again - because it isn't helpful as you incorporate fractions, and negative numbers.
> As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again.
The standard manual method of multiplication of large numbers relies on leveraging heavily:
(1) the fact that my multiplication is equivalent to repeated addition for nonnegative integers,
(2) the fact that shifting digit positions are equivalent to multiplication by the base (usually 10) or it's multiplicative inverse, depending on direction, and
(3) memorization of multiplication tables for single digits in the base.
So, no, I don't think the “multiplication is repeated addition” thing is something people are exposed to and then never use.
The standard manual method of multiplication of large numbers relies on leveraging heavily:
(1) the fact that my multiplication is equivalent to repeated addition for nonnegative integers,
(2) the fact that shifting digit positions are equivalent to multiplication by the base (usually 10) or it's multiplicative inverse, depending on direction, and
(3) memorization of multiplication tables for single digits in the base.
So, no, I don't think the “multiplication is repeated addition” thing is something people are exposed to and then never use.
>nonnegative integers,
Why, prey tell, did you make a distinction between positive and negative integers? Is it that repeated addition starts to fail as conceptual model there?
>the fact that my multiplication is equivalent to repeated addition for nonnegative integers, ..
Your claim seems to be that in some abstract way any algorithmic approach to multiplication relies on the fact that multiplying non-negative integers (there's that distinction again .. nasty stuff those negative integers are for your argument) can be rewritten as a repeated addition ... but that's not how people (not even children learning multiplication) do it. That's not how computers do it either.
So no, algorithmically multiplying numbers is not the same thing as 'using' the concept of 'multiplication is repeated addition' in any sense, colloquial or otherwise. Also it starts feeling like you're begging the question when you claim that.
Why, prey tell, did you make a distinction between positive and negative integers? Is it that repeated addition starts to fail as conceptual model there?
>the fact that my multiplication is equivalent to repeated addition for nonnegative integers, ..
Your claim seems to be that in some abstract way any algorithmic approach to multiplication relies on the fact that multiplying non-negative integers (there's that distinction again .. nasty stuff those negative integers are for your argument) can be rewritten as a repeated addition ... but that's not how people (not even children learning multiplication) do it. That's not how computers do it either.
So no, algorithmically multiplying numbers is not the same thing as 'using' the concept of 'multiplication is repeated addition' in any sense, colloquial or otherwise. Also it starts feeling like you're begging the question when you claim that.
Better served by the notion of cardinality of the categorical product.
Except this isn't a pedantic falsehood. Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. Understanding ratio is key to understanding a lot of the physical sciences and you can build a really good intuitive understanding of a lot of simple physical concepts if you can just do dimensional analysis. But if you think of multiplication and division as a kind of addition then it all breaks down and you can't reason about much at all with those tools. In fact you can't even get over the hurdle of doing dimensional analysis in chemistry or physics in the first place. It doesn't make any sense. And you'll have been set up to fail by your teachers.
They're not saying "we should be teaching kindergarteners using Euclid" they're suggesting that focusing on the process of doing multiplication hides a lot of very useful insights about why someone would do multiplication in the first place.
The biggest problem with math as it was taught to me in grade school is that it was focused on rote memorization of a procedure. Which is fine if you need to calculate how long to cut 2x4s or stair stringers but it doesn't help you at all if you need to understand why that works the way it does. And god help you if nobody ever shows you the exact method of calculation for those two things in the first place. You're set up to fail if you only focus on process.
They're not saying "we should be teaching kindergarteners using Euclid" they're suggesting that focusing on the process of doing multiplication hides a lot of very useful insights about why someone would do multiplication in the first place.
The biggest problem with math as it was taught to me in grade school is that it was focused on rote memorization of a procedure. Which is fine if you need to calculate how long to cut 2x4s or stair stringers but it doesn't help you at all if you need to understand why that works the way it does. And god help you if nobody ever shows you the exact method of calculation for those two things in the first place. You're set up to fail if you only focus on process.
> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related.
I just finished studying Galois Fields, which LITERALLY redefines multiplication and addition operators to study new forms of math (IE fields in particular)
In all fields and rings, multiplication and addition are related by the distributive property.
A(b + c) is equal to Ab + Ac, in literally every form of math you can every think of. A and b could be matrices, vectors, polynomials, prime numbers, integers, rational, complex, real, or Galois polynomials over a weird modulus ring thingy inside of a vector inside of a matrix. Doesn't matter, addition and multiplication are and always defined in that manner.
I just finished studying Galois Fields, which LITERALLY redefines multiplication and addition operators to study new forms of math (IE fields in particular)
In all fields and rings, multiplication and addition are related by the distributive property.
A(b + c) is equal to Ab + Ac, in literally every form of math you can every think of. A and b could be matrices, vectors, polynomials, prime numbers, integers, rational, complex, real, or Galois polynomials over a weird modulus ring thingy inside of a vector inside of a matrix. Doesn't matter, addition and multiplication are and always defined in that manner.
Yes, they are two binary operations and depending on the sets you consider and which properties you impose for those operations you have different algebraic structures. (This used to be taught at school before "modern mathematics" were considered harmful, maybe they were but at least they were correct).
The thing is that as you can write m (let it be a positive integer) as m=1+...+1 (m-times), you can write n·m=n·(1+...+1), invoke the distributive property for · wrt + and express it as: n·m=n+...+n (m-times), so it looks like "repeated addition" for integers in this case. But it's not a good idea to let ourselves get carried away, we still have two binary operations going on. At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.
The thing is that as you can write m (let it be a positive integer) as m=1+...+1 (m-times), you can write n·m=n·(1+...+1), invoke the distributive property for · wrt + and express it as: n·m=n+...+n (m-times), so it looks like "repeated addition" for integers in this case. But it's not a good idea to let ourselves get carried away, we still have two binary operations going on. At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.
> At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.
n * 0 = n * (1 + (-1)) = n + (-n) = 0.
-----
The 0-element in a Galois Field works identically btw. In GF(5), the 0 element is 5 (5 mod 5 == 0).
n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0 mod5.
For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0.
2 * (1 + 4) == 2 + 8 == 2 + 3 (mod 5) == 5 mod 5 == 0. Etc. etc.
-------
This property literally holds in all fields and rings (but not groups).
n * 0 = n * (1 + (-1)) = n + (-n) = 0.
-----
The 0-element in a Galois Field works identically btw. In GF(5), the 0 element is 5 (5 mod 5 == 0).
n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0 mod5.
For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0.
2 * (1 + 4) == 2 + 8 == 2 + 3 (mod 5) == 5 mod 5 == 0. Etc. etc.
-------
This property literally holds in all fields and rings (but not groups).
> n * 0 = n * (1 + (-1)) = n + (-n) = 0
You don't think it's easier to say "n·5 = n·(5+0) = n·5 + n·0"?
You don't think it's easier to say "n·5 = n·(5+0) = n·5 + n·0"?
The (1 + (-1)) thing was very precisely written.
All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is:
--------------
As such, the concept of "5" does not necessarily exist in all possible Ring-systems. "5" exists in GF(5) for example, but not really in GF(3). Case in point, what does "5" mean in 2x2 Matrix Linear Algebra over GF(2)?
--------
"-A" is called the additive inverse of A, which also exists in all rings. A - A = 0.
In GF(5), -1 is 4 for example. In 2x2 Linear Algebra, -1 is [-1 0; 0 -1]. In Real Numbers, -1 is... well... -1.
----------
Anyway, the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings. (And then later, I did an example in the GF(5) system as a specific example).
All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is:
[ 0 0
0 0 ]
[ 1 0
0 1 ]
--------------
As such, the concept of "5" does not necessarily exist in all possible Ring-systems. "5" exists in GF(5) for example, but not really in GF(3). Case in point, what does "5" mean in 2x2 Matrix Linear Algebra over GF(2)?
--------
"-A" is called the additive inverse of A, which also exists in all rings. A - A = 0.
In GF(5), -1 is 4 for example. In 2x2 Linear Algebra, -1 is [-1 0; 0 -1]. In Real Numbers, -1 is... well... -1.
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Anyway, the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings. (And then later, I did an example in the GF(5) system as a specific example).
> As such, the concept of "5" does not necessarily exist in all possible Ring-systems.
Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared.
> the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings.
To repeat myself:
Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared.
> the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings.
To repeat myself:
n·5 = n·(5+0) [definition of 0]
n·(5+0) = n·5 + n·0 [multiplication is distributive over addition]
n·0 = 0 [definition of 0]
[deleted]
[deleted]
I see what you're saying now. So to answer your original question...
> You don't think it's easier to say...
No. I disagree, your way of thinking is harder for me to think. :-)
You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, although we tweaked the formulas to look like the simplest ways for our own brains.
> You don't think it's easier to say...
No. I disagree, your way of thinking is harder for me to think. :-)
You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, although we tweaked the formulas to look like the simplest ways for our own brains.
Fair enough.
I do like your format though. So I'll borrow it for my side of the discussion.
n·0 = 0 [definition of 0]
n·(1 + -1) = 0 [definition of additive inverse]
n·(1 + -1) = n + (-n) [multiplication is distributive over addition]
n + (-n) = 0 [definition of additive inverse]I feel like I'm not understanding something. Isn't "n·0 = 0" the demonstrandum? I would have written something like
n·0 = 0 [we seek to show this]
-------
n·0 = n·(1 + -1) [definition of additive inverse]
n·(1 + -1) = n + (-n) [multiplication is distributive...]
n + (-n) = 0 [definition of additive inverse]
QED
Repeated addition that ain't...
It boggles my mind that you've been studying Galois theory yet somehow try to reduce the algebraic structures associated to two binary operations to playing with one of them.
It boggles my mind that you've been studying Galois theory yet somehow try to reduce the algebraic structures associated to two binary operations to playing with one of them.
The algebraic structures multiplication brings is different from the algebraic structures from addition. That's the point of rings (as opposed to groups).
Nonetheless, in a ring (and all fields are rings), multiplication must and always is related to addition, through the distributed property (which I argue, the distributed property IS the mathematical term for "repeated addition").
Without the distributed property, you have no ring. You at best only have a group. Therefore, all multiplication operators ever defined (or more precisely, all rings) must have multiplication related to addition: (A * (B+C) == AB+AC)
Nonetheless, in a ring (and all fields are rings), multiplication must and always is related to addition, through the distributed property (which I argue, the distributed property IS the mathematical term for "repeated addition").
Without the distributed property, you have no ring. You at best only have a group. Therefore, all multiplication operators ever defined (or more precisely, all rings) must have multiplication related to addition: (A * (B+C) == AB+AC)
Without the distributive property you have two magmas in general for the same set that don't see each other. I'm not sure what your point is. When you have two binary operations you need some sort of distributive property to build a structure.
All this is trivial, if you consider a ring, you get a·0=0 as a property, if your starting point is the Peano axioms for the arithmetic of natural numbers that's one of them, for the latter seeing it as "repeated addition" makes no sense, for the former, well you have a ring, you have two binary operations, not one, and of course you have some form of distributive property or else you'd be studying this set with just one binary operation at a time.
I'd like to see how "repeated addition" works in polynomial rings.
All this is trivial, if you consider a ring, you get a·0=0 as a property, if your starting point is the Peano axioms for the arithmetic of natural numbers that's one of them, for the latter seeing it as "repeated addition" makes no sense, for the former, well you have a ring, you have two binary operations, not one, and of course you have some form of distributive property or else you'd be studying this set with just one binary operation at a time.
I'd like to see how "repeated addition" works in polynomial rings.
> I'd like to see how "repeated addition" works in polynomial rings.
Consider the following polynomial: x0 * b^0 + x1 * b^1 + x2 * b^2 ... xn * b^n, where "n" goes to both positive infinity and negative infinity.
When "b = 10" and when "x" can be numbers from [0-9], we have the so called base-10 set of real numbers, do we not? IIRC, if b = sqrt(-1) * 10, we then have the set of complex numbers (a non-intuitive result. I may have made a mistake somewhere, but I assure you there's a surprising property along those lines).
That's the funny thing about real numbers and complex-numbers. Real numbers and even complex-numbers ARE polynomials, and therefore a polynomial ring. 3.1415926 == 3 * 10^0 + 1 * 10 ^-1 + 4 * 10 ^-2 ...
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I'm using a lot of words here. But all I'm saying is once again: Pi * 3 == 3.14... * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3 + ... == 9.42...
We can evaluate 3 * Pi by splitting Pi up into a set of additions (3 + 0.1 + 0.04 + 0.001...), even if that set of additions is infinite. Then evaluate 3*(each component). This is possible because Pi is easily represented as a polynomial X0 * 10^0 + X1 * 10^-1 + ... Xn * 10^-n.
There's a reason why polynomial multiplication is usually called "Carry-free multiplication". Because Real-numbers are just polynomials where you have that annoying "carry the one" property to keep track of. Remove the "carry the one" property (in say: Galois extension fields), and all the math still works.
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Hmmmm... I probably could have said all that in fewer words. TL;DR: "Real numbers ARE a polynomial ring". (And complex numbers are probably a polynomial ring, I just forgot how to prove that factoid)
> I'd like to see how "repeated addition" works in polynomial rings.
EDIT: Just thought of a cute and simple retort. You ever do a CRC32 check? There ya go.
Consider the following polynomial: x0 * b^0 + x1 * b^1 + x2 * b^2 ... xn * b^n, where "n" goes to both positive infinity and negative infinity.
When "b = 10" and when "x" can be numbers from [0-9], we have the so called base-10 set of real numbers, do we not? IIRC, if b = sqrt(-1) * 10, we then have the set of complex numbers (a non-intuitive result. I may have made a mistake somewhere, but I assure you there's a surprising property along those lines).
That's the funny thing about real numbers and complex-numbers. Real numbers and even complex-numbers ARE polynomials, and therefore a polynomial ring. 3.1415926 == 3 * 10^0 + 1 * 10 ^-1 + 4 * 10 ^-2 ...
------------
I'm using a lot of words here. But all I'm saying is once again: Pi * 3 == 3.14... * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3 + ... == 9.42...
We can evaluate 3 * Pi by splitting Pi up into a set of additions (3 + 0.1 + 0.04 + 0.001...), even if that set of additions is infinite. Then evaluate 3*(each component). This is possible because Pi is easily represented as a polynomial X0 * 10^0 + X1 * 10^-1 + ... Xn * 10^-n.
There's a reason why polynomial multiplication is usually called "Carry-free multiplication". Because Real-numbers are just polynomials where you have that annoying "carry the one" property to keep track of. Remove the "carry the one" property (in say: Galois extension fields), and all the math still works.
--------
Hmmmm... I probably could have said all that in fewer words. TL;DR: "Real numbers ARE a polynomial ring". (And complex numbers are probably a polynomial ring, I just forgot how to prove that factoid)
> I'd like to see how "repeated addition" works in polynomial rings.
EDIT: Just thought of a cute and simple retort. You ever do a CRC32 check? There ya go.
A polynomial is a polynomial, a decimal representation of a real number is a decimal representation of a real number, and your representation of complex numbers has funny properties once you begin exponentiating that.
I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
> I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 * (1-x^3)) ??
The polynomial itself gives us the means at which we logically split up the multiplication into component parts.
Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we move onto polynomials, we do the same exact thing. EDIT: remember, ALL REAL NUMBERS ARE POLYNOMIALS with a base of 10.
Or to put it another way: when x == 10, your polynomial of (1 + x + x^2) * (1-x^3) == 111 * (-999). That is to say: real numbers are simply polynomials where "x" has been defined to be a particular number, instead of an abstract entity. We call that number the radix-base.
If you instead defined the base to be x = 16 (hexadecimal numbers), you'd get 111 * (-FFF), which you'll find will satisfy similar properties. Now leave x-undefined (since it could be 10 or 16), and what do you get?
Polynomial math. Or so called "Carry-less multiplication" (https://en.wikipedia.org/wiki/Carry-less_product). We don't have a ring yet though: we still need to perform a modulus on all those polynomials to return to a proper ring (and if the modulus is irreducable, we have a Galois field). But we can already see how polynomials and the Reals are so closely related.
Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 * (1-x^3)) ??
The polynomial itself gives us the means at which we logically split up the multiplication into component parts.
Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we move onto polynomials, we do the same exact thing. EDIT: remember, ALL REAL NUMBERS ARE POLYNOMIALS with a base of 10.
Or to put it another way: when x == 10, your polynomial of (1 + x + x^2) * (1-x^3) == 111 * (-999). That is to say: real numbers are simply polynomials where "x" has been defined to be a particular number, instead of an abstract entity. We call that number the radix-base.
If you instead defined the base to be x = 16 (hexadecimal numbers), you'd get 111 * (-FFF), which you'll find will satisfy similar properties. Now leave x-undefined (since it could be 10 or 16), and what do you get?
Polynomial math. Or so called "Carry-less multiplication" (https://en.wikipedia.org/wiki/Carry-less_product). We don't have a ring yet though: we still need to perform a modulus on all those polynomials to return to a proper ring (and if the modulus is irreducable, we have a Galois field). But we can already see how polynomials and the Reals are so closely related.
Precisely, you're not repeating p(x) q(x)-times, you've used that p(x) is a linear combination of monomials and then the distributive property of Z[x].
Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.
Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.
> At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.
Are you kidding? This is the exact opposite of the truth; the nature of multiplication as repeated addition is the entire reason why multiplying by 0 gives the additive identity. It's exactly the same as how exponentiating by 0 gives the multiplicative identity, since exponentiation is just repeated multiplication. And this is so fundamental that 1 is frequently referred to by this property, as "the empty product".
Are you kidding? This is the exact opposite of the truth; the nature of multiplication as repeated addition is the entire reason why multiplying by 0 gives the additive identity. It's exactly the same as how exponentiating by 0 gives the multiplicative identity, since exponentiation is just repeated multiplication. And this is so fundamental that 1 is frequently referred to by this property, as "the empty product".
Could you sketch a proof starting from some definition of the operation product of integers as "repeated addition" without using the distributive property, which would imply that we already have another binary operation besides the sum?
I was thinking about the Peano axioms when I wrote that (hence the working backwards thing). Obviously if you start with a ring, you don't have to impose it, you get that as a property. I think I mentioned that in a later post.
I was thinking about the Peano axioms when I wrote that (hence the working backwards thing). Obviously if you start with a ring, you don't have to impose it, you get that as a property. I think I mentioned that in a later post.
If you want to work up to it conceptually, then I'd say consider the meaning of something like
At that point you notice that when you say 5·3 means "add 3 five times", you forgot to say what you were adding it to. You're adding it to the identity, 0.
And finally we say that starting from 0 and adding something zero times leaves you where you started, at 0, so we can observe that 0·n = 0.
If you formalize that, you'll end up either deriving or postulating the distributive property, depending on what you start with. But it isn't arbitrary; it's not a coincidence that (as I mentioned above) exponentiation by 0 gives the multiplicative identity, and (as I haven't mentioned yet) exponentiation distributes over multiplication the same way multiplication distributes over addition.
(Asymmetry does start creeping in; aggregate exponentiation isn't as nice since exponentiation doesn't have the nice properties that addition and multiplication do.)
5·3 + 2·7
5·3 + 2·7 + 0·4
At that point you notice that when you say 5·3 means "add 3 five times", you forgot to say what you were adding it to. You're adding it to the identity, 0.
And finally we say that starting from 0 and adding something zero times leaves you where you started, at 0, so we can observe that 0·n = 0.
If you formalize that, you'll end up either deriving or postulating the distributive property, depending on what you start with. But it isn't arbitrary; it's not a coincidence that (as I mentioned above) exponentiation by 0 gives the multiplicative identity, and (as I haven't mentioned yet) exponentiation distributes over multiplication the same way multiplication distributes over addition.
(Asymmetry does start creeping in; aggregate exponentiation isn't as nice since exponentiation doesn't have the nice properties that addition and multiplication do.)
> exponentiation distributes over multiplication
things that irks me
things that irks me
a**b * a**c == a**(b + c) != a**(b * c)
Distributivity of exponentiation over multiplication looks like
(ab)^x = (a^x)(b^x)
> commutativity
yes my mistake
yes my mistake
> But it's not a good idea to let ourselves get carried away, we still have two binary operations going on. At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.
Why is this not a good idea?
Can’t we just accept/postulate that the additive identity is different from the multiplicative identity?
And still define a relationship between the addition and multiplication?
Maybe I misunderstand the issue..
(I’m not trying to be pedantic, but my math background has some holes :)
Why is this not a good idea?
Can’t we just accept/postulate that the additive identity is different from the multiplicative identity?
And still define a relationship between the addition and multiplication?
Maybe I misunderstand the issue..
(I’m not trying to be pedantic, but my math background has some holes :)
> In all fields and rings, multiplication and addition are related by the distributive property.
That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.
That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.
pi(3 + 5) == 3pi + 5pi == (Pi + Pi + Pi) + (Pi + Pi + Pi + Pi + Pi) == 8-pi.
Am I missing something here?
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This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related.
Am I missing something here?
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This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related.
Pi * ([ 1 0 ; + [ 1 0 ;
0 1 ] 2 1 ])
Pi * [ 1 0 ; + Pi * [ 1 0 ;
0 1 ] 2 1 ]
[ Pi 0 ; + [ Pi 0 ;
0 Pi ] 2Pi Pi ]
[ 2Pi 0 ;
2Pi 2Pi ]
Pi * ( 1 0 + 0 0 + 1 0 + 0 0 + 0 0 )
0 0 0 1 0 0 2 0 0 1
I think the issue is more about:
3 * pi = pi + pi + pi
but how do you represent the other distribution, where 3 is added together pi times?
3 * pi = pi + pi + pi
but how do you represent the other distribution, where 3 is added together pi times?
Break-up pi in whatever means you think is reasonable.
3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...)
Aka: 9.4245...
You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)
3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...)
Aka: 9.4245...
You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)
> Break-up pi in whatever means you think is reasonable.
You can't. Pi is irrational.
You can't. Pi is irrational.
> You can't. Pi is irrational.
You just need to break it up infinitely times. We usually call the sequence 31415926...
You just need to break it up infinitely times. We usually call the sequence 31415926...
> You just need to break it up infinitely times.
I've responded to this elsewhere in the thread: I don't think adopting increasingly perverse interpretations of "repeated addition" as you try to include more and more numbers is a useful way to teach multiplication.
I've responded to this elsewhere in the thread: I don't think adopting increasingly perverse interpretations of "repeated addition" as you try to include more and more numbers is a useful way to teach multiplication.
what they described is identical to long multiplication. That's not perverse, it's how most people multiply numbers.
You extend to the rationals in the natural way, and then use continuity to define what happens for irrationals. (Of course, in a discrete context that doesn't apply. But the article's author is a grade school teacher.)
> Am I missing something here?
Sorry, I should have said substitute b = pi or c = pi. Or see the sibling post by iio8999.
Sorry, I should have said substitute b = pi or c = pi. Or see the sibling post by iio8999.
pi isn't a counting number like you teach to children.
Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name.
The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation is expected to leave that property intact.
This is not unique to multiplication. Modern mathematics is all about extracting fundamental properties that make something true, to generalize results while keeping the original simpler statement true. Starting from the generalisations without explaining the intellectual process it took us to get there means just giving facts without explaining them.
The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation is expected to leave that property intact.
This is not unique to multiplication. Modern mathematics is all about extracting fundamental properties that make something true, to generalize results while keeping the original simpler statement true. Starting from the generalisations without explaining the intellectual process it took us to get there means just giving facts without explaining them.
"Multiplication is literally repeated addition"
How do you repeat something a negative number of times? What does it mean to multiply two negatives, if multiplication is repeated addition? This is not an overly generalized concept, we teach negative numbers to elementary school students.
How do you repeat something a negative number of times? What does it mean to multiply two negatives, if multiplication is repeated addition? This is not an overly generalized concept, we teach negative numbers to elementary school students.
I don't know what you mean by "overly". Multiplication by negative numbers is a generalisation of the earlier concept of multiplication by positive numbers. In a sense even multiplying by 1 is a generalisation (at least if you read Euclid he doesn't really define multiplying by 1).
I don't claim that we should not teach generalisations to children. What I claim is that of we teach generalisations we at least should be aware that they are generalisations, and maybe even tell them about it.
I don't claim that we should not teach generalisations to children. What I claim is that of we teach generalisations we at least should be aware that they are generalisations, and maybe even tell them about it.
That sounds a bit backwards to me. If anything, negative numbers are the generalization (as is 0), and we should have a concept of multiplication that works for such generalizations. Euclid's approach to arithmetic always seemed strained to me, something motivated by a religious view of math where a compass and straightedge were the only tools anyone should need.
Repeated addition is just a technique for multiplication. It is one of many techniques we teach children, and we should leave it at that -- a technique. When we get to multiplication by negative numbers or fractions, we teach children other techniques, and we do not feel any need to try to "define" multiplication in terms of those techniques. Why should we "define" multiplication in terms of repeated addition, and then do circles around ourselves trying to generalize that "definition?" We can just say that multiplication is one operation we can do on numbers; addition is another, and they are related by the distributive law (and it from the distributive law that we can derive the various techniques we use for multiplication).
Repeated addition is just a technique for multiplication. It is one of many techniques we teach children, and we should leave it at that -- a technique. When we get to multiplication by negative numbers or fractions, we teach children other techniques, and we do not feel any need to try to "define" multiplication in terms of those techniques. Why should we "define" multiplication in terms of repeated addition, and then do circles around ourselves trying to generalize that "definition?" We can just say that multiplication is one operation we can do on numbers; addition is another, and they are related by the distributive law (and it from the distributive law that we can derive the various techniques we use for multiplication).
Define π times e in terms of repeated addition.
I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.
I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.
Maybe it is not literally addition but it certainly reduces to addition in all cases: https://news.ycombinator.com/item?id=26329855
Nope. I'll reply there.
I am afraid I did not make myself clear. I don't mean that multiplying by irrational numbers is repeated addition. What I mean is that multiplication by irrational (or even non-integers) was defined to generalize the earlier notion of multiplying by positive integers.
Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").
That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.
Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").
That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.
We can't really talk about what multiplication "is" or "isn't" independently of context. It's an operation on two objects and the context its used in is necessary for defining the operation.
Though it can be helpful to think of multiplication as scaling or rotation in certain contexts, or as repeated addition in others, none of those are a universal truth.
To say it's not related to addition at all is also too broad to be true. They're related in many useful ways that other commenters have pointed out, in addition to the obvious way that in some situations you can define one in terms of the other. Even at the cutting edge, our inability to prove the Goldbach Conjecture might have something to do with not fully understanding the deepest relationships between the operations.
Though it can be helpful to think of multiplication as scaling or rotation in certain contexts, or as repeated addition in others, none of those are a universal truth.
To say it's not related to addition at all is also too broad to be true. They're related in many useful ways that other commenters have pointed out, in addition to the obvious way that in some situations you can define one in terms of the other. Even at the cutting edge, our inability to prove the Goldbach Conjecture might have something to do with not fully understanding the deepest relationships between the operations.
> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related.
My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?
My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?
When you multiply two negative numbers.
As long as we have
(I guess we would need
-1 * -1 = 1
-x * -y = -1 * -1 * x * y = 1 * x * y = x * y
(I guess we would need
x * 0 = 0
Well, first, why should (-1) * (-1)= 1? Also, you take for granted that -x = (-1) * x, and why should that be the case?
You had asked when the idea that multiplication is repeated addition breaks down. What does (-1) * (-1) mean if multiplication is repeated addition?
Really all these statements, including x * 0 = 0, are about the relationship between addition and multiplication. 0 is the additive identity, so why should we expect anything special about it when it comes to multiplication? The answer is that addition and multiplication are related by the distributive law:
0 * x + x = 0 * x + 1 * x = (0+1) * x = x
Thus (0 * x) is an identity element for addition, but the additive identity is unique (this can be proved using only the properties of addition) and therefore 0 * x = 0. From here we can show:
(-1)*x + x = (-1)*x + 1* x = (-1 + 1) * x = 0 * x = 0
Thus (-1)*x is an additive inverse of x, and again, we can show that additive inverses are unique and therefore (-1)*x = -x. Now we have everything needed for your proof.
Again, the idea that multiplication is repeated addition plays no role here. What actually matters is the distributive law, the meaning of "0", "1", and of "negative," and that multiplication and addition are closed (i.e. the sum or product of two numbers is a number). Those properties are part of the definition of addition and multiplication (along with the associative and commutative laws, and some form of cancellation for multiplication / multiplicative inverses) and can be taken as axioms.
You had asked when the idea that multiplication is repeated addition breaks down. What does (-1) * (-1) mean if multiplication is repeated addition?
Really all these statements, including x * 0 = 0, are about the relationship between addition and multiplication. 0 is the additive identity, so why should we expect anything special about it when it comes to multiplication? The answer is that addition and multiplication are related by the distributive law:
0 * x + x = 0 * x + 1 * x = (0+1) * x = x
Thus (0 * x) is an identity element for addition, but the additive identity is unique (this can be proved using only the properties of addition) and therefore 0 * x = 0. From here we can show:
(-1)*x + x = (-1)*x + 1* x = (-1 + 1) * x = 0 * x = 0
Thus (-1)*x is an additive inverse of x, and again, we can show that additive inverses are unique and therefore (-1)*x = -x. Now we have everything needed for your proof.
Again, the idea that multiplication is repeated addition plays no role here. What actually matters is the distributive law, the meaning of "0", "1", and of "negative," and that multiplication and addition are closed (i.e. the sum or product of two numbers is a number). Those properties are part of the definition of addition and multiplication (along with the associative and commutative laws, and some form of cancellation for multiplication / multiplicative inverses) and can be taken as axioms.
The original commenteer I responded to (not you), said:
>Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related.
So thank you for explaining the actual relationship!
>Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related.
So thank you for explaining the actual relationship!
These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.
You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication?
This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.
You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication?
This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.
> These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.
That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square.
In fact, I have a high school textbook that makes that derivation.
That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square.
In fact, I have a high school textbook that makes that derivation.
"That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square."
That is one way to derive it, but there are others, including my personal favorite, the resolvents method (which also works for deriving cubic and quartic formulas).
That is one way to derive it, but there are others, including my personal favorite, the resolvents method (which also works for deriving cubic and quartic formulas).
We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.
That's not the norm?
I don't think so. I was on the Calculus track in high school so we derived it in...Pre-Calculus.
Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from.
It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation. And there it was. The quadratic formula!
Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from.
It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation. And there it was. The quadratic formula!
> Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation.
How is this not ruining the surprise? The only possible outcomes of doing that are that (1) you make a mistake; or (2) you get a formula for solving quadratic equations. Quadratic equations have the same solutions regardless of your methodology, so there's only one formula you can get.
How is this not ruining the surprise? The only possible outcomes of doing that are that (1) you make a mistake; or (2) you get a formula for solving quadratic equations. Quadratic equations have the same solutions regardless of your methodology, so there's only one formula you can get.
She did not tell us we were deriving the quadratic formula. I had no idea where she was going when she started the exercise.
[deleted]
No, it's not. It depends a bit on your teachers but also on which "level" of math you're in (in the US). Lower level but still algebra/geometry classes tend to teach facts, not derivations from foundational concepts. Those are the classes aimed at non-Honors and maybe non-College Prep students (2 of the 3 typical "tracks" students end up in the US, names may vary by state and decade).
The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.
The point of the Geometry course in that curriculum is to teach the student how to do proofs. The Geometry itself is secondary.
Unfortunately this is typically done in a counterproductive way.
See http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%...
See http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%...
Beliefs 1–4 on page 151 are exactly what I thought about geometry until later in college. Thanks for the article, it's fascinating.
> Instead of learning that you can use shapes to do useful calculations like square roots
Huh?
You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.
Huh?
You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.
All calculation is inherently imperfect. But if you want to have something of length sqrt(2) you can.
I'd say the opposite; the drawing is guaranteed to be imperfect, but the calculation won't be.
Calculation must always be imperfect, though. You can never perfectly calculate sqrt(2).
Geometric constructions do presume that you have a perfectly straight edge and perfectly rigid compass, though.
Geometric constructions do presume that you have a perfectly straight edge and perfectly rigid compass, though.
Can you perfectly calculate 1/6? sqrt(2) has a similar exact representation as a periodic "sequence"; its continued fraction is [1;2,2,2,2,2,...].
That's of limited use if you're trying to measure something, but in that case, conceptual representations are out (where the symbol √ and a picture of a square are equally valid), and you're choosing between a geometric approach where your accuracy is limited by the quality of your tools, or a symbolic approach where your accuracy is limited by how much accuracy you want.
That's of limited use if you're trying to measure something, but in that case, conceptual representations are out (where the symbol √ and a picture of a square are equally valid), and you're choosing between a geometric approach where your accuracy is limited by the quality of your tools, or a symbolic approach where your accuracy is limited by how much accuracy you want.
Continued fractions require and infinite number of operations to be worked out, while periodic decimals are both finite in different bases and do not require any operations to decode.
There's no contradiction here.
You can refine your real-world realization of a geometric construction by eg executing it bigger and bigger. (Or doing something more clever about the errors.)
Similarly, you can keep working on your calculation of sqrt(2) as a decimal number, and keep adding digits.
You can refine your real-world realization of a geometric construction by eg executing it bigger and bigger. (Or doing something more clever about the errors.)
Similarly, you can keep working on your calculation of sqrt(2) as a decimal number, and keep adding digits.
The US education system varies tremendously. I don't think you can credibly claim that this teaching method is "standard" in the US. At least my experience was much better than you described. Indeed it wasn't until university-level mathematics that I started to get into the "stop trying to understand how they work and just memorize these formulae". Fortunately my engineering classes provided a through-line for understanding how those formulae worked, and I was much stronger for it than my math-major contemporaries.
The best math class I ever had was a drafting class called "Descriptive Geometry". In that college class we used a drafting table to solve math problems.
An easy example would be the length of line of the corner seam of a hip roof. Sadly I have forgotten what the more complex problems were. Fantastic class that isn't offered anymore.
An easy example would be the length of line of the corner seam of a hip roof. Sadly I have forgotten what the more complex problems were. Fantastic class that isn't offered anymore.
Btw, the game Euclidea on iOS is fun for that. You "discover" and prove concepts and progress through higher-orders of geometry proofs.
Oh god, no. Not at all, especially in the US.
I can tell you how great my math education was. I don't even recall the quadratic formula or completing the square is. I'm almost certain they were part of the curriculum at some point.
I recall in later math course, notably calculus, the teacher assumed we we're familiar with some concept because we we're supposedly taught it the prior year, yet not a single student in the class could recall it having been taught before.
I recall in later math course, notably calculus, the teacher assumed we we're familiar with some concept because we we're supposedly taught it the prior year, yet not a single student in the class could recall it having been taught before.
Are you sure? My teacher did that.
It doesn't take very long, and I can definitely see someone forgetting all about it.
It doesn't take very long, and I can definitely see someone forgetting all about it.
I distinctly remember "accidentally" deriving it when I forgot the formula on a test; perhaps my proudest math moment (though really I was just scrambling apply any rule I could think of that got me closer to the vague form I remembered)
And then I got really annoyed that no one ever told me to do that before and started discounting teachers for years onwards.. probably to my own detriment.
And then I got really annoyed that no one ever told me to do that before and started discounting teachers for years onwards.. probably to my own detriment.
Yeah, we got it as bonus homework, to derive it ourselves.
Exactly. Probably every discovery in math starts out with something that's only close to the truth (like multiplication-is-repeated-addition) but generalizing it to open up whole new areas.
This part of math is so exciting and compelling, I feel like it should be more of a focus in schools.
This part of math is so exciting and compelling, I feel like it should be more of a focus in schools.
> You can definitely see multiplication as repeated addition.
Doesn't work in every context where multiplication is defined.
> It's even a useful thing to do, how else would you define multiplication?
Perhaps something along the lines of: A multiplication is an endomorphism on an additive semigroup.
Doesn't work in every context where multiplication is defined.
> It's even a useful thing to do, how else would you define multiplication?
Perhaps something along the lines of: A multiplication is an endomorphism on an additive semigroup.
> how else would you define multiplication?
With my six year old I told them to say "groups of" instead of "times". So 3 X 2 isn't read as "three times two" it is "three groups of two" which I think has been helpful.
With my six year old I told them to say "groups of" instead of "times". So 3 X 2 isn't read as "three times two" it is "three groups of two" which I think has been helpful.
> You can definitely see multiplication as repeated addition.
Only for rational numbers. Doesn't work for real and complex numbers.
> It's even a useful thing to do, how else would you define multiplication?
Axiomatically, not algorithmically.
Only for rational numbers. Doesn't work for real and complex numbers.
> It's even a useful thing to do, how else would you define multiplication?
Axiomatically, not algorithmically.
Multiplication of real and complex numbers is typically defined by starting from repeated addition, extending this notion to rationals, and then extending that notion to reals by taking limits.
How exactly are you going to present multiplication of real numbers axiomatically without essentially including an axiom that bootstraps everything from repeated addition?
I suppose you can try defining the reals as "the unique complete ordered field" or the complex numbers as "the unique algebraically closed field of characteristic zero with cardinality c," but I don't think either of those are pedagogically useful to someone who is still learning what multiplication is.
How exactly are you going to present multiplication of real numbers axiomatically without essentially including an axiom that bootstraps everything from repeated addition?
I suppose you can try defining the reals as "the unique complete ordered field" or the complex numbers as "the unique algebraically closed field of characteristic zero with cardinality c," but I don't think either of those are pedagogically useful to someone who is still learning what multiplication is.
Multiplication of the Surreals is a recursive operation using sums (addition and subtraction on the left and right sets). Since the Reals are a strict subfield of the Surreals one can define multiplication of the reals using only the same recursive formula and restricting both operands to be Reals.
Its probably worth emphasising that the recursion you need to "construct" the Surreals is infinite, in other words this does not give a reasonable algorithm to (for example) add two real numbers, you need S_omega in order to have even all rational numbers.
The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, in particular define a particular "real number" to be the set of all rational numbers less than it, for example sqrt(2) is defined to be the set of all rationals p/q such that p^2/q^2 < 2. We can "recursively" define addition of these real numbers in terms of addition of rational numbers because to add two reals you "just" have to add all the rationals in their respective sets.
In general there is no sensible algorithm to do anything in the real numbers, since most real numbers aren't even computable (there is no way to represent an arbitrary real number on a Turing machine).
The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, in particular define a particular "real number" to be the set of all rational numbers less than it, for example sqrt(2) is defined to be the set of all rationals p/q such that p^2/q^2 < 2. We can "recursively" define addition of these real numbers in terms of addition of rational numbers because to add two reals you "just" have to add all the rationals in their respective sets.
In general there is no sensible algorithm to do anything in the real numbers, since most real numbers aren't even computable (there is no way to represent an arbitrary real number on a Turing machine).
Of course, once you decide to iterate over uncountable sets, infinity starts to appear all the time.
This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.
This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.
Multiplying algebraics is trivial. A rectangle with sides sqrt(3) and sqrt(2) has area sqrt(6), which can be approximated by a decimal if needed.
There are countable/computable/constructable subsets of the reals where multiplication has a finite algorithm and is it repeated addition.
One example is the algebraics, as well as extensions the including a few special constants like pi. These are the subsets of the reals most commonly used for math and science. So in a wide range of problems areas, multiplication is not just repeated addition.
There are countable/computable/constructable subsets of the reals where multiplication has a finite algorithm and is it repeated addition.
One example is the algebraics, as well as extensions the including a few special constants like pi. These are the subsets of the reals most commonly used for math and science. So in a wide range of problems areas, multiplication is not just repeated addition.
I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "algorithmic" description of real addition or multiplication is probably a bad idea.
I wasn't even attempting to give an "algorithmic" description. Just a formulaic one, that depends on the axiom of infinity (and accepting transfinite induction as a valid process). Since even at least one of the usual process for constructing the Reals (Dedekind cuts) needs this I don't feel it's much of a stretch in reasoning. And the Surreals have a nicer recursive formula for multiplication that eventually turns into repeated sums, and they're a strict superset of the Reals, so it does apply there.
If you want an algorithmic (no possible need an infinite number of steps) explicit construction of anything on the Reals you're going to be disappointed. They're an infinite set.
If you want an algorithmic (no possible need an infinite number of steps) explicit construction of anything on the Reals you're going to be disappointed. They're an infinite set.
You can get algorithmic explicit constructions of (for example) integer addition, rational addition and multiplication and even addition and multiplication of algebraic numbers on a Turing machine (all algebraic numbers are computable). All of these sets are infinite. The reals are particularly "badly behaved" even as far as infinite sets go.
If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.
Not algorithms. There will be infinite addition involved, and algorithms are finite.
Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots.
This metaphor is just training wheels. At some point you should lose it.
Together with the notion that "multiplication is repeated addition" comes the notion that numbers are quantities. Only some of them are, and this isn't really what makes them numbers. Now what exactly gets repeated, when you don't have quantities?
Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots.
This metaphor is just training wheels. At some point you should lose it.
Together with the notion that "multiplication is repeated addition" comes the notion that numbers are quantities. Only some of them are, and this isn't really what makes them numbers. Now what exactly gets repeated, when you don't have quantities?
Multiplication on the naturals is repeated addition.
You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...
You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...
Number theorists work with integers, not naturals. And they too will agree with me, as they mostly work with polynomials, which show exactly that.
I also didn't say natural numbers are training wheels, just this metaphor, which to use HN lingo: doesn't scale.
I also didn't say natural numbers are training wheels, just this metaphor, which to use HN lingo: doesn't scale.
Number theorists work with complex numbers and irrational numbers too
https://www.britannica.com/science/Riemann-zeta-function
https://www.britannica.com/science/Riemann-zeta-function
> There will be infinite addition involved, and algorithms are finite.
People are just bit sloppy, and say algorithm when they mean something slightly different.
See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion
Basically, you don't want an 'algorithm' here to produce the whole number.
All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).
People are just bit sloppy, and say algorithm when they mean something slightly different.
See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion
Basically, you don't want an 'algorithm' here to produce the whole number.
All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).
Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite precision aren't a problem".
Not quite.
Consider the following real number made of binary digits:
Enumerate all Turing machines and all possible inputs, iff the i-th machine/input combination holds, the i-th binary digit in our number is 0, otherwise 1.
This number is well-defined (once you fix your enumeration scheme).
But there's no finite algorithm to produce approximations in your sense.
What I was after were what's also called Computable numbers (https://en.wikipedia.org/wiki/Computable_number). But I used the more general term of co-recursion, that also applies to arbitrary other data-structures like infinite lists, or with some generalization, infinite event-loops where the important condition is that each run through the body of the loop only takes finite time.
Consider the following real number made of binary digits:
Enumerate all Turing machines and all possible inputs, iff the i-th machine/input combination holds, the i-th binary digit in our number is 0, otherwise 1.
This number is well-defined (once you fix your enumeration scheme).
But there's no finite algorithm to produce approximations in your sense.
What I was after were what's also called Computable numbers (https://en.wikipedia.org/wiki/Computable_number). But I used the more general term of co-recursion, that also applies to arbitrary other data-structures like infinite lists, or with some generalization, infinite event-loops where the important condition is that each run through the body of the loop only takes finite time.
Algorithms can work on symbolic formulas, and symbols can represent anything; infinite objects, operations on infinite objects, infinite sets of operations on infinite objects, and so on.
Algorithms aren’t necessarily finite, eg “while True, print 1”.
Also, an irrational exponent is the product of component factors.
b = Prod(0,inf) a^[x_i * 10^(-i)] = a^x
So even with irrational numbers, operations can be decomposed - such as exponentiation into multiplication of integer exponents and roots.
Which makes sense, because in the sequence definition of reals you need a way to generate the resulting sequence from the two original sequences.
I think you’re trying to claim more than is true.
Also, an irrational exponent is the product of component factors.
b = Prod(0,inf) a^[x_i * 10^(-i)] = a^x
So even with irrational numbers, operations can be decomposed - such as exponentiation into multiplication of integer exponents and roots.
Which makes sense, because in the sequence definition of reals you need a way to generate the resulting sequence from the two original sequences.
I think you’re trying to claim more than is true.
Algorithms are not finite. E.g. Newton's method.
First sentence on Wikipedia[1]:
> In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ (About this soundlisten)) is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation.
Newton's method is finite too. You perform finitely many iterations. It doesn't calculate roots. It calculates their approximations.
[1]: <https://en.wikipedia.org/wiki/Algorithm>
> In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ (About this soundlisten)) is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation.
Newton's method is finite too. You perform finitely many iterations. It doesn't calculate roots. It calculates their approximations.
[1]: <https://en.wikipedia.org/wiki/Algorithm>
If you use a termination condition that has to do with convergence of iterates instead of a number of iterations (often the case), then you generally don't know beforehand the length of the finite sequences. Maybe you know a bound, but in general you might not even have that.
In an important sense, it only becomes a finite algorithm. It isn't one. You cannot write the finite sequence of instructions down. It's got loops.
To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers.
In an important sense, it only becomes a finite algorithm. It isn't one. You cannot write the finite sequence of instructions down. It's got loops.
To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers.
> To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers.
Not really, since approximations, no matter how accurate, don't preserve algebraic properties. You only get to know what it's bigger/smaller than.
Not really, since approximations, no matter how accurate, don't preserve algebraic properties. You only get to know what it's bigger/smaller than.
I think I understand what you mean, so let me dial back "anything involving the real numbers".
If you are representing or thinking of "sqrt(2)" as "the positive solution to x^2 = 2", then you preserve algebraic properties. But you generally (correct me if I'm wrong) don't get to know whether it's bigger or smaller than something else of the form "the _choose_uniquely_ solution to _some_equation_" unless you rely on an argument where you invoke approximations.
If you are representing or thinking of "sqrt(2)" as "the positive solution to x^2 = 2", then you preserve algebraic properties. But you generally (correct me if I'm wrong) don't get to know whether it's bigger or smaller than something else of the form "the _choose_uniquely_ solution to _some_equation_" unless you rely on an argument where you invoke approximations.
Well, no, not really. The standard definition of the reals is as the unique nontrivial totally-ordered, Dedekind-complete, Archimedean field up to isomorphism.
So what you would really need is a uniqueness proof, with addition and multiplications "provided" by the hypothesis.
And how do you prove that a totally-ordered, Dedekind-complete, Archimedean field does not lead to contradiction besides constructing it explicitly by bootstrapping from natural numbers?
Nothing in the construction requires you to show an algorithm that given x, y \in R allows you to compute x+y and xy. You would make the usual Dedekind construction and show it satisfies the axioms of such a field. (as a matter of fact, no such algorithm exists in full generality!)
It's probably a tomato/tomato kind of thing, but I'm only objecting to the 'algorithm' part of parent's comment.
It's probably a tomato/tomato kind of thing, but I'm only objecting to the 'algorithm' part of parent's comment.
>Only for rational numbers. Doesn't work for real and complex numbers.
Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication.
To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.
Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication.
To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.
Yep, it is a clickbait
Funny, my teachers in middle and high school frowned on using the quadratic formula until we could demonstrate we knew how to complete the square.
>These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.
Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case.
When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics.
Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method.
The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.
Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case.
When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics.
Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method.
The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.
To make this more explicit, mathematics (even at the primary school level) is arguably more about “proving” things than it is about computation.
Obviously you’re not writing proofs in 3rd grade, but the emphasis should be on students being able to say why their answer is correct rather than just being able to produce the correct answer.
Obviously you’re not writing proofs in 3rd grade, but the emphasis should be on students being able to say why their answer is correct rather than just being able to produce the correct answer.
> how else would you define multiplication?
In Church arithmetic, multiplication is defined as function composition; that is, the dot operator `.` in Haskell-like languages. Addition is considerably more complex; it is defined as `lift (.)`. Exponentiation is even simpler; it is function application, so the `id` function - except that we write exponentiation backwards, so actually `flip id`.
In Church arithmetic, multiplication is defined as function composition; that is, the dot operator `.` in Haskell-like languages. Addition is considerably more complex; it is defined as `lift (.)`. Exponentiation is even simpler; it is function application, so the `id` function - except that we write exponentiation backwards, so actually `flip id`.
Interesting, what about tetration then?
Good question: you can define m↑↑n as:
\m n -> n (exp m) 1'
flip flip id . flip id . flip id
flip flip id . flip id . flip flip id . flip id . flip idAnyone that teaches this way should find something else to do. Memorizing a formula is easy but if you want to do well later you must be able to complete the square.
I never learned to complete the square, and I managed to do well enough later to get a degree in math.
> You can definitely see multiplication as repeated addition
Only if you restrict to whole numbers. But multiplication does not just apply to whole numbers. How would you see multiplication by pi as repeated addition?
> how else would you define multiplication?
As a separate operation with its own properties. The subject article gives several examples of properties of multiplication that are simply different from properties of addition.
Only if you restrict to whole numbers. But multiplication does not just apply to whole numbers. How would you see multiplication by pi as repeated addition?
> how else would you define multiplication?
As a separate operation with its own properties. The subject article gives several examples of properties of multiplication that are simply different from properties of addition.
Multiplication by rational fractions p/q is just multiplication by whole p, followed by multiplication by 1/q. 1/q is just the number that when multiplied by q gives 1. So you can cash that all out as repeated addition.
Multiplication by irrationals is just an infinite sum of multiplication by rationals, no? x times pi = x times 3, plus x times 1 / 10, plus....
I don't see there's any conceptual issue here. Mathematicians feel free to correct me.
Multiplication by irrationals is just an infinite sum of multiplication by rationals, no? x times pi = x times 3, plus x times 1 / 10, plus....
I don't see there's any conceptual issue here. Mathematicians feel free to correct me.
> multiplication by 1/q
Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.
Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.
You can do it like this:
First define a*b for integers by addition.
Define the the rational numbers as ordered pairs of integers where the last number may not be 0. Consider the rational number (a,b) to be equivalent to (c, d) if a*d = b*c
(Exercise for the reader: Prove that this is an equivalence relation)
Now define (a, b) + (c, d) = (ad + bc, bd), (a, b) * (c, d) = (ac, bd)
(Exercise for the reader: Prove that the subset of rationals (n, 1) behaves just like the integers)
First define a*b for integers by addition.
Define the the rational numbers as ordered pairs of integers where the last number may not be 0. Consider the rational number (a,b) to be equivalent to (c, d) if a*d = b*c
(Exercise for the reader: Prove that this is an equivalence relation)
Now define (a, b) + (c, d) = (ad + bc, bd), (a, b) * (c, d) = (ac, bd)
(Exercise for the reader: Prove that the subset of rationals (n, 1) behaves just like the integers)
Hmm, so just go about it differently: x * p/q with p,q integers is (x*p) / q. The first bit is repeated addition. The second bit means, find the number r such that r * q = x * p. Even if r is not integer, q is integer, so we can try different numbers, add them to themselves q times, and close in on the answer.
So I still think conceptually it's fine to think of it as repeated addition? It might be algorithmically a bad way to do it ("increasingly perverse", taking limits when the number is irrational etc.). I don't know how computers actually implement multiplication - though wikipedia (https://en.wikipedia.org/wiki/Binary_multiplier) says it works via shifts and adds.
Or maybe, it's bad to teach it to kids this way? But then I think we need evidence from educationalists.
So I still think conceptually it's fine to think of it as repeated addition? It might be algorithmically a bad way to do it ("increasingly perverse", taking limits when the number is irrational etc.). I don't know how computers actually implement multiplication - though wikipedia (https://en.wikipedia.org/wiki/Binary_multiplier) says it works via shifts and adds.
Or maybe, it's bad to teach it to kids this way? But then I think we need evidence from educationalists.
> The first bit is repeated addition
So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.
Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of addition.
So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.
Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of addition.
I don't think so. Look more closely. (x * p) is repeated addition with p integer - we can do this no multiplication involved. Then dividing by q involves finding r such that r * q = x * p. Here, again, we can do this by finding r such that r, added to itself q times, equals x * p. Since q is integer, again, this again involves repeated addition. (It does need an algorithm to find the right value of r. I suppose that trying an arbitrary value, then increasing or decreasing it, would work.)
> (x * p) is repeated addition with p integer
Yes, that's the "first bit" that I was not raising an issue with.
> dividing by q involves finding r such that r * q = x * p
Which, as you note, requires a trial and error algorithm to find r. But in any case, r is the answer being sought, not the thing we're supposed to be adding to itself some number of times to get the answer.
Yes, that's the "first bit" that I was not raising an issue with.
> dividing by q involves finding r such that r * q = x * p
Which, as you note, requires a trial and error algorithm to find r. But in any case, r is the answer being sought, not the thing we're supposed to be adding to itself some number of times to get the answer.
OK. It sounds like we agree that my definition doesn't secretly involve multiplication and isn't circular?
How would you see multiplication by pi as repeated addition?
By taking 3 times as usual and then another .14 approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28. (edit: if you’re confused by /100, read it as “two places next to a point”)
If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:
By taking 3 times as usual and then another .14 approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28. (edit: if you’re confused by /100, read it as “two places next to a point”)
If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:
> By taking 3 times as usual and then another .14 approximately
First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition".
Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times.
> If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:
Add 1 to the 3 and keep the part to the right of the decimal point the same. Addition is commutative.
Basically, your argument is that multiplication is repeated addition as long as we're willing to adopt increasingly perverse interpretations of "repeated addition" as we expand the scope of the numbers we use. Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition".
Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times.
> If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:
Add 1 to the 3 and keep the part to the right of the decimal point the same. Addition is commutative.
Basically, your argument is that multiplication is repeated addition as long as we're willing to adopt increasingly perverse interpretations of "repeated addition" as we expand the scope of the numbers we use. Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
But what is that number? You only described an algorithm that relies on a vague definition of “part to the right”. See, you can’t even write it down (unlike 1 + 3.14 = 4.14), because pi is not really a number, in a sense. It is an infinite calculation that happens to converge between 3.14 and 3.15. You can never get rid of “pi” in your calculations unless it cancels out naturally, so for practical reasons it’s 3.14 and for theoretical reasons it’s “pi”.
Let’s make it clear: you cannot add pi or take pi times at all. You only can add to factors to the left of it (same as a+bi). That’s why you cannot multiply by pi by repeated addition, not because the numeric method is wrong.
Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
For generalized reasoning, yes. For teaching, not sure. Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that? When they learn, they ask themselves why and it’s a rabbit hole. You should stop somewhere before turning into maths professor, if that’s not your goal.
Let’s make it clear: you cannot add pi or take pi times at all. You only can add to factors to the left of it (same as a+bi). That’s why you cannot multiply by pi by repeated addition, not because the numeric method is wrong.
Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
For generalized reasoning, yes. For teaching, not sure. Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that? When they learn, they ask themselves why and it’s a rabbit hole. You should stop somewhere before turning into maths professor, if that’s not your goal.
> what is that number?
Pi minus 3. If you are saying you can't actually perform that operation, great! That means you are agreeing with me. See below.
> you cannot multiply by pi by repeated addition
Thank you for agreeing with my main point. See below.
> not because the numeric method is wrong.
If someone is going to claim that multiplication is repeated addition, then their definition of multiplication in terms of repeated addition must cover all cases. You are saying it doesn't cover pi, which means it doesn't cover all cases. So the definition is wrong.
> For generalized reasoning, yes. For teaching, not sure.
Why not? What's wrong with telling kids, multiplication in general is a distinct primitive operation, we are teaching you how to multiply whole numbers using repeated addition, but be aware that this method will not generalize to all cases?
> Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that?
A student who has been told what I just described above is not in either of these positions, so you are arguing against a straw man.
> When they learn, they ask themselves why and it’s a rabbit hole.
No, it's natural human curiosity which should be encouraged, not stomped on. At some point they'll either realize that they're up against material they're not yet ready for, and put it aside for later, or they'll end up being a math prodigy. Both outcomes are better ones than them just being told "this is it, don't ask any questions".
Pi minus 3. If you are saying you can't actually perform that operation, great! That means you are agreeing with me. See below.
> you cannot multiply by pi by repeated addition
Thank you for agreeing with my main point. See below.
> not because the numeric method is wrong.
If someone is going to claim that multiplication is repeated addition, then their definition of multiplication in terms of repeated addition must cover all cases. You are saying it doesn't cover pi, which means it doesn't cover all cases. So the definition is wrong.
> For generalized reasoning, yes. For teaching, not sure.
Why not? What's wrong with telling kids, multiplication in general is a distinct primitive operation, we are teaching you how to multiply whole numbers using repeated addition, but be aware that this method will not generalize to all cases?
> Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that?
A student who has been told what I just described above is not in either of these positions, so you are arguing against a straw man.
> When they learn, they ask themselves why and it’s a rabbit hole.
No, it's natural human curiosity which should be encouraged, not stomped on. At some point they'll either realize that they're up against material they're not yet ready for, and put it aside for later, or they'll end up being a math prodigy. Both outcomes are better ones than them just being told "this is it, don't ask any questions".
I think what they are saying is that if we can't say that multiplication is repeated addition due to the reason you propose, then addition itself is ill-defined as a concept, which is a bigger problem.
That by using your argument we move the whole debate one level down the fundamental operations and now you no longer have a way to simply explain addition or subtraction.
That by using your argument we move the whole debate one level down the fundamental operations and now you no longer have a way to simply explain addition or subtraction.
> not in the elementary school student's understanding of "repeated addition".
Pi isn't particularly well understood by elementary school students either.
Pi isn't particularly well understood by elementary school students either.
I don't understand.
The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant.
Furthermore, being able to compute/define multiplication through repeated addition doesn't prevent you from looking at the special properties of this new operator.
The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant.
Furthermore, being able to compute/define multiplication through repeated addition doesn't prevent you from looking at the special properties of this new operator.
I have a PhD in physics and more maths qualifications than I can shake a stick at; to me, multiplication is repeated addition.
I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”.
I also happen to have an 8-year-old going through third grade right now, and when we were talking through his homework, it was quite clear that using simple concepts he already knew (addition & subtraction) to explain slightly more complex things that he was learning (multiplication and division) was really useful to him. As I recall it being to me.
[aside] I think the maths schedule is more advanced now than it was in my day anyway - he only did multiplication and division this year, but he also did algebra and simultaneous linear equations now, as in:
a + b + 8 = 24
a - b = 4
“Solve for a and b”
Pretty sure I only did that in senior school (11 and up), not at age 8. No powers as yet (presumably they’ll come after the multiplication/division stuff), so no quadratic formula, but still...
[/aside]
I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”.
I also happen to have an 8-year-old going through third grade right now, and when we were talking through his homework, it was quite clear that using simple concepts he already knew (addition & subtraction) to explain slightly more complex things that he was learning (multiplication and division) was really useful to him. As I recall it being to me.
[aside] I think the maths schedule is more advanced now than it was in my day anyway - he only did multiplication and division this year, but he also did algebra and simultaneous linear equations now, as in:
a + b + 8 = 24
a - b = 4
“Solve for a and b”
Pretty sure I only did that in senior school (11 and up), not at age 8. No powers as yet (presumably they’ll come after the multiplication/division stuff), so no quadratic formula, but still...
[/aside]
I think the useful distinction is that, when you teach multiplication as a mechanical computation (arithmetic) it's useful to talk about it as repeated addition.
As you get to negative numbers, rational/irrational numbers, complex numbers, matrices, etc. it becomes more useful to think about multiplication in more abstract ways, among which repeated addition is still often a useful way to look at it.
I also think it's not particularly useful to talk about those other ways to think about multiplication until you actually need to.
It's too easy once you've mastered the concepts to forget how beginners look at them and struggle to understand them. I think the author isn't remembering what it's like to try to understand multiplication as a new concept - I certainly can't remember.
As you get to negative numbers, rational/irrational numbers, complex numbers, matrices, etc. it becomes more useful to think about multiplication in more abstract ways, among which repeated addition is still often a useful way to look at it.
I also think it's not particularly useful to talk about those other ways to think about multiplication until you actually need to.
It's too easy once you've mastered the concepts to forget how beginners look at them and struggle to understand them. I think the author isn't remembering what it's like to try to understand multiplication as a new concept - I certainly can't remember.
I think the heart of the issue is whether it's more useful to teach children how to multiply two abstract numbers together as a kind of "mathematical procedure" that they need to memorize, or whether it's more useful to teach children that if they measure two sides of a square with a measuring tape, they can "multiply" the measurement and that the result is now in "square inches" rather than regular inches. And the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and other people believe that the procedural memorization does kids a disservice by divorcing mathematical thinking from the concrete world entirely.
And the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and other people believe that the procedural memorization does kids a disservice by divorcing mathematical thinking from the concrete world entirely.
In my very limited teaching experience, I think the answer differs based on the student. At the individual level I don't think there's much controversy. Just align with the student's learning style. At scale, I have no idea and do not have the data/experience to have a well-formed opinion.
I was perfectly happy to focus on mechanical mastery of the multiplication rituals well before I had any concrete reasons to use them. I know plenty of others didn't work that way.
In my very limited teaching experience, I think the answer differs based on the student. At the individual level I don't think there's much controversy. Just align with the student's learning style. At scale, I have no idea and do not have the data/experience to have a well-formed opinion.
I was perfectly happy to focus on mechanical mastery of the multiplication rituals well before I had any concrete reasons to use them. I know plenty of others didn't work that way.
Your last two paragraphs are, I believe, the crux of my own argument. Sure, matrices aren’t even commutative, and complex numbers have their own quirks because of i^2 == -1, but these concepts build on the earlier axioms the kid has learnt. Our entire education system is built on “lies-to-children”, and as you progress they point out that what you comfortably believed was a gross simplification. This is no different.
Speaking of “lies to children”, and since you are a physicist, it was only in my last year studying EE that Feynman’s QED came my way and just imagine my surprise on finding that photons do not travel in a straight line.
(I do agree with your pedagogical position.)
(I do agree with your pedagogical position.)
I have got almost exactly the same situation: PhD, physics, forgotten more math than most folks ever learn, etc., and with an 8 year old learning the same level of mathematics as you describe.
I think the only difference might be I used "iterated" rather than "repeated" when helping him. Anyone who is just learning multiplication likely lacks the depth of experience necessary to make use of the "correct" jargon and abstract concepts as a starting point. "Repeated addition" is a useful aid in learning the operation to build that experience.
I think the only difference might be I used "iterated" rather than "repeated" when helping him. Anyone who is just learning multiplication likely lacks the depth of experience necessary to make use of the "correct" jargon and abstract concepts as a starting point. "Repeated addition" is a useful aid in learning the operation to build that experience.
Another physics PhD chiming in here. I have never before noted the difference between "multiplier" and "multiplicand". The whole article has me rolling my eyes.
In fact, I would argue that multiplication being associative shows that this distinction is meaningless.
In fact, I would argue that multiplication being associative shows that this distinction is meaningless.
Surely you knew subtrahend and minuend from vector subtraction? Or divisor:dividend?
Since you're a physician, do you think that helps for multiplicative relationships in real world laws ? It took me decades .. sadly, to be comfy with handling U = RI formulas. On the algebraic level it's stupid simple, but for real world physics the meaning is more bidirectional coupling of ratios and amplitudes and taps into a different part of my brain.
Physicist, not physician, but really - I haven't used my physics knowledge directly in a few decades now... I've been a software engineer for most of my life :)
As for V=IR (I had to google U=RI, maybe U is the more modern version, but it was always V=IR when I were a lad), I don't really have a problem with ratios. When I was learning equations, the simple rule is "do unto one side whatever you do to the other", so ...
V = IR, divide by R -> V/R = I
I was happy with either representation, and I didn't think of it as multiplying, dividing, adding or subtracting, it's just "do the same thing" on each side. The problems I had were more "when do you apply Kirchoff's laws to figure something out, and when do you apply Ohm's law; that sort of thing you just get by experience, I think.
As for V=IR (I had to google U=RI, maybe U is the more modern version, but it was always V=IR when I were a lad), I don't really have a problem with ratios. When I was learning equations, the simple rule is "do unto one side whatever you do to the other", so ...
V = IR, divide by R -> V/R = I
I was happy with either representation, and I didn't think of it as multiplying, dividing, adding or subtracting, it's just "do the same thing" on each side. The problems I had were more "when do you apply Kirchoff's laws to figure something out, and when do you apply Ohm's law; that sort of thing you just get by experience, I think.
hum sorry about the physician.. freudian slip
what I meant about the ratio part is that physicists don't care much about the computation they care about whenever you know one piece, you know the other will variate through a third factor in a multiplicative manner.. maybe it makes sense, but it's not at all a structural notion like (x) === iterate(+,n)
what I meant about the ratio part is that physicists don't care much about the computation they care about whenever you know one piece, you know the other will variate through a third factor in a multiplicative manner.. maybe it makes sense, but it's not at all a structural notion like (x) === iterate(+,n)
Agreed. Multiplier and multiplicand are different words but the commutative property says their values can swap equivalently, so . . . what was the author's point again?
If, for a moment, you conceptualize of multiplication on non-negative whole numbers as repeated addition, then this is the algorithm:
> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
The point with bringing up dimensional analysis is that the above algorithm doesn't work because what does it mean to do `for i = 1 to 3 chairs/row`? You might think of it like
procedure product(multiplier, multiplicand)
acc := 0
for i = 1 to multiplier
acc := acc + multiplicand
end
return acc
end
> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
The point with bringing up dimensional analysis is that the above algorithm doesn't work because what does it mean to do `for i = 1 to 3 chairs/row`? You might think of it like
procedure product(multiplier, multiplicand)
acc := "0" # an "absolute" zero that cooperates with any dimension
each single_multiplier in multiplier
acc := acc + (multiplicand * single_multiplier)
end
return acc
end
This was exactly my thought, I don't understand why those two word exist since it's commutative. Perhaps all this confusion is purely semantic.
Multiplication over the reals is commutative. Matrix multiplication of non-square matrices isn't. Multiplication in a Ring isn't necessarily commutative. Other algebraic structures also have non-commutative multiplication.
One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use of the product operation "multiplication", at least in cases where there's only one product operation to use. EG Geometric Algebra has Inner, Outer, and Geometric products, so calling them "multiplication" seems less common IME.
One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use of the product operation "multiplication", at least in cases where there's only one product operation to use. EG Geometric Algebra has Inner, Outer, and Geometric products, so calling them "multiplication" seems less common IME.
I really don't get the point you're making. If we're going to pull out random examples, monoids aren't guaranteed to be abelian; strings and concatenation form a monoid that's not abelian.
If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue.
The original article was contrasting addition and multiplication on the basis that addends are called the same while factors are supposed to be called differently, which not only makes no sense (it's just a naming convention), but it also breaks down when you have more than two factors: what is the "c" in a x b x c called? Or we're talking about non-associative operations now?
If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue.
The original article was contrasting addition and multiplication on the basis that addends are called the same while factors are supposed to be called differently, which not only makes no sense (it's just a naming convention), but it also breaks down when you have more than two factors: what is the "c" in a x b x c called? Or we're talking about non-associative operations now?
My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not necessarily a useful distinction, and in the usual use of multiplication it's utterly useless and only adds confusion.
But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math.
Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!
But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math.
Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!
Sure, my counterpoint was just that the same reasoning technically applies to addition as well, but as you note "matrix multiplication" is taught in high school, while my example wouldn't come up.
I agree not calling it multiplication would probably help, perhaps something like "linear transformation composition" might encourage students to keep it separated from real multiplication, I jsut found the argument in the original article kind of ridiculous to be honest.
I agree not calling it multiplication would probably help, perhaps something like "linear transformation composition" might encourage students to keep it separated from real multiplication, I jsut found the argument in the original article kind of ridiculous to be honest.
Yea, math tends to reuse terms and notation across disciplines in ways that adds confusion rather than clarity. It’s much better to think of infinity for example as multiple independent concepts than assume it’s all the same idea.
They exist to distinguish the element being operated on (the LHS) and what it is operated on by (the RHS).
Total technicality, but I could see myself using the term multiplicand/multiplier in my code if I had to implement e.g. a stack-based parser for arithmetic expressions.
Total technicality, but I could see myself using the term multiplicand/multiplier in my code if I had to implement e.g. a stack-based parser for arithmetic expressions.
> to distinguish the element being operated on (the LHS)
I'm not sure what you mean, they are both operated on, it is a binary operator and commutative, there is literally no difference.
I'm not sure what you mean, they are both operated on, it is a binary operator and commutative, there is literally no difference.
I agree that it's a generally pointless distinctions, although it might be useful in some cases such as number systems where the multiplication isn't commutative, or for a particular implementation where the distinction matters.
After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2.
To return to TFA I think the author is talking from a pedagogical standpoint, that teaching that multiplication is just a bunch of additions under a trench coat is not the best way to go. I'm not sure that I agree personally.
In particular this bit regarding multiplier/multiplicand makes zero sense to me:
>Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
Of course 3 rows and 2 rows aren't the same, but what does it have to do with the order of the multiplication? Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2.
To return to TFA I think the author is talking from a pedagogical standpoint, that teaching that multiplication is just a bunch of additions under a trench coat is not the best way to go. I'm not sure that I agree personally.
In particular this bit regarding multiplier/multiplicand makes zero sense to me:
>Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
Of course 3 rows and 2 rows aren't the same, but what does it have to do with the order of the multiplication? Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
Suppose your friend Alice arranges your wedding. You ask her to arrange the lawn chairs in an arrangement of 2 rows and 3 columns. But she misinterprets your request as 3 rows and 2 columns. Oops. Now a particular family can't all sit in a single row without rearranging the chairs.
If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" may encode information that "6" does not.
If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" may encode information that "6" does not.
Except we are talking about scalar multiplication, that was the premise, not matrices.
The point isn't "matrix multiplication != scalar multiplication". The point is "the process of evaluating a reducible expression inherently discards information about the original expression", which is a fact about evalution rather than any specific operator. The fact that "the information discarded is of little consequence to the compressed result" is a quirk specific to scalar multiplication. Thus, the commutative property distracts from explicitly modeling the "AST" so that the student understands what multiplication represents under the hood, beyond the rote memorization of scalar multiplication tables.
Perhaps an analogous situation would be: Suppose a teacher wanted to introduce the notion of limits to a calculus curriculum. "That makes zero sense. The only things a student needs to know are the shortcuts for each parent function, e.g. that (d/dx x^2) reduces to (2x) via handwavey magic." But what if an engineer needs to integrate over an arbitrary curve? Can students solve the problem without being comfortable with Riemann Sums? Maybe 1st-year calc students should rederive the shortcuts from scratch? "Except we're talking about a math course, not an engineering course."
> In particular this bit regarding multiplier/multiplicand makes zero sense to me.
> Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
It's bizarre to simias (and you, I assume) because y'all can't imagine performing the operation without thunking. (Don't get me wrong, I think "repeated addition" is the best method. I'm just attempting to explain the opposite perspective so that it feels less bizarre.)
Perhaps an analogous situation would be: Suppose a teacher wanted to introduce the notion of limits to a calculus curriculum. "That makes zero sense. The only things a student needs to know are the shortcuts for each parent function, e.g. that (d/dx x^2) reduces to (2x) via handwavey magic." But what if an engineer needs to integrate over an arbitrary curve? Can students solve the problem without being comfortable with Riemann Sums? Maybe 1st-year calc students should rederive the shortcuts from scratch? "Except we're talking about a math course, not an engineering course."
> In particular this bit regarding multiplier/multiplicand makes zero sense to me.
> Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
It's bizarre to simias (and you, I assume) because y'all can't imagine performing the operation without thunking. (Don't get me wrong, I think "repeated addition" is the best method. I'm just attempting to explain the opposite perspective so that it feels less bizarre.)
I address that point specifically in my comment. What you say is that the units of the operands matter, which I agree, but the order doesn't. 2rows x 3chairs and 3chairs x 2rows is the same thing.
It distinguishes these two rectangles.
*** **
*** **
**If we are just looking at numbers with no units attached there is no difference. As soon as there are units, a difference arises.
I will collect 12 apples 5 times. The 12 apples are the multiplicand and the 5 times is the multiplier.
The answer retains the units from the multiplicand.
I will collect 12 apples 5 times. The 12 apples are the multiplicand and the 5 times is the multiplier.
The answer retains the units from the multiplicand.
Still don't see the difference.
Density times volume is mass. Volume times density is mass. Distance times force is work/energy...
"5 times I will connect 12 apples"
Density times volume is mass. Volume times density is mass. Distance times force is work/energy...
"5 times I will connect 12 apples"
You're correct. The GP in incorrect about their dimensional analysis. 5 apples 12 times yields apples because it's apples times a dimensionless scalar (count). Newtons times meters is always Newton-meters, never Newtons or meters. Units are never magically dropped in dimensional analysis.
Not all multiplication is commutative, e.g. matrices.
> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
The author has it all backwards. Basically that you can turn this arrangement by 90 degrees (and turning the chairs, if you do not modell them by points by 90 degrees) is the reason why multiplication is commutative. It is non trivial that 5+5+5+...5 (100 times) is the same as 100+100+100+100+100
The author has it all backwards. Basically that you can turn this arrangement by 90 degrees (and turning the chairs, if you do not modell them by points by 90 degrees) is the reason why multiplication is commutative. It is non trivial that 5+5+5+...5 (100 times) is the same as 100+100+100+100+100
Not defending the article, but how would you compute:
\pi*\pi using repeated addition?
You first define multiplication of natural numbers to be repeated addition, then define multiplication of rationals in terms of multiplication and addition of naturals, then define multiplication of reals in terms of multiplication of Cauchy sequences of rationals ;)
You are making a mistake that many students unfamiliar with abstract algebra make, which is to confuse a particular "encoding" or "implementation" with the algebraic structure.
The concept of a real closed field [0] (and its categorical second-order version, the Dedekind-complete ordered field) stands on its own, without multiplication being defined in terms of repeated addition. It is completely independent of whether you happen to encode the reals as Cauchy sequences, Dedekind cuts, or something else. The (equivalence classes of) Cauchy sequences are not the same thing as the real numbers, even if we sometimes abuse terminology in this way for expediency. The distinction becomes increasingly important as you delve into more exotic algebraic structures.
Another illustrative example is the Hessenberg product [1]. Even the ordinary product cannot really be reduced to "repeated addition", because you have to use the infinitary concept of a limit. And not just your everyday limit [2], but a limit on a proper-class sized domain [3]!
See also the category-theoretic product [4].
[0] https://en.wikipedia.org/wiki/Real_closed_field
[1] https://en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_ope...
[2] https://en.wikipedia.org/wiki/Limit_of_a_sequence
[3] https://en.wikipedia.org/wiki/Order_topology#Ordinal-indexed...
[4] https://en.wikipedia.org/wiki/Product_(category_theory)
The concept of a real closed field [0] (and its categorical second-order version, the Dedekind-complete ordered field) stands on its own, without multiplication being defined in terms of repeated addition. It is completely independent of whether you happen to encode the reals as Cauchy sequences, Dedekind cuts, or something else. The (equivalence classes of) Cauchy sequences are not the same thing as the real numbers, even if we sometimes abuse terminology in this way for expediency. The distinction becomes increasingly important as you delve into more exotic algebraic structures.
Another illustrative example is the Hessenberg product [1]. Even the ordinary product cannot really be reduced to "repeated addition", because you have to use the infinitary concept of a limit. And not just your everyday limit [2], but a limit on a proper-class sized domain [3]!
See also the category-theoretic product [4].
[0] https://en.wikipedia.org/wiki/Real_closed_field
[1] https://en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_ope...
[2] https://en.wikipedia.org/wiki/Limit_of_a_sequence
[3] https://en.wikipedia.org/wiki/Order_topology#Ordinal-indexed...
[4] https://en.wikipedia.org/wiki/Product_(category_theory)
I think its easy to get too hung up on particular definitions, you're free to define the reals to be the isomophism class of Dedekind-complete ordered fields if you like, I'm free to define the reals by some construction (Cauchy sequences, Dedekind cuts or whatever). As long as the chosen definitions are isomorphic it doesn't matter at all.
In general in a field I have addition and a multiplicative identity 1, as well as the distributive property, which means that for all elements x satisfying x = 1 + 1 + .... 1 for some number of ones I can write a.x = a.(1 + 1 + .... 1) = a + a + .... a, so there is always a subset for which multiplication works like repeated addition.
In general in a field I have addition and a multiplicative identity 1, as well as the distributive property, which means that for all elements x satisfying x = 1 + 1 + .... 1 for some number of ones I can write a.x = a.(1 + 1 + .... 1) = a + a + .... a, so there is always a subset for which multiplication works like repeated addition.
You are confusing a theory [0] with a model [1] of that theory. These are two different things.
> there is always a subset for which multiplication works like repeated addition
A subset, yes.
[0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic)
[1] https://en.wikipedia.org/wiki/Model_theory
> there is always a subset for which multiplication works like repeated addition
A subset, yes.
[0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic)
[1] https://en.wikipedia.org/wiki/Model_theory
Na, I'm not - I'm explaining how to "compute: \pi*\pi using repeated addition". Step 1 - find a convenient Cauchy sequence that converges to \pi.
> Na, I'm not
Yes, you are. Cauchy sequences are a model of the real numbers (which are the Dedekind-complete ordered field).
> I'm explaining how to "compute: \pi*\pi using repeated addition"
How will you define π times π as a repeated addition, i.e. something of the form x + x + … + x?
Yes, you are. Cauchy sequences are a model of the real numbers (which are the Dedekind-complete ordered field).
> I'm explaining how to "compute: \pi*\pi using repeated addition"
How will you define π times π as a repeated addition, i.e. something of the form x + x + … + x?
There is not one Dedekind-complete ordered field, for every two Dedekind-complete ordered fields there is a unique isomorphism between them. For example the nxn diagonal matrices with real entries with all the entries along the diagonal equal are a Dedekind-complete ordered field.
I don't know why you are conflating the words "compute" and "define", do you understand how these are different words? I was responding to how to "compute: \pi*\pi using repeated addition", this is rather different to defining \pi*\pi as a repeated addition.
I don't know why you are conflating the words "compute" and "define", do you understand how these are different words? I was responding to how to "compute: \pi*\pi using repeated addition", this is rather different to defining \pi*\pi as a repeated addition.
> There is not one Dedekind-complete ordered field
Yes there is.
> For example the nxn diagonal matrices with real entries
That's a model, not the theory.
> I don't know why you are conflating the words "compute" and "define"
This whole discussion is about definition.
Your own comment started with "You first define..."
And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by computation you mean a finitary algorithmic process.
Yes there is.
> For example the nxn diagonal matrices with real entries
That's a model, not the theory.
> I don't know why you are conflating the words "compute" and "define"
This whole discussion is about definition.
Your own comment started with "You first define..."
And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by computation you mean a finitary algorithmic process.
Ok, I think your definition of "field" is different to mine. Mine starts "A field is a set F with binary operations + and x" and then goes on to list some properties they have to have, yours seems to be doing something different. If you start with mine you get a whole bunch of different complete ordered fields, but you can easily show they're all isomorphic (e.g. Spivak does this IIRC).
>This whole discussion is about definition
No, it isn't this whole discussion is answering the question about computing pi * pi. Maybe I was slightly sloppy in my first answer to that question, I was only attempting to sketch the method.
I don't really want to compute products of arbitrary reals in that way (or any way), but the method I sketched works for computable reals which is sufficient to cover the case of \pi. By computation I mean something that runs on a Turing machine, but I don't assume finitary (I'm happy for my Turing machine to keep producing more and more bits of precision forever).
>This whole discussion is about definition
No, it isn't this whole discussion is answering the question about computing pi * pi. Maybe I was slightly sloppy in my first answer to that question, I was only attempting to sketch the method.
I don't really want to compute products of arbitrary reals in that way (or any way), but the method I sketched works for computable reals which is sufficient to cover the case of \pi. By computation I mean something that runs on a Turing machine, but I don't assume finitary (I'm happy for my Turing machine to keep producing more and more bits of precision forever).
Exactly. I've scrolled through hundreds of comments here now and it really is beyond me how the question of whether you can define x·y in terms of addition for x and y being arbitrary reals is even a matter of debate.
The question is not whether you can define it in terms of addition in some abstract way, but whether you can define it in terms of repeated addition, i.e. something of the form x + x + … + x.
> The question is not whether you can define it in terms of addition in some abstract way
This is splitting hairs. The reals themselves are defined "in some abstract way". For instance, what does addition of reals even mean? How do you add two arbitrary real numbers? Exactly, by adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) and the only abstraction involved here is the abstraction due to the way the reals are constructed in the first place.
This is splitting hairs. The reals themselves are defined "in some abstract way". For instance, what does addition of reals even mean? How do you add two arbitrary real numbers? Exactly, by adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) and the only abstraction involved here is the abstraction due to the way the reals are constructed in the first place.
> This is splitting hairs.
Nope.
The question addressed by the article is precisely whether multiplication is repeated addition.
> adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication)
Nope.
See https://news.ycombinator.com/item?id=26336108.
Nope.
The question addressed by the article is precisely whether multiplication is repeated addition.
> adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication)
Nope.
See https://news.ycombinator.com/item?id=26336108.
pi + pi + pi + (.141592... of pi) ~= 9.8696
You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.
You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.
Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.
Sure it is, as long as you're willing to repeat in increments of real numbers.
And that's my point, basically - By the time we're discussing real numbers, we need multiplication as an operator, because we're discussing ratios already.
And that's my point, basically - By the time we're discussing real numbers, we need multiplication as an operator, because we're discussing ratios already.
This is an interesting statement.
While it's true, it obscures the fact that pi is defined as a fraction: Circumference/Diameter.
That this fraction cannot be represented as a numeric fraction is one of the great insights of early mathematics.
While it's true, it obscures the fact that pi is defined as a fraction: Circumference/Diameter.
That this fraction cannot be represented as a numeric fraction is one of the great insights of early mathematics.
I wouldn't call the fact that pi is irrational an insight of "early mathematics." We knew that sqrt(2) was irrational around 500 BC if not earlier. We didn't know pi was irrational until around the time of the American revolution.
Interesting how computers (which only understand ‘1’ and ‘0’ can do multiplication of reals, then. Unless you’re intel, of course... (no, I will never let it go :)
Can we not consider that the algorithm taught for multiplication of real numbers is repeated multiplication of natural numbers which can be seen as repeated addition of natural numbers so we could define an addition only algorithm for multiplication of real numbers.
It's not cheating. It's precisely the point. There are contexts where multiplication is not simply repeated addition.
However, pi is already a ratio…
Right, that's why it's cheating. He's asking about a number set that requires the concept of ratios, and ratios are multiplication.
3.14159 + 3.14159 + 3.14159 + (3.14159*0.14159)
How would you compute \pi+\pi using addition?
Irrationals break all our intuitions.
Irrationals break all our intuitions.
I think the author's argument becomes more clear when you consider multiplication in rings other than the integers, for example (square) matrices. The product of two matrices A*B does not correspond to repeated addition, and it is not commutative (A*B does not equal B*A in general).
I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning about mathematics. However, this is true for a lot of basic properties that hold over the integers and not in other domains, so I'm not really convinced that it's actually wrong to teach kids about multiplication this way.
I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning about mathematics. However, this is true for a lot of basic properties that hold over the integers and not in other domains, so I'm not really convinced that it's actually wrong to teach kids about multiplication this way.
The better line of thought here might be "A*B is not multiplication, it's function composition".
It literally looks nothing like multiplication when carried out manually.
It literally looks nothing like multiplication when carried out manually.
It looks exactly like scalar multiplication when your matrices are diagonal.
Then: all matrices look like the identity matrix if applied to a suitably rescaled eigenvector.
Not the zero matrix, or any non-invertible matrix over a field with 0 characteristic.
The Jordan-Chevalley decomposition makes the difference precise -- the nilpotent part (off diagonal 1's in the jordan normal form)
The Jordan-Chevalley decomposition makes the difference precise -- the nilpotent part (off diagonal 1's in the jordan normal form)
We're talking about matrix/matrix multiplication here, not matrix/vector, right?
I was talking about matrix/matrix composition (rather than multiplication) at first. Then I talked about matrix application.
A matrix is a function. Not all functions can be represented by matrices (although all smooth functions can be represented by a Taylor series that sums over matrices (and "the sum C=A+B" really means "the function C such that Cx = Ax+Bx for all x in the range of both A and B"))
A matrix is a function. Not all functions can be represented by matrices (although all smooth functions can be represented by a Taylor series that sums over matrices (and "the sum C=A+B" really means "the function C such that Cx = Ax+Bx for all x in the range of both A and B"))
I rather dislike this way of saying things. A matrix is a matrix, its a table of numbers, and one can usefully define operations like addition and multiplication on them.
A linear map is a linear map, it maps between vector spaces and it obeys some nice axioms. You can define addition and composition as operations on them.
It is a quite interesting and non-trivial theorem that if you fix a particular choice of basis then you get "for free" a bijection between linear maps and matrices. The bijection between matrices and linear maps is completely dependent on the basis you choose, however, and there certainly isn't a canonical way to choose the basis.
Often it is natural to change basis to make it easier solve some particular problem, and then the matrix that represents a particular linear map will change, but the properties of the linear map won't change (for example its rank, kernel, eigenvectors/values etc).
A linear map is a linear map, it maps between vector spaces and it obeys some nice axioms. You can define addition and composition as operations on them.
It is a quite interesting and non-trivial theorem that if you fix a particular choice of basis then you get "for free" a bijection between linear maps and matrices. The bijection between matrices and linear maps is completely dependent on the basis you choose, however, and there certainly isn't a canonical way to choose the basis.
Often it is natural to change basis to make it easier solve some particular problem, and then the matrix that represents a particular linear map will change, but the properties of the linear map won't change (for example its rank, kernel, eigenvectors/values etc).
The mathematical term is "product". It's a product.
It's just that matrix multiplication is not the same thing as scaler multiplication. I think you need to understand that no matter how you define or think of scaler multiplication.
Hadamard multiplication of matrices is a scalar multiplication.
https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition.
For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases and all of the more advanced concepts the author (and the related articles) discuss can be taught later.
For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases and all of the more advanced concepts the author (and the related articles) discuss can be taught later.
> I'm at a loss how you would teach it without explaining that it is repeated addition.
How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition?
Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't always, it only is for whole numbers. Others might be better served by telling them up front that multiplication is a separate operation, but can be "emulated" for whole numbers by repeated addition. (As a child I was in the latter group; I hated it when teachers told me one thing in one grade, and then said "well, what you were taught before isn't actually correct..." in a later grade. How many more times would they change the rules?)
How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition?
Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't always, it only is for whole numbers. Others might be better served by telling them up front that multiplication is a separate operation, but can be "emulated" for whole numbers by repeated addition. (As a child I was in the latter group; I hated it when teachers told me one thing in one grade, and then said "well, what you were taught before isn't actually correct..." in a later grade. How many more times would they change the rules?)
The approach you describe definitely does not work for all children. The vast majority (that I've taught) are better served by a scaffold of increasingly complex metaphors and definitions. The fact that the metaphor of repeated addition wears thin actually highlights the idea that different sets of mathematical objects have different operators for most students in a way that just telling them that ahead of time does not.
The changing of the rules is good preparation for scientific study where our understanding actually changes over time and we have to internalize new models regularly. I taught my chemistry students that quarks only combine in pairs or triplets, until one year Fermilab released evidence that quarks combine in quartets and quintets while I was teaching the unit. How could I avoid telling my students that what I'd taught them before was incorrect? Literally no one knew. So the practice of revising our ideas has utility.
The changing of the rules is good preparation for scientific study where our understanding actually changes over time and we have to internalize new models regularly. I taught my chemistry students that quarks only combine in pairs or triplets, until one year Fermilab released evidence that quarks combine in quartets and quintets while I was teaching the unit. How could I avoid telling my students that what I'd taught them before was incorrect? Literally no one knew. So the practice of revising our ideas has utility.
Maybe I've been doing too much computer-assisted proofs recently, but I'm not understanding what all the fuss is. At some point you need a `Nat.mul` that's concretely defined, and probably the only reasonable definition is recursively using repeated addition (and if your natural numbers are in unary, which is theoretically the simplest, then even `Nat.add` is going to be defined recursively by repeated incrementing! like taking a bead from a pile one at a time and putting it into another).
Then you can define an interface with abstract operations like addition and subtraction, then any given number system can try to implement this interface. Naturals, integers, rationals, reals, polynomials, rational functions, complex numbers, and many others can implement at least the semiring interface. But what's important, I think, is that there's no universal multiplication operation -- it's just a word that fills in for whatever is the right multiplication operation at the moment. It's some ad-hoc polymorphism.
Unless you're accepting the real line as being axiomatic (maybe from geometry, where you can define all the basic operations using ruler-and-compass constructions), all the definitions of multiplication for all the above number systems are going to be, eventually, based on the one for the naturals.
Then you can define an interface with abstract operations like addition and subtraction, then any given number system can try to implement this interface. Naturals, integers, rationals, reals, polynomials, rational functions, complex numbers, and many others can implement at least the semiring interface. But what's important, I think, is that there's no universal multiplication operation -- it's just a word that fills in for whatever is the right multiplication operation at the moment. It's some ad-hoc polymorphism.
Unless you're accepting the real line as being axiomatic (maybe from geometry, where you can define all the basic operations using ruler-and-compass constructions), all the definitions of multiplication for all the above number systems are going to be, eventually, based on the one for the naturals.
> At some point you need a `Nat.mul` that's concretely defined
And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation.
But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever. The latter claim is the claim I'm arguing against.
> Unless you're accepting the real line as being axiomatic
Your computer can't manipulate real numbers. It can only implement finite operations on numbers with a finite bit size, which, as above, is a set of numbers isomorphic to the integers. So if you want to work with the real numbers themselves, abstractly, then you can't expect what the computer does to carry over.
And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation.
But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever. The latter claim is the claim I'm arguing against.
> Unless you're accepting the real line as being axiomatic
Your computer can't manipulate real numbers. It can only implement finite operations on numbers with a finite bit size, which, as above, is a set of numbers isomorphic to the integers. So if you want to work with the real numbers themselves, abstractly, then you can't expect what the computer does to carry over.
My computer can manipulate and prove things about real numbers just fine. I think it can do at least everything I can do normally, though much more laboriously.
What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences), and then you can embed the whole numbers inside and pretend that that was where they were the whole time.
I do get that you were giving different pedagogical approaches, and this was just one of them, but I think this one is at least as bad of a misrepresentation as saying "multiplication is repeated addition" without specifying "for whole numbers." That is, unless you're like the Greek geometers and accept a continuum from the start.
(There are other number systems that the integers sit inside, though, that don't themselves sit inside the real or complex numbers, which is a reason I would hesitate to think about whole numbers this way in general.)
What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences), and then you can embed the whole numbers inside and pretend that that was where they were the whole time.
I do get that you were giving different pedagogical approaches, and this was just one of them, but I think this one is at least as bad of a misrepresentation as saying "multiplication is repeated addition" without specifying "for whole numbers." That is, unless you're like the Greek geometers and accept a continuum from the start.
(There are other number systems that the integers sit inside, though, that don't themselves sit inside the real or complex numbers, which is a reason I would hesitate to think about whole numbers this way in general.)
> My computer can manipulate and prove things about real numbers just fine.
If you mean it can, with appropriate software, do symbolic manipulations of general formulas that are valid for real numbers, yes, of course. But that's not the same as doing specific concrete computations with them.
> What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences)
Ah, I see. As far as I know the axiomatic reasoning involved can go either way. But in any case, that wasn't what I was trying to get at with the term "emulating"; I'm sorry if my use of that term caused confusion, and I agree with you that telling a child something like "the whole numbers are inside the real numbers" without qualification would also be a misrepresentation. My point was simply that teaching a child a specific computational procedure, repeated addition, in order to get an answer to particular multiplication problems does not require telling the child that multiplication is repeated addition, without qualification. The two things are distinct, and I am fine with the former; I only object to the latter.
If you mean it can, with appropriate software, do symbolic manipulations of general formulas that are valid for real numbers, yes, of course. But that's not the same as doing specific concrete computations with them.
> What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences)
Ah, I see. As far as I know the axiomatic reasoning involved can go either way. But in any case, that wasn't what I was trying to get at with the term "emulating"; I'm sorry if my use of that term caused confusion, and I agree with you that telling a child something like "the whole numbers are inside the real numbers" without qualification would also be a misrepresentation. My point was simply that teaching a child a specific computational procedure, repeated addition, in order to get an answer to particular multiplication problems does not require telling the child that multiplication is repeated addition, without qualification. The two things are distinct, and I am fine with the former; I only object to the latter.
> If you mean it can, with appropriate software, do symbolic manipulations of general formulas that are valid for real numbers, yes, of course. But that's not the same as doing specific concrete computations with them.
No, I don't mean computer algebra systems, if that's what you mean. In something like Lean, you can set up an actual construction of the system of real numbers (in mathlib it's the Cauchy completion of the rationals, which in turn are a given by a numerator and positive nonzero denominator that are coprime; an integer is essentially a natural number and a sign bit, and natural numbers are inductively constructed from zero and taking successors). Maybe this will convince you that these are "the" real numbers: you can prove there are uncountably many of them, and more importantly that it is the unique complete ordered field (up to isomorphism).
No, I don't mean computer algebra systems, if that's what you mean. In something like Lean, you can set up an actual construction of the system of real numbers (in mathlib it's the Cauchy completion of the rationals, which in turn are a given by a numerator and positive nonzero denominator that are coprime; an integer is essentially a natural number and a sign bit, and natural numbers are inductively constructed from zero and taking successors). Maybe this will convince you that these are "the" real numbers: you can prove there are uncountably many of them, and more importantly that it is the unique complete ordered field (up to isomorphism).
> In something like Lean, you can set up an actual construction of the system of real numbers
I'm not familiar with Lean, but from what I can gather, it's a theorem proving system. That is included in what I was referring to. Proving general theorems is still not the same as doing specific concrete computations.
I'm not familiar with Lean, but from what I can gather, it's a theorem proving system. That is included in what I was referring to. Proving general theorems is still not the same as doing specific concrete computations.
Yes, it's a theorem proving system, and it's based on foundations that are powerful enough to represent all of (formal) math. Including something like Lean excludes math itself from being able to do "specific concrete computations," which is absurd.
Here are some examples to give you an idea of concrete calculations. Lean is not a CAS, so it needs some help in the form of tactic proofs (which are unreadable without an IDE, but their structure sort of mirrors work you'd do on paper):
(A meta-comment: I'm not sure why if you're not familiar with Lean you'd tell me what it can and cannot do. Maybe someone told you at some point "theorem prover systems prove general theorems" or "theorem provers can only prove things about computable reals" and you had no reason to believe that they weren't telling you the whole story? Don't take this too seriously, I'm just riffing on your original point.)
p.s. Looking back to the beginning of this thread, I realize my saying "I'm not understanding what all the fuss is" could be misinterpreted -- I didn't meant to refer to you in particular. This was meant to be in the context of all the comments and the original article, and somehow I ended up replying to yours. Maybe one thing that caught my eye was the idea that there was much to unlearn here, but I ended up not addressing that point because I didn't have anything cogent to say. I think the sibling comment put it well, that there's value in learning to unlearn.
Here are some examples to give you an idea of concrete calculations. Lean is not a CAS, so it needs some help in the form of tactic proofs (which are unreadable without an IDE, but their structure sort of mirrors work you'd do on paper):
import data.real.basic
import data.real.sqrt
import tactic
-- that 2^2 - 2 = 0
example : (real.sqrt 2)^2 - 2 = 0
:= by norm_num
-- that if x solves x^2 - 2x + 1, then x = 1.
example (x : ℝ) (h : x^2 - 2*x + 1 = 0) : x = 1
:= begin
have : x^2 - 2*x + 1 = (x - 1)*(x - 1), ring,
rw this at h,
cases eq_zero_or_eq_zero_of_mul_eq_zero h with h' h',
repeat {
convert_to x - 1 + 1 = 1,
norm_num,
rw h',
norm_num,
},
end
(A meta-comment: I'm not sure why if you're not familiar with Lean you'd tell me what it can and cannot do. Maybe someone told you at some point "theorem prover systems prove general theorems" or "theorem provers can only prove things about computable reals" and you had no reason to believe that they weren't telling you the whole story? Don't take this too seriously, I'm just riffing on your original point.)
p.s. Looking back to the beginning of this thread, I realize my saying "I'm not understanding what all the fuss is" could be misinterpreted -- I didn't meant to refer to you in particular. This was meant to be in the context of all the comments and the original article, and somehow I ended up replying to yours. Maybe one thing that caught my eye was the idea that there was much to unlearn here, but I ended up not addressing that point because I didn't have anything cogent to say. I think the sibling comment put it well, that there's value in learning to unlearn.
[deleted]
I tend to think of multiplication as a transformation. "repeated additional" almost makes it sounds like we're merely telescoping an interval within a single dimension. Whereas "transformation" (to me), evokes more of a Cartesian Product or Quadrature sort of mental image.
You can teach (some intuition for) multiplication without numbers by multiplying lengths to get areas. Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows.
Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.
Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.
> Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows.
How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?
How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?
Using a calculator.
Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.
Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.
So teaching multiplication ends up as "whatever the calculator does when I press this button"?
Having been teaching programming I can see an immediate problem with it, whilst it works great as a starter tying it too hard in the mind will require more un-doing when it comes to more advanced concepts where it isn't an appropriate level.
On the other hand knowing the relation is a good hint at how to explain exponentiation.
In the end teachers should be aware of what will be needed at higher grades to use tools such as it as a starting point but not ingrain it too much and try to push students onto thinking of multiplication as it's own functional primitive once they get the hang of basic tables and start moving on to long multiplication.
On the other hand knowing the relation is a good hint at how to explain exponentiation.
In the end teachers should be aware of what will be needed at higher grades to use tools such as it as a starting point but not ingrain it too much and try to push students onto thinking of multiplication as it's own functional primitive once they get the hang of basic tables and start moving on to long multiplication.
You start by teaching rings, duh.
But seriously, her first objection is
> To define multiplication as repeated addition is to make multiplication a sub-species of addition.
To me, that means the author is wrong from the get-go. Not all cases of multiplication might be easily seen as repeated addition, but to get hung up on a deliberate linguistic misunderstanding? And to add insult to injury, she finishes by proposing to see multiplication as the answer to "how many or how much OF the unit", and shows a diagram with _repeating_ units.
Oh, never mind, the answer is at the end: she's got books to sell, and probably tries to improve her sales by appealing to parents with children that can't multiply 83 by 17 when they're 8.
But seriously, her first objection is
> To define multiplication as repeated addition is to make multiplication a sub-species of addition.
To me, that means the author is wrong from the get-go. Not all cases of multiplication might be easily seen as repeated addition, but to get hung up on a deliberate linguistic misunderstanding? And to add insult to injury, she finishes by proposing to see multiplication as the answer to "how many or how much OF the unit", and shows a diagram with _repeating_ units.
Oh, never mind, the answer is at the end: she's got books to sell, and probably tries to improve her sales by appealing to parents with children that can't multiply 83 by 17 when they're 8.
Lego is a great tool for this.
I also taught prime numbers by handing out (round) 1x1 pieces and told him to figure out which ones could be made into squares.
The teacher was surprised when - in the middle of a more convoluted explanation for primes - he just said "primes are numbers that can't be arranged as rectangles".
I also taught prime numbers by handing out (round) 1x1 pieces and told him to figure out which ones could be made into squares.
The teacher was surprised when - in the middle of a more convoluted explanation for primes - he just said "primes are numbers that can't be arranged as rectangles".
I don’t think the aruments made here are compelling. Multiplication can be defined in this way. Saying that multiplication is repeated addition is not the same as saying that multiplication IS addition. Addition relates to multiplication as multiplication relates to powers.
>> Multiplication requires different units.
>> 2 baskets × 3 apples per basket = 6 apples
No it doesn't. In that example the "baskets" and "per baskets" (ie x/baskets) are cancelling each other. 2 baskets x 3 apples/baskets = 2 x 3 apples = 6 apples. Once you remove the unnecessary terms the actual multiplication is still basic repeated addition.
2 hours x 3 miles per hour = 2 x 3 miles = 3miles+3miles = 6 miles.
No it doesn't. In that example the "baskets" and "per baskets" (ie x/baskets) are cancelling each other. 2 baskets x 3 apples/baskets = 2 x 3 apples = 6 apples. Once you remove the unnecessary terms the actual multiplication is still basic repeated addition.
2 hours x 3 miles per hour = 2 x 3 miles = 3miles+3miles = 6 miles.
Yes, that was the most glaring mistake in the article, in my eyes. The runner up strange argument was the one about multiplication and addition having different identities and inverses. But why would the operations "Addition" and "Repeated Addition" have exactly the same properties? The operations are not the same.
I think the real issue is about how to interpret the adjective "Repeated" when we turn to more general domains than the natural numbers. But that's not a huge issue, it's mostly a matter of terminology.
I think the real issue is about how to interpret the adjective "Repeated" when we turn to more general domains than the natural numbers. But that's not a huge issue, it's mostly a matter of terminology.
Learning isn't possible without unlearning whatever temporary construct we used as a crutch, but it doesn't mean using that temporary construct is wrong even if it's not technically correct.
More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:
From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.
From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.
Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.
More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:
From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.
From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.
Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.
The author is speaking from personal experiences teaching children K-10 (ages 5-16). It sounds like she is documenting her experience teaching these various temporary constructs, and that her experience is that teaching multiplication as repeated addition is a less effective way of teaching it.
> Ironically, this article wants to be very normative about which mathematical intuition is better…
No, the article is not being normative about “which intuition is better,” this is an incorrect reading of the article.
The article is giving advice about how to teach multiplication. Advice that is apparently based on years of experience teaching multiplication.
It’s not uncommon for people to experience teachers who prescribe specific intuitions about math rather than accommodate different intuitions—but the author is not doing that. What she’s doing here is outlining the various ways in which one particular intuition may fail you.
> Ironically, this article wants to be very normative about which mathematical intuition is better…
No, the article is not being normative about “which intuition is better,” this is an incorrect reading of the article.
The article is giving advice about how to teach multiplication. Advice that is apparently based on years of experience teaching multiplication.
It’s not uncommon for people to experience teachers who prescribe specific intuitions about math rather than accommodate different intuitions—but the author is not doing that. What she’s doing here is outlining the various ways in which one particular intuition may fail you.
(Unrelated to pedagogy, but related to when multiplication is not repeated addition.)
When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:
What if it were that 2*2 = 5 ?
Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4
1. Associative
2. Commutative
3. Multiplicative Identity
4. Increasing ( If i,j > 1, then m(i,j) > i,j )
5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )
6. Uniqueness of Prime Factorization
An easy example is the odd numbers.
It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.
When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:
What if it were that 2*2 = 5 ?
Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4
2*2 = (1+1) * (1+1) = 1+1+1+1 = 4
1. Associative
2. Commutative
3. Multiplicative Identity
4. Increasing ( If i,j > 1, then m(i,j) > i,j )
5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )
6. Uniqueness of Prime Factorization
An easy example is the odd numbers.
1,2,3,4,5, ... n
1,3,5,7,9, ... odd(n)
odd(n) = 2n-1
inverse_odd(x) = (x+1)/2
m(i,j) = inverse_odd( odd(i) * odd(j) ) = 2ij - i - j + 1
It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.
My favourite way of thinking about these operations comes from category theory:
* addition: the cardinality of the disjoint union of two sets
* multiplication: the cardinality of the direct product of two sets
* exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a
* addition: the cardinality of the disjoint union of two sets
* multiplication: the cardinality of the direct product of two sets
* exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a
That is exactly how I explained multiplication to my 5 years old, right after getting him into programming by playing with profunctor optics
I learned adding from counting disjoint unions of dots.
I learned multiplication from counting squares of dots.
But agreed: I learned about exponentiation first via repeated multiplication.
I learned multiplication from counting squares of dots.
But agreed: I learned about exponentiation first via repeated multiplication.
That is very cool. I prefer geometric/visual approaches. Like move that many steps along the line, figure out area of a rectangle and so on
This is just set theory, not category theory.
Neither is more foundational than the other. So I think the use of "just" is not justified here.
https://mathoverflow.net/questions/360578/category-theory-an...
https://mathoverflow.net/questions/360578/category-theory-an...
Yeah but your comment didn't use the concept of a "category" at all, whereas you used "set" three times. Its definitely more natural to say these definitions come from set theory.
To me, "disjoint union", "direct product" and "set of maps" (as a special case of "internal hom") are the central keywords in what I wrote: Concepts from category theory.
Those keywords all appeared in the context of set theory long before category theory was thought of.
Even ignoring that as a historical accident (we invented set theory before category theory), those ideas are all natural, meaningful and interesting to examine in the context of sets even if one has never heard of a category.
Even ignoring that as a historical accident (we invented set theory before category theory), those ideas are all natural, meaningful and interesting to examine in the context of sets even if one has never heard of a category.
Yes, there's no need to invoke any category theory concepts here. Everything is happening in the category of sets.
First, that's not what your link says. (See, e.g., the comments to the first answer.)
Second, I find that claim highly dubious. To do metamathematics and talk about the relative strength of various axiomatic systems, you need to talk about large cardinals. So that, at least to me, seems like a good argument that set theory is foundational to mathematics in a way that category theory is not, since the latter has no non-eliminable place in the (study of) contemporary foundations of mathematics.
Second, I find that claim highly dubious. To do metamathematics and talk about the relative strength of various axiomatic systems, you need to talk about large cardinals. So that, at least to me, seems like a good argument that set theory is foundational to mathematics in a way that category theory is not, since the latter has no non-eliminable place in the (study of) contemporary foundations of mathematics.
[deleted]
This seems a bit restricted. How do you go about extending it to real and complex numbers?
From what I know: This is still an open question.
Here's a list of some approaches.
https://mathoverflow.net/questions/310004/categorifications-...
Here's a list of some approaches.
https://mathoverflow.net/questions/310004/categorifications-...
Can this perspective accommodate fractional exponents, like 2^(0.5)?
No (At least not yet: open research question). But neither can "repeated multiplication". How would you multiply 2 with itself "a half" times?
Operationally, unsure. But if there was an answer, it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once.
I think it was Gerolamo Cardano who in a book of worked exercises came up with a problem like "obtain two numbers that add to 10 and multiply to 40" and arrived at the solution had to involve square roots of -1. Then he goes "this makes no sense, but let's calculate nevertheless..."
I think it was Gerolamo Cardano who in a book of worked exercises came up with a problem like "obtain two numbers that add to 10 and multiply to 40" and arrived at the solution had to involve square roots of -1. Then he goes "this makes no sense, but let's calculate nevertheless..."
> it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once
from what i recall that's the way to go:
You can use this to extend the definition of exponentials as "repeated addition" from allowing only natural numbers expoonents to allowing positive rational numbers. Afterwards you can "complete" this to allow real exponents.
But this requires some hard maths to do rigorously.
from what i recall that's the way to go:
You can use this to extend the definition of exponentials as "repeated addition" from allowing only natural numbers expoonents to allowing positive rational numbers. Afterwards you can "complete" this to allow real exponents.
But this requires some hard maths to do rigorously.
You're absolutely right about rigor: I'm not claiming that Cardano discovered complex analysis or anything...
I like this article, but I feel like it misses the value of teaching analysis to students. It's true that it's not-quite-right to say multiplication is repeated addition, but it's also a really nice demonstration of the analytical/generalization approach anyone could use to "invent" multiplication on their own.
This same pattern can be seen driving discoveries in math. - Integrals are generalizations of Riemann sums. - Fractional exponents generalize taking square roots. - The gamma function sorta generalizes factorials. It's okay to start with something not-quite-right and explore how it generalizes.
This same pattern can be seen driving discoveries in math. - Integrals are generalizations of Riemann sums. - Fractional exponents generalize taking square roots. - The gamma function sorta generalizes factorials. It's okay to start with something not-quite-right and explore how it generalizes.
I'll tell you what's wrong: the mentality of the author of this text
Multiplication has started as repeated addition. That's where the idea came from.
> It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication.
Nobody. Literally nobody said that
Of course, you'll need to forget a bit the idea of repeated multiplication when you get into the rationals/reals/complex numbers, but even there it kinda makes sense
So no, I think this is the kind of teacher that makes the students even more confused and prone to hating math
Multiplication has started as repeated addition. That's where the idea came from.
> It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication.
Nobody. Literally nobody said that
Of course, you'll need to forget a bit the idea of repeated multiplication when you get into the rationals/reals/complex numbers, but even there it kinda makes sense
So no, I think this is the kind of teacher that makes the students even more confused and prone to hating math
Now that's an interesting point, did it start as repeated addition?
It is just as conceivable that people were faced with a problem like 'each person needs 2 apples, we have 5 people, so we need 10 apples'?
In this case repeated addition is a perfectly fine algorithm to calculate the product but the product itself is not defined as repeated addition, it's the solution to a particular type of problem.
It is just as conceivable that people were faced with a problem like 'each person needs 2 apples, we have 5 people, so we need 10 apples'?
In this case repeated addition is a perfectly fine algorithm to calculate the product but the product itself is not defined as repeated addition, it's the solution to a particular type of problem.
Your problem just reinforces the notion that multiplication is repeated addition. "So we need 2 for him and 2 for her and 2 for him and 2 for him and 2 for her. 2+2+2+2+2 = 5 x 2 = 10"
I disagree, translating from "2 for (him + her + him + him + her)" to "2 for him + 2 for her + 2 for him + 2 for him + 2 for her" is using the distributive property of multiplication.
The sentence "2 for him and 2 for her and 2 for him and 2 for him and 2 for her" is unnatural and not the way people generally think (unless they're just tallying up but then we're not even talking about multiplication anymore, a tally rarely consists of all equal numbers)
The sentence "2 for him and 2 for her and 2 for him and 2 for him and 2 for her" is unnatural and not the way people generally think (unless they're just tallying up but then we're not even talking about multiplication anymore, a tally rarely consists of all equal numbers)
The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may.
I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach multiplication when we got to that point (which we did about a year later).
I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.)
I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her.
With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller.
It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?
I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach multiplication when we got to that point (which we did about a year later).
I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.)
I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her.
With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller.
It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?
Math PhD here. You're a fantastic dedicated parent and you did no harm.
First, as to the merits of your approach --- I think of multiplication as scaling myself. That's totally valid.
Second, whether it's even possible to do harm here --- worst case scenario, the metaphor doesn't make sense to your kid and she doesn't use it in her own thinking about mathematics. Everyone has to develop their own intuitions --- like you had your epiphany --- and she'll develop hers even if you set her on a track that doesn't work, just as part of her learning and doing mathematics.
First, as to the merits of your approach --- I think of multiplication as scaling myself. That's totally valid.
Second, whether it's even possible to do harm here --- worst case scenario, the metaphor doesn't make sense to your kid and she doesn't use it in her own thinking about mathematics. Everyone has to develop their own intuitions --- like you had your epiphany --- and she'll develop hers even if you set her on a track that doesn't work, just as part of her learning and doing mathematics.
I appreciate the reply. I didn't feel too terribly off-base, but seeing the discourse here gave me cause to question.
I suspect that the problem isn't introducing a given model, rather the problem is then not moving on to additional models.
Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.
Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.
I will say what we did work thru some multiplication of integers as repeated addition and scaling, just to show that the answers "come out the same". I used the scaling by a fraction to show that the repeated addition method doesn't work on all classes of numbers. (We've talked some about the difference between integers and real numbers.)
The gist of my statement was something like: Adding repeatedly is a fine method to solve this kind of problem but it doesn't for all classes of numbers. We can't say, like we can w/ addition and subtraction, that multiplication implies a specific direction of movement (towards or away from zero) on the number line.
The gist of my statement was something like: Adding repeatedly is a fine method to solve this kind of problem but it doesn't for all classes of numbers. We can't say, like we can w/ addition and subtraction, that multiplication implies a specific direction of movement (towards or away from zero) on the number line.
I wasn't really trying to analyze your approach, more speculating that getting in front of the teacher probably doesn't matter. It's the insight that multiplication isn't strictly equivalent to addition that matters, and illustrating that they are often equivalent probably doesn't block that insight.
The new new math isn't just multiplication is addition, it's addition and substraction.
39 x 40 = 40 x 40 - 40
39 x 40 = 40 x 40 - 40
That seems like a reasonable way to do that computation, from a purely practical perspective. Simple "tricks" to arrive at the correct answer quickly don't at all seem like a bad idea. When we were doing addition of two digit integers and working with place value there was a lot of adding the ones separately from the tens, then adding those together. Later, when addition with carry / regrouping came up we used both strategies to solve the problems and demonstrated how they really were the same thing, just following a different "recipe" to get the result.
One of the best concise guides I've encountered to the relationship between the various algebraic operations -- and how they circle back to one another -- is chapter 22 "Algebra" of volume I of the Feynman Lectures on Physics. For some reason this tied everything together in my head in a way that made so much sense, when the pedagogy I head leading up to it simply did not supply me the same vantage point.
https://www.feynmanlectures.caltech.edu/I_22.html
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.
https://www.feynmanlectures.caltech.edu/I_22.html
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.
As someone with a pragmatic, visual mind, I rely entirely on this kind of reasoning in order to understand and apply math. For example:
- Multiplication is repeated addition.
- Power is repeated multiplication.
- The square of a number can be visualized as a geometric square when you duplicate a row of x items x number of times.
- The cube of a number can be visualized as a geometric cube when you duplicate a row of x items x number of times and then you take the resulting geometric square and duplicate it x times along the orthogonal axis.
- n to the power of p can be visualized as a tree with height p such that each branch splits up into n branches at each level.
- The logarithm base x of y is the height 'number of levels' of the tree when each branch splits into x branches at each level until the number of tips is equal to y.
- A factorial can be visualized as a tree whose branches split up in such a way that there is one fewer branch at each level until the branches cannot be split anymore.
- Multiplication is repeated addition.
- Power is repeated multiplication.
- The square of a number can be visualized as a geometric square when you duplicate a row of x items x number of times.
- The cube of a number can be visualized as a geometric cube when you duplicate a row of x items x number of times and then you take the resulting geometric square and duplicate it x times along the orthogonal axis.
- n to the power of p can be visualized as a tree with height p such that each branch splits up into n branches at each level.
- The logarithm base x of y is the height 'number of levels' of the tree when each branch splits into x branches at each level until the number of tips is equal to y.
- A factorial can be visualized as a tree whose branches split up in such a way that there is one fewer branch at each level until the branches cannot be split anymore.
I strongly believe that helping younger students gain strong intuition for these operators pays dividends towards their later success in maths.
I've always run into the following problem: I try to motivate multiplication as repeated addition, which does help with intuition, but then things totally fall apart when we move on from integers into fractional values.
1/2 * 1/2 -> 1/4.
Sure you can teach someone to simply multiple the numerator and denominator, but it doesn't necessarily help them make clear sense of what's going on.
I've always run into the following problem: I try to motivate multiplication as repeated addition, which does help with intuition, but then things totally fall apart when we move on from integers into fractional values.
1/2 * 1/2 -> 1/4.
Sure you can teach someone to simply multiple the numerator and denominator, but it doesn't necessarily help them make clear sense of what's going on.
I think it's sort of an illusion (though certainly a useful one) that numbers are all part of the same system. Multiplication of natural numbers is defined to be repeated addition, and I'm not really sure how you could define natural number multiplication in any other way. From the natural numbers you can go on to define the integers, rationals, then reals, and each has its own definition of multiplication, though they each depend on the definition of multiplication from the previous system.
There's a standard way of lifting each type of number to the next type, and this lift is compatible with all the basic operations (the lift is a "homomorphism"), so it's easy to pretend that the real numbers (or complex numbers if you want) are the universal system.
So with your example of going to fractional values, you're right, repeated addition falls apart -- but I'd say that's because it's not the definition for multiplication of rational numbers! Multiplying numerators and denominators is the usual definition, but that gives about as much intuition as does the definition for multiplying natural numbers. Sort of "the point" of multiplication of naturals, I think, is that it represents how many things you have if you arrange them in an n by m grid. Rational numbers show up in geometry with similar shapes (scaling), and for a few reasons you'd want multiplication to represent by how much something scales after a composition of scalings; maybe "the point" of rational number multiplication (at least algebraically) is that you can defer dividing until later, i.e. (a/b) * (c/d) is (a*c)/b / d.
There's a standard way of lifting each type of number to the next type, and this lift is compatible with all the basic operations (the lift is a "homomorphism"), so it's easy to pretend that the real numbers (or complex numbers if you want) are the universal system.
So with your example of going to fractional values, you're right, repeated addition falls apart -- but I'd say that's because it's not the definition for multiplication of rational numbers! Multiplying numerators and denominators is the usual definition, but that gives about as much intuition as does the definition for multiplying natural numbers. Sort of "the point" of multiplication of naturals, I think, is that it represents how many things you have if you arrange them in an n by m grid. Rational numbers show up in geometry with similar shapes (scaling), and for a few reasons you'd want multiplication to represent by how much something scales after a composition of scalings; maybe "the point" of rational number multiplication (at least algebraically) is that you can defer dividing until later, i.e. (a/b) * (c/d) is (a*c)/b / d.
Fractional numbers make sense to me as an extension, but it also requires an intuition of division on the same lines.
Take number n and multiply it by number x/y. To do this, you have to split number n into y parts and take x number of them. So to multiply 8 by 3/4, you split 8 into 4 parts (2 + 2 + 2 + 2) and then take 3 of those parts. This ends up being 2 + 2 + 2 = 6.
For multiplying two fractions, you have to extend it to n/m * x/y. Since you can multiply the top and bottom of a fraction by the same number, you can write n/m as ny/my. Then you can have n/my be your “equal part”, and take x of them. So 1/2 * 1/2, you take 2/4 and split it into 1/4 + 1/4 and 1 of them, so the answer is 1/4.
To me at least, this makes sense as an extension of multiplication is repeated addition. It’s when you get to irrationals that it starts to fall apart, and even then the intuitions the above way of thinking led me to have served me well.
Take number n and multiply it by number x/y. To do this, you have to split number n into y parts and take x number of them. So to multiply 8 by 3/4, you split 8 into 4 parts (2 + 2 + 2 + 2) and then take 3 of those parts. This ends up being 2 + 2 + 2 = 6.
For multiplying two fractions, you have to extend it to n/m * x/y. Since you can multiply the top and bottom of a fraction by the same number, you can write n/m as ny/my. Then you can have n/my be your “equal part”, and take x of them. So 1/2 * 1/2, you take 2/4 and split it into 1/4 + 1/4 and 1 of them, so the answer is 1/4.
To me at least, this makes sense as an extension of multiplication is repeated addition. It’s when you get to irrationals that it starts to fall apart, and even then the intuitions the above way of thinking led me to have served me well.
As someone with an almost solely intuitive understanding of mathematics, I still find the "repeated addition" idea useful when it comes to fractions. If 4 * 4 is 4 repeated 4 times (4 + 4 + 4 + 4). Then 1/2 * 1/2 is a half repeated half times, or a half of a half, which is a quarter. The numbers get hard to work with but the intuitive idea is still there.
The nicest formal definition of multiplication is "the operation that distributes over addition". When defining operations on abelian groups that might be called "multiplication", it's the distributive property that makes the term useful, and brings us into the categories of ring, semiring, [weird math prefix here]-ring, etc.
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?
> Is there really a difference between multiplication and repeated addition, or am I tilting at windmills here?
You are tilting at windmills
> Is it even necessary for teachers to define multiplication?
No, at least not in formal terms
> Or is the teacher’s job to provide plenty of examples of multiplication in action?
I think so
> Should we let the students intuit their own definition(s)?
Yes
> Will it help students if we change our focus from “how to get the answer” and teach them to identify the multiplicand, the “this per that” unit? Or will that introduce new difficulties I haven’t considered?
It will introduce new difficulties
> Or do we already teach this way, only in different words?
We don't, thankfully
> If you are an elementary teacher, how do you teach multiplication to your students?
I will let elementary teacher answer
> Are some students clueless because, no matter how we explain it, they just don’t pay attention?
While we can't always blame teachers when students don't pay attention, if it is systematic, maybe there is a problem with your teaching.
> Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?
I will let teachers respond
Ok, I hope you are still there and didn't just click the "downvote" button.
I think the author raises good points for at least high-school level students in STEM fields. But in reality, the important thing in elementary school is that we want people to know how to count. 5 apples at $0.50 each is $2.50, this kind of thing.
Dimensional analysis is very important in engineering, and I definitely think is should be taught, as well as everything in the article, but at college level, not in elementary schools. Teaching such abstract concepts too soon is a recipe for disaster, I know because it has been tried. The expectation was that it will make better engineers in the future, the reality is that while it may have helped a tiny minority get college degrees, it mostly produced kids who couldn't do simple arithmetic.
You are tilting at windmills
> Is it even necessary for teachers to define multiplication?
No, at least not in formal terms
> Or is the teacher’s job to provide plenty of examples of multiplication in action?
I think so
> Should we let the students intuit their own definition(s)?
Yes
> Will it help students if we change our focus from “how to get the answer” and teach them to identify the multiplicand, the “this per that” unit? Or will that introduce new difficulties I haven’t considered?
It will introduce new difficulties
> Or do we already teach this way, only in different words?
We don't, thankfully
> If you are an elementary teacher, how do you teach multiplication to your students?
I will let elementary teacher answer
> Are some students clueless because, no matter how we explain it, they just don’t pay attention?
While we can't always blame teachers when students don't pay attention, if it is systematic, maybe there is a problem with your teaching.
> Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?
I will let teachers respond
Ok, I hope you are still there and didn't just click the "downvote" button.
I think the author raises good points for at least high-school level students in STEM fields. But in reality, the important thing in elementary school is that we want people to know how to count. 5 apples at $0.50 each is $2.50, this kind of thing.
Dimensional analysis is very important in engineering, and I definitely think is should be taught, as well as everything in the article, but at college level, not in elementary schools. Teaching such abstract concepts too soon is a recipe for disaster, I know because it has been tried. The expectation was that it will make better engineers in the future, the reality is that while it may have helped a tiny minority get college degrees, it mostly produced kids who couldn't do simple arithmetic.
In some settings, "repeated addition" is the only sensible way to think of multiplication. Such as multiplying an elliptic curve point with a scalar.
Multiplication is a shorthand for 'repeated addition'. Much like power is short hand for 'repeated multiplication'. Even with imaginary numbers. You can create an in infinite series of additions that looks a lot like the imaginary number.
The argument they are making is kind of strange. But not 'wrong' per se. If you mix 2 types of number systems you get a different number system out of it. It is a similar argument I make about fractions. It is basically physically impossible to have a fraction of some things. If I have a car and divide it in half. I no longer have a car. I have two of something else. They are clearly not a car because they can not act like a car anymore.
The argument they are making is kind of strange. But not 'wrong' per se. If you mix 2 types of number systems you get a different number system out of it. It is a similar argument I make about fractions. It is basically physically impossible to have a fraction of some things. If I have a car and divide it in half. I no longer have a car. I have two of something else. They are clearly not a car because they can not act like a car anymore.
Imagine explaining the content of this post to a five year old instead of that multiplication is repeated addition.
The more methods one studies, the better one understands. There is no one true method.
I like how 3blue1brown presents it in terms of animated number line translations
https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365
https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365
> Is that really how we want our students to think? Multiplication is not a mere sub-species of addition. Multiplication is its own animal, an independent operation.
Wtf, why not? Why wouldn't you want people to draw connections between different parts of math. The whole point of math is to find the patterns and interconnections.
Wtf, why not? Why wouldn't you want people to draw connections between different parts of math. The whole point of math is to find the patterns and interconnections.
I think the author mixes variables with numbers. If you add units in to the mix then you have to think like it variables so for example that’s taken “2 baskets × 3 apples per basket = 6 apples” would correspond to 2x3y/x=6y removing the numbers and the equation becomes clear xy/x=y or we the numbers for them selves 23=6, That is because numbers are like their own variable but with connection to other numbers(variables). So if we would define 1 as x then 2 would be 1+x or we could call it y but y would still be 1+x in relation to 1.
And also”3 cm + 3 cm = 6 cm” as 3x+3x=6x and removing the numbers it become 3x/3+3x/3=6x/3 => x+x=2x it makes more sense. And”2 cm × 3 cm = 6 cm^2″ can be then seen as 2x*3x=6x^2.
Multiplication doesn’t change anything. We must see what we add, in this case the unit and calculate accordingly both for addition as for multiplication.
And also”3 cm + 3 cm = 6 cm” as 3x+3x=6x and removing the numbers it become 3x/3+3x/3=6x/3 => x+x=2x it makes more sense. And”2 cm × 3 cm = 6 cm^2″ can be then seen as 2x*3x=6x^2.
Multiplication doesn’t change anything. We must see what we add, in this case the unit and calculate accordingly both for addition as for multiplication.
I feel like people here have a weird definition of "truth". Platonism aside, there is no outside ground truth on what "addition" or "multiplication" is. It is an abstract symbol defined solely by what it does to numbers. As long as your definition describes its behaviour, your definition is no more or less right than any other definition.
If you want to define multiplication as repeated addition (and addition as repeated invocation of the succesor function), by all means do so. Its not wrong.
Sure you could say it doesn't work for complex numbers, but usually when you introduce complex numbers you say something like, multiplication works as normal except i×i=-1. I hardly see that as a problem with the original definition of multiplication. Its more about the definition of i than the definition of multiplication.
If you want to define multiplication as repeated addition (and addition as repeated invocation of the succesor function), by all means do so. Its not wrong.
Sure you could say it doesn't work for complex numbers, but usually when you introduce complex numbers you say something like, multiplication works as normal except i×i=-1. I hardly see that as a problem with the original definition of multiplication. Its more about the definition of i than the definition of multiplication.
Why not just say:
There is the symbol '*'. And when there's two integers it does one thing, and you can think about it as repeated addition. But when there's something different, it works differently. But now we're just going to concentrate on integers.
There is the symbol '*'. And when there's two integers it does one thing, and you can think about it as repeated addition. But when there's something different, it works differently. But now we're just going to concentrate on integers.
I think what the author is missing is the word "scalar". I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.
In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless.
Of course, we should start teaching immediately that you can't add apples and pears. Maybe we should just do this explicitly there, that 5 times 3 apples is fine, as 5 is scalar. Explain that immediately and then add the unit analysis in physics.
In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless.
Of course, we should start teaching immediately that you can't add apples and pears. Maybe we should just do this explicitly there, that 5 times 3 apples is fine, as 5 is scalar. Explain that immediately and then add the unit analysis in physics.
> I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.
"Scalar" is too broad; irrational numbers are scalars, but multiplication by an irrational number is not repeated addition.
"Scalar" is too broad; irrational numbers are scalars, but multiplication by an irrational number is not repeated addition.
I know how addition is done in hardware. I've programmed multiplication and division on computers that only had shift and add instructions. I've implemented IEEE 754 floating point packages, too.
https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
Multiplication is repeated addition.
Yes, I know about roundoff errors, overflows, etc.
https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
Multiplication is repeated addition.
Yes, I know about roundoff errors, overflows, etc.
A few years ago, I was teaching a class and decided to record a short series on the fundamentals of mathematics, for complete "beginners" - whether they be kids or adults.
This is the first video in the series Thinking Mathematically (after the introductory video, "Why think Mathematically?") which I put on YouTube under a channel of the same name. It proceeds through the sets of numbers, N -> Z -> Q -> R -> C and yes it's for beginners. Would love some feedback:
https://www.youtube.com/watch?v=Wd1lzqR3xW0
The other videos you'll find on that channel are:
https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
(PS: I am generalizing this approach in our upcoming app, https://teaching.app)
This is the first video in the series Thinking Mathematically (after the introductory video, "Why think Mathematically?") which I put on YouTube under a channel of the same name. It proceeds through the sets of numbers, N -> Z -> Q -> R -> C and yes it's for beginners. Would love some feedback:
https://www.youtube.com/watch?v=Wd1lzqR3xW0
The other videos you'll find on that channel are:
1. Why think mathematically?
2. Numbers and Algebra
3. Sets and Infinity
4. Logic and Probability
https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
(PS: I am generalizing this approach in our upcoming app, https://teaching.app)
I think the author is trying to say that _from a teaching point of view_, thinking of multiplication as a kind of addition is suboptimal.
Because you can indeed multiply through repeated addition (which she acknowledges), repeated addition is one valid way to look at multiplication. But it might not be the maximally helpful framework for students.
Because you can indeed multiply through repeated addition (which she acknowledges), repeated addition is one valid way to look at multiplication. But it might not be the maximally helpful framework for students.
This is more confusing then it helps.
Why mix concepts of Linear Algebra/Physics with basic Arithmetics for no good reason?
The multiplication, that is just repeated addition, is very possible in dimensional analysis and is called scalar multiplication:
The outright worst part is this weird complication:
2 baskets × 3 apples per basket = 6 apples
of 2 x 3 apples = 6 apples.
Just why? This is just a scalar operation.
Staying in strictly mathematical concepts, the operation described in OP are closer to the cross product of orthogonal vectors, then multiplication in my opinion.
[4 0 0] × [0 3 0] = [0 0 12]
https://en.wikipedia.org/wiki/Scalar_multiplication
https://en.wikipedia.org/wiki/Cross_product#Geometric_meanin...
Why mix concepts of Linear Algebra/Physics with basic Arithmetics for no good reason?
The multiplication, that is just repeated addition, is very possible in dimensional analysis and is called scalar multiplication:
The outright worst part is this weird complication:
2 baskets × 3 apples per basket = 6 apples
of 2 x 3 apples = 6 apples.
Just why? This is just a scalar operation.
Staying in strictly mathematical concepts, the operation described in OP are closer to the cross product of orthogonal vectors, then multiplication in my opinion.
[4 0 0] × [0 3 0] = [0 0 12]
https://en.wikipedia.org/wiki/Scalar_multiplication
https://en.wikipedia.org/wiki/Cross_product#Geometric_meanin...
To me the "So What’s the Problem?" section has a lot of irrelevant stuff.
The problem with "multiplication is repeated addition" is that the concept breaks down once you move on from integers.
2 X 3: "add 2 together, three times" - works well
½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.
I can see why teachers who aren't that comfortable with math might be getting confused about this though. Repeated addition is a reasonable algorithm when you've got one operand that is a relatively small integer.
If you spend a long time teaching small integer multiplication you may forget that "You can solve these multiplication problems with repeated addition" does not mean "Multiplication is repeated addition."
The problem with "multiplication is repeated addition" is that the concept breaks down once you move on from integers.
2 X 3: "add 2 together, three times" - works well
½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.
I can see why teachers who aren't that comfortable with math might be getting confused about this though. Repeated addition is a reasonable algorithm when you've got one operand that is a relatively small integer.
If you spend a long time teaching small integer multiplication you may forget that "You can solve these multiplication problems with repeated addition" does not mean "Multiplication is repeated addition."
> ½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.
On the contrary, continuing to conceptualize fractional multiplication exactly as your "1/2 X 1/2" as addition example served me well. My wife—with a higher measured IQ than mine, FWIW—evidently can't see it that way. She basically stopped learning math when multiplication of fractions was introduced (it's interesting that one can continue to earn decent math grades, regardless) as all the results felt arbitrary and magical to her, and still do in middle-age. This is actually a very common point for students to stop following WTF is going on in their math classes and never really get back on track (yes, all the way down in, what, 2nd or 3rd grade) from what I've seen; factoring is another, later on.
"1/2, added 1/2 times" is precisely how I think of it. If I can't just pattern-match or rule-follow my way to a solution (because I've forgotten the rules, say) that's still my line of reasoning to figure out what to do to get the solution.
I do exactly the same sort of thing to come back to my senses if I get lost or forget exactly what is happening in fractional division. "How many times does 1/2 fit in 1/4? 1/2. How many times does 1/4 fit in 1/2? 2." Even if I have to manipulate some things to figure out the result with uglier fractions, that's absolutely how I think about what I'm doing as I do it, and it keeps me focused on the ultimate purpose of the calculation. I imagine that's also wrong, according to the author.
In short I can vouch that yes, that exact thing was very helpful to this particular person, including for multiplication of fractions. I truly don't know how else I might have understood those problems, to avoid joining the ranks of the mathematically-lost as early as lower elementary school.
On the contrary, continuing to conceptualize fractional multiplication exactly as your "1/2 X 1/2" as addition example served me well. My wife—with a higher measured IQ than mine, FWIW—evidently can't see it that way. She basically stopped learning math when multiplication of fractions was introduced (it's interesting that one can continue to earn decent math grades, regardless) as all the results felt arbitrary and magical to her, and still do in middle-age. This is actually a very common point for students to stop following WTF is going on in their math classes and never really get back on track (yes, all the way down in, what, 2nd or 3rd grade) from what I've seen; factoring is another, later on.
"1/2, added 1/2 times" is precisely how I think of it. If I can't just pattern-match or rule-follow my way to a solution (because I've forgotten the rules, say) that's still my line of reasoning to figure out what to do to get the solution.
I do exactly the same sort of thing to come back to my senses if I get lost or forget exactly what is happening in fractional division. "How many times does 1/2 fit in 1/4? 1/2. How many times does 1/4 fit in 1/2? 2." Even if I have to manipulate some things to figure out the result with uglier fractions, that's absolutely how I think about what I'm doing as I do it, and it keeps me focused on the ultimate purpose of the calculation. I imagine that's also wrong, according to the author.
In short I can vouch that yes, that exact thing was very helpful to this particular person, including for multiplication of fractions. I truly don't know how else I might have understood those problems, to avoid joining the ranks of the mathematically-lost as early as lower elementary school.
> ½ X ½:
Or just re-arrange the terms to be (1*1)÷(2*2). No need to add half, half a time, just understand that multiplication distributes over fractions.
That said i think the concept would make sense to most school children as a way of extending the intuitive concept of multiplication of natural numbers.
Or just re-arrange the terms to be (1*1)÷(2*2). No need to add half, half a time, just understand that multiplication distributes over fractions.
That said i think the concept would make sense to most school children as a way of extending the intuitive concept of multiplication of natural numbers.
Is addition a shortcut for counting, or is that wrong, too?
Many commenters have already said it: the article is probably not helpful when teaching kids. Or is it? I recall having asked myself these questions at school: if lines are infinitely thin, how is it repeated addition that defines surface areas? What is "inside" the variables we use as units?
Measure theory (for your volume) and the structure of dimensional quantities [1] are non-trivial topics, and while probably not giving good answers, I think the article is certainly asking good questions.
[1] https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...
Measure theory (for your volume) and the structure of dimensional quantities [1] are non-trivial topics, and while probably not giving good answers, I think the article is certainly asking good questions.
[1] https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...
To put it in terms of what the audience here can relate to:
multiplication is an overloaded operation, or in more modern terms, it is doing multiple dispatch, depending on whether the input is a whole number or integer or rational or real or complex
multiplication is an overloaded operation, or in more modern terms, it is doing multiple dispatch, depending on whether the input is a whole number or integer or rational or real or complex
I just found this interesting article. If you are interested, it is a review of the following "conversation of articles" (which are reachable from the OP but I guess it's useful enough to list):
- https://www.maa.org/external_archive/devlin/devlin_09_07.html
- https://www.maa.org/external_archive/devlin/devlin_06_08.html
- https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/
- (the link I posted that creates this thread)Clickable:
- https://www.maa.org/external_archive/devlin/devlin_09_07.htm...
- https://www.maa.org/external_archive/devlin/devlin_06_08.htm...
- https://denisegaskins.com/2008/07/01/if-it-aint-repeated-add...
(Sorry, it just bothers me when people don't use clickable links here.)
- https://www.maa.org/external_archive/devlin/devlin_09_07.htm...
- https://www.maa.org/external_archive/devlin/devlin_06_08.htm...
- https://denisegaskins.com/2008/07/01/if-it-aint-repeated-add...
(Sorry, it just bothers me when people don't use clickable links here.)
I wanted to post the same links, you can find the same articles and others about that subject by googling "Keith Devlin addition multiplication"
It doesn't make sense to say:
2 apples + 3 apples = 5 apples
This confuses a lot of people when they later learn algebra and believe that it is equivalent to:
2a + 3a = 5a
Both of these equations look correct at face value but the critical difference is that an apple is not a variable. Do we mean the weight of the apple? Or the price of the apple? Or its energy content? Apple is too concrete to be used as a variable and it's not quantifiable. The word 'apples' invites people to think of the variable as a concrete object instead of a placeholder for a quantifiable property of an object.
2 apples + 3 apples = 5 apples
This confuses a lot of people when they later learn algebra and believe that it is equivalent to:
2a + 3a = 5a
Both of these equations look correct at face value but the critical difference is that an apple is not a variable. Do we mean the weight of the apple? Or the price of the apple? Or its energy content? Apple is too concrete to be used as a variable and it's not quantifiable. The word 'apples' invites people to think of the variable as a concrete object instead of a placeholder for a quantifiable property of an object.
Does anyone have an intuition for why Presburger arithmetic is decidable, but once you introduce x, it is not? If + can be rewritten as repeated succ, (which is decidable) and x can be rewritten as repeated +, then it seems logical that x should be decidable, but as any undergraduate CS student knows, Peano arithmetic is undecidable. What is so special about x exactly?
Its probably worth pointing out that the system with only multiplication and no addition is also decidable (Skolem arithmetic). Things get interesting when you have both multiplication and addition defined and the reason for that is basically the fundamental theorem of arithmetic which gives you unique prime factorisation and directly the Gödel encoding (writing out statements about integers as integers).
The fundamental theorem of arithmetic says for any non-zero natural number x there is a unique finite sequence of integers a_i such that x = 2^a_0 . 3^a_1 . 5^a_3 ... p_n^a_n and that given x the sequence a_i is computable (similarly given a_i you can easily construct their x).
This basically gives you a (bijective) mapping between integers and tuples of integers x <-> (a_0, a_1, ... a_n) which lets you build up more complicated structures and eventually construct the sentence you need for an incompleteness theorem (if you're as smart as Gödel).
The fundamental theorem of arithmetic says for any non-zero natural number x there is a unique finite sequence of integers a_i such that x = 2^a_0 . 3^a_1 . 5^a_3 ... p_n^a_n and that given x the sequence a_i is computable (similarly given a_i you can easily construct their x).
This basically gives you a (bijective) mapping between integers and tuples of integers x <-> (a_0, a_1, ... a_n) which lets you build up more complicated structures and eventually construct the sentence you need for an incompleteness theorem (if you're as smart as Gödel).
That’s a great explanation, thank you!
> but as any undergraduate CS student knows, Peano arithmetic is undecidable
We went to very different universities it seems. Neither Peano arithmetic nor decidability were ever mentioned
We went to very different universities it seems. Neither Peano arithmetic nor decidability were ever mentioned
It's a variable number of addition operations.
Isn’t + a variable number of succ’s? Why does x break decidability, but + does not? It seems to me you should be able to define a second order Presburger system with + replacing succ, and x replacing + to restore decidability. Or is there some issue with countability or assumption that is broken?
In mathematics, sometimes a wrong approach can be used as a teaching tool until you get the concept, at which the correct definition may be disclosed. Example: derivative calculus is easier to teach with infinitesimals, even though the derivative is correctly defined in terms of limits.
The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?
> but is 2 cm × 3 cm the same thing as 2 × 3?
This made me think too, in particular the teacher's example. So; here's what I think it should be calculated.
3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm).
What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm".
However 3 cm + 3 cm != 2 cm X 3 cm as the author states which is what leads them to wrong conclusion. In other words 2 cm X 3 cm leads to an entirely new thing in geometry which is "area". So while 3 cm + 3 cm is still length (and hence it should be translated to 2 X 3 cm i.e., one scalar unitless number and other number with unit) 2 cm x 3 cm is not length. And the resultant area is correctly written as 6 cm^2; by multiplying units as well as the numbers. Now, how does one multiply numbers (2 X 3)? It's indeed by repeated addition!
The author, however is onto something. Which is when you multiply a something with unit (like 3Kg, 2Cm) with a unitless number that resultant product is still something (that weight, length) that you began with. However when you multiply two things both of which have units the resultant product is something entirely new (i.e., area as opposed to length, pressure as opposed to weight etc.,)
This made me think too, in particular the teacher's example. So; here's what I think it should be calculated.
3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm).
What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm".
However 3 cm + 3 cm != 2 cm X 3 cm as the author states which is what leads them to wrong conclusion. In other words 2 cm X 3 cm leads to an entirely new thing in geometry which is "area". So while 3 cm + 3 cm is still length (and hence it should be translated to 2 X 3 cm i.e., one scalar unitless number and other number with unit) 2 cm x 3 cm is not length. And the resultant area is correctly written as 6 cm^2; by multiplying units as well as the numbers. Now, how does one multiply numbers (2 X 3)? It's indeed by repeated addition!
The author, however is onto something. Which is when you multiply a something with unit (like 3Kg, 2Cm) with a unitless number that resultant product is still something (that weight, length) that you began with. However when you multiply two things both of which have units the resultant product is something entirely new (i.e., area as opposed to length, pressure as opposed to weight etc.,)
Another way to say it is that 2cm * 3cm is shorthand for (2)(1cm)(i) * (3)(1cm)(j). We are complecting 3 semantics: measure and unit (which compose a module, in vector terms) and “meaning” (dimension, in vector terms).
This is why in high school physics you are taught to always write down the units of each number: typing them helps avoid a big amount of errors.
Note that “typing” and “vector terms” are not essential, just helps me express it with the HN crowd easily.
This is why in high school physics you are taught to always write down the units of each number: typing them helps avoid a big amount of errors.
Note that “typing” and “vector terms” are not essential, just helps me express it with the HN crowd easily.
The author is confusing math with physics.
Math: 3x3 = 3^2 physics: cmxcm = cm^2
3cm x 2cm = (3x2)(cm x cm) = (math)(physics)
The context gives the meaning.
Or in computers, how many bits would the representation of a matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory is linear)
Math: 3x3 = 3^2 physics: cmxcm = cm^2
3cm x 2cm = (3x2)(cm x cm) = (math)(physics)
The context gives the meaning.
Or in computers, how many bits would the representation of a matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory is linear)
Yeah I don't think they are the same. You can of course also have 2 cm * 3, which still fits with the repeated addition idea, as long as you frame it as 2 cm + 2 cm + 2 cm. I'm not an educator by any stretch, but that seems like an intuitive way to introduce units without causing confusion about previous number teachings.
Arguably, what they're saying is that 2 x 3 should be understood as a unit-less "scaler" 2 that multiplies a number (of things) "3".
Both examples use the same operator, but we resolve them to different functions. Mentally, we use the tuple of argument types to select the appropriate function. Addition does not provide equivalent functions for all multiplication cases. Hence, multiplication is not repeated addition.
You need to stop conflating the tyoe mumbo jumbo with multiplication.
We don't define it based on what machines do with it, we define it based on what WE do with it. In the context of teaching, dimensional analysis comes far later in the process of education.
The pedants answer is that there is more than one multiplication operator taught. It's most basic incarnation is, for all intents and purposes, repeated addition.
It's more advanced forms, which include extra auditing for dimensional analysis isn't merely repeated addition. Though I'd challenge that too.
Dimensional analysis is an extra aspect stapled onto and seperate from the arithmetic operation. Therefore makes up an additional level of auditing seperately as a whole, and therefore inappropriate to define as an inherent property of the operator. You multiply numbers.
People need to stop trying to teach unrelated things bundled by some programmer or language writer to kids/adults. It doesn't make one seem smart. It just confuses people by blurring orthogonal aspects of problem solving, which puts people off of it.
We don't define it based on what machines do with it, we define it based on what WE do with it. In the context of teaching, dimensional analysis comes far later in the process of education.
The pedants answer is that there is more than one multiplication operator taught. It's most basic incarnation is, for all intents and purposes, repeated addition.
It's more advanced forms, which include extra auditing for dimensional analysis isn't merely repeated addition. Though I'd challenge that too.
Dimensional analysis is an extra aspect stapled onto and seperate from the arithmetic operation. Therefore makes up an additional level of auditing seperately as a whole, and therefore inappropriate to define as an inherent property of the operator. You multiply numbers.
People need to stop trying to teach unrelated things bundled by some programmer or language writer to kids/adults. It doesn't make one seem smart. It just confuses people by blurring orthogonal aspects of problem solving, which puts people off of it.
I distinctly remember when I realized that the times in "4 times 3" is 3 added 4 times. That was the end of trying to memorize the times tables. If I forgot what "6 times 9" was I just thought "6 times 10 minus 6" and there it was.
Calculator multiplication can be performed by what is essentially repeated addition.
Probably the author would argue that what calculators do is neither addition nor multiplication because of precision loss. In practice it works pretty well though.
Probably the author would argue that what calculators do is neither addition nor multiplication because of precision loss. In practice it works pretty well though.
As software developers we can defenetly point the many flaws in this article:
1. IS A CLICKBAIT
2. Course you can decopose multiplications in a series of repeated additions.
3. Extra points for the flawed argument of adding different, well you can't because they are different.
1. IS A CLICKBAIT
2. Course you can decopose multiplications in a series of repeated additions.
3. Extra points for the flawed argument of adding different, well you can't because they are different.
In 2п, п can be though of as added to itself; on the other hand, it is not immediately obvious how to think about 2 being added to itself п times.
He makes a good case for multiplication being a special subset of (repeated) addition, whilst semantically trying to argue otherwise :D
It's not circular reasoning - it's bootstrapping the rational implementation atop the integer implementation.
The euclidean algorithm can compute the GCD of the factors of the naive rational form, repeated subtraction (down to zero) of the GCD from the denominator and numerator can be used to find them in their reduced forms.
A bit meandering and muddled in some respects, but agree with the gist: Multiplication is not just repeated addition.
Proof: The derivative of x^2 is x.
x^2 = x * x = x + x ... + x (x times)
Therefore,
d/dx(x^2) =
d/dx(x * x) =
d/dx (x + x + ... + x (x times)) =
d/dx(x) + d/dx(x) + ... + d/dx(x) (x times) =
1 + 1 + ... + 1 (x times) =
x
x^2 = x * x = x + x ... + x (x times)
Therefore,
d/dx(x^2) =
d/dx(x * x) =
d/dx (x + x + ... + x (x times)) =
d/dx(x) + d/dx(x) + ... + d/dx(x) (x times) =
1 + 1 + ... + 1 (x times) =
x
By the way, why doesn't exponentiation have units in physics?
E.g. why don't we ever have something like kg^s?
E.g. why don't we ever have something like kg^s?
I think it would have to play well with Taylor series -- for example e^s would be 1 + s + s^2/2 + s^3/6 + s^4/24 + ... -- and that would be mixing units. That's not to say it's meaningless, but rather you'd have to accept measurements like 4s + 5s^2, with incompatible units hanging around. (There's something similar in geometric algebra, I think, where expressions contain terms of different dimensions added together.)
Interesting point. But I think that the Taylor series is just a mathematical tool, and as such it hides the units that exist in these expressions. From a physics point of view, every term in the Taylor series needs to be augmented with a unit that follows from the definition of the Taylor series.
This is purely a guess, but I think it's because all the exponents are scalar. You get exponents when the derivative of a function is proportional to itself:
Example: population growth is exponential.
d/dx (f) = k*f
Example: population growth is exponential.
d/dt(p) = (ln(2)/r)*p
p(t) = p0 * 2^(t/r)
p : population
p0: population
t : days
r : days (time it takes for the population to double)
t/r : scalarIt may be wrong, but it is good enough to run all of our computation at the transistor level :)
The fascinating thing is that now these transistors are helping make new math proofs beyond human comprehension.... how does that come out of addition?
The fascinating thing is that now these transistors are helping make new math proofs beyond human comprehension.... how does that come out of addition?
AFAIK hardware multipliers are not quite as dumb and work more like this though:
5*11 = 5*(8+2+1) = 5*8 + 5*2 + 5*1 = (5<<3) + (5<<1) + (5<<0)
...and then doing all this in parallel.
Instead of:
5*11 = 5+5+5+5+5+5+5+5+5+5+5
;)
5*11 = 5*(8+2+1) = 5*8 + 5*2 + 5*1 = (5<<3) + (5<<1) + (5<<0)
...and then doing all this in parallel.
Instead of:
5*11 = 5+5+5+5+5+5+5+5+5+5+5
;)
Philosophical mathematicians trying to confuse the plebs.
I think it's okay to say: "It's useful to visualize (or think of) multiplication as repeated addition, in contexts where that makes sense." Properly qualified the statement seems true and unbojectionable, and I think it's a useful way to help students understand multiplication. (It's not the only way.)
The reservation I'd have with the statement "Multiplication is repeated addition" is that the use of the word "is" implies an identification, or a necessary derivation of one from the other. The operations aren't identical, and one is not necessarily derived from the other. As operations, they're distinct.
(The issue also has nothing per se to do with units. When you introduce units of length or area, you're using numbers in a particular applied context. Outside of that context, numbers don't have units.)
In everyday usage "addition" and "multiplication" are conventional designations. Conventions always have ambiguities and edge cases.
The most common formal structure in which you have both addition and multiplication is in a ring. A ring is a set with two operations, which we call "addition" and "multiplication" --- but those are just names. We could just as well have used "foo" and "bar". We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention/convenience. (Note also that those operations are not unique.)
Rings include the integers, rationals, reals, complex numbers - but also (for instance) polynomials with real coefficients, 42 x 42 matrices with real entries, quaternions, or finite fields.
Now consider what happens as you list the axioms for a ring. You say that addition is associative, addition has an identity element (conventionally denoted "0"), every element has an additive inverse, and addition is commutative. That's addition.
Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted "1").
At the moment, you have two independently defined operations, which have nothing to do with one another. "Multiplication" is therefore not identical to (or defined as) repeated addition. But it's not too useful to have two independent unrelated operations. You connect the two operations by introducting the distributive axiom: For all a, b, c in your ring,
Once the distributive property is available to connect the operations, it explains why you can "think of" multiplication as repeated addition: For instance,
Or consider one of the standard illustrations given to kids in grade school:
But what happened here? We relied on our physical intuition to "know" that putting the original 6 dollars into 2 bags in the second step didn't change the number of dollars. But formally, it is
(BTW regrouping the dollars again into 3 groups of 2 is a standard way of motivating commutativity of multiplication, since 3 groups of 2 is visibly "the same as" 2 groups of 3.)
(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don't know math history well enough to say, but historical derivation doesn't imply identity or logical derivation.)
So I think saying "Multiplication is repeated addition" is a little sloppy in the use of the word "is", but we can agree to disagree about how much sloppiness is okay. The statement is fine as a way of giving students one (of many) ways to think about multiplication. (Pictures are important, too!)
Here's something to think about. (I don't have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as "7 i added to itself (3 + 4 i) times". If you don't like the sound of that ... why? You might say "You can't have '3 + 4 i' things." Well, in the real world discrete "things" come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that "-3 things" means "I'm missing 3 things" or "you owe me 3 things". Nothing stops us from making agreements about the use of words. So maybe we could "agree" that "(3 + 4 i) times" means exactly an occurrence of the expression "3 + 4 i" in a context like this one. Is there any harm in that? :-)
The reservation I'd have with the statement "Multiplication is repeated addition" is that the use of the word "is" implies an identification, or a necessary derivation of one from the other. The operations aren't identical, and one is not necessarily derived from the other. As operations, they're distinct.
(The issue also has nothing per se to do with units. When you introduce units of length or area, you're using numbers in a particular applied context. Outside of that context, numbers don't have units.)
In everyday usage "addition" and "multiplication" are conventional designations. Conventions always have ambiguities and edge cases.
The most common formal structure in which you have both addition and multiplication is in a ring. A ring is a set with two operations, which we call "addition" and "multiplication" --- but those are just names. We could just as well have used "foo" and "bar". We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention/convenience. (Note also that those operations are not unique.)
Rings include the integers, rationals, reals, complex numbers - but also (for instance) polynomials with real coefficients, 42 x 42 matrices with real entries, quaternions, or finite fields.
Now consider what happens as you list the axioms for a ring. You say that addition is associative, addition has an identity element (conventionally denoted "0"), every element has an additive inverse, and addition is commutative. That's addition.
Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted "1").
At the moment, you have two independently defined operations, which have nothing to do with one another. "Multiplication" is therefore not identical to (or defined as) repeated addition. But it's not too useful to have two independent unrelated operations. You connect the two operations by introducting the distributive axiom: For all a, b, c in your ring,
a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c.
a * (b + c) = a * b + a * c.
Once the distributive property is available to connect the operations, it explains why you can "think of" multiplication as repeated addition: For instance,
3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1 = 3 + 3.
Or consider one of the standard illustrations given to kids in grade school:
3 * 2 is $ $ $ which is [ $ $ $ ]
$ $ $ [ $ $ $ ]
But what happened here? We relied on our physical intuition to "know" that putting the original 6 dollars into 2 bags in the second step didn't change the number of dollars. But formally, it is
3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1,
(BTW regrouping the dollars again into 3 groups of 2 is a standard way of motivating commutativity of multiplication, since 3 groups of 2 is visibly "the same as" 2 groups of 3.)
(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don't know math history well enough to say, but historical derivation doesn't imply identity or logical derivation.)
So I think saying "Multiplication is repeated addition" is a little sloppy in the use of the word "is", but we can agree to disagree about how much sloppiness is okay. The statement is fine as a way of giving students one (of many) ways to think about multiplication. (Pictures are important, too!)
Here's something to think about. (I don't have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as "7 i added to itself (3 + 4 i) times". If you don't like the sound of that ... why? You might say "You can't have '3 + 4 i' things." Well, in the real world discrete "things" come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that "-3 things" means "I'm missing 3 things" or "you owe me 3 things". Nothing stops us from making agreements about the use of words. So maybe we could "agree" that "(3 + 4 i) times" means exactly an occurrence of the expression "3 + 4 i" in a context like this one. Is there any harm in that? :-)
tl;dr: "Nothing"
You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be defined in any or all of these contexts if one enjoys doing so. There are far ranging context to define an object and less far ranging. I am quite sure that in grade school it is quite unhelpful to look for the furthest ranging context in which one could define multiplication. E.g, let us start in grade 1 with the definition of an algebra and derive everything from that.