Can 1/3 and 1/3 = 2/6? It seemed so(marilynburnsmathblog.com)
marilynburnsmathblog.com
Can 1/3 and 1/3 = 2/6? It seemed so
http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/
307 comments
You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard!
> The confusion comes because no one calls out that they're talking about fractions of different things.
But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking about the two tables together."
Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:
"1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at both tables"
"1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the first table"
Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition much more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.
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The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.
> The confusion comes because no one calls out that they're talking about fractions of different things.
But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking about the two tables together."
Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:
"1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at both tables"
"1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the first table"
Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition much more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.
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The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.
> You're right, but the tricky part is, how do you explain that to a group of children who is just being introduced to the concept of adding fractions, without leading them off track? That's hard!
She's already introduced bottles of water and pencils. So one interesting question is, "If you take three bottles of water, and add three pencils, do you now have all of the bottles of water?"
She's already introduced bottles of water and pencils. So one interesting question is, "If you take three bottles of water, and add three pencils, do you now have all of the bottles of water?"
This is interesting. I wonder if kids can then take that and transfer it over. Bottles and Pencils are obviously different but kids at table A and kids at table B are both kids. Even more problematically, the equation appears to work. I suspect there'll be some amount of dissatisfaction here and the kids won't understand.
Not sure if this is at the level of comprehension of these kids, but I'd explain it this way: "kids at table A" and "kids at table B" are both "kids", but different amount of kids. You can treat them as the same only if you have a conversion factor. So, if you know that there are 12 kids at table A, and 20 kids at table B, you can multiply your variables by these amounts, and now both expressions have the unit "kids", and you can add them together. But if you don't know the conversion factor, they're like Bottles and Pencils.
> you can multiply your variables by these amounts, and now both expressions have the unit
In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.
In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.
Since numbers are usually introduced to children in terms of quantities of things (e.g. 3 apples), I think it might suffice to stress the importance of keeping these things always attached to those numbers. In general, numbers very rarely go alone.
Sure. Might point is that you’re whole statement is way beyond the math comprehension of grade schoolers.
I agree. I think many people are thinking the main obstacle here is the mathematical mistake, when the main obstacle here seems to be explaining the mathematical mistake to 4th graders who are learning fractions for the first time. I'm also not sure what the best way to do that would be. If you start with one type of fraction (say, half circles), it's probably easier to intuit things like 5 halves and how much that is, but other things are probably going to be more difficult.
So yes, telling kids that there are more complications when approaching certain problems that you can explain later might be a good approach.
So yes, telling kids that there are more complications when approaching certain problems that you can explain later might be a good approach.
Not only to 4th graders.
I have a PhD in physics and the more I think about that, the less obvious it becomes (or at least makes you seriously think about it).
The average person is probably in the easiest situation because they learned how to add fractions without further philosophy and they can live happily after.
I have a PhD in physics and the more I think about that, the less obvious it becomes (or at least makes you seriously think about it).
The average person is probably in the easiest situation because they learned how to add fractions without further philosophy and they can live happily after.
I think we really need to start teaching people paying attention to units. Even in math classes.
1/3 + 1/3 is not a correct mathematical description of the problem.
1/3 [students at table A] + 1/3 [students at table B] is.
Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3".
Or, in other words, it's best to "keep your attention on what the whole is" by keeping it explicitly written out in the equation.
1/3 + 1/3 is not a correct mathematical description of the problem.
1/3 [students at table A] + 1/3 [students at table B] is.
Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3".
Or, in other words, it's best to "keep your attention on what the whole is" by keeping it explicitly written out in the equation.
Units is a real eye opener once you do it right. I remember when i studied and could almost verify what I did was correct, I would just check if the resulting unit was what I expected
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The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and it's still 1:3 or 2:6. You haven't changed the ratio for this at all by showing it as fractions, it's simply being represented and presented without correct context.
Isn't this why you end up seeing these "silly" units like kg/kg in chemistry? So that, while the value is technically dimensionless, it doesn't get added to another dimensionless value (e.g. l/l) that's a ratio of values of a different dimension?
This hits the nail on the head:
1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.
1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.
And then it's easy to make them see how her answer was right in its own way by adding another unit, (person at table a+b and people at table a+b) and show why it may be harder to works like that for now.
I heard so many people complains that each year in maths they would essentially learn that everything they learned the year before was wrong... can we fix that please?!
I heard so many people complains that each year in maths they would essentially learn that everything they learned the year before was wrong... can we fix that please?!
Why? Each is a pure number without units. Since the units cancel in each term.
Another way of labeling this:
: = Out Of
A = Table A
B = Table B
E = Every Table Added (A + B)
1:3 * A + 1:3 * B = (1:3 + 1:3) * E
2 out of 6 among Every Table AddedSomewhat aside, but in grade school I recall learning that ratios were written [group-A]:[group-B], not [group-A]:[total]. So the girl-boy ratio at the table would be 1:2. Different for you?
It really depends on the two things being compared. The ratio of boys to girls is 1:2, but the ratio of boys (implicitly: to the total) is 1:3.
I don't think I agree with everything here.
Glossing over critical mental models while explaining new concepts is what creates the confusion. Absolutely using drawings would have made things easier. That's a good way to establish those correct mental models.
"No you can't do that" is probably not the right approach.
The kid's instinct was spot on. The calculation they did was correct, just out of context. Rather than telling them they did it wrong and disrupt their correct mental model, I think it makes way more sense to talk about how what they did is different.
Glossing over critical mental models while explaining new concepts is what creates the confusion. Absolutely using drawings would have made things easier. That's a good way to establish those correct mental models.
"No you can't do that" is probably not the right approach.
The kid's instinct was spot on. The calculation they did was correct, just out of context. Rather than telling them they did it wrong and disrupt their correct mental model, I think it makes way more sense to talk about how what they did is different.
I'd absolutely agree if you were tutoring the child one-on-one. My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it.
Ignoring it does feel wrong, but I don't know that it is. There's only so much time in a class period, etc.
That said, I really liked where mnsc went with it: https://news.ycombinator.com/item?id=23312266
Ignoring it does feel wrong, but I don't know that it is. There's only so much time in a class period, etc.
That said, I really liked where mnsc went with it: https://news.ycombinator.com/item?id=23312266
My observation of points at which even very smart people fall off the "math train" k-12:
- Fractions
- Factoring
So yeah, I'd say great care is warranted in teaching kids fractions. Screw that up and there's a good chance you've lost them for life.
- Fractions
- Factoring
So yeah, I'd say great care is warranted in teaching kids fractions. Screw that up and there's a good chance you've lost them for life.
> My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it.
Except that the kids were all agreeing with her. Whether she did made that statement or not, they still believed that made sense.
I agree with you about the solution from mnsc, that was a good way to skip that without confusion.
Except that the kids were all agreeing with her. Whether she did made that statement or not, they still believed that made sense.
I agree with you about the solution from mnsc, that was a good way to skip that without confusion.
The right response could then be expanded...
"No, you can't do that, because those kids are at a different table. If you want to add Jack and Brad you have to go back and see their fraction _of both tables_. They are each 1/6 so together they are 1/6 + 1/6 = 2/6"
Then you could talk about if the ratios are same at both tables 1/3. Adding both tables together keeps the ratio. 1/3 + 1/3 = 1/3.
"No, you can't do that, because those kids are at a different table. If you want to add Jack and Brad you have to go back and see their fraction _of both tables_. They are each 1/6 so together they are 1/6 + 1/6 = 2/6"
Then you could talk about if the ratios are same at both tables 1/3. Adding both tables together keeps the ratio. 1/3 + 1/3 = 1/3.
Ah, I like that! Yes, re-frame the wrong statement into a correct one, then keep going. Feels better than ignoring it. (I wouldn't introduce ratios though, that's a totally new concept.)
This is closer to what the teacher actually did—but then she became focused on explaining why the original supposition was wrong, as opposed to moving on to clearer examples.
This is closer to what the teacher actually did—but then she became focused on explaining why the original supposition was wrong, as opposed to moving on to clearer examples.
I like the units explanation a lot more. Saying you can't put two tables together when you clearly can in the real world is seems deeply unsatisfying. Kids hate being told they can't do something.
mnsc's explanation shows that you can put the tables together, you just need to take both tables into account from the outset.
1/3 of one table = 1 student.
1/3 of two tables = 2 students.
1/3 of one table + 1/3 of one table = 2/3 of one table.
2/3 of one table = 1/3 of two tables.
1/3 of one table + 1/3 of one table = 1/3 of two tables.
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It would take a lot longer to explain than to write, but that's how I'd be tempted to proceed.
As far as learning the basics, it sounds like this class was just getting them introduced to the concept of fractions, while the arithmetic rules would come later...
1/3 of two tables = 2 students.
1/3 of one table + 1/3 of one table = 2/3 of one table.
2/3 of one table = 1/3 of two tables.
1/3 of one table + 1/3 of one table = 1/3 of two tables.
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It would take a lot longer to explain than to write, but that's how I'd be tempted to proceed.
As far as learning the basics, it sounds like this class was just getting them introduced to the concept of fractions, while the arithmetic rules would come later...
It’s 1/3 + 1/3 = 2/3, and 1:3 + 1:3 = 2:6 = 1:3.
You add slices of pie thus /, and win a vote by ratios thus :.
You add slices of pie thus /, and win a vote by ratios thus :.
I think she could have continued by adding more tables until they got to 4/3. That is obviously wrong - so it is easier to convince the kids that something is wrong. Then back track and explain what went wrong and explain the rules for adding fractions.
There is no track!!! Of course there is, because what we call mathematics education is broken beyond repair, but that's a crime.
Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do.
Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing anomalies is the best habit you want the next generation of engineers or statespeople to learn as a reflex from an early age?
Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do.
Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing anomalies is the best habit you want the next generation of engineers or statespeople to learn as a reflex from an early age?
there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves.
this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves.
there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning experience in general.
now, if you are homeschooling a child or somehow in a 1 on 1 (or at least, working with a very small teacher:student ratio), then perhaps a more freeform exploration of fractional arithmetic might be a wonderful thing.
doing so with a general class of kids? i strongly suspect you're wrong. they won't even understand that there is an anomaly, because they don't even understand the basic concepts that make it an anomaly. why shouldn't adding two fractions (never seen those before!) result in different answers? in fact, why is 2/3 different from 2/6 anyway? and so on.
this is not about shutting down curiosity IMO. it's about nurturing the basic concepts so that curiosity can grow amidst them in the (near) future.
this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves.
there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning experience in general.
now, if you are homeschooling a child or somehow in a 1 on 1 (or at least, working with a very small teacher:student ratio), then perhaps a more freeform exploration of fractional arithmetic might be a wonderful thing.
doing so with a general class of kids? i strongly suspect you're wrong. they won't even understand that there is an anomaly, because they don't even understand the basic concepts that make it an anomaly. why shouldn't adding two fractions (never seen those before!) result in different answers? in fact, why is 2/3 different from 2/6 anyway? and so on.
this is not about shutting down curiosity IMO. it's about nurturing the basic concepts so that curiosity can grow amidst them in the (near) future.
I have a hard time accepting this. My daughter is in 3rd grade and seems to have a reasonable grasp of fractions. They teach her about them at (public) school and we've discussed them at home. Sure, "1/3 + 2/7" is outside her reach, but simpler stuff, things she can visualize, are well within her grasp.
> why is 2/3 different from 2/6 anyway
Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces of a pie that you cut into 6 pieces. She understands that 4/8 == 2/4 == 1/2. Sometimes she needs to think about it a bit, but she does "get it".
> why is 2/3 different from 2/6 anyway
Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces of a pie that you cut into 6 pieces. She understands that 4/8 == 2/4 == 1/2. Sometimes she needs to think about it a bit, but she does "get it".
No disrespect, but you read HN and think that your daughter's take on math and the home context you provide for it is a sensibly representative starting point?
My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I wish they were - and hey, here I am reading HN too :)
My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I wish they were - and hey, here I am reading HN too :)
I think the homework her school gives out is a sensibly representative starting point. Why wouldn't it be?
That homework expects a reasonable understanding of fractions. Enough that the child doing them can understand the difference between the number of slices in a pie being representative of the bottom number of a fraction.
Sure, I do math exercises with my daughter that aren't representative of what they teach in school (square roots, the fact that parallel lines _do_ meet at the vanishing point in the real world, etc). But those things aren't what I'm basing my assumptions are; the expectations the school has of her are.
That homework expects a reasonable understanding of fractions. Enough that the child doing them can understand the difference between the number of slices in a pie being representative of the bottom number of a fraction.
Sure, I do math exercises with my daughter that aren't representative of what they teach in school (square roots, the fact that parallel lines _do_ meet at the vanishing point in the real world, etc). But those things aren't what I'm basing my assumptions are; the expectations the school has of her are.
There's incredibly strong evidence that the way to teach mathematics and the only way to learn it is to foster the environment in which you can rediscover the key insight for yourself. Obviously (to the younger students, to me, Lockhart, Dewey, and others, but not to most teachers, administrators, voters) there is no track and no curriculum for this.
They are not stumbling into it by themselves! They have a guide! The guide's job is to point out the works in the museum, and to make sure the kids don't get lost. It's not to stand in front of the art so the kids can't even see it and lecture about it!
There's nothing complex or sophisticated about recognizing that "+" has a meaning that we chose, and we could have chosen others, and in some cases other choices would be more natural.
You don't think students learning fractions know that there is one right answer, or that 2/3 and 1/3 can't both be right? Their short little lives have already been filled with enough test-taking to teach them that, at least.
And if they don't understand that 1/3 and 2/3 represent something different about the real world, and that 2/6 and 1/3 are different in a different way from 1/3 and 2/3, then why are we going on to teach them even more complex sophisticated concepts before they have grasped these most basic of basics?
There's never been a pedagogical program to shut down the curiosity of primary students. We do it by accident, by "nurturing the basic concepts" (the ones you learned at that age) so that curiosity can grow amidst them "in the future," which means, maybe, after they finish calculus, if they are lucky.
They are not stumbling into it by themselves! They have a guide! The guide's job is to point out the works in the museum, and to make sure the kids don't get lost. It's not to stand in front of the art so the kids can't even see it and lecture about it!
There's nothing complex or sophisticated about recognizing that "+" has a meaning that we chose, and we could have chosen others, and in some cases other choices would be more natural.
You don't think students learning fractions know that there is one right answer, or that 2/3 and 1/3 can't both be right? Their short little lives have already been filled with enough test-taking to teach them that, at least.
And if they don't understand that 1/3 and 2/3 represent something different about the real world, and that 2/6 and 1/3 are different in a different way from 1/3 and 2/3, then why are we going on to teach them even more complex sophisticated concepts before they have grasped these most basic of basics?
There's never been a pedagogical program to shut down the curiosity of primary students. We do it by accident, by "nurturing the basic concepts" (the ones you learned at that age) so that curiosity can grow amidst them "in the future," which means, maybe, after they finish calculus, if they are lucky.
You're stretching what I said. I should note that I'm a lifelong fan of radical education (particularly John Holt), and I don't think that the way we teach children most things has a lot to recommend it.
However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.
Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.
In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.
However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.
Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.
In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.
Yes. Every child I've ever met is curious about something, and that's what they should be learning. The idea that in year X every student should learn Y during hour Z every day is one of the reasons I said the system was irreparably broken. That is how schooling gets in the way of education.
Does every student learn to understand fractions under the current regime? Not in my experience.
Edit: And yes, you should say something about the platypus, because understanding that models are simplifications and incomplete can be enough of an escape hatch for the smart kids (the ones that usually hate math class) to notice that even the teacher knows that there's always more, and that can be enough to keep them from throwing it all away in disgust as a useless mishmash of arbitrary, conflicting, and incorrect rules.
Does every student learn to understand fractions under the current regime? Not in my experience.
Edit: And yes, you should say something about the platypus, because understanding that models are simplifications and incomplete can be enough of an escape hatch for the smart kids (the ones that usually hate math class) to notice that even the teacher knows that there's always more, and that can be enough to keep them from throwing it all away in disgust as a useless mishmash of arbitrary, conflicting, and incorrect rules.
You are right, I think before jumping to different things, kids should learn on simple examples, that do not include different types, and then when they grasp one concept they should be gradually introduced to the other concept.
table/tables would just confuse them even more.
> The more I think about it, the more I think the right response might have been "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.
Yea, the typing (or w/e) is the problem here. I like another posters examples of apples + oranges = fruits.
Otherwise you could further blow their minds. Where 1/3 + 1/3 == 2/6 and 2/6 is equal to 1/3 so 1/3 + 1/3 == 1/3. Hah. Understanding the incompatibility, or at least difference between the first, second and resulting fractions seems essential to understand.
Yea, the typing (or w/e) is the problem here. I like another posters examples of apples + oranges = fruits.
Otherwise you could further blow their minds. Where 1/3 + 1/3 == 2/6 and 2/6 is equal to 1/3 so 1/3 + 1/3 == 1/3. Hah. Understanding the incompatibility, or at least difference between the first, second and resulting fractions seems essential to understand.
> You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard!
I think it's the opposite: if you are taught the procedure for adding fractions without understanding what's going wrong here, then you just know a procedure, not a concept. Taking the detour into the detailed discussion of what 'the whole' is takes longer, but cements that crucial idea, and so lays the foundation for understanding addition of fractions more generally.
I think it's the opposite: if you are taught the procedure for adding fractions without understanding what's going wrong here, then you just know a procedure, not a concept. Taking the detour into the detailed discussion of what 'the whole' is takes longer, but cements that crucial idea, and so lays the foundation for understanding addition of fractions more generally.
Just a missing step in the explanation:
"1/3 of the students at the first table + 1/3 of the students at a second table =
1/3 of 1/2 of all the students at both tables [ie. those students at the first table] + 1/3 of another 1/2 of all the students at both tables [ie. those at a second table] =
1/3 of the students at both tables"
In US math education I noticed there is a lot emphasis on specific patterns/formulas/etc., kind of "deus ex machina", and a lot of excercises to mechanize their application, yet too little is spend on where it comes from, manual derivation/proval of formulas, establishing logical connections, etc.
"1/3 of the students at the first table + 1/3 of the students at a second table =
1/3 of 1/2 of all the students at both tables [ie. those students at the first table] + 1/3 of another 1/2 of all the students at both tables [ie. those at a second table] =
1/3 of the students at both tables"
In US math education I noticed there is a lot emphasis on specific patterns/formulas/etc., kind of "deus ex machina", and a lot of excercises to mechanize their application, yet too little is spend on where it comes from, manual derivation/proval of formulas, establishing logical connections, etc.
The core concept kids need to know when adding fractions together is that the denominators need to be the same before you can add the numerators. Your use of units is on the right track, but I think you need to use units for both the top and bottom of the fraction:
(1 girl at table A)/(3 students at table A) + (1 girl at table B)/(3 students at table B) = (2 girls at tables A & B) / (6 students at tables A & B)
vs
(1 boy at table A)/(3 students at table A) + (1 boy at table A)/(3 students at table A) = (2 boys at table A) / (3 students at table A)
(1 girl at table A)/(3 students at table A) + (1 girl at table B)/(3 students at table B) = (2 girls at tables A & B) / (6 students at tables A & B)
vs
(1 boy at table A)/(3 students at table A) + (1 boy at table A)/(3 students at table A) = (2 boys at table A) / (3 students at table A)
Ah, yes, that would definitely be a better way to write it if you were to go forward with the unit approach.
This. What the student said was 'wrong' and they need to look at the whole. This "1/3 + 1/3 = 2/6" problem would be a great example to use for a lesson on units though.
What they said wasn't wrong. Their mental model was absolutely correct.
"One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.'
Their written representation of the mental model was incorrect because their instruction was focused only on abstract numbers instead of concrete labels. If those fractions (or ratios or whatever) are labeled properly the equation is completely correct.
"One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.'
Their written representation of the mental model was incorrect because their instruction was focused only on abstract numbers instead of concrete labels. If those fractions (or ratios or whatever) are labeled properly the equation is completely correct.
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Separate from the discussion of what is or is not counterproductive / educative, the student's mental model was wrong.
The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.
The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.
No. The mental model was exactly correct. Their assignment of notation to the model was non-standard when they wrote "+" for the operation of merging the wholes.
The confusion of what is wrong with what is correct but non-standard is in fact quite standard and quite wrong.
The confusion of the mental model with the notation is also a standard mistake.
The confusion of what is wrong with what is correct but non-standard is in fact quite standard and quite wrong.
The confusion of the mental model with the notation is also a standard mistake.
Math is about learning how mathematical language is applied to problems. She is incorrect about how the math is applied. Ergo she is wrong. Her intuition is on the right track.
The only "whole" in this case is the entire set of students (or seats or whatever it is that is being talked about).
The only "whole" in this case is the entire set of students (or seats or whatever it is that is being talked about).
Yup, it's a typing problem. Do it with different things in the two sets and it becomes clearer: 1/3 apples + 1/3 oranges = 2/6 fruits.
This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.
This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.
It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.
You definitely can add 1:2 apples and 1:2 oranges, you just need to do so using a common base type (such as fruits or objects).
Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 becomes 1/3), but you can't do that here and still support the mediant operation. https://en.wikipedia.org/wiki/Mediant_(mathematics)
Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 becomes 1/3), but you can't do that here and still support the mediant operation. https://en.wikipedia.org/wiki/Mediant_(mathematics)
You can't perform operations with apples, oranges and fruits unless you use them completely interchangeably ,i.e. 1 apple = 1 orange. Otherwise, you will break algebraic equalities.
You can convert apples to fruit and oranges to fruit and then do operations on fruit, but you can't go back from the result to apples and/or oranges. For example, 1/7 of (3 apples + 4 oranges) is 1 fruit, but we can't tell if it's one orange, or 1 apple, or 1/3 apple and 2/3 orange.
You can convert apples to fruit and oranges to fruit and then do operations on fruit, but you can't go back from the result to apples and/or oranges. For example, 1/7 of (3 apples + 4 oranges) is 1 fruit, but we can't tell if it's one orange, or 1 apple, or 1/3 apple and 2/3 orange.
The grandparent's note about units and your mention of ratios get at the same problem for me:
The lesson mixes two of the physical world concepts that are often modeled by fractions.
One is an incomplete whole (the six pack) and the other is the proportions of a mixture (table demographics).
Fractions always have the same rules, but the rules map differently to the different territory being modeled.
The lesson mixes two of the physical world concepts that are often modeled by fractions.
One is an incomplete whole (the six pack) and the other is the proportions of a mixture (table demographics).
Fractions always have the same rules, but the rules map differently to the different territory being modeled.
abstract class Fruit:
prop name: string
prop rotten: bool
class Apple (Fruit):
prop name: string = "apple"
prop rotten: bool = False
class Orange (Fruit):
prop name: string = "orange"
prop rotten: bool = False
func add (a: List<Fruit>, b: List<Fruit>) -> List<Fruit>:
return foldl(lambda l, f: l.append(f), a, b)
func filter (p: Func<[Fruit], bool>, l: List<Fruit>) -> List<Fruit>:
return [f for f in l if p(f)]
func is_rotten (f: Fruit) -> bool:
return f.rotten
a1 = Apple()
a2 = Apple()
a3 = Apple()
a3.rotten = True
apples = [a1, a2, a3]
o1 = Orange()
o2 = Orange()
o3 = Orange()
o3.rotten = True
oranges = [o1, o2, o3]
fruits = add(apples, oranges)
print(len(filter(is_rotten, apples)), "/", len(apples))
> "1/3"
print(len(filter(is_rotten, oranges)), "/", len(oranges))
> "1/3"
print(len(filter(is_rotten, fruits)), "/", len(fruits))
> "2/6"This does nothing to elucidate the problem. 1/3 apples + 1/3 oranges could be 2/6 fruits, or 2/781 fruits, or any other number you want. In normal mathematical notation, 1/3 is interpreted as 1/3 of 1, and in that case 1/3 apple + 1/3 orange is 2/3 fruit.
And of course this only works if 1 apple = 1 orange = 1 fruit, which means that they are the same unit of measure, or, equivalently they are of substitutable types. It's even debatable of it's correct to say that 'if I have 1/3 of an apple and 1/3 of an orange, I have 2/3 of a fruit', so depending on what you want to do with your apples, oranges and fruit, your hierarchy may in fact break Liskov substitution. For example, if 1 apple + 1 orange = 2 fruit, so 2 fruit - 1 apple = 1 orange, so 1 apple = 1 orange, which is not true.
And of course this only works if 1 apple = 1 orange = 1 fruit, which means that they are the same unit of measure, or, equivalently they are of substitutable types. It's even debatable of it's correct to say that 'if I have 1/3 of an apple and 1/3 of an orange, I have 2/3 of a fruit', so depending on what you want to do with your apples, oranges and fruit, your hierarchy may in fact break Liskov substitution. For example, if 1 apple + 1 orange = 2 fruit, so 2 fruit - 1 apple = 1 orange, so 1 apple = 1 orange, which is not true.
You're being dishonest in your reasoning.
> In normal mathematical notation, 1/3 is interpreted as 1/3 of 1
Why would you deny the possibility of 1 being 1 list of fruits rather than 1 fruit?
In the original example, it was 1 table of students, not 1 student.
Everything you say after that is based on this broken assumption.
> In normal mathematical notation, 1/3 is interpreted as 1/3 of 1
Why would you deny the possibility of 1 being 1 list of fruits rather than 1 fruit?
In the original example, it was 1 table of students, not 1 student.
Everything you say after that is based on this broken assumption.
you can. It's what you do in fruit salad.
Well, in fruit salad 1/3 apple + 1/3 orange = 1/3 apple + 1/3 orange. I can say 3 * (1/3 apple + 1/3 orange) = 1 apple + 1 orange.
And then by way of analogy of course, (e apples)^(pi oranges) + 1 apple = 0.
And then by way of analogy of course, (e apples)^(pi oranges) + 1 apple = 0.
I think you both are saying the same thing, in this context at least.
There is a very important difference between types and quantities.
For example, 1/3 + 1/3 = 1 is true, if I mean 1/3 of 3 + 1/3 of 0. 1/3 + 1/3 = 2/6 can only be true if we add some quantities, we can't make it true by choosing the right types (we could fix it by choosing a different definition of +, or =, though).
For example, 1/3 + 1/3 = 1 is true, if I mean 1/3 of 3 + 1/3 of 0. 1/3 + 1/3 = 2/6 can only be true if we add some quantities, we can't make it true by choosing the right types (we could fix it by choosing a different definition of +, or =, though).
Interesting - I never thought of CS as being something to help with this. Instead, I’d think of other engineering fields where there is a heavy reliance on correct units and unit conversion. Pound of feathers vs pound of gold , etc...
F# offers a nifty "unit of measure" language feature to help with that. You might have
[<Measure>] type lbFeather
[<Measure>] type lbGold
[<Measure>] type lbLead
let convertGoldToLead ( weight : float<lbGold> ) = 1.0<lbLead> / 1.0<lbGold> * weight
// works
let barOfGold = 1.0<lbGold>
let barOfLead = convertGoldToLead barOfGold
// doesn't work
let bagOfFeathers = 1.0<lbFeather>
let leadDuck = convertGoldToLead bagOfFeathers
// Error reported: mismatched units.One of the article's comments:
"Janelle Schorg says:
"This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my opinion, ratios should not be written like fractions until later after students have conceptual understanding and fractions should never be taught with sets in the 3-5 work. Many teachers are not even aware of this difference and misconception we are creating in student understanding."
"Janelle Schorg says:
"This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my opinion, ratios should not be written like fractions until later after students have conceptual understanding and fractions should never be taught with sets in the 3-5 work. Many teachers are not even aware of this difference and misconception we are creating in student understanding."
That's totally wrong. It's not OK to add ratios. It's OK to add populations.
You can't add 1/3 and 1/3 to get 2/6 if the first 1/3 was reduced from 3 of 9 and th second was 1/3. Well, you can, but that only works in the degenerate case where the items you add (actually, average) are equal and there's no point in adding in the first place.
You can't add 1/3 and 1/3 to get 2/6 if the first 1/3 was reduced from 3 of 9 and th second was 1/3. Well, you can, but that only works in the degenerate case where the items you add (actually, average) are equal and there's no point in adding in the first place.
One could explain the mistake to have different units on both side without fractions at all:
1+1=1
One shoe plus one shoe equals one pair of shoes.
Once they grasp that explaining the fraction issue should be easier.
1+1=1
One shoe plus one shoe equals one pair of shoes.
Once they grasp that explaining the fraction issue should be easier.
This is a great baseline for the units answer.
I think it then needs an addendum on things that are the same dimension but expressed in different units.
Like, "students at table A" and "students at table B" are both of the same dimension as "students", but are different units. Like meters and miles. You can add them together, but only if you know the factors needed to convert them to a common unit. In this case, the conversion factors are, how many students are at table A and how many at table B.
Like, "students at table A" and "students at table B" are both of the same dimension as "students", but are different units. Like meters and miles. You can add them together, but only if you know the factors needed to convert them to a common unit. In this case, the conversion factors are, how many students are at table A and how many at table B.
Or a different way to look at it is that if you put two things together, the mathematical operation is not always "+"; it totally depends on the things (and how you put them together). You use plus if you put together fractions of the same thing (e.g. fractions of the same box of crayons), but potentially some totally other operation if you put together different things.
Someone mentioned on this thread using CS to teach math. What you have just described here is the fact that the + operator can be overloaded.
Wonder if the kids can grasp that concept though.
Wonder if the kids can grasp that concept though.
The units aren’t the issue IMO. You said:
> What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).
That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.
Explaining this in a classroom setting may be quite challenging indeed.
> What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).
That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.
Explaining this in a classroom setting may be quite challenging indeed.
I think you guys actually agree more than you think --
You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).
You can only add two things if they have the same units, as per "dimensional analysis" [1].
So this is an entirely meaningless statement:
[students]/[seats at table 1] + [students]/[seats at table 2]
But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):
([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])
Which simplifies to:
[students]/[total seats] + [students]/[total seats]
Now that's a statement with meaning!
Since I know that
[seats at table 1]/[total seats] = 1/2
[seats at table 2]/[total seats] = 1/2
I've just derived the calculation that I really wanted to do:
(1/3)(1/2) + (1/3)(1/2) = (2/6)
[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...
You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).
You can only add two things if they have the same units, as per "dimensional analysis" [1].
So this is an entirely meaningless statement:
[students]/[seats at table 1] + [students]/[seats at table 2]
But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):
([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])
Which simplifies to:
[students]/[total seats] + [students]/[total seats]
Now that's a statement with meaning!
Since I know that
[seats at table 1]/[total seats] = 1/2
[seats at table 2]/[total seats] = 1/2
I've just derived the calculation that I really wanted to do:
(1/3)(1/2) + (1/3)(1/2) = (2/6)
[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...
I'm not sure we can really guess what the student meant, but I do know for sure that she was wrong.
If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.
The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).
This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.
This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.
Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.
If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.
The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).
This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.
This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.
Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.
They sneakily replaced scalar addition by point-wise vector addition.
[1,3] <+> [1,3] = [2,6]
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This was what I came up with in the moment, but with drawing. Draw two circles on the board, each divided into thirds and compare it with a single circle divided into sixths. It demonstrates that the total has grown.
Also a fantastic way to represent this. The assumption in fractional arithmetic is that you're always performing arithmetic on things with the same type/unit.
The two formulas on the board are essentially:
1/3x + 1/3y = 2/6z
1/3x + 1/3x = 2/3x
Both are correct, but without units labeled you wouldn't know that.
The two formulas on the board are essentially:
1/3x + 1/3y = 2/6z
1/3x + 1/3x = 2/3x
Both are correct, but without units labeled you wouldn't know that.
As I said elsewhere, this is not a problem of units or types. If it were, then the computation wouldn't make sense.
It is a problem of implicit refernces. The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).
The correct equation would have been 1/3 * 3 + 1/3 * 3 = 2/6 * 6. Note that 3, 3 and 6 have the same unit. If they didn't, then this would be meaningless. 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals) ...).
It is a problem of implicit refernces. The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).
The correct equation would have been 1/3 * 3 + 1/3 * 3 = 2/6 * 6. Note that 3, 3 and 6 have the same unit. If they didn't, then this would be meaningless. 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals) ...).
> The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).
That is, precisely, the problem of units. Each object is its own unit here.
> The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.*
That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't, so you can't add like that.
> 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals)
That's the point (but it's 1/3, not 2/6). Also, 1/3 of a meter, + 1/3 foot = 1/3 (1 meter + 1 foot). Different units, but same dimension, so if you know the conversion factor (here, 1 meter = 3.3 feet), you can change it into (1/3 meter * 3.3 feet/meter) + 1/3 foot = 1.1 foot + 1/3 foot = 33/30 feet + 10/30 feet = 43/30 feet = 1.43(3) feet.
You can do the same math with students at tables.
That is, precisely, the problem of units. Each object is its own unit here.
> The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.*
That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't, so you can't add like that.
> 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals)
That's the point (but it's 1/3, not 2/6). Also, 1/3 of a meter, + 1/3 foot = 1/3 (1 meter + 1 foot). Different units, but same dimension, so if you know the conversion factor (here, 1 meter = 3.3 feet), you can change it into (1/3 meter * 3.3 feet/meter) + 1/3 foot = 1.1 foot + 1/3 foot = 33/30 feet + 10/30 feet = 43/30 feet = 1.43(3) feet.
You can do the same math with students at tables.
You are trying to look at a different problem. It was absolutely correct that 1/3 of the students at one table of 3 plus another 1/3 of the students at another table of 3 is the same number as 2/6 of the 6 students sitting at the two tables. This is not disputable.
The way you can write this observation mathematically is as I did: ((1/3) × 3) + ((1/3) × 3) = ((2/6) × 6), or 1 + 1 = 2, after computing the fractions. The student's observation was perfectly correct, but he was missing the proper explanation, as it is not about the addition of fractions (it is almost a coincidence that the fractions used on one side of the equation happen to have the sum of their numerator and the sum of their denominators equal to the numerator and denominator of the fraction on the other side - this only happens because we are multiplying the fractions by their denominators).
Sure, you can express this in terms of units and dimensions of you really choose to. You can also express it in terms of different definitions of +, or even of =. It is pretty unnatural to me to invent an ad-hoc measurement unit N1, "number of people at 1 table" and a different measurement unit, N2, "number of people at 2 tables", with the relation 1N2 = 2N1, and then correct the student's formula to 1/3N1 + 1/3N1 = 2/6N2. It is correct, but it is extremely artificial to me.
By far the most natural way to explain it is using the correct mathematical interpretation of the phrase "one third of the 3 people" - (1/3) × 3.
Inventing measurement units to describe exact quantities reminds me of a silly joke from Portal: "computer: 2 + 2 = 10 <pause to wonder if the computer is broken> ... in base 4". You can always find a way to make the formula direct by adding assumptions.
The way you can write this observation mathematically is as I did: ((1/3) × 3) + ((1/3) × 3) = ((2/6) × 6), or 1 + 1 = 2, after computing the fractions. The student's observation was perfectly correct, but he was missing the proper explanation, as it is not about the addition of fractions (it is almost a coincidence that the fractions used on one side of the equation happen to have the sum of their numerator and the sum of their denominators equal to the numerator and denominator of the fraction on the other side - this only happens because we are multiplying the fractions by their denominators).
Sure, you can express this in terms of units and dimensions of you really choose to. You can also express it in terms of different definitions of +, or even of =. It is pretty unnatural to me to invent an ad-hoc measurement unit N1, "number of people at 1 table" and a different measurement unit, N2, "number of people at 2 tables", with the relation 1N2 = 2N1, and then correct the student's formula to 1/3N1 + 1/3N1 = 2/6N2. It is correct, but it is extremely artificial to me.
By far the most natural way to explain it is using the correct mathematical interpretation of the phrase "one third of the 3 people" - (1/3) × 3.
Inventing measurement units to describe exact quantities reminds me of a silly joke from Portal: "computer: 2 + 2 = 10 <pause to wonder if the computer is broken> ... in base 4". You can always find a way to make the formula direct by adding assumptions.
If you are omitting the units, it would be a fair assumption that the values all belong to the same unit, unless stated otherwise. You're representing the fractions related to each other and you can't do that if the fractions are of two separate things.
I could say that 1+1 = 2 or 1+1 = 10 but it wouldn't be right to say that 1+1=2 && 1+1=10 because while both are true if we're talking about decimals and binary, we're omitting the units and everything loses its' meaning if we do that.
I could say that 1+1 = 2 or 1+1 = 10 but it wouldn't be right to say that 1+1=2 && 1+1=10 because while both are true if we're talking about decimals and binary, we're omitting the units and everything loses its' meaning if we do that.
> The units are missing
> The student came up and wrote 1/3 + 1/3 = 2/6.
Units in your example are unclear. Also results interpretation depend on question.
Here is my example:
Student should wrote 2 articles, but wrote 1 of 3 pages for the 1st article and 1 of 3 pages for the 2nd article.
Q: How many pages student wrote? - A: 2/6.
Q: What part of the whole task is done? - A: 1/3.
Q: How many tasks partially completed? - A: 2/2.
Q: How many tasks fully completed? - A: 0/2.
> The student came up and wrote 1/3 + 1/3 = 2/6.
Units in your example are unclear. Also results interpretation depend on question.
Here is my example:
Student should wrote 2 articles, but wrote 1 of 3 pages for the 1st article and 1 of 3 pages for the 2nd article.
Q: How many pages student wrote? - A: 2/6.
Q: What part of the whole task is done? - A: 1/3.
Q: How many tasks partially completed? - A: 2/2.
Q: How many tasks fully completed? - A: 0/2.
I can't see the page (slashdotted), but that sounds just like the puzzle from an old kids' math show we used to watch:
https://www.youtube.com/watch?v=bCoGMYV3UPk&t=2m29s
https://www.youtube.com/watch?v=bCoGMYV3UPk&t=2m29s
Yes, 1/3x + 1/3x = 2/6(x + x).
Units are absolutely necessary, and 'fractions' don't make sense without them.
'1/3' without units is not a 'fraction' but instead is 0.3333, a number on the number line, which is why it looks so wrong to see 1/3 + 1/3 = 2/6.
Units are absolutely necessary, and 'fractions' don't make sense without them.
'1/3' without units is not a 'fraction' but instead is 0.3333, a number on the number line, which is why it looks so wrong to see 1/3 + 1/3 = 2/6.
I would do piecewise addition like this:
Girl A is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.
Girl B is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.
So girls represent 1/6 + 1/6 = 2/6 of the total.
Girl A is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.
Girl B is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.
So girls represent 1/6 + 1/6 = 2/6 of the total.
Exactly what I was thinking. Maybe it could be formalized this way? (with t = 1 table)
1/3 * (1 table) + 1/3 * (1 table) = 2/6 * (2 tables)
or
1/3 * t + 1/3 * t = 2/6 * 2 * t
but 2/6 * 2 = 2/3 so you get
1/3 * t + 1/3 * t = 2/3 * t
which is correct.
1/3 * (1 table) + 1/3 * (1 table) = 2/6 * (2 tables)
or
1/3 * t + 1/3 * t = 2/6 * 2 * t
but 2/6 * 2 = 2/3 so you get
1/3 * t + 1/3 * t = 2/3 * t
which is correct.
or if students are in both tables
This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students.
Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students.
It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates are.
Some other lovely examples:
3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input.
x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.
\sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell.
There are many more.
Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students.
It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates are.
Some other lovely examples:
3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input.
x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.
\sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell.
There are many more.
Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand.
A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects.
This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.
A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects.
This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.
As you say, the artistic use of cross-object synthesis and analogy definitely distinguishes the advanced students from the mechanical. Advanced students develop a sort of dialect that harmonizes the mathematical objects they encounter in a way that illuminates them all.
More than any of that, though, I think what you can see (even here in this thread!) is that, while we pretend that mathematics is a single cultural practice of rational humans communicating with other rational humans, it's really many smaller communities of mathematicians, all of varying skills trying to communicate with each other. My mathematical language as a teacher of 12-18 year olds is very different from my mathematical language when I did computational geometry for a living.
Because you have many communities of mathematicians producing new notation, you end up with dialects that all sort of meld together in the same way that reading Shakespeare is very different from reading Hemingway or Eco (in translation).
The closest programming analogue would be C++, where you have several mutually unintelligible dialects spoken by different communities of programmers with different concerns.
More than any of that, though, I think what you can see (even here in this thread!) is that, while we pretend that mathematics is a single cultural practice of rational humans communicating with other rational humans, it's really many smaller communities of mathematicians, all of varying skills trying to communicate with each other. My mathematical language as a teacher of 12-18 year olds is very different from my mathematical language when I did computational geometry for a living.
Because you have many communities of mathematicians producing new notation, you end up with dialects that all sort of meld together in the same way that reading Shakespeare is very different from reading Hemingway or Eco (in translation).
The closest programming analogue would be C++, where you have several mutually unintelligible dialects spoken by different communities of programmers with different concerns.
I think it's a lot less beautiful. I think it has to do more with historical accidents. We've stumbled our way forward in mathematics; there is no grand plan that unifies our notation. Just look at calculus for a prime example - df/dx and f'(x) come from two different lineages and get used interchangeably; the df/dx notation can be intensely misleading when students think that (for example) they should be able to use normal fraction rules.
Of course, there is no grand plan, and yes, a lot of notation is historical accidents. But my point is that attempting to "fix" it will probably yield a better notation, but not the "perfect" notation.
As an aside, df/dx treated a fraction is used as soon as second or third semester of uni when they learn how to solve differential equations using separation of variables, and physicists/chemists start using total derivatives for thermodynamics. I am not aware of notation different from df/dx where these subjects would be just as clear.
As an aside, df/dx treated a fraction is used as soon as second or third semester of uni when they learn how to solve differential equations using separation of variables, and physicists/chemists start using total derivatives for thermodynamics. I am not aware of notation different from df/dx where these subjects would be just as clear.
Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)).
I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always.
Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.
I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always.
Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.
Does f^2(x) in any scenario mean (f(x))^2? Usually the square of a function, with named arguments is written as f(x)^2. So, not clear on the confusion.
Yes, in trigonometry sin^2(x) is commonly written to mean (sin(x))^2
> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.
That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.
That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.
It's not because 'function powers' make sense, yhey are just iterated function application. That's how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse.
Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.
Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.
The idea that "when you extend that" is something human beings can do is an absolute revelation to young mathematicians. The idea that our notations are, at least in some sense, choices that we make that come with tradeoffs is a huge point of mathematical maturity for them, and usually causes my students to look back with either deep awe or deep suspicion when they realize that even our choice of base 10 is a choice among many and we can make other choices.
As von Neumann said, very much something to get used to.
As von Neumann said, very much something to get used to.
> It's not because 'function powers' make sense, yhey are just iterated function application.
To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!
To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!
Indeed!
They’re both inverses: one with respect to multiplication, the other with respect to function composition.
In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value.
There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.
In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value.
There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.
> one with respect to multiplication, the other with respect to function composition
And, of course, multiplication is a function so one is just a specific instance of the other.
Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.
And, of course, multiplication is a function so one is just a specific instance of the other.
Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.
Multiplication is a function of numbers, and function composition is a function of functions, so neither is an instance of the other (unless you take the unusual perspective of thinking of numbers themselves as functions, but I don’t think that’s what you meant).
> I don’t think that’s what you meant
Why not?
Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
Why not?
Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
In both of these instances, I think that you are using composition to mean iteration. Iterating is repeated self-composition, so it is composition, but it seems likely that you meant the more specific term. (For example, "the successor function is not a composition of anything" is definitely false in the literal sense of composition. It's also false in the literal sense of iteration, but there it's clear what you meant: there's no natural operation which we iterate some number of times greater than 1 to get the successor function.)
> you meant the more specific term
No, I meant what I said. The fact that iteration is a kind of composition is true, but it's a tangent from the point I was trying to make, which is that infix notation is a Really Bad Idea. No one in their right mind would use it if they were not indoctrinated into it.
> "the successor function is not a composition of anything" is definitely false
That's news to me. I genuinely thought that successor could legitimately be considered a primitive. What is successor a composition of?
No, I meant what I said. The fact that iteration is a kind of composition is true, but it's a tangent from the point I was trying to make, which is that infix notation is a Really Bad Idea. No one in their right mind would use it if they were not indoctrinated into it.
> "the successor function is not a composition of anything" is definitely false
That's news to me. I genuinely thought that successor could legitimately be considered a primitive. What is successor a composition of?
> Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
> the point I was trying to make, which is that infix notation is a Really Bad Idea
Not to be snarky, but, reading these two (from your two thread-successive posts https://news.ycombinator.com/item?id=23312725 and https://news.ycombinator.com/item?id=23314776) in succession, I still can't see anything about the first one that indicates the point that infix notation is a bad idea. Not that I'm disagreeing with the point, just that I can't find it in the first post. Could you clarify the connection?
> What is successor a composition of?
It's a composition of, for example, itself and the identity function, like everything else; or you could view it as a composition (-1) . (+2). Those kind of silly solutions are why I thought you meant 'iteration'.
Iterative solutions are easy if you don't restrict yourself to natural numbers—for example, (+1) is (+(1/2)) composed with itself—but that's clearly not what you meant. (As soon as you leave the natural numbers, even the idea that addition is iterated successor becomes false.)
If you do so restrict yourself, then it becomes true that the successor is not a non-trivial iterate. (I just skated the edge of claiming the opposite in my post, but avoided error by not specifying what domain I meant. That's just luck, though; I meant a particular thing, and I was wrong. To prove it, supposing you start your natural numbers at 0 and that f is a function such that f^{\circ k} is the successor function for some k > 1, then note that f is injective (because a composition power is). If f(0) = 0, then succ(0) = f(f(0)) = f(0) = 0, which is a contradiction. Put n = f(0) and note that f^{\circ n k}(0) = succ^n(0) = n, but n k > 1.)
By the way, you quoted (https://news.ycombinator.com/item?id=23314776):
> > you meant the more specific term
Just to be clear, what I said (https://news.ycombinator.com/item?id=23314530) was "I think you meant the more specific term". And I was wrong, but I intentionally didn't just assume I knew you what you meant!
> the point I was trying to make, which is that infix notation is a Really Bad Idea
Not to be snarky, but, reading these two (from your two thread-successive posts https://news.ycombinator.com/item?id=23312725 and https://news.ycombinator.com/item?id=23314776) in succession, I still can't see anything about the first one that indicates the point that infix notation is a bad idea. Not that I'm disagreeing with the point, just that I can't find it in the first post. Could you clarify the connection?
> What is successor a composition of?
It's a composition of, for example, itself and the identity function, like everything else; or you could view it as a composition (-1) . (+2). Those kind of silly solutions are why I thought you meant 'iteration'.
Iterative solutions are easy if you don't restrict yourself to natural numbers—for example, (+1) is (+(1/2)) composed with itself—but that's clearly not what you meant. (As soon as you leave the natural numbers, even the idea that addition is iterated successor becomes false.)
If you do so restrict yourself, then it becomes true that the successor is not a non-trivial iterate. (I just skated the edge of claiming the opposite in my post, but avoided error by not specifying what domain I meant. That's just luck, though; I meant a particular thing, and I was wrong. To prove it, supposing you start your natural numbers at 0 and that f is a function such that f^{\circ k} is the successor function for some k > 1, then note that f is injective (because a composition power is). If f(0) = 0, then succ(0) = f(f(0)) = f(0) = 0, which is a contradiction. Put n = f(0) and note that f^{\circ n k}(0) = succ^n(0) = n, but n k > 1.)
By the way, you quoted (https://news.ycombinator.com/item?id=23314776):
> > you meant the more specific term
Just to be clear, what I said (https://news.ycombinator.com/item?id=23314530) was "I think you meant the more specific term". And I was wrong, but I intentionally didn't just assume I knew you what you meant!
> I still can't see anything about the first one that indicates the point that infix notation is a bad idea.
I didn't make a very good argument for it. I really intended that to be more of a throwaway rant than a serious critique. But since you ask...
There are two problems with infix:
1. It's hard to parse. It requires precedence rules which are not apparent in the notation. In actual practice, the precedence rules vary from context to context and this causes real problems. It's an unnecessary cognitive burden that pays very little in the way of dividends (a few less pen strokes or key strokes).
2. It obscures the fact that infix operators are just syntactic sugar for function applications. It leads people to think that there is something fundamentally different about a+b that distinguishes it from sum(a,b) and this in turn leads to a ton of confusion.
> that's clearly not what you meant
Indeed not. I meant the successor operator as defined in the Peano axioms.
> you quoted
Yeah, sorry about that. When I first replied, I thought you were the same person who posted the grandparent comment. My first draft response turned out to be completely inappropriate when I realized you were a different person, but some of my initial mindset apparently leaked into the revised comment. My apologies.
I didn't make a very good argument for it. I really intended that to be more of a throwaway rant than a serious critique. But since you ask...
There are two problems with infix:
1. It's hard to parse. It requires precedence rules which are not apparent in the notation. In actual practice, the precedence rules vary from context to context and this causes real problems. It's an unnecessary cognitive burden that pays very little in the way of dividends (a few less pen strokes or key strokes).
2. It obscures the fact that infix operators are just syntactic sugar for function applications. It leads people to think that there is something fundamentally different about a+b that distinguishes it from sum(a,b) and this in turn leads to a ton of confusion.
> that's clearly not what you meant
Indeed not. I meant the successor operator as defined in the Peano axioms.
> you quoted
Yeah, sorry about that. When I first replied, I thought you were the same person who posted the grandparent comment. My first draft response turned out to be completely inappropriate when I realized you were a different person, but some of my initial mindset apparently leaked into the revised comment. My apologies.
[deleted]
I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the other [function composition]”.
Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said.
As for multiplication being “composed” addition, addition being “composed” succession, etc.: The multiplication function (at least of integer arguments) is composed of addition. Function composition is a function that returns a composition of two other functions. The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing.
Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). The function which does function composition is not itself a composed function, and the act of multiplication is the act of a composed function, but not an act of function composition.
Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said.
As for multiplication being “composed” addition, addition being “composed” succession, etc.: The multiplication function (at least of integer arguments) is composed of addition. Function composition is a function that returns a composition of two other functions. The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing.
Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). The function which does function composition is not itself a composed function, and the act of multiplication is the act of a composed function, but not an act of function composition.
> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing.
Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. That is the whole point.
Numbers and addition and multiplication happen to be particularly important functions, but they are not structurally different from any other functions. Giving them special notation, especially when you are first introduced to them, obscures this fact. This kind of mental damage is very hard to recover from in later life. I believe it's one of the reasons so many people think they hate math. Math can be beautiful and elegant, but the standard notation used for school-book arithmetic is arbitrary and perverse, a bizarre accident of history with no actual merit.
IMHO of course.
[UPDATE:]
> Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)).
Function composition is a function, no different from any other function. There is no more reason to use infix notation for it than there is for any other function. In fact, if you drop the infix notation it immediately becomes obvious how ubiquitous and non-special function composition actually is:
Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. That is the whole point.
Numbers and addition and multiplication happen to be particularly important functions, but they are not structurally different from any other functions. Giving them special notation, especially when you are first introduced to them, obscures this fact. This kind of mental damage is very hard to recover from in later life. I believe it's one of the reasons so many people think they hate math. Math can be beautiful and elegant, but the standard notation used for school-book arithmetic is arbitrary and perverse, a bizarre accident of history with no actual merit.
IMHO of course.
[UPDATE:]
> Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)).
Function composition is a function, no different from any other function. There is no more reason to use infix notation for it than there is for any other function. In fact, if you drop the infix notation it immediately becomes obvious how ubiquitous and non-special function composition actually is:
compose(f,g)(x) = compose(f)(g)(x) = f(g(x))
[deleted]
You seem to be jumping around in terms of what position you're taking. Is times(a, b) function composition because a and b are functions, or is times(a, b) function composition because it's composition of the addition function, or is times(a, b) function composition because multiplication is a function? Those are very different assertions, with very different implications to the original discussion, and so far as I can tell you’ve made all three of them. Maybe they're all true, maybe one or two of them is true, maybe it's a matter of perspective which are true. But I don't think you're being clear or consistent in which assertion your position is based on.
And no matter what, there's still a difference between composing a with b, and composing b with itself a times, which is what I mean by the distinction between composition and iteration (sort of like the distinction between a brick and a brick wall).
At this point I feel we're going around in circles, so I'll bow out.
And no matter what, there's still a difference between composing a with b, and composing b with itself a times, which is what I mean by the distinction between composition and iteration (sort of like the distinction between a brick and a brick wall).
At this point I feel we're going around in circles, so I'll bow out.
> Those are very different assertions
But they are not mutually exclusive. That's the whole point.
But they are not mutually exclusive. That's the whole point.
> I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.
Maybe they did.
Maybe they did.
A good example of this can be found in an incredible algorithm with a terrible name: Fast inverse square root.
It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0]
Related to this: the way a superscript '2' next to a function means the function shall be applied twice (that is, composition with itself)... unless it's a trigonometric function, in which case it means the square of the result. [1] [2]
[0] https://en.wikipedia.org/wiki/Fast_inverse_square_root
[1] https://en.wikipedia.org/wiki/Function_composition#Functiona...
[2] https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...
It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0]
Related to this: the way a superscript '2' next to a function means the function shall be applied twice (that is, composition with itself)... unless it's a trigonometric function, in which case it means the square of the result. [1] [2]
[0] https://en.wikipedia.org/wiki/Fast_inverse_square_root
[1] https://en.wikipedia.org/wiki/Function_composition#Functiona...
[2] https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...
That one is clear to me, although it might be for a reason you find displeasing: if the thing being computed was an "inverse square root" in the sense of a functional inverse, then you'd be computing a square and it would make much more sense to call it that instead.
There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}.
Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead.
Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be.
I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.
Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead.
Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be.
I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.
> I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.
Indeed - I wish we could conceal those notations until the redemptive context became more apparent, in the same way that Latin teachers can conceal Sallust's inconcinitas until students are ready for it.
Indeed - I wish we could conceal those notations until the redemptive context became more apparent, in the same way that Latin teachers can conceal Sallust's inconcinitas until students are ready for it.
But the inverse function is the "multiplicative inverse" in the group of functions with composition as "multiplication". In that way, it makes a ton of sense. It's only a problem because you are mixing together two group operations.
To me it always seemed pretty natural. \frac{1}{f(x)} would be (f(x))^{-1}, by analogue to x^{-1}+1 vs (x+1)^{-1}. I would have been very surprised if raising just the 'f' part of the function application expression to some power were to mean raising the whole expression to that power.
I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).
I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).
> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell.
I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.
I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.
Are you sure it wasn't sin (30 pi / 180), because that actually makes sense.
Absolutely terrifying. How did he communicate with other peer mathematicians?
I honestly don't know - it was an intro level physics class.
Mathematicians understand how to read definitions of notation.
If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
Is that commonly done?
It is not. Though it wouldn't be too hard to adapt, I think, if it was noted at the beginning of a text. For it to take, it would have to power some influential result, sort of like how the summation convention in tensor notation works (something that infuriated me as a kid).
Allow me to introduce modern theoretical physics, where comments like "X is obvious in context" never is, except to the person that wrote it. Don't worry though, they helpfully add that "When X isn't what it seems in context, this will be called out". Okay. "Sometimes it may not be." Wat?
Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated.
If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group.
I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.
Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated.
If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group.
I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.
I also remember having constant confusion of what were assumed associations in written bigger formulas especially involving the functions and the practice of not writing the braces, and before that, if 3 1/5 meant three times 1/5 or 3 wholes and 1/5th.
I thought it was only on these low levels of educations that people settle on such confusing practices.
Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.
I thought it was only on these low levels of educations that people settle on such confusing practices.
Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.
I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.
Right.
Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices.
It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.
Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices.
It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.
Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.
And so it is.
edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use.
But I think we often thrown them in unintentionally because we're so used to it.
edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use.
But I think we often thrown them in unintentionally because we're so used to it.
You used too little context when arguing with an audience who isn't used to that style of writing...
Also, there are actual quite reasonable rules to know which 3rd declension nouns are i-stems, so it doesn't seem too right too just say "follow your gut".
Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^
Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^
x(3) is a function "x" being applied to the number 3, because "three times x" is 3x.
Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)
Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)
I'm sure some would take it that way as "x" isn't commonly used in mathematical notation as a function name, the norm is to use f, g, etc.
However, the notation is relatively clear notwithstanding. It almost has to be x as a function with an input of 3.
However, the notation is relatively clear notwithstanding. It almost has to be x as a function with an input of 3.
I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.
It depends on the context. As an intermediate step I definitely write things like x (3) meaning multiplication, as it can more clearly indicate what's just happened (see my other comment in this thread).
There's other context too, based on what is known. Up to a certain point in first year at my university, most engineering students haven't ever seen functions named x and y, and so they'd mostly interpret x(3) as multiplication. Then we show them parametric curves, and suddenly x(3) looks like the x coordinate of a point on the (x(t), y(t)) curve.
There's other context too, based on what is known. Up to a certain point in first year at my university, most engineering students haven't ever seen functions named x and y, and so they'd mostly interpret x(3) as multiplication. Then we show them parametric curves, and suddenly x(3) looks like the x coordinate of a point on the (x(t), y(t)) curve.
This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f(x)=fx=f×x while f[x] means “apply the function f to the argument x”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.
It's not devoid of ambiguity, because f[A] usually means the image of the set A under f.
Not at all. f[x] is clearly the xth component of the array f :-D
On the blackboard in the mathematics department it’s not.
But I get your point.
I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.
But I get your point.
I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.
Square brackets can also denote the floor function (especially in old works where '⌊' and '⌋' were typographically unavailable).
An array is merely a function from N to some set.
If you meant "function", would you generally write 𝑓(3) instead of f(3)?
(I have no mathematics background past K-12)
(I have no mathematics background past K-12)
So, have been any sufficient effort in design a new syntax of maths that could help the regular folk?
If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...
If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...
There are some gentle efforts in that direction (tau radians for example), but mathematical notation is old (except the parts that are new). And it's really hard to strike the right balance between concision and ambiguity.
I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time.
When you see something like.
When you see something like.
f(x) = summation(x^n, n=0, 10)
> We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 0 in this context.
What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it.
So, what do you think the summation actually is when expanded?
What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it.
So, what do you think the summation actually is when expanded?
Sorry you're right, it should be 0^0 = 1.
An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w.
You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.
An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w.
You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.
It makes perfect sense, everyone knows:
0^0 = 0
1^0 = 1
0^1 = 1
1^1 = 0
0^0 = 0
1^0 = 1
0^1 = 1
1^1 = 0
That's just lazy, and I haven't ever read a good text use summation where any of the terms would be outright undefined if evaluated.
You've never seen a textbook present the series form of e^x?
Fair, infinite series are pretty esoteric. How about derivatives. The power rule:
Eh maybe not, programmers don't use calculus that often. But surely statistics!
exp(x) = summation(x^n/n!, n=0, inf)
Fair, infinite series are pretty esoteric. How about derivatives. The power rule:
d/dx x^n = n x^(n-1)
Eh maybe not, programmers don't use calculus that often. But surely statistics!
(1 + x)^n = summation((n choose k)x^k, k=0, n)
(1 + x)^n = (1 + 0)^0 = 1 = 0^0 = x^0 = x^0 + ... + x^n
Ok, fair, I think I have seen those. Personally I would prefer noting the special case even when using the fairly standard degenerate case of 0^0=1, but I agree that a lot of people don't.
What isn't really true?
The difficulty isn't with fractions. It's about understanding what "+" means.
Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie").
His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way?
Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway.
Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).
Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie").
His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way?
Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway.
Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).
That's my favourite explanation so far. That's how I would put it to the student:
Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples:
* At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table.
* Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables.
The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3"
The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6"
What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.
Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples:
* At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table.
* Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables.
The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3"
The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6"
What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.
We explicitly encourage the "let's invent name for this similar but unnamed thing" with our students, and use whatever name they come up with for the thing for the rest of their time with us.
Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)
Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)
I would say the issue with symbols is why I had trouble with linear algebra in college, specifically dot and cross products. The fact that they used the same symbols as scaler multiplication and are also called products confused my mind so much.
As others have already pointed out, the question is ambiguous without more information. Always include the unit; bare magnitudes could mean anything! The student was describing the mediant[1], which was the correct solution when the problem is interpreted with forbiddenvoid'a units[2].
"Adding" fractions with the mediant leads to fun things like the Farey sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the Farey sequence)
[1] https://en.wikipedia.org/wiki/Mediant_%28mathematics%29
[2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6 (of the students in the room)"
[3] https://en.wikipedia.org/wiki/Farey_sequence
[4] https://en.wikipedia.org/wiki/Ford_circle
[5] https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree
[6] https://www.youtube.com/watch?v=0hlvhQZIOQw
"Adding" fractions with the mediant leads to fun things like the Farey sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the Farey sequence)
[1] https://en.wikipedia.org/wiki/Mediant_%28mathematics%29
[2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6 (of the students in the room)"
[3] https://en.wikipedia.org/wiki/Farey_sequence
[4] https://en.wikipedia.org/wiki/Ford_circle
[5] https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree
[6] https://www.youtube.com/watch?v=0hlvhQZIOQw
[deleted]
I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging.
It's not as bad of a nightmare as trying to explain the answer to the missing dollar riddle: https://en.m.wikipedia.org/wiki/Missing_dollar_riddle. That's an absolute nightmare.
It's not as bad of a nightmare as trying to explain the answer to the missing dollar riddle: https://en.m.wikipedia.org/wiki/Missing_dollar_riddle. That's an absolute nightmare.
I never understood why the Missing Dollar Riddle ever confuses people. As soon as they say "Add the $2 to the $27" I say, "But why are you adding something someone has to a total that people paid?"
That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean.
[0] https://mystudentvoices.com/how-old-is-the-shepherd-the-prob...
That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean.
[0] https://mystudentvoices.com/how-old-is-the-shepherd-the-prob...
The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project.
I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
As I recall, that study was debunked/retracted, so you may have trouble finding it.
The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf
Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf
Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
It's a bit more complicated than "debunked/retracted", as that second document explains:
> Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor.
The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability.
Which i think does line up with wcarey's point:
> The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
> Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor.
The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability.
Which i think does line up with wcarey's point:
> The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
Agreed.
I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.
I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.
Excerpt from the retraction:
> In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true.
A sad story :(
> In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true.
A sad story :(
Thanks! I hadn't seen the retraction. That's really interesting.
This reminds me of this problem which made the rounds recently[0], where a word problem essentially makes a lot of people commit a fencepost error
[0] https://math.stackexchange.com/questions/379927/how-long-wil...
[0] https://math.stackexchange.com/questions/379927/how-long-wil...
Adding numerators and denominators to find the "mediant"
is sometimes called "doing a freshman sum". In some countries I gather this is introduced as a separate operation (see page 23 in the translation of a Soviet primary school textbook by Gelfand linked below).
https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator.
And hi Bjørn!
And hi Bjørn!
> I'm surprised there is no such operator for averaging.
You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d).
Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders.
What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.
You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d).
Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders.
What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.
Well, you're right so long as 1/3 = 2/6. This is a mathematical equivalence... Fractions are a 2d projective space: they are pairs (n, d) where (mn, md) is equivalent to (n, d). The suggested operation is totally fine if you are working in ambient (that is, non-projective) space... ie, if you continually keep track of the 'size' of the thing being measured.
The equivalence operation is hard to master, and makes the arithmetic complicated.
The equivalence operation is hard to master, and makes the arithmetic complicated.
[deleted]
> It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging.
This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.
This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.
Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2.
You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages!
No way someone would have come up with that approach on the fly, though.
You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages!
No way someone would have come up with that approach on the fly, though.
Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...
And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples:
1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules.
1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular.
15% + 20% - who knows? 15% of what? 20% of what?
(1,2) + (2,4) - can't be done.
a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.
It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules.
1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular.
15% + 20% - who knows? 15% of what? 20% of what?
(1,2) + (2,4) - can't be done.
a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.
It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
> (1,2) + (2,4) - can't be done.
are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors
> a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.
what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)
are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors
> a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.
what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)
Indeed. And if they're cartesian vectors, you're good. But if that second number is an angle measured in radians, you use yet a different set of rules for the addition.
The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.
I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".
The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.
I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".
How did I go all of my life so far without hearing the super useful word "monomial"? It's such an obvious concept to have a meaningful name, and yet I don't recall anyone ever having said that word.
I also didn't encounter it until I was relatively older. Our school has been pushing to introduce more rigorous language and definition in our 7-12th math program. For some students, it really seems to help. For others, it's really hard.
> 15% + 20% - who knows? 15% of what? 20% of what?
That's just notational sugar on fractions though (15/100, 20/100) - so why is that less troubling than the general case?
That's just notational sugar on fractions though (15/100, 20/100) - so why is that less troubling than the general case?
I mean, the general case is certainly trouble for kids. But 15% and 20% look very similar to 15x and 20x. That the two (sometimes!) operate using different rules causes confusion for many students. The implicitness of the fractions conceals something important that's explicit if you write out the fractions. For example, multiplying percents does not do what most students intuitively think it does.
15% of 8 times 20% of 10 isn't 35% of any nice arithmetic combination of 8 and 10. That's hard to communicate to many students.
15% of 8 times 20% of 10 isn't 35% of any nice arithmetic combination of 8 and 10. That's hard to communicate to many students.
The main thing I find myself explaining with percentages (IANA educator) is that 'grossing up' != adding the same percentage back, e.g. 80% * 1.2 != 100%, which looks pretty obvious like that, but it's a common mistake among adults talking about real life percentages like taxes.
Another common, and I suppose related, one (but that I don't bother correcting often) is 'percent' != 'percentage point'. Talk of 47% of something being '3% less than' half of it really winds me up - and it's stupidly common among journalists - but it's too common to bother pointing out IMO. Live and let get wrong.
Another common, and I suppose related, one (but that I don't bother correcting often) is 'percent' != 'percentage point'. Talk of 47% of something being '3% less than' half of it really winds me up - and it's stupidly common among journalists - but it's too common to bother pointing out IMO. Live and let get wrong.
I think this just adds to the confusion. Is "adding" ratios really the same as averaging them? (I would just say that adding ratios is simply not defined.) Can you only average things by moving from fractions to ratios and then back again?
I think better to address the problem directly in fractions by saying that they're two different ways of combining them, addition and averaging, like the parent comment says. Feel free to make up a different symbol for the averaging operator, just let the kids know that it's not standard.
I think better to address the problem directly in fractions by saying that they're two different ways of combining them, addition and averaging, like the parent comment says. Feel free to make up a different symbol for the averaging operator, just let the kids know that it's not standard.
There's a lot more than 2 ways to combine things!
Arithmetic average or geometric average, to start.
Arithmetic average or geometric average, to start.
Unlike 95% of the other answers here, this may have been received And been processed by children who are just learning fractions.
I think the most basic answer to the “what do you do next” is unfortunately to explain they can’t do 1/3+1/3=1/6 but that why will be a future lesson.
The article gets a “fair” stopping point for the day at being mindful of the whole. And that you just can’t always add them.
I think the most basic answer to the “what do you do next” is unfortunately to explain they can’t do 1/3+1/3=1/6 but that why will be a future lesson.
The article gets a “fair” stopping point for the day at being mindful of the whole. And that you just can’t always add them.
> ... to explain they can’t do 1/3+1/3=1/6 ...
Except that 1/3 + 1/3 = 2/3, not 1/6.
You probably just mis-typed, but in this context it's an interesting mistake.
Except that 1/3 + 1/3 = 2/3, not 1/6.
You probably just mis-typed, but in this context it's an interesting mistake.
>> they can’t do 1/3+1/3=1/6
> Except that 1/3 + 1/3 = 2/3, not 1/6.
That's why they can't do it.
> Except that 1/3 + 1/3 = 2/3, not 1/6.
That's why they can't do it.
The confusion comes from the fact that we attach the same label "3" to two different sets in the case of bottles, so we end up saying "one out of three plus one out of three". It should be "one out of group 1 + one out of group 2", and then as you said, we'll need a special operator for combining groups.
And the "missing dollar riddle" can be exploited as the "change raising scam" https://www.youtube.com/watch?v=1Uaw88H0AmE
You do this by including units. 1/3 + 1/3 = 2/3. And here, 1/3x + 1/3y = 2/6(x+y).
A great opportunity to introduce multiplications of fractions!
1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too.
When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents.
The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important!
To do this, we need to multiply the fractions together like so:
(Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls).
So in this case we would have:
(1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3).
This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing:
(1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).
1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too.
When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents.
The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important!
To do this, we need to multiply the fractions together like so:
(Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls).
So in this case we would have:
(1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3).
This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing:
(1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).
I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still feels like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation.
Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it).
Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.
Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it).
Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.
I would suggest to replace 1/2 with 3/6 since the number of students at each table is what matters, not the number of tables, which only yields the correct result if each table has the same number of students.
I think this is a perfect example of why people find math hard. Not because there's something inherently difficult, but because we have teachers like this one in charge of open and curious minds at a young age. You get a teacher who doesn't quite understand what it is she's trying to teach, and this confusion is multiplied over and over again for each batch of students that pass through. Not too long afterwards you get kids who think they don't understand math, when in fact it was a failing of the teachers who couldn't correctly explain it in the first place.
Something I missed the first time around—in the comments below the article, the teacher expands on what she actually did. This was originally one long paragraph, but I've added paragraph breaks for readability:
---
> At the time, I let it go so I could think and regroup myself. There was more than fractions to think about: Is it possible for both equations to be true: 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6. This was a long discussion that kept their interest, in which I learned that many (most?) of the students didn’t think of fractions as numbers. That was another hurdle. I kept going back to what we know about adding whole numbers.
> Then I took another approach, once I was sure that they understood that 1/3 and 2/6 are equivalent, so how can I add a number to itself and wind up with a sum that’s the same as one of them? In all of these discussions, students would change their thinking. We looked at adding on a number line. I used pattern blocks to explore the same problem. I kept talking about keeping our attention on what was 1 whole.
> My, this all takes time, and the time is important for students to develop, cement, and extend their understanding. What I didn’t do, that I’ve been thinking about now, is to make it part of a writing workshop on persuasive writing and have them choose a conjecture and write. I think I could spend most of the year on this with students.
---
> At the time, I let it go so I could think and regroup myself. There was more than fractions to think about: Is it possible for both equations to be true: 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6. This was a long discussion that kept their interest, in which I learned that many (most?) of the students didn’t think of fractions as numbers. That was another hurdle. I kept going back to what we know about adding whole numbers.
> Then I took another approach, once I was sure that they understood that 1/3 and 2/6 are equivalent, so how can I add a number to itself and wind up with a sum that’s the same as one of them? In all of these discussions, students would change their thinking. We looked at adding on a number line. I used pattern blocks to explore the same problem. I kept talking about keeping our attention on what was 1 whole.
> My, this all takes time, and the time is important for students to develop, cement, and extend their understanding. What I didn’t do, that I’ve been thinking about now, is to make it part of a writing workshop on persuasive writing and have them choose a conjecture and write. I think I could spend most of the year on this with students.
I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express.
The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."
It's hard when the language required to express the ideas they're having is just a little to advanced. And nothing about this stops when you get older. The "just a little outside your knowledge" keeps stretching on forever.
The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."
It's hard when the language required to express the ideas they're having is just a little to advanced. And nothing about this stops when you get older. The "just a little outside your knowledge" keeps stretching on forever.
>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."
I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.
I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.
> Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.
That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 != 2/6. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.
The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
Now, the example from the classroom is more subtle, because it deals with an improper translation of the English into mathematical notation. If I say 'one third of the students at table 1 are girls', that should be translated to 1/3 * 3, not 1/3. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously correct.
That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 != 2/6. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.
The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
Now, the example from the classroom is more subtle, because it deals with an improper translation of the English into mathematical notation. If I say 'one third of the students at table 1 are girls', that should be translated to 1/3 * 3, not 1/3. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously correct.
My only point is that analogies are flawed. The abstraction that they provide hides complexity that at some point will leak out.
>There is an intuitive reason why 1/3 + 1/3 != 2/6.
You and I have different ideas of what 'intuitive' means. It's 'intuitive' once you understand the rules of fractions and what they mean. It's not so easy to derive this rule if you're working in the space of real world things.
And sure, I agree you can ad hoc extend the analogy of tables to bring it in line with the underlying mathematical rules, but then your analogy is no longer as simple as it was. The complexity is leaking out of the abstraction you had it under.
>The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents
Sure, using abstractions and analogies is a powerful way of teaching. All I did was point out that analogies have limits and at some point they can become detrimental to understanding the fundamental concepts.
This is a common complaint by Physicists when doing public lectures on Quantum Mechanics and then having people extrapolate from the metaphors to derive incorrect physical rules (e.g. faster-than-light communication from a shallow understanding of quantum entanglement).
>because it deals with an improper translation of the English into mathematical notation.
It isn't just about the improper translation to English. It is also about the improper mapping of fractions to real-world things. 2 girls out of 6 kids in a table maps nicely to the fraction 2/6. But even though 2/6 is equivalent to 80/240, the latter is a little harder to map to a table of 6 kids and 2 girls - don't you think?
>The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
I disagree with that in context of learning how fraction operators work. The fourth-grader logically extended the analogy that they were given because conceivably, there could have been an operator defined that matched their intuition, for example, let's call it '@' and define it (not rigorously) as "a/b @ c/d = (a+c)/(b+d)". This operator, if existed, would work very well for combining tables of boys and girls and getting the fraction of girls to match the fourth grader's intuition. The fourth-grader is learning fractions for the first time, and that operation could have conceivably existed - so the only reason they were wrong is that they haven't been told what the rules of fraction addition are and NOT that they misunderstood the analogy. The problem is that the "+" operator does not work that way because it isn't defined this way as per axioms for fractions.
>There is an intuitive reason why 1/3 + 1/3 != 2/6.
You and I have different ideas of what 'intuitive' means. It's 'intuitive' once you understand the rules of fractions and what they mean. It's not so easy to derive this rule if you're working in the space of real world things.
And sure, I agree you can ad hoc extend the analogy of tables to bring it in line with the underlying mathematical rules, but then your analogy is no longer as simple as it was. The complexity is leaking out of the abstraction you had it under.
>The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents
Sure, using abstractions and analogies is a powerful way of teaching. All I did was point out that analogies have limits and at some point they can become detrimental to understanding the fundamental concepts.
This is a common complaint by Physicists when doing public lectures on Quantum Mechanics and then having people extrapolate from the metaphors to derive incorrect physical rules (e.g. faster-than-light communication from a shallow understanding of quantum entanglement).
>because it deals with an improper translation of the English into mathematical notation.
It isn't just about the improper translation to English. It is also about the improper mapping of fractions to real-world things. 2 girls out of 6 kids in a table maps nicely to the fraction 2/6. But even though 2/6 is equivalent to 80/240, the latter is a little harder to map to a table of 6 kids and 2 girls - don't you think?
>The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
I disagree with that in context of learning how fraction operators work. The fourth-grader logically extended the analogy that they were given because conceivably, there could have been an operator defined that matched their intuition, for example, let's call it '@' and define it (not rigorously) as "a/b @ c/d = (a+c)/(b+d)". This operator, if existed, would work very well for combining tables of boys and girls and getting the fraction of girls to match the fourth grader's intuition. The fourth-grader is learning fractions for the first time, and that operation could have conceivably existed - so the only reason they were wrong is that they haven't been told what the rules of fraction addition are and NOT that they misunderstood the analogy. The problem is that the "+" operator does not work that way because it isn't defined this way as per axioms for fractions.
I still don't agree. Sure, there are limits to particular intuitions, but all of the rules make perfect sense with real world quantities.
For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange, not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number of 1kg of flour, so it does break down at some point, but unless you go overboard with quantities, all of the rules for fractions are in fact intuitive, and important for day to day things like cooking and money management. In fact, fractions and their operations are probably older than the idea of abstract rules, because they are fundamentally useful things.
The child in this example wasn't even making the mistake of thinking the rule for + is the rule for your @ operation. They were confused because they were trying to apply the intuition they had built up for how to translate real-world problems into fractions in the wrong way. Their result was in fact physically true: it was true that 1/3 of the children at one table + 1/3 of the children at the other table was equal to 2/6 of the children at both tables. This was confusing them because it suggested a different way of manipulating the numbers than they had just been shown.
The right solution, again, was to teach them how to translate 'a fraction of something' to rational numbers - that is, to multiply the fraction by the something, with only a special notational case when that something is 1. If they had known to do this, their intuition would have translated directly into the correct algebraic formula. No need to learn the abstract rules yet.
For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange, not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number of 1kg of flour, so it does break down at some point, but unless you go overboard with quantities, all of the rules for fractions are in fact intuitive, and important for day to day things like cooking and money management. In fact, fractions and their operations are probably older than the idea of abstract rules, because they are fundamentally useful things.
The child in this example wasn't even making the mistake of thinking the rule for + is the rule for your @ operation. They were confused because they were trying to apply the intuition they had built up for how to translate real-world problems into fractions in the wrong way. Their result was in fact physically true: it was true that 1/3 of the children at one table + 1/3 of the children at the other table was equal to 2/6 of the children at both tables. This was confusing them because it suggested a different way of manipulating the numbers than they had just been shown.
The right solution, again, was to teach them how to translate 'a fraction of something' to rational numbers - that is, to multiply the fraction by the something, with only a special notational case when that something is 1. If they had known to do this, their intuition would have translated directly into the correct algebraic formula. No need to learn the abstract rules yet.
You can't just ignore the fact that people didn't conceive of fractions as the smallest field containing an integral domain with a specific equivalence relation. They are an intuitive idea that has had its grammar formalized. It is absolutely beautiful to see fractions from the ten thousand foot abstractions but you can't skip to the end.
Also, looking at maths from this pure beautiful abstract lens is something that some people (hiii!) enjoy but it's also the common language for modeling problems in the real world. And sure, we do sometimes intentionally confuse the fact that modeling systems with maths is supposed to be descriptive but we have no choice because otherwise we're betraying the fact that the attempt to describe these systems is where the notation came from in the first place.
All this is to say that teaching math is hard and that the abstract "rules" view doesn't do it justice. It's practical for sure but it's also the reason you have lots of students who are good at pushing symbols around but not much else.
Also, looking at maths from this pure beautiful abstract lens is something that some people (hiii!) enjoy but it's also the common language for modeling problems in the real world. And sure, we do sometimes intentionally confuse the fact that modeling systems with maths is supposed to be descriptive but we have no choice because otherwise we're betraying the fact that the attempt to describe these systems is where the notation came from in the first place.
All this is to say that teaching math is hard and that the abstract "rules" view doesn't do it justice. It's practical for sure but it's also the reason you have lots of students who are good at pushing symbols around but not much else.
How would you explain that successfully to fourth graders? (Or, for that matter, seventh graders?)
I would avoid multiplication entirely and say that if you are combining tables, you need to update the bottom with the new number of total students. So something like:
(T1 girls) + (T2 girls) 1 + 1
_______________________ = _____
(T1 total) + (T2 total) 3 + 3
This is definitely a good explanatory strategy (and one that I'd use). I might label the units of the right side as well. Would you attempt to explain the abstract rules of fraction arithmetic as an system?
In the comments, an educator references an excellent article related to this confusion, "When Can you Meaningfully Add Rates, Ratios, and Fractions" that implicitly suggests some pedagogical approaches: https://flm-journal.org/Articles/11019C10CF34E90DC5866E53E90...
When you want to add fractions of a whole together.
1 1 2
--- + --- = ---
3 3 3
1 + 1 2
--------- = ---
3 + 3 6From a 'not currently in front of the class' perspective, it's pretty clear what the student meant by 1/3 + 1/3 = 2/6. They were taking + to mean something like a general 'and' or combining action, not in the strict sense of standard fraction addition. It even has a name in mathematics, 'Farey addition'.
The kids clearly want to write it in shorthand, so maybe the thing to do is to come up with another symbol for this similar but distinct operation. For example, ⊕.
The kids clearly want to write it in shorthand, so maybe the thing to do is to come up with another symbol for this similar but distinct operation. For example, ⊕.
I happen to have a math teacher masters though I myself do not teach (but I do help with the program of a tiny, tiny reform school). This teacher here had painted themselves into a corner and it's hard to get out of it. Do not explain fractions with things you can't change the denominator of.
Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other table has five kids and 5 coins then 1/5 means when splitting five coins a kid gets one. Figure out together what happens with splitting 10, 15, 20 coins. Now putting together the two tables we want to calculate 1/3+1/5, how many coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12 ... but can you tell what the fifth of nine coins are? You can't ... Then find 15 and then show them 1/3+1/5 means 8/15. This is all play and very smooth.
To answer the question posed in the blog post: I would plan the class carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two operations together and it does not even make sense. We can restore sanity but that has its own rules.
Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other table has five kids and 5 coins then 1/5 means when splitting five coins a kid gets one. Figure out together what happens with splitting 10, 15, 20 coins. Now putting together the two tables we want to calculate 1/3+1/5, how many coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12 ... but can you tell what the fifth of nine coins are? You can't ... Then find 15 and then show them 1/3+1/5 means 8/15. This is all play and very smooth.
To answer the question posed in the blog post: I would plan the class carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two operations together and it does not even make sense. We can restore sanity but that has its own rules.
One of the commenters gave a great suggestion:
> I would use bar models to show how the whole changes. First, draw one bar (table) with 3 students inside; label the bar one whole, star one student and label that student as one third of that whole. Do the same thing beside the first bar model, again showing the bar as one whole, starring one student and labelling that student as one third. Then, push the two bar models together (I’d use a Smart Board) to show a new whole: the two bars together with 6 students in the one bar is labeled as the new whole. Point out that this is a NEW whole, with a different # of pieces. Now they would see the 2 starred students in the one new bar made up of 6 total students, so the “old” 1/3 student becomes the “new” 1/6 student once the whole changes. This is like a name change, once the size of the whole changes. You can also show them that by cutting each “student” in half, you would get an equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6, not two 1/3’s = 2/6.
http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seeme...
> I would use bar models to show how the whole changes. First, draw one bar (table) with 3 students inside; label the bar one whole, star one student and label that student as one third of that whole. Do the same thing beside the first bar model, again showing the bar as one whole, starring one student and labelling that student as one third. Then, push the two bar models together (I’d use a Smart Board) to show a new whole: the two bars together with 6 students in the one bar is labeled as the new whole. Point out that this is a NEW whole, with a different # of pieces. Now they would see the 2 starred students in the one new bar made up of 6 total students, so the “old” 1/3 student becomes the “new” 1/6 student once the whole changes. This is like a name change, once the size of the whole changes. You can also show them that by cutting each “student” in half, you would get an equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6, not two 1/3’s = 2/6.
http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seeme...
Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality.
From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions contribute to the new whole.
From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions contribute to the new whole.
Perhaps an argument for arithmetic followed by geometry using Nicomachus and Euclid. We sit on that until 9th grade, but I wonder how young you could go with it?
Introducing line segments as alternative representations of numbers at this point feels very natural, and is already implied by most circulum with the standard 'number line'. As you say, we don't do anything with that until much later.
Let's leverage that early on!
Let's leverage that early on!
"If we reduce 2/6, we get 1/3, so 1/3 + 1/3 = 1/3, that seems weird, what did we do wrong?"
This is why I taught my kids to label units.
( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be added, because the denominators are different.
To fix that, you can multiply by the factors ( 3 students at table L / 6 students at both tables ) and ( 3 students at table R / 6 students at both tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students at both tables ) = ( 2 girls / 6 students at both tables ).
To hammer it home, you point to the table that has 2 or 4 students at it, and ask someone how to add the proportion of girls at that table to the two already under consideration.
( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be added, because the denominators are different.
To fix that, you can multiply by the factors ( 3 students at table L / 6 students at both tables ) and ( 3 students at table R / 6 students at both tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students at both tables ) = ( 2 girls / 6 students at both tables ).
To hammer it home, you point to the table that has 2 or 4 students at it, and ask someone how to add the proportion of girls at that table to the two already under consideration.
I think the problem is the teacher moved from one type of "fraction problem" to another kind.
The example of drinking 4 bottles of water to only have 2/6 left of the original pack is doing an integral problem and wrapping up the answer as a fraction. Same deal with the pack of 12 pencils.
The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6). In other words, there is only one "whole" in the problem.
When you're adding the fractions of desks filled with students, the fraction is counter-intuitive because both desks have to have the same number of students for it to make sense. (1/3 of a 5000-student round table is different from a 3-student table). And to say "the units are wrong" is kind of a limited way of explaining this. The units also need to have the table capacity as part of the unit identity (i.e. the units would have to be 3-student-table). That's a pretty sophisticated way of thinking about units.
I think that after the pencil/water examples, transitioning from the pencil/water pack examples to a more "pure" fraction example would be better. e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of a pizza. Now you can throw away one of the original pizza boxes and put the two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).
The example of drinking 4 bottles of water to only have 2/6 left of the original pack is doing an integral problem and wrapping up the answer as a fraction. Same deal with the pack of 12 pencils.
The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6). In other words, there is only one "whole" in the problem.
When you're adding the fractions of desks filled with students, the fraction is counter-intuitive because both desks have to have the same number of students for it to make sense. (1/3 of a 5000-student round table is different from a 3-student table). And to say "the units are wrong" is kind of a limited way of explaining this. The units also need to have the table capacity as part of the unit identity (i.e. the units would have to be 3-student-table). That's a pretty sophisticated way of thinking about units.
I think that after the pencil/water examples, transitioning from the pencil/water pack examples to a more "pure" fraction example would be better. e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of a pizza. Now you can throw away one of the original pizza boxes and put the two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).
I would say it was a mistake to introduce the concept of adding fractions from two different "wholes". Instead, teach that in order to add fractions, you first have to get everything into the same "whole".
Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But adding the boys from one table to another is quite a different thing.
Instead, you should teach that you should first make the fractions with both tables as the whole. Only then are you allowed to add. This could come as a later concept
Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But adding the boys from one table to another is quite a different thing.
Instead, you should teach that you should first make the fractions with both tables as the whole. Only then are you allowed to add. This could come as a later concept
Use pie graphs to show the original tables split up into thirds of one girl two boys.
Then show two ways of combining two tables.
(>-) combined with (>-) is a table of six: 2 girls, 4 boys show all boys and girls moving from tables with 3 seats to the same table with 6 seats.
Now show a table with three seats, one occupied by a girl. Show another table with same configuration. One girl moves from one table to join the first table. 1/3 full + 1/3 full (at the same table) is 2/3 full.
Then show two ways of combining two tables.
(>-) combined with (>-) is a table of six: 2 girls, 4 boys show all boys and girls moving from tables with 3 seats to the same table with 6 seats.
Now show a table with three seats, one occupied by a girl. Show another table with same configuration. One girl moves from one table to join the first table. 1/3 full + 1/3 full (at the same table) is 2/3 full.
[deleted]
That's what happens when you try to "build an intuition" for a concept instead of defining it.
Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!!
The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)."
Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)
Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!!
The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)."
Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)
As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult.
I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens.
About a third work their way through it fine.
About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.
I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens.
About a third work their way through it fine.
About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.
> About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2
...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too.
A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so.
I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.
...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too.
A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so.
I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.
This is fantastic material. I'm going to present this quandary to my kids. Some of my kids struggle at math, so it will help them understand that math is a language and, like any language, it has abbreviations that sometimes lead you down a confusing path until you spell out more. Some of my kids are good at math and it will help them relate with others.
I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction?
While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define a new inclusive set.
My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms.
Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.
While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define a new inclusive set.
My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms.
Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.
Bad idea!
They tried exactly that in France in the 60s and 70s. It was called Maths modernes (New maths). I think it was the same in most of the western world. It was a disaster and they abandoned it in the 80s. The problem was that while it may have helped the best students understand advanced concepts later on, it produced a generation of people who lacked practical counting skills.
EDIT: By "bad", I don't mean "stupid". The people who came up with new maths were experts in their field, with the goal of having a more scientifically literate population. No doubt very smart people, but maybe a little too smart for their own good. Turned out these concepts were too much for most young kids, and too remote from the way math is used on a day to day basis.
They tried exactly that in France in the 60s and 70s. It was called Maths modernes (New maths). I think it was the same in most of the western world. It was a disaster and they abandoned it in the 80s. The problem was that while it may have helped the best students understand advanced concepts later on, it produced a generation of people who lacked practical counting skills.
EDIT: By "bad", I don't mean "stupid". The people who came up with new maths were experts in their field, with the goal of having a more scientifically literate population. No doubt very smart people, but maybe a little too smart for their own good. Turned out these concepts were too much for most young kids, and too remote from the way math is used on a day to day basis.
I solved this by then introducing the kids to slices in a pie. Visually, the kids can see that 1/3 of a pie plus 1/3 of a pie is not the same as 2/6. Fractions are often introduced as "parts of a set" which is great for the first day. It's easy for kids to see and understand, but kids are more familiar with trying to slice the birthday cake for all the kids at the party. After the kids realize there are two kinds of fractions, we spend time at each problem thinking through is this a set (which we'll return to in ratios) or a whole (which can't be expanded) and visually representing fractions both ways to check their answers. Adding units is absolutely important, but when I teach it, that's what I add in fourth or fifth grade when I'm teaching them to reduce.
> What would you do now???
I would probably start crying, to be honest.
But in all seriousness, this is the kind of thing that makes me dread teaching. I mean, I know what is wrong, I just can't wrap my head around how to explain it, specially without going into stuff that would be too advanced for these particular students.
I would probably start crying, to be honest.
But in all seriousness, this is the kind of thing that makes me dread teaching. I mean, I know what is wrong, I just can't wrap my head around how to explain it, specially without going into stuff that would be too advanced for these particular students.
It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction:
3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7
Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d).
On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.
3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7
Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d).
On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.
2/4 + 1/3 would clearly be 5/12.
You halve each to put them in terms of the two wholes you're combining (which you must be doing, since you're adding them—else where do those go?) then combine those. Same process that gives you 1/3 + 1/3 = 1/3.
Worked out (conventional meaning of + for clarity; we'd need a new operator otherwise):
1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12
You halve each to put them in terms of the two wholes you're combining (which you must be doing, since you're adding them—else where do those go?) then combine those. Same process that gives you 1/3 + 1/3 = 1/3.
Worked out (conventional meaning of + for clarity; we'd need a new operator otherwise):
1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12
My take: Adding proportions is kinda "semantically incorrect". It's well defined as a value, it just doesn't mean what you think it means, akin to thinking that union of two sets is written A + B (which would be the set of pair-wise additions of elements). Now, dovetailing the equation with units that signify concrete quantities (as suggested by others) circumvents this by making it meaningful, at the cost of no longer working with the abstract concept of proportion.
An example of semantically meaningful manipulation of proportions is the weighted combination of proportions. This here's a more verbose version of what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the weighted average of the proportions give the final proportion)
An example of semantically meaningful manipulation of proportions is the weighted combination of proportions. This here's a more verbose version of what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the weighted average of the proportions give the final proportion)
1/3 + 1/3 does equal 2/6.
1/3 of one table + 1/3 of one table = 2/6 of two tables.
1/3 of one table + 1/3 of one table = 2/6 of two tables.
I would express this in a diagram with a dotted line around the set under consideration. When dealing with thirds, make a diagram containing three objects with their respective properties, and end by drawing a dotted line around the set. When switching to sixths, add three more objects, erase the line around the first three, and draw a new dotted line around the whole set of six. Then go on to sevenths by adding one more object, erasing the dotted line for the six and drawing it around all seven.
If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.
If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.
[deleted]
It’s 1/3rd of something. what that something is matters. Fractions express proportions of a whole, so you can’t add and subtract them without taking into account what whole they refer to.
Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning that is probably common among children who are just learning fractions that will still arrive at the same answer.
In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4. The erroneous intuition of simply adding the groups and their subsets together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8
In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4. The erroneous intuition of simply adding the groups and their subsets together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8
And now I feel stupid for obvious reasons
As the concept of fairness is innate in primates including humans in the 4th grade I wouldn't stop and correct the students but try and re-frame the problem in terms of a problem where fairness was involved and then immediately the ratios and the units would come into more clear focus. As an example I might say that if I promised to give you 1/3 of the pizza as a reward for cleaning your room. And then how much should you get if I had 1 pizza. How about 2 pizzas? Things like that.
Her first example, "proportion of a six-pack of water" is actually very complex, with a lot of potentially distracting details. Nowhere does she mention that's what her fractions are measuring, either! The whole lecture was simply a mass of detail with no reference to the underlying concept it was putatively intended to explain. No wonder the children were confused. Why not start with cutting a pie? Guess that doesn't lead to nearly as much busy-work.
Not that I would have thought of this on the spot, but ...
The numerator and denominator represent different things. The numerator is "how many things", and the bottom is "of how many spaces". Asking what fraction of the table is girls is dangerous, because the answer is (1 girl) / (1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are filled by girls" might have helped show the idea that the number of spaces is fixed.
The numerator and denominator represent different things. The numerator is "how many things", and the bottom is "of how many spaces". Asking what fraction of the table is girls is dangerous, because the answer is (1 girl) / (1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are filled by girls" might have helped show the idea that the number of spaces is fixed.
I would have said that a whole table is 3/3 and that both tables together are 6/3 and not 6/6 (maybe they'll get it that 6/3 equals to 2). Probably a good idea to explain that the "whole" isn't all pens and bottles but the group of 3 or 6 or 12 or whatever.
Maybe start with explaining from the start what happens if there are 7 bottles, when bottles are counted in the groups of six-packs. How many six-packs can you make from 15 bottles?
Maybe start with explaining from the start what happens if there are 7 bottles, when bottles are counted in the groups of six-packs. How many six-packs can you make from 15 bottles?
Don't use tables of the same size:
1/3 + 1/4 = 2/X
There would be no confusion as to what X should be.
And as someone else said, use units:
1/3 of table A + 1/4 of table B = 2/7 of both tables.
1/3 + 1/4 = 2/X
There would be no confusion as to what X should be.
And as someone else said, use units:
1/3 of table A + 1/4 of table B = 2/7 of both tables.
http://iconicmath.com
We probably haven't come up with the best visual abstractions yet, eh?
We probably haven't come up with the best visual abstractions yet, eh?
That’s the same as saying: if you put 1/3rd pizza slice on top of an other 1/3rd pizza slice, you get 2/6 pizza. Layered math.
Wow, I had no idea this blog existed - I loved Marilyn Burns's children's math books as a kid, awesome to see her still in action.
When you put two tables together, each table has 1/2 of the total number of kids. By having 2 tables, now you have 6 kids, so:
the original 1/3 in the first table becomes 1/6
the original 1/3 in the second table becomes 1/6
so adding the two tables means 2/6 are girls
She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an example where she joins two units (two tables)
the original 1/3 in the first table becomes 1/6
the original 1/3 in the second table becomes 1/6
so adding the two tables means 2/6 are girls
She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an example where she joins two units (two tables)
Have the kids stand up and count out numbers.
Now they know how many kids are in the room.
Then have them go and stand in two different corners. Count one corner.
Then have them stand in three different corners. Count one corner.
Finally the four corners. Count one corner.
Then have the two tables distribute their students to the corners and figure out these elusive fractions.
Have each corner be a Pokemon or something. That should do the trick.
Now they know how many kids are in the room.
Then have them go and stand in two different corners. Count one corner.
Then have them stand in three different corners. Count one corner.
Finally the four corners. Count one corner.
Then have the two tables distribute their students to the corners and figure out these elusive fractions.
Have each corner be a Pokemon or something. That should do the trick.
This reminds me of a homework problem I did in numerical methods where I made a small mistake in the program but still got an answer that was super close the the actual answer out of pure luck. The professor spent a while looking at it knowing it had to be wrong before he ran out the decimals and showed that it didn't converge.
I would try to explain semantics of addition (without the word “semantics” of course). I would say, in the first group 1/3 is the girl so if we take out one boy and replace it with another 1/3 (another girl), the resulting proportion is 2/3. Once we mix both groups, that is not what “plus” is.
I would explain that the "+" sign is not in the right place in the equation and write
(1+1)/(3+3) = 2/6
instead.
(1+1)/(3+3) = 2/6
instead.
Many students at beginner university level math courses do the same kind of mistakes, albeit with larger numbers: 10/13 + 7/21 = 17/34. Decimals are much easier to deal with imo, but mathematicians does not like them because they are inexact.
The moment you add three more students, the first girl is no longer one girl out of a group of three students, she is one girl (out of the two girls) out of a group of six students.
Same with the second girl.
So, once you add 3 more students, 1/3 becomes 1/6.
1/6 + /6 = 1/3.
Same with the second girl.
So, once you add 3 more students, 1/3 becomes 1/6.
1/6 + /6 = 1/3.
"+" is the wrong mathematical operation for combining fractions of distinct sets.
The right operation is a weighted sum:
1/3×1/2 + 1/3×1/2 = 1/3
The "1/2" is there because each of the two sets that we are combining is the same size.
The right operation is a weighted sum:
1/3×1/2 + 1/3×1/2 = 1/3
The "1/2" is there because each of the two sets that we are combining is the same size.
[deleted]
Really great anecdote! That first plus sign that Addison wrote is actually more of an "And" instead of an arithmetic "plus"
My favourite:
I got 1/3 on the first test and 1/3 on the second test. So in total I got 2/6. So 1/3 + 1/3 = 2/6.
This is a great example of the downside of over-relying on metaphors and analogies to teach mathematical concepts. The reality is that fractions are not like groups of pencils or tables of girls and boys. They are rigorously defined mathematical constructs that occasionally can be mapped to real-world things (and usually with severe constraints). 1/3 + 1/3 isn't 2/6 because it doesn't follow from the underlying axioms that define rational numbers - and not because of anything else.
Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere.
The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working.
Much better (imho) than teaching math as a purely formal and transcendent topic that happens to apply to real-world problems, and start with axiomatic definitions (which is the way maths are often taught).
Note : i'm sure you're not advocating for that as well, and i don't mean to contradict you. Just that i think it's better to start with a partially broken metaphor, then fix it using formal definition, than not try at all.
The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working.
Much better (imho) than teaching math as a purely formal and transcendent topic that happens to apply to real-world problems, and start with axiomatic definitions (which is the way maths are often taught).
Note : i'm sure you're not advocating for that as well, and i don't mean to contradict you. Just that i think it's better to start with a partially broken metaphor, then fix it using formal definition, than not try at all.
>We didn't invent fraction out of nowhere.
But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy.
>The moment where you have to stop relying on intuition is a pretty delicate matter
I didn't argue that metaphors and analogies shouldn't be used. I argued that analogies and metaphors are intrinsically flawed and this article provides a great example. At some point, you have to give up on the analogy and fallback on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.
But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy.
>The moment where you have to stop relying on intuition is a pretty delicate matter
I didn't argue that metaphors and analogies shouldn't be used. I argued that analogies and metaphors are intrinsically flawed and this article provides a great example. At some point, you have to give up on the analogy and fallback on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.
> You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.
It is pedagogically superior to choose the route implied by the comments about this being a type error.
That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.)
For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following:
1. "1/3+1/3=2/6" doesn't have any units, but it must somehow map to operations with units.
2. If unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table.
3. 2/3 does not equal 2/6, so unitless math can't map to both operations.
4. macspoofing said that 2/6 is wrong.
5. Therefore, unitless fraction addition implies addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy.
If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless <-> unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.)
I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors.
Edit: clarification
It is pedagogically superior to choose the route implied by the comments about this being a type error.
That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.)
For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following:
1. "1/3+1/3=2/6" doesn't have any units, but it must somehow map to operations with units.
2. If unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table.
3. 2/3 does not equal 2/6, so unitless math can't map to both operations.
4. macspoofing said that 2/6 is wrong.
5. Therefore, unitless fraction addition implies addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy.
If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless <-> unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.)
I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors.
Edit: clarification
>That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.),
Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out.
Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.
Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out.
Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.
"But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world"
This seems totally absurd to me. The concepts of proportion, ratio, etc preexisted any kind of formal definition of fraction.
They may have invented a definition of fraction, with the correct notation, and the correct set of rules after a long series of trial and errors (as with a lot of mathematical "rigourous definition"), but they always had in mind that this concept they tried to define should "work" when manipulating ratios / proportions, etc.
I may be wrong, but it seems to me that purely mathematical concepts spawning out of pure mathematical world exploration is a very modern (aka 19th century max) concept.
This seems totally absurd to me. The concepts of proportion, ratio, etc preexisted any kind of formal definition of fraction.
They may have invented a definition of fraction, with the correct notation, and the correct set of rules after a long series of trial and errors (as with a lot of mathematical "rigourous definition"), but they always had in mind that this concept they tried to define should "work" when manipulating ratios / proportions, etc.
I may be wrong, but it seems to me that purely mathematical concepts spawning out of pure mathematical world exploration is a very modern (aka 19th century max) concept.
Fractions are like groups of pencils or tables of girls and boys. Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.
>Fractions are like groups of pencils or tables of girls and boys.
No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1?
>Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.
I think you have it reversed. The formal definition is the pure definition, which is ancillary to the application to the real-world but covers much much more.
No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1?
>Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.
I think you have it reversed. The formal definition is the pure definition, which is ancillary to the application to the real-world but covers much much more.
You're taking a strange ahistorical perspective. Most Natural numbers, Integers, Rationals/Fractions, and even some Irrational numbers are much older than the concept of formal mathematics.
Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests.
All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered.
The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it.
There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability.
> I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls?
It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".
Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests.
All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered.
The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it.
There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability.
> I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls?
It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".
>The modern formal definition is a post-hoc formalization of an existing intuitive concept
Sure, but we're aiming to teach those 'post-hoc' formalizations. What fourth-graders are learning is supposed to be a foundation for future abstract mathematics. And we are using the operators and syntax of the modern formalizations as well.
Again we're not just teaching kids how to think about ratios and do practical arithmetic. If that's all we wanted, we could just give them an algorithm and an abacus and that's all they would ever need. Fractions can be used to think about ratios and things like that but they come with WAY more baggage because they are more than that. They are abstract concepts that underpin further abstract mathematics.
>It's a bit contrived, but say I ask the pupils at the table to send someone away. They
You can always add an ad-hoc extension to whatever the analogy you're using. I'm not saying that's wrong. I'm making no judgement call on how to teach mathematics to fourth-graders either (I don't know what the best way is). But I do want to point out that your extension to the analogy (negative fractions being equivalent to sending a student away) is because you've reached the limits of your initial abstraction. It's not a clean mapping either. For one thing, you just assumed the numerator is negative (student being away), but it is just as valid that the denominator may be negative (or both!). And it isn't clear that a negative numerator and a positive denominator is equivalent to a positive numerator and negative denominator. You either have to prove that from first principles using formal methods or you simply have to assert that to the fourth-grader and justify it with "it's just the way things are"
Sure, but we're aiming to teach those 'post-hoc' formalizations. What fourth-graders are learning is supposed to be a foundation for future abstract mathematics. And we are using the operators and syntax of the modern formalizations as well.
Again we're not just teaching kids how to think about ratios and do practical arithmetic. If that's all we wanted, we could just give them an algorithm and an abacus and that's all they would ever need. Fractions can be used to think about ratios and things like that but they come with WAY more baggage because they are more than that. They are abstract concepts that underpin further abstract mathematics.
>It's a bit contrived, but say I ask the pupils at the table to send someone away. They
You can always add an ad-hoc extension to whatever the analogy you're using. I'm not saying that's wrong. I'm making no judgement call on how to teach mathematics to fourth-graders either (I don't know what the best way is). But I do want to point out that your extension to the analogy (negative fractions being equivalent to sending a student away) is because you've reached the limits of your initial abstraction. It's not a clean mapping either. For one thing, you just assumed the numerator is negative (student being away), but it is just as valid that the denominator may be negative (or both!). And it isn't clear that a negative numerator and a positive denominator is equivalent to a positive numerator and negative denominator. You either have to prove that from first principles using formal methods or you simply have to assert that to the fourth-grader and justify it with "it's just the way things are"
From my own experience as a pupil, teaching purely formal systems with no help in building the intuition of why the formal rules are what they are is a recipe for disaster. It invites thinking of mathematics as a game, and when you forget the rules, you tend to invent new ones.
My belief is that is exactly how you end up with students doing fraction addition as a/b + c/d = (a+c)/(b+d); or the infamous shepherd's age problem[0] - they forgot the actual rules, and picked a different rule that makes just as much sense to them.
And regarding analogies, I think that the best approach is to pick a real-world problem, and translate that into math, as a starting step for explaining the formal rules and building this intuition. Doing thing the other way around is much more likely to lead to contrived examples. But math rules have good intuitive reasons for existing, and explaining these as you introduce the rules is likely to help rather than hinder.
Of course, I wouldn't advocate for having students go back to the analogy while solving more advanced exercises with the rules that they have internalized. But having lots of exercises initially that try to drive home the intuition behind the rules is going to be very helpful in my opinion.
[0] "A shepherd has 25 goats and 53 sheep. How old is the shepherd?" A lot of kids will give you an answer: if they do, they will probably say that it's 25+53 or 53-25, since they may apply some common-sense reasoning after they "do the math", but at the wrong end of the problem.
My belief is that is exactly how you end up with students doing fraction addition as a/b + c/d = (a+c)/(b+d); or the infamous shepherd's age problem[0] - they forgot the actual rules, and picked a different rule that makes just as much sense to them.
And regarding analogies, I think that the best approach is to pick a real-world problem, and translate that into math, as a starting step for explaining the formal rules and building this intuition. Doing thing the other way around is much more likely to lead to contrived examples. But math rules have good intuitive reasons for existing, and explaining these as you introduce the rules is likely to help rather than hinder.
Of course, I wouldn't advocate for having students go back to the analogy while solving more advanced exercises with the rules that they have internalized. But having lots of exercises initially that try to drive home the intuition behind the rules is going to be very helpful in my opinion.
[0] "A shepherd has 25 goats and 53 sheep. How old is the shepherd?" A lot of kids will give you an answer: if they do, they will probably say that it's 25+53 or 53-25, since they may apply some common-sense reasoning after they "do the math", but at the wrong end of the problem.
I think that one key issue is that the students are confusing addition with the more general concept of considering two things together.
1+1=50.8
My forth grader understood that this equation makes the same mistake. Hint: units of inches and mm must be made explicit.
related: notation as a tool of thought
https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=tr...
1/3 relates to one of the tables
2/6 relates to both tables
https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=tr...
1/3 relates to one of the tables
2/6 relates to both tables
If you make it extreme, then its easier to see the folly / error -
500/1500 students + 1/3 students = ?
500/1500 students + 1/3 students = ?
You're taking a weighted average, not adding the fractions.
It takes a while to learn what that is used for.
It takes a while to learn what that is used for.
No cat has nine tails. Every cat has one more tail than no cat. Ergo, every cat has 10 tails.
Half a loaf is greater than nothing. Nothing is greater than God. Therefore, half a loaf is greater than God.
> It’s hard to think and teach at the same time!
So true...
You do not see anything when you are at the blackboard.
So true...
You do not see anything when you are at the blackboard.
The and operation could be logical so the result of anding with one self is a mop
This kind of thing fascinates me. I've always felt that you don't really understand a thing, even a simple thing like fractions, until you can find and resolve all the paradoxes that arise in using it naively.
My take when reading the 1/3 + 1/3 = 2/6 equation is that Addison was right but the "+" here is a different operation than the "+" of 1/3 + 1/3 = 2/3. In one case we are adding disjoint parts of the same whole and counting how much of the parts of the whole are now included, and get 2/3, and in the other case we are adding two wholes and counting how much of the new whole is included.
People who pass primary education without internalizing this distinction will have to do so later if they learn statistics or probability theory.
The answer "what would you do now?" is the same one I would try to ask the students. First I would stop, confused, and write 1/3 + 1/3 = ? elsewhere on the board, and then wait for the students to see why I'm confused. They already understand that we want mathematics to be consistent, and getting different answers means we were wrong at least once, but they will have a hard time to find out or explain why 2/6 and 2/3 both are intuitively and obviously right. You have to compare the intuitions directly to each other to see that in one case you're adding parts of the same whole (set or group or table of kids) and in another case adding the wholes.
From what I know of elementary school, we focus a lot on fractions and the formal manipulation of them as unitless values, and drill into kids that 2/4, 1/2, and 0.5 are all "the same thing". Bullshit! It's no wonder they have problems with word problems! Two-out-of-four apples and one-out-of-two apples and a half apple are very different things, as every elementary student knows well. How can we expect them to believe something that's obviously false unless we've introduced the narrow context in which it's true? When used as unitless "coefficients" or bare numbers that exist only to scale another value that represents a real quantity of something, then 1/2 and 2/4 are the same. This is what we should be teaching when we teach fractions, not some mindless rote symbol manipulation like cross-multiplying or what have you without any ability to intuitively derive those rules yourself.
I'd spend the rest of the class time letting them work out all the different ways of looking at it, and the connections to everything else fractions and ratios and addition are good for.
Of course we want to be able to have "+" be well-defined, especially in elementary school, and the students understand that! We've now found a perfectly intuitive meaning of addition for which our standard "+" definition does not fit! This gives an opportunity to explore the connections between mathematics and intuition and the symbols and the real world operations they stand for. It gives us an excellent chance to, as Polya puts it, "introduce a suitable notation" for the setwise addition of "marked" sets that we have just discovered. Then we can explore the properties of this operation and the ones between it and the familiar addition operation. We can show that the plus sign and the addition operation we use it to represent are distinct, and our use of one for the other is a choice, and one we could make differently.
I saw some mentions of units in the comments, and yet 1/3 + 1/3 = ? is still ill-defined if we know only that "1/3" means "one out of three students". What we need to know is which students they are! So if one out of Alice, Bob, and Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair, how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made the question challenging even for adults if we ask it that way! And now you can introduce uncertainty and degrees of belief. (What if three people tell you they have one red-headed friend, in a class of 23 students? Do they all have the same friend, or is there more than one redhead?)
Of course what most of us will actually do as the teacher in this situation is freeze and think "oh no, I've made a mistake and the teacher (in your mind) is going to point it out and the class is going to laugh at me!". Because that's what we've learned happens in these situations. If we interpret confusion and being wrong as painful and embarrassing, little wonder that mathematics seems a long and painful road that we avoid whenever we can.
My take when reading the 1/3 + 1/3 = 2/6 equation is that Addison was right but the "+" here is a different operation than the "+" of 1/3 + 1/3 = 2/3. In one case we are adding disjoint parts of the same whole and counting how much of the parts of the whole are now included, and get 2/3, and in the other case we are adding two wholes and counting how much of the new whole is included.
People who pass primary education without internalizing this distinction will have to do so later if they learn statistics or probability theory.
The answer "what would you do now?" is the same one I would try to ask the students. First I would stop, confused, and write 1/3 + 1/3 = ? elsewhere on the board, and then wait for the students to see why I'm confused. They already understand that we want mathematics to be consistent, and getting different answers means we were wrong at least once, but they will have a hard time to find out or explain why 2/6 and 2/3 both are intuitively and obviously right. You have to compare the intuitions directly to each other to see that in one case you're adding parts of the same whole (set or group or table of kids) and in another case adding the wholes.
From what I know of elementary school, we focus a lot on fractions and the formal manipulation of them as unitless values, and drill into kids that 2/4, 1/2, and 0.5 are all "the same thing". Bullshit! It's no wonder they have problems with word problems! Two-out-of-four apples and one-out-of-two apples and a half apple are very different things, as every elementary student knows well. How can we expect them to believe something that's obviously false unless we've introduced the narrow context in which it's true? When used as unitless "coefficients" or bare numbers that exist only to scale another value that represents a real quantity of something, then 1/2 and 2/4 are the same. This is what we should be teaching when we teach fractions, not some mindless rote symbol manipulation like cross-multiplying or what have you without any ability to intuitively derive those rules yourself.
I'd spend the rest of the class time letting them work out all the different ways of looking at it, and the connections to everything else fractions and ratios and addition are good for.
Of course we want to be able to have "+" be well-defined, especially in elementary school, and the students understand that! We've now found a perfectly intuitive meaning of addition for which our standard "+" definition does not fit! This gives an opportunity to explore the connections between mathematics and intuition and the symbols and the real world operations they stand for. It gives us an excellent chance to, as Polya puts it, "introduce a suitable notation" for the setwise addition of "marked" sets that we have just discovered. Then we can explore the properties of this operation and the ones between it and the familiar addition operation. We can show that the plus sign and the addition operation we use it to represent are distinct, and our use of one for the other is a choice, and one we could make differently.
I saw some mentions of units in the comments, and yet 1/3 + 1/3 = ? is still ill-defined if we know only that "1/3" means "one out of three students". What we need to know is which students they are! So if one out of Alice, Bob, and Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair, how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made the question challenging even for adults if we ask it that way! And now you can introduce uncertainty and degrees of belief. (What if three people tell you they have one red-headed friend, in a class of 23 students? Do they all have the same friend, or is there more than one redhead?)
Of course what most of us will actually do as the teacher in this situation is freeze and think "oh no, I've made a mistake and the teacher (in your mind) is going to point it out and the class is going to laugh at me!". Because that's what we've learned happens in these situations. If we interpret confusion and being wrong as painful and embarrassing, little wonder that mathematics seems a long and painful road that we avoid whenever we can.
Apologies but I don't see the mystery here?
A fraction describes a proportion ie. the amount of something(s) in relation to another amount of something(s) thus, if we double all of the things in a fraction the proportion and hence the fraction is equivalent.
A fraction describes a proportion ie. the amount of something(s) in relation to another amount of something(s) thus, if we double all of the things in a fraction the proportion and hence the fraction is equivalent.
1/3 and 1/3 is 1/6 ;)
This is pretty simple. Put two tables together, 0/6 girls. Now put only one girl: 1/6 put the other girl: 2/6
One third of one half, plus one third of the other half, is one third of the whole?
Yeah the writer didn't have the words "of one half" or "of the other half", therefore it seemed a lot more confusing than it should have.
This is exactly why third-level teaching is so much easier.
I don't understand why this simple 2nd grade math is featured on hacker news. I guess the next topic will be a long discussion what would be 1/2 divided by 2/3.
One of the things I've realized in life is the best teachers are people who were not very good at something and then became very good at it through sheer practice and grit.
People who are naturally gifted at something because they can't understand the reason for someone else's failure can't teach well.
I worked for a year of work study as a math tutor at a community college. "How to do integration by parts[1]?" one student asked? That's easy to explain. My hardest day was when I was trying to help this one woman understand that when you multiply two negative numbers you get a positive numbers. "Wouldn't that make it more negative?" she asked.
I decided I would have been a terrible math teacher even though I was very good at math. And reading this article was interesting to me because it explained to me why I would be a terrible math teacher.
[1] https://en.wikipedia.org/wiki/Integration_by_parts
People who are naturally gifted at something because they can't understand the reason for someone else's failure can't teach well.
I worked for a year of work study as a math tutor at a community college. "How to do integration by parts[1]?" one student asked? That's easy to explain. My hardest day was when I was trying to help this one woman understand that when you multiply two negative numbers you get a positive numbers. "Wouldn't that make it more negative?" she asked.
I decided I would have been a terrible math teacher even though I was very good at math. And reading this article was interesting to me because it explained to me why I would be a terrible math teacher.
[1] https://en.wikipedia.org/wiki/Integration_by_parts
A lot of people are at home right now trying to teach their kids who are also at home.
Bingo. To me, this is the key point of the article:
> It’s hard to think and teach at the same time!
This teacher seems pretty bright. I'm sure if she had a few quiet minutes to think and regroup, she could come up with a stellar explanation. But instead, she has to proceed with the lesson, manage the classroom, and tackle an interesting puzzle at the same time.
At home, merely trying to manage my work and two kids with their remote learning often puts me in the same spot. I will often do some tutoring with them during a work break. But my brain power is already silently depleted, so if any kind of obstacle or creative question arises, I struggle to even begin thinking of an approach to it.
> It’s hard to think and teach at the same time!
This teacher seems pretty bright. I'm sure if she had a few quiet minutes to think and regroup, she could come up with a stellar explanation. But instead, she has to proceed with the lesson, manage the classroom, and tackle an interesting puzzle at the same time.
At home, merely trying to manage my work and two kids with their remote learning often puts me in the same spot. I will often do some tutoring with them during a work break. But my brain power is already silently depleted, so if any kind of obstacle or creative question arises, I struggle to even begin thinking of an approach to it.
I'm surprised I haven't been seeing more discussions about education recently on HN and everywhere else.
I suspect we skew young enough on here that most of us don't (yet?) have kids.
I very strongly suspect that’s not true.
It would be interesting to see a demographic survey of HN commenters/contributors. Wonder if one has ever been done?
After I read the article, I thought the article had less to do with the math involved than the way kids see things compared to adults. So I thought it was pretty interesting.
Javascript:
console.log((0.1 + 0.2) === 0.3) // => false
C#:
(0.1m + 0.2m) == 0.3m // => true
see here: https://dotnetfiddle.net/zVnrNQ
Javascript is such a poor language, there is no even a good way to work with currencies or exact numbers and you have to multiply numbers with a factor to avoid such problems. The worst language I ever used.
console.log((0.1 + 0.2) === 0.3) // => false
C#:
(0.1m + 0.2m) == 0.3m // => true
see here: https://dotnetfiddle.net/zVnrNQ
Javascript is such a poor language, there is no even a good way to work with currencies or exact numbers and you have to multiply numbers with a factor to avoid such problems. The worst language I ever used.
Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions.
The student came up and wrote 1/3 + 1/3 = 2/6.
What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).
The teacher then demonstrates an entirely different formula: 1/3 (of the students at a table + 1/3 (of the students at a table) = 2/3 (of the students at a table).
The confusion comes because no one calls out that they're talking about fractions of different things.
Edit: There are a whole range of exploratory questions you can follow on from here as well.
Imagine if the tables have different numbers of students or if there are more than two tables. Helping students navigate these types of ratio transformations is why keeping track of units is so important. Otherwise, things can get hairy for the students very quickly.