"I can't do math" (2001)(faculty.umf.maine.edu)
faculty.umf.maine.edu
"I can't do math" (2001)
http://faculty.umf.maine.edu/michael.molinsky/public.www/iaq/mmm001.html
My favorite professor's take on the ever-increasing "I can't do math" statement.
132 comments
I see his larger point, but the conflation of "understanding math" and "being able to do simple arithmetic" is something of a pet peeve. The latter may be a useful life skill, but is not very closely related to understanding mathematics. In fact it's not only non-technical office workers who can be proud of not being able to do it: if you walk into any math department you will find accomplished mathematicians almost gleefully recounting how bad they are at doing simple arithmetic. They can prove things about topological spaces but can't balance a checkbook!
That makes the comparison to literacy more complex. Clearly mathematicians think mathematics is important, or they wouldn't be studying it. But many of them consider ability to do arithmetic, as a skill, essentially irrelevant, to the point where they don't even learn it themselves. Then a question might be: what is the relevant literacy-like skill? I think it comes closer to logic and analytical thinking than specifically mathematics, although understanding of statistical arguments fits into that category.
That makes the comparison to literacy more complex. Clearly mathematicians think mathematics is important, or they wouldn't be studying it. But many of them consider ability to do arithmetic, as a skill, essentially irrelevant, to the point where they don't even learn it themselves. Then a question might be: what is the relevant literacy-like skill? I think it comes closer to logic and analytical thinking than specifically mathematics, although understanding of statistical arguments fits into that category.
I think being able to interpret common mathematical writing in your area of expertise is a valuable skill. I think this is what is meant by "understanding math".
I don't know much about anything but signal processing, but I am able to read and understand vector/matrix notation and differential equations. This enables me to read most papers published in signal processing. Just like being able to at least roughly read most C-derived programming languages, this is a valuable skill.
I have seen coworkers "not understanding" my work because it involved problem descriptions in mathematic notation. Frankly, it annoys the hell out of me.
I don't know much about anything but signal processing, but I am able to read and understand vector/matrix notation and differential equations. This enables me to read most papers published in signal processing. Just like being able to at least roughly read most C-derived programming languages, this is a valuable skill.
I have seen coworkers "not understanding" my work because it involved problem descriptions in mathematic notation. Frankly, it annoys the hell out of me.
I understand mathematical notation, but I do think it generally is best only used when necessary in day-to-day work. Whenever something is of sufficient complexity, trying to keep track of the 10+ variables in a mathematical notation is generally harder than just reading a description.
For a recent example, my thought process yesterday: "So let's see, it says to multiply all of the 'F' functions, from 'm' to 'n'.. 'F' is, what, the base array? OK, so that makes 'm' and 'n'.. hmm. Let's come back to that. OK, so we multiply the product of those 'F's times the product of all 'alpha's of 'm,n' over 'm'.. so alpha is, let's see, the.. message? And... uh.."
Long story short, I finally deciphered it just meant "Multiply the base array of values by the value of each message that comes in that shares a parameter," only it took about 5 minutes instead of 10 seconds.
For a recent example, my thought process yesterday: "So let's see, it says to multiply all of the 'F' functions, from 'm' to 'n'.. 'F' is, what, the base array? OK, so that makes 'm' and 'n'.. hmm. Let's come back to that. OK, so we multiply the product of those 'F's times the product of all 'alpha's of 'm,n' over 'm'.. so alpha is, let's see, the.. message? And... uh.."
Long story short, I finally deciphered it just meant "Multiply the base array of values by the value of each message that comes in that shares a parameter," only it took about 5 minutes instead of 10 seconds.
Think vector notation. Multiply arrays with matrices to get vectors. Add some sum notations and integrals. All that stuff is tedious to express in code but trivial to express in mathematics.
Also, many more scientific concepts such as gradient descent are most easily expressed using the appropriate mathematic operators like the nabla operator for gradients.
Also, many more scientific concepts such as gradient descent are most easily expressed using the appropriate mathematic operators like the nabla operator for gradients.
You're right, I don't want to give the impression that it's never warranted. Just that it's important to give thought to if a simple description or even a graph might be more efficient in each case.
I try to stay mindful of the limits of short term brain storage.. if there's over 7 unique variables or interval indicators, it might be tough for someone unfamiliar with what's going on to pick it up without wasting extra time noodling on it. When you've got a sum over 'm', a sum over 'm,n', a product sum over 'n,x' and another over 'x', etc.. sometimes it's just easier to bite the bullet and give a visual aid so the reader's not trying to envision all these overlapping regions in his head while still trying to remember what each is supposed to represent in real life.
I try to stay mindful of the limits of short term brain storage.. if there's over 7 unique variables or interval indicators, it might be tough for someone unfamiliar with what's going on to pick it up without wasting extra time noodling on it. When you've got a sum over 'm', a sum over 'm,n', a product sum over 'n,x' and another over 'x', etc.. sometimes it's just easier to bite the bullet and give a visual aid so the reader's not trying to envision all these overlapping regions in his head while still trying to remember what each is supposed to represent in real life.
How about I write my comments in Turkish? It's your fault you don't understand them, after all.
The art of programming, and of comments, is the ability to communicate to another human, and incidentally to computers. (I believe that was a paraphrase of Knuth.) If the terminology gets in the way of their understanding, then use a different terminology.
(I can fight my way through a mathematics paper, myself. But why be obtuse in your communications?)
The art of programming, and of comments, is the ability to communicate to another human, and incidentally to computers. (I believe that was a paraphrase of Knuth.) If the terminology gets in the way of their understanding, then use a different terminology.
(I can fight my way through a mathematics paper, myself. But why be obtuse in your communications?)
Mathematics offers a standardized, internationally accepted notation for its various problem domains. Turkish is not the standard language of software comments. So I don't think the analogy holds.
Neither is math. My guess is that you will find many more Chinese programmers who understand English comments than those that understand high-school mathematics symbols.
I would venture that any such programmers ought not work on a math-intensive domain like signal processing. Sure, not every programmer has to know high school math. You can, for example, do great work with HTML, CSS, and JavaScript without knowing high school math. But there are certain fields where math is unavoidable. Signal processing is one of them. And the internationally accepted notation for such math is part of what you're expected to know.
BWT, I assume you're not saying China uses different mathematical notation. I wouldn't know about that. If so, then that does undermine my argument.
BWT, I assume you're not saying China uses different mathematical notation. I wouldn't know about that. If so, then that does undermine my argument.
"If the terminology gets in the way of their understanding, then use a different terminology."
Literally the reason mathematics exists.
Literally the reason mathematics exists.
That's because mathematicians need notational flexibility, because more formal (unambiguous) syntax would hurt their ability to digest huge chunks of data at once. At least that's what I read somewhere in defence of mathematical, cryptic and ambiguous notation.
It sounds suspiciously similar to what Lispers say about macros and s-exps, btw ;)
It sounds suspiciously similar to what Lispers say about macros and s-exps, btw ;)
I think that's an uninformed metaphor because it assumes both math is totally foreign and should be able to be read by anyone. if you write code for a server that handles HTTP requests you wouldn't explain the guts of HTTP in your comments, you'd just assume if the reader wants to know more they'll figure it out themselves.
I think sometimes mathematical definitions, especially advanced ones that rely on simpler definitions, are too complicated/long to recant every time you use them, which you have to. If I want to prove something relating or relying on uniform continuity I don't want to state the definition every time both because its repetitive and because definitions often use the same notation, so I don't want my uniform continuity deltas and epsilons getting confused with my normal continuity epsilons and deltas. Try proving advanced things about field extensions and galois theory using induction like that and you may actually run out of symbols.
That being said it is hard to read real math writing/research which can be both dense and obtuse even if you're used to it. Maybe that's why you need a PhD to consider entering the field.
I think sometimes mathematical definitions, especially advanced ones that rely on simpler definitions, are too complicated/long to recant every time you use them, which you have to. If I want to prove something relating or relying on uniform continuity I don't want to state the definition every time both because its repetitive and because definitions often use the same notation, so I don't want my uniform continuity deltas and epsilons getting confused with my normal continuity epsilons and deltas. Try proving advanced things about field extensions and galois theory using induction like that and you may actually run out of symbols.
That being said it is hard to read real math writing/research which can be both dense and obtuse even if you're used to it. Maybe that's why you need a PhD to consider entering the field.
It merely assumes that his co-workers, who did not fluently speak mathematical notation, should somehow learn a foreign language to keep up. I find the idea of being obtuse toward a co-worker who does not speak the same language someone I would have reservations about working with, myself.
It's just a theory, but I think some mathematicians intentionally 'unlearn' or 'forget' arithmetic when they start to understand what math is about because people who don't understand math will ask them if they add and multiply all day. Theoretical mathematics is an entirely different beast than what most people think it is, so mathematicians get a lot of dumb questions about it. When someone asks if they add and subtract all day, if they say 'I can't even do arithmetic', people may be more inclined to rethink that stereotype.
My wife is one of those 'math people' that struggles with simple arithmetic. She has calculators and excel for things like that. But for her it wasn't a case of unlearning it. She struggled in school with arithmetic and didn't start to excel until they started introducing variables and all of the other aspects that separate math from arithmetic.
"ask them if they add and multiply all day"
A good example, but you should have thrown in an analogy to the general population thinking anyone who does anything with electronics is automatically a microsoft windows admin/personal trainer with special focus on virus removal.
A good example, but you should have thrown in an analogy to the general population thinking anyone who does anything with electronics is automatically a microsoft windows admin/personal trainer with special focus on virus removal.
> the conflation of "understanding math" and "being able to do simple arithmetic" is something of a pet peeve
Agreed. I'm horrible at simple arithmetic -- I've just never gotten the knack for it, despite clear life benefits to being good at it. But I'm decent at math-with-letters.
If I'm in a social setting and someone wants me to do the arithmetic on something, I'll happily say "I can't do math" because it makes more sense than saying the above. I bet there are others like me out there.
Agreed. I'm horrible at simple arithmetic -- I've just never gotten the knack for it, despite clear life benefits to being good at it. But I'm decent at math-with-letters.
If I'm in a social setting and someone wants me to do the arithmetic on something, I'll happily say "I can't do math" because it makes more sense than saying the above. I bet there are others like me out there.
> what is the [maths] relevant literacy-like skill?
I'm deeply convinced that the answer to this is easy - the ones that programmers use.
I'm deeply convinced that the answer to this is easy - the ones that programmers use.
> Then a question might be: what is the relevant literacy-like skill?
How about "I'm terrible with names."
How about "I'm terrible with names."
People who say "I can't do math" are lazy? Sorry, but I think that's a gross oversimplification of the issue (which is also evidenced by the comparison to physical exercise) and the two assumptions fundamental to that opinion are just plain wrong. Namely:
> So what do people actually mean when they say that they can’t “do” math? Usually they are really stating one of two things. First, that they don’t like mathematics. Secondly, that mathematics is more difficult for them than other subjects, and that it takes a great deal more effort on their part to learn it.
He conveniently chose the two reasons that are easiest to dismiss (irony, anyone?). What he fails to account for is the fact that math education is so poor that many people don't truly understand what math is. Beyond arithmetic and algebra, they think it's some really complicated stuff with big numbers and funny symbols that geeky people with glasses do -- it's practically a foreign language to them, except it has a reputation for being much harder.
Why is this? If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach. Those who attempt to do so often do it very poorly, which leads students to the false belief that math is extremely difficult. On the other hand, it's very easy to teach someone to memorize formulas and plug in numbers, so that's what we're most often taught in math class. That's good enough to get us through the standardized test so we can graduate from high school, but memorizing formulas and plugging in numbers is not "doing math". So I believe many people are completely justified in saying "I can't do math".
What's more, people who "can do math" should be taking the blame for those who say "I can't do math" rather than using pointless semantics to wag a finger at them.
> So what do people actually mean when they say that they can’t “do” math? Usually they are really stating one of two things. First, that they don’t like mathematics. Secondly, that mathematics is more difficult for them than other subjects, and that it takes a great deal more effort on their part to learn it.
He conveniently chose the two reasons that are easiest to dismiss (irony, anyone?). What he fails to account for is the fact that math education is so poor that many people don't truly understand what math is. Beyond arithmetic and algebra, they think it's some really complicated stuff with big numbers and funny symbols that geeky people with glasses do -- it's practically a foreign language to them, except it has a reputation for being much harder.
Why is this? If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach. Those who attempt to do so often do it very poorly, which leads students to the false belief that math is extremely difficult. On the other hand, it's very easy to teach someone to memorize formulas and plug in numbers, so that's what we're most often taught in math class. That's good enough to get us through the standardized test so we can graduate from high school, but memorizing formulas and plugging in numbers is not "doing math". So I believe many people are completely justified in saying "I can't do math".
What's more, people who "can do math" should be taking the blame for those who say "I can't do math" rather than using pointless semantics to wag a finger at them.
> If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach.
I think it's even worse than that -- I suspect that many teachers in American public schools are terrified of math themselves, and they transmit that terror to their students. It's going to be very hard for a student to learn to view mathematics as reason if their teachers don't see it that way.
There was a point when I was in grade school and we were learning formulas for the area of different shapes. When a trapezoid came up and I noted that it could be decomposed into a square and two triangles, I was admonished to just use the formula from the handout. Don't get in the habit of trying to think while doing math, it'll just get you in trouble.
This was from an otherwise excellent teacher, but when it came to math we were to turn the thinking switch to "off". This fear of math seemed to not been exceptional, even among my high school math instructors. I may have had a bad run (public school in California in the 80s), but I've been told similar stories by most everyone I've met who eventually managed to figure math out on their own.
I think it's even worse than that -- I suspect that many teachers in American public schools are terrified of math themselves, and they transmit that terror to their students. It's going to be very hard for a student to learn to view mathematics as reason if their teachers don't see it that way.
There was a point when I was in grade school and we were learning formulas for the area of different shapes. When a trapezoid came up and I noted that it could be decomposed into a square and two triangles, I was admonished to just use the formula from the handout. Don't get in the habit of trying to think while doing math, it'll just get you in trouble.
This was from an otherwise excellent teacher, but when it came to math we were to turn the thinking switch to "off". This fear of math seemed to not been exceptional, even among my high school math instructors. I may have had a bad run (public school in California in the 80s), but I've been told similar stories by most everyone I've met who eventually managed to figure math out on their own.
Stories like this make me glad I was homeschooled. Mom handed me a Saxon math textbook and I just read it and did the problems. No fear, no classroom consensus that "math is haaaard".
I can support this from the other side. Coming out of a maths degree at a reasonably good university, a lot of the mathematicians who became teachers were those who struggled and pretty much gave up at some point in undergrad. Pushing many of those who will become teachers to mathematical breaking point seems like a bad way of doing things, and I think contributes to this. It's hard to think of a good alternative that still produces enough teachers though.
Only thing I can think of is to pay maths teachers really well, which probably means increasing class-sizes, so is a no-go at least in the current political clime of the US.
I recall a math teacher I had in grade 9 - she was normally a phys. ed. teacher, who had been voluntold to teach math when the usual math teacher had a nervous breakdown and took a year off. She freely admitted that she had studied phys. ed. because she "loved to teach", and wanted her students to have a better time than she did, but she had struggled in English and Math, so she went for one of the easier specialties. Not exactly a shining role model there. All the other students loved her, of course, but I thought it would be better if she were a babysitter, not a teacher.
What he fails to account for is the fact that math
education is so poor that many people don't truly
understand what math is. Beyond arithmetic and algebra,
they think it's some really complicated stuff with big
numbers and funny symbols that geeky people with glasses
do -- it's practically a foreign language to them,
except it has a reputation for being much harder.
In fact, if I'm totally honest, I'm not 100% I completely understand the sine function now. And it wasn't just math. In physics, current, voltage, resistance etc. were taught as inputs to formulas. I know it must be challenging to teach about these kinds of principles that lack concrete macroscopic analogs, but I can't help but feel they could have done a better job than they did. In chemistry too, I remember being taught about valency and how you could work out the valency of an element by its position on the periodic table. I asked what valency actually was, either didn't understand or wasn't satisfied with the answer, asked again, and the teacher brushed off my question and carried on the with the lesson. "Oh well," I thought, "I guess I don't understand chemistry." That was when I was about 12 years old, and I didn't study chemistry after that. I studied biology until I was 16 because I had a teacher who took the time to actually explain things.
The worst part is, I went to a pretty good school. It must be absolutely dreadful at bad schools.
Most of this happened before I had regular access to the internet and the chance to learn about these things for myself. I can't help but feel the whole course of my schooling and advanced education might have been different had I had better (or at least different) teachers of hard science and math at an early age.
It's the phrase, "I'm just no good at math" that breaks my heart. Most of the time, the student is taking the blame that belongs to the education industry.
Last week, my son had a "Chapter 9" math test here in a top-ranked Silicon Valley public school. His teacher pointed us to an official study guide PDF, which we went over carefully. I was not at all surprised to find that it covered a random grab bag of unrelated topics: sorting a half-dozen fractions, each with different denominators, two different silly algorithms for multidigit multiplication, how many $2.30 widgets can you buy for $9.00, and a few others.
This incoherent, random presentation of unrelated topics within a single chapter is totally characteristic of the "reform math" so beloved by our "progressive educators." They despise the approach of methodically working through a small number of carefully sequenced topics, making sure that the foundation of layer N is solid before getting to work building the closely related layer N+1 on top of it. They call it, "drill and kill," "soul-crushing," and "creativity destroying."
Instead of mastering a few closely-related concepts each year and systematically building expertise, they prefer "exposing" kids briefly to lots of unrelated math ideas, trusting that some kids will get some of it, and telling the rest to "trust the spiral," meaning trust that when they hop, skip, and jump over multiple topics the following year and the year after that, most of them will eventually "get" most of the stuff.
The result is that many parents just teach their kids real math outside of school. Many in our neighborhood send them to Chinese school, which teaches them math in addition to Chinese. The Chinese school buses line up in front of all of our local elementary schools at the end of each school day. (A lot of blond kids board those buses.) Some send them to Kumon, which is getting to be as common a sight around here as McDonalds or Starbucks.
I teach mine myself, using non-US curricula (Chinese, Japanese, and Singaporean in my case.) I feel terrible for the kids who don't have parents doing the schools' job for them, whose math skills are limited to what they can pick up from their classmates in "group discovery" sessions, since the "professional educators" have now decided that kids learn best what they discover for themselves and now serve merely as "guides on the side" in edu-speak.
My son took his Chapter 9 test and reported to me that, with the exception of testing the two different, useless multiplication algorithms, the test was a DIFFERENT grab bag of unrelated math topics, bearing little resemblance to the study guide. Totally typical of "reform math." He did fine, but only because he had learned all of it outside school. His friends who rely on what they learn at school think he's a genius.
So kids go through this ridiculous joke of a math education and can't do math. The school points at their friends who did just fine (because--shh!--they learned math elsewhere), the school takes credit for having taught them so well and tells the others and their parents, "well, not all kids are equally good at math, but many of your classmates learned quite well," clearly implying that the kids who didn't are somehow defective.
The result is that those kids will soon be saying, "I'm just no good at math." What a disgrace.
Last week, my son had a "Chapter 9" math test here in a top-ranked Silicon Valley public school. His teacher pointed us to an official study guide PDF, which we went over carefully. I was not at all surprised to find that it covered a random grab bag of unrelated topics: sorting a half-dozen fractions, each with different denominators, two different silly algorithms for multidigit multiplication, how many $2.30 widgets can you buy for $9.00, and a few others.
This incoherent, random presentation of unrelated topics within a single chapter is totally characteristic of the "reform math" so beloved by our "progressive educators." They despise the approach of methodically working through a small number of carefully sequenced topics, making sure that the foundation of layer N is solid before getting to work building the closely related layer N+1 on top of it. They call it, "drill and kill," "soul-crushing," and "creativity destroying."
Instead of mastering a few closely-related concepts each year and systematically building expertise, they prefer "exposing" kids briefly to lots of unrelated math ideas, trusting that some kids will get some of it, and telling the rest to "trust the spiral," meaning trust that when they hop, skip, and jump over multiple topics the following year and the year after that, most of them will eventually "get" most of the stuff.
The result is that many parents just teach their kids real math outside of school. Many in our neighborhood send them to Chinese school, which teaches them math in addition to Chinese. The Chinese school buses line up in front of all of our local elementary schools at the end of each school day. (A lot of blond kids board those buses.) Some send them to Kumon, which is getting to be as common a sight around here as McDonalds or Starbucks.
I teach mine myself, using non-US curricula (Chinese, Japanese, and Singaporean in my case.) I feel terrible for the kids who don't have parents doing the schools' job for them, whose math skills are limited to what they can pick up from their classmates in "group discovery" sessions, since the "professional educators" have now decided that kids learn best what they discover for themselves and now serve merely as "guides on the side" in edu-speak.
My son took his Chapter 9 test and reported to me that, with the exception of testing the two different, useless multiplication algorithms, the test was a DIFFERENT grab bag of unrelated math topics, bearing little resemblance to the study guide. Totally typical of "reform math." He did fine, but only because he had learned all of it outside school. His friends who rely on what they learn at school think he's a genius.
So kids go through this ridiculous joke of a math education and can't do math. The school points at their friends who did just fine (because--shh!--they learned math elsewhere), the school takes credit for having taught them so well and tells the others and their parents, "well, not all kids are equally good at math, but many of your classmates learned quite well," clearly implying that the kids who didn't are somehow defective.
The result is that those kids will soon be saying, "I'm just no good at math." What a disgrace.
Any chance you could link me to the non-US curricula materials you use to teach your kid?
I found a lot of the things you talked about in this school's approach to education: http://www.russianschool.com/about-us/our-approach
I found a lot of the things you talked about in this school's approach to education: http://www.russianschool.com/about-us/our-approach
I can't link you to the Chinese or Japanese materials, because I bought them in Shanghai and Tokyo. Also, they aren't written in English. The Googlers next door use Russian materials from Moscow, also not written in English.
It's hard to do better than Singaporean materials, which are in English and modified (not in a bad way) for the US market, which you can find at SingaporeMath.com. Their Primary Mathematics series is superb. I use the Standards Edition, which is said to track the California State Math Standards. That sounds ominous, but actually the state standards are excellent. The districts essentially ignore them by using a ridiculous "reform" curriculum that, being "a mile wide and an inch deep," will always include a checkmark every year for any topic you can think of, thereby covering anything mentioned in the state standards (superficially and in random order).
Note that for these Asian curricula, you REALLY need to know how to teach the math. The textbooks only provide visual aids and example problems, not the tutorial text (paragraphs of explanation) typical in US books. If you go for Singapore Math, you should get the Home Instructors Guide (at least for a few levels), which teaches you how to teach it.
And DON'T start a kid at too high a level. Use the placement tests downloadable from singaporemath.com to decide where to start. It's all about carefully building up from the bottom, mastering each level before moving on.
It's hard to do better than Singaporean materials, which are in English and modified (not in a bad way) for the US market, which you can find at SingaporeMath.com. Their Primary Mathematics series is superb. I use the Standards Edition, which is said to track the California State Math Standards. That sounds ominous, but actually the state standards are excellent. The districts essentially ignore them by using a ridiculous "reform" curriculum that, being "a mile wide and an inch deep," will always include a checkmark every year for any topic you can think of, thereby covering anything mentioned in the state standards (superficially and in random order).
Note that for these Asian curricula, you REALLY need to know how to teach the math. The textbooks only provide visual aids and example problems, not the tutorial text (paragraphs of explanation) typical in US books. If you go for Singapore Math, you should get the Home Instructors Guide (at least for a few levels), which teaches you how to teach it.
And DON'T start a kid at too high a level. Use the placement tests downloadable from singaporemath.com to decide where to start. It's all about carefully building up from the bottom, mastering each level before moving on.
Thank you very much!
We were taught sine, cosine and tangent in the context of how they could be used to derive angles from other angles, not what they were and how they worked. They were presented as tools that could be used in particular ways that had to be memorized.
This has to be the worst things you could do to a student in a math class. In engineering they call it "plug and chug" -- students must plug numbers into a formula they've memorized and come up with an answer.
By the way, we learned trigonometry with the unit circle. If we forgot a formula, we'd just draw a little circle and derive it. I'm always grateful for that teacher.
This has to be the worst things you could do to a student in a math class. In engineering they call it "plug and chug" -- students must plug numbers into a formula they've memorized and come up with an answer.
By the way, we learned trigonometry with the unit circle. If we forgot a formula, we'd just draw a little circle and derive it. I'm always grateful for that teacher.
same here, though i learnt it during my university years not at the high school.
> In fact, if I'm totally honest, I'm not 100% I completely understand the sine function now.
I've found this .gif does wonders for explaining sine and cosine to people:
http://www.butlercc.edu/mathematics/math_courses/ma140/SineC...
Sine is horizontal, cosine is vertical.
I've found this .gif does wonders for explaining sine and cosine to people:
http://www.butlercc.edu/mathematics/math_courses/ma140/SineC...
Sine is horizontal, cosine is vertical.
I share many of your thoughts here - we learn so much about using mathematical tools but we don't know why they were invented in the first place. I was wondering if it would be useful to understand the history and basis of concepts like the sine function - would that help me connect to them better?
Your post is insightful. Imagine trying to teach philosophy like American schools teach math. That would be unimaginably awful. Which makes me thankful that (most?) American high schools don't offer philosophy class. Superficially it would be good for the students if done right, but since it will not be done right...
Even worse imagine history taught that way. Whoops that's pretty much how they do teach history and (coincidentally?) that doesn't work too well either.
Even worse imagine history taught that way. Whoops that's pretty much how they do teach history and (coincidentally?) that doesn't work too well either.
Well... I don't comment often, but as someone with a B.A. in mathematics who minored in philosophy, let me follow up on your comment:
Would a philosophy class really be good for high school students? Broadly speaking, I mean, not those 1 out of 100 students who are reading Sartre or Nietzsche (or even Dostoyevsky or Kafka) on their own, already, anyway.
A survey course, I mean, a 101-type of course, like you would see at a University. My feeling is that a course in introductory logic is the #1 most useful course that's missing in high school right now.
And again, there are going to be a few students for whom this would be redundant, but something like 99/100 students do not understand the machinery of thinking. If there's one thing my university education taught me, it's the machinery of thinking, carefully and rigorously. Most people never learn this, and I feel like this then precludes any understanding of anything advanced and even slightly abstract--and that includes both philosophy and (of course!) mathematics.
Would a philosophy class really be good for high school students? Broadly speaking, I mean, not those 1 out of 100 students who are reading Sartre or Nietzsche (or even Dostoyevsky or Kafka) on their own, already, anyway.
A survey course, I mean, a 101-type of course, like you would see at a University. My feeling is that a course in introductory logic is the #1 most useful course that's missing in high school right now.
And again, there are going to be a few students for whom this would be redundant, but something like 99/100 students do not understand the machinery of thinking. If there's one thing my university education taught me, it's the machinery of thinking, carefully and rigorously. Most people never learn this, and I feel like this then precludes any understanding of anything advanced and even slightly abstract--and that includes both philosophy and (of course!) mathematics.
> that a course in introductory logic
This is barely touched in standard geometry text books. 2-column proofs and all that.
But most people don't notice it, and it deserves far more treatment. Many of the smartest people I know attended special gifted programs in K-12 that did teach formal logic and informal critical thinking skills.
(correlation != causation, though)
This is barely touched in standard geometry text books. 2-column proofs and all that.
But most people don't notice it, and it deserves far more treatment. Many of the smartest people I know attended special gifted programs in K-12 that did teach formal logic and informal critical thinking skills.
(correlation != causation, though)
Philosophy is mandatory as part of the IB curriculum. The course I'm thinking of is typically named "Theory of Knowledge", and while formally an epistemology course, it can be taught quite broadly.
I can't speak to anyone else's experience, but I found being exposed to Philosophy while still in high school to be quite meaningful, and went on to dual-major in it while in college.
I can't speak to anyone else's experience, but I found being exposed to Philosophy while still in high school to be quite meaningful, and went on to dual-major in it while in college.
As another IB student I can concur that I found the Theory of Knowledge class very rewarding. And the things I learnt in that class have probably given more than the stuff I learnt in math or physics class, despite going off to study physics at university and then switching to math.
That being said I also remember being one of the few students who felt the course interesting and worthwhile. Most of the other students saw it as a waste of time and focused mainly on how to get a good enough grade to not affect their overall average while doing the least amount of work possible.
That being said I also remember being one of the few students who felt the course interesting and worthwhile. Most of the other students saw it as a waste of time and focused mainly on how to get a good enough grade to not affect their overall average while doing the least amount of work possible.
> What's more, people who "can do math" should be taking the blame for those who say "I can't do math" rather than using pointless semantics to wag a finger at them.
You're invoking a false dichotomy here by assuming that one of the two groups (if they are even well defined at all) should be assigned blame and the other should be held blameless.
It's debatable whether blame should be assigned at all. Many people who do not suffer math phobia live lives where advanced mathematics is rarely, if ever, needed. These people "can do math" but simply find little practical need for it. If their lives are no worse in the absence of serious mathematics, I see no reason to intervene. That said, I do think we would be better off with a more mathematically literate society.
In recent years it seems like there has been a great deal of collective guilt and introspection by the technically literate. It probably has a lot to do with the rapidly increasing difference in one's quality of life that deep technical knowledge of various kinds can produce for individuals. It will never be productive to launch crusades with mottos like "everyone can program!" or "everyone can do math!" because these crusades presume that everyone who can do X should do X. A far more productive use of our time and energies is to expose children to these disciplines early in their lives and be honest with them about the potential rewards (practical, personal, and aesthetic) they can bring. There is no need to blame anyone or try to make anyone feel guilty.
You're invoking a false dichotomy here by assuming that one of the two groups (if they are even well defined at all) should be assigned blame and the other should be held blameless.
It's debatable whether blame should be assigned at all. Many people who do not suffer math phobia live lives where advanced mathematics is rarely, if ever, needed. These people "can do math" but simply find little practical need for it. If their lives are no worse in the absence of serious mathematics, I see no reason to intervene. That said, I do think we would be better off with a more mathematically literate society.
In recent years it seems like there has been a great deal of collective guilt and introspection by the technically literate. It probably has a lot to do with the rapidly increasing difference in one's quality of life that deep technical knowledge of various kinds can produce for individuals. It will never be productive to launch crusades with mottos like "everyone can program!" or "everyone can do math!" because these crusades presume that everyone who can do X should do X. A far more productive use of our time and energies is to expose children to these disciplines early in their lives and be honest with them about the potential rewards (practical, personal, and aesthetic) they can bring. There is no need to blame anyone or try to make anyone feel guilty.
>>Many people who do not suffer math phobia live lives where advanced mathematics is rarely, if ever, needed. These people "can do math" but simply find little practical need for it. If their lives are no worse in the absence of serious mathematics, I see no reason to intervene.
This is highly debatable. We had an economic meltdown just a few years ago, and one of the (many) reasons for it was that people were taking loans that they could not afford to pay off later, given their income, assets and expenses. Many of those people were victims of predatory lending because their math knowledge was so poor.
This is highly debatable. We had an economic meltdown just a few years ago, and one of the (many) reasons for it was that people were taking loans that they could not afford to pay off later, given their income, assets and expenses. Many of those people were victims of predatory lending because their math knowledge was so poor.
I would be careful to assume poor math knowledge where greed could just as easily explain the housing bubble.
Hence my phrasing. I said "many of those people were victims." Others may simply have been greedy.
Commenters in this thread might not think compound interest is advanced mathematics. But you're right, the majority don't have enough confidence in maths to attempt basic finanicial planning, and this is an issue for society.
I think you're using different definitions of math. He clearly shows that he includes basic arithmetic in his definition. And yet, you started off by saying that arithmetic and algebra don't count as "doing math". I'm not really surprised that you came to a different conclusion by defining words differently, but I'm unsure as to why that's relevant.
> He clearly shows that he includes basic arithmetic in his definition.
He mentioned arithmetic just to show that he's not being 100% literal about the "I can't do math" statement.
He mentioned arithmetic just to show that he's not being 100% literal about the "I can't do math" statement.
This reminds me of the comment PG made about people with the most negative rebuttal to the argument getting up-voted.
The argument wasn't entirely about how you perceive math and more about being socially accepted, even proud, of being able to say "I can't do math". It's a very North American perspective and it would be similar to saying "I can't read" in most of Europe.
The argument wasn't entirely about how you perceive math and more about being socially accepted, even proud, of being able to say "I can't do math". It's a very North American perspective and it would be similar to saying "I can't read" in most of Europe.
There are also learning disabilities. I've struggled with Math my entire life, from simple arithmetic in elementary school all they way up into high school mathematics. I believe I suffer from "math anxiety", although I've never been diagnosed. When doing math problems it's almost like dyslexia for me, the numbers get jumbled in my head... I know the basic processes I need to come to an answer, but everything in between gets twisted and mixed up along the way.
How is it my fault (as someone who "can do math") that someone else can't/won't do math? What do you believe I am morally obligated to do that I'm not doing with regard to someone else's preference/understanding of mathematics?
That's mainly directed at the person who wrote the essay (a math professor) and others like him who blame the students rather than themselves. I think he should be examining the state of math education in this country rather than chastizing the people it has failed.
It definitely comes down to the educational issue. I'm a mechanical engineer and have gotten pretty good with "math tricks" (e.g. 103x7 is difficult, but I can compute 100x7+3*7 much easier and know to rewrite the problem to be done in my head).
A trick like that should be pretty simple to figure out, but grade school math is taught in such a rigid fashion that students (who later become full grown adults) don't think to try it. Think about the last time you went to dinner with friends. How many calculators did it take to figure out the bill? Here in New Jersey, tax is 7% and 18% gratuity is pretty standard. Add up your meal and add 25% which you should be able to do in your head since you just need to divide by 4. Yet, last time I went out to dinner, the lawyer, accountant, and two physical therapists (i.e. 3 years of grad school) all pulled out their iPhones and then looked at me with confusion when I tried to explain the 25% solution. I'm not sure they need to be able to do the math in their heads (part of my job involves doing quick math in my head, so I have more practice), but the logic behind it shouldn't confuse them.
I often hear people talk about the need for high school classes that teach people how to balance a checkbook and other questionably useful skills. If you have to teach someone to balance a checkbook, you've already failed them. You've missed the part of education that should teach and develop the logic to make the checkbook lesson take 1 minute.
A trick like that should be pretty simple to figure out, but grade school math is taught in such a rigid fashion that students (who later become full grown adults) don't think to try it. Think about the last time you went to dinner with friends. How many calculators did it take to figure out the bill? Here in New Jersey, tax is 7% and 18% gratuity is pretty standard. Add up your meal and add 25% which you should be able to do in your head since you just need to divide by 4. Yet, last time I went out to dinner, the lawyer, accountant, and two physical therapists (i.e. 3 years of grad school) all pulled out their iPhones and then looked at me with confusion when I tried to explain the 25% solution. I'm not sure they need to be able to do the math in their heads (part of my job involves doing quick math in my head, so I have more practice), but the logic behind it shouldn't confuse them.
I often hear people talk about the need for high school classes that teach people how to balance a checkbook and other questionably useful skills. If you have to teach someone to balance a checkbook, you've already failed them. You've missed the part of education that should teach and develop the logic to make the checkbook lesson take 1 minute.
>Why is this? If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach. Those who attempt to do so often do it very poorly, which leads students to the false belief that math is extremely difficult. On the other hand, it's very easy to teach someone to memorize formulas and plug in numbers, so that's what we're most often taught in math class. That's good enough to get us through the standardized test so we can graduate from high school, but memorizing formulas and plugging in numbers is not "doing math"
This got me thinking a bit: What if it's the opposite?
I know I'm really bad at memorizing formula, and I need to really figure out everything for it to stick in my head. You say it's easier to get people to do that, but loads of people have bad experiences with math, so maybe it's because we teach it that way and not in spite of?
Trying to apply memorized formula to a problem is a form of pattern matching, and it might be harder because of all the doubts from the abstraction. This might end up being less efficient than we would originally think (oh I memorised all this, but I have no confisdence in using it...). In the end we have just displaced the difficulty to something a lot less tangible.
Maybe we should try lowering the scope of what is taught, but really try to make sure people can use what they learn, even if it's small.
This got me thinking a bit: What if it's the opposite?
I know I'm really bad at memorizing formula, and I need to really figure out everything for it to stick in my head. You say it's easier to get people to do that, but loads of people have bad experiences with math, so maybe it's because we teach it that way and not in spite of?
Trying to apply memorized formula to a problem is a form of pattern matching, and it might be harder because of all the doubts from the abstraction. This might end up being less efficient than we would originally think (oh I memorised all this, but I have no confisdence in using it...). In the end we have just displaced the difficulty to something a lot less tangible.
Maybe we should try lowering the scope of what is taught, but really try to make sure people can use what they learn, even if it's small.
> If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach.
I don't think critical thinking is very difficult to teach.
I think critical thinking is, despite being foundational, not prioritized in most educational curricula (and, particularly, not in most of the high-stakes testing regimes which we use to evaluate students, schools, teachers, etc.), and consequently insufficient effort is put into teaching it.
I don't think critical thinking is very difficult to teach.
I think critical thinking is, despite being foundational, not prioritized in most educational curricula (and, particularly, not in most of the high-stakes testing regimes which we use to evaluate students, schools, teachers, etc.), and consequently insufficient effort is put into teaching it.
And I'm going to go one step further:
Critical thinking is not prioritized because it is hard to evaluate.
Due to the demand for teacher accountability, the insane level of competition for college entry, and the political games surrounding education policy, modern public education is entirely centered around examination and evaluation.
Not only is critical thinking challenging to evaluate, but, more importantly, people—read, parents—do not accept evaluations that report bad critical thinking skills. If a child can't answer 2 + 2 or who President Washington was, then they clearly didn't know. But if you ask a question that truly challenges critical thinking skills, and the child receives a bad score, the parents will be marching into an administrator's office with complaints of "trick questions" and "unfair grading". And fear of parent backlash drives American public school administration's decision making.
Critical thinking is not prioritized because it is hard to evaluate.
Due to the demand for teacher accountability, the insane level of competition for college entry, and the political games surrounding education policy, modern public education is entirely centered around examination and evaluation.
Not only is critical thinking challenging to evaluate, but, more importantly, people—read, parents—do not accept evaluations that report bad critical thinking skills. If a child can't answer 2 + 2 or who President Washington was, then they clearly didn't know. But if you ask a question that truly challenges critical thinking skills, and the child receives a bad score, the parents will be marching into an administrator's office with complaints of "trick questions" and "unfair grading". And fear of parent backlash drives American public school administration's decision making.
> Critical thinking is not prioritized because it is hard to evaluate.
I don't think that's the root cause for why its never been considered a core skill and treated (when treated at all) as sort of an optional additional skill usually addressed, if at all, late in schooling as part of the English curriculum.
But I do think that's an additional challenge to getting it treated as a core focus in today's testing-obsessed public education context.
I don't think that's the root cause for why its never been considered a core skill and treated (when treated at all) as sort of an optional additional skill usually addressed, if at all, late in schooling as part of the English curriculum.
But I do think that's an additional challenge to getting it treated as a core focus in today's testing-obsessed public education context.
I think critical thinking is the opposite of what most schools want to teach. Take Khan Academy founder's book [1] which describes the mainstream "Prussian Model":
"The idea was not to produce independent thinkers, but to churn out loyal and tractable citzens who would learn the value of submitting to the authority of parents, teachers, church, and ultimately, king. The Prussian philosopher and political theorist Johann Gottlieb Fichte, a key figure in the development of the system, was perfectly explicit about its aims. 'If you want to influence a person,' he wrote, 'you must do more than merely talk to him; you must fashion him, and fashion him in such a way that he simply cannot will otherwise than what you wish him to will.'"
Chomsky further discusses the important role of "stupidity" in the educational system (like stupid assignments), in teaching obedience: (http://www.youtube.com/watch?v=pFf6_0T2ZoI)
[1] Salman Khan, "The One World School House"
"The idea was not to produce independent thinkers, but to churn out loyal and tractable citzens who would learn the value of submitting to the authority of parents, teachers, church, and ultimately, king. The Prussian philosopher and political theorist Johann Gottlieb Fichte, a key figure in the development of the system, was perfectly explicit about its aims. 'If you want to influence a person,' he wrote, 'you must do more than merely talk to him; you must fashion him, and fashion him in such a way that he simply cannot will otherwise than what you wish him to will.'"
Chomsky further discusses the important role of "stupidity" in the educational system (like stupid assignments), in teaching obedience: (http://www.youtube.com/watch?v=pFf6_0T2ZoI)
[1] Salman Khan, "The One World School House"
"I can't do informal semantics."
I had a trig/pre-calc teacher in high school who I really respected. He had a catchphrase in his classes, especially in the earlier math classes. "In my class, there are only two kinds of people: those who love math, and those who are going to learn to love math." I only had the one class with him, but he left a huge impression on my life. He was convinced that computers were going to become essential to education, so he spent his department's entire budget for a year on two spankin' new Apple ][+ computers. Just set them on carts in his classroom and didn't touch them. He knew he didn't need to. Eventually, a handful of us approached him to ask if he could teach us to use them. "No," he responded. "I don't know anything about them myself... but they came with a tutorial, so maybe you could teach yourself how to write programs, and then maybe help me out."
That changed the course of my life. I was planning to be a lawyer, catering to artists and performers. My time spent learning to write programs led to my taking a number of computer science classes in my undergrad, for the easy grades they represented, which then led to majoring in CS, and, eventually, opting for a career in software development.
All of which is just to say that, if someone feels that they cannot "do" math, maybe they missed out on the right math teachers; those who understand kids and know how to motivate them and who have their own love for the subject matter and are able to infect others with it. If their math teacher(s) couldn't be bothered to make personal connections with their students, nor find any way at all to make math relevant to their lives, it shouldn't come as a surprise if they wind up being uninterested in "doing" math.
That changed the course of my life. I was planning to be a lawyer, catering to artists and performers. My time spent learning to write programs led to my taking a number of computer science classes in my undergrad, for the easy grades they represented, which then led to majoring in CS, and, eventually, opting for a career in software development.
All of which is just to say that, if someone feels that they cannot "do" math, maybe they missed out on the right math teachers; those who understand kids and know how to motivate them and who have their own love for the subject matter and are able to infect others with it. If their math teacher(s) couldn't be bothered to make personal connections with their students, nor find any way at all to make math relevant to their lives, it shouldn't come as a surprise if they wind up being uninterested in "doing" math.
What a wonderful person. I sometimes take the time to contact teachers that have made an impact in my professional/personal life and thank them for setting me on the right course. It's a really simple gesture but they are always thrilled to hear they've made a difference.
The key problem with math is that the learning keeps going and at some point, every single person fails to either have the ability or time to learn more math. Everyone becomes "bad at math" relative to other people. This same thing doesn't happen with reading because once you are a pretty good reader you feel like you have mastered it.
3 examples: 1) Kid who never completes pre-calc. "can't do math" 2) Adult who gets through calc, but doesn't go any further "can't do math"... like an engineer 3) Math PhD. who gets stuck on something they can't solve.
Person 1 and 2 are both stating they can't do math, and both people are relatively right. Relatively being the key word.
If we rephrased the way math is taught, it would never be labeled as math at all. This would box people in to saying things like "I don't do calculus" which would be correct in many cases, whereas it would be alarming if someone said "I can't add".
Really, whether people say "I don't do math" or "my bad" doesn't really matter and being stressed over it is probably just as ridiculous as saying those things.
3 examples: 1) Kid who never completes pre-calc. "can't do math" 2) Adult who gets through calc, but doesn't go any further "can't do math"... like an engineer 3) Math PhD. who gets stuck on something they can't solve.
Person 1 and 2 are both stating they can't do math, and both people are relatively right. Relatively being the key word.
If we rephrased the way math is taught, it would never be labeled as math at all. This would box people in to saying things like "I don't do calculus" which would be correct in many cases, whereas it would be alarming if someone said "I can't add".
Really, whether people say "I don't do math" or "my bad" doesn't really matter and being stressed over it is probably just as ridiculous as saying those things.
This can't be overemphasized. My favorite (apocryphal) Einstein anecdote is the one where a society matron approaches him at a party and exclaims, "I don't know how you can deal with all that math. I have so much trouble with math." Einstein replies, "Lady, you can't imagine how much trouble I have with it."
s/math/computers, and that line comes in very handy in my own social life.
s/math/computers, and that line comes in very handy in my own social life.
I think you have hit this point on the head. It is very hard to compare mathematics education between countries. Eastern Bloc countries seem to go a lot further with the secondary level mathematics education, my point of caparison is to Scotland cavet emptor.
I am constantly embarrassed by my level of mathematics, I like to play with 3D graphics on the weekend.
I am constantly embarrassed by my level of mathematics, I like to play with 3D graphics on the weekend.
I have a Masters in applied math, but I still put myself in the Bad At Math camp compared to many of my class mates, especially those who went on to pure math PhD's. And even among that group, I know a couple who would also classify themselves as Bad At Math because they couldn't pull off the more hard core PhD research topics and had to drop down to easier problems.
Indeed there is something wrong with someone saying "I can't do math..." but it is not what the author identifies.
In blunt summary, the author says "you may not be able to do the specific math you claim, but don't worry I'll teach you." In my experience this is not at all what the complaint is saying and the solution "I'll teach you" is not at all what they are looking for. The comparison to illiteracy is completely off-target.
In my experience, "I can't do math" is simply "I don't know" in disguise. Ask most children: "what's 300 times 248" and you get a knee-jerk "I don't know". Ask also "what's a balloon made of"; "I don't know". Same goes for many other questions that appear to them to have a definite answer. We excuse children for saying this, but it becomes less and less acceptable as an answer because we learn of tools for finding the correct answers.
The real lesson that needs to be taught is:
Your worldview of can/can't do math is wrong. Doing math is learning what tools to use after we've broken down our question to its core...kind of like everything else.
In blunt summary, the author says "you may not be able to do the specific math you claim, but don't worry I'll teach you." In my experience this is not at all what the complaint is saying and the solution "I'll teach you" is not at all what they are looking for. The comparison to illiteracy is completely off-target.
In my experience, "I can't do math" is simply "I don't know" in disguise. Ask most children: "what's 300 times 248" and you get a knee-jerk "I don't know". Ask also "what's a balloon made of"; "I don't know". Same goes for many other questions that appear to them to have a definite answer. We excuse children for saying this, but it becomes less and less acceptable as an answer because we learn of tools for finding the correct answers.
The real lesson that needs to be taught is:
Your worldview of can/can't do math is wrong. Doing math is learning what tools to use after we've broken down our question to its core...kind of like everything else.
Saying “I don't know” is a fantastic skill that children are born with but have it beaten out of them into adulthood and the beating never stops. I have deep respect for people who can say “I don't know.” That person is one step away from learning something new.
It's great when adults who do not usually say "I don't know" can admit to not knowing something. Other than that, I cannot agree with anything else you say.
My favorite story about how Ernst Kummer [1] did arithmetic, from Hoffman's The Man Who Loved Only Numbers[2]:
"One story has him standing before a blackboard, trying to compute 7 times 9. "Ah," Kummer said to his high school class, "7 times 9 is eh, uh, is uh...." "61," one of his students volunteered. "Good," said Kummer, and wrote 61 on the board. "No," said another student, "it's 69." "Come, come, gentlemen," said Kummer, "it can't be both. It must be one or the other." (Erdos liked to tell another version of how Kummer computed 7 times 9: "Kummer said to himself, 'Hmmm, the product can't be 61 because 61 is a prime, it can't be 65 because that's a multiple of 5, 67 is a prime, 69 is too big-that leaves only 63.' ") "
[1] https://en.wikipedia.org/wiki/Ernst_Kummer
[2] http://www.amazon.com/The-Man-Loved-Only-Numbers/dp/B004R6HX...
"One story has him standing before a blackboard, trying to compute 7 times 9. "Ah," Kummer said to his high school class, "7 times 9 is eh, uh, is uh...." "61," one of his students volunteered. "Good," said Kummer, and wrote 61 on the board. "No," said another student, "it's 69." "Come, come, gentlemen," said Kummer, "it can't be both. It must be one or the other." (Erdos liked to tell another version of how Kummer computed 7 times 9: "Kummer said to himself, 'Hmmm, the product can't be 61 because 61 is a prime, it can't be 65 because that's a multiple of 5, 67 is a prime, 69 is too big-that leaves only 63.' ") "
[1] https://en.wikipedia.org/wiki/Ernst_Kummer
[2] http://www.amazon.com/The-Man-Loved-Only-Numbers/dp/B004R6HX...
> Erdos liked to tell another version of how Kummer computed 7 times 9: "Kummer said to himself, 'Hmmm, the product can't be 61 because 61 is a prime, it can't be 65 because that's a multiple of 5, 67 is a prime, 69 is too big-that leaves only 63.'
That's how I do problems like that, too (I am not a genius mathematician), and it is exactly the sort of thinking that kids should be doing all through K-12. Estimation, intuitive reasoning, analogy, pattern matching, logic, etc.
That's how I do problems like that, too (I am not a genius mathematician), and it is exactly the sort of thinking that kids should be doing all through K-12. Estimation, intuitive reasoning, analogy, pattern matching, logic, etc.
> Estimation, intuitive reasoning, analogy, pattern
> matching, logic, etc.
It is not that people can't do math. It is that other people (who supposedly can but in fact cannot OR will not do them properly) insist on them doing the math they do not need (like computing the second derivative of the unemployment rate) or doing wrong the math they DO need (like with their mortgages).
As long as people can add, subtract, multiply and basically understand what a division is, they can do math. The problem is they are scared at what other people tell them 'math' is.
I am a mathematician, a professor of mathematics and witness to what I have said.
Of course people cannot do 'math' when 'math' means being able to compute multiple integrals or roots of third degree polynomials. That is not 'math' that is UTTER RUBBISH EXCEPT FOR PROFESSIONALS in all caps. Really.
You do not expect the average man to be able to write a sonnet with alliteration, second order metaphors and in iambic, do you? And they call themselves "literate".
People are not lazy, they are scared. An the blame falls on BAD MATHS TEACHERS, in all caps. Really.
As long as people can add, subtract, multiply and basically understand what a division is, they can do math. The problem is they are scared at what other people tell them 'math' is.
I am a mathematician, a professor of mathematics and witness to what I have said.
Of course people cannot do 'math' when 'math' means being able to compute multiple integrals or roots of third degree polynomials. That is not 'math' that is UTTER RUBBISH EXCEPT FOR PROFESSIONALS in all caps. Really.
You do not expect the average man to be able to write a sonnet with alliteration, second order metaphors and in iambic, do you? And they call themselves "literate".
People are not lazy, they are scared. An the blame falls on BAD MATHS TEACHERS, in all caps. Really.
Hi, I'm a former (high school & middle school) math teacher. I taught various ages, but was only ever responsible for curriculum with age 10-11.
We studied a lot of fractions, decimals, percentages & converting between them. There was a bit of angle stuff mixed in there, too.
I love mathematics, so I think I'd be the sort of teacher you could see eye-to-eye with. What should I be doing differently?
We studied a lot of fractions, decimals, percentages & converting between them. There was a bit of angle stuff mixed in there, too.
I love mathematics, so I think I'd be the sort of teacher you could see eye-to-eye with. What should I be doing differently?
Hi! That is exactly what they need: fractions are just a way to explain division and it helps them a lot (actually they are the basis for the "rule of three" which is the "fifth rule").
Proportions. Trying to start with square (?) triangles and the idea of similarity of triangles so that they can later (when 12-13) understand trigonometry (the basics) which, once again, is PROPORTIONALITY. There is little more to 'maths' than that.
What I object to is the unnecessary abstraction. Getting 10-11s to perform correct computations is hard but exactly what they need: lots of exercises (no sweat no learn or whatever).
You are a HERO. Really. In all caps My respect. I teach undergrads and this is way easier.
Proportions. Trying to start with square (?) triangles and the idea of similarity of triangles so that they can later (when 12-13) understand trigonometry (the basics) which, once again, is PROPORTIONALITY. There is little more to 'maths' than that.
What I object to is the unnecessary abstraction. Getting 10-11s to perform correct computations is hard but exactly what they need: lots of exercises (no sweat no learn or whatever).
You are a HERO. Really. In all caps My respect. I teach undergrads and this is way easier.
Aw! This was super kind, and I like the idea of stressing proportionality. It's nice to be told that the abstraction can wait - I've felt like I should be abstracting more at times, but it's not my natural instinct, so it's nice to be told I'm doing things right on that front! I feel the same way about anyone who can teach older (or younger!) mathematicians - keep on keeping on.
[A night later]
Oh, I really mean it. Teachers to children (and especially maths teachers) are essential for our society, and have one of the hardest job.
Focusing on proportions you can teach almost anything: from basic triangle geometry, including elements of what later they will know as 'trigonometry', to interest rates -even letting the best get the scent of 'compound interests'-, to areas & volumes to the notion of 'speed' as a ratio, to how to save money for the future... There is little more a normal 'literate' person needs to know, as I see it.
However, it takes quite an effort getting them to actually perform the computations. This is where 'good' -appealing- exercises and problems are required, and this is where the teacher's craftmanship comes into play. A good craftman will find the correct and 'fancyful' exercises, according to the class, the student, the time... You know, this is where the 'heroism' takes place.
All the best.
Oh, I really mean it. Teachers to children (and especially maths teachers) are essential for our society, and have one of the hardest job.
Focusing on proportions you can teach almost anything: from basic triangle geometry, including elements of what later they will know as 'trigonometry', to interest rates -even letting the best get the scent of 'compound interests'-, to areas & volumes to the notion of 'speed' as a ratio, to how to save money for the future... There is little more a normal 'literate' person needs to know, as I see it.
However, it takes quite an effort getting them to actually perform the computations. This is where 'good' -appealing- exercises and problems are required, and this is where the teacher's craftmanship comes into play. A good craftman will find the correct and 'fancyful' exercises, according to the class, the student, the time... You know, this is where the 'heroism' takes place.
All the best.
Really, the most important thing you can teach is the context behind the maths. Find examples in real life of where the maths is applicable; spend 3/4 of a lesson explaining the backstory as to why we do things the way we do. Students lose their way because maths is presented as an endless series of facts to rote learn, with no context. There's a thriving backstory of human ingenuity behind the numbers and operations which sadly gets little or no time in the classroom.
Read Bill Bryson's book "a short history of nearly everything" for an example of the style I wish my maths classes had been taught in. It's a survey of science book, but mostly focuses on the human interaction behind the discoveries & theories, and makes fascinating reading because of it. We need to teach maths (and all hard technical subjects) closer to this approach.
Read Bill Bryson's book "a short history of nearly everything" for an example of the style I wish my maths classes had been taught in. It's a survey of science book, but mostly focuses on the human interaction behind the discoveries & theories, and makes fascinating reading because of it. We need to teach maths (and all hard technical subjects) closer to this approach.
The underlying problem is two-fold in my opinion.
#1 We focus on teaching procedures rather than understanding. Most people therefore view math as a list of memorized fixed procedures, and that's intrinsically very hard to remember and become good at.
#2 Math is simple in a way our brains are not wired to be good at, yet we have a mistaken belief that simple is easy. It is not. Adding 1000 numbers by hand and getting the right answer is very simple, but hard. Recognizing my voice is very complex, but easy. Be aware that your brain is working in a way it is not designed to work and have patience with it. Otherwise you'll get frustrated, and mistake "It took me this long to understand something THIS simple?" for, "I'm stupid!"
The result is that most people understand math in a way that is hard, and their experience of math is an experience of repeatedly confirming the message that they are stupid. Is there any wonder that they take that frustration out on the entire subject of mathematics?
#1 We focus on teaching procedures rather than understanding. Most people therefore view math as a list of memorized fixed procedures, and that's intrinsically very hard to remember and become good at.
#2 Math is simple in a way our brains are not wired to be good at, yet we have a mistaken belief that simple is easy. It is not. Adding 1000 numbers by hand and getting the right answer is very simple, but hard. Recognizing my voice is very complex, but easy. Be aware that your brain is working in a way it is not designed to work and have patience with it. Otherwise you'll get frustrated, and mistake "It took me this long to understand something THIS simple?" for, "I'm stupid!"
The result is that most people understand math in a way that is hard, and their experience of math is an experience of repeatedly confirming the message that they are stupid. Is there any wonder that they take that frustration out on the entire subject of mathematics?
I only think the author forgot that there are people that REALLY cannot do math.
It might sound bizarre and weird, but I met more than once person, that did their best to learn, and were intelligent with many other things (one of these persons had a degree in law, another in international relations, and was doing a masters in international law), yet could not do 29/3 without a calculator...
Or even worse, I knew people (in that case usually working with more low level work, like burger flipping) that even trying, or even if needed (ie: cashiers) cannot do it right even with a calculator, their grasp of math is so weak (even if they want to have a grasp) that they cannot even use the correct operations.
Also the same apply to many other fields, I knew intelligent people that could not read, or that could not grasp history, or geography, and so on...
People keep forgetting that brains CAN be very specialized, and be great with something, and terrible with other, and I personally believe that the old way of teaching professions (ie: throw the kid to work with a Master in that profession) was better because of that, currently you throw kids on the school, and the ones that might excel at some things that are not on school (like Music) reach adulthood thinking they are dump and they don't make a effort even in thinks they do have a talent to do.
It might sound bizarre and weird, but I met more than once person, that did their best to learn, and were intelligent with many other things (one of these persons had a degree in law, another in international relations, and was doing a masters in international law), yet could not do 29/3 without a calculator...
Or even worse, I knew people (in that case usually working with more low level work, like burger flipping) that even trying, or even if needed (ie: cashiers) cannot do it right even with a calculator, their grasp of math is so weak (even if they want to have a grasp) that they cannot even use the correct operations.
Also the same apply to many other fields, I knew intelligent people that could not read, or that could not grasp history, or geography, and so on...
People keep forgetting that brains CAN be very specialized, and be great with something, and terrible with other, and I personally believe that the old way of teaching professions (ie: throw the kid to work with a Master in that profession) was better because of that, currently you throw kids on the school, and the ones that might excel at some things that are not on school (like Music) reach adulthood thinking they are dump and they don't make a effort even in thinks they do have a talent to do.
> It might sound bizarre and weird, but I met more than once person, that did their best to learn, and were intelligent with many other things (one of these persons had a degree in law, another in international relations, and was doing a masters in international law), yet could not do 29/3 without a calculator...
There are multiple ways to learn even something like arithmetic.
Some people can naturally juggle a lot of numbers in their head and can basically brute force problems.
Others can break problems down into component pieces, work out each individual piece, and reassemble them into an answer. This requires a different type of mental juggling than the above.
And then there are those who just have to memorize a lot of problems.
When I was in school, we started off with memorization (multiplication tables and such), which requires a large time investment that I am sure many students did not make. (My peer group tended to stay inside during recces and practice our multiplication!) After that I think we were supposed to "naturally" progress to breaking problems down into parts, but that was never really covered all that well. From what I understand, other countries make this part of learning arithmetic very explicit.
A good deal of this involves training ones working memory. Right at the end of college my working memory for numbers was amazing, I could do 3 digit divides in my head, and at one point I could even do a binary search to find logarithms down to a decimal place or two!
But as with many other skills, they degrade from a lack of use.
29/3? I have a minor on mathematics. If you give me that problem, I'd honestly type "win-r calc 29/3 enter".
Now days I have problems just adding up large strings of numbers, I play a bunch of D10 games and I have to actually do math rather than it coming to me instantly!
> People keep forgetting that brains CAN be very specialized, and be great with something, and terrible with other,
Well yes of course, but we choose what to specialize in! I really do believe that anyone can learn math if they put the time and effort into it. My math classes took 2-3 hours a day of studying a good 4 days a week in order for me to completely grasp the concepts being taught.
Repetition of hundreds of problems, as much as I hated it, was the only real way to burn technique into my head, and even then most of those techniques have fallen by the way side! A few still bounce around inside my skull, but it has been a good 8 years since my last math class, so the amazing feats of mental gymnastics I could perform are long gone.
On the flip side, ask me to design a test infrastructure for code sometime, and I'm right on it! How about a custom memory allocation scheme? No problem! Specialization indeed.
There are multiple ways to learn even something like arithmetic.
Some people can naturally juggle a lot of numbers in their head and can basically brute force problems.
Others can break problems down into component pieces, work out each individual piece, and reassemble them into an answer. This requires a different type of mental juggling than the above.
And then there are those who just have to memorize a lot of problems.
When I was in school, we started off with memorization (multiplication tables and such), which requires a large time investment that I am sure many students did not make. (My peer group tended to stay inside during recces and practice our multiplication!) After that I think we were supposed to "naturally" progress to breaking problems down into parts, but that was never really covered all that well. From what I understand, other countries make this part of learning arithmetic very explicit.
A good deal of this involves training ones working memory. Right at the end of college my working memory for numbers was amazing, I could do 3 digit divides in my head, and at one point I could even do a binary search to find logarithms down to a decimal place or two!
But as with many other skills, they degrade from a lack of use.
29/3? I have a minor on mathematics. If you give me that problem, I'd honestly type "win-r calc 29/3 enter".
Now days I have problems just adding up large strings of numbers, I play a bunch of D10 games and I have to actually do math rather than it coming to me instantly!
> People keep forgetting that brains CAN be very specialized, and be great with something, and terrible with other,
Well yes of course, but we choose what to specialize in! I really do believe that anyone can learn math if they put the time and effort into it. My math classes took 2-3 hours a day of studying a good 4 days a week in order for me to completely grasp the concepts being taught.
Repetition of hundreds of problems, as much as I hated it, was the only real way to burn technique into my head, and even then most of those techniques have fallen by the way side! A few still bounce around inside my skull, but it has been a good 8 years since my last math class, so the amazing feats of mental gymnastics I could perform are long gone.
On the flip side, ask me to design a test infrastructure for code sometime, and I'm right on it! How about a custom memory allocation scheme? No problem! Specialization indeed.
Exactly, this is the point, not a bad or good education. This teacher forget that each brain is different.
The "kitchen" for doing complex maths is in a specific area of the brain. Some people have this area not specially developped, or damaged, and compensate this developping other areas. The 90% of the great geniuses of the history are really plain stupid in other fields of knowledge.
I understand the frustration of the teacher, but to spread that everybody can do complex math or either is a lazy/dumb people, is exactly the same idea as to spread that everybody can be Mozart or that everybody "can grow a cut leg if trying enough" (salamanders can do, we are much smarter, so why not?) if you prefer substitute leg by brain.
There is a biological/chemical/physical structure subjacent. Sometimes it can be changed, sometimes can not be recovered in a reasonable time, and often simply don't need to be changed. The people find a new way to achieve the solution, or simply ask the solution to a machine or other people. Nothing wrong with this.
The "kitchen" for doing complex maths is in a specific area of the brain. Some people have this area not specially developped, or damaged, and compensate this developping other areas. The 90% of the great geniuses of the history are really plain stupid in other fields of knowledge.
I understand the frustration of the teacher, but to spread that everybody can do complex math or either is a lazy/dumb people, is exactly the same idea as to spread that everybody can be Mozart or that everybody "can grow a cut leg if trying enough" (salamanders can do, we are much smarter, so why not?) if you prefer substitute leg by brain.
There is a biological/chemical/physical structure subjacent. Sometimes it can be changed, sometimes can not be recovered in a reasonable time, and often simply don't need to be changed. The people find a new way to achieve the solution, or simply ask the solution to a machine or other people. Nothing wrong with this.
I don't think the author meant that people aren't walking calculators. I believe he implies that people don't want to understand how to evaluate a math problem. Most people that "can do" math still use a calculator; additionally, they understand what's happening and they can identify an answer that is absurd (because of a key-press mistake).
I don't think the author forgot that, he seems to refer a lot to 'most people' and 'usually' enough to acknowledge that his assertion is not a truism. Besides that, it's a rant, those are often filled with generalities that need to be pared down before we can get to the real discussion.
Seymour Papert (http://www.papert.org/), inventor of Logo in his seminal work Mindstorms (1980) did a very good job on analyzing this phenomena.
Not only he did a great analysis on children but also came up with Logo, one of the best paradigm-changing environments for teaching math.
On his second book, The Children's Machine (1991), almost 11 years later he went deeper into the issue of computer's at schools and what should or shouldn't be taught at school on math lectures. There's a nice section on an experiment involving hat he called "Kitchen Math" which served to evidence that a constructivist approach is inevitably better than memorizing formulas, rules and formal stuff.
I recommend both books to anyone interested in this topic.
Here is a link to an essay written by Dr. Papert in 1996: http://papert.org/articles/AnExplorationintheSpaceofMathemat...
Not only he did a great analysis on children but also came up with Logo, one of the best paradigm-changing environments for teaching math.
On his second book, The Children's Machine (1991), almost 11 years later he went deeper into the issue of computer's at schools and what should or shouldn't be taught at school on math lectures. There's a nice section on an experiment involving hat he called "Kitchen Math" which served to evidence that a constructivist approach is inevitably better than memorizing formulas, rules and formal stuff.
I recommend both books to anyone interested in this topic.
Here is a link to an essay written by Dr. Papert in 1996: http://papert.org/articles/AnExplorationintheSpaceofMathemat...
I went through this exact issue with a junior employee moved to my team to do some analysis that required a basic understanding of medians/means and a little intro stats.
It took about 8 weeks of tasks with incremental mathematical challenges to overcome this, and at least reach the point where he could do his work, and apply the fundamentals learned to solving more complex problems.
In the end he "could do math", the real issue was he had never "put any work into math".
It took about 8 weeks of tasks with incremental mathematical challenges to overcome this, and at least reach the point where he could do his work, and apply the fundamentals learned to solving more complex problems.
In the end he "could do math", the real issue was he had never "put any work into math".
Telling people they can do math because they can add is disingenuous at best.
"Ciphering" is the term that used to be used for figuring sums and such. Nobody pretended that was math. At the school house it was called "arithmetic". It was one of the three "R's" not something with it's own Phd granting departments at universities.
People who say they cannot do math are speaking with the knowledge that they are looking up toward a massive intellectual edifice containing ideas they do not understand - whether they are looking up to high-school trigonometry or differential equations or Bayesian statistics is irrelevant.
What is relevant is that they are not looking up to ciphering as unobtainable - at least not those people likely to be conversing causally or academically with a tenure track academic.
"Ciphering" is the term that used to be used for figuring sums and such. Nobody pretended that was math. At the school house it was called "arithmetic". It was one of the three "R's" not something with it's own Phd granting departments at universities.
People who say they cannot do math are speaking with the knowledge that they are looking up toward a massive intellectual edifice containing ideas they do not understand - whether they are looking up to high-school trigonometry or differential equations or Bayesian statistics is irrelevant.
What is relevant is that they are not looking up to ciphering as unobtainable - at least not those people likely to be conversing causally or academically with a tenure track academic.
When `she' said "I hate math".
Read as: I hate the public school/university mathematics rote droning pretentious pedagogy ecosystem.
I personally still retain a enthusiastic thread of the childlike wonder and delight 15 contiguous years in classroom minefields attempted to lame, but its `opportunity cost' has been expensive.
Today's increasing autodidact free materials eliminates this cost.
Read as: I hate the public school/university mathematics rote droning pretentious pedagogy ecosystem.
I personally still retain a enthusiastic thread of the childlike wonder and delight 15 contiguous years in classroom minefields attempted to lame, but its `opportunity cost' has been expensive.
Today's increasing autodidact free materials eliminates this cost.
It traces back to cultural and educational issues.
Math has typically been taught devoid of any context or relevance to our everyday lives. Even in college, you just mindlessly plug and chug formulas to get by. Some students don't see the use in it, beyond perhaps the parts needed for basic financial literacy (which many do not have, either).
Better and more engaging and relevant and effective ways to teach math have already been developed, such as Realistic Mathematics Education.
But people actually say things like "I can't do math" in regards to all forms of literacy, I believe. It depends what standards you are mentally comparing yourself to, I guess, such as professional vs. social standards.
"I'm not good with money, or I'm not good with the business stuff" - financial literacy
"I can't write" - as in, I can't write books or novels or easily do other professional writing tasks. Again, there are pedagogical techniques that can help students write more, write better, and have more interest and confidence in writing.
"I don't read" - most folks nowadays don't really read books or novels anymore, what with TV, movies, and the Internet.
"I don't know computers" - how often do we hear that - that's computer literacy. I hear it less and less though nowadays.
To tell you the truth, maybe it will be nice when one day people feel forced to admit "I can't code." Because that would mean that programming and computational literacy is something taught in most schools.
Math has typically been taught devoid of any context or relevance to our everyday lives. Even in college, you just mindlessly plug and chug formulas to get by. Some students don't see the use in it, beyond perhaps the parts needed for basic financial literacy (which many do not have, either).
Better and more engaging and relevant and effective ways to teach math have already been developed, such as Realistic Mathematics Education.
But people actually say things like "I can't do math" in regards to all forms of literacy, I believe. It depends what standards you are mentally comparing yourself to, I guess, such as professional vs. social standards.
"I'm not good with money, or I'm not good with the business stuff" - financial literacy
"I can't write" - as in, I can't write books or novels or easily do other professional writing tasks. Again, there are pedagogical techniques that can help students write more, write better, and have more interest and confidence in writing.
"I don't read" - most folks nowadays don't really read books or novels anymore, what with TV, movies, and the Internet.
"I don't know computers" - how often do we hear that - that's computer literacy. I hear it less and less though nowadays.
To tell you the truth, maybe it will be nice when one day people feel forced to admit "I can't code." Because that would mean that programming and computational literacy is something taught in most schools.
My wife is a public librarian, and occasionally comes home with stories of people she had to help with the computer. Patrons are allotted one hour of computer time per day, and she says it often takes that much time just to create a gmail account, compose an email to <government office> and send it. The mouse seems to give a lot of people trouble, which blows my mind--I first encountered a mouse when I was a college sophomore and one of the first Macs appeared on campus. "What's this thing?" I wondered, reaching for it. When I touched it, I noticed the arrow on the screen moving. It didn't take but a second or two to realize that moving the mouse caused the pointer to move in similar fashion. It seemed obvious to click the buttons on the mouse to see what happened, and discover, in about a minute and a half, that a single click selected whatever the pointer was touching, and a double click activated it. How can people not grasp that connection intuitively?
But the sad fact is, there are a lot of people who don't grasp that simple cause and effect relationship, and you know what? Those are the people in your neighborhood... they're the people that you meet, when you're walking down the street. They're the people that you meet each day!
But the sad fact is, there are a lot of people who don't grasp that simple cause and effect relationship, and you know what? Those are the people in your neighborhood... they're the people that you meet, when you're walking down the street. They're the people that you meet each day!
I completely agree that it traces back to educational issues. In fact, i recently wrote an article about this problem - http://aditgupta.me/post/what-is-17x13
> most folks nowadays don't really read books or novels anymore, what with TV, movies, and the Internet.
Reading off of a screen is still reading. I don't know why some people think paper is magical.
Reading off of a screen is still reading. I don't know why some people think paper is magical.
Sadly, most of what is on the internet counts as reading the same as watching a spongebob episode and "The Stand" both count as watching a 'show.'
It's telling when writing a 3-4 paragraph article requires a ;TLDR section.
Usually when a person says they are a "reader," or the like, are specifically talking about long works, not articles in Maxim magazine.
It's telling when writing a 3-4 paragraph article requires a ;TLDR section.
Usually when a person says they are a "reader," or the like, are specifically talking about long works, not articles in Maxim magazine.
> It's telling when writing a 3-4 paragraph article requires a ;TLDR section.
You mean humans are still lazy mammals as opposed to being industrious insects? Say it isn't so!
The rest of your post has nothing to do with the Internet. People read Maxim both online and off. We haven't changed.
You mean humans are still lazy mammals as opposed to being industrious insects? Say it isn't so!
The rest of your post has nothing to do with the Internet. People read Maxim both online and off. We haven't changed.
My post was, in context, not about the Internet, it was in response to your comment in reference to:
"most folks nowadays don't really read books or novels anymore, what with TV, movies, and the Internet."
Comparing reading a 3-4 paragraph article on HN (or elsewhere) as being the same as reading a book or novel, the "rest of my post" was an anology.
True, those people that don't read ebooks or longer articles propably were not reading books or novels to begin with, so it is a win for literacy, but its just as likely they just shifted to that because it's easier to read an article off their phone while shitting than turning the pages of a physical magazine.
So TLDR; Reading articles online isn't the same level of effort as reading a book, which your comment seemed to support.
"most folks nowadays don't really read books or novels anymore, what with TV, movies, and the Internet."
Comparing reading a 3-4 paragraph article on HN (or elsewhere) as being the same as reading a book or novel, the "rest of my post" was an anology.
True, those people that don't read ebooks or longer articles propably were not reading books or novels to begin with, so it is a win for literacy, but its just as likely they just shifted to that because it's easier to read an article off their phone while shitting than turning the pages of a physical magazine.
So TLDR; Reading articles online isn't the same level of effort as reading a book, which your comment seemed to support.
You can read books on a screen, too - they are called ebooks
This reminds me of my experience with people in China when I was there. My friends there told me that many young people in China do understand English. But every time I approached them with "Do you speak English?" They usually took two steps backward and replied "no". They hesitate speaking English (because they find it difficult) despite knowing enough to serve my purpose.
I have had similar experience with "Do you know maths?" in the US.
I have had similar experience with "Do you know maths?" in the US.
Really good book on the subject that deals with why/ramifications of a society that's okay with "I can't do math" as a reasonable thing for an otherwise educated person.
Innumeracy: Mathematical Illiteracy and Its Consequences John Allen Paulos http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Its...
Innumeracy: Mathematical Illiteracy and Its Consequences John Allen Paulos http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Its...
One fun parallel - not to literacy, but to another skill at which artsy types are often better than mathematicians - is the oft-repeated, and completely acceptable, "I can't read minds".
Lots of people can read minds. Not literally, of course, but they've put in the work (and perhaps it was work that they found easy and pleasant, much as many programmers found math) to be able to essentially tell what people want, don't want, are implying, will be offended by, etc. Yet I've certainly heard plenty of people, and especially STEM-types, making a point of pride about lacking this skill: "I'm a straight-shooter" and the like. There's aspergers and there's dyscalculia, but many STEM types who happily admit to lacking this skill are just like those whom the author bemoans - they find it difficult and uninteresting. And that's okay! But it's no truer, really, than "I can't do maths".
Lots of people can read minds. Not literally, of course, but they've put in the work (and perhaps it was work that they found easy and pleasant, much as many programmers found math) to be able to essentially tell what people want, don't want, are implying, will be offended by, etc. Yet I've certainly heard plenty of people, and especially STEM-types, making a point of pride about lacking this skill: "I'm a straight-shooter" and the like. There's aspergers and there's dyscalculia, but many STEM types who happily admit to lacking this skill are just like those whom the author bemoans - they find it difficult and uninteresting. And that's okay! But it's no truer, really, than "I can't do maths".
When I hear, "I can't do math...", it gets immediately translated in my head to, "My math skill are on the level of a typical 5th grader, and I don't want to be asked to do math because it will expose an embarrassing ignorance." The author's approach to people that feel this way is likely to humiliate them further.
This sounds like an extension of the recently widely covered practice of not telling your kids they're smart. Basically, studies have shown that praising your child for intelligence as opposed to hard work can cause them to undervalue effort. They start to separate subjects into things they are "good" at and things they are "bad" at. They then avoid the things they've labeled themselves as bad at doing.
On the contrary, praising a child for their hard work seems to not be associated with this same negative behavior. http://abcnews.go.com/blogs/lifestyle/2012/02/why-you-should...
On the contrary, praising a child for their hard work seems to not be associated with this same negative behavior. http://abcnews.go.com/blogs/lifestyle/2012/02/why-you-should...
Yeah, I didn't like this article. It's overly harsh and doesn't take into account other situations. I have dyscalculia, the math version of dyslexia. People generally don't say "I can't do English" if they have dyslexia, but they can say, "I can't spell" which I wonder would be assessed as harshly by the author as this article.
Also, it was pointed out that what math is can be fuzzy - I always thought I was BAD at math, when I was young and it was still adding numbers together and stuff, which I only can do with great difficulty. Once I got into algebra, I was already devising my own ways to solve equations and acing the class (although they told me to stop solving the equations in my own way...)
Also, it was pointed out that what math is can be fuzzy - I always thought I was BAD at math, when I was young and it was still adding numbers together and stuff, which I only can do with great difficulty. Once I got into algebra, I was already devising my own ways to solve equations and acing the class (although they told me to stop solving the equations in my own way...)
I feel like the sentiment of "I can't do math" stems mostly from the way math is taught. To contrast we can look at how reading is taught to elementary students. Students read books at their "level." Once they are comfortable with the words and concepts in a given level then the difficulty of the content is scaled up appropriately. In math, students are often pulled along with the rest of the class. It is very difficult to understand division when you are still struggling to understand subtraction. This happens for a number of reasons but mostly because it is more difficult to teach math on an individual level the way reading can be though. I think that the solution to issues like this will/would look similar to the reverse model that the folks at KahnAcademy are creating. It's clear that it is difficult to teach math of every student is at a different point and understands a different amount of math. By transferring the teaching onto prerecorded videos the students are able to learn at their own pace and rewatch the lessons for more difficult concepts without exasperating the teacher. Salman Kahn has stated that his cousin (that he first started making videos for) told him that she liked learning from him in his videos more because he wouldn't get frustrated when she didn't understand a concept. She could just watch it again. This also frees the teacher to help the students who are learning at a slower rate or have gotten stuck with a difficult concept.
We seem to accept what's essentially illiteracy in all the subjects that require more precise thinking as the status quo. Most people don't seem to understand much about chemistry, physics, logic or computers either, and that doesn't get them in for a lot of stick.
If people are poorly taught and have gaps in their understanding that are never adequately addressed, then I can see how they'd think that the problem was them - that they couldn't do maths. Couldn't make the connections that were expected of them and portrayed as normal functioning for maths.
Maths is different to history, and many other subjects, in that regard. If you don't understand something in history, it probably doesn't have a massive list of dependencies that you're going to fail a lot of stuff in the future on. If you don't understand something in English, the worst that's going to happen is you have an esoteric interpretation of the text. You can still do those subjects if you don't really understand them, as long as you use the right buzzwords and hook them off the right things.
I suspect part of the answer may just be that admitting you can't do history or the like is different to admitting that you can't do maths in that it's unclear what someone would even mean by claiming that they can do history. Certainly the claim that someone knows a lot about the broad strokes of history would rarely be justified these days.
If people are poorly taught and have gaps in their understanding that are never adequately addressed, then I can see how they'd think that the problem was them - that they couldn't do maths. Couldn't make the connections that were expected of them and portrayed as normal functioning for maths.
Maths is different to history, and many other subjects, in that regard. If you don't understand something in history, it probably doesn't have a massive list of dependencies that you're going to fail a lot of stuff in the future on. If you don't understand something in English, the worst that's going to happen is you have an esoteric interpretation of the text. You can still do those subjects if you don't really understand them, as long as you use the right buzzwords and hook them off the right things.
I suspect part of the answer may just be that admitting you can't do history or the like is different to admitting that you can't do maths in that it's unclear what someone would even mean by claiming that they can do history. Certainly the claim that someone knows a lot about the broad strokes of history would rarely be justified these days.
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I do like the author's comparison to illiteracy. Language, just like mathematics, tries to solve the problem of communication but while language is more about the expression of emotions, thoughts, and feelings, mathematics is meant to describe precise ideas and abstractions in a quantified way. While language can be ambiguous in how it relays our true intentions, mathematics is anything but. It is the only method humanity has that, through experiment and observation, can help us reach for and perhaps even attain absolute truth.
Just like reading/writing, you don't learn mathematics (real maths, not arithmetic) to be able to scribble symbols on a piece of paper. You learn it to develop a way of thinking that promotes certainty and helps you develop and understand complex abstract ideas that describe how the world works. When you dive deep into math (or programming for that matter), you don't just learn a subject, your brain actually reconfigures itself and fundamentally changes how you act and think. This effect, called neuroplasticity, functions pretty much until you die so it's never too late to learn math.
When someone says that they can't "do" math it says nothing about their intellect. All it says is that they didn't like the mathematical equivalent of "The Cat in the Hat" taught in primary school and are now going to live (many quite proudly) without the mental faculties to express and grok abstract and complicated ideas and systems. Sadly, now more than ever, we need each and every human to have these faculties if we are going to survive and thrive as individuals and as a species.
Just like reading/writing, you don't learn mathematics (real maths, not arithmetic) to be able to scribble symbols on a piece of paper. You learn it to develop a way of thinking that promotes certainty and helps you develop and understand complex abstract ideas that describe how the world works. When you dive deep into math (or programming for that matter), you don't just learn a subject, your brain actually reconfigures itself and fundamentally changes how you act and think. This effect, called neuroplasticity, functions pretty much until you die so it's never too late to learn math.
When someone says that they can't "do" math it says nothing about their intellect. All it says is that they didn't like the mathematical equivalent of "The Cat in the Hat" taught in primary school and are now going to live (many quite proudly) without the mental faculties to express and grok abstract and complicated ideas and systems. Sadly, now more than ever, we need each and every human to have these faculties if we are going to survive and thrive as individuals and as a species.
> But many people feel the same way about, for example, physical exercise: they don’t enjoy doing it and find it to be very hard work when they try. But it would be strange indeed to say that you can’t “do” exercise. You may not want to do it. You may be unwilling to invest the time and effort into doing it. But you aren’t incapable of doing it. And the same is true, in my experience, in mathematics.
I think a more apt comparison might be to sports, rather than to exercise. Everyone can get reasonably good at "exercise" i.e. be able to run a mile in a reasonable time, but I think we can all agree that there are certainly many people who simply "can't do" a sport.
For example, I am pretty bad at baseball. I was a terrible batter and an even worse fielder. I "persevered" and played Little League Baseball for several years, eventually realized I didn't have much if any aptitude for it, and stopped playing altogether. Does this mean I'm lazy or that the Little League coaching system is broken? No, it just means I'm not that good at baseball and probably never will be, and I don't think any amount of great coaching would have changed that.
Are there some people who could have been good at a given sport but "slip through the cracks" due to laziness or bad teaching? Maybe, but I doubt it's a significant number.
It's easy to take mathematics ability for granted on a focused forum like HN, but perhaps being good at math is no different from being a great sports athlete - Some people are really good at it, but most aren't and it's pretty unrealistic to expect everyone to change their expectations and opinions on the matter.
I think a more apt comparison might be to sports, rather than to exercise. Everyone can get reasonably good at "exercise" i.e. be able to run a mile in a reasonable time, but I think we can all agree that there are certainly many people who simply "can't do" a sport.
For example, I am pretty bad at baseball. I was a terrible batter and an even worse fielder. I "persevered" and played Little League Baseball for several years, eventually realized I didn't have much if any aptitude for it, and stopped playing altogether. Does this mean I'm lazy or that the Little League coaching system is broken? No, it just means I'm not that good at baseball and probably never will be, and I don't think any amount of great coaching would have changed that.
Are there some people who could have been good at a given sport but "slip through the cracks" due to laziness or bad teaching? Maybe, but I doubt it's a significant number.
It's easy to take mathematics ability for granted on a focused forum like HN, but perhaps being good at math is no different from being a great sports athlete - Some people are really good at it, but most aren't and it's pretty unrealistic to expect everyone to change their expectations and opinions on the matter.
Totally relevant:
http://www.phdcomics.com/comics/archive.php?comicid=1356
> But many people feel the same way about, for example, physical exercise: they don’t enjoy doing it and find it to be very hard work when they try. But it would be strange indeed to say that you can’t “do” exercise. You may not want to do it. You may be unwilling to invest the time and effort into doing it. But you aren’t incapable of doing it. And the same is true, in my experience, in mathematics.
The way physical exercise is often presented to people, it's no surprise they give up on it. A lot of gimmicky exercise programs and contraptions designed to take money from you in exchange for no visible results.
And even time tested exercise programs like P90X won't teach you the fundamentals. You'll spend a lot of time doing crunches which is the least effective and most time consuming way of working abs.
The same is certainly true for math and when people say they can't do any moderately complex math, they're saying "fuck you" to the establishment that wasted their time while teaching them nothing. They also do just fine without complex math, just like most people who never exercise.
The way physical exercise is often presented to people, it's no surprise they give up on it. A lot of gimmicky exercise programs and contraptions designed to take money from you in exchange for no visible results.
And even time tested exercise programs like P90X won't teach you the fundamentals. You'll spend a lot of time doing crunches which is the least effective and most time consuming way of working abs.
The same is certainly true for math and when people say they can't do any moderately complex math, they're saying "fuck you" to the establishment that wasted their time while teaching them nothing. They also do just fine without complex math, just like most people who never exercise.
Neil Degrasse Tyson made this point.
He basically said that imagine if instead of people saying at cocktail parties "I can't do math" they said "I can't read..."
He basically said that imagine if instead of people saying at cocktail parties "I can't do math" they said "I can't read..."
Sure, but "I can't do math" is a shorthand for "I can't do math beyond arithmetic, and even then, I often can't do that in my head." That is not equivalent to "I can't read." It is more equivalent to "I can't read Proust without getting bored and confused."
The author in the OP acknowledges this substitution of "I can't" for "I currently struggle with." It's complaint about semantics that misses the point, I think.
Rather than complaining when people say they can't do math, we should find out why they feel that way. I suspect a large part of the reason is that math education is pedantic and boring (memorizing axioms and doing rote calculations, when we should prove them, for example, or apply math in ways that doesn't involve trains leaving stations) and when people do speak their minds about their difficulties with mathematics, they often face snooty responses like this one in the OP, rather than a lighter touch.
The author in the OP acknowledges this substitution of "I can't" for "I currently struggle with." It's complaint about semantics that misses the point, I think.
Rather than complaining when people say they can't do math, we should find out why they feel that way. I suspect a large part of the reason is that math education is pedantic and boring (memorizing axioms and doing rote calculations, when we should prove them, for example, or apply math in ways that doesn't involve trains leaving stations) and when people do speak their minds about their difficulties with mathematics, they often face snooty responses like this one in the OP, rather than a lighter touch.
One of the OP's main gripes is that many people will gleefully, happily tell you that they can't do math. Nobody will ever admit to a stranger at a cocktail party that they're illiterate but being bad at math is, for some reason, a conversation starter.
There are three other defense mechanisms at play:
1) I already made up my mind based on emotion or tradition or the salesperson was cute or whatever, and I was hoping the math proves my select decision is correct, but its not looking good and/or I've found a easier / better justification so "I can't do math"
2) I'm getting totally financially reamed over this (housing bubble / car lease / rent / mortgage / tuition / credit card) but its less painful to say I can't do math than admit I have awful financial judgment.
3) I suspect the result of this decision is going to be a minefield, and the other party is more traditionally mathematical than my side's ethnic / school major / gender / job title / whatever so I automatically come out ahead regardless of result by not making any decision and defer via "I can't do math" because first I don't have to put in the effort and secondly I can't be blamed.
1) I already made up my mind based on emotion or tradition or the salesperson was cute or whatever, and I was hoping the math proves my select decision is correct, but its not looking good and/or I've found a easier / better justification so "I can't do math"
2) I'm getting totally financially reamed over this (housing bubble / car lease / rent / mortgage / tuition / credit card) but its less painful to say I can't do math than admit I have awful financial judgment.
3) I suspect the result of this decision is going to be a minefield, and the other party is more traditionally mathematical than my side's ethnic / school major / gender / job title / whatever so I automatically come out ahead regardless of result by not making any decision and defer via "I can't do math" because first I don't have to put in the effort and secondly I can't be blamed.
> So what do people actually mean when they say that they can’t 'do' math?
That I "can't". That my brain is differently "shaped", simply, and that this a common variation.
Personally I think in spatial 3d structures, classify images, read superfast recognising words by the shape of the contour of the letters, learn at a good rate and can accurately drawn what I see in the real world...
But in the other hand I find deadly boring to express, in a unnecesarily complicated formulae drawn in 2d with many arcane symbols, a simple concept that could be introduced and expressed instead in two or three simple phrases... or logically, graphically...
The "atoms" of my thinking process are shapes and relationships between shapes, not abstract quantities. Is as simply as this. Other people "think in sounds" and are very good at music, and other "think in mathematical structures". I'm not blaming nobody, not excuses... I'm just a perfectly normal human with an intelligence basically visual
(... And I prefer not spend much time with this when a machine can do the math part for me in the blink of an eye).
That I "can't". That my brain is differently "shaped", simply, and that this a common variation.
Personally I think in spatial 3d structures, classify images, read superfast recognising words by the shape of the contour of the letters, learn at a good rate and can accurately drawn what I see in the real world...
But in the other hand I find deadly boring to express, in a unnecesarily complicated formulae drawn in 2d with many arcane symbols, a simple concept that could be introduced and expressed instead in two or three simple phrases... or logically, graphically...
The "atoms" of my thinking process are shapes and relationships between shapes, not abstract quantities. Is as simply as this. Other people "think in sounds" and are very good at music, and other "think in mathematical structures". I'm not blaming nobody, not excuses... I'm just a perfectly normal human with an intelligence basically visual
(... And I prefer not spend much time with this when a machine can do the math part for me in the blink of an eye).
I believe that lots of people really can't do math. Most people are fine until fractions. Many people never gain any intuitive feel for what "2/5 * 7/8" even means. It only gets worse with algebra. They can often learn the mechanics of fraction arithmetic and solving equations, but they will never understand how to apply the knowledge later in life.
"I can't do math" actually means "I both don't like to and don't believe I really need to do any math, so I'm going to say 'I can't do math' to cut you off before you propose I do any."
Of course college-educated people can do a little math - but they don't like to, so they don't.
Of course college-educated people can do a little math - but they don't like to, so they don't.
I sympathize with the OP on all he says, but what does he mean by this, aren't these expressions synonymous, the only difference being the tone?:
> although saying “my bad” when you mean “my mistake” comes close
...and going further on the linguistic ambiguity route, most people saying "can't do" actually mean simply "I hate it / I'm not good at it, and because it's so much effort for met o do it I'd rather not have to do it"... and the only bad thing in it is the uberannoying implied "I don't want to learn it, don't try and teach it to me"...
> although saying “my bad” when you mean “my mistake” comes close
...and going further on the linguistic ambiguity route, most people saying "can't do" actually mean simply "I hate it / I'm not good at it, and because it's so much effort for met o do it I'd rather not have to do it"... and the only bad thing in it is the uberannoying implied "I don't want to learn it, don't try and teach it to me"...
'My bad' is a foolish thing to get upset about people saying. People have said it for thousands of years and they're going to keep saying it. See: "Mea Culpa"
I think these people mean "I can't do arithmetic" which is much more acceptable, in my opinion. If you asked me to find a derivative, I could probably do it faster than I could find 123*26. It's not because the multiplication is harder, but because I don't have to do it in my head that often anymore. I have a calculator, and I use it. However, I have to do derivatives in math class everyday and at this point I'm probably faster at it. Not being able to do arithmetic is fine by me, I can't do it.
That's just because there are fewer steps for finding a simple derivative.
If you evaluate 12326 in your head, most people probably end up thinking through something like this
Although I could ask you to take the derivative of 123x^26.
If you evaluate 12326 in your head, most people probably end up thinking through something like this
123 * 20 + 123 * 6
123 * 20
123 * 10 * 2
1230 * 2
2460 Now remember this part
123 * 6
100*6 + 20*6 + 3*6
100*6 = 600
20*6 = 120
3*6 = 18
600+120+18
738
2460 + 738 = 3198
Although I could ask you to take the derivative of 123x^26.
There is an idea that pervades their rant that all people are more or less equally educable in all subjects. In my experience this is simply not true.
I think part of the problem is we don't teach Math in a way that is visual/obvious enough, which means early math is experienced as 'mechanics' not 'understanding'.
For example, we should explain commutative rule [ a(b+c)=ab+ac ] by drawing the rectangles.
My efforts to help this in some way : GridMaths.com [ sample pics + blurb : quantblog.wordpress.com ]
For example, we should explain commutative rule [ a(b+c)=ab+ac ] by drawing the rectangles.
My efforts to help this in some way : GridMaths.com [ sample pics + blurb : quantblog.wordpress.com ]
Math is just another word for structured thought. When people say they're not good at math, what they really mean is that they attempted learn something without understanding the underlying principles, and therefore found it difficult.
They're not lazy, they're just ignorant to the process because they were taught incorrectly.
They're not lazy, they're just ignorant to the process because they were taught incorrectly.
Yeah, it's interesting that people will proudly admit "I can't do math" but you don't nearly as often hear them say "I don't understand politics, government, or world affairs." People will do their damnedest to appear knowledgeable about those things.
That's true. “I don't really understand national politics in any deep way,” is something I've said many times and has been responded to often with “me neither”, but I'm quite sure had I not said it, the person would've kept quiet about their ignorance and feigned insight in other conversations.
Maybe they should teach it as 'modelling' instead of 'mathematics.' When you start thinking of it in terms of interpreting real world dynamics and relationships, curiosity takes over and it becomes a much more compelling subject to learn. IMO.
Surely if effort and attention to detail were applied most of these people could work their way through whatever math task was set before them. What they should instead be saying is; "I don't do math".
But If a person says "I can't \"do\" mathematics" also could imply that the person would like to like it, or maybe the person at least respect the subject as important.
how did these added slashes slip in?
Java/C++ programmer who observed that English lacks Perl's qq operator or python's r''
I never learned how to do long division by hand. That doesn't mean I can't do math, though. I could, of course, learn how, but I haven't needed to.
"I can't do math [at a university level]..."
Indeed, it is often beneficial to state your preferences up front instead of sputtering nonsense trying to be "diplomatic".
Everyone can interpret what I have to say about text editors much more usefully if I state up front that I love Emacs. The same goes for programming languages, frameworks, operating systems and math.
Everyone can interpret what I have to say about text editors much more usefully if I state up front that I love Emacs. The same goes for programming languages, frameworks, operating systems and math.
I wish i could do math