Math education: It’s not about numbers, it’s about learning how to think • Hacker News Board

Re: Math education: It’s not about numbers, it’s about learning how to think

#51

Post by santoriv » Tue, Jul 11, 2017, 3:18 PM UTC

Koshkin wrote: post #11

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…

"The reason is that mathematics creates its own universe of concepts and ideas"

This also applies to reading music and understanding music theory. Written music is a map of events/time - which is totally different than any other math I studied. I've noticed that a lot of musicians pick up programming pretty easily. I've often wondered if learning more symbolic systems that are different from natural language makes it easier to pick up additional systems like programming languages.

Re: Math education: It’s not about numbers, it’s about learning how to think

#52

Post by kungito » Tue, Jul 11, 2017, 3:20 PM UTC

simias wrote: post #29

I completely agree. I think we start all wrong too, the first memories I have of maths at school was learning how to compute an addition, a subtraction and later a multiplication and division. Then we had to memorize by heart the multiplication tables. That can be useful of course (especially back then when we didn't carry computers in our pockets at all times) but I think it sends some pupils on a bad path with rega…

A friend's kid did not know how to do subtraction with just numbers but when you said it like he had some currency in a shop then he understood it. That was peculiar although the kid was very young (5-6 I guess) so it's interesting how kids develop ability to do abstract thinking as they age

Re: Math education: It’s not about numbers, it’s about learning how to think

#53

Post by throwawayjava » Tue, Jul 11, 2017, 3:22 PM UTC

jharger wrote: post #50

Ntrails wrote: post #49

Earlier quoted context omitted.

> I never had any problems with math until I went to university So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above? I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial…

I saw what looked like this sort of thing with my daughter. She struggled in her high-school algebra 1 class, but whenever I would sit down with her and she would work out problems, she had no trouble (but she also had no confidence that she could do it).. She ended up having to repeat the class, which ruined her confidence even more, and she hated it even more the second time through. At the start of middle school,…

Encourage her to give it another try when she gets to late high school or college. I know a lot of math phds who fell out of love with math around middle school and then fell back in love during their first proof-based course in college.

Re: Math education: It’s not about numbers, it’s about learning how to think

#54

Post by jdmichal » Tue, Jul 11, 2017, 3:24 PM UTC

hasenj wrote: post #24

jdmichal wrote: post #18

Earlier quoted context omitted.

Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".

I find this sentiment somewhat surprising. Limits is incredibly easy to link to real physical things: movement, speed, acceleration. If speed is distance over time, what's the speed of a moving object at a very specific instant?

Are you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve.

Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.

Re: Math education: It’s not about numbers, it’s about learning how to think

#55

Post by mbaha » Tue, Jul 11, 2017, 3:24 PM UTC

jfaucett wrote: post #36

Koshkin wrote: post #11

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…

"all these notions are so different from what we have to deal with every day that learning them takes a lot of training" I disagree in the general sense. I think most of mathematics with which "normal people" (by that I mean people who don't get a masters or phd in mathematics) have problems is directly correlated to problems and situations in the real world. The problem is that we teach mathematics as if we were com…

Henri Poincaré once said: "Mathematics is the art of giving the same name to different things"

I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands.

Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about projectiles, you end up ignoring that quadratic equations are used... heavily... everywhere.

You may see them in chemistry, crypto or nuclear physics. But if your brain only remembers projectiles, it doesn't have the ability to abstract new problems the right way.

Re: Math education: It’s not about numbers, it’s about learning how to think

#56

Post by dahart » Tue, Jul 11, 2017, 3:25 PM UTC

taneq wrote: post #2

As my old boss once said, "never confuse mathematics with mere arithmetic ."

I've always liked that line, but it's clearly being cheeky and there are times it's applicable and times it's not.

All arithmetic is math, but not all math is arithmetic.

Except that if you dig deep enough, all math is arithmetic under the hood. Algebra is doing the arithmetic on variables before they're numbers. Calculus is just useful shorthand for compound arithmetic; the derivative is a divide and the integral is a sum. Exponentiation is repeated multiplication, which is repeated addition. Linear algebra is arithmetic on blocks of numbers.

So we abstract over the individual arithmetic operations for our own convenience, there are too many of them. But it's all still fancy arithmetic.

Maybe we should call math "fancy arithmetic" to help all the people who are intimidated by math...

Re: Math education: It’s not about numbers, it’s about learning how to think

#57

Post by adamnemecek » Tue, Jul 11, 2017, 3:31 PM UTC

redial wrote: post #40

dahart wrote: post #34

I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English? I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't…

Symbols are symbols, are symbols. Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).

Let's add Chinese, Amharic and Arabic characters. Why not, expanding is expanding.

Re: Math education: It’s not about numbers, it’s about learning how to think

#58

Post by jdmichal » Tue, Jul 11, 2017, 3:32 PM UTC

Koshkin wrote: post #32

jdmichal wrote: post #18

Earlier quoted context omitted.

Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".

Even simple objects, like vectors, for example, can very quickly become so abstract (higher dimensions; complex vector spaces) that the physical intuition stops being useful and can even lead to mistakes. This is the point when new kinds of mathematical intuition becomes necessary, and building them is very difficult and takes time.

I remember doing only simple (two-dimensional Euclidean) vectors before calculus -- length, scaling, addition, subtraction... Agreed that they obviously can get pretty hairy, but I don't think that side is exposed to people before calculus? Complex vector math was calculus III for myself.

Re: Math education: It’s not about numbers, it’s about learning how to think

#59

Post by thrownblown » Tue, Jul 11, 2017, 3:35 PM UTC

jharger wrote: post #50

Ntrails wrote: post #49

Earlier quoted context omitted.

> I never had any problems with math until I went to university So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above? I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial…

I saw what looked like this sort of thing with my daughter. She struggled in her high-school algebra 1 class, but whenever I would sit down with her and she would work out problems, she had no trouble (but she also had no confidence that she could do it).. She ended up having to repeat the class, which ruined her confidence even more, and she hated it even more the second time through. At the start of middle school,…

Teachers are important. I had consistently bad grades in science until I my 8th grade science teacher that liked to blow stuff up.

I really like him and I wanted to impress him with how smart I was and I remember putting in a lot of time doing all my homework. I got my only "A" in science for the first two quarters. Then he got fired and became a harley mechanic. Back the C's and D's

Re: Math education: It’s not about numbers, it’s about learning how to think

#60

Post by BeenThere888 » Tue, Jul 11, 2017, 3:38 PM UTC

throwawayjava wrote: post #53

jharger wrote: post #50

Earlier quoted context omitted.

I saw what looked like this sort of thing with my daughter. She struggled in her high-school algebra 1 class, but whenever I would sit down with her and she would work out problems, she had no trouble (but she also had no confidence that she could do it).. She ended up having to repeat the class, which ruined her confidence even more, and she hated it even more the second time through. At the start of middle school,…

Encourage her to give it another try when she gets to late high school or college. I know a lot of math phds who fell out of love with math around middle school and then fell back in love during their first proof-based course in college.

Yeah, this.

I hated math in middle-school and early high-school. It was boring and rote.

But proofs based math, and learning how to think of math as a language, instead of a collection of overly specific "solutions" crammed into my head by teachers, was the key.