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

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

#141

Post by dahart »

bjl wrote: post #83
dahart wrote: post #56

Earlier quoted context omitted.

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…

Actually, the opposite is true. If you dig deep enough, it turns out that arithmetic is far more complicated than most other areas.

Then maybe instead of calling math fancy arithmetic, we should call arithmetic "hard math"? :P

Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?

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

#142

Post by lordnacho »

jack_h wrote: post #99
taneq wrote: post #2

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

As someone who went to university for mathematics (and computer science) I can tell you this is the most common misconception. People think that I spent years studying how to become a calculator. I love talking about both subjects I went to university for actually and both can be difficult to get people interested in. There are massive misconceptions about both.

Yes, this is one of the most annoying things about people who don't know anything about math.

- They see a shopping bill or a phone number and ask you if you can add them.

- They see random equations, often in the background, and ask you if you understand what it means.

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

#143

Post by kaitai »

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,…

Accelerated math could have been a solution -- I know that when I was bored in math I did worse, and when the problems got more interesting I did better. Now I have a math PhD, even though I got so bored in one class earlier on that I had to drop it in order to avoid failing.

There are a lot of social factors in US schools, too. I notice a lot of young women doing poorly in algebra and precalc even though they're perfectly capable if you sit down with them and ask them to simply write down the steps of solving the problem. The social benefits of being "bad at math" can be seductive in the short terms and the long-term benefits of being good at math are entirely invisible to many young women. Guys at least see some male nerds on TV making good money; girls who want to be like nerdy tv stars know that they have to go into forensics and don't know that math is very handy for figuring out the rate of cooling of bodies and decay of tissue.

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

#144

Post by mamon »

Jtsummers wrote: post #89
g9yuayon wrote: post #78

Is this a US thing? Why would people still think that math is about numbers? Math is about patterns, which got drilled into us by our teachers in primary school. I really don't understand how US education system can fuck up so badly on fundamental subject like math.

It is a US thing. And it's because until middle school (age 11-13) your teacher is likely not that familiar with math (they have taken math courses, but likely little formal math beyond geometry and trigonometry, perhaps some courses on math pedagogy). So for them, math is all about the numbers, they barely got to the more abstract concepts. In middle school in the US you start getting teachers specialized in their t…

What kind of education is required from teachers in US? In Poland it is Bachelor's degree in relevant topic (for elementary and middle school) or Master's degree (for high schools) plus a course in pedagogy. Most teachers however get Master's degree even if not required (they get salary increase for that). It is hard for me to imagine math teacher not good in match themselves.

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

#145

Post by dahart »

Jtsummers wrote: post #135
dahart wrote: post #66

Earlier quoted context omitted.

I'm not sure what that means. Are you saying the use of Greek letters is a good thing, and that we need more of them, not less? Or do you mean that all symbols are equally easy to learn for beginners, whether they're English or Greek or Chinese or abstract? Or do you mean that we should do away with symbols that don't mean anything and use real language to describe math instead? The symbols we use in math aren't to b…

∆ - delta - d as in difference ∑ - sigma - s as in sum ∫ - long s - s as in sum It's not totally arbitrary.

I could use alpha - a as in add.

I could use sigma - s as in subtract.

I could use iota - I as in integrate.

Of course it's arbitrary.

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

#146

Post by kaitai »

acjohnson55 wrote: post #110
gusmd wrote: post #75

I studied Mechanical Engineering, and it was my experience that several professors are only interested in having the students learn how to solve problems (which in the end boil down to math and applying equations), instead of actually learning the interesting and important concepts behind them. My wife went to school for Architecture, where she learned "basic" structural mechanics, and some Calculus, but still cannot…

Nailed it. I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep up with all the new things I was expected to compute: radicals, synthetic division, logarithms, etc. It was a major dent in my math confidence, which up until that point had be reinforced by being told I was "good at math". I didn't start regaining that confidence until Chemistry, when I learned what pH actually meant. That spark…

This is compounded in the weird American class called "precalculus," which is just a random grab bag of math crap that might maybe needed before calculus but might not. In most precalc classes I've encountered, not only is there no explanation of why the concepts will be used in calculus, but there is little attempt to relate the concepts covered in the class. Let's go from linear equations to complex numbers to polar coordinates! And let's not mention that complex numbers and polar coordinates are related!

To top it all off, a lot of young college students taking precalculus are taking it as a terminal math class -- they'll never take calculus! It's just for a credit! So you've got some poor kid who hit that wall in high school and is repeating precalc and going through some unit on polar coordinates who will never take another math class again and will somewhat justifiably think this whole thing is stupid.

In theory, I strongly support "math for liberal arts"-type classes instead in college. When done well they might cover voting systems and how you can prove there isn't a perfect one, gerrymandering and how you can use math to get several solutions, perspective drawing and its relation to projective geometry and Renaissance art, a bit of number theory and basic cryptography....

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

#147

Post by dahart »

eeZah7Ux wrote: post #94
dahart wrote: post #66

Earlier quoted context omitted.

I'm not sure what that means. Are you saying the use of Greek letters is a good thing, and that we need more of them, not less? Or do you mean that all symbols are equally easy to learn for beginners, whether they're English or Greek or Chinese or abstract? Or do you mean that we should do away with symbols that don't mean anything and use real language to describe math instead? The symbols we use in math aren't to b…

> unfamiliar and foreign looking The whole point of education is to approach and embrace unfamiliar concepts. By the way, arabic numbers are... arabic.

Agreed. But learning multiple unfamiliar concepts at once takes longer than learning one unfamiliar concept at once.

It's a fact that some people are intimidated by Greek letters and highly symbolic math, I'm not sure why that's being pushed back on. It's an open question, not a statement on my part, whether using more familiar symbols would speed up learning. Are you absolutely certain it would not? Why? Have you tried?

Lots of people on HN believe music education would be improved by updating the notation to be more "modern", and complain that music is more difficult to learn than it could be.

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

#148

Post by yequalsx »

eeZah7Ux wrote: post #88
yequalsx wrote: post #82

Earlier quoted context omitted.

In theory it shouldn't matter but it does. We brainwash students to think in terms of y being a function of x. y = f(x) If you give a problem with x = f(y) it really confuses people. If I label the horizontal axis y and the vertical axis x in the cartesian plane then confusion ensues. t = b(a) where t is the velocity and a is time That's confusing. I suppose it's analogous to me saying "The male, fluffy, white, large…

If this is enough to "really confuse" someone than there's a serious problem with understanding VS rota learning.

Here's something else to consider. There used to be obfuscated C programming contests. If symbols are just symbols and using them in nonstandard ways isn't confusing then shouldn't this mean that there is no such thing as obfuscated C programs? Does the existence of obfuscated C programming contests imply that programmers have a serious problem with understanding vs rote learning?

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

#149

Post by bjl »

dahart wrote: post #141
bjl wrote: post #83

Earlier quoted context omitted.

Actually, the opposite is true. If you dig deep enough, it turns out that arithmetic is far more complicated than most other areas.

Then maybe instead of calling math fancy arithmetic, we should call arithmetic "hard math"? :P Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?

Well, I was trying to make two points.

1) Arithmetic (number theory) is far harder than most people realise, to the point that its considered one of the most complicated mathematical disciplines.

2) Most subject areas in math bear little to no resemblance to arithmetic, so it doesn't make sense to call all math 'Fancy Arithmetic'.

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

#150

Post by steverb »

kungito wrote: post #52
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

That's the way my kids were able to grasp basic math. For fractions we used pizzas. Being able to hang the abstract concept on something that they were used to dealing with made it work for them.
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