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

#231

Post by agumonkey » Tue, Jul 11, 2017, 10:03 PM UTC

d3ckard wrote: post #19

Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…

My experience differs. My brain was intuitively ok until high school. The abstraction shift around cross references between log / exp / S 1/x went right above my head; same for discrete math basis such as prime relationships.

My conclusion is that every one will have strong and weak spots, and it's very important to stay alert to this and not beat around the bush thinking it's not important.

Took me more than a decade to get back into intuitive thinking, where you can play mentally with symbols and ideas just like LEGOs (as I was around high school).

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

#232

Post by hinkley » Tue, Jul 11, 2017, 10:04 PM UTC

brightball wrote: post #216

Balgair wrote: post #209

Earlier quoted context omitted.

Anecdote time: I was a late bloomer in math too. From about 6th to 10th grade, I was terrible at math. I think it was because of inapplication problems. Yes, you get a word problem about Jack and Kwanzeeah trying to fill up a pail or something, it was nonsense to me. Why are they filling up a pail? Why are they using a bucket that is 173/9th's the size of the hose? Why do they have to not let it overflow? Why do I ha…

I'd agree with this completely. It's less about teaching what, how and when than it is teaching WHY you need to know this. In high school, if you can convince a single person that they need to know calculus, more power to you. 99% of people don't and it's literally a waste of time. Teach people finance though...suddenly we have math that people will know is important.

I was so pissed in college when I found out that all of those equations we had to memorize in physics were just the integrals.

Why did I have to memorize all that shit if there's a simple way to derive the formula? Jesus.

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

#233

Post by Chinjut » Tue, Jul 11, 2017, 10:05 PM UTC

steanne wrote: post #229

taco_emoji wrote: post #86

Earlier quoted context omitted.

I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi wit…

the beginning of the end of my math studies was eighth-grade algebra. i GOT algebra, it was easy, and i enjoyed it. ..until the quadratic equation was introduced. "hey, that's really neat! how does it work?" "oh, you'll learn that in calculus, which we won't allow you to take for another four years."

It's common not to explain where the quadratic formula comes from (which is silly, it's straightforward enough to show, but standard math education curricula are shot through with this blind formula memorization nonsense), but… do they really tell you you'll learn it in calculus? It's got nothing to do with calculus.

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

#234

Post by agumonkey » Tue, Jul 11, 2017, 10:13 PM UTC

Balgair wrote: post #209

Merovius wrote: post #159

Earlier quoted context omitted.

> Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I indeed believe you are wrong. I started getting interested in STEM when I was 16 and now have a masters in mat…

Anecdote time: I was a late bloomer in math too. From about 6th to 10th grade, I was terrible at math. I think it was because of inapplication problems. Yes, you get a word problem about Jack and Kwanzeeah trying to fill up a pail or something, it was nonsense to me. Why are they filling up a pail? Why are they using a bucket that is 173/9th's the size of the hose? Why do they have to not let it overflow? Why do I ha…

I often think that kid psychology is probably extremely well suited to kinetic problems. Things that involve their own natural desires, all kids want to move, jump, interact with the world. Connect their eagered mind with a way to denote the world and let them abuse math to communicate and reason their own desires.

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

#235

Post by agumonkey » Tue, Jul 11, 2017, 10:18 PM UTC

deftturtle wrote: post #219

d3ckard wrote: post #189

Earlier quoted context omitted.

So basically you're saying that it is a preferred solution to teach kids math poorly, let them develop insecurity and aversion towards it and then spent a lot of money to let them overcome it than just do it right in the first place? Yeah, I know that's not what you mean, but you're missing a point. The premise of the article is that US has a STEM problem. I merely suggested that they're allocating resources wrong to…

Went to college with a guy. Freshman year calculus. He was really slow with it. I helped him, and we did homework together. Now he's a math major. Totally surprised me, but he's now majoring in a subject that he wasn't super good at. People need good teachers at all ages. I'm glad his college professors helped him out enough to lead him to major in math.

Mind and persons are complex, they can shift entirely. I've seen a girl that sucked at math she had to take lessons in HS. In college she flipped around and got a Master in Math.

On my side I was good at math but crashed in college, so much I avoided it, it's the reason I went into programming. Until 2 or 3 years ago I clicked again, I started to see through abstract math and now I'm back to math/phys because it's beautiful again.

If your buddy finds pleasure and can walk the path I'm not surprised he's still walking :)

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

#236

Post by dahart » Tue, Jul 11, 2017, 10:25 PM UTC

julian_1 wrote: post #210

dahart wrote: post #163

Earlier quoted context omitted.

Sure, understood and I agree. I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, so it does make some sense to call all math 'fancy arithmetic'. There is no math at all without the concept of addition sitting somewhere under it. But I'm not actually proposing that, just sharing a cheeky point of view to contrast @taneq's boss, so don't take me too seri…

> I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, I am a newb at this stuff, but I believe (basic) number theory actually builds on recursive structures, and then uses induction to make statements about behavior. https://en.wikipedia.org/wiki/Peano_axioms#Addition What is interesting is how the same approach works for analogous structures like lists…

This is interesting! I'm not sure, but I think this may reinforce what I said, not contradict it. I'm suggesting that math operations we know all build on top of addition, not that addition is necessarily the atom. Addition can build on top of something else, and still sit under all higher level math.

But that said, I didn't learn addition in terms of monoids, and I don't know anyone else who did. This might be a grand unified theory of math under which addition fits, but physical addition has no recursive monoid analogue. You can add two weights together to get a measurable sum, and it does not depend on a recursive structure that uses induction. You just put two separate things side by side on the scale.

Just because we can explain addition using monoids and induction does not mean that addition is made of monoids and induction.

Also, I probably wouldn't use Peano theory as the starting place to teach addition. As a pedagogical tool, this is advanced math, and I guess would be a bigger impediment to learning than Greek symbols, for beginners. Right?

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

#237

Post by agumonkey » Tue, Jul 11, 2017, 10:25 PM UTC

kungito wrote: post #43

d3ckard wrote: post #19

Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…

To this point, I have been doing a plenty of 1on1 lessons with kids trying to catch up with Math in high school/college/matura exams and one this I have noticed is that things which are problematic to these people are things which were covered quite well already in elementary school. Later the curriculum only expands on these problems so it's even harder and harder to catch up. I have been working with college freshm…

Oh reminds me of something peculiar. I gave math lessons to a 16yo guy. Mainly quadratic root formula. He understood the "high level" bits, but he failed so often at basic algebra after plugging the coefficient into the formula that he felt completely demotivated. It was odd to try to explain to him that he got the idea right, but failed at the basics. I told him that to mean "you can do the hardest, you will be able to get the easiest" but he interpreted it at "whats the point if you cant do the low level bits" ...

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

#238

Post by wolfgang42 » Tue, Jul 11, 2017, 10:39 PM UTC

bitwize wrote: post #98

taco_emoji wrote: post #86

Earlier quoted context omitted.

I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi wit…

> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables. There's just no way around the fact that drilling is key to early mathematical lea…

> drilling is key to early mathematical learning.

I was told to memorize the times table, and tried but never managed to succeed. Instead, I found that I got along just as well by memoizing them instead; that is, I would compute the parts I needed on the fly in the margins of the paper. (Example: Say I need to find 37. I happen to know that 33=9, which I can double to get 36=18, plus 3 to get 37=21. These figures would be written down, so when I later needed 47 I could easily add another 7 to get 28. I had similar tricks for various other numbers, and could generally get the figure I needed--if it wasn't already written down--in a few hops.)

These contortions don't seem to have significantly affected my mathematical development, but they did* improve my logic and reasoning skills (or possibly merely showed that I had them). Particularly now that nearly everyone I know carries a calculator in their pocket, I don't see why we would continue to focus on rote learning over actually understanding how the underlying principles work.

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

#239

Post by jacobolus » Tue, Jul 11, 2017, 10:54 PM UTC

inanutshellus wrote: post #85

throwawayjava wrote: post #69

Earlier quoted context omitted.

Tying students' progress through the curriculum to their age is silly. If you think about it for a single second, the whole thing is totally absurd. Vague pre-puberty/tween/teenager distinctions make sense, but within each grouping/building? totally unmotivated. Yearly age-to-competency distinctions continue by sheer force of tradition, and are harmful to all but the totally mythological "average" student.

But it works at-scale, which is where the focus is. Bang-for-the-taxpayer-buck. Further, socially-speaking, being with your actual peers is extremely valuable, obviously. But yes, the ideal would be a tailored-to-each-child education. Personally, having seen the power of a good Montessori education, I think it's frickin' genius and should be the template for all education (specifically the grouping of 3 yrs together…

I went to a public elementary school which had only mixed-grade classes (2–3 grades at a time), and it was just fine, and didn’t cost any more than having one grade per class with kids all precisely the same age.

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

#240

Post by jacobolus » Tue, Jul 11, 2017, 10:56 PM UTC

emilecantin wrote: post #205

inanutshellus wrote: post #85

Earlier quoted context omitted.

But it works at-scale, which is where the focus is. Bang-for-the-taxpayer-buck. Further, socially-speaking, being with your actual peers is extremely valuable, obviously. But yes, the ideal would be a tailored-to-each-child education. Personally, having seen the power of a good Montessori education, I think it's frickin' genius and should be the template for all education (specifically the grouping of 3 yrs together…

My kids are homeschooled, and I'm in charge of teaching Math, because my wife is one of those "I suck at math" people. Her parents focused exclusively on memorization when they helped her with homework, and then she got a few bad teachers, so her mental models are completely off. She's getting better, but that's not really my point. The math program I use with my kids is this one: http://www.defimath.ca/ecole-maison/…

They learn about negative numbers and multiplication before they learn to count past ten? Or they just learn counting without learning how to write the numbers?