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

#311

Post by Kluny » Wed, Jul 12, 2017, 2:19 PM UTC

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…

> when I learned what pH actually meant

potential (for attracting) Hydrogen. I had to look it up.

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

#312

Post by bb611 » Wed, Jul 12, 2017, 2:21 PM UTC

ashark wrote: post #63

kungito wrote: post #43

Earlier quoted context omitted.

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…

Schools (at least around here) do not hold kids back anymore. No matter what. Some of them get behind in one or more subjects (math, say) as early as 1st or 2nd grade and all their later teachers are stuck trying to give them the remedial math help the desperately need, while teaching their other students on-grade-level material, and also trying to teach the remedial kids enough of the new stuff that their scores on…

There is plenty of research that holding kids back isn't actually an effective intervention, for example: http://www.sciencedirect.com/science/article/pii/S0022440506...

Retained students (the current terminology for "held back") are much more likely to stop attending school early, and on average do not see larger academic gains in their year retained than their equivalent peers who are not retained.

The ONLY effective intervention is dramatically increased services for that student, delivered rapidly once the deficiency is detected. Many high performing charter school networks actually use this systematically, but only by overworking their teachers and generally burning them out within 5 years (retention rates at 5 years are frequently <10%, compared to 50% in the broader profession). Most teachers in mainstream district schools are covered by union contracts that limit their required working hours, so for the vast majority of schools even this isn't an option. Actually delivering these services in a sustainable manner is a financial burden American taxpayers appear largely unwilling to shoulder.

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

#313

Post by emilecantin » Wed, Jul 12, 2017, 2:22 PM UTC

jacobolus wrote: post #240

emilecantin wrote: post #205

Earlier quoted context omitted.

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?

The latter. My daughter actually learned it because we mixed different learning methods early on (we don't anymore, it confused her), but you have to admit that writing numbers greater than 10 kinda involves multiplication (by powers of ten, but still), so it makes sense to learn multiplication first. Negative numbers were seen while learning addition / subtraction (the concept was basically adding a negative number), which made a lot of sense too.

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

#314

Post by mindcrime » Wed, Jul 12, 2017, 2:34 PM UTC

sdrothrock wrote: post #298

mindcrime wrote: post #265

This part really resonates with me as well: "You read all the time, right? We constantly have to read. If you're not someone who picks up a book, you have to read menus, you've got to read traffic signs, you've got to read instructions, you've got to read subtitles -- all sorts of things. But how often do you have to do any sort of complicated problem-solving with mathematics? The average person, not too often." From…

I wanted to quote this passage for another reason -- it seems like you're agreeing with it, but I completely disagree with it, especially this: > But how often do you have to do any sort of complicated problem-solving with mathematics? There are a LOT of chances to do that kind of complicated problem solving, especially if you're shopping or comparison shopping on your own. It's not that people don't have the chances…

I mean, in a certain sense I agree with you. There are chances to do that stuff, yes. To a point. Basic algebra could come up in a shopping scenario for example. But even then, people aren't going to be doing that stuff every day, or even close to it. And when you get even slightly more esoteric, like, say, exponents... how often are people (especially kids) really going to be using exponents, or logarithms, or the quadratic formula, etc. in their daily lives?

Sure, if you go out of your way to actively look for reasons to find that stuff, you can find them. But my point is that when you're a kid learning math, it's summer vacation and you're busy playing with your friends, you aren't out actively looking for reasons to apply the quadratic formula, etc. At least not for most people, from what I've seen.

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

#315

Post by mindcrime » Wed, Jul 12, 2017, 2:35 PM UTC

mindcrime wrote: post #258

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…

But once I needed to use math, it was trivially easy. This is close to my experience as well. Math, for me, wasn't super difficult, but it didn't necessarily come naturally either, and I got really bored with the tedious, mechanical aspects of it in algebra, college algebra, trig, etc. Geometry and Calc I were mildly interesting, but I was handicapped a bit in Calc I because my basic algebra skills weren't as strong…

I'm not working

should read

I'm now working...

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

#316

Post by findoer » Wed, Jul 12, 2017, 2:40 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…

Anecdotally, I am also not in agreement with you on this. Another poster mentioned not 'getting' algebra until 10th grade--this didn't happen to me until roughly my second year of college. Anyway, I eventually became very motivated to excel at math. I ended up changing majors, transferring to a school with a more serious math program, and doing so well under a very heavy courseload that I won some department honors and scholarships. I basically went from not knowing high school (middle school?) algebra to demolishing my Rudin course in about 3 years. Honestly I think most people could do this and probably a lot more if they had the desire and access to a university/books

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

#317

Post by [deleted] » Wed, Jul 12, 2017, 2:40 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…

[deleted]

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

#318

Post by sethammons » Wed, Jul 12, 2017, 2:46 PM UTC

watwut wrote: post #165

bitwize wrote: post #98

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. 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…

Memorizing time tables and similar rote learnign means that precisely kids who could be good at math hate it. Meanwhile, those who have good memory and sux at problem solving think they are good at it. People who were taught by memorising go into outrage a new type of exercise is introduced. Suddenly thinking is needed and that is bad in their eyes. Not exactly success. Meanwhile, you can memorize time tables in late…

Blind memorizing is a problem. Memorizing things that you have learned concepts behind makes a student faster. As an example, the way I've taught my kids multiplication starts with: let's count by 2s! Great, now you know that, let's do 3's! 4's, ... 12's. We got to the point where we could count fluently (speed + accuracy), and incorporated fingers to help. Then we started saying things like, "if we count by fours five times, what do we get?" and then we changed our words to "four times five." All my kids have done well so far with this method. There is some memorizing to become fluent, but it is fully backed by understanding.

The next steps are to start doing things backwards, "how many times do we count by four to get to twenty?" And we start to introduce notation. Bingo, simple division. This leads directly to simple fractions. This opens up conversations on adding and subtracting fractions, then multiplying and dividing them.

My oldest kid, now in college, could add, subtract, multiply, and divide fractions by second grade and understood them. Oddly, she had a horrendous time learning decimals. Her mental model of numbers was fractions and decimals were "weird." She would have to change things like 5.045 to 5 45/1000 to understand it, and wanted to work with it as a fraction. It took a long time for her to get comfortable working with decimals.

One time that I think it paid off. I told her that 0.999... is equal to 1. She said false. I said, no, it is true. Can you tell me why? At this time, she was in algebra, and I was expecting to show her how to prove it using algebra. She had a much better way of looking at it. In about a couple of seconds, she said, "well, 1/9 is 0.111... and 9/9 would be 0.999... and that is also 1." Her answer was much better than mine. :)

An example of when memorizing is bad (ie, when the underlying knowledge is skipped) was her 7th grade algebra teacher. In teaching the laws of exponents, he said "anything to the 1st power it itself and anything to the 0th power is 1. We don't know why, it is just one of those math things." Teaching like this is why we have students who, later in high school, can't do x^0 or x^1 because they think, "it is either 1 or 0 or itself, I don't remember." As opposed to applying mental models and patters to see that 3^3 -> 3^2 -> 3^1 -> 3^0 is just dividing by 3 each time. These students know 3^2 is 9. So they should know that the next is 9/3 = 3 and that the next is 3/3 => 1.

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

#319

Post by Balgair » Wed, Jul 12, 2017, 2:59 PM UTC

digitalshankar wrote: post #305

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…

Hi Balgair, your comment made my day, i was feeling the same as you at that age.

Glad to have done so!

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

#320

Post by findoer » Wed, Jul 12, 2017, 3:04 PM UTC

jeffdavis wrote: post #97

My theory is that math anxiety is really anxiety about a cold assessment. In other subjects you can rationalize to yourself in various ways: the teacher doesn't like me, or I got unlucky and they only asked the history questions I didn't know. But with math, no rationalization is possible. There's no hope the teacher will go easy on you, or be happy that you got the gist of the solution. Failure in math is often (but…

Funnily enough, this goes away to some as you move along. Say I'm taking an exam and don't know how to do a proof. I pray and think that some equation is important, so I write it down with a little context. If I've done a great job up to that point, my professor will be more likely to think that I hit upon the key idea and give me 3/4 credit. If not, I'm probably more likely to just be marked wrong. You can also lose points for stylistic reasons if your professor is opinionated about them (-1 with "it is clear that" underlined and "don't write this ever again" in red ink :) ). Some professors will fail you if you get too technically correct with logical symbols everywhere instead of writing mathematical English. If you show too many steps you're wasting everyone's time; if you don't show enough or you don't seem confident you might get marked down. Etc