Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

nwaonline.com

Post reply on HN
Print view View original

251–260 of 343 posts

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

#251

Post by jacobolus »

throwawayjava wrote: post #158
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…

> This was tried. It was called "New Math". No it wasn't, at least writ large in the USA. And that's not what New Math was. Explaining what multiplication is doesn't demand an introduction to set theory. If we read Feynman's CRITICISM of New Math, we actually find that he ADVOCATES for exactly what your parent is suggesting ("cobbled together.. in order to solve real-life problems"). So clearly, your parent isn't des…

Also Kline spends a good part of his book shredding the standard math curriculum (especially rote memorization and mindless application of standard algorithms) and agreeing that it needs reform, at the end advocating constructivist alternatives, with word problems, use of physical manipulatives, and motivation via applications to other fields.

His beef with the New Math is with an emphasis on axioms, deductive reasoning, rigorous abstract logic, linguistic purity, and symbol manipulation, rather than with teaching conceptually or letting students think for themselves. He also doesn’t like the specific content of the New Math (set theory, inequalities, alternate number bases, boolean algebra, modular arithmetic). [I haven’t studied the New Math curriculum enough for myself to know how fair these arguments are.]

His key criticism: “Psychologically the teaching of abstractions first is all wrong. Indeed, a thorough understanding of the concrete must precede the abstract. Abstract concepts are meaningless unless one has many and diverse concrete interpretations well in mind. Premature abstractions fall on deaf ears.”

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

#252

Post by Chris2048 »

yequalsx wrote: post #103
Chris2048 wrote: post #87

Earlier quoted context omitted.

There's a nuance here too. in programming, y = f(x) = x^2 is one way; 'y' is a label for a function of one variable. But in math there is no fixed typing, or distinctions. The symbolic equation represents a relationship. Hence: y(x) = x^2 x(y) = (+/-) sqrt(y) the domain and domain can be switched and the entire equation rearrange so that a 'variable' becomes a 'function'.

Here is an interesting observation on what you wrote: x(y) = (+/-) sqrt(y) is not valid notation. On the left hand side you are stating that x is a function of y. But on the right hand side your usage of (+/-) indicates that y can go to two different values. Hence x is not a function of y. To be a function one must have each input going to a distinct output.

Yep, that was on purpose somewhat. A choice of +/- will essentially make a restricted codomain, to positives say.

That means switching the variables around switched domain/codomain.

An example where they are both the same:

y(x) = x + 1 x(y) = y - 1

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

#253

Post by monic_binomial »

I was a math teacher for 10 years. I had to give it up when I came to realize that "how to think" is about 90% biological and strongly correlated to what we measure with IQ tests.

This may be grave heresy in the Temple of Tabula Rasa where most education policy is concocted, but nonetheless every teacher I ever knew was ultimately forced to chose between teaching real math class with a ~30% pass rate or a watered-down math Kabuki show with a pass rate just high enough to keep their admins' complaints to a low grumble.

In the end we teachers would all go about loudly professing to each other that "It's not about numbers, it's about learning how to think" in a desperate bid to quash our private suspicions that there's actually precious little that can be done to teach "how to think."

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

#255

Post by gertef »

Merovius wrote: post #159
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…

> 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…

Are you also suggesting that we shouldn't teach kids to read until they are in high school and decide they are interested?

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

#256

Post by gertef »

autokad wrote: post #160
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…

that sounds utterly dreadful. computer science is often taught like this; to its folly. imagine a kid learning to speak 'stop just memorizing words, you need to understand how language was derived before you learn how to speak it'. its completely counter intuitive to how creatures learn. learn easy things, especially those that relate to problems we deal with, and then get deeper into the subject if necessary.

the pillars of science are theory and experiment.

memorization is a side-effect of these, not a foundation.

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

#257

Post by hasenj »

hasenj wrote: post #84
jdmichal wrote: post #54

Earlier quoted context omitted.

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 r…

Will, speed breaks when you consider a specific instant in thing because there is no time span nor a distance the be traveled.

lol sorry for the typos! Wrote that from my mobile phone at night.

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

#258

Post by mindcrime »

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…

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 as they needed to be... because I found basic algebra to be mind-numbingly boring.

Fast forward 20+ years and I'm not working on re-learning some lower level math + leveling up on higher level calculus and linear algebra. This time, it seems easier (although I might not go so far as to say "trivial") and I attribute that largely to the fact that I'm much more motivated now. Now I have specific reasons for wanting to learn this math (mostly to do with machine learning and AI) and that seems to make all the difference in the world.

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

#259

Post by jeffdavis »

mbizzle88 wrote: post #204
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…

Marking a math test is only objective when an answer is entirely correct. What mark do you give a student who makes a minor arithmetic error on a single step of a multi-step problem?

Objective tests for math ability are pretty good. Arguing that they are only 99% objective doesn't really change my point.

People aren't afraid of making tiny mistakes. They are afraid of looking at a problem similar to a dozen they saw in lecture; and having no idea what to do, and then being told under no incertain terms that they did not succeed.

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

#260

Post by Mz »

Well, I actually liked math and took kind of a lot of it in K-12. I was in my 30s before I knew there were actual applications for some of the things I memorized my way through without really understanding.

When I homeschooled my sons, I knew this approach would not work. My oldest has trouble with numbers, but he got a solid education in the concepts. He has a better grasp of things like GIGO than most folks. We also pursued a stats track (at their choice) rather than an algebra-geometry-trig track.

Stats is much more relevant to life for most people most of the time and there are very user-friendly books on the topic, like "How to lie with statistics." If you are struggling with this stuff, I highly recommend pursuing something like that.

Post reply on HN