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

#261

Post by jacobolus » Wed, Jul 12, 2017, 1:29 AM UTC

monic_binomial wrote: post #253

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-dow…

Unfortunately your realization is empirically unsound, and the problems of “every teacher you know” have as much to do with lack of available expert teacher attention per student and standard expected teaching styles and school structure with substantial friction/latency in feedback and huge amounts of wasted time as with inherent student ability.

Start here http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf but then there is a vast literature (thousands of studies and other research papers) exploring the general topic of math pedagogy. Some approaches and some teachers are radically more successful than others, given comparable students.

If you hand me one of your 20th percentile students who is struggling but willing to learn, and give us the same amount of time which would otherwise be spent in a classroom for a year of one-on-one face-to-face time, with the student also spending a typical amount of time working independently, I can have them outperforming your 80th percentile students by the end of the year, without issue. The difficulty of course is that direct mentoring by an expert tutor is too expensive for society to be willing to pay for at scale.

The main reasons that your teacher friends are stuck is because (a) many if not most of their students are unprepared before they arrive in any particular course, which is largely down to structural social factors and school scheduling inflexibility, and (b) the combination of lectures and independent work with inadequate feedback are for the average student a terribly inefficient and ineffective way to learn. Neither of those has all too much to do with biological predestination or whatever.

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

#262

Post by wisty » Wed, Jul 12, 2017, 1:31 AM 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…

"School is about the advanced stuff, not the basics" is the call of good students, often ones who learnt the basics from their parents. It's also kind of popular with bad students, who think that they might suck at the basics but have untapped powers if they were doing something else.

It's nice, vague, and populist. Everything thinks they know how to do critical thinking. Everyone thinks they know how it could be taught. I suspect most people think that they have above average critical thinking skills. As long as they don't have to agree on how or what critical thinking is, everyone can agree it's a good thing.

The other one that's dumb is "you learn best when you teach yourself". Sure, if you are really super-interested in something then you learn better, and tend to teach yourself a bit, but correlation is not causation.

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

#263

Post by monic_binomial » Wed, Jul 12, 2017, 1:43 AM UTC

jacobolus wrote: post #261

monic_binomial wrote: post #253

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-dow…

Unfortunately your realization is empirically unsound, and the problems of “every teacher you know” have as much to do with lack of available expert teacher attention per student and standard expected teaching styles and school structure with substantial friction/latency in feedback and huge amounts of wasted time as with inherent student ability. Start here http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf but th…

Heh. You can cite all the old wives' tales, superstitious folklore, and education research papers you want. None of that makes it more likely that a 95 IQ student will ever understand precalculus.

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

#264

Post by jacobolus » Wed, Jul 12, 2017, 1:59 AM UTC

monic_binomial wrote: post #263

jacobolus wrote: post #261

Earlier quoted context omitted.

Unfortunately your realization is empirically unsound, and the problems of “every teacher you know” have as much to do with lack of available expert teacher attention per student and standard expected teaching styles and school structure with substantial friction/latency in feedback and huge amounts of wasted time as with inherent student ability. Start here http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf but th…

Heh. You can cite all the old wives' tales, superstitious folklore, and education research papers you want. None of that makes it more likely that a 95 IQ student will ever understand precalculus.

I actually have direct experience getting a (I would guess) 95 IQ student to understand precalculus, tutoring for a few months back when I was a high school student ~15 years ago. She got an A, after having had a string of Cs in every prior high school math course, and not being properly fluent with 9th grade algebra at the beginning of the year. [I don’t say this to brag; I expect any competent tutor can achieve the same thing with a typical below-average student if the student is motivated and they get lots of 1-on-1 time.]

However, I can well believe that a 95 IQ student who was dramatically unprepared at the beginning of the year and believes himself to be an irredeemable failure will have a lot of trouble in a standard lecture + homework drills format high school precalculus course taught by a teacher with no time to help him catch up on remedial material at his current level of understanding, give him any special attention, or make the class material engaging enough to give him reason to care. In that context, failure (or scraping by with an undeserved C) is the obvious outcome, but can’t fairly be blamed on the student’s effort during that year or some kind of mental deficiency.

(Personally, I skipped half of precalculus and somewhat wish I had skipped the other half; I found it to be an uninspired grab-bag of disconnected topics, with much too rote a focus.)

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

#265

Post by mindcrime » Wed, Jul 12, 2017, 1:59 AM UTC

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 this, two deductions:

• Having trouble remembering the quadratic equation formula doesn't mean you're not a "numbers-person."

• To remember your math skills, use them more often.

What I remember from high-school and college was this: I'd take a given math class (say, Algebra I) and learn it reasonably well. Then, summer vacation hits. Next term, taking Algebra II, all the Algebra I stuff is forgotten because, well, who uses Algebra I over their summer vacation? Now, Algebra II is harder than it should be because it builds on the previous stuff. Lather, rinse, repeat.

This is one reason I love Khan Academy so much. You can just pop over there anytime and spend a few minutes going back over stuff at any level, from basic freaking fractions, up through Calculus and Linear Algebra.

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

#266

Post by alistproducer2 » Wed, Jul 12, 2017, 2:04 AM UTC

I can't agree more. Math is about intuition of what the symbols are doing. In the case of functions, intuition about how the symbols are transforming the input. I've always thought I was "bad at math." It wasn't until my late 20's when I took it upon myself to get better at calculus and I used "Calculus Success in 20 Minute a Day[0]" did I finally realize why I was "bad" at it; I never understood what I was doing.

That series of book really put intuition at the forefront. I began to realize that the crazy symbols and formulas were stand-in for living, breathing dynamic systems: number transformers. Each formula and symbol represented an action. Once I understood Math as a way to encode useful number transformation, it all clicked. Those rules and functions were encoded after a person came up with something they wanted to do. The formula or function is merely a compact way of describing this dynamic system to other people.

The irony was I always thought math was boring. In retrospect it was because it was taught as if it had no purpose other than to provide useless mental exercise. Once I started realizing that derivatives are used all around me to do cool shit, I was inspired to learn how they worked because I wanted to use them to do cool shit too. I went through several years of math courses and none of them even attempted to tell me that math was just a way to represent cool real world things. It took a $10 used book from amazon to do that. Ain't life grand?

[0]:https://www.amazon.com/Calculus-Success-20-Minutes-Day/dp/15...

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

#267

Post by kaitai » Wed, Jul 12, 2017, 2:08 AM UTC

gnaritas wrote: post #195

kaitai wrote: post #191

Earlier quoted context omitted.

Disagree. If you read the common core standards, they are very sensible and make no curricular or pedagogical recommendations. "Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph." CCSS.MATH.CONTENT.HSF.IF.B.6. The way people are teaching math in the US is baffling and weird, but Pearson's textbook…

I think you misunderstand, "New Math"[1] was a thing from the 60's, he clearly took it as an adjective talking about common core, the current "new math", but not the same as the "New Math" bitwize was saying had been tried and was a failure. [1] https://en.wikipedia.org/wiki/New_Math

I know New Math from the 1960s; the sibling comment to my post indicates that ghettoCoder is talking about Common Core today and I disagree with his characterization of Common Core. New Math as practiced in the 1960s is a fascinating and very different failure case, although it probably had similar problems in implementation (poor teacher training and shoddy repackaged curricula).

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

#268

Post by mindcrime » Wed, Jul 12, 2017, 2:09 AM UTC

wolfgang42 wrote: post #238

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…

> 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 3 7. I happen to know that 3 3=9, which I can double to get 3 6=18, plus 3 to get 3 7=21. These figures…

Hah... glad to see I'm not the only one who does that. I never did truly memorize the times table, but I can work out most small multiplications easily enough.

I still feel a certain tinge of guilt though, over not memorizing that stuff.

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

#269

Post by andrepd » Wed, Jul 12, 2017, 2:10 AM UTC

alistproducer2 wrote: post #266

I can't agree more. Math is about intuition of what the symbols are doing. In the case of functions, intuition about how the symbols are transforming the input. I've always thought I was "bad at math." It wasn't until my late 20's when I took it upon myself to get better at calculus and I used "Calculus Success in 20 Minute a Day[0]" did I finally realize why I was "bad" at it; I never understood what I was doing. Th…

But math isn't about intuition, much to the contrary, it's about rigorous reasoning. In math "intuition" and "common sense" are worth nothing. Only objective proof matters. It's a valuable lesson too, learning not to take anything on a hunch and only take as true what what you carefully and logically prove.

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

#270

Post by metaobject » Wed, Jul 12, 2017, 2:19 AM UTC

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…

For me, at least, understanding math was all about understanding how to translate symbols to concepts.

Seeing an integral sign and understanding what it means, for example, can help simplify seemingly complicated mathematical expressions and make them easy to understand.