How our 1,000-year-old math curriculum cheats America's kids • Hacker News Board

Re: How our 1,000-year-old math curriculum cheats America's kids

#91

Post by j2kun » Tue, Mar 04, 2014, 4:05 PM UTC

arbitrage wrote: post #60

j2kun wrote: post #27

> By hiding math's great masterpieces from students' view, we deny them the beauty of the subject. The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.

Wow, that's a really bold statement. Do you have a proof of that?

This comes from my interactions with colleagues at my own undergraduate institution. So it's biased evidence, but my institution was also one of California's largest math teaching credential programs, so I think it's a reasonable sample.

I can't say how many times my 'teaching concentration' colleagues would look at a basic linear algebra problem (which requires a proof) and whine saying things like, "I don't care about this stuff, it's so stupid and pointless, I just want to get my degree so I can go teach high school math."

All irony aside, these are the students who actually enjoyed the structure of high school math, and were rudely awakened when they were asked to apply their minds to reason. As a result, 'teaching concentrations' in undergraduate and graduate school usually provide mathematically watered down courses and automatic passing grades, and the teachers-in-training don't learn anything new. They certainly don't learn, for example, any mathematics that has been done since 1900, or any open problems that aren't Millenium Prize Problems. And god forbid they ask and try to answer an original question (as they expect their students to do when they're confused).

This is what they mean by the "blind leading the blind."

Re: How our 1,000-year-old math curriculum cheats America's kids

#92

Post by j2kun » Tue, Mar 04, 2014, 4:13 PM UTC

Pxtl wrote: post #52

bane wrote: post #33

I was a terrible math student in school. Looking back, it wasn't because the material was hard, or boring, but because it was completely unmotivated. They may as well have been asking me to organize piles of toothpicks or count ceiling tiles to fill up the class periods. There was simply no sense to it. I think this is partially understood, and the attempted solution is to try to find applications for the math you're…

The problem is that if you don't fill the period with busywork? The students don't build up the intuitive, gut-level skills they need to handle the higher-level stuff. Math requires building muscles in your head, and muscles take reps. Lots of reps. Let's look at the elementary level: it's not enough to simply introduce BEDMAS or least-common-denominator and write a few problems that get the student to check off "oka…

The problem with your comment is that busywork is not (and never has been) mathematical weight lifting. You don't gain thinking skills by memorizing multiplication tables or practicing adding fractions, you gain it by solving problems you don't know how to solve.

Of course it's not enough to state something, but making the students reiterate what you've stated a hundred times is only marginally better.

Another problem is that no student really learns ANYTHING until they need to use it to do something else. No student learns fractions until they need to use it to do algebra, and they don't learn algebra until they need it to do calculus. And they don't learn calculus until they need it to do engineering/physics or higher mathematics (real analysis, which they don't really learn until they do measure theory...).

Re: How our 1,000-year-old math curriculum cheats America's kids

#93

Post by j2kun » Tue, Mar 04, 2014, 4:20 PM UTC

Pxtl wrote: post #67

katbyte wrote: post #65

Earlier quoted context omitted.

personally, busy work just never sticks for me, i don't remember almost any of the math i did in HS and uni that was just "do these problems over and over till you remember it" because it just wasn't all that interesting, and this goes for any subject. However some concepts, limits, derivatives and integrals i remember perfectly well (5+ years later) because my math teacher explained how the all fit together, what th…

It's the difference between skills and knowledge. Limits/derivatives/integrals are interesting knowledge so you can discuss them verbally, even if you don't have extensive practice in using them. The basic, core stuff is the opposite. Fractions, bedmas, etc. you don't care enough to talk about, but if I handed a problem that used those concepts you could do it in your head, because they're skills that have been devel…

There is quite a lot to say about 'common factoring.'

For example, nobody knows whether there is an efficient algorithm to factor integers (where efficient depends on the number of digits in the integer). However, there is an efficient algorithm to factor polynomials, and you can write all irreducible polynomials of a certain degree as factors of one mother polynomial.

Moreover, being able to factor integers efficiently would allow you to break a lot of encryption schemes.

Re: How our 1,000-year-old math curriculum cheats America's kids

#94

Post by j2kun » Tue, Mar 04, 2014, 4:23 PM UTC

ghaff wrote: post #28

j2kun wrote: post #21

Earlier quoted context omitted.

This is the viewpoint of an engineer. The salient difference in thought is that the goal is rarely to be right (initially), but to get insight about mathematical stuff. Mathematicians do (and students should) focus on proofs over computations, and there is an extreme amount of subjectivity there. If you produce a false proof, it can nevertheless be beautiful and yield fantastic insights. Likewise, a correct proof can…

I guess I have an engineering viewpoint as well--which may not be surprising because I, well, was one. I never especially connected to geometry with its proofs in high school. What I did learn was lots of basic trig and algebra and the beginnings of calculus (which I took more of in college). The result is that while I've never had much of an interest or flair for advanced math, I did develop most of the math skills…

The question is what are you trying to develop in your students: factual knowledge or mathematical thinking skills? I think it's the latter, and if that's the case then proofs are by far the best way to do it.

And do you really think students would be any more motivated to learn about accounting than they are about current math? I mean, come on, nobody wants to do that stuff. It's a chore even for adults.

Re: How our 1,000-year-old math curriculum cheats America's kids

#95

Post by j2kun » Tue, Mar 04, 2014, 4:24 PM UTC

teach wrote: post #34

j2kun wrote: post #16

Earlier quoted context omitted.

Low standards for math teachers is a huge concern, and Lockhart addresses it too, I believe. Some possible ways to help: give teachers less class time and more time to prepare interesting lessons, pay teachers more, raise the standards of training and hiring.

And that's the real problem: those things all cost money. Quite a bit, actually.

I can't think of a more worthy cause, and it would be chump change compared to what we spend on bailouts/wars/petty politics.

Re: How our 1,000-year-old math curriculum cheats America's kids

#96

Post by j2kun » Tue, Mar 04, 2014, 4:32 PM UTC

dalke wrote: post #41

j2kun wrote: post #23

Earlier quoted context omitted.

Any attempt to force a particular mathematical subject matter on students will undoubtedly be ruined by a committee.

Such cynicism. What do you have against Feynman's presence in the California committee to select, among other things, math books? Ahh, I see. If you are the person behind "Math ∩ Programming" then I can see what you have against that committee. Feynman wrote, arguing against new math: > It will perhaps surprise most people who have studied these textbooks to discover that the symbol ∪ or ∩ representing union and inte…

The reason new math failed is because it was organized and administered by committees (comprised of people who don't do math). So your example is humorously evidence to my claim.

What does a committee do when they are tasked with designing a curriculum or standards? They identify what should be tested, and then they pick the order that the topics are taught, likely outlining a time constraint on how long should be spent on each topic.

This is the problem. If you're looking for things to test your students on then you will invariably add pointless content.

Physicists are no exception (they invented the elaborate and pointless notation for vectors, the bra and ket). I don't propose we test students on mathematical notation, not at all. I propose we actually have them do mathematics (craft proofs, make conjectures), and I assert that any committee trying to craft a curriculum would ruin that. They have already ruined proofs in high school geometry.

Re: How our 1,000-year-old math curriculum cheats America's kids

#97

Post by dalke » Tue, Mar 04, 2014, 7:53 PM UTC

j2kun wrote: post #96

dalke wrote: post #41

Earlier quoted context omitted.

Such cynicism. What do you have against Feynman's presence in the California committee to select, among other things, math books? Ahh, I see. If you are the person behind "Math ∩ Programming" then I can see what you have against that committee. Feynman wrote, arguing against new math: > It will perhaps surprise most people who have studied these textbooks to discover that the symbol ∪ or ∩ representing union and inte…

The reason new math failed is because it was organized and administered by committees (comprised of people who don't do math). So your example is humorously evidence to my claim. What does a committee do when they are tasked with designing a curriculum or standards? They identify what should be tested, and then they pick the order that the topics are taught, likely outlining a time constraint on how long should be sp…

"evidence to my claim"? But it's an example of a committee rejecting new math, not proposing it. And do you seriously want to argue that Feynman doesn't "do math"? Actually, it looks like you do. Why should only a certain type of math practitioner determine the curriculum? Do we only let people like Ramanujan set the curriculum? 'Cause that won't work.

What you're complaining about is called "democracy." Yes, democracy is messy. The current movement towards high-stakes testing in the US is anti-democratic in part because it coalesces more power into fewer people. If you don't want a committee then what do you want? A so-called "education czar" with absolute power to make things work the way you think it should?

Re: How our 1,000-year-old math curriculum cheats America's kids

#98

Post by sitkack » Tue, Mar 04, 2014, 10:15 PM UTC

SamuelMulder wrote: post #80

sitkack wrote: post #75

Earlier quoted context omitted.

Is your work done in the public? It would be nice to put it on a public repo and get others feedback and help.

I'm still struggling with how to handle this. On the one hand, it makes a lot of sense for it to be public. On the other, I've given up a lot to spend time developing it (including going part time on my day job), and I would love to be able to devote more time to it, which would require making some money off of the material in some way. I've been thinking about doing a kickstarter, but I'm interested in exploring oth…

Just checked out your site, it is like you are brother from different mother. Awesome stuff. I have a project I am working on that I will ping you with to get your feedback, I think you will like it.

Re: How our 1,000-year-old math curriculum cheats America's kids

#99

Post by SamuelMulder » Mon, Mar 10, 2014, 4:16 AM UTC

sitkack wrote: post #98

SamuelMulder wrote: post #80

Earlier quoted context omitted.

I'm still struggling with how to handle this. On the one hand, it makes a lot of sense for it to be public. On the other, I've given up a lot to spend time developing it (including going part time on my day job), and I would love to be able to devote more time to it, which would require making some money off of the material in some way. I've been thinking about doing a kickstarter, but I'm interested in exploring oth…

Just checked out your site, it is like you are brother from different mother. Awesome stuff. I have a project I am working on that I will ping you with to get your feedback, I think you will like it.

I just realized it might not be that easy to find my contact info. I added it to the website, but it is my username here, no caps, at gmail.com