Comment by ArtWomb | Hacker News Reader

Comment by ArtWomb | Hacker News Reader

This is wonderful. I love when memes breed philosophy. We're teaching math wrong. It's as plain as day to me. Or rather we teach the wrong maths to the wrong students. If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better.

As to the epistemological quandary, I always return to Geometry or Visual Mathematics. Consider the infinite ways of calculating the digits of PI. In "Playing Pool with PI", Galperin demonstrated an analog computer that could calculate the digits of PI using only the perfectly elastic collisions of two billiard balls on a pool table. It works for all integral base number systems, and can even be extended to irrational numbers. And recently Google Brain's Adam Brown made an equally stunning observation: Galperin's bouncing billiard balls look identical to the Grover Quantum Search algorithm!

Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search

https://arxiv.org/abs/1912.02207

Personal anecdote: I never understood what math "was," and this totally demotivated me to study it. I got A's all the way through high school mathematics because I was a systematic and logical thinker who had no problem following rules or algorithms, but I never once saw the point. When I reached college, I had already tested out of Calculus and never took a formal math course again.

Fast-forward ten years. I am a self-taught programmer and now find mathematics absolutely fascinating and beautiful, and I see its applications. So I'm taking math classes on Udemy, buying hundreds of dollars of textbooks, and trying to teach myself.

I can't help feeling like I got a little robbed. I understand that teaching is hard, and some kids just don't care to learn certain things and it's best for them and society to force some absolute fundamentals down their throats. But at a certain point around middle or high school, I think there needs to be a hard pivot to focusing on motivating students. If you do that well, they will have the ability to teach themselves and learn for the rest of their lives. If you don't they either end up like me, playing catch up a decade too late, or they just leave entire fields of knowledge untapped forever. With some subjects that's inevitable and probably ok, but math doesn't seem like it should be one of those subjects.

What do you think would convince you in the high school that math was important and worth studying?

I think three things would have done it:

1. A clear explanation of a symbol system and how math is a symbol system to deal with issues of quantity, shape, structure, and logic.

2. Frequently converting between diagrams/drawings and symbols (show "x x x" at times and "3" at other times). I actually have a concept for a game I want to program that does this.

3. Showing multiple concrete applications of different ideas, and lots of word problems. In retrospect I think this was always reversed in the American system. We would learn the symbolic and algorithmic aspects of a math concept, then use it to solve some word problems. Once I went back and started teaching myself some of this stuff, I read that the Russian system is quite the opposite: lots of word problems, multi-faceted problems. I suspect I would have learned from that style a lot better.

EDIT: One more point I should make. I scored in the 85th percentile on my SAT math. It is probably not a great sign for our system of education that I could do so well with nothing but the most rudimentary understanding of what I was doing.

Interestingly, much of what you describe is present in old mathematical texts. For example, Fibonacci's Liber Abaci is essentially a long succession of story problems. Diophantus's Arithmetic is a succession of concrete problems -- no story -- but it's concrete prior to abstract as you suggest.

Euclid II also develops most of Algebra geometrically, so there's much conversion between shape and symbol baked it. His method of computing the product of binomials is far superior to what's commonly taught.

So there's a good tradition of the thing you're looking for!

I liked math in high school and usually saw the underlying meaning of the concepts. Teachers generally aren't idiots (or at least mine weren't), they try to explain it best they can, and start with that explanation. I remember explaining the meaning to friends that had only tried to memorize the rules without bothering to consider what they meant. They were surprised to learn these meaningful explanations, but in fact I was just repeating what the teacher had told us. I think the difference is I came into the class knowing what I should, while they had gaps in their prior teaching that they had to compensate for with memorization. So I was ready to hear the motivation at the start, but others weren't ready yet until they had memorized the concepts as a kind of crutch.

What you want is methodology like "just in time learning" or "problem based learning", which is very powerful but also very slow. You have to greatly reduce the content you can cover in a class. It's good for some students, boring for the smart ones (who solve your fancy word problems instantly), and the worst ones don't pay attention or do homework either way.

I'm not really convinced that's the case. As I mentioned, the Russian system is apparently very much like this, and they don't seem to have a problem producing large numbers of advanced quantitative thinkers. I also take a little exception to the implication that I wasn't one of the "smart ones." I had no problem doing what I was taught, I got good grades at at an advanced prep school, and I scored just fine on standardized tests. I basically succeeded according to every standard set by the system, and yet I wouldn't say I really learned very much. To me that signifies a real problem. I certainly deserve some of the blame, but definitely not all of it.

You're not convinced of what exactly? That alternative methods reduce content? That is precisely what you are instructed to do by pedagogy researchers giving advice to teachers.

Asian education systems like India, China, Japan are very rigid and based on rote practice and memorization. "Prep school" there means evening schools where they just go do yet more memorization and rote practice to help pass entrance exams. Yet they produce many brilliant mathematicians too. Of course the students are motivated by parents and society, not love of cramming.

The ideal would be teaching that is individualized and does the best it can to get through to each student (there are plenty of efforts and even "edTech" that tries to do this, especially at lower grades). Without that it's always a trade-off in which students get the most value for the time spent.

You said it's "boring for the smart ones," which I took to mean that if we go by the methodology I'm suggesting, we'll bore all the smart students, they'll have less knowledge, and we'll produce fewer good ones. Maybe you didn't mean all of that, but if you did, my point was that the Russian system, which seems to operate this way, seems to do just fine in keeping good students motivated and producing lots of talented mathematicians. I'm not convinced volume of content = engagement and better outcomes, either for "the smart ones" or the rest.

What convinced me was the book Goedel, Escher, Bach, by Douglas Hofstadter. It was my first exposure to the more abstract and playful side of math, and motivated me to become a math major in college.

I was "good at math" in high school, to the extent that I could crunch through any problem in the math or physics textbooks. School math existed mainly to serve the science and engineering students. Now I was happy to use math, for instance in my electronics and programming hobbies, but that wasn't math as an end unto itself.

But proofs are what really made math come alive for me. I don't think this makes me a freak. A lot of people I know from my generation said that their favorite high school math course was geometry, which was heavily proof oriented at the time.

Sadly, the contemporary K-12 math curriculum is sorely lacking in proofs.

Now, what about the students who would be more motivated by usefulness than by abstract theory? For those students, I suggest looking at how people in the so called "real world" actually do math, and work backwards from there. My days of abstract math are behind me, but I am still one of the "math people" at my workplace, in an industrial R&D department. When I do math, I'm never far from a computer. Yet school math is still taught by mainly pencil-and-paper methods, even if those have been translated to online forms.

For many of my math tasks, I start by playing with numbers, e.g., computing a function and graphing it in a Jupyter notebook. It brings tears to my eyes that K-12 students are not exposed to this. For one thing, it's fun, and it's an honest portrayal of how people actually work. For another, you can brute force your way through a problem even if you've forgotten some particular formula, making it more likely for someone to use it later on. Graphing calculators are of course a thing, but they hardly go far enough.

Interestingly (I'm one of ther guys self-teaching as an adult), this is exactly how I'm learning. I'm doing a course that uses Jupyter to cover all of the major areas up to linear algebra. I also aced my Geometry class in high school exactly because of what you said: it gave me some of the flavor of proofs and "real math," which was fun and made me motivated. I remember at the time being sorely disappointed when no other math classes were like that again and figured it was some kind of lucky fluke.

I think you're onto something, because teaching myself programming was precisely my backdoor into math. I remember a line in the Clause Shannon biography "A Mind at Play" that said he was essentially the same way: he wanted to apply his mathematics and understand the proofs/structure behind it, but essentially (as the title suggests) it was a form of play.

I'm glad you wrote this comment since I think this happens to almost everyone in the current system to some degree. It's sad that even those who see the beauty in mathematics are basically forced to rote learn algorithms to recite them later. It's no wonder that maths, school, and boredom are synonymous for a lot of people.

Which courses/books are you buying? I am on a similar path and would like to know more.

> If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better.

Sorry, but following this logic we will end up with a conclusion that almost no school knowledge is "useful". How useful is playing volleyball? Reading poetry? Painting? Studying history? Biology would be reduced to first aid and basic medicine. I have those discussions with my daughters who learn how to do long multiplication. How is it useful if even I who know how to do it would rather use a calculator on my mobile instead? We would reduce math to basic operations on a calculators, probably nothing more would be considered useful by the majority of the population. The same apply to all other subjects.

I think "application" is the wrong focus. The real thing to help students understand is this exact question: what is math? That remains a mystery to most students. It gets conflated with the problem of application, but I think the real problem is most educations don't make clear the idea of abstraction, symbol systems, and logic. When you don't understand something, especially why it might be useful or at least intrinsically interesting, it's hard to get motivated about it.

But this exact question has no answer or has many different, contradictory answers as this discussion shows. I doubt that such discussion would be interesting for someone who struggles with algebra. Moreover the same questions could be asked about all subjects. The point of the school is to show kids different things even if they cannot see their usefulness right away. One person will enjoy math, other poetry or history. But all will have a chance to try different things and to choose what to study deeper. Of course it's nice to give some motivation and her teachers failed if she cannot see applications of basic algebra.

I responded to the comment above with some examples of the kind of thing I'm hoping for. I don't think a deep philosophical discussion about math is necessary. Just reinforce the idea that math is a symbol system for dealing with questions of shape, quantity, and logic, and give multiple examples of applications of the same concept. Also convert between symbols and diagrams/drawings frequently (for example, I'm willing to bet an absurd number of Americans can't understand how a virus spreads exponentially; if you draw a pyramid of infected people, I bet they'd get it immediately; but instead we use curves and lines that don't connect with people).

Reply on news.ycombinator.com