What makes teaching math special

post by Viliam · 2023-12-17T14:15:01.136Z · LW · GW · 27 comments

Contents

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Related:

• Arguments against constructivism (in education)? [LW · GW]
• Seeking PCK (Pedagogical Content Knowledge) [LW · GW]

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Designing good math curriculum for elementary and high schools requires one to have two kinds of expertise: deep understanding of math, and lot of experience teaching kids. Having just one of them is not enough. People who have both are rare (and many of them do not have the ambition to design a curriculum).

Being a math professor at university is not enough, now matter how high-status that job might be. University professors are used to teaching adults, and often have little patience for kids. Their frequent mistake is to jump from specific examples to abstract generalizations too quickly (that is, if they bother to provide specific examples at all). You can expect an adult student to try to figure it out on their own time; to read a book, or ask classmates. You can't do the same with a small child.

(Also, university professors are selected for their research skills [LW · GW], not teaching skills.)

University professors and other professional mathematicians suffer from the "curse of knowledge". So many things are obvious to them than they have a problem to empathize with someone who knows nothing of that. Also, the way we remember things is that we make mental connections with the other things we already know. The professor may have too many connections available to realize that the child has none of them yet.

The kids learning from the curriculum designed by university professors will feel overwhelmed and stupid. Most of them will grow up hating math.

On the other hand, many humanities-oriented people with strong opinions on how schools should be organized and how kids should be brought up, suck at math. More importantly, they do not realize how math is profoundly different from other school subjects, and will try to shoehorn mathematical education to the way they would teach e.g. humanities. As a result, the kids may not learn actual mathematics at all.

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How specifically is math different?

First, math is not about real-world objects. It is often inspired by them, but that's not the same thing. For example, natural numbers proceed up to... almost infinity... regardless of whether our universe is actually finite or infinite. Real numbers have infinite number of decimal places, whether that makes sense from the perspective of physics or not. The Euclidean space is perfectly flat, even if our universe it not. No one ever wrote all the possible sequences of numbers from 1 to 100, but we know how many they would be.

If you want to learn e.g. about Africa, I guess the best way would be to go there, spend 20 years living in various countries, talking to people of various ethnic and social groups. But if you can't do that... well, reading a few books about Africa, memorizing the names of the countries and their capitals, knowing how to find them on the map... technically also qualifies as "learning about Africa". This is what most people (outside of Africa) do.

You cannot learn math by second-hand experience alone. Imagine someone who skimmed the Wikipedia article about quadratic equations, watched a YouTube channel about the history of people who invented quadratic equations, is really passionate about the importance of quadratic equations for world peace and ecology, but... cannot solve a single quadratic equation, not even the simplest one... you probably wouldn't qualify this kind of knowledge as "learning quadratic equations".

The quadratic equation is a mental object, waiting for you somewhere in the Platonic realm, where you can find it, touch it, explore it from different angles, play with it, learn to live with it. Only this intimate experience qualifies as actually learning quadratic equations. (Even if you are completely ignorant about who invented them in what century. That is an interesting trivia, but ultimately irrelevant.)

Second, math is not about social conventions and historical accidents. We can decide (individually, or collectively as a civilization) somewhat arbitrarily which mathematical objects we pay attention to, but once we made that choice, the rest is given and we need to figure it out. Once you decide to use natural numbers, 2+2 is already 4, even if no one knows it yet.

The only way to learn that Paris is the capital of France, is to obtain that information from a credible source. The facts that Paris or France exist are historical accidents; they may not exist in parallel universes. The fact that Paris is the capital is also a historical accident; given different history, it could have been some other city. Even the fact that we have the concept of "capital city" is a social convention.

The only way to learn how "cheese" is spelled, is to obtain that information from a credible source. Yes, it comes from Latin "caseus", but that just passes the buck, and also doesn't explain why the spelling has changed this way rather than some other way. Or why by a historical accident English didn't import some other nation's word for cheese -- perhaps there was no sufficiently impressive cheese empire nearby, but in a parallel world that could have happened.

This is why I wouldn't trust a geography teacher, or an English teacher, to design a math curriculum. They would probably unconsciously import the assumptions valid in their subjects, which would be wrong for math. I would expect them to teach mathematical theorems as arbitrary rules to be memorized, rather than show why they are inevitable given the nature of the mathematical object.

By the way, did you memorize $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ or did you derive it? If you forgot the equation, could you re-derive it [LW · GW] without the help of internet? I am asking because this was the point when my personal nullius in verba hit the wall at high school. I despaired: "oh no, here comes the point when I stopped understanding math; from now on I will only be able to memorize it, just like everyone else does". Luckily, it turned out that the following math lessons made sense again, and a few years later I re-derived that equation. And maybe this specific equation is not so important in a larger picture, but generally, memorizing instead of deriving is a bad habit. If you memorize too many things, you will forget them soon after you stop using them actively. Also, each memorized theorem locks the door to something else; if you can't solve the quadratic equation (and you have internalized that as "not necessary"), you are not going to solve the cubic equation either. How many other doors have you already locked?

(I am not commenting on subjects like physics, because I am not an expert there. It seems to me that physics has a lot of real-life knowledge and a lot of mathematical models, so designing a good physics curriculum would probably be even more difficult. Or maybe easier, because you could choose one or the other for elementary school, and leave the rest for the older students? I honestly don't know. Anyway, university professors of physics will probably have the same problems as university professors of mathematics.)

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So, here comes the love-hate relationship with constructivism.

On one hand, some constructivist approach in education is necessary, because learning math is what constructivism in the sense of Piaget and Vygotsky was about, probably more so than other subjects. You internalize the mathematical objects. Merely looking at them (memorizing the definitions and theorems) is not enough; you need to digest them (explore the mathematical objects in different contexts, find their relations to other mathematical objects you know, explore different cases, explore the what-ifs).

Notice how LLMs suck at math, by the way. IT industry has created a humanities polymath before it could create a machine with 3-rd grade math knowledge, LOL. One of the reasons is that mathematics is not about knowing the facts, but using the tools.

On the other hand, constructivism seems hopelessly infested by the plague that calls itself "radical constructivism", or more frequently just "constructivism". The kind where people believe that 2+2 is a mere social convention, and might equal to anything else, if different people invented math instead. The current discourse seems to focus on the false dilemma between something called personal constructivism and social constructivism, where the former seems to assume that kids can derive all knowledge without any help from outside, and the latter seems to assume that all conclusions are arbitrary and therefore the kids can only learn them by memorizing. (Or they should be brave enough to reject social conventions and choose their own truth, because there is no such thing as truth anyway.) I keep saying "seems" because I don't really understand most of this, and the more I find out, the more it makes me depressed. Asking GPT-3.5 to summarize their position, radical constructivism:

• rejects the idea of an objective reality that exists independently of the observer
• reality is considered to be a mental construct
• In education, the emphasis is on fostering an awareness of the subjective nature of knowledge. It suggests that educators should recognize and respect individual interpretations and avoid imposing a single, objective understanding.

Ok, fuck that. You definitely do not want those guys to design a math curriculum. (Or anything else.) But it seems inevitable that they would insert themselves into the debate. It seems also inevitable that you will be associated with them once people sense that you have something to do with constructivism (even in its Piagetian/Vygotskian form).

But, you know, constructivism (Piagetian/Vygotskian) is basically this:

• learning by actively solving problems, thinking and talking about them
• using specific tools (think: Montessori) that help us understand abstract ideas
• students create mental models [? · GW] of the world, and try to fit the knowledge there
• students can figure out things within small inferential distance [? · GW] on their own
• so the textbooks should be arranged accordingly
• learning is supposed to be a pleasant experience

And by negating these things, you get:

• students learn by memorizing abstract statements they do not really understand
• and as a consequence, they are stressed and feel stupid

And I think most people who are good at math would agree that it should be like the former, and most people who hate math would agree that their lessons at school actually were more like the latter.

So we just cannot ignore this topic.

As of now, the state of the art of debating math education seems to consist of mutual yelling between math professors who say "the abstractions that I work with in my daily job are the real, high-status math, and if the kids don't get it, it's their fault, not mine; I don't mind if most of them grow up hating math, as long as a few talented students get it anyway (frankly, usually from sources other than school), and those can become my PhD students", and those who should properly be called New-Age Math activists who say "dude, math isn't even real, so how could you teach it?".

We need to have a different kind of talk.

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Let's address the false dilemma between "personal" and "social" more closely. Math as such, is impersonal. (You could solve equations alone on Mars.) But the process of learning math is inevitably social. (A child couldn't learn math alone on Mars, without teachers, without textbooks.) You learn about most mathematical concepts because other people care about them. Only a few people invent their own math, and even that only comes after they learned the existing one.

Furthermore, some aspects of how-humans-do-math are grounded in our social experience. Consider the notion of "proof". Now that you have sufficiently internalized it, it can be a solitary activity of checking something thoroughly. But the psychological base this was built on is "words that would convince another human". First you learn to prove things to others, and only after you get good at it, you can reliably prove them to yourself even when no one is around. (More generally, as Vygotsky would say, first you learn to talk, then you learn to think.) You cannot reinvent the math without the social experience of doing math together. And if you are doing things together, but those things aren't math, you cannot reinvent it either. But if you learn the math together with others, you can continue to explore more of it alone. And the results you derive -- if your math was correct -- will be the same as the results anyone else derives (no matter their race or gender; sorry for dragging politics into this, but other people already do so).

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Perhaps the idea of "learning math" is itself a bit confused. There is no such thing as knowing all math -- even Terence Tao doesn't know everything about math, no offense meant. Once we accept that everyone's knowledge of math is limited, we can talk about the trade-offs such as "knowing more things, superficially" and "knowing fewer things, deeply". (Though we can agree that knowing very little, superficially, is bad.)

This is in contrast with e.g. language education, where there are natural Schelling points; for example, once you learn to speak Spanish as well as a native Spanish speaker, we might consider your learning of Spanish complete. (Even if there are still some obscure words you don't know, and you could still study etymology of everything, etc.) Reading comprehension, similarly: once you can read a story and you understand it completely, you are done.

There are no such natural limits in math. Categories of "elementary-school math" and "high-school math" are arbitrary. Should you learn the basics of calculus at high school? I guess, if your country teaches it that way, you will probably say "yes", and if your country doesn't teach it that way, you will feel that calculus is clearly university math. We can agree that addition and multiplication belong to elementary schools, but at the boundaries it gets blurry.

I am saying this because no matter what curriculum you design, there will always be someone complaining that you could have added an extra topic or two, if you only didn't waste so much time practicing the trivial topics. And therefore, your curriculum sucks.

But of course, if you comply with that, people will notice that your students learn a lot, but they also forget a lot, quickly. And they are stressed, because there is so much information, and so little time to understand it. Therefore, your curriculum sucks, too.

Saying "we could test the mathematical knowledge" just passes the buck. Does your test include a lot of superficial knowledge from diverse parts of math? Or does it torture you with complicated examples from a specific part of math, and ignores everything else? (Who decided which part of math, and why?) This is how the tests can drive the curriculum.

And of course there is correlation between math knowledge and success at test. The problem is that at the extremes the tails come apart. Kids that are better at multiplication will be better at the tests. But the kids who score maximum at the tests and the kids who win math olympiads, are probably two distinct groups. I am saying this as a former kid who did well at math olympiads, but often got B's at high school, because of some stupid numeric mistake. The opposite example, I imagine, is some poor kid whose parents insist he or she must drill dozen textbook exercises before breakfast, so the kid always gets the A's but will probably never have a single original thought about math.

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Okay, time to conclude this rant. Is there a conclusion to be made?

I think if you want to fix math education, you need to start with lower grades first. Because the higher grades are constrained by what knowledge the kids already come with. So you should fix the elementary-school math first, then the high-school math, and maybe then the university math.

You should be looking for people who:

• have a PhD in math, or won a math olympiad, and
• were teaching (1) math (2) at an elementary school (3) for the last ten years, and (4) their pupils don't hate them -- all four points are non-negotiable

You probably won't find many of them. They might all fit into one room, which is exactly what you should do next.

Then you should pay them 10 years of generous salary to produce a curriculum and write model textbooks. You need both of that. (If you let someone else write the textbook, the priors say that the textbook will probably suck, and then everyone will blame the curriculum authors. And you, for organizing this whole mess.) They should probably also write model tests.

The system should have a lot of slack:

• the curriculum should only cover 80% of the available time, so that the teachers can make their own decisions how to use the remaining 20% (whether it means learning something outside the curriculum, or repeating the topics where the students struggle most)
• the first half of the first year at high school (and wherever else your school system has lots of students moving from one institution to another) should be spent practicing the stuff the students were supposed to learn during the previous grades (because chances are many of them actually didn't)
• tests should be written in such way that learning 80% of the curriculum gives you 100% at the test

Everyone else needs to shut up. There will be many university professors who never taught little kids who will provide "useful suggestions" (such as making the math more abstract, or not wasting time giving too many specific examples). There will be many self-proclaimed educational experts who will provide "useful suggestions" (such as memorizing more, and reminding the kids that math is just a social convention invented by the evil white men). These should not be allowed to disturb the people in the room.

The authors should be provided books written by Piaget and Vygotsky (and Montessori, Papert, etc.), but ultimately, the decision is on them. If something doesn't feel right, just ignore it. Only use the books for inspiration.

Of course, you need to test the lessons with actual kids.

If it seems like the kids are having too much fun, you are going in the right direction. Keep going. (As long as it is still math; but given the selection of the people in the room this should not be a concern.)

You do not need to hurry. Build your house on solid foundations. Time is not wasted by making sure that students actually understand the lesson before moving on to the next one.

At the end, you win if kids following these lessons will understand the math and love it.

And then, of course, the new curriculum and textbooks should be tested on a randomly selected sample of all schools.

Oh, and don't expect too much praise. Most people will complain anyway, either because this is not the math education as they know it, or because this is not the math education as they would have designed it. Expect to get a lot of complaints, often mutually contradictory. (Your new curriculum is simultaneously too easy and too hard. It either only works for the gifted kids and not for the average ones, or vice versa. It is simultaneously an irresponsible experiment and "just common sense, nothing special".)

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Some ideas were taken from Ladislav Kvasz's Princípy genetického konštruktivizmu (Principles of genetic constructivism). The article is in Slovak; as far as I know there is no English translation. All mistakes and exaggerations are my own anyway.

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