Learning algebra in my 60s

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1. Although when I think about educational malpractice, I remember a program I was teaching in where the director asked me about why the numerator and denominator were called what they were and I explained the denominator identified what kind of fraction (and made the analogy to denominations of currency) and the numerator was how many (the “number”) of that kind of fraction. She then, that same day, gave a talk to the assembled teachers saying that she’d asked me and I’d said that’s just what they were called. I nearly stood up and screamed in anger. If I hadn’t desperately needed the money, I would have quit on the spot.

For the others you'd have to do the mental work to append a 1 regardless, might as well stick with decimals

This is pretty easy to do for a polynomial term, but much harder for x raised to a non-integer power[1], or sin(x). There is no one way to "derive the formula for a derivative"; different functions have to be analyzed differently.

[1] Wikipedia suggests that the simplest way to derive this formula is to begin by establishing that exp(x) is its own derivative. But that's not an approach I'd be likely to take with students new to calculus.

Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people.

If we skip the medieval theories, one of the relatively modern ones was called "associationism". The theory was that human mind is basically a set of associations. We are born with zero associations; as we observe the world, we learn to associate this with that; and after many years we have learned to associate things properly and now we are smart adults. For extra nuance, some of us form new associations faster than others, probably for biological reasons; that is what intelligence is. Anyway, associations are all there is.

Building an education theory on associanism is quite easy. You need a teacher who understands the subject (has the correct associations, a lot of them). Then the teacher stands in front of the classroom and keeps talking. The more he talks, the more associations the students can make. That's all there is. -- The order of lessons is not relevant; ultimately, after the teacher mentions everything, all associations will be properly connected into one large network; until then, you have to memorize. You don't wait until the students "understand", that would be a waste of time (there is no such thing as "understanding", you either have the right associations or you don't); the more you keep talking, the more associations the students can make. Of course you can (and should) repeat the facts, that's how the associations are deepened. But after a while, if some students don't get it, they are just hopeless: they had the opportunity to make the right associations, and yet they failed. It is their fate to remain farmers.

This is a bit of a strawman, and yet many teachers follow this method intuitively, even without knowing the underlying theory. And their students complain that they don't get it. And the teachers reply that yeah, some students are just talented and some are not, "the camel has two humps", et cetera.

In psychology, the next step after associationism was Piaget's "genetic epistemology". Where "genetic" is an adjective for "genesis", not the DNA. In modern language, we would probably call it "developmental epistemology", i.e. the study of the ontogenetic origins of understanding. The revolutionary approach was to watch how kids actually learn, rather than trying to shoehorn everything into a simplistic framework. One of the interesting findings was that kids actually do not make linear progress from "zero associations" to "correct understanding", but the process often takes a detour through a phase of magical thinking or some other kind of wrong understanding. Instead of "no opinion -> correct opinion", it is often "no opinion -> wrong opinion -> correct opinion". There are specific examples, not important now. Also, Piaget got some things wrong; this was later improved by Vygotsky. The important thing is the idea that the child is making mental models of the world. It is not just associations floating in a vacuum; the child has a paradigm, and tries to fit the new knowledge in that paradigm, and sometimes it doesn't work and the paradigm changes into a better one.

An educational theory built on this, originally called "constructivism", says that the teacher should not just keep saying random true facts, but also check that the students have the right models. This is achieved on one hand by making the models explicit, saying the facts in proper order, putting them in the right context... and on the other hand by checking the students' models, finding the problems and fixing them. If you find out that many students keep making the same mistake, you should adjust your way of teaching accordingly: make it obvious at the beginning that it is X not Y, maybe change the order of lessons so that making the correct model becomes easier. Keep checking the students' models regularly, because the sooner you find the mistake, the easier it is to fix it. Etc.

This is what many good tutors do intuitively, because if you teach 1:1, there is more interaction, and it is easier to catch the mistakes right when they happen and ask "why did you do this?" You do not wait until the student makes the same mistake hundred times to declare him a failure without talent; you notice when the mistake happens for the first time and keep "debugging" until the mistake is fixed. This is easy to do when tutoring; more difficult to make it scale to a classroom full of kids.

And then... there is another thing, I do not know if it has a proper name in psychology, but "postmodernism" is what some people use (and other people object to this usage)... the "edgy" idea that knowledge transfer is actually impossible, everyone lives in their own different reality, trying to teach something is an oppression, if only we left the kids alone they would reinvent the civilization and make it much better (Rousseau's "Emile"). -- For stupid political reasons ("there are exactly two sides of the story, not more, not less", "the enemy of my enemy is my friend"), these people are typically associated with the constructivists, because they both oppose rote memorization. But although the opponent may be the same, the proposed solutions are quite different ("teaching better" vs "not teaching at all").

To increase the confusion, the educational theory build on this was called "radical constructivism", misleadingly suggesting that this might be "something like the famous Piaget, only much more so", when if fact it is something completely different. The kids taught using this philosophy are left alone to reinvent the math... and fail predictably! Or sometimes they are taught dozen different methods how to do addition (bonus point if the method was used by some indigenous population, because, you know, "noble savage", doesn't matter if the specific method only works for adding 7+8 and 8+9), hoping that this will kickstart their math thinking so now they will develop the rest of the math independently. Predictably, that also never happens.

So, how is this related to the "new math"? The curriculum of the "new math" was based exactly on this "radical constructivist" thought, except the authors did not emphasise the "radical" part enough and often just called it "constructivism". So you had the math curriculum that didn't work at all, and was a complete disaster. And when finally people got angry and returned to the traditional math education, the lesson everyone remembered was that "constructivism has been debunked".

So now, whenever someone proposes to teach math in a way that emphasizes understanding over memorization, the kneejerk reaction is "haha, that sounds like constructivism... yeah, we tried that but that didn't work at all", plus a link to some web page that criticizes "new math". And if you try to explain how this is completely unrelated to Piaget, you are dismissed with "yeah right, the true constructivism has never been tried, comrades, hahaha".

And then, ironically, people keep quoting the Feynman's story about how actually understanding physics is better than mere memorization that "light is waves". And the helpless (Piagetian) construstivist is like "yeah guys, that's exactly what I was trying to tell you all the time", but no one cares, and when it comes back to adding some understanding to the lessons, someone inevitably comes with the condescending "haha, but constructivism has been debunked" and a link to Ten Facts Why New Math Sucks.

Possible solution: perhaps the brand of "constructivism" has been thoroughly poisoned, and we need to reinvent it and call it "Feynmanism". Then you can go and say "nope, I am totally not proposing a constructivist curriculum, that has already been tried and debunked, haha, what I am proposing instead is the Feynmanist curriculum", and then people will go online and say "wow, I hated math at school, but then we got a new teacher who used this new Feynmanist method, and the math finally started make sense and now I love it".

See Wikipedia: https://en.wikipedia.org/wiki/Bloom%27s_2_sigma_problem

Anyone who tried both teaching and tutoring knows that the difference is just incredibly large. A part of it is that 1:1 you can pay more attention individually; a similar argument can be used in favor of smaller classes. But the other part is that as a tutor, you are free to actually use your best judgment, while in school it is more of "yeah, I know that I should do X, but the rules say that I have to do Y instead".

The school is just insanely ineffective, for various reasons. You need to follow a predetermined schedule, whether it makes sense for the given classroom or not. Your students are expected to already have some knowledge from their previous grade, and if they don't (which happens quite often) you don't get any extra time to catch up. There are all kinds of disruptions, like students who never pay attention and interrupt you and their classmates during lessons, but you must proceed at the speed that was designed for a hypothetical classroom without disruptions. The school inspection randomly checks whether you follow the latest fad, usually based on some pseudoscience, like whether your lessons are okay for both visual and kinesthetic learners, or whether your math lessons are sufficiently decolonialized.

So the same teacher who fails to teach her class fractions at school, may be a successful tutor during the afternoon and explain the fractions properly.