The Nonlinear Library LW - What makes teaching math special by Viliam
Dec 18, 2023
17:07
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: What makes teaching math special, published by Viliam on December 18, 2023 on LessWrong.
Related:
Arguments against constructivism (in education)?
Seeking PCK (Pedagogical Content Knowledge)
*
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, 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.
*
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 quadrati...
Related:
Arguments against constructivism (in education)?
Seeking PCK (Pedagogical Content Knowledge)
*
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, 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.
*
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 quadrati...