Did a PhD and post-doc in maths and also have a 10 and 12 year old so have been interested in watching them and their peers learning mathematics at school and have even been in to help at school a little bit.
I think understanding and problem-solving are different, although the former is a prerequisite for the latter. I can imagine understanding areas of mathematics but not having a good ability to solve novel problems.
Abstraction is a massive part of mathematics. Some people seem to find abstraction much easier than others. I don’t know to what extent it is a learnable skill, but I think without it more advanced mathematics is almost impossible.
Also some people (and I see this in children) seem to have real blocks with aspects of logic. The difference between ‘if’ and ‘only if’ seems to not exist for some people and yet is obvious for others. The idea of mathematical induction likewise I remember really stumping some of the other students in my class. Again, I don’t know how learnable this is!
I also suspect there are certain mental models related to number and geometry one acquires early on that either make aspects of maths very easy or very hard depending on whether one acquires them. Jo Boaler’s work in this is interesting—I like her ‘Mindset Mathematics’ activities for school-level mathematics.
I suspect most people who don’t get some of these things naturally lose motivation for mathematics making it hard to know how learnable they are. There are examples though of successful research mathematicians who weren’t like that, Mirzakhani being an obvious example. Mathematicians do gravitate towards different fields as well—I do know some mathematicians who don’t like geometry.
I do find it interesting how from a very early age, there are huge differences in how much different children ‘get’ mathematics. Either there is something genetic going on or is there are critical things one can do in those first few years that make a difference! I’m sure working memory helps, but I’m inclined to suspect that it is not just that. I have a reasonably good working memory but it wasn’t one of my strong points. I was fine at mental arithmetic but not the best person in the class for example, but by the time I was 16-17, I was way better at mathematics than anybody else at school.
Did a PhD and post-doc in maths and also have a 10 and 12 year old so have been interested in watching them and their peers learning mathematics at school and have even been in to help at school a little bit.
I think understanding and problem-solving are different, although the former is a prerequisite for the latter. I can imagine understanding areas of mathematics but not having a good ability to solve novel problems.
Abstraction is a massive part of mathematics. Some people seem to find abstraction much easier than others. I don’t know to what extent it is a learnable skill, but I think without it more advanced mathematics is almost impossible.
Also some people (and I see this in children) seem to have real blocks with aspects of logic. The difference between ‘if’ and ‘only if’ seems to not exist for some people and yet is obvious for others. The idea of mathematical induction likewise I remember really stumping some of the other students in my class. Again, I don’t know how learnable this is!
I also suspect there are certain mental models related to number and geometry one acquires early on that either make aspects of maths very easy or very hard depending on whether one acquires them. Jo Boaler’s work in this is interesting—I like her ‘Mindset Mathematics’ activities for school-level mathematics.
I suspect most people who don’t get some of these things naturally lose motivation for mathematics making it hard to know how learnable they are. There are examples though of successful research mathematicians who weren’t like that, Mirzakhani being an obvious example. Mathematicians do gravitate towards different fields as well—I do know some mathematicians who don’t like geometry.
I do find it interesting how from a very early age, there are huge differences in how much different children ‘get’ mathematics. Either there is something genetic going on or is there are critical things one can do in those first few years that make a difference! I’m sure working memory helps, but I’m inclined to suspect that it is not just that. I have a reasonably good working memory but it wasn’t one of my strong points. I was fine at mental arithmetic but not the best person in the class for example, but by the time I was 16-17, I was way better at mathematics than anybody else at school.