The bit at the end with his description of the various math classes was pretty funny:
PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.
CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim
I'm a huge fan of Paul's, and his book Measurement is easily the most beautiful introduction to geometry out there.
I appreciate his focus on how to maintain engagement with students who are predisposed to math, but I think it's equally important to consider how to teach students who are deeply uninterested in math but still need a working knowledge to live life. Granted, the current system seems to be failing both kinds of students.
This paper is what got me interested in mathematics. I happened to read it when I was 14 in geometry class, and it fundamentally reshaped math from something that I happened to be good at to something I was deeply fascinated by and motivated to do more of
> The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.
> ..The fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.
I run a microschool and also teach maths there. I love this essay, and book, but by focusing on maths as art, it focuses on only one aspect of meaning. The larger issue with the way maths is taught, and especially maths, is that it is all mechanics. Now for a certain set of students, those mechanics are incredibly beautiful. Prime numbers have a seduction to them! A few more can be introduced to this beauty and aesthetics, but many are still not into it, and that's entirely okay. What's needed for kids to see and start accepting it is meaning. And meaning comes from connection to the real world. This is why kids who are unschooled but work in shops become great at arithmetic. It's part of their daily web of life. No amount of sanitized exposure to abstract aspects, however aesthetically dressed up will help most. The mind needs to answer "why do i need to care about this" to be open to learning. And this is not something that can come with instruction. It is lived experience, culture. To fix maths education,and any education, you need to bring in meaning before mechanics. Unfortunately the curriculum, assesment, and therefore teaching inevitably end up with focus on the mechanics, which is the wrong end.
It is meaning > motivation > mechanics > measurement
I also love A Mathameticians Lament. I did not know about Benezet so thanks for that. I found your blog post interesting too.
A lot of this reflects the approach I took to teaching my kids maths (and everything else), both before/alongside school and after I took them out of school. They are both very good at maths and enjoy it.
Both at about 8 or 9. I taught them a lot before that age because of unsatisfactory schools - especially when we were living (provincial Sri Lanka) at the time I started. The older one went in an out for a bit though. Especially with maths I did a lot of fun stuff with them because I did not like what (little!) the schools were doing.
if you are familiar with the UK system they were both home educated up to GCSEs (taken at 16, end of compulsory school age here), then went to college (school for 16+) for A levels. The younger one is 18 and just finishing college - and her A level choices included maths, which is quite a turn around given she hated maths when she left school. Her older sister works in power electronics R & D.
The main part about mathematics is to understand how it works and practice, which is the language of the universe. This paper really got me into mathematics even more.
The bit at the end with his description of the various math classes was pretty funny:
PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.
CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim
I'm a huge fan of Paul's, and his book Measurement is easily the most beautiful introduction to geometry out there.
I appreciate his focus on how to maintain engagement with students who are predisposed to math, but I think it's equally important to consider how to teach students who are deeply uninterested in math but still need a working knowledge to live life. Granted, the current system seems to be failing both kinds of students.
See an excerpt here: https://www.hup.harvard.edu/file/feeds/PDF/9780674284388_sam...
This paper is what got me interested in mathematics. I happened to read it when I was 14 in geometry class, and it fundamentally reshaped math from something that I happened to be good at to something I was deeply fascinated by and motivated to do more of
> The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.
> ..The fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.
I run a microschool and also teach maths there. I love this essay, and book, but by focusing on maths as art, it focuses on only one aspect of meaning. The larger issue with the way maths is taught, and especially maths, is that it is all mechanics. Now for a certain set of students, those mechanics are incredibly beautiful. Prime numbers have a seduction to them! A few more can be introduced to this beauty and aesthetics, but many are still not into it, and that's entirely okay. What's needed for kids to see and start accepting it is meaning. And meaning comes from connection to the real world. This is why kids who are unschooled but work in shops become great at arithmetic. It's part of their daily web of life. No amount of sanitized exposure to abstract aspects, however aesthetically dressed up will help most. The mind needs to answer "why do i need to care about this" to be open to learning. And this is not something that can come with instruction. It is lived experience, culture. To fix maths education,and any education, you need to bring in meaning before mechanics. Unfortunately the curriculum, assesment, and therefore teaching inevitably end up with focus on the mechanics, which is the wrong end. It is meaning > motivation > mechanics > measurement
School inverts this. And that's a tragedy.
Wrote about it here. https://blog.comini.in/p/schooling-has-a-meaning-crisis-para...
I also love A Mathameticians Lament. I did not know about Benezet so thanks for that. I found your blog post interesting too.
A lot of this reflects the approach I took to teaching my kids maths (and everything else), both before/alongside school and after I took them out of school. They are both very good at maths and enjoy it.
Thanks! How early did you take them out of school? How old are they now.
Both at about 8 or 9. I taught them a lot before that age because of unsatisfactory schools - especially when we were living (provincial Sri Lanka) at the time I started. The older one went in an out for a bit though. Especially with maths I did a lot of fun stuff with them because I did not like what (little!) the schools were doing.
if you are familiar with the UK system they were both home educated up to GCSEs (taken at 16, end of compulsory school age here), then went to college (school for 16+) for A levels. The younger one is 18 and just finishing college - and her A level choices included maths, which is quite a turn around given she hated maths when she left school. Her older sister works in power electronics R & D.
Thanks for sharing this. So lovely to hear and hope they find their joy and meaning in whatever they do!
The main part about mathematics is to understand how it works and practice, which is the language of the universe. This paper really got me into mathematics even more.