Teaching and [Re]Learning Mathematics
Resources for Parents, Teachers, and Students Who Want to Understand Math
This is going to be a “live” post, that I will try to update regularly with new discoveries and rediscoveries. I may at some point refine and re-organize it better, too. If you think I have missed anything or should take something away, please send me a message or drop a comment on this post.
Two main sections
I talk a lot about specific and general problems with mathematics today, and I give general advice for fixing it.
But I rarely write about specific ways to teach mathematics well.
There are a couple reasons for this:
1. I am not confident in what I know
This is because what I think I know about these things is always changing.
For this reason, this post is a LIVE post — I am going to be updating it as frequently as necessary to reflect any new things I find. Please give recommendations.
2. Each student is different.
Every kid who struggles with math struggles differently. Same human nature, different obstacles.
No textbook or curriculum can fix this. One-size-fits-all solutions don't exist.
What works…maybe? A toolbox approach: try different things until something clicks.
The teacher's real job isn't following the script—it's diagnosing what's blocking each individual child.
For this reason, I propose some puzzles, investigations, one-off lesson plans, and other things that have guided my way of approaching math. It’s a shotgun approach.
3. Teachers/parents teach the way they were taught.
This means that any change in math education needs to start with teachers and parents re-learning math a better right way.
I am working through that myself.
Parents and teachers, including myself, were all taught math a certain way. For the most part, our society has “succeeded” in largely homogenizing the content and pedagogy of math instruction.
But if society got it wrong, we’ve got a problem.
The problem is bigger than any curricula or textbooks.
That would be like putting a band-aid on a bullet wound to a major organ.
For this reason, I propose reading materials for parents and teachers to help them re-learn mathematics.
For Parents and Teachers
First, Read “A Mathematician’s Lament” by Paul Lockhart
A Mathematician’s Lament by Paul Lockhart is by far the best exposition of what is wrong with the way we teach mathematics in school today. It offers a scathing critique of mathematics education and gives some insight into how things could be different.
Second, Study Great Books
Studying “Great Books” of mathematics are the surest way of internalizing mathematics. Most of us were deprived of this in our own education. We need to internalize mathematics if we are to pass on this to our children/students.
Introduction to Arithmetic by Nicomachus of Geresa. This book really gets into the ancient study of Arithmetic. It takes time to work through, but is worth the while.
The Analytic Art by Viete on algebra and analytic geometry
Extra resources for parents
Mathematics for Human Flourishing by Francis Su.
Books by Paul Lockhart:
Measurement: Geometry
Arithmetic: Logistical side of arithmetic
The Mending of Broken Bones: Algebra
The History of Mathematics volume 1 and 2, by David Eugene Smith.
For The Kiddos
Number Stories of Long Ago by David E. Smith; here is a digital copy. I think the ideal is probably to read these aloud to kids, showing pictures as you go. Paul Lockhart probably drew inspiration for Arithmetic from this book or one like it.
String, Straight-Edge, & Shadow by Julia E Diggins. Depending on whether the children read it alone or with help, this book is good for students from 6th through 8th grade (suggestion courtesy of Greg William).
The 24 Game (and its variants). Growing up, this game is why I fell in love with mathematics.
Prime Climb is an awesome game to explore prime numbers. It was gifted to me twice by different parents at my school. I’ll break it out sometimes in class or let kids play it after school on a rainy day.
Math puzzles are probably the most important element in teaching and learning mathematics well. In the words of Paul Lockhart
“A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity.” (A Mathematician’s Lament)
IMHO, the best puzzles are physical:
I have a bunch of these puzzles out on my desk at school, and kids really enjoy trying to figure them out.
Beast Academy is an excellent curriculum that utilizes comics to teach elementary mathematics. They also have online classes. The books are a bit advanced, so I recommend using the books as supplemental material.
Morris Kline’s “Calculus: An Intuitive & Physical Approach”: long preview and Amazon link.
Mathnasium Learning Centers are phenomenal. If you live near one, I encourage you to check them out. I worked at one for a year, and it was a transformational experience. They do an impressive job of helping children to fall in love with math. This is an awesome resource for a parent or teacher, too: work there for a year!
Montessori School: I’m becoming more and more convinced that Montessori does things right when it comes to school structure and exposure to mathematical concepts, especially at a young age.
Singapore Math: to be honest, I have never taught it and was never taught by it. BUT I have seen teachers who dislike teaching math fall in love with teaching math because they committed themselves to using Singapore.
Having tutored others in mathematics, from arithmetic to ODEs, I have some observations.
1. The first problem is cultural, as it relates to educational institutions, especially in North America.
Children are not allowed to fail, and can thus not develop resilience. By contrast, my one cousin (South Africa) failed grade 1, which was a life long signal that he would have to work much harder to get somewhere. Some years ago he got his matric. Granted that this was white education, but the principle is general.
2. Coupled with this is a rejection of substantial homework, and the idea of studying for the test, rather than for mastery.
I recall getting four hours of homework per day in grade six (standerd four, note that the spelling is correct...).
I recently tutored someone in trigonometry, and gave twelve hours of homework, to be completed over two weeks. I made myself available should questions arise. None of it was done. During the next lesson, I thus started by getting the student (adult) to start working through the homework, as it was substantially prerequisite for following material. After that, the student didn't return.
3. The worship of the "math genius," and neglect of the rest inhibits mathematical interest. I recall in grade six that we once nearly rioted as the teacher didn't assign enough (math) homework, thereby denying us the opportunity to substantially internalize and master the material.
In North America, the "geniuses" are shunted along to the next topic without internalization, while the rest do not master.
To my thought, having those students who show apparent mastery early on assist other students benefits both the quick learners, as teaching requires greater mastery than superficial learning, and the slow learners, as they can get personalized attention and assistance.
Excellent compilation…. Vedic math book addition will develop great reasoning skills