How to Teach Math?
The only way to learn mathematics is to do mathematics—Paul Halmos
Angie Hodge is an Associate Professor of mathematics at Northern Arizona University. Her interests as stated on her website are focused mainly on education, mentoring, and equity in the STEM disciplines.
Today, Ken and I thought we would discuss the issue of teaching and learning math and complexity theory.
I am now retired from Georgia Tech. But I continue to be interested in how we can teach math. Dually how we can learn math. I still try to learn math—not for formal classes, nor in formal classes—but to help advance my own research. Even now I find that I need to learn some topics to help write a post or to solve a problem.
On this blog with Ken I am also interested in still helping others learn. Some call that teaching. This is therefore the topic for today.
First An Issue
One thing that drives me nuts about learning new math is what I will call the
“It is too obvious to state” principle.
What I mean is that when you start to learn material from some corner of math you get the key definitions and the main results. What I do not always get is some totally obvious ideas. Does this make any sense?
Here is an example. Consider the class of sequences that are defined by linear recurrences. That is: a recursion that defines a sequence as a linear combination of earlier terms. One of the most famous is the Fibonacci numbers:
And 
The property we are interested in is:
Is a product of two such sequences also of the same form?
This is a basic question, but it is not trivial to find the answer.
How To Teach?
This is an ancient question that we all probably have thought about from time to time. It is complicated by the recent issue of most classes being done via online. Hodge has some nice stuff on teaching, especially on Inquiry-Based Learning (IBL).
Ken’s Similar Question
Ken writes: I am interested in manipulating logistic curves. Such curves not only undergird the chess rating system and the theory of standardized tests, they relate directly to the design of chess programs which I use to take my data. This decade’s neural chess-playing programs, following on from AlphaZero, express values directly in terms of the likelihood of winning, as numbers between 0 and 1, rather than traditional evaluation schemes built on counting 1.00 for a pawn, 3–3.5 for a knight or bishop, and so on.
The new likelihood numbers follow a logistic curve. I especially want to convert from the evaluation numbers to them. After doing this conversion for various major chess programs, I would like to average their values as input to my predictive model. This involves taking averages of logistic curves. The curves can be generalized to the form named for Francis Richards:
for constant parameters 
When does a linear combination of logistic curves belong to the same family? And if it doesn’t belong, how close to a member of the family does it come?
My point is that I have not been able to find an easy source for answers. This strikes me as exactly the kind of question for which sites like MathOverflow and StackExchange exist. But it is also a nice instructional exercise for training students in both the grit and adventure of mathematical research.
One can also pose literally the same as Dick’s question above: how about a product of two curves 
Open Problems
The following video shows how not to teach math: The Kettles do math.
[some format and word tweaks]
The big problem with teaching is that you can’t force students to be curious or want to figure something out.
In the us we made a decision (perhaps via inaction) that all college hopeful students at good schools should be taking advanced algebra if not calculus in high school and some of those students are going to hate and resent math. A great teacher can maybe turn around a few of those students but at the end of the day you need to decide what you do with the students who aren’t interested in actively engaging and figuring out the material.
Problem is that rote work is the only way you can ensure that students who won’t actively engaged but are willing to do work pass and do well. That’s not what we should be valuing but it’s what teachers get pressured to do.
There are all sorts of interesting teaching technique discussions to have but until we decide that there isn’t value in making students with no desire to be there memorize rules that don’t help them understand and will never use. Let’s not sacrifice students who could learn and love the subject for students who aren’t interested in learning.
Dear Dick & Ken,
Here’s a backgrounder on the pragmatic theory of inquiry from Aristotle thorough Peirce to Dewey, with reflections on its implications for educational practice and automated research and teaching tools.
Interpretation as Action : The Risk of Inquiry
Additional papers and presentations on inquiry based approaches to research and teaching, along with its implications for information technology, the university, and society, can be found here.
Okay, I’ll bite. You wrote, “What I do not always get is some totally obvious ideas. Does this make any sense? Here is an example.” How is what you say about recurrences an example of a totally obvious idea, or of not getting a totally obvious idea? I don’t understand.
Here’s an example that I encountered while teaching Grover’s algorithm from our textbook. We say how the reflection around the all-1 vector is given by 2J – I, where I is the NxN identity matrix and J is the all-1s matrix divided by N. We show that the reflection around the (orthogonal complement of the) solution vector, as given by the Grover oracle, is feasible—given that it is feasible to verify solutions. But we never actually say why 2J – I is feasible. We present that as “totally obvious” but maybe it’s not—those are exponential sized matrices, after all.
Have you considered writing a blog post about TeXmacs and giving your opinion on whether TCS conferences and journals should accept TeXmacs submissions (i.e., without exporting to LaTeX)?
Ah, I think I see. You mean like the old joke about the professor who has to think for half an hour before declaring something to be obvious? Things that are passed over without comment because they seem trivial, but may or may not be trivial if you look closely? In terms of teaching math, I wonder if proof assistants such as Lean could help. Kevin Buzzard has an interesting blog post where he talks about how mathematicians often assume without proof that certain constructions depend only on the isomorphism class of the object they are studying. Usually they are right, but they can underestimate how much they are sweeping under the rug unless a proof assistant forces them to face the music.
https://xenaproject.wordpress.com/2019/06/02/equality-part-3-canonical-isomorphism/comment-page-1/
Search for “could not be omitted” and “Imagine” if you want to skip to the punchline.
Indeed, worthy of note, and deeper than the kind of error I’ve encountered of thinking one equivalence relation is the same as another (or has polynomial index when it is really exponential). We did post about Lean last December: https://rjlipton.wpcomstaging.com/2020/12/10/the-future-of-mathematics/
The sum of sigmoids is not in general sigmoid. For example, https://math.stackexchange.com/a/9601
Approximate numerical forms have been explored, for example “Equivalent Analytical Functions of Sums of Sigmoid like Transcendental Functions”, Jenõ Takács, Applied Mathematics and Nonlinear Sciences, July 2018.
I think if you found very close approximations, it could have significant implications for the computational efficiency of deep neural network machine learning models.
Ah, thank you very much! Noted after a week of dealing with another spate of instances of such sigmoids being very much distended… I actually did know about being able to get multiple inflection points at the time of my posts in 2018, but forgot that I knew it.
Chess, like some other games, teaches patience, strategy, and observation, three things that help with reading and mathematics.