Math education: It’s not about numbers, it’s about learning how to think

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically. The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day that learning them takes a lot of training, years of intensive experience dealing with mathematical structures of one kind or another, so it should come as no surprise that people have difficulty learning math.

> if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?"

Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time.

The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules.

It's as if when learning programming, you would learn fragments of various languages here and there but never connect the various pieces into anything meaningful.

Lesson 1: boolean variables in Java

Lesson 2: If statements in python

Lesson 3: Three-part for-loop in C

Mid-term exam:

Q1: In the C language, how do you separate the components of a for loop? (correct answer: semicolons)

Q2: What can you put in the right hand side of the '=' symbol in a boolean variable declaration in Java? (correct answer: an expression that evaluates to either true or false).

Q3: You are supplied a function that can take any number and determine if it's a Crastomarian number. We have 20 foot ball players each wearing a T-shirt with its own number. The numbers go from 5 to 24. We want to find how many players have Crastomarian numbers. The program to find the correct answer will use a for loop. Write the header part of the for loop. (correct answer: `for(i = 5; i ...

And this is supposed to be how you learn programming.

Also note: there's almost no way to know the answers without rote memorization. Of course you will never absorb these concept by osmosis because your day to day activity will not involve writing programs that do things. Instead, you're just memorizing the rules given to you and doing exercises that help you recall which rule to use for which situation.

If you haven't memorized the for syntax and written 25 different variations of it, you will not be able to know it.

And when I say that you write 25 different variations, I mean you write them as part of pointless exercises of course because these are not in the context of a program.

It's more like just repeating these lines until the pattern somehow hopefully sinks into your mind.

for(x = y; x for(y = 10; y .. etc

I don't understand why this point seems so important to people. Seems like a semantic point, if we're talking about math education in schools.

I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" label. For first year undergraduates, they would probably need to take most of the same classes.

I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics.

Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem involves perhaps a dozen simple algebraic manipulations.

These basic things have to be drilled and practiced A LOT before moving on to more advanced topics. There is no way around that. No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late.

I never had any problems with math until I went to university, so I was merely a passive observer of everyday struggle for some people. I honestly believe that foundations are the key. Either you're taught to think critically, see patterns and focus on the train of thought, or you focus on numbers and memorization.

The latter obviously fails at some point, in many cases sufficiently late to make it really hard to go back and relearn everything.

Math is extremely hierarchical and I believe schools do not do enough to make sure students are on the same page. If we want to fix teaching math, I would start there, instead of working on motivation and general attitude. Those are consequences, not the reasons.