Learning algebra in my 60s

Unfortunately, I don't have an easy answer. My son recently graduated from my high school and I noticed that while his school used Pearson's website to drill him on the mechanics. My son struggled with word problems precisely because he did not have the fundamental concepts.

For example, I argued that division is more intuitive when we multiply using fractions. - Whole numbers have an implicit denominator of one. E.g., 7 is equal to 7/1. - It is more intuitive to think of division as multiplying by fractions. E.g., 7/2 is equal to (7/1) x (1/2). We multiply the numerators together. And multiply the denominators together. - Where this is helpful is when we deal with complex arithmetic like (7/2)/2. We have 7 pizzas. Divide by 2. And divide by 2 again. If you break it into fractions, this is equal to (7/1) x (1/2) x (1/2). The numerator is 7x1x1. While the denominator is 1x2x2. Putting it together, it is (7x1x1) / (1x2x2) which is equal to 7/4.

This discussion usually leads to the question of what is multiplication (again)? I argued that multiplication can be thought of as a shortcut for addition or can be thought of as set theory. I have two bags of marbles and each bag contains 10 marbles each. How many marbles do I have? You can either write 10+10 (multiplication as a shortcut for addition) or you can write 2x10 (set theory). Taking it back to division. I have two bags of marbles and each bag contains 10 marbles and I have to share my marbles with my sibling (divide by two), how many marbles do I have?. You can write either (10+10)/2 or you can write 2 x 10 x (1/2).

I think those of us who are good at math greatly underestimate the amount of rote needed to reach competence-- first, how much we actually needed, and second, how much more of it most people need.

A good math education in arithmetic, algebra, and calculus is a combination of concepts and drill. This was a concept that I rejected back in school. Now I'm teaching-- not mathematics, but I have tutored and helped students catch up who have had problems.

There's a fair number of students struggling in pre-calc who have all the concepts just fine, from the bottom to the top. But when they're dealing with lots of terms and keeping a higher level goal in mind, their performance on a few simpler things, like fractions and quotient properties, falls apart. Maybe they missed a week or two in 6th grade when this was really solidified and practiced.

The terrible thing that tends to happen, once you stumble in math: the amount of concept content you have falls. The focus moves even more to rote, but focused on the "more difficult" stuff--- leaving whatever core deficit there is intact. There's solid reasons for this, but the outcomes are not great.

I also have a kid who is going into AP Calculus pretty young. I'm kinda nervous that he just has not had enough reps of practice, even though he scores very high on placement tests.

Scientists can't agree to what extent human intelligence is inherited vs. acquired, and so asking if doing some activity x will result in an increase in personal smartness y will result in a wide variety of answers. (sort of a differential equation isn't it? Change in intelligence over time as a function of doing algebraic manipulations every day).

There are some plausible benefits - it's likely to help with internal mental organization, for example. However, some thought should be given to how to set to work, in particular avoiding bad habits that eventually thwart future progress (this can happen in say, learning to play a musical instrument).

Learning algebra (or any other field) conceptually is like constructing a connected map of various islands, although from reading this article it isn't clear whether that was accomplished. There's no mention of concepts like associativity, distribution, commutation, how this applies to the order of operations when you start mixing up + * - /, and the rather strange but also fundamental notion of identity. For example:

https://byjus.com/maths/commutative-property/

This is a bit sad because understanding these concepts opens the doors to fun higher math. Matrix multiplication is generally non-commutative, and this property makes it useful for quantum mechanics calculations. Symmetry operations - rotations and reflections and so on - are not generally commutative, but are associative. This is all very important in things like protein crystallography. A solid grasp of these ideas also allows for the introduction of the concept of groups as a way of looking at algebra. Here's a great series on that whole business:

https://www.socratica.com/lesson/groups-motivation-for-defin...

A good rule is to spend at least as much time understanding fundamental concepts and abstractions as on working out the results of specific explicit examples.

> you lose the ability to understand math as anything other than a human attempting to be a computer

I get what you're saying and it's important, but I also disagree with how you're framing it. It's desirable to get to the point where you can automatically do purely mechanical symbol transformations as you work through a problem, thus saving your higher reasoning for the conceptual problem of searching for which transformations will lead to a solution.

Frankly, I find the continuing concern with "intuition" with respect to purely mechanical transformations to be rather medieval. It is slowly changing though as the influence of computing science osmoses back into more traditional mathematics. Using Mathematica or similar tools makes it abundantly clear how distinct the mechanical and conceptual challenges are.

Most of pure math does have practical applications - to pure maths! My favorite professors never launched into a subject without a motivating example, even if that motivation was often "Look at x, y, and z. Aren't they awfully similar?". My first exposure to Abstract Algebra started with a little number theory, moved on to rings, then to ideals, and only then to groups. Many people I've talked to about it are surprised we took axioms away rather than adding them, but the way we learned motivated each step. Indeed, groups themselves were introduced with permutations. Similarly, I found measure theory was best introduced by showing how handy cardinality was for finite sets. A "practical" application would have been probability, so perhaps this wasn't exactly application focused, but we certainly didn't start from the definition and work our way out.