[csa]

> My third grader does simple equations like this in math.

Most third graders cannot consistently do the following:

10 - ? = 3

10 - ? (- 10) = 3 (- 10)

- ? = - 7

(-) ? = (-) 7

? = 7

This is absolute voodoo magic to most third graders. They may be able to memorize specific patterns, but they won’t be able to manipulate equations consistently and accurately.

Fwiw, at 3rd grade, some of the smarter kids might be able to understand and manipulate these abstractions, but those kids aren’t the norm.

[mturmon]

You can't quite do it the way you have shown, but take a look at:

https://dragonbox.com/products/algebra-5

My kid tore through almost the whole set of problems in the app at about the recommended age (5yo).

[csa]

Sure. The way I presented it was not an accident (abstract with little scaffolding).

The question with the software that I have is how much is actually understood. Specially, how much of what was done can be applied in a different context. Ideally consistently, accurately, and without any scaffolding. I’m guessing the answer might be “a lot” for a typical HNer’s child (or maybe not), but it rounds to zero for the average or lower 5yo.

I would also be curious about how much success can be had with just rapid trial and error rather than learning and applying. 5 year olds can be great at the trial and error part while not actually developing and retaining much understanding. I could be very off the mark with this speculation, but a lot of my experience in this area makes me think I’m not.

[mturmon]

This sounds like the difference between basic understanding of the idea, and mastery. Mastery would always take a lot of repetition and diverse problem-solving -- true even for algebra.

Basic understanding is a lower bar, and I'd suggest that most students leave Algebra 1 with just basic understanding. Hopefully a little better than Dragon Box.