> I understood how to do arithmetic for numbers with multiple digits before I was taught a "procedure"
What did you understand, exactly? You understood how to "count" using "numbers" that you also memorized? You intuitively understood that addition was counting up and subtraction was counting down, or did you memorize those words and what they meant in reference to counting?
> Also, I am not even sure what you mean by "memorization is how you learned fractions". What is there to memorize?
The procedure to add or subtract fractions by establishing a common denominator, for instance. The procedure for how numerators and denominators are multiplied or divided. I could go on.
Fractions is exactly an area of mathematics where I learned by understanding the concept and how it was represented and then would use that understanding to re-reason the procedures I had a hard time remembering.
I do have the single digit multiplication table memorized now, but there was a long time where that table had gaps and I would use my understanding of how numbers worked to to calculate the result rather than remembering it. That same process still occurs for double digit number.
Mathematics education, especially historically, has indeed leaned pretty heavily on memorization. That does mean thats the only way to learn math, or even a particularly good one. I personally think over reliance on memorization is part of why so many people think they hate math.
> Fractions is exactly an area of mathematics where I learned by understanding the concept and how it was represented and then would use that understanding to re-reason the procedures I had a hard time remembering.
Sure, I did that plenty too, but that doesn't refute the point that memorization is core to understanding mathematics, it's just a specific kind of memorization that results maximal flexibility for minimal state retention. All you're claiming is that you memorized some core axioms/primitives and the procedures that operate on them, and then memorized how higher-level concepts are defined in terms of that core. I go into more detail of the specifics here:
I agree that this is a better way to memorize mathematics, eg. it's more parsimonious than memorizing lots of shortcuts. We call this type of memorizing "understanding" because it's arguably the most parsimonious approach, requiring the least memory, and machine learning has persuasively argued IMO that compression is understanding [1].
What did you understand, exactly? You understood how to "count" using "numbers" that you also memorized? You intuitively understood that addition was counting up and subtraction was counting down, or did you memorize those words and what they meant in reference to counting?
> Also, I am not even sure what you mean by "memorization is how you learned fractions". What is there to memorize?
The procedure to add or subtract fractions by establishing a common denominator, for instance. The procedure for how numerators and denominators are multiplied or divided. I could go on.