[aleph_minus_one]

> There is evidence that a person’s ability to understand and succeed in algebra is mostly determined by whether or not they can do arithmetic with fractions. Number sense is important in my opinion.

My opinion differs a lot here. I would not say that I have a good number sense (I guess that people who have to do "numeric calculations" or "back-of-the-envelope calculations" as daily part of their job have a much better number sense than me). On the other hand, I find it rather easy to learn really abstract algebraic concepts (think Grothendieck-style algebraic geometry or similarly abstract mathematical topics), which many people (most of them with much better number sense than me) tend to find insanely difficult.

[skhunted]

The number sense I talk of is not being able to do numerical calculations easily or in your head but rather understanding how to operate with numbers and their different representations. A person who can understand algebraic geometry doesn’t have trouble understanding things like simplifying x + 5/3 x. People workout any number sense have a hard time with this. Knowing that 8/3 is just a different way of writing 1+5/3 is confusing to them.

[aleph_minus_one]

Textbooks about "abstract nonsense" rarely require you to do such routine calculations/simplifications - they rather require you to be capable of making sense of definitions that are (at a first glance) insanely far removed from anything you have seen in your real life: I would rather liken it to taking strong, dangerous hallucinogenic drugs, and making sense of the world that you now see (which is something that only some people are capable of); by the way: I don't understand why hallucinogenic drugs are illegal, but textbooks about very abstract math are not. :-D

On the other hand, textbooks about, say, analysis and mathematical physics (both in a broader sense) - which can also be very complicated - have a tendency to demand a lot of (also long, tedious) "routine" calculations from the reader (often to do by his own). For these areas of mathematics your argument surely makes sense.

[skhunted]

I studied commutative algebra in graduate school which is an adjacent subject to algebraic geometry. People capable of understanding Hartshorne have number sense.

[aleph_minus_one]

I just wrote down how I feel about this topic.

Textbooks about particular areas, in particular specific topics in physics (including mathematical physics), teach me a lot about number sense (and let me feel that mine is not really good or perhaps badly trained). On the other hand, these very abstract topics feel like a quite different activity to me that is only barely related to number sense.

> People capable of understanding Hartshorne have number sense.

This can also be explained by the hypothesis that people with a strong number sense love to feel themselves challenged - thus they attempt to understand this nontrivial textbook (even though understanding it may in particular require different skills).