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.
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.
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).
Evidence that it's causative? That would be utterly bizarre and I'd love to see a citation, because doing algebra has nothing to do with fractions. I'd think it's far more likely that there's a strong correlation between the two because they're both determined by the ability to understand and follow the rules of an abstraction/notation system, and if you taught people algebra first and then fractions afterwards you'd say that ability to understand fractions was determined by whether they could do algebra.
Doubt. Do you have any evidence at all for this claim?
> It would be bizarre to teach someone how to add rational functions before they can add fractions.
Sure, rational functions obviously sit at the intersection of algebra and fractions and require both. But they're hardly some deep foundational piece of algebra; I'm not sure my classes even covered them.
Only anecdotal evidence. I’ve taught beginning algebra courses at a community college for 23 years. Students who don’t know fractions have a very hard time in algebra. Those who can’t understand that x + 5/3 x is 8/3 x have a hard time understanding that 2xy+ay is (2x + a)y.
Understanding rational functions helps to understand what vertical asymptotes are and as such are a fundamental source of examples when learning limits. They also aid in understanding why tan(x) has vertical asymptotes where cos is 0. Every complete algebra curriculum includes rational functions. I say complete because algebra is usually broken up into 3 courses (2 at the pre-college level).
Sure - but that's just as true in reverse.
> Understanding rational functions helps to understand what vertical asymptotes are and as such are a fundamental source of examples when learning limits. They also aid in understanding why tan(x) has vertical asymptotes where cos is 0. Every complete algebra curriculum includes rational functions.
Meh. x^-1 is a good example of some things, sure, but I don't remember ever doing addition of rational functions which is what you originally talked about, and I went through an extremely reputable maths degree.
No middle school, and I very much doubt it. Searching I can see them mentioned in a further maths GCSE (which is something most schools including the one I went to don't offer, and rather suggests they're not on the regular maths GCSE, which would match my memory).
> Most students who know that x + 5/3 x is 8/3 x have trouble, initially, with understanding that 2xy+ay is (2x + a)y.
Who know that first or who have been taught it? I genuinely would like to see any actual evidence that the latter is objectively more difficult than the former.