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. Relying always on the calculator or a CAS leaves students confused and befuddled. I see this all the time in calc classes that I teach. The CAS loving students just don’t understand as well.
> 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.
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.
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.
I studied commutative algebra in graduate school which is an adjacent subject to algebraic geometry. People capable of understanding Hartshorne have number 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).
> 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.
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.
Whether it is causative or not it is still the case that someone who doesn’t know fractions will have a hard time in algebra. It would be bizarre to teach someone how to add rational functions before they can add fractions.
> Whether it is causative or not it is still the case that someone who doesn’t know fractions will have a hard time in 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).
> 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.
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.
You learned about rational functions in high school or middle school (most likely given your use of “maths”). I can tell you have very little experience with teaching. 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. There is a reason for the order in which topics are taught.
> You learned about rational functions in high school or middle school
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.
I can tell you have very little experience with teaching. But surely your thoughts on the topic must be on par or superior to those with training and experience. My wife is a doctor and lots of people like to tell her how the body works and why she must be wrong. They think reading a blog post on vaccines is equivalent to 4 years of med school. The same phenomenon occurs in education. Lots of people think that since they went to school they know about teaching and how it should be done.
An introspective person would wonder why it is so obvious to others that they have no experience with teaching in the classroom based solely on their views of teaching.
There's no deep mystery to that; anyone who questions the dogma in any of the fields I mentioned is also obviously an outsider. The fact you went so quickly to attacking my credentials rather than giving any real rationale is not a sign that your field is full of legitimate knowledge; quite the opposite.
