[lmm]

> 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.

[skhunted]

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.

[lmm]

> 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.

[skhunted]

Do you have any evidence at all for this claim?

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).

[lmm]

> 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.

[skhunted]

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.

[lmm]

> 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.