ps - the whole concept of "show your work in multiple ways" in early math was just sooooo non-motivating for my kids (and would have been for me too, and I have a degree in math!). DS in particular thought it was ridiculous, why should he have to do it? My MG dd even thinks it's silly. I have to stop typing now or I'll proceed into a full-on rant about our school district's elementary math curriculum! Anyway, fwiw, once my ds go to pre-algebra and higher math courses, he was so much happier - that's where gifted kids who are strong at math concepts really start to soar.
I have a math degree, too.
There are so many kinds of math. Most of what is taught until Algebra is just arithmetic. Geometry, if taught using proofs, is the first REAL math class. It is shame kids have to wait until 8th grade to get to it when many are clearly ready for "proofs lite" by the 3rd grade.
What the GT kids get excited about is number THEORY and algebraic theory - things THEIR minds can sense but which ALL elementary teachers are totally ignorant of. If a kid can grasp series, summations, and then limits, then they are ready for Calculus as well.
Kids can be slow computationalists, BUT they get the THEORY and get it spades.
So, the teachers saying show your work, or picking on hash marks, are really forcing compliance because there IS NOTHING ELSE THEY HAVE TO OFFER.
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IMHO, Logic, Euclidian Geometry, a good abstract algebra and another in analysis should be taught to GT kids FIRST with hints to the clerical parts, then double back for the clerical aspects in each - i.e traditional algebra, analytical geometry, linear algebra, and then calculus. You could combine algebra/analysis in one course and linear/calculus in another. The latter courses would reinforce theory and also introduce areas where ideas break down. Essentially the student would get the first three years of a BS in math completed.
So,
Sem 1 - Logic
Sem 2 - Proofs
Sem 3 - Geometry & Abstract Algebra
Sem 4 - Analysis 1
Sem 5 - Analysis 2 and AA 2
Sem 6 - Traditional Algebra I/Geometric Analysis
Sem 7 - Calculus I /Linear Algebra
Sem 8 - Calc II/III
Sems 9-12 - Electives + physics/chem
