I tutored algebra students and I reccommend the following approach with Khan academy/Openstax resources.

Order of Operations(No Variable)

Order of Operations(One Variable)

Advanced Order of Operations (One Variable)*optional

- Logs, Roots, Exponentials

- Polynomial multiplication/Factorization

Change the variable to random symbols

- Greek characters, shapes, animals

Order of Operations(Multiple Variables)

Introduce functions

- f(x) = mx + b

Connect Order of Operations with Functions

- Simplify an equation and plot as a function

Introduce units (basic physics equations)

- Simplify to formula and Plot

The goal in formal classes is to introduce order of operations with variables, the concept of functions, and how to manipulate those functions. If you are stem oriented, then this is the perfect time to introduce dimensional analysis(units) and physics-based functions. Instilling a sense of confidence and comfort with algebraic manipulation is critical. Prepping her for physics and working with functions is just smart.

Breaking things down to their basic axioms and rebuilding them with proofs is arguably quite an advanced approach. By Terry Tao's three stages of of rigour classification [0], this perspective isn't typical until undergraduate study.

It doesn't seem like a bad idea to give a taste of it significantly earlier, but I don't think it's surprising if someone without a more mature relationship with mathematics doesn't find much value in 'going back to basics.' If your daughter is doing well with a more intuitive approach it sounds like a good idea to stick with it for now.

[0] https://terrytao.wordpress.com/career-advice/theres-more-to-...

This feels like the math equivalent of learning to program by reading an assembly spec document, to be sure.

Art of Problem Solving is great - I think they take credit for the US Math olympiad team's performance. I use books from the same organization at elementary school level and find them very good and fun to teach. I would recommend going a level lower among the books published by them [1] to see if that is a good starting point for your child. Their website also has some placement tests etc. to evaluate the right level for a student.

[1]https://artofproblemsolving.com/

There's two hurdles for the pre-algebra student: numbers being generically represented as letters and the idea of axioms being generic rules for numbers. Check to see which issue or both is misunderstood. If it's numbers as letters, show them any interactive programming environment and demonstrate the idea of letters as generic numbers. From there, axioms are just more interactive demonstration.

I remember my mom struggling to teach me pre-algebra ahead of the school curriculum. My uncle (a math teacher) mailed us a bunch of "part-is-to-whole-as-part-is-to-whole" word problems of increasing complexity and various forms, and working through those made a lot of things click. The concreteness helped, and it also made me understand the equivalence between proportions, fractions, decimals, percents, and their real-world concepts, plus it has some simple solve-for-x attached to it. And it's learning-by-doing instead of learning-by-reading.

Use a basic text book, TAoPS series is intended to be quite advanced.