I'm home-schooling my daughter in math because Zoom and it not agreeing with her. She did great last semester and tests several years ahead and has a very intuitive sense of why things work the way they do, I'd like to keep her interest.
However, we're now on to pre-algebra and using this book [0], "Prealgebra: The Art of Problem Solving". The first chapter is all about axioms, proofs of some sort, breaking down "obvious" conclusions back to their constituent proof-from-first-principles and it's not agreeing with her at all. Since this is the first week I'm struggling to find a way to have it all make sense, and I've concluded we need to kind of skip much of this first chapter (we'll look at the summary) and get to the later content which is more intuitive and applies "obvious" principles, and come back periodically to revisit the more mechanistic content in the first chapter. I think the parent post's description of exploring a topic matches how my daughter will come to understand the whole "algebra stack." (Wish me luck.)
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.
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.
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.
However, we're now on to pre-algebra and using this book [0], "Prealgebra: The Art of Problem Solving". The first chapter is all about axioms, proofs of some sort, breaking down "obvious" conclusions back to their constituent proof-from-first-principles and it's not agreeing with her at all. Since this is the first week I'm struggling to find a way to have it all make sense, and I've concluded we need to kind of skip much of this first chapter (we'll look at the summary) and get to the later content which is more intuitive and applies "obvious" principles, and come back periodically to revisit the more mechanistic content in the first chapter. I think the parent post's description of exploring a topic matches how my daughter will come to understand the whole "algebra stack." (Wish me luck.)
[0] https://www.amazon.com/Prealgebra-Richard-Rusczyk/dp/1934124... , Amazon: "Prealgebra: The Art of Problem Solving"