It is quite usual in Algebra books to introduce variables early — they seem to be what makes algebra Algebra, that is. However, a good amount of early stages of manipulating expressions and equations is not dependent on having variables. I am wondering about possibly spending longer than usual on concrete number manipulation with problems like–
Isolate the 253:
253 + 23 = 276
Isolate the 3:
2(3 + 4) + 5 = 18 + 1
Show why the two sides of the equation are equivalent:
6 * 4 = 3 * 4 + 3 * 4
Theoretically, one could provide a strong enough background in manipulating expressions and equations to allow variables to be introduced while hitting the ground running, with students quickly able to isolate variables in multi-step equations.
This has the advantages of:
1. It’s clear that (using the first problem as an example) 253 = 253 is the goal. The point becomes the process, and the usual student tactic of writing the answer without explanation (I did it in my head!) would look silly.
2. Students who have “symbol overload” can use their concrete sense to help focus solely on operations, a la Bret Victor (sort of).
3. The meaning of the equality sign is reinforced; it’d appear much odder to a student to perform an operation on only one side of the equation when starting with 20 + 1 = 21. Also, the restrictions to operations on an expression would make more sense when starting with, say, 5 rather than 5 + 2x.
4. Answers would have a built-in sanity check.
Unfortunately, it means applications would be forestalled. While some of the manipulations represent tricks for mental arithmetic, I don’t know if that would be enough motivation to carry a struggling student through.
Has anyone tried something like this, even on a small scale?