Algebra is typically taught using a letter x with no distinction as to whether x is natural, rational or real. Keith Devlin illustrates this thinking by advocating teaching the field axioms instead of the progression from naturals to rationals to reals.
However, we miss an opportunity to help students who are still struggling with rational numbers. In fact, this is all of them, since rationals are not explained well from a foundations of math point of view. This includes mixing in geometry and number concepts, another mistake widely practiced and illustrated by Keith Devlin’s comments on multiplication and scaling.
When we teach algebra, we should first teach it on naturals to naturals, i.e. functions from naturals to naturals. This moves us away from variables as existing on their own and on purely symbolic thinking. Functions as sets of ordered pairs are conceptual and full of meaning.
So we start with y = a + bx where a,b,x and y all have to be natural numbers. These are without sign. So we do naturals prior to the integers, which have sign built in.
So y = 1 + 2*x is a function we teach defined for natural number inputs x.
So the ordered pair (1,3) is in the function, i.e. one of the ordered pairs that constitute the function or the graph of the function as it is sometimes said.
Suppose we want
y = b x
so that (1,3) is in the graph. Fine, we take b=3.
Now we want (3,1) to be in the graph for some b.
We can’t do it. This should be taught just this way. This teaches many concepts and sets up the need for rational numbers.
Now we consider functions from rationals to rationals. We can now have b as 1/3. So we can have the ordered pair (3,1)
within the class of functions with no intercept.
This helps teach us to distinguish functions from intercepts from those without.
Going back to natural functions only, with intercepts we can
write y = 1 + 0*x. Now we have the ordered pair (3,1) in the function. Is there any other way to get a natural linear function with the ordered pair (3,1)? For the class of natural linear functions with no intercept, none of them contain the ordered pair (3,1)?
This is an unfamiliar way to raise these issues. But it helps teach functions, ordered pairs, natural numbers and rational numbers. This is much more productive than jumping into letters without paying attention to the sets of numbers that are inputs and outputs. It also emphasizes the ordered pair nature of functions. It reinforces the need for rational numbers and it links the need for slopes of functions with no intercept to the need for rational numbers.