Get the Cart Behind the Horse

Many times during the flash of time when I was in the classroom (even though sometimes it felt like an eternity), I questioned what I was doing as I was doing it, and in the process it seemed like I was living the distinction between school and education.

One thing that always bugged me was that I had to repeatedly refer to the commutative property of addition before students had adequately internalized the facts that in addition the order of the two numbers is irrelevant, but with subtraction order matters. Similarly with multiplication and division. In my school, students would become versed in arithmetic well before getting bogged down in the legalese vocabulary of the laws of arithmetic. (Just as we expect people to be able to drive legally and safely without needing to quote government statutes on transportation)

As another example, I refer to the distance between two points on the coordinate plane. For those non-mathematically inclined, that amounts to placing two dots on a grid such as shown here, and determining the distance between them using the grid's scale.

Quite frequently you would see a textbook begin by giving the formula, then include a short dissertation on where it came from (normally making reference to the Pythagorean theorem), and then jump into an application or two.

Just like the real world! First the formula is found, then it is verified to be true, then it is used! That is exactly how life goes, isn't it?  Of course not!

In my personal school I would never make mention of the distance formula until the time came to compare the distance method to that for slope. As a matter of fact, I would introduce the formulas together, AFTER students already had a solid grasp of calculating distances and slopes on the coordinate grid.  Prior to that I would make copious use of the phrases "change in y", and "change in x" interlaced with references to "rise" and "run" until the students picked out their seeming interchangeability, at which point we would shorten them to Δx and  Δy, at which point there would be discussion of the Greek letter delta (any connection to a river's delta?).

Somewhere buried withing the classroom conversation would be the discovery that in the question of distance the signs of Δx and  Δy were irrelevant, but in the case of slope they were very relevant. To help us with this issue, we would establish a tradition of referring to one of the points as (x_1,y₁) and (x_2, y₂). Only then, when appropriate, would we actually convert our methods for distance and slope into actual formulas. The formulas would appear as short hand for what they were already doing, not as a prescriptive rule to be mindlessly followed. (Could you imagine if a student driver's first lesson was all about the cruise control buttons?)