I often forget just how dysfunctional a relationship many of my weaker students have with mathematics.

It’s been a long time since I’ve been surprised at the extent to which such students harbour an unproductive and damaging belief that mathematics is nothing more than a mishmash of symbols and voodoo procedures. After all, this is understandable, what with students being taught that memorizing templates of questions and plugging memorized formulas into their fucking graphing calculators is homologous with “doing mathematics”.

What surprised me for a long time after – at least the first fifty times I encountered the phenomenon – was how resistent these same students are to seeing mathematics as anything *other* than a collection of disconnected formulas and calculator algorithms.

The other week, I found myself teaching introductory graphing to a handful of students. Partway through a lesson, one student asked me – how do I graph the line in this question? Do I find two points and join them, or should I just find one point and the slope and then graph it that way?

Giddy with delight at this hint of outside-the-box thinking, I replied: you can do it either way you want! It’s your choice! Both of these options are *totally valid methods* of graphing the line! Two points, point slope, *it’s up to you*! In fact, you can graph it one way, and then if you want to check your work, you can graph it the other way, and ISN’T MATHEMATICS SUPER?

Pregnant pause. Hesitation. The barely-perceptible tremours of a worldview beginning to collapse unto itself.

*There are two ways to do this question?*

Yes! Not one, but two (2) ways to achieve the goal of graphing a straight line! Pick one! It’s entirely up to you!

*But which way should WE do it?*

EITHER way! The easy way! The quick way! Try ‘em both for practice, and then on your homework you can do the graphing questions whichever way you prefer, unless you’re explicitly instructed otherwise!

Facial expression indictating flicker of hope. *Oh, so sometimes you’ll tell us which way to do it?*

Well, yes, sometimes, because I want to make sure you understand both methods, but in general I’ll –

*But which is the right way THIS time?*

Do it both ways, and if you did it right, you’ll get the same line both times!

Shock and awe. *Oh, we get the same line? No matter which way we do it?*

Yes! That’s the point – these two methods are *different ways of answering the same question correctly!* Remember last class, when we talked about how we can think of a line as a path, and the two methods of being different ways of giving directions? I can say “start at 8th and Burrard and head north”, which is like a point – 8th and Burrard – and a slope – the direction “north”. Or I can say “start at 8th and Burrard, go to 7th and Burrard, and keep going in the same direction”, which is like two points. These are two ways of describing the exact same route. Just like the two points, or the point-slope method, will give you the exact same line.

Pause that gets pregnant, gives birth, and raises twins to adulthood. *But which method is BETTER?*

SWEET JESUS GOD, THE POINT-SLOPE METHOD, ARE YOU HAPPY NOW?

Remind me why I bother again? Give them freedom, and they beg for a dictator.