It’s Time to Let Go of Antiquated Edicts in the Mathematics Classroom
Nothing squelches joy more than an edict handed down from on high without rationale or context. Yet so much of mathematics pedagogy took such a turn to this approach over the past two centuries. With a focus on pencil-and-paper computation, to be done with facility and speed, it may well have served students in their daily lives to learn mathematics calculation with drill and push-ups. (We expected shop clerks to compute change for $6.37 from a ten-dollar bill by quickly pulling out pencil and paper and performing a quick long-subtraction algorithm, right?)
Do we really want this for our students today?
We have a generation of mighty flexible and agile thinkers heading into a world seeking innovative problem-solvers. I believe we can teach all we’ve taught before, and expect more, and moreover, do so with joy and natural ease for student and teacher alike.
One first and easy step is to let go of edicts and the mindset that accompanies them. The biggest culprit here is the standalone word “simplify.” The US Common Core State Standards have eschewed use of this word as command, as have I in all my decades of teaching. Without context, it means nothing.
Consider, for example, this command.
My personal answer is: It looks just fine to me. Leave it as it is! Of course, textbooks expect students to rewrite the quantity as twice the square root of five. But really. Does that look simpler?
Of course, the issue is not the looks of this quantity — mathematics does not have an innate preferred presentation — but instead the task you might want to do next with the quantity. There needs to be a next! For instance, if I want to know between which two integers the square root of twenty lies, then I would leave the expression as is — it is the simplest way to see that the number lies between 4 (the square root of sixteen) and 5 (the square root of twenty-five). If, on the other hand, I am wondering if a square of area 20 and a square of are 5 will fit side-by-side in a square of area 40, then I might indeed want to rewrite the quantity as twice root five.
Don’t get me wrong, we can and should teach our fabulous students to rewrite radicals in multiple ways and it is okay to have exercises that promote algebraic fluency. But without context first, those exercises are likely to be joyless boot-camp drills and nothing more.
The mathematics curriculum is rife with arithmetic and algebraic edicts that propagate the image that mathematics is remote, distant, secret, and context-less: one must rationalize the denominator, one must reduce fractions, improper fractions are improper, we should seek a common denominator, set quadratic equations equal to zero, cross multiply, collect like terms first, and so on. These might be good rules of thumb, but they are not edicts and we need to change the nature of these adages.
Try, for instance, solving the following questions. Better yet, try to playing with each of the edicts in your school curriculum and have your students create contexts for moving either direction with them: to “simplify,” to not simplify, or even to un-simplify! Let’s not settle for uni-directional facility. Let’s go for students having full ownership of their mathematics.
* In my day we did not have calculators but instead booklets of values of standard quantities: square roots, cube roots, logarithmic values, trigonometric values, and the like. If we were required to find the decimal expansion of 1/sqrt(2), say, we’d look up sqrt(2)=1.414 (to three decimal places) and use pencil and paper to then compute 1/sqrt(2). Can you see why we were encouraged to rationalize the denominator? This was good advice pre-calculator days!