### An equation is a relationship among quantities.

That’s the sentence that I utter every time I present a word problem on the blackboard in my precalculus class. Then I slowly, *slowly* outline what quantities we’re working with, which are known, and which are unknown, and how they’re related. Then I work through the problem, and when I’m done, I go over how - and why - we set up the equations that we did.

Today, I graded 30ish tests, and saw that I might as well have spent the above time reinforcing the prevailing view that equations are voodoo amalgamations of letters and numbers, for all the good my attempts did. (Boy, am I ever feeling Rudbeckia Hirta’s frustration today.) Answers to a “rowing with the current/rowing against the current” question revealed that while close to a quarter of my students seem to know that speed=distance/time, about as many are under the impression that speed=time/distance, and many others hold fast to the view that speed=distance*time. I’m sure there’s a bad scifi novel in here somewhere; I’m equally sure that if I asked a random student “if I drive 100 km in 2 hours, what’s my [constant] speed?” they’d reply “50 km/h”. Anyway, this may have happened because I sneakily gave the time in *minutes* not hours, so the time/distance formula had the advantage of giving a whole number instead of one of those tricky fractions. Another student opted to forgo multiplication and division altogether for this problem, and wrote down the formula “speed of current + 50 minutes = 10 kilometers”. The problem involved the current on the return trip being half of its original value, so some students squished all of the numbers in the original problem together in some haphazard fashion, threw in an *x* so they’d have something to solve for, and then multiplied the whole thing by one half.

A supply/demand problem had a number (not zero) of students finding that the equilibrium occurred when widgets were sold for negative thirty bucks a pop. No one appeared to bat an eye over this one; they just stated their conclusion and moved onto the next problem. On a question about finding the dimensions of a structure with given area and given amount of fencing, a plurality of students faithfully parrotted the formula that perimeter=2*length+2*width, apparently not noticing (or caring) that the fencing of the figure in question (I provided a diagram) did not surround a rectangle. Nor did it surround a triangle, but many students seemed eager to show off their knowledge of the Pythagorean Theorem.

The class average, despite all that, wasn’t that bad; it’s just that the poorly-done tests were *really* poorly done, and many of my students seem to have no idea what mathematics is. There was no point in writing comments on some of the papers; so profound is the disconnect between my weakest students and the subject matter.

I have no idea what I’m going to say when I hand these tests back tomorrow. Right now, I’m mostly *angry* with my students - haven’t they been *listening* to what I’ve been saying all term? If they were, then they clearly didn’t understand me - why didn’t they ask, in private during office hours if not in the classroom? Much of my frustration is justified - attendance in that class sucks, no one showed up to my office hours during the week before the test - but many of my students are trying. They just don’t have the background right now, and I don’t think I can provide it. But that’s not a productive way to give back the test papers. Students who bombed a test don’t need to be told that I’m disappointed in them; they’re more upset than I am as it is.

Fellow high school/college teachers: what do you say to your students when you hand back a terrible test? I don’t want to upset them, but at the same time - I ain’t scaling these test marks, and the course isn’t going to get any easier.