- “So we’re just supposed to KNOW that 7x^2 is like 7 TIMES x^2?”
- “There’s no solution [to the equation x^2=3.52534231], because you can’t take the square root of a decimal.”
- “Okay, I see how you got 2x=1.46, but how did you get x=0.73 from that?”
- “Last week’s test was totally unfair – you never gave us a homework problem where we had to graph a rational function where the denominator was like x^2-4x – it was always like x^2-4x plus some NUMBER.”
- “Can we just plug those numbers into our calculators instead? Because, like, I’m not very good with fractions.”
- “Miss, I think you made a mistake grading my test. I had (x(x-2)+3x(x+4))/((x-2)(x+4)) , and then I cancelled out the x-2 and the x+4 to get x+3x=4x, but you marked that wrong.”
These are my students who desperately need grades of at least C+ in my class in order to be permitted to take calculus next year. What’s sad is that many of them seem to be under the impression that I am the one thing standing between them and success in university math classes – if only I’d just make my tests easier, they could advance, and surely their calculus prof next year won’t expect them to be able to solve any math problems that they haven’t seen word for word before.
Ideally I’d have taught them well enough that they’d have earned their C+’s and above – but I’m just not a good enough teacher that I can take kids who can’t do grade school math and mold capable university pupils out of them in one semester. As far as I know, there’s only one teacher who was ever able to do that, and they made a movie about him. So, having given up on educating my weakest students, I’ve committed myself to doing the next best thing: ensuring that they don’t get the grades to take calculus next term.
And this is what I’m keeping in mind in designing the final exam. I’m testing them on the material – graphing functions, solving various flavours of equations, factoring polynomials – but I’m specifically designing the questions so as to trip up everyone who hasn’t learned grade 9 math. I could ask them to complete the square of a quadratic with integer coefficients, but I like fractions.
I could ask them to solve a problem that reduces to finding the positive and negative square roots of 7 – but I prefer the number 7.5. I could give them a problem in which they’re given the distance in kilometers and the time in hours – but I prefer minutes. I could ask them to sketch the graph of f(x)=2^(x+3), but I rather like the graph of g(x)=-2^x – which many calculators interpret as g(x)=(-2)^x.
The students who belong in university math classes will do fine; the exam is easier than my tests. Those who memorized the problems I did in class without bothering to learn any grade school math will do miserably.