Ifeoma sat in the third row and almost never finished a test. Not because she was slow at arithmetic — she was quick — but because she would not write down anything she was not sure of. Blank space where the rest of the class had confident nonsense. Her scores were poor. By every measure the school cared about, she was struggling. It took me most of a term to work out that she was the strongest mathematical thinker I had. What the marking scheme rewards Consider what a multiple choice question actually measures. A student who knows the answer scores. A student who eliminates two options and picks between the rest also scores, half the time. A student who recognises the shape of the question from a worked example and pattern-matches to it scores too, often. And a student who notices that the question is ambiguous, or that it depends on an assumption nobody stated, sits there — because the honest answer is "it depends" and there is no box for that. We spend twelve years training people to produce a plausible answer under time pressure, and then we are surprised when adults cannot say "I do not know". The turn I started giving her problems with no clean answer. A question about the surface area of a real object that was not quite a cylinder. A rate problem where the rate was not constant and the data did not say how it varied. Things with a genuine "it depends" at the centre. She was transformed. She would come back with three pages: here is what I assumed, here is what follows, here is where it breaks, here is what I would need to measure to decide. That is not a school exercise. That is what applied mathematics is. What I changed I did not stop preparing them for the exam. That would have been a betrayal — the exam is real and it decides things about their lives, and a teacher who sacrifices his students’ results to his own pedagogical principles is doing something indefensible. But I added two habits, and they cost almost nothing. A confidence mark next to every answer: sure, fairly sure, guessing. Ungraded. It made the difference between knowing and guessing visible to them, which it usually is not. One question per test with no determinate answer, marked entirely on the reasoning. The strong pattern-matchers hated it at first. Ifeoma’s scores came up, eventually, once she made peace with the exam as a game with rules rather than a description of her mind. She reads engineering now. But I think about the version of that term where I never worked out what the blank spaces meant, and simply recorded her as weak — and I wonder how many of those I have already missed.
