Thinking Through Geometry - Free Lessons, Notes, and Tasks for Thinking Classrooms.
Lesson 0.1 – When Intuition Isn’t Enough
Focus: specializing, testing, and the danger of confirmation bias
This is the first day of the course, and the most important thing that happens today has nothing to do with the mathematics. What matters is that students experience, immediately and concretely, that this class is different — that they are expected to think before they are told how, that their ideas are worth testing, and that intuition, however confident it feels, is not the same as knowledge.
The Warehouse Discount Problem is the entry point. It looks simple — deceptively so. Most students feel certain they know the answer before they have tested anything, and a significant number will have compelling-sounding arguments for why one order must be better than the other. Let them make those arguments. Let the room split. Then ask them to check.
What they discover — that the order does not matter, and that this can be shown for any starting price by treating the process as multiplication by three numbers whose order is irrelevant — is mathematically satisfying. But the more important discovery is the one about themselves: they were confident, and they were either right for the wrong reasons or wrong entirely. That experience is the foundation of everything that follows in this course.
Guess My Rule, which closes the lesson, sharpens this. Leonard Mlodinow uses a version of this game in The Drunkard’s Walk to illustrate how science advances: the rule represents nature’s true law, and each sequence a student proposes represents an experiment. Nature — like the teacher in this game — can only say yes or no. It never tells you whether you have found the truth. The only way to distinguish between a hypothesis that is probably right and one that is certainly wrong is to look for evidence that would disprove it.
The Drunkard’s Walk: How Randomness Rules Our Lives — Leonard Mlodinow (Pantheon, 2008)
The Guess My Rule game appears here as a metaphor for scientific inquiry. Worth reading for its treatment of how humans systematically misread randomness and evidence — themes that connect directly to the epistemological thread of this course.
Most students will propose only sequences that confirm what they already believe the rule is. When the true rule — simply that the numbers are in increasing order — is revealed, the reaction is almost always the same: disbelief, then recognition, then something close to embarrassment about the process they used. That embarrassment is productive. Channel it toward the question: what would you have had to ask to figure this out sooner?
On confirmation bias: a useful analogy
If students struggle to see why only looking for confirming evidence is a problem, try this: a doctor who has already decided a patient has a tension headache will ask questions that confirm tension headache — are you stressed, have you been sleeping poorly — and stop there. A hundred confirming answers do not make the diagnosis more reliable. One anomalous symptom, noticed and pursued, might reveal something else entirely.
The Guess My Rule game works the same way. Students accumulate confirming cases and feel increasingly certain. But certainty built on confirmation alone is a feeling, not a proof. The sequence that would have disproved their hypothesis was available to them the whole time — they never thought to ask for it.
Karl Popper put the underlying principle precisely: ‘The dividing line between science and pseudoscience is whether advocates of a hypothesis deliberately search for evidence that could falsify it and accept the hypothesis only if it survives.’ This is not just a scientific principle. It is the intellectual posture this course is asking students to develop.