Lesson 0.3

Convincing Yourself and Others

Focus: constructing convincing arguments; the difference between showing and proving

This lesson introduces the most important distinction in the course: the difference between showing that something appears to be true and showing that it must be true.

The Chessboard Paths task makes this distinction unavoidable. To show that a path exists from a given square, one example is sufficient — you draw it, and the question is answered. To show that no path exists, a single failed attempt proves nothing. A hundred failed attempts prove nothing. What is needed is an argument that explains why, structurally and necessarily, no path can ever exist. Students feel this frustration directly, and it is exactly the frustration you want them to feel.

The coloring argument — that a 5×5 grid has 25 squares, the path must alternate colors with every move, an odd-length path must start and end on the same color, and the arithmetic forces an impossible final step from certain starting squares — is not obvious. It requires a way of seeing the grid that students do not arrive at immediately. Give them time. The argument is worth finding slowly.

I frame this task as a story about James Tanton, who shares in his Teaching Company Geometry course that he spent a long time as a student lying in bed tracing paths on a ceiling grid — a puzzle he carried unresolved before the final argument eventually arrived. He was a student at Princeton when the insight came to him, walking to class one morning. I tell this story not because Princeton is impressive but because it establishes that serious mathematical thinking takes time, that even capable mathematicians carry unresolved questions, and that the goal here is persistence, not speed. Students who have not solved the puzzle by the end of class are not behind. They are doing what mathematicians do.

The Socrates moment is worth including if the conversation allows it. When students grow frustrated with being asked why the squares that don’t work can’t work, it is genuinely useful to acknowledge the annoyance — and to note that Socrates was so relentless about asking why we believe what we believe that his fellow Athenians eventually made him drink hemlock. The lesson is not that asking why is fatal. The lesson is that it has always been uncomfortable, and that the discomfort is a sign the question matters.

On organization as a mathematical tool

One reflection from teaching this lesson: I forgot to emphasize that shading in the squares that worked on the chessboard grid is itself an act of mathematical organization. How we arrange our findings determines what patterns we are able to see. A student who checks squares randomly may find many that work and not notice any structure. A student who organizes results spatially — by shading, by coloring, by listing in order — begins to see the pattern that makes the argument possible.

Point this out explicitly during consolidation. Some students noticed symmetry and rotational structure in the working squares and used that to reduce their work. That is a mathematically significant observation, and it should be named as such.

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