What Math Teaching Taught Me About AI Training and Evaluation

As a former classroom teacher, I was fascinated by how students think about mathematics. I never expected my daily classroom moves—(1) how I framed questions, (2) when I let learners struggle, and (3) how I “listened” to their reasoning to converge on the same fundamental question both fields now share: how do we nurture environments where complex reasoning can emerge?

Below, I connect these classroom principles to recent AI breakthroughs, weaving together the evidence to argue that intelligence, whether human or artificial, thrives when it is not simply optimized for efficiency, but allowed to explore, struggle, and adapt.

Yes, we need to make one small but important fix to the piece since the Webb reference was removed.

The original sentence in the "From 'How Many?' to 'What Can You Tell Me?'" section used both Webb and Kapur to support a claim. Since we found that the Webb source is not verifiable, we must remove it from the sentence. The corrected Kapur reference is sufficient to support the point about the benefits of open-ended tasks.

Here is the revised paragraph:

1. From "How Many?" to "What Can You Tell Me?"

Question Framing Shapes What Reasoning Emerges

Consider the pyramid pattern shown here. I could ask one of two very different questions:

Option A: “How many blocks are in Case 10?”
Option B: “What can you tell me about Case 10?”

The first question has a single correct answer (121 blocks) but reveals almost nothing about how the student thinks. The second question, however, opens a universe of possibilities. It unleashes multiple, equally valid lines of attack—some students attend to the center stack, others to height symmetry, others to the quadratic rule. In math-education studies, tasks framed like Option B predict 2–3× larger gains in transfer tests (Kapur, 2010) because they reveal a richer, multi-pathway understanding.

AI analogue. This principle has recently produced a stunning breakthrough. Google DeepMind's Advanced Gemini with Deep Think achieved a gold medal at the International Mathematical Olympiad by "generating multiple solution candidates" and exploring different proof pathways for each problem. Crucially, this wasn’t a change to the model's core training objective. Instead, at inference time, the system mirrored the diversity of student pathways elici...

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