Christopher Welch

Design problem: Completing the square is usually taught as a memorized recipe, so the −(b/2)² correction feels arbitrary and errors follow.

Learner & objective: Algebra students meeting the technique for the first time (GCSE / US Algebra 1–2). Objective: connect each algebraic move to the geometry so the correction term is understood, not recited.

Key decisions: A four-stage reveal (pieces → rearrange → see the gap → complete it) with the algebra line synced to the figure, and a slider over b so the structure generalizes instead of being one worked example.

How I would evaluate it: Before the final stage, ask the learner to predict the correction term for an unseen b; success = correct prediction and a one-sentence reason referencing the corner square.

Design problem: The sign rule is memorized without justification, so it is misapplied under pressure and erodes trust that mathematics makes sense.

Learner & objective: Middle-grades learners (US grades 6–8). Objective: replace “because the rule says so” with a pattern the learner extends themselves, then a two-line proof from facts they already accept.

Key decisions: Predict-before-reveal stepping (the learner commits to the next line before seeing it), then an optional distributivity proof kept behind a click so the pattern comes first and the formalism second.

How I would evaluate it: Ask the learner to rebuild the argument with different numbers; success = the pattern and the proof both reproduced without prompting.

Design problem: Learners meet the derivative as a rule to apply and never see the limit process it names, which surfaces later as brittle understanding in applications.

Learner & objective: Precalculus / early calculus students (A-Level, US grades 11–12). Objective: experience the secant-to-tangent limit as an observable process in graph and symbols simultaneously.

Key decisions: One control drives the graph and numerical readout together. The visualization makes the approach observable; the algebra establishes the exact limit.

How I would evaluate it: Pause near h = 0 and ask the learner to state the exact slope the secant slopes are approaching and explain why the difference quotient is evaluated only for nonzero h. Success means identifying the limiting slope and distinguishing the limit process from direct substitution at h = 0.

Design problem: Proof is usually introduced symbolically, which excludes learners who could reason structurally before they can write induction.

Learner & objective: Any learner meeting proof for the first time. Objective: build the conviction that the result must hold — a bridge into formal proof, not a replacement for it.

Key decisions: Each odd number is drawn as a colored L-shaped layer wrapping the previous square, with the running sum synced beneath, so the identity is visible layer by layer.

How I would evaluate it: Ask why the next layer always adds an odd count of cells; success = an answer referencing the two sides plus the corner (2n−1), which is the seed of the formal argument.