Pascal’s triangle through path-counting: from mysterious identities to visual inevitability

We report on an instructional sequence where prospective mathematics teachers discovered binomial identities—including Vandermonde’s sum-of-squares formula—through systematic engagement with a path-counting model of Pascal’s triangle. Students progressed from proving known identities to independently formulating new ones within 2 weeks. The cognitive architecture enabling this discovery aligns with Minsky’s uniframing theory: students constructed mental structures linking spatial paths, symbolic expressions and combinatorial interpretations. When one student called Vandermonde’s identity ‘beautiful’ and immediately asked about generalizations, we knew the approach had succeeded. Our findings demonstrate that mathematical creativity emerges not from exceptional talent but through carefully sequenced experiences that position outcome sets as primary objects of investigation, invert traditional formula-first approaches and make ‘mysterious’ algebraic results appear as inevitable consequences of visual structure.