Seeing an exercise as a single mathematical object: using variation to structure sense-making

Author: Anne Watson, John Mason

Year: 2006

Paper Description:

The paper discusses an exercise in taxi-cab geometry designed by Krause and how it can provide valuable insights into learners’ responses. The exercise involves calculating the shortest distance between points on a coordinate grid using only horizontal and vertical movements. Participants’ reactions to the task were observed and recorded in various settings, providing a comprehensive understanding of their responses.

Key Takeaway 1:

Participants reported becoming more fluent, motivated, and accurate with each successive point. They dealt with non-integer coordinates by interpolating from the initial points. Some learners independently created additional points that met the same criteria before being asked to do so. The exercise encouraged learners to seek similarities, make conjectures, and think mathematically about taxi-cab geometry.

Key Takeaway 2:

The paper discusses the role of variation in promoting mathematical sense-making. It emphasizes the importance of structuring exercises to facilitate learning, allowing learners to discern patterns and make conjectures. The sequence of tasks and their variations influence learners’ experiences, promoting conceptual understanding. The author highlights the need for careful task design and reflective abstraction to optimize learning outcomes, acknowledging that teaching and learning interactions can be complex and context-dependent.

Standout Quote

“The structure of the exercise as a whole, not the individual items, promotes individual disturbance and common mathematical sense-making.”


variation theory, variation, pedagogy, mathematics, task design,