What constitutes mathematical thinking? I am teaching a course on mathematical thinking without defining the term for the students or myself. I want a working definition to arise from the explorations we undertake during the course.
After the 2nd session—and without consulting texts that might provide a well-thought-out definition for me—I am thinking that mathematical thinking might have three components:
- a prediction or projection, e.g., what is 3 +4? The prediction or projection is straightforward here; the answer is 7. What will be the population of the earth by 2050?
- systematic, repeatable steps to move from the initial materials (e.g., 3, 4, +) to the prediction or projection.
- patterns in the steps so different predictions or projections can be seen as instances of a general class (e.g., addition as moving on the number line from the first number a distance given by the second number).
Extensions of mathematical thinking—which we might call critical mathematical thinking—would be when we identify alternatives to a given set of steps or pattern in those steps that indicates that our thinking is constrained by relying on the resulting prediction or projection. (The contrasting implications of population growth in two islands described here indicates that standard projections of population growth obscure the fact that “the analysis of causes and the implications of the analysis change qualitatively if uniform units are replaced by unequal units subject to further differentiation as a result of their linked economic, social and political dynamics.”)
Comments welcome to nudge me to develop these definitional efforts further.