# What constitutes mathematical thinking?

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:

1. 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?
2. systematic, repeatable steps to move from the initial materials (e.g., 3, 4, +) to the prediction or projection.
3. 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.

## 3 thoughts on “What constitutes mathematical thinking?”

1. terylcartwrightblog

The neat thing about mathematical creativity is that it is so different from the scientific. It often doesn’t solve the problem, it solves the other problems first. Some examples of this are how the solutions to the sum of infinity and the distance between prime numbers were found. Like scientists though, mathematicians are appreciated if they show their work while artists are appreciated if they don’t.
Patterns are important, building the bridge between is equally critical. I like this video about how the indirect approach to problem-solving works (minute 21 of video if you don’t want to see the whole show) https://www.youtube.com/watch?v=OT0AlFmAkv0

1. Peter J. Taylor Post author

Could studying prime gaps (https://en.wikipedia.org/wiki/Prime_gap) fit under my definition? There are predictions (e.g., about the average gap between primes increases in relation to the natural logarithm of the integer) and steps to establish that prediction–or improve on it (say by exposing deeper patterns). But I would need to check whether people who study prime numbers think in ways that match well the 3 components I define.

2. cquarrington

I think it would be an interesting exercise to try and draw a visual representation of what mathematical thinking is. I too have been struggling to arrive at a concise explanation or definition of mathematical thinking. I did a Google image search of “mathematical thinking” and saw some interesting concepts presented – though still more word-based than graphics/illustrations – but perhaps this would help in the evolution of your understanding of this topic?