*‘Mathematics is the music of reason’* – James Joseph Sylvester

Mathematical thinking is a lot more than just being able to do arithmetic or solve algebra problems. It is a whole way of looking at things, stripping them down to their essentials, whether it’s numerical, structural or logical and then analyzing the underlying patterns. Math is about patterns. When we are teaching a mathematical method, we are showing something that happens all the time, something that happens in general. Getting students to see these underlying structures, whether it’s in a math problem, in society, or in nature, is one of the reasons that studying mathematics is so worthwhile. It transforms math from drudgery to artistry.

As Steven Strogatz says, ‘Math is not just what we heard about in high school, the known and straightforward part of the subject. There are all sorts of interesting theoretical and applied problems out there.’ Helping students to see the beauty and wonder in math will go a long way in getting them to think more mathematically. Let them know that what makes mathematicians is the quality of their creativity and the quality of their technique. Combining these two is what leads to great mathematical thinking.

The problem lies in the fact that students usually have no idea about their strengths and weaknesses in these two areas. Schools emphasize the procedural side of math. There is the misconception that it only involves mechanical, robotic thinking. What pervades is a ‘we didn’t cover that’ mentality when they encounter a novel problem.

Once you have identified a task or situation to explore, mathematical thinking involves these steps that are often done together and simultaneously:

- break task down into components
- identify similar tasks that may help
- identify appropriate knowledge and skills
- identify assumptions
- select appropriate strategy
- consider alternative approaches
- look for a pattern or connection
- generate examples

This all leads to formulating a conjecture (solution, generalization or relationship) that can then be tested for counter examples or special cases. Similar to the design process, if it doesn’t work, it’s time to iterate and try again. Proving the solution, making connections, considering limitations, and extending it further to ‘what if’ questions are additional aspects of mathematical thinking. Spending the time to think of and ask really good questions is at the heart of math.

John Mason’s questions and prompts provide a nice framework for ensuring and promoting students’ mathematical thinking. The verbs across the top of the chart highlight what is involved when we ask students to think about math. The specific examples are practical and useful.