What is Mathematical Thinking and why is it important?

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What is Mathematical Thinking

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.

Identified four fundamental processes, in two pairs, and showed how thinking mathematically very often proceeds by alternating between them:
  • Specializing – trying special cases, looking at examples
  • Generalizing - looking for patterns and relationships
  • Conjecturing – predicting relationships and results
  • Convincing – finding and communicating reasons why something is true.

Mathematical thinking is important as a way of learning mathematics

  • Recognize that all reasoning depends on assumptions. Scientists seek truth. Mathematicians don’t. 
They just seek truth relative to starting assumptions. They know that angles on a triangle add up to 180 degrees only if you assume that you’re on a flat plane — on a sphere, they can add up to 270 degrees or more. When you’re reasoning (or critiquing reasoning) about the world, always question the starting assumptions. Different assumptions lead to different results.

  • Believe you could be wrong. When a mathematician sets out to prove a conjecture, there are two possible results. 
One is that it’s provable. The other is that there is a counterexample, showing that the conjecture is wrong. (The third possibility — that you can’t prove it either way, we try not to talk about.) If the result is unknown, you must work both sides — find a proof, or find a counterexample. Pursuing both possibilities creates clarity and progress. You’ve got to research how you might be wrong.

  • Value intuition and ideas. People think mathematicians are about logic. 
They’re not. Mathematicians have big ideas in their head that inspire what they research — their proofs spring from that intuition. (One prof I had used to write the intuitive story on a separate blackboard, so we could see where the ideas behind the proofs came from.) There is no contradiction between powerful ideas and locked-down reasoning — you need the ideas to inspire you, and the reasoning to prove you are right

  • Question numbers. The relationship between mathematics and the real world is fuzzy. 
Mathematical concepts have nice clear answers, while real-world results have uncertainty. When you connect the two, you must know where the uncertainty comes from and what its consequences are. Bring the same skepticism to any numbers you read or measure. Polls are not truth. Economic estimates are not truth. And nothing is exact.

  • Model things. I found an amazing thing once I landed in business. 
Using nothing but a spreadsheet, with formulas based only on addition, subtraction, multiplication, division, and percentages, I could model the real-world. Those models generated insights that weren’t available any other way. So now I model everything from consumer segments to Supreme Court n
Location: Nashua, NH, USA

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