# Mathematical Thinking and Problem Solving

Mathematics is

a way of thinkingthat can help make muddy relationships clear. It is a language that allows us to translate the complexity of the world into manageable patterns. In a sense, it works like turning off the house lights in a theater the better to see a movie. Certainly, something is lost when the lights go down; you can no longer see the faces of those around you or the inlaid patterns on the ceiling. But you gain a far better view of the subject at hand. — K.C. Cole bold added, p.2

Cole, in her wonderful book, *The Universe and the Teacup,* describes two ways of describing mathematics.

“William Thurston, the director of the Mathematical Sciences Research Institute… calls math a kind of ‘mindware.’ It — allows us to see and articulate concepts we can’t handle in any other way.

Ingrid Daubechies …– Princeton mathematician who resurrected wavelet analysis (a tool for doing everything from storing fingerprints to seeing stars) — says “it’s akin to poetry: a way of taking a big idea and condensing and honing it until it communicates exactly the right information.” ( 2)

Cole gives her own description or metaphor. She was flying near Boston and looked out the window.

She “saw islands that were clearly connected under the shallow water by strips of land. On the ground, these links would have been invisible, the islands completely unconnected. From the air, the paths between them were laid out as clearly as road maps. … In the same way, the tools of mathematics allow one to see otherwise invisible patterns and connections.”(7)

She continues to name some of those patterns and hidden trends. “HIV infection, new kinds of matter (quarks, dark matter, antimatter), and crucial correlations (between smoking and lung cancer).”

## How can you use Mathematical Thinking?

Obviously, if you are studying math, science or business, you will need skills in Mathematical Thinking. Other students often celebrate when they completed their last math class, claiming they would never need to use math again.

Well, of course they will need to make change, balance a checkbook, use a budget, calculate Income tax, plan for your retirement and that sort of thing. What else will they need?

In many fields of study, basic research is presented using statistics and a variety of graphs. To really understand this material (or to write that thesis), it is often important to consider statistical information. You need to be able to interpret the graphs accurately. You need to understand articles describing the probability of something happening. When you use critical thinking, you need to understand the difference between 4.5 and 4.500 and decide if the person writing research understands the difference between exact numbers and measured numbers.

Even reading and understanding the newspaper, including the advertising, requires an understanding of statistics, percentages, probability as well as very large numbers – as in national debt – and very small numbers such as parts per million. You might rely on your pocket calculator to do your calculations but you need to understand basic mathematical concepts to understand and solve problems.

## How can we think more mathematically?

After you take the Quick Test and check your answers, this section will continue to describe Nine Ways of Mathematical Thinking.

Mathematicians reading this may want to suggest numerous additions. Many of these categories were adapted from *Curriculum and Evaluation Standards for School Mathematics.*(1989)