# What is Mathematical Thinking?

Mathematical Thinking does not mean making top grade in math classes. It isn’t always found in those who have taken many years studying math in college. Many of your math teachers do not understand mathematical thinking.

Mathematical Thinking isn’t about what you KNOW but how you THINK about math. It includes four characteristics:

1. A habit of questioning

A child might ask “What is the biggest number? Why isn’t there a biggest number? Why do we count big things and little things the same way. This glass has water that is higher than the others. Why doesn’t that mean it has the most water?

Mathematicians through the ages have explored methods of measurement. How can we measure the distance around the earth …. the distance to the moon, sun, or stars, or the size of atoms. Great mathematical questions are questions with no known answer. Just as some people enjoy doing crossword puzzles or jigsaw puzzles, the mathematical thinker finds great joy in a mathematical questions that others have not been able to solve.

2. Creative Problem Finding

The mathematical thinker doesn’t wait for someone else to ask the question. They look for opportunities to apply mathematics to all areas of life. It could be in economics, tracking economic trends, in geology, studying the rates of change in the earth’s crust, or perhaps in the study of weather and global warming. Everywhere they look, they search for patterns and ways to use math to understand the world in new ways.

3. Inventive Problem Solving

They don’t just use basic problem solving methods. They look for a new and creative strategies for solving problems. Sometime, to solve an important problem, they invent an entirely new branch of mathematics.

4. Use mathematical methods of proof

We have seen movies or pictures where a mathematician has worked on a problem, covering an entire blackboard with his calculations. It might take years to discover the way to begin with what we know to mathematically prove a new idea. To prove a science theory, we use experiments, but even the most careful experiments cannot prove an idea beyond all doubt. New evidence mightbe found and scientists must re-think their ideas. In Math, the proof is absolute.

## How does this apply to Students in High School or College?

1. Ask questions about mathematics.

Ask why we use a certain method to solve a problem. Ask how mathematical procedures were discovered. Ask how you might apply the procedures you are learning in solving practical problems.

2. Be a problem finder.

When you play a game, watch the team play football or other sports, use a recipe to make a new dessert, visit the doctor’s office, or watch the evening news, think about ways math is being used in these areas, and new ways it might be applied in the future.

3. Use different strategies for problem solving.

After you understand how your teacher expects you to solve a problem, look for other methods that might work. You aren’t likely to discover new and better methods, but when you can do a problem in several ways, you can begin with one method and check your work with another. If you can’t remember how to solve a problem using algebra, you might use the “guess and check” method.

4. Practice Mathematical Reasoning.

When you learn something new, try to understand the problems, the methods used, and why they work. As you look at the problems on the next page and my explanations on the page after that, you will see how I often explain things that might not have made sense to you before. If you do understand all the problems, you are already using mathematical reasoning.

You can use your math notebook to write your own explanations for new ideas you learn in math. Understanding math is much more important that making straight A’s in math. Making straight A’s is high school, especially if you forget much of what you had studied, won’t be much help in college. Understanding math, learning mathematical reasoning, will help you in college and for the rest of your life.

5. Use Mathematical Thinking in other subjects. You might evaluate statistics used as evidence or at probablity improperly used. Too many people are mathematically illiterate.

## Test Yourself: A Tests of Mathematical Reasoning

The next page has 15 questions about math. You don’t need high school math to answer these questions. Middle school students knows everything they need to know. This is a test of how well you understand and can reason on problems based on elementary school math.

The test of Mathematical Reasoning

You might also want to read: Study Math and Improve Math Skills: 12 Tips for Learning Math