Mathematical Habits of the Mind.

One advantage to having internet access is that one can explore various Master's and PhD papers and dissertations.  I love finding research to help me learn to become a better teacher.

This paper "Five Processes of Mathematical Thinking" by Toni Scusa caught my attention because I'd never considered the processes connected with mathematical thinking.

The author identified the five key areas as representation, reasoning and proof, communication, problem solving and connections.  These are also referred to as "Mathematical Habits of the Mind."

Let's look at each area in more detail and why it's important.  When a student is able to create representations designed to show mathematical concepts or relationships, they have gained tools to expand their ability to model and interpret a variety of situations.  Furthermore it shows they understand the material better.  Representations can range from simple drawings to the use of manipulative including tiles, and Legos.

When a student is able to reason well, they find it easier to think analytically so they can "see" patterns, and structure in both real world and mathematical situations.  With good reasoning skills, students can determine is patterns occur regularly or if they happened by accident.  Furthermore, they are able to create and investigate conjectures, and create and argue for their proofs.  They know why.

As far as problem solving, students need to have persistence because if one method does not work, they need to stand back, review, and try again.  When a student knows multiple ways to problem solve, it allows them to be curious about things and it builds confidence to attack problems both in and out of the classroom.

When students are able to communicate clearly they are able to use mathematical language concisely in either verbal or written form.  These four lead to the student being able to connect prior knowledge to the current situation, concepts to the abstract application, and connect similarities among the processes used to solve problems such as the distributive property and its application to multiplying binomials in Algebra.

All of these processes of mathematical thinking or mathematical habits of the mind allow students to think about mathematics in the same way that professional mathematicians do.  The nice thing about habits is they are automatic which from a brain point of view means students are able to focus more on the concept rather than simple calculations.

Another thing about habits is that they can be good or bad.  Some bad habits mathematically is a person looks for a fast answer, gives up quickly because its seen as "too hard", prefers to memorize things rather than have true understanding, guesses a lot.  On the other hand, good mathematical habits are a willingness to stick to a problem until the answer is found, a desire to explore a curiosity, visualizes, makes comparisons, and noticing the patterns, etc around us.

A simple paper sent me off to an exploration of a topic.  I'm currently exploring speech to text for mathematical equations, etc.  I hope to report on this next week because I'd like to have students talk out solving a problem and have it show up in written form.  Let me know what you think about either topic, I'd love to hear.  Have a great day.