# Habits of a Mathematician

The purpose of the Habits of a Mathematician is for students and teachers to have a common language when talking about “doing mathematics” (as opposed to, for example, developing procedural fluency).
Look for Patterns

Mathematicians are always on the hunt for patterns and regularity.

They bring a skeptical eye to pattern recognition.

They look for reasons why a pattern exists.

Start Small

Mathematicians start with small cases.

They take a large, complex problem and work first with a simpler case.

They build on their success with small cases.

Be Systematic

Mathematicians experiment methodically and systematically.

They make small changes to look for change and permanence.

Take Apart and Put Back Together

Mathematicians look for structure and repetition.

They use structure/repetition to break complexity into a number of simpler problems.

Conjecture and Test

Mathematicians ask “What if…?”

They invent numbers to test relationships.

They use specialized cases to test their ideas.

They modify conjectures to continue/deepen exploration.

Stay Organized

Mathematicians use charts and other methods to organize information.

They organize their work in a way that can be referenced by others.

Describe and Articulate

Mathematicians visualize.

They draw pictures and diagrams to represent a situation.

They invent notation or representations to facilitate exploration.

Seek Why and Prove

Mathematicians seek to understand why things are the way they are.

They look for clues that help them understand why.

They create logical arguments that prove their ideas.

Be Confident, Patient and Persistent

Mathematicians have faith in their abilities.

They are willing to take risks.

They will explore even when they are unsure about where to start.

They do not give up easily.

Collaborate and Listen (with Modesty)

Mathematicians listen carefully to and respect the ideas of others.

They effectively use strengths of people around them.

They ask people for their thoughts when they seem hesitant.

They don't dominate a conversation.

Generalize

Mathematicians look to understand all cases.

They invent notation that helps them generalize.

They algebraically represent the structure they see.