Habits of Math

This is an aspirational framework outlining what we see as the values and characteristics associated with being a mathematician, whether the mathematician is a five-year-old playing with chalk or Sophie Germain corresponding with Carl Gauss. The values of empowering, collaborating, and persevering form the cornerstones of engaging in the subject. Mathematicians play with structure and rules by appreciating, applying, tinkering, inventing, pattern sniffing and experimenting. Mathematicians then make sense of unfamiliar or novel structures and rules, using strategies such as organizing, communicating, reasoning, proving, representing, connecting, and reflecting.

CULTURE OF MATHEMATICS

EMPOWERS

Everyone has the ability and right to engage in deep mathematical thinking, regardless of identity markers, age, experience, and current content knowledge. We acknowledge that math has been and continues to be used as a tool of oppression, but see math as a tool for justice: a mechanism to critique social constructs, promote positive change, and empower ourselves and others.

COLLABORATES

Math is a social and collaborative endeavor. Mathematicians share, encourage, build-on, and connect with others' ideas. We value different ways of being good at math. We act as skeptical, yet respectful peers. We openly invite and encourage others to join us in playing with and making sense of mathematics, limiting barriers for access as much as possible.

PERSEVERES

Engaging in mathematics necessitates being comfortable with struggle, both in recovering from being stuck and/or making mistakes. We thoughtfully balance perseverance and the use of resources and appropriate tools. We value challenge, depth over breadth, and risk-taking.

PLAY PHASE

How do mathematicians explore, investigate, and create?

APPRECIATES AND APPLIES MATHEMATICS

You see math as a creative endeavor, an art form, a language to describe the world, a set of tools and structures for solving problems, a process for modeling the world, a historical, cultural subject, and a tool for social justice. You find joy in finding solutions, being surprised by a result or connection, and uncovering patterns.

TINKERS AND INVENTS

You play with existing problems and problem spaces, changing the constraints as needed to make them interesting for you. You create mathematical problems, language, algorithms, rules, or models of your own, connected to your own interests and curiosities.

PATTERN SNIFFS AND EXPERIMENTS

You are always on the lookout for patterns that may lead to conjectures or sense-making, including looking for and creating shortcuts and procedures. From discovered patterns you make guesses, celebrating the surprise when these guesses turn out to be wrong.

SENSE-MAKING PHASE

How do mathematicians understand and communicate?

ORGANIZES AND COMMUNICATES

You clearly and concisely organize your results, conjectures, arguments, processes, proofs, questions, and reflections to uncover underlying order. You aim to communicate succinctly and elegantly; clearly communicating process and reasoning through multiple modes; and effectively targeting communication to an appropriate audience.

REASONS AND PROVES

You use estimation to make sense of problems or to create lower and upper bounds. You create and test conjectures based on mathematical structure instead of data. You work from first principles. You prove conjectures using flexible techniques. You can identify mistakes or holes in proofs proposed by others. You can explain both how and why your process works.

REPRESENTS AND CONNECTS

You connect different representations. You don’t start from scratch, applying and connecting old skills and concepts to new material and identifying and exploiting similarities within and between problems. You create analogies. You determine whether a problem can be broken up into simpler pieces. You move between contextualized and abstracted representations, and use pictures and manipulatives when helpful.

REFLECTS

When introduced to a problem, you determine explicit and implicit constraints of a problem, along with useful and superfluous information. During the process of exploring and/or solving a problem, you consider whether your current path is fruitful. After coming to a conclusion, solution, or answer, you ask whether your answer makes sense and consider whether there are alternative paths.