Mathematical Habits of Mind

The widespread utility and effectiveness of mathematics come not just from mastering specific skills, topics, and techniques, but more importantly, from developing the ways of thinking—the habits of mind—used to create the results. —Al Cuoco, Paul Goldenberg, and June Mark, Habits of Mind: An Organizing Principle for Mathematics Curriculum

In our work, we define mathematical habits of mind (MHoM) to be the specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians. Understanding the MHoM and general purpose tools underlying the various topics and techniques of secondary mathematics content can bring parsimony and coherence to teachers’ mathematical thinking and to their work with students. In this sense, we envision MHoM as a critical component of mathematical knowledge for teaching, i.e., the knowledge necessary to carry out the work of teaching mathematics.
 
At this time, we have focused our research on algebraic habits of mind, and we have also narrowed to three categories of mathematical habits that we think are particularly important to secondary teachers:

  • Engaging with one's experiences (EXPR),
  • Making use of structure to solve problems (STRC), and
  • Using mathematical language precisely (LANG)

 
Engaging with one's experiences (EXPR)
This habit is about what one does when faced with new ideas. Its purpose includes (1) understanding the meaning, context, and purpose of a given situation, and (2) finding general, unifying principles that provide insight and explanatory power. The habit has two main components, distinguished according to how understanding is sought. 

  • One component is using language to acquire clarity and understanding of one's experience. Language can serve to both unpack and compress complex ideas. Thus, the process of clearly describing "what's going on" facilitates sense-making by organizing and framing one's experience. In a nutshell, if you can say it precisely, then you are close to understanding it.
  • Another critical component is experimenting. Examples of the experimental process may include (a) working with smaller or special cases, (b) seeking regularity and/or coherence in repeated calculations, (c) finding and explaining patterns, and (d) generalizing from examples.

We also note that the process of engaging often requires immersing oneself in ideas long enough to see structure within them.
 
Making use of structure to solve problems (STRC)
This habit entails taking advantage of the underlying structure of a given situation to facilitate problem solving. Examples of this habit may include (a) creating, choosing, and/or using representations, (b) writing expressions in equivalent forms to solve problems, (c) interpreting and making use of the structure of expressions, and (d) "chunking" to delay or avoid thinking about certain details in order to see the big picture.
This habit is closely related to EXPR. Broadly speaking, EXPR may be viewed as being about understanding structure, while STRC is about using structure. But they are not mutually exclusive, and mathematicians typically use both habits in conjunction.
 
Using mathematical language precisely (LANG)
Also known as Exercising appropriate mathematical hygiene, the purpose of this habit is to communicate precisely to others. Its examples may include (a) using mathematical notations correctly, (b) defining variables before using them, (c) understanding and appropriately using mathematical terminologies, and (d) giving precise descriptions of the steps in a process.
 
Note these habits are closely related to the following Common Core Standards for Mathematical Practice:

  • MP2: Reason abstractly and quantitatively
  • MP6: Attend to precision
  • MP7: Look for and make use of structure
  • MP8: Look for and express regularity in repeated reasoning