Mathematical Mindset#

Related: mathematical-mindsets - vague notes from reading Boaler (2015)

Definitions#

Boaler (2015, p. 34) describes a mathematical mindset this way

Successful math users have an approach to math, as well as mathematical understanding, that sets them apart from less successful users. They approach math with the desire to understand it and to think about it, and with the confidence that they can make sense of it. Successful math users search for patterns and relationships and think about connections. They approach math with a mathematical mindset, knowing that math is a subject of growth and their role is to learn and think about new ideas. ... When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing. When students see mathematics as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have a mathematical mindset.

The issue being that most common approaches to mathematical education has the result of producing fixed, procedural mindsets e.g.

When students see math as a series of short questions, they cannot see the role for their own inner growth and learning. They think that math is a fixed set of methods that either they get or they don’t. (Boaler, 215, p. 34)

Examples#

Example given of number-sense

Contributing factors#

Boaler (2015) discusses math-facts and the common practice that such are taught in isolation. Creating the impression that math-facts are the essence of mathematics.

Methods#

Illustrate definitions through multiple examples, including non-examples
Avoid practice-approach-to-mathematics
Prefer a conceptual-approach-to-mathematics

Daly et al (2019) summarises Boaler's (2015b) recommendations as

In general, the MM recommendations for writing a mathematical problem can be briefly summarised as follows (see Chapter 5 in [11]for both this list and a more detailed discussion). - Open up the task so that there are multiple methods, pathways, and representations. - Include inquiry opportunities. - Ask the problem before teaching the method. - Add a visual component and ask students how they see the mathematics. - Extend the task to make it lower floor and higher ceiling. - Ask students to convince and reason; be skeptical.

Questions arising#

What are the important concepts in mathematics? What is their relationship with other concept and with the methods/algorithms/rules that are taught/learnt?

References#

Boaler, J. (2015). Creating Mathematical Mindsets: The Importance of Flexibility with Numbers. In Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching (pp. 33--56). John Wiley & Sons, Incorporated.

Boaler, J. (2015b). Rich Mathematical Tasks. In Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching (pp. 57--92). John Wiley & Sons, Incorporated.

Daly, I., Bourgaize, J., & Vernitski, A. (2019). Mathematical mindsets increase student motivation: Evidence from the EEG. Trends in Neuroscience and Education, 15, 18--28. https://doi.org/10.1016/j.tine.2019.02.005