What habits of mind do teachers and students of maths need?
Maths is a mindset.
You might be ‘good’ at maths but do you have the essential habits of mind and dispositions to think like a mathematician?
Habits of mind were introduced by Cuoco, Goldenberg, and Mark (1996) as an organising principle for mathematics curricula in which high-school students and college students think about mathematics the way mathematicians do. They asserted,
“The goal is … to help high school students learn and adopt some of the ways that mathematicians think about problems. … A curriculum organized around habits of mind tries to close the gap between what the users and makers of mathematics do and what they say. … It is a curriculum that encourages false starts, calculations, experiments, and special cases.”
In the brilliant report Thinking Like An Engineer, the Royal Academy of Engineering adapts Cuoco et al 1996 and states students who think like mathematicians should be:
1. Pattern sniffers
Always on the lookout for patterns and the delight to be derived from finding hidden patterns and then using shortcuts arising from them in their daily lives.
2. Experimenters
Performing experiments, playing with problems, performing thought experiments allied to a healthy scepticism for experimental results.
3. Describers
Able to play the maths language game, for example, giving precise descriptions of the steps in a process, inventing notation, convincing others and writing out proofs, questions, opinions and more polished presentations.
4. Tinkerers
Taking ideas apart and putting them back together again.
5. Inventors
Always inventing things – rules for a game, algorithms for doing things, explanations of how things work, or axioms for a mathematical structure.
6. Visualisers
Being able to visualise things that are inherently visual such as working out how many windows there are on the front of a house by imagining them, or using visualisation to solve more theoretical tasks.
7. Conjecturers
Making plausible conjectures, initially using data and increasingly using more experimental evidence.
8. Guessers
Using guessing as a research strategy, starting with a possible solution to a problem and working backward to achieve the answer.
Cuoco has advocated making mathematical habits of mind a key component of any syllabus because “without explicit attention to mathematical ways of thinking, the goals of intellectual sophistication and higher order thinking skills will remain elusive.
Mathematicians think big and talk small and think small and talk big. They look at things from multiple points of view and are receptive to new ideas. They demonstrate a desire to learn and a passion for discovery.
Other ‘habits of mind’ models are available. It depends where you look!