Mathematical Culture – Being Mathematicians
We would like our students to leave our classrooms having experienced the intellectual satisfaction of overcoming mathematical challenges, with a belief in their potential and their capability to improve, with enquiring minds, and with the capacity to function as independent mathematicians.
We believe that in order for students to make the most of the opportunities that we offer them, we need to address the factors that influence our students' attitudes to learning and to mathematics. These might include the classroom ethos, atmosphere, rubric, practices, and ways of working.
Listed below are some of the areas that we might wish to influence if we are to foster a culture that allows our students to develop these attributes. So what can we do as teachers to make a difference? Eventually we would like a bank of resources expanding on each of these areas, collecting ideas and illustrating strategies that can be shared and implemented in schools.
Perhaps an integral part of any suite of resources developed with these ideas in mind should include case studies/videos/cameos of people who work mathematically, showing that they get stuck, produce messy work, have an internal dialogue while problem-solving, reflect, discuss their problems with peers... and thus offering positive mathematical role models to young people who have a very limited
idea of what a mathematician actually looks like!
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Revised listing after meeting on 16 January 2013
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First level:
Fixed intelligence versus flexible intelligence [Carol Dweck], [Gareth Malone]
Teachers can actively shape the norms of classroom participation
Safe to learn
Concern for students as individuals
Recognising debilitating effect of "maths anxiety"
Second level:
Models of teaching mathematics
Exploring before codifying (Ken Ruthven)
Challenging rather than spoonfeeding (Wigley)
Scaffolding and fading
Embracing 'being in the pit' (James Nottingham)
'Designing & Using Tasks' (That is also from Tarquin!)
Third level:
Motivation
- Exploiting natural curiosity
'Questions first'
Students asking questions
Homeworks to provoke thinking
"What might a mathematician ask next?"
Teachers showing curiosity
Loose ends
- Looking for connections/expecting coherence and structure
Linking new topics to our existing knowledge
Using new techniques to solve old problems
Checking consistency with existing knowledge
Interactions
- Independent work and collaborative work
Time for independent thought before discussion
- Interactions between students and other students
Collaborative
Supportive
Mutual respect
Critical
Jo Boaler's Railside ways of working
- Interactions between teachers and students
"The most important single factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly." [Ausubel]
"whole-class and small-group interactions between pupils, and responsive teacher adaptation to the thinking elicited" [epiSTEMe]
Scaffolding: "Only do for students what they cannot yet do for themselves." [Mason?]
Fading: promoting self-sufficiency
Jo Boaler's Railside ways of working
David Wheeler "The role of the teacher"
- Feedback
Getting stuck
- Springboard to learning
Resilience
Messiness
Loose ends
Rough edges
Bumpiness
Hitting brick walls
Going down blind alleys
Feeling lost [Gareth Malone]
Understanding the problem [Simon Singh "Fermat's Last Theorem" Horizon programme]
- Asking for help
Explain what you have tried
Explain where you are stuck
- Offering help
Consider when help is required
Advice that is helpful in the long term as well as in the immediate context
Students helping each other
- Getting stuck when working by oneself
Reflecting on what has worked in the past
"What would my teacher ask now?"
Mathematical behaviour
- Being mathematical with and in front of students
Curiosity
Excitement [Gareth Malone]
Surprise
Wanting to be sure -- seeking convincing justifications
Asking questions -- 'What if ...?', 'Always?', 'Never?', 'Are there other approaches?'
Making conjectures
Looking back on solutions
Drawing attention to examples of good mathematical behaviour
Using analogies
- Mathematical rigour
Constructing reasoned arguments and justifications
Appreciating the value of proof
Critically assessing arguments
- Looking for connections/expecting coherence and structure
Linking new topics to our existing knowledge
Using new techniques to solve old problems
Checking consistency with existing knowledge
Historical context
Links between topics
Links between representations
Use of analogies
- Communication via written work
Clarity
- Responses to progress and 'success' (from both students and teachers)
Excitement
"What enabled me to make progress this time?" "What would I do differently next time?"
"What have I learned from this problem?"