Mathematical Culture – being mathematicians

9749 /www/nrich/html/content/id/9749/ 3 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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! *************************** Revised listing after meeting on 16 January 2013 *************************** <h3> First level:</h3> <h4 style="text-align: center;"><a href="/8068">Our Philosophy</a> </h4> Fixed intelligence versus flexible intelligence [Carol Dweck], [Gareth Malone] <div>Teachers can actively shape the norms of classroom participation</div> <h4 style="text-align: center;">Safe to learn</h4> Concern for students as individuals Recognising debilitating effect of "maths anxiety" <h3>Second level:</h3> <h4 style="text-align: center;">Models of teaching mathematics</h4> 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!) <h3>Third level:</h3> <h4 style="text-align: center;">Motivation </h4> <h4>- Exploiting natural curiosity</h4> 'Questions first' Students asking questions Homeworks to provoke thinking "What might a mathematician ask next?" Teachers showing curiosity Loose ends <h4>- Looking for connections/expecting coherence and structure</h4> Linking new topics to our existing knowledge Using new techniques to solve old problems Checking consistency with existing knowledge <h4 style="text-align: center;">Interactions</h4> <h4>- Independent work and collaborative work</h4> Time for independent thought before discussion <h4>- Interactions between students and other students</h4> Collaborative Supportive Mutual respect Critical <a href="/7011">Jo Boaler's Railside ways of working</a> <h4>- Interactions between teachers and students</h4> "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 <a href="/7011">Jo Boaler's Railside ways of working</a> David Wheeler "The role of the teacher" <h4>- Feedback</h4> <h4 style="text-align: center;">Getting stuck</h4> <h4>- Springboard to learning</h4> 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] <h4>- Asking for help</h4> Explain what you have tried Explain where you are stuck <h4>- Offering help</h4> Consider when help is required Advice that is helpful in the long term as well as in the immediate context Students helping each other <h4>- Getting stuck when working by oneself</h4> Reflecting on what has worked in the past "What would my teacher ask now?" <h4 style="text-align: center;">Mathematical behaviour</h4> <h4>- Being mathematical with and in front of students</h4> 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 <h4>- Mathematical rigour</h4> Constructing reasoned arguments and justifications Appreciating the value of proof Critically assessing arguments <h4>- Looking for connections/expecting coherence and structure</h4> 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 <h4>- Communication via written work</h4> Clarity <h4>- Responses to progress and 'success' (from both students and teachers)</h4> Excitement "What enabled me to make progress this time?" "What would I do differently next time?" "What have I learned from this problem?"</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> A lot of our time is often spent in crafting problems/tasks that will offer students opportunities to explore/gain insight/apply knowledge. We recently got talking about mathematical resilience and the qualities that students may need to have in order to take full advantage of these opportunities. We'd like to develop some resources that can go alongside our usual offering, offering ideas and illustrating strategies that can be shared and implemented in schools.... Use of the word 'disposition', and connect with Klpatrick et al" Kilpatrick, J. Swafford, J. & Findell, B. (Eds) (2001). Adding It Up: Helping Children Learn Mathematics. Mathematics Learning Study Committee. Washington DC, USA: National Academy Press.) with 'surprise', include Movshovits-Hadar : Movshovits-Hadar, N. (1988) 'School Mathematics Theorems - an Endless Source of Surprise' For the Learning of Mathematics 8(3) pp. 34-39 Especially important at the beginning of the school year; maximally effective if negotiated amongst staff so that there is something common as students progress from year to year. John P.S. Also recommended: 'Learning & Doing' (That is from Tarquin now) </mdoxml> 0 0 0 0 1 1 1 Mathematical Culture – being mathematicians How can we influence our students' attitudes to learning and to mathematics? sfh10 Steve - Development Mathematics Education and Research Pedagogy Mathematics Education and Research Mathematical Thinking Mathematics Education and Research Learning mathematics