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Outline. What is collaborative learning? Why is collaborative learning rare? Why is it important? How can we make it happen?.
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Outline • What is collaborative learning? • Why is collaborative learning rare? • Why is it important? • How can we make it happen?
The best teaching gave a strong sense of the coherence of mathematical ideas; it focused on understanding mathematical concepts and developed critical thinking and reasoning. Careful questioning identified misconceptions and helped to resolve them, and positive use was made of incorrect answers to develop understanding and to encourage students to contribute. Students were challenged to think for themselves, encouraged to discuss problems and to work collaboratively. Effective use was made of ICT. (OFSTED, 2006) Ofsted think it’s important….
Collaborative learning is when students • take active roles in the classroom • are responsible for their own learning and the learning of others • discuss (rather than just talk) together • share and explain their own reasoning • listen to, reflect on, and challenge the reasoning of others • ‘argue’ and resolve disagreements and misconceptions • take joint responsibility for a shared outcome
.. meanwhile in many classrooms.. • XXX teaching • Procedural agendarather than concept-focused • Passive learning (listen and imitate) • Unimaginative resources (worksheets)
A ‘Transmission’ culture Mathematics is seen as • a body of knowledge and procedures to be ‘covered’ Learning is seen as: • an individual activity based on listening and imitating Teaching is seen as: • structuring a linear curriculum for the learner • giving explanations and checking these have been understood through practice questions • ‘correcting’ misunderstandings when students fail to ‘grasp’ what is taught
‘Collaborative, challenging’ culture Mathematics is seen as • a network of ideas which teacher and students construct together Learning is seen as • a social activity in which students are challenged and arrive at understanding through discussion Teaching is seen as • non-linear dialogue in which meanings and connections are explored • recognising misunderstandings, making them explicit and learning from them
The illusion of ‘coverage’ One of the major pressures I feel is the obligation to cover everything in the GCSE specification and complete everything in the scheme of work. It is difficult to get through everything in under three hours a week. I recall a staffroom conversation in which we sounded like we were competing to see who had managed to ‘cover’ trigonometry in the shortest time possible. Is this effective teaching and learning? When I ‘speed teach’, I sometimes ask myself who is covering GCSE maths? Is it the students or just me?
Low expectations of learners It is important to remember that they may never grasp certain concepts and for some learners we are talking about maintaining skills rather than making progress.
What is involved in teaching maths? • Fluencyin recalling facts, performing skills • Interpretationsfor concepts and representations • Strategiesfor investigation and problem solving • Awarenessof the nature and values of the educational system • Appreciationof the power of mathematics in society
Collaboration produces useful learning • Textbook exercises or discussion of mathematical ideas are not merely vehicles for developing knowledge, they shape the forms of knowledge produced. • Learners who work through textbook exercises, find it difficult to use mathematics in applied or discussion-based situations. • Learners who had engaged in collaborative work develop relational forms of knowledge that are more useful in a range of different situations (including traditional examination questions). (from Boaler,1997)
1980s - “Diagnostic teaching” • Explore existing ideas through tests and interviews, before teaching. • Expose existing concepts and methods • Provoke ‘tension’ or ‘cognitive conflict’ • Resolve conflict through discussion and formulate new concepts and methods. • Consolidate learning by using the new concepts and methods on further problems.
Fractions and Decimals Write these numbers in order of size, from smallest to largest: 0.75 0.4 0.375 0.25 0.125 0.04 0.8 0.375 0.125 0.75 0.25 0.04 0.4 0.8 I know this because they work like fractions, 0.4 is like a quarter.
‘Diagnostic Teaching’ Research Reflections Rates Decimals
‘Expository’ v ‘Conflict and discussion’ Topic: Graphs (Brekke, 1986) Used same booklets with different teachers. Content • the misconception that a graph is a picture of a situation; • the ability to coordinate the information relating to two variables • the ability to discriminate between different types of variation when graph sketching • the interpretation of intervals and gradients
How does the speed of the ball vary? Sketch a speed v time graph
Expository approach Short introduction: Introduces worksheet. Extended period of groupwork: Students work in groups at their own pace. Teacher intervenes when they get answers wrong: This cannot be right because the ball then would have had its greatest speed at the top. The graph must be like this because it starts off with zero speed, then it picks up speed because it is hit by the club, as it travels up in the air it will slow down, and as it is dropping it will pick up speed because of the gravity. Short final discussion Teacher gives class a fresh problem and leads them to the correct answer.
Conflict and discussion approach Introduction Introduces the activity and the way of working: (1) Think about the problem alone; (2) Discuss the problem with your group; (3) Write about the problem; (4) Sketch the graph; (5) Interpret the graph back into words; (6) Is it the same as the problem? (If no, return to (1)); (7) Discuss with the whole class; (8) Does everyone agree? (If no, return to (1)). Students work in pairs, then groups After 20 minutes teacher reminds them how to work.
Conflict and discussion approach Long final discussion Resolving errors. Which of these is right and why?What common errors do people make?
Results The conflict discussion approach was felt to be much more interesting, demanding and effective. Students needed to learn how to discuss.
Outcomes of research (on algebra) When collaborative activities were used: • Teachers’ beliefs about teaching and learning changed.
Outcomes of research (on algebra) When collaborative activities were used: • Teachers’ beliefs about teaching and learning changed. • Learning increased. • Learner-centred was more effective than teacher-centred. • Significant (but small) improvement in self-efficacy. When collaborative activities were not used: • Learners regressed in confidence and motivation • Increase in passive learning and anxiety about algebra.
Activities that develop thinking… • Evaluating mathematical statements • Classifying mathematical objects • Interpreting multiple representations • Creating and solving problems • Analysing reasoning and solutions
1. Evaluating mathematical statements • Learners decide whether given statements are always, sometimes or never true. • They are encouraged to develop: • rigorous mathematical arguments and justifications; • examples and counterexamples to defend their reasoning.
2. Classifying mathematical objects Learners examine and classify mathematical objects according to their different attributes. They create and use categories to build definitions, learning to discriminate carefully and to recognise the properties of objects. They also develop mathematical language.
Why might each be the ‘odd one out’? Percentage Fraction Decimal
3. Interpreting multiple representations • Learners match cards showing different representations of the same mathematical idea. • They draw links between different representations and develop new mental images for concepts.
4. Creating and solving problems • Learners devise their own mathematical problems for other learners to solve. • Learners are creative and ‘own’ the problems. • While others attempt to solve them, learners take on the role of teacher and explainer. • The ‘doing’ and ‘undoing’ processes of mathematics are exemplified.
5. Analysing reasoning and solutions Learners • compare different methods for doing a problem, • organise solutions and/ or • diagnose the causes of errors in solutions. They recognise that there are alternative pathways through a problem, and develop their own chains of reasoning.
