Outline What is collaborative learning? Why is collaborative learning rare? Why is it important? How can we make it happen?

Ofsted think it’s important…. 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)

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 agenda rather than concept-focused Passive learning (listen and imitate) Unimaginative resources (worksheets)

Most common learning strategies Statements are rank ordered from most common to least common 1 = almost never, 2 = occasionally, 3 = half the time, 4= most of the time; 5 = almost always. Source: Swan (2005) Mean (n=779) I listen while the teacher explains. 4.28 I copy down the method from the board or textbook. 4.15 I only do questions I am told to do. 3.88 I work on my own. 3.72 I try to follow all the steps of a lesson. 3.71 I do easy problems first to increase my confidence. 3.58 I copy out questions before doing them. 3.57 I practise the same method repeatedly on many questions. 3.42 I ask the teacher questions. 3.40 I try to solve difficult problems in order to test my ability. 3.32 When work is hard I don’t give up or do simple things.

Least common learning strategies Statements are rank ordered from most common to least common 1 = almost never, 2 = occasionally, 3 = half the time, 4= most of the time; 5 = almost always. Source: Swan (2005) Mean (n=779) I discuss my ideas in a group or with a partner. 3.25 I try to connect new ideas with things I already know. 3.20 I am silent when the teacher asks a question. 3.16 I memorise rules and properties. 3.15 I look for different ways of doing a question. 3.14 My partner asks me to explain something. 3.05 I explain while the teacher listens. 2.97 I choose which questions to do or which ideas to discuss. 2.54 I make up my own questions and methods. 2.03

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

Drawing connections…

Why is collaborative learning rare? Time pressures “ It’s a gallop to the main exam.” “ Learners will waste time in social chat.” Control “ What will other teachers think of the noise?” “ How can I possibly monitor what is going on?” Views of learners “ My learners cannot discuss.” “ My learners are too afraid of being seen to be wrong.” Views of mathematics “ In maths, answers are right or wrong –nothing to discuss.” “ If they understand it there’s nothing to discuss. If they don’t, they are in no position to discuss anything.” Views of learning “ Mathematics is a subject where you listen and practise.” “ Mathematics is a private activity.” Non-tangible outcomes “ If we have discussions they won’t have a neat set of notes”

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? Fluency in recalling facts, performing skills Interpretations for concepts and representations Strategies for investigation and problem solving Awareness of the nature and values of the educational system Appreciation of the power of mathematics in society

The principles… Uses cooperative small group work Exposes/discusses common misconceptions Builds on knowledge learners already have Uses higher-order questions Creates connections between topics Uses rich, collaborative tasks Encourages reasoning not “answer getting” Uses technology appropriately

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 Each question labelled -2 to +2. Higher numbers are better. Expository (n=29) Discussion (n=27) How interesting? -0.48 +1.29 How hard did you work? +0.62 +1.03 How much did you learn? -0.59 +1.18 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. Perceived self-efficacy is defined as people's beliefs about their capabilities

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. Perceived self-efficacy is defined as people's beliefs about their capabilities

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.

Always, sometimes or never true? Pay rise Max gets a pay rise of 30%. Jim gets a pay rise of 25%. So Max gets the bigger pay rise. Sale In a sale, every price was reduced by 25%. After the sale every price was increased by 25%. So prices went back to where they started. Area and perimeter When you cut a piece off a shape you reduce its area and perimeter Right angles A pentagon has fewer right angles than a rectangle Birthdays In a class of ten learners, the probability of two learners being born on the same day of the week is one. Lottery In a lottery, the six numbers 3, 12, 26, 37, 44, 45 are more likely to come up than the six numbers 1, 2, 3, 4, 5, 6.

Always, sometimes or never true? Bigger fractions If you add the same number to the top and bottom of a fraction, the fraction gets bigger in value. Smaller fractions If you divide the top and bottom of a fraction by the same number, the fraction gets smaller in value. Square roots The square root of a number is less than or equal to the number Add a nought To multiply by ten, you just add nought on the right hand end of the number. Algebra

Evaluating mathematical statements

Evaluating mathematical statements It doesn’t matter which way round you multiply, you get the same answer. a x b = b x a It doesn’t matter which way round you divide, you get the same answer. a ÷ b = b ÷ a If you add a number to 12 you get a number greater than 12. 12 + a > 12 If you divide 12 by a number the answer will be less than 12. 12 ÷ a < 12 The square root of a number is less than the number. √a < a The square of a number is greater than the number. a2 > a

Evaluating mathematical statements When you roll a fair six-sided die, it is harder to roll a six than a four. Scoring a total of three with two dice is twice as likely as scoring a total of two. In a lottery, the six numbers 3, 12, 26, 37, 38, 40 are more likely to come up than the numbers 1, 2, 3, 4, 5, 6. There are three outcomes in a football match, win lose or draw. The probability of winning is therefore 1/3 If a family has already got four boys, then the next baby is more likely to be a girl than a boy. In a ‘true or false’ quiz with ten questions, you are certain to get five right if you just guess.

Evaluating mathematical statements

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’?

Why might each be the ‘odd one out’? Percentage Fraction Decimal

Classifying using 2-way tables Large Area Small area Large perimeter Small perimeter

Classifying using 2-way tables No rotational symmetry Rotational symmetry No lines of symmetry One or two lines of symmetry More than two lines of symmetry

Classifying using 2-way tables Factorises with integers Does not factorise with integers Two x intercepts No x intercepts Two equal x intercepts Has a minimum point Has a maximum point y intercept is 4

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.

“Bog standard” exam question

An open template for a new question

Doing and undoing processes The problem poser… Undoing: The problem solver… generates an equation step-by-step, ‘doing the same to both sides’. solves the resulting equation. draws a rectangle and calculates its area and perimeter. tries to draw a rectangle with the given area and perimeter. writes down an equation of the form y=mx+c and plots a graph. tries to find an equation that fits the resulting graph.

Doing and undoing processes The problem poser… Undoing: The problem solver… expands an algebraic expression such as (x+3)(x-2). factorises the resulting expression: x2+x-6. writes down a polynomial and differentiates it. integrates the resulting function. writes down five numbers and finds their mean, median and range. tries to find five numbers with the given mean, median and range.

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.

Discussing common errors

Qualitative comparisons More effective Offer challenge before help. Discuss ways of working. Ask questions eliciting interpretations and methods. Pause after questions and after answers. Listen before intervening. When students are unable to explain, asked other students in the group to contribute, or asked them to explain something else. Left discussions unresolved. Less effective Offer help before challenge. Tell them what to do. Ask questions eliciting facts and numerical answers. Give answers if none immediately forthcoming. Intervene before listening. Predetermine the agenda. When student are unable to explain, explain for them. Ensured all discussions are resolved.

Ground rules for learners Talk one at a time. Share ideas and listen to each other. Make sure people listen to you. Follow on. Challenge. Respect each other’s opinions. Enjoy mistakes. Share responsibility. Try to agree in the end.

Teacher’s role Make the purpose of the task clear. Keep reinforcing the ‘ground rules’. Listen before intervening. Join in, don’t judge. Ask learners to describe, explain and interpret. Do not do the thinking for learners. Don’t be afraid of leaving some discussions unresolved.