Jane Imrie Deputy Director Inquiry-based learning in mathematics: Who’s doing the maths? What is the sum of ? Jane Imrie Deputy Director

Jane Imrie Deputy Director Inquiry-based learning in mathematics: Who’s doing the maths? What is the most boring number between 1 and 1000? Not sure if you noticed the subtle change there ... Both could generate discussion in their own way – what’s the same and what is different about them? Jane Imrie Deputy Director

www.ncetm.org.uk www.ncetm.org.uk - Please register and explore!

? Think about a lesson/moment/person that inspired you – tell someone Was it maths? Subtitle because, for me, this has become quite an obsession in recent years. I think inquiry in the mathematics classroom is more about helping pupils behave like mathematicians than about engaging in real problems – though clearly there is an overlap. So, when observing maths lessons, or even thinking of my own, I try to analyse who is doing the maths? What is the balance between teachers and pupils doing the maths

Most common learning strategies (GCSE classes) Statements are ranked from most 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

Least common learning strategies (GCSE classes) Statements are ranked from most 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) When work is hard I don’t give up or do simple things. 3.32 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

Most and 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 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 Data from a sample of 120 teachers working with 779 learners following GCSE resit courses. 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

Most and least common teaching methods 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=120) Learners start with easy questions and work up to harder questions. 4.26 I tell learners which questions to tackle. 4.02 I teach the whole class at once. 3.90 Learners learn through discussing their ideas. 2.53 I jump between topics as the need arises. 2.51 I find out which parts learners already understand and don’t teach those parts. 2.44 I teach each learner differently according to individual needs. 2.43 Learners compare different methods for doing questions. 2.24 More details in ‘Collaborative Learning in Mathematics’ Malcolm Swan 2006

“....Mathematics teaches you a valuable way of thinking – you know, the skills you learn 'at the back of your head' which apply to any situation that needs some hard thinking." Factors influencing progression to A Level Mathematics (NCETM 2008)

“Many pupils refer frequently to prompts provided by the teacher about how to carry out a technique, but such methods, memorised without understanding, often later become confused or forgotten, and subsequent learning becomes insecure.” ‘Mathematics: understanding the score (Ofsted 2008)

“... most pupils recognised the difference between just getting answers right and understanding the work. Nevertheless, many of those observed were content to have the right answers in their books when they did not know how to arrive at them. This view that mathematics is about having correct written answers rather than about being able to do the work independently, or understand the method, is holding back pupils’ progress.” Mathematics: Understanding the Score (Ofsted 2008)

“They contrasted this with occasional lessons they enjoyed where they did investigations, tackled puzzles, sometimes working in groups, and used ICT independently. Often such lessons happened at the end of term and were regarded as end-of-term activities rather than being ‘real maths.” Recent post on NCETM forum: I have yet to see a maths lesson where the hairs stand up on the back of my neck-- yet this happens in art and history quite easily. The pupils are so engaged it can be  awesome to watch and it is only natural to grade these lessons as outstanding. Achieving this in maths is extremely difficult -- the nature of our subject I think. It is not impossible but much, much harder. Portal post Sept 2011 (Outstanding lessons) Mathematics: Understanding the Score (Ofsted 2008)

"What do you think makes a good teacher of Mathematics?" “The answer given in nearly all cases was that of exhibiting the dual professionalism of being good at their subject and having a concern about effective pedagogy. Good teachers of Mathematics were expected to have very high expectations of their pupils and to communicate those expectations in ways that encouraged self-confidence in the subject. Pupils had a high regard for the abilities of their teachers, spoke warmly about their approachability and were confident of receiving help and support in their learning.” Factors influencing progression to A Level Mathematics (NCETM 2008)

Subject knowledge Many secondary teachers The best teachers Classroom practice Pedagogy Many primary teachers Mathematics: Understanding the Score (Ofsted 2008)

Maths is.... Maths Café, NCETM portal ... a way of learning involving numbers and letters to solve equations and a wide variety of real life problems. ... used to quantify and explain the real world. ... a global language and provides the tools for our societies in using and developing science, technology, economics etc. ... applying taught methods to solve given problems in life. ... producing a strategy to solve a problem, with or without applying a known technique. ... a logical and unique way of looking at the world. It can tell us how an aeroplane flies or explain the beauty of a flower. Trainee teachers Maths Café, NCETM portal

Maths Café, NCETM portal “My current thinking is that relationships and patterns are there naturally. The ratio of the circumference of a circle to its diameter isn't a human construct, but nor is it maths! I think that maths is the process you go through in order to notice the relationship between these things, to see and understand the patterns, to try to make sense of what's around us and beyond. Without people there could be no maths because, to me, maths is a process, not a result.“ With a range of views like this, it is no wonder we have such a range of attitudes in our students. Maths Café, NCETM portal

Note the balance of the lilac bars and the dominance of ‘fluency’ in the current curriculum That’s not to suggest that fluency is a bad thing. It’s important for some things to become second nature in order that they don’t get in the way of solving problems further down the line. However, there are ways of teaching fluency which don’t perpetuate this:

Teaching is more effective when it: Builds on the knowledge learners already have Exposes and discusses common misconceptions and other surprising phenomena Uses higher-order questions Makes appropriate use of whole-class interactive teaching, individual work and cooperative small group work Encourages reasoning rather than ‘answer getting’ Uses rich, collaborative tasks Creates connections between topics both within and beyond mathematics and with the real world Uses resources, including technology, in creative and appropriate ways Confronts difficulties rather than seeks to avoid or pre-empt them Develops mathematical language through communicative activities Recognises both what has been learned and also how it has been learned https://www.ncetm.org.uk/files/309231/Mathematics+Matters+Final+Report.pdf Mathematics Matters, NCETM 2008

Improving learning in mathematics “Our first aim is to make mathematics teaching more effective by challenging learners to become more active participants. We want them to engage in discussing and explaining their ideas, challenging and teaching one another, creating and solving each other’s questions and working collaboratively to share their results. They not only improve in their mathematics; they also become more confident and effective learners.” http://www.nationalstemcentre.org.uk/elibrary/collection/282/improving-learning-in-mathematics

Range of activity types Classifying mathematical objects Multiple representations Evaluating statements Creating and solving problems Analysing reasoning

Discussion in mathematics

A group activity Work in twos and three and consider the statements on the cards. Decide whether each is always, sometimes or never true. If you think a statement is ‘always true’ or ‘never true’, then explain how you can be sure. If you think a statement is ‘sometimes true’, describe all the cases when it is true and all the cases when it is false. Stick your statement on a poster and write your explanation next to it. These statements don’t constitute one lesson – but they are part of a PD session to help teachers understand how discussion might be stimulated in the classroom – as well as how evaluating statement like this can bring out misconceptions and understandings. We’ll return to this tomorrow.

Always, sometimes or never true? Numbers with more digits are greater in value. The square of a number is greater than the number. When you cut a piece off a shape, you reduce its area and perimeter. A pentagon has fewer right angles than a rectangle. Quadrilaterals tessellate.

Always, sometimes or never true? If a right-angled triangle has integer sides, the incircle has integer radius. If you square a prime number, the answer is one more than a multiple of 24. If you add n consecutive numbers together the result is divisible by n. If you double the lengths of the sides, you double the area. Continuous graphs are differentiable. If the sequence of terms tends to zero, the series converges.

Reflect on your discussion Who talked the most? Who spoke the least? What was their role in the group? Did everyone feel that all views were taken into account? Did anyone feel threatened? If so, why? How could this have been avoided? Did people tend to support their own views, or did anyone take up and improve someone else's suggestion? Has anyone learnt anything? If so, how did this happen? Can you suggest possible other statements for primary teachers based on common misconceptions.

Why is discussion rare in mathematics? 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 mathematics, answers are either right or wrong – there is nothing to discuss.” “ If they understand it there is 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.”

What kind of talk is most helpful? Cumulative talk Speakers build positively but uncritically on what each other has said. Repetitions, confirmations and elaborations. Disputational talk Disagreement and individual decision-making. Short exchanges, assertions and counter-assertions. Exploratory talk Speakers elaborate each other’s reasoning. Collaborative rather than competitive atmosphere. Reasoning is audible; knowledge is publicly accountable. Critical, constructive exchanges. Challenges are justified; alternative ideas are offered.

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.

Managing a discussion How might we help learners to discuss constructively? What is the teacher’s role during small group discussion? What is the purpose of a whole group discussion? What is the teacher’s role during a whole group discussion?

Teacher’s role in small group discussion 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 discussions unresolved.

Purposes of whole group discussion Learners present and report on the work they have done. The teacher recognises ‘big ideas’ and gives them status and value. The learning is generalised and linked to other ideas and the wider context.

Teacher’s role in whole group discussion Mainly chair or facilitate. Direct the flow and give everyone a say. Do not interrupt or allow others to interrupt. Help learners to clarify their own ideas. Occasionally be a questioner or challenger. Introduce a new idea when the discussion is flagging. Follow up a point of view. Play devil’s advocate; ask provocative questions. Don’t be a judge who… assesses every response with ‘yes’, ‘good’ etc; sums up prematurely

Planning a discussion session How should you: organise the furniture? introduce the task ? introduce the ways of working on the task? allocate learners to groups? organise the rhythm of the session? conclude the session?

Make a poster Make a poster showing all you know about one of the following. Decimal numbers Shapes Time Show all the facts, results and relationships you know. Show methods and applications. Select only the most important and interesting facts at a basic and more advanced level.

Many of the activities can be made into posters – how often do you see a display of mathematics work? Also some activities focus on the poster itself. These slides are posters produced by primary school children who were asked to “make a poster showing all you know about Numbers Show all the facts, results and relationships you know. Show methods and applications. Select only the most important and interesting facts at a basic and more advanced level.

Range of activity types Classifying mathematical objects Multiple representations Evaluating statements Creating and solving problems Analysing reasoning

1. Classifying mathematical objects Learners devise their own classifications for mathematical objects, and apply classifications devised by others. They learn to discriminate carefully and recognise the properties of objects. They also develop mathematical language and definitions

1. 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

2. 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.

2. Using multiple representations

3. 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? p + 12 = s + 12 3 + 2y = 5y n + 5 is less than 20 4p > 9+p 2(x + 3) = 2x + 3 2(3 + s) = 6 + 2s

Always, sometimes or never true? a x b = b x a It doesn’t matter which way round you multiply, you get the same answer. a ÷ b = b ÷ a It doesn’t matter which way round you divide, you get the same answer. 12 + a > 12 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. √a < a The square root of a number is less than the number. a2 > a The square of a number is greater than the number.

True, false or unsure? When you roll a fair six-sided dice, it is harder to roll a 6 than a 4. Scoring a total of 3 with two dice is twice as likely as scoring a total of 2. In a lottery, the six numbers 3, 12, 26, 37, 38, 40 is more likely to come up than the numbers 1, 2, 3, 4, 5, 6. In a true or false quiz, with 10 questions, you are certain to get 5 right if you just guess. If a family has already got four boys, then the next baby is more likely to be a girl than a boy. The probability of getting exactly 3 heads in 6 coin tosses is 1/2.

Always, sometimes or never true?

Generalisations made by learners Area of rectangle ≠ area of triangle If you dissect a shape and rearrange the pieces, you change the area.

Generalisations made by learners 0.567 > 0.85 The more digits, the larger the value. 3÷6 = 2 Always divide the larger number by the smaller. 0.4 > 0.62 The fewer the number of digits after the decimal point, the larger the value. It's like fractions. 5.62 x 0.65 > 5.62 Multiplication always makes numbers bigger.

Some more limited generalisations What other generalisations are only true in limited contexts? In what contexts do the following generalisations work? If I subtract something from 12, the answer will be smaller than 12. All numbers can be written as proper or improper fractions. The order in which you multiply does not matter.

4. Creating and solving problems Learners devise their own problems or problem variants for other learners to solve. They are creative and ‘own’ problems. While others attempt to solve them, they take on the role of teacher and explainer. The ‘doing’ and ‘undoing’ processes of mathematics are exemplified.

4. Developing an exam question

4. Developing an exam question Make up your own question

4. Doing and undoing processes Kirsty created an equation, starting with x = 4. She then gave it to another learner to solve.

4. 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.

4. 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 adds together 3 numbers tries to find the three numbers writes down five numbers and finds their mean, median, range tries to find five numbers with the given mean, median and range.

5. Analysing reasoning Learners compare different methods for doing a problem, organise solutions and/ or diagnose the causes of errors in solutions. They begin to recognise that there are alternative pathways through a problem, and develop their own chains of reasoning.

5. Analysing reasoning and solutions Comparing different solution strategies Paint prices 1 litre of paint costs £15. What does 0.6 litres cost? Chris: It is just over a half, so it would be about £8. Sam: I would divide 15 by 0.6. You want a smaller answer. Rani: I would say one fifth of a litre is £3, so 0.6 litres will be three times as much, so £9. Tim: I would multiply 15 by 0.6.

5. Analysing reasoning Correcting mistakes in reasoning If you can produce 20% more milk per cow, you can decrease your herd by 20% to produce the same amount of milk. (Observer Magazine)

5. Analysing reasoning Putting reasoning in order

“....Mathematics teaches you a valuable way of thinking – you know, the skills you learn 'at the back of your head' which apply to any situation that needs some hard thinking." Factors influencing progression to A Level Mathematics (NCETM 2008)

?

jane.imrie@ncetm.org.uk www.ncetm.org.uk