Jane ImrieDeputy Director

Inquiry-based learning in mathematics: Who’s doing the maths?

What is the sum of ?

Jane ImrieDeputy Director

Inquiry-based learning in mathematics:Who’s doing the maths?

What is the most boring number between 1 and 1000?

www.ncetm.org.uk

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Most common learning strategies(GCSE classes)

Statements are ranked from most to least common1 = 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 common1 = 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 commonlearning strategies

Statements are rank ordered from most common to least common1 = 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

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

“....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.”

Mathematics: Understanding the Score (Ofsted 2008)

“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.”

"What do you think makes a good teacher of Mathematics?"

Factors influencing progression to A Level Mathematics (NCETM 2008)

Subjectknowledge

Classroompractice Pedagogy

The bestteachers

Many primary teachers

Manysecondaryteachers

Mathematics: Understanding the Score (Ofsted 2008)

► ... 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.

Maths is....

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.“

Maths Café, NCETM portal

Builds on the knowledge learners already haveExposes and discusses common misconceptions and other

surprising phenomenaUses higher-order questionsMakes appropriate use of whole-class interactive teaching, individual

work and cooperative small group work Encourages reasoning rather than ‘answer getting’Uses rich, collaborative tasksCreates connections between topics both within and beyond

mathematics and with the real worldUses resources, including technology, in creative and appropriate

waysConfronts difficulties rather than seeks to avoid or pre-empt themDevelops mathematical language through communicative activitiesRecognises both what has been learned and also how it has been

learned

Teaching is more effective when it:

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.”

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.

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.

Quadrilateralstessellate.

ab>a+ b

2

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?

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 ofwhole 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.

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

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

1. Classifying using 2-way tables

n n nn

n n nnnn

3n2

9n 2 (3n)2

Square n then

multiply your answer

by 3

Multiply n by 3 then

square your answer

Square n then

multiply your answer

by 9

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

aa

1

3

a - b = b - aIt doesn't matterwhich way round yousubtract, you get thesame answer.

12 - a < 12If you take a numberaway from 12, youranswer will be lessthan 12.

12 + a > 12If you add a numberto 12, your answer willbe greater than 12.

12a > 12If you multiply 12 by anumber, the answer willbe greater than 12.

12 ÷ a < 12If you divide 12 by anumber, the answerwill be less than 12.

a ÷ b = b ÷ aIt doesn't matterwhich way round youdivide, you get thesame answer.

¦a < aThe square root of anumber is less than thenumber.

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 ornever 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 ornever true?

a x b = b x aIt doesn’t matter which way

round you multiply, you get the same answer.

a ÷ b = b ÷ aIt doesn’t matter which way

round you divide, you get the same answer.

12 + a > 12If you add a number to 12 you

get a number greater than 12.

12 ÷ a < 12If you divide 12 by a number the

answer will be less than 12.

√a < aThe square root of a number is

less than the number.

a2 > aThe 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?

A

When you cut a piece off a shapeyou reduce its area and perimeter.

A

What happens to the area and perimeter with these cuts?

aa

Generalisations made by learners

Area of rectangle ≠ area of triangle

If you dissect a shape and rearrange the pieces, you change the area.

AB C

AB

C

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 > 0.62The fewer the number of digits after the decimal point, the larger the value. It's like fractions.

5.62 x 0.65 > 5.62Multiplication 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

4. Doing and undoingprocesses

Kirsty created an equation, starting with x = 4.

She then gave it to another learner to solve.

Doing: 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 undoingprocesses

Doing: 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.

4. Doing and undoingprocesses

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 strategiesPaint 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