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Jane Imrie Deputy Director

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Jane ImrieDeputy Director

What is the sum of ?

Inquiry-based learning in mathematics:

Who’s doing the maths?

Jane ImrieDeputy Director

Inquiry-based learning in mathematics:

Who’s doing the maths?

What is the most boring number between 1 and 1000?

?

“....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.”

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)

- ... 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 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 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

Mathematics Matters, NCETM 2008

- “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.”

- Classifying mathematical objects
- Multiple representations
- Evaluating statements
- Creating and solving problems
- Analysing reasoning

- 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.

- 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?

- 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.

- 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?

- 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.

- 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.

- 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

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 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.

- Classifying mathematical objects
- Multiple representations
- Evaluating statements
- Creating and solving problems
- Analysing reasoning

- 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

- Learners match cards showing different representations of the same mathematical idea.
- They draw links between different representations and develop new mental images for concepts.

- 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.

- Area of rectangle ≠ area of triangle
- If you dissect a shape and rearrange the pieces, you change the area.

- 0.567 > 0.85The more digits, the larger the value.
- 3÷6 = 2Always 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.

- 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.

- 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

Kirsty created an equation, starting with x = 4.

She then gave it to another learner to solve.

- 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.

Comparing different solution strategies

- 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)

- 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)

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www.ncetm.org.uk