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Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale [email protected]

Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale [email protected]

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Page 1: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Learner-centred Education in Mathematics

If you want to build higher,dig deeper

Charlie Gilderdale [email protected]

Page 2: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Initial thoughts

• Thoughts about Mathematics

• Thoughts about teaching and learning Mathematics

Page 3: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Five ingredients to consider

• Starting with a rich challenge: low threshold, high ceiling activities

• Valuing mathematical thinking

• Purposeful activity and discussion

• Building a community of mathematicians • Reviewing and reflecting

Page 4: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Starting with a rich challenge:Low Threshold, High Ceiling activity

To introduce new ideas and develop understanding of new curriculum content

Page 5: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Making use of a Geoboard environment

Page 6: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Why might a teacher choose to use this activity in this way?

Page 7: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Some underlying principles

Mathematics is a creative discipline, not a spectator sport

Exploring → Noticing Patterns

→ Conjecturing → Generalising

→ Explaining→ Justifying

→ Proving

Page 8: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Tilted Squares

The video in the Teachers' Notes shows how the problem was introduced

to a group of 14 year old students:

http://nrich.maths.org/2293/note

Page 9: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Some underlying principles

Teacher’s role• To choose tasks that allow students to explore new

mathematics• To give students the time and space for that

exploration• To bring students together to share ideas and

understanding, and draw together key mathematical insights

Page 10: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Give the learners something to do, not something to learn; and if the doing is of such a nature as to demand thinking; learning naturally results.

John Dewey

Page 11: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

The most exciting phrase to hear in science, the one that heralds new discoveries,

is not Eureka!, but rather, “hmmm… that’s funny…”

Isaac Asimov

mathematics

Page 12: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

There are many more NRICH tasks that make excellent starting points…

Page 13: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Number and Algebra

Summing Consecutive Numbers

Number Pyramids

What’s Possible?

What’s It Worth?

Perimeter Expressions

Seven Squares

Attractive Tablecloths

Page 14: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Geometry and Measures

Painted Cube

Changing Areas, Changing Perimeters

Cyclic Quadrilaterals

Semi-regular Tessellations

Tilted Squares

Vector Journeys

Page 15: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Handling Data

Statistical Shorts

Odds and Evens

Which Spinners?

Page 16: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

…and for even more, see the highlighted problems on the

Curriculum Mapping Document

Page 17: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Time for reflection

• Thoughts about Mathematics

• Thoughts about teaching and learning Mathematics

Page 18: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Morning Break

Page 19: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Valuing Mathematical Thinking

What behaviours do we value in mathematics and how can we encourage them

in our classrooms?

Page 20: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

As a teacher, do I value students for being…

• curious – looking for explanations – looking for generality – looking for proof

• persistent and self-reliant• willing to speak up even when they are uncertain• honest about their difficulties• willing to treat ‘failure’ as a springboard to new learning

Page 21: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

… and do I offer students sufficient opportunities to develop these “habits for success” when I set tasks

• to consolidate/deepen understanding• to develop fluency• to build connections

Page 22: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

We could ask…

Area = ?

Perimeter = ?

or we could ask …

6cm

4cm

We could ask:

Page 23: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Perimeter = 20 cm Area = 24 cm² = 22 cm = 28 cm = 50 cm = 97 cm = 35 cm

and we could ask …

Page 24: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

• Think of a rectangle

• Calculate its area and perimeter

• Swap with a friend – can they work out the length and breadth of your rectangle?

or we could ask …

…students to make up their own questions

Page 25: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Can you find rectangles where the value of the area is the same as the value of the perimeter?

QuickTime™ and a decompressor

are needed to see this picture.

Page 26: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Why might a teacher choose to use these activities in this way?

Page 27: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

We could ask students to find…

(x + 2) (x + 5)(x + 4) (x - 3)…

or we could introduce them to…

Page 28: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Pair Products

Choose four consecutive whole numbers, for example, 4, 5, 6 and 7.

Multiply the first and last numbers together.

Multiply the middle pair together…

What might a mathematician do next?

Page 29: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

We could ask students to…

Identify coordinates and straight line graphs

or we could introduce them to…

Page 30: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Route to Infinity

Will the route passthrough (18,17)?

Which point will it visit next?

How many points will it pass through before (9,4)?

Route to Infinity

Page 31: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

We could ask students to…

List the numbers between 50 and 70 that are

(a) multiples of 2(b) multiples of 3(c) multiples of 4(d) multiples of 5(e) multiples of 6

or we could ask students to play…

Page 32: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

The Factors and Multiples Game

A game for two players.

You will need a 100 square grid.

Take it in turns to cross out numbers, always choosing a number that is a factor or multiple of the previous number that has just been crossed out.

The first person who is unable to cross out a number loses.

Each number can only be crossed out once.

Page 33: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Why might a teacher choose to use these activities?

Page 34: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Some underlying principles

Consolidation should address both content and process skills.

Rich tasks can replace routine textbook tasks, they are not just an add-on for students who finish first.

Page 35: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

There are many more NRICH tasks that offer opportunities for consolidation…

Page 36: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Number and Algebra

What Numbers Can We Make?

Factors and Multiples Game

Factors and Multiples Puzzle

Dicey Operations

American Billions

Keep It Simple

Temperature

Painted Cube

Arithmagons

Pair Products

What’s Possible?

Attractive Tablecloths

How Old Am I?

Page 37: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Geometry and Measures

Isosceles Triangles

Can They Be Equal?

Translating Lines

Opposite Vertices

Coordinate Patterns

Route to Infinity

Pick’s Theorem

Cuboid Challenge

Semi-regular Tessellations

Warmsnug Double Glazing

Page 38: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Handling Data

M, M and M

Which List is Which?

Odds and Evens

Which Spinners?

Page 39: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

…and for even more, see the highlighted problems on the

Curriculum Mapping Document

Page 40: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Time for reflection

• Thoughts about Mathematics

• Thoughts about teaching and learning Mathematics

Page 41: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Lunch

Page 42: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Promoting purposeful activity and discussion

‘Hands-on’ doesn’t mean ‘brains-off’

Page 43: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

The Factors and Multiples Challenge

You will need a 100 square grid.

Cross out numbers, always choosing a number that is a factor or multiple of the previous number that has just been crossed out.

Try to find the longest sequence of numbers that can be crossed out.

Each number can only appear once in a sequence.

Page 44: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

We could ask…

3, 5, 6, 3, 3

Mean = ?Mode = ?

Median = ?

or we could ask…

Page 45: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

M, M and M

There are several sets of five positive whole numbers with the following properties:

Mean = 4 Median = 3 Mode = 3

Can you find all the different sets of five positive whole numbers that satisfy these conditions?

Page 46: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Possible extension

How many sets of five positive whole numbers are there with the following property?

Mean = Median = Mode = Range = a single digit number

Page 47: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

What’s it Worth?

Each symbol has a numerical value.

The total for the symbols is written at the end of each row and column.

Can you find the missing total that should go where the question mark has been put?

Page 48: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Translating Lines

Each translation links a pair of parallel lines.

Can you match them up?

QuickTime™ and a decompressor

are needed to see this picture.

Page 49: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Why might a teacher choose to use these activities?

Page 50: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Rules for Effective Group Work

• All students must contribute:no one member says too much or too little

• Every contribution treated with respect:listen thoughtfully

• Group must achieve consensus:work at resolving differences

• Every suggestion/assertion has to be justified:arguments must include reasons

Neil Mercer

Page 51: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Developing Good Team-working Skills

The article describes attributes of effective team work and links to "Team Building" problems that can be used to

develop learners' team working skills.

http://nrich.maths.org/6933

Page 52: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Time for reflection

• Thoughts about Mathematics

• Thoughts about teaching and learning Mathematics

Page 53: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Afternoon Break

Page 54: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Build a community of mathematicians by:

Creating a safe environment for learners to take risks

Promoting a creative climate and conjecturing atmosphere

Providing opportunities to work collaboratively

Valuing a variety of approaches

Encouraging critical and logical reasoning

Page 55: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Multiplication square

X 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

Page 56: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

The Challenge

• To create a climate in which the child feels free to be curious

• To create the ethos that ‘mistakes’ are the key learning points

• To develop each child’s inner resources, and develop a child’s

capacity to learn how to learn

• To maintain or recapture the excitement in learning that was

natural in the young child

Carl Rogers, Freedom to Learn, 1983

Page 57: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

There are many NRICH tasks that encourage students to work as a

mathematical community…

Page 58: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Making Rectangles

What’s it Worth?

Steel Cables

Odds and Evens

M, M and M

Odds, Evens and More

Evens

Tilted Squares

Pair Products

What’s Possible?

Cyclic Quadrilaterals

How Old Am I?

Factors and Multiples Game

Page 59: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

…and for even more, see the highlighted problems on the

Curriculum Mapping Document

Page 60: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Enriching mathematics websitewww.nrich.maths.org

The NRICH Project aims to enrich the mathematical experiences of all learners by providing free resources designed to develop subject knowledge and problem-solving skills.

We now also publish Teachers’ Notes and Curriculum Mapping Documents for teachers:http://nrich.maths.org/curriculum

Page 61: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

What next?

Secondary CPD Follow-up on the NRICH site:

http://nrich.maths.org/7768

Page 62: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Reviewing and reflecting

There should be brief intervals of time for quiet reflection – used to organise what has been gained in periods of activity.

John Dewey

Page 63: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

“If I ran a school, I’d give all the average grades to the ones who gave me all the right answers, for being good parrots. I’d give the top grades to those who made lots of mistakes and told me about them and then told me what they had learned from them.”

Buckminster Fuller, Inventor

Page 64: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Time for us to review…

Page 65: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Five strands of mathematical proficiency

NRC (2001) Adding it up: Helping children learn mathematics

Page 66: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

• Conceptual understanding - comprehension of mathematical concepts, operations, and relations

• Procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

• Strategic competence - ability to formulate, represent, and solve mathematical problems

• Adaptive reasoning - capacity for logical thought, reflection, explanation, and justification

• Productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Page 67: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Alan Wigley’s challenging model(an alternative to the path-smoothing model)

• Leads to better learning – learning is an active process

• Engages the learner – learners have to make sense of what is offered

• Pupils see each other as a first resort for help and support

• Scope for pupil choice and opportunities for creative responses provide motivation

Page 68: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

“…the ability to know what to do when they don’t know what to do”

Guy Claxton

Page 69: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Guy Claxton’s Four Rs

• Resilience: being able to stick with difficulty and cope with

feelings such as fear and frustration

• Resourcefulness: having a variety of learning strategies and

knowing when to use them

• Reflection: being willing and able to become more strategic about

learning. Getting to know our own strengths and weaknesses

• Relationships: being willing and able to learn alone and with

others

Page 70: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

What Teachers Can Do

• aim to be mathematical with and in front of learners

• aim to do for learners only what they cannot yet do for themselves

• focus on provoking learners to

– use and develop their (mathematical) powers

– make mathematically significant choices

John Mason

Page 71: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Reflecting on today: the next steps

Two weeks with the students or it’s lost……

• Think big, start small

• Think far, start near to home

• A challenge shared is more fun

• What, how, when, with whom?

Page 72: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

… a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.

Polya, G. (1945) How to Solve it

Page 73: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

I don't expect, and I don't want, all children to find mathematics an engrossing study, or one that they want to devote themselves to either in school or in their lives. Only a few will find mathematics seductive enough to sustain a long term engagement. But I would hope that all children could experience at a few moments in their careers...the power and excitement of mathematics...so that at the end of their formal education they at least know what it is like and whether it is an activity that has a place in their future.

David Wheeler

Page 74: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Recommended Reading Deep Progress in Mathematics: The Improving Attainment in Mathematics Project – Anne Watson et al, University of Oxford, 2003

Adapting and extending secondary mathematics activities: new tasks for old. Prestage, S. and Perks, P. London: David Fulton, 2001

Thinking Mathematically. Mason, J., Burton L. and Stacey K. London: Addison Wesley, 1982.

Mindset: The New Psychology of Success. Dweck, C.S. Random House, 2006

Building Learning Power, by Guy Claxton; TLO, 2002

Page 75: Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale cfg21@cam.ac.uk

Final thoughts

• Thoughts about Mathematics

• Thoughts about teaching and learning Mathematics