Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale...

Learner-centred Education in Mathematics

If you want to build higher,dig deeper

Charlie Gilderdale [email protected]

Initial thoughts

• Thoughts about Mathematics

• Thoughts about teaching and learning Mathematics

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

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

To introduce new ideas and develop understanding of new curriculum content

Making use of a Geoboard environment

http://nrich.maths.org/2883

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

Some underlying principles

Mathematics is a creative discipline, not a spectator sport

Exploring → Noticing Patterns

→ Conjecturing → Generalising

→ Explaining→ Justifying

→ Proving

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


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

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

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

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

Number and Algebra

Summing Consecutive Numbers

Number Pyramids

What’s Possible?

What’s It Worth?

Perimeter Expressions

Seven Squares

Attractive Tablecloths




Geometry and Measures

Painted Cube

Changing Areas, Changing Perimeters

Cyclic Quadrilaterals

Semi-regular Tessellations

Tilted Squares

Vector Journeys

Handling Data

Statistical Shorts

Odds and Evens

Which Spinners?

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

Curriculum Mapping Document

Time for reflection



Morning Break

Valuing Mathematical Thinking

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

in our classrooms?

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

… 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

We could ask…

Area = ?

Perimeter = ?

or we could ask …

6cm

4cm

We could ask:

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

and we could ask …

• 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

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.

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

We could ask students to find…

(x + 2) (x + 5)(x + 4) (x - 3)…

or we could introduce them to…

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?

We could ask students to…

Identify coordinates and straight line graphs

or we could introduce them to…

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

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…

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.

Why might a teacher choose to use these activities?


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.

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

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?

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

Handling Data

M, M and M

Which List is Which?

Odds and Evens

Which Spinners?

Time for reflection



Promoting purposeful activity and discussion

‘Hands-on’ doesn’t mean ‘brains-off’

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.

We could ask…

3, 5, 6, 3, 3

Mean = ?Mode = ?

Median = ?

or we could ask…

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?

Possible extension

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

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

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?

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.

Why might a teacher choose to use these activities?

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

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.


Time for reflection



Afternoon Break

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

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

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

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

mathematical community…

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

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

What next?

Secondary CPD Follow-up on the NRICH site:


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

“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

Time for us to review…

Five strands of mathematical proficiency

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

• 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.

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

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

Guy Claxton

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

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

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?

… 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

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

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

Final thoughts



Learner-centred Education in Mathematics If you want to build higher, dig deeper Charlie Gilderdale...

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