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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
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
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
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
• Thoughts about Mathematics
• Thoughts about teaching and learning Mathematics
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?
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.
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?
…and for even more, see the highlighted problems on the
Curriculum Mapping Document
Time for reflection
• Thoughts about Mathematics
• Thoughts about teaching and learning Mathematics
Lunch
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.
http://nrich.maths.org/6933
Time for reflection
• Thoughts about Mathematics
• Thoughts about teaching and learning Mathematics
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
…and for even more, see the highlighted problems on the
Curriculum Mapping Document
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:
http://nrich.maths.org/7768
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
• Thoughts about Mathematics
• Thoughts about teaching and learning Mathematics