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Title: Learner-centred Education in Mathematics If you want to build higher, dig deeper


1
Learner-centred Education in Mathematics If you
want to build higher,dig deeper
  • Charlie Gilderdale cfg21_at_cam.ac.uk

2
Initial thoughts
  • Thoughts about Mathematics
  • Thoughts about teaching and learning Mathematics

3
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

4
Starting with a rich challengeLow Threshold,
High Ceiling activity
  • To introduce new ideas and develop understanding
    of new curriculum content

5
Making use of a Geoboard environment
6
Why might a teacher choose to use this activity
in this way?
7
Some underlying principles
  • Mathematics is a creative discipline, not a
    spectator sport
  • Exploring ? Noticing Patterns
  • ? Conjecturing
  • ? Generalising
  • ? Explaining
  • ? Justifying
  • ? Proving

8
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

9
Some underlying principles
  • Teachers 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

10
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
11
The most exciting phrase to hear in science, the
one that heralds new discoveries, is not
Eureka!, but rather, hmmm thats
funny Isaac Asimov
mathematics
12
There are many more NRICH tasks that make
excellent starting points
13
Number and Algebra
  • Summing Consecutive Numbers
  • Number Pyramids
  • Whats Possible?
  • Whats It Worth?
  • Perimeter Expressions
  • Seven Squares
  • Attractive Tablecloths

14
Geometry and Measures
  • Painted Cube
  • Changing Areas, Changing Perimeters
  • Cyclic Quadrilaterals
  • Semi-regular Tessellations
  • Tilted Squares
  • Vector Journeys

15
Handling Data
  • Statistical Shorts
  • Odds and Evens
  • Which Spinners?

16
and for even more, see the highlighted problems
on the Curriculum Mapping Document
17
Time for reflection
  • Thoughts about Mathematics
  • Thoughts about teaching and learning Mathematics

18
Morning Break
19
  • Valuing Mathematical Thinking
  • What behaviours do we value in mathematics and
    how can we encourage them in our classrooms?

20
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

21
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

22
We could ask
We could ask
6cm
  • Area ?
  • Perimeter ?
  • or we could ask

4cm
23
  • Perimeter 20 cm Area 24 cm²
  • 22 cm
  • 28 cm
  • 50 cm
  • 97 cm
  • 35 cm and we could ask

24
students to make up their own questions
  • 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

25
Can you find rectangles where the value of the
area is the same as the value of the perimeter?
26
Why might a teacher choose to use these
activities in this way?
27
We could ask students to find
  • (x 2) (x 5)
  • (x 4) (x - 3)
  • or we could introduce them to

28
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?

29
We could ask students to
  • Identify coordinates and straight line graphs
  • or we could introduce them to

30
Route to Infinity
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)?

31
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

32
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.
33
Why might a teacher choose to use these
activities?
34
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.

35
There are many more NRICH tasks that offer
opportunities for consolidation
36
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 Whats
Possible? Attractive Tablecloths How Old Am I?
37
Geometry and Measures
  • Isosceles Triangles
  • Can They Be Equal?
  • Translating Lines
  • Opposite Vertices
  • Coordinate Patterns

Route to Infinity Picks Theorem Cuboid
Challenge Semi-regular Tessellations Warmsnug
Double Glazing
38
Handling Data
  • M, M and M
  • Which List is Which?
  • Odds and Evens
  • Which Spinners?

39
and for even more, see the highlighted problems
on the Curriculum Mapping Document
40
Time for reflection
  • Thoughts about Mathematics
  • Thoughts about teaching and learning Mathematics

41
Lunch
42
  • Promoting purposeful activity and discussion
  • Hands-on doesnt mean brains-off

43
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.
44
We could ask
  • 3, 5, 6, 3, 3
  • Mean ?
  • Mode ?
  • Median ?
  • or we could ask

45
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?

46
Possible extension
  • How many sets of five positive whole numbers are
    there with the following property?
  • Mean Median Mode Range a single digit
    number

47
Whats 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?

48
Translating Lines
  • Each translation links a pair of parallel
    lines.
  • Can you match them up?

49
Why might a teacher choose to use these
activities?
50
Rules for Effective Group Work
  • All students must contributeno one member says
    too much or too little
  • Every contribution treated with respectlisten
    thoughtfully
  • Group must achieve consensuswork at resolving
    differences
  • Every suggestion/assertion has to be
    justifiedarguments must include reasons
    Neil Mercer

51
  • 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

52
Time for reflection
  • Thoughts about Mathematics
  • Thoughts about teaching and learning Mathematics

53
Afternoon Break
54
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
55
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
56
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 childs inner resources, and
    develop a childs 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

57
There are many NRICH tasks that encourage
students to work as a mathematical community
58
  • Making Rectangles
  • Whats it Worth?
  • Steel Cables
  • Odds and Evens
  • M, M and M
  • Odds, Evens and More Evens

Tilted Squares Pair Products Whats
Possible? Cyclic Quadrilaterals How Old Am
I? Factors and Multiples Game
59
and for even more, see the highlighted problems
on the Curriculum Mapping Document
60
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 teachershttp//nrich.maths
    .org/curriculum

61
What next?
  • Secondary CPD Follow-up on the NRICH site
  • http//nrich.maths.org/7768

62
  • 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

63
  • If I ran a school, Id give all the average
    grades to the ones who gave me all the right
    answers, for being good parrots. Id 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

64
  • Time for us to review

65
Five strands of mathematical proficiency
NRC (2001) Adding it up Helping children learn
mathematics
66
  • 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 ones own efficacy.

67
Alan Wigleys 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

68
  • the ability to know what to do when they dont
    know what to do
  • Guy Claxton

69
Guy Claxtons 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

70
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

71
Reflecting on today the next steps
  • Two weeks with the students or its lost
  • Think big, start small
  • Think far, start near to home
  • A challenge shared is more fun
  • What, how, when, with whom?

72
  • 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

73
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
74
Recommended Reading Deep Progress in
Mathematics The Improving Attainment in
Mathematics Project Anne Watson et al,
University of Oxford, 2003Adapting and
extending secondary mathematics activities new
tasks for old. Prestage, S. and Perks, P. London
David Fulton, 2001Thinking 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
75
Final thoughts
  • Thoughts about Mathematics
  • Thoughts about teaching and learning Mathematics
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