Title: Learner-centred Education in Mathematics If you want to build higher, dig deeper
1Learner-centred Education in Mathematics If you
want to build higher,dig deeper
- Charlie Gilderdale cfg21_at_cam.ac.uk
2Initial thoughts
- Thoughts about Mathematics
- Thoughts about teaching and learning Mathematics
3Five 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
4Starting with a rich challengeLow Threshold,
High Ceiling activity
- To introduce new ideas and develop understanding
of new curriculum content
5Making use of a Geoboard environment
6Why might a teacher choose to use this activity
in this way?
7Some underlying principles
- Mathematics is a creative discipline, not a
spectator sport - Exploring ? Noticing Patterns
- ? Conjecturing
- ? Generalising
- ? Explaining
- ? Justifying
- ? Proving
8Tilted 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
9Some 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
10Give 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
11The most exciting phrase to hear in science, the
one that heralds new discoveries, is not
Eureka!, but rather, hmmm thats
funny Isaac Asimov
mathematics
12There are many more NRICH tasks that make
excellent starting points
13Number and Algebra
- Summing Consecutive Numbers
- Number Pyramids
- Whats Possible?
- Whats It Worth?
- Perimeter Expressions
- Seven Squares
- Attractive Tablecloths
14Geometry and Measures
- Painted Cube
- Changing Areas, Changing Perimeters
- Cyclic Quadrilaterals
- Semi-regular Tessellations
- Tilted Squares
- Vector Journeys
15Handling Data
- Statistical Shorts
- Odds and Evens
- Which Spinners?
16and for even more, see the highlighted problems
on the Curriculum Mapping Document
17Time for reflection
- Thoughts about Mathematics
- Thoughts about teaching and learning Mathematics
18Morning Break
19- Valuing Mathematical Thinking
- What behaviours do we value in mathematics and
how can we encourage them in our classrooms?
20As 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
22We 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
24students 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
25Can you find rectangles where the value of the
area is the same as the value of the perimeter?
26Why might a teacher choose to use these
activities in this way?
27We could ask students to find
- (x 2) (x 5)
- (x 4) (x - 3)
-
- or we could introduce them to
28Pair 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?
29We could ask students to
- Identify coordinates and straight line graphs
- or we could introduce them to
30Route 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)?
31We 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
32The 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.
33Why might a teacher choose to use these
activities?
34Some 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.
35There are many more NRICH tasks that offer
opportunities for consolidation
36Number 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?
37Geometry 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
38Handling Data
- M, M and M
- Which List is Which?
- Odds and Evens
- Which Spinners?
39and for even more, see the highlighted problems
on the Curriculum Mapping Document
40Time for reflection
- Thoughts about Mathematics
- Thoughts about teaching and learning Mathematics
41Lunch
42- Promoting purposeful activity and discussion
- Hands-on doesnt mean brains-off
43The 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.
44We could ask
- 3, 5, 6, 3, 3
- Mean ?
- Mode ?
- Median ?
-
- or we could ask
45M, 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?
46Possible extension
- How many sets of five positive whole numbers are
there with the following property? - Mean Median Mode Range a single digit
number
47Whats 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?
48Translating Lines
- Each translation links a pair of parallel
lines. - Can you match them up?
49Why might a teacher choose to use these
activities?
50Rules 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
52Time for reflection
- Thoughts about Mathematics
- Thoughts about teaching and learning Mathematics
53Afternoon Break
54Build 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
55Multiplication 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
56The 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
57There 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
59and for even more, see the highlighted problems
on the Curriculum Mapping Document
60Enriching 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
61What 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 65Five 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.
67Alan 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
69Guy 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
71Reflecting 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
73I 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
74Recommended 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
75Final thoughts
- Thoughts about Mathematics
- Thoughts about teaching and learning Mathematics