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Problem solving in Mathematics eNRICHing students’ learning experience
Charlie Gilderdale
University of Cambridge
Sri Lanka
3 December 2014
Initial thoughts
Thoughts about Mathematics
Thoughts about teaching and learning Mathematics
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.
Four parts to the day
• Engaging learners
• Valuing mathematical thinking
• Building a community of mathematicians
• Reviewing and reflecting
Engaging learners
Consolidating with rich tasks to:
Develop fluency
Deepen understanding
Build connections
Dicey operations
Find a partner and a 1-6 dice,
or preferably a 0-9 dice.
Each of you draw an addition grid.
Take turns to throw the dice and
decide which of your cells to fill either fill in each cell as you throw the
dice or collect all your numbers and
then decide where to place them.
Throw the dice nine times each until all the cells are full.
Whoever has the sum closest to 1000 wins.
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
The Factors and Multiples Challenge
• This time, try to find the longest sequence of numbers
that can be crossed out.
• Again, choose a number that is a factor or multiple of
the previous number that has just been crossed out.
• Each number can only appear once in a sequence.
Morning break
Valuing mathematical thinking
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
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?
Some (quick) ways to (probably) make
consolidation tasks more interesting
• reverse the question
• seek all possibilities
• greater generality (what if…?)
• look at/for alternative methods
We could ask…
6cm
Area = ?
Perimeter = ?
4cm
or we could ask …
Perimeter = 20 cm
= 22 cm
= 28 cm
= 50 cm
= 97 cm
= 35 cm
and we could ask students to…
…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?
Why might a teacher choose to
use these activities?
Some (quick) ways to (probably) make
consolidation tasks more interesting
• reverse the question
• seek all possibilities
• greater generality (what if…?)
• look at/for alternative methods
Isosceles Triangles
Draw some isosceles triangles with an
area of 9 cm2 and a vertex at (20, 20).
If all the vertices have whole number
coordinates, how many is it possible to
draw?
Can you explain how you know that
you have found them all?
Can you find five positive whole numbers
that satisfy the following properties:
Mean = Mode = Median = Range
Can you find all the different sets of five positive
whole numbers that satisfy these conditions?
Mean = Mode = Median = Range = 40
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
Some (quick) ways to (probably) make
consolidation tasks more interesting
• reverse the question
• seek all possibilities
• greater generality (what if…?)
• look at/for alternative methods
We could ask…
Can you find five positive whole numbers
that satisfy the following properties:
Mode < Median < Mean
Mode < Mean < Median
Mean < Mode < Median
Mean < Median < Mode
Median < Mode < Mean
Median < Mean < Mode
Four positive whole numbers? Six?
Why might a teacher choose to
use this activity?
Some (quick) ways to (probably) make
consolidation tasks more interesting
• reverse the question
• seek all possibilities
• greater generality (what if…?)
• look at/for alternative methods
Temperature
The freezing point of water is
0°C and 32°F.
The boiling point of water is
100°C and 212°F.
Is there a temperature at which
the Celsius and Fahrenheit
readings are the same?
Can they be equal?
Can you find rectangles where the value of the area is the
same as the value of the perimeter?
Other examples
Cuboid Challenge
Cut a square from
each corner and fold
up the flaps.
What volumes are
possible for different
sizes of cut-out
squares?
Warmsnug
Double Glazing
Route to Infinity
Which point will it
visit after (18,17)?
How many points
will it visit before
reaching (9,4)?
Why might a teacher choose to
use these activities?
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
Some underlying principles
Consolidation tasks 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.
Time for reflection
Thoughts about Mathematics
Thoughts about teaching and learning Mathematics
Lunch
Build a community of mathematicians
By:
Creating a safe environment for learners to take risks
Providing opportunities to work collaboratively
Valuing a variety of approaches
Encouraging critical and logical reasoning
Mathematics is a creative discipline,
not a spectator sport
Exploring
→ Noticing Patterns
→ Conjecturing
→ Generalising
→ Explaining
→ Justifying
→ Proving
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
Tilted Squares
Can you find a quick and easy method to
work out the areas of tilted squares?
Making use of a Geoboard environment
Why might a teacher choose to use
this activity in this way?
Some underlying principles
Teacher’s role:
• To choose tasks that allow students to explore
new mathematics
• To give students the time and space to explore
• To bring students together to share ideas and
understanding, and draw together key
mathematical insights
Enriching mathematics website
www.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
Time for reflection
Thoughts about Mathematics
Thoughts about teaching and learning Mathematics
Afternoon break
Time for us to review…
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
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
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
Reciprocity: being willing and able to learn alone and
with others
What can we offer learners?
• Low threshold, high ceiling tasks
• Opportunities to exhibit their thinking and refine their understanding
• A conjecturing culture where it is OK to make mistakes
• A careful use of guiding questions and prompts
• Opportunities to practice skills in an engaging way: HOTS not MOTS
• Frequent opportunities for talk (about maths)
• Teachers who model mathematical behaviour
• Teachers who emphasise mathematical behaviours that they wish to
promote
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
Take a topic you’ve just taught,
or are about to teach,
and look for opportunities to
• reverse questions
• list all possibilities
• search for generality
• consider alternative methods
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?
What next?
Secondary CPD Follow-up
on the NRICH site:
http://nrich.maths.org/7768
… 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
Recommended reading
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
Adapting and extending secondary mathematics activities: new tasks
for old. Prestage, S. and Perks, P. London: David Fulton, 2001
Deep Progress in Mathematics: The Improving Attainment in
Mathematics Project – Anne Watson et al, University of Oxford, 2003
http://www.atm.org.uk/reviews/books/bookspix/DeepProgressEls.pdf
Final thoughts
Thoughts about Mathematics
Thoughts about teaching and learning Mathematics
Learn more!
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