Charlie Gilderdale

University of Cambridge

Sri Lanka

3 December 2014

Problem solving in Mathematics - eNRICHing students’ learning experience

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…

• 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 Game

• 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.

The Factors and Multiples Challenge

Morning break

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

Valuing mathematical thinking

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…

Area = ?

Perimeter = ?

or we could ask …

6cm

4cm

Perimeter = 20 cm

= 22 cm

= 28 cm

= 50 cm

= 97 cm

= 35 cm

and we could ask students to…

Area = 24 cm²

Think of a rectangle

Calculate its area and perimeter

Swap with a friend – can they work out the length and breadth of your rectangle?

…students to make up their own questions

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 < MeanMode < Mean < Median

Mean < Mode < MedianMean < 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

mathematics

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 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

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

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, 2003http://www.atm.org.uk/reviews/books/bookspix/DeepProgressEls.pdf

Recommended reading

Final thoughts

Thoughts about Mathematics

Thoughts about teaching and learning Mathematics

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