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Why solve

problems?

Although problem

solving is a phrase

currently on the

lips of many maths

teachers, people

have been solving

problems since the

beginning of man,

albeit by trial and

error.

In his 1998 paper

Problem Solving,

Kevin Dunbar

gives focuses on a

number of

important issues in

problem solving

research, along

with an overview of

developments in

problem solving

research. Dunbar

describes two

crucial features of

problem solving:

“First, a problem

exists when a goal

must be achieved

and the solution is

not immediately

obvious. Second,

problem solving

often involves

attempting different

ways of solving the

problem.”

www.mei.org.uk

Problem solving in mathematics

education

How and when does problem solving fit into mathematics education?

In her nrich article Problem Solving and the New Curriculum author Lynne McClure asks

“What's the point of doing maths?” McClure says: “What children should be doing is solving problems, their own as well as those posed by others. Because the whole point of learning maths is to be able solve problems. Learning those rules and facts is of course important, but they are the tools with which we learn to do maths fluently, they aren’t maths itself.”

MEI's response to the draft primary national curriculum (July

2012) states that:

“Incorporating a problem solving cycle into the national curriculum is not, of itself, a sufficient means of improving classroom teaching. Fundamental improvements in classroom practice, supported by appropriate professional development and

resources, are needed.”

In his MEI Conference 2014 session,

Using problem solving to develop

mathematical thinking in post GCSE

students, Phil Chaffé suggests that the

following benefits will be enjoyed by

teachers and students:

More engaged students

Students with a better

understanding of mathematics,

who rise above the symbol

manipulation

Students who appreciate

mathematics techniques as tools

rather than endpoints

Students who are better prepared

for HE and careers

More possibilities to spot

mathematical talent

Better results

In the following pages we will examine

the impact on maths education of people

involved in problem solving: George

Pólya, Derek Holton, Charles Lovitt,

Edward de Bono and Marylin vos Savant.

Curriculum Update

GCSE Mathematics for first teaching

2015, AQA, Edexcel Pearson and OCR

are all accredited.

Teaching of the level 3 Certificate in

Quantitative Methods (MEI) has

started. Resources are freely

available online

DfE and Ofqual consultations about A

level Mathematics and Further

Mathematics have now closed. The

outcome of the consultations is

expected Oct/Nov.

George Pólya

Teaching

resources

Carol Knights has adapted resources from three problem-solving sessions delivered by MEI staff at the MEI Conference 2014:

Carol’s resulting teaching and learning resource is included at the end of this newsletter. As usual, you can download this resource in its original file formats, from the MEI Monthly

Maths web page.

to mathematical discovery”. He plays a

guessing game with a group of students,

explaining that guessing is the important

beginning of solving a problem, and that

looking at a simpler version of a problem

will help to solve a more complex

problem.

Although the film’s picture quality is quite

dark, it is worth watching and listening to

follow Pólya’s encouraging style of

engagement with the group, as they work

together towards a solution to the problem

using reasonable guessing, then

observation, analogy and generalisation –

the process of induction*.

Pólya advises the students that they

should not hold back from guessing;

however they should not believe their own

guesses, but test them. They should

recognise the difference between a fact

and a guess. By taking the students

through a simple problem Pólya is able to

steer them gently towards a more

complex problem. You can observe the

students becoming increasingly engaged

and questioning during this process.

*For more information see Pólya’s 1953 article in A. Bogomolny, INDUCTION AND ANALOGY IN MATHEMATICS - Preface from Interactive Mathematics

Miscellany and Puzzles.

George Pólya

George Pólya (1887 – 1985) was a Hungarian mathematician and professor of mathematics at ETH Zürich and at Stanford

University. His work on heuristics (general problem solving strategies) and pedagogy has had lasting influence on mathematical education. In 1945 Pólya published the book How To Solve It,

which sold over one million copies and has been translated into 17 languages. In this book he identifies four basic

principles of problem solving:

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Look back

In the Scholastics Teachers Resources section, the 4 Steps to Problem Solving page details Pólya’s problem solving steps as adopted by Billstein, Libeskind and Lott in their book A Problem Solving Approach to Mathematics for Elementary

School Teachers.

The one-hour 1965 film Pólya Guessing starts with

Pólya explaining his attitude to teaching:

“Teaching is giving students the opportunity to discover things by

themselves.”

“First guess then prove.”

“Finished mathematics consists of proofs…but mathematics in the making

consists of guesses.”

In the film Pólya demonstrates his

“extraordinary ability to stimulate a group

to guess intelligently, to make reasonable

conjecture, a process which is essential

Clare Parsons

Kevin Lord

Phil Chaffé

Derek Holton

Problem solving resources

The New Zealand Maths website that hosts the 400 Problem lesson resource (see right) also includes Problem Solving Information. This provides practical guidance about how to implement problem solving in a maths programme (referenced to The New Zealand Curriculum but much of it can be related to other curricula) as well as some of the philosophical ideas behind problem solving. The Problem Solving section of the nzmaths website provides

problem-solving lessons that cover Levels 1 to 6 (ages 5-15) of the New Zealand Curriculum.

Although written with a focus on mathematics at university level, Holton’s article Mathematics: What? Why? How? (page 21 of the Community for Undergraduate Learning in the Mathematical Sciences Newsletter No.1, July 2010) includes an attempt to describe the structure of the creative

process in mathematics.

Holton says that as mathematics teachers

“We shy away from setting questions that

are, in some sense ‘open’, and we avoid

‘natural questions’ while we move down

set paths through traditional courses.” He

stresses that the second type of question

should be used more frequently: “Solving

problems is not just about solving

problems that everyone knows how to

solve”.

In Holton’s experience with bright

secondary students they really enjoy the

more open questioning approach, the

human side of mathematics, as opposed

to a set of things

to be learned.

Experiencing

problem solving

in this way will

also prepare

them to tackle

new problems in their future careers.

Derek Holton

British-born and Australian-

educated Derek Holton,

former Professor of Pure

Mathematics at the

University of Otago, New

Zealand, has a special

interest in problem solving. He has written

several books on the subject, including a

2013 book, More problem solving: the

creative side of mathematics, published

by the Mathematical Association. The

abstract states that the “the underlying

aim of this book is to show that

mathematics is more than a collection of

results; there is also a creative, people-

side to the subject”, and that the book will

demonstrate how the ordinary can

become extraordinary when viewed

through a mathematical prism”.

Now retired and living in Melbourne, Australia, Holton continues to write about the importance of problem solving in mathematics. In his article What? No Moses? (in his blog Del’s Disturbances on the Casio Edu Australia website),

Holton tells us about mathematics:“ there was no one person who brought down the precepts of the subject from a mountain...But if there are any precepts in mathematics, at least at the macro level,

they are problems, people and proof.”

The Mathematics Association publishes the magazine Mathematics in School; in the March 2006 issue there is an article

by Holton where he describes and discusses ‘The 400 Problem’: “This problem really does contain experimenting, conjecturing, proving, extending and generalizing. All these are

the mathematician’s tools in trade”.

Charles Lovitt

nrich problem solving resources

In Jennifer Piggott’s excellent article, A Problem Is a Problem for All That, she

advises:

Give learners space Value their differences Learn from what they do and help them to make connections Use the inherent richness of opportunities to highlight interesting mathematics nrich also offers this post-16 investigation : Maths Problem - Twisty Logic. “Sometimes mathematical setups which appear to be straightforward can lead to circular of self-contradictory or 'paradoxical' logic. Give your brain a workout by thinking about these scenarios”.

1. Find an interesting (meaningful/

worthwhile) problem.

2. Informally explore, unstructured ‘play’

which generates data.

3. From patterns in the data, create

hypotheses, conjectures, theories.

4. Invoke problem solving strategies to

prove or disprove any theories.

5. Apply any basic skills I know as part

of this proof process.

6. Extend and generalise the problem –

what else can I learn from it?

7. Publish (or perish).

8. Go back to step 1.

The basic skills mentioned in Stage 5

would include mathematical skills such as

algorithms, graphing techniques,

algebraic modelling, solution methods for

equations, etc. How can we equip our

students with the problem solving

strategies mentioned in Stage 4?

Lovitt suggests that a separate lesson

could be devoted to developing thinking

skills to create a toolbox of thinking

strategies.

In the next pages we look at Edward de

Bono, a proponent of the teaching of

thinking as a subject in schools. As a

young teacher in the early 1980s I taught

some of de Bono’s thinking skills in a

lesson a week to my tutor group. This

was as part of a wide humanities-based

curriculum, but the thinking skills could be

applied to any curricular subject, including

mathematics.

Charles Lovitt

Charles Lovitt has been

involved in Mathematics

Professional

Development for many

years. Now retired, he

was the director of a host

of initiatives and networks

in Australia and beyond, including RIME

(Reality in Mathematics Education).

In his keynote presentation for an

Australian Association of Mathematics

Teachers Virtual Conferences,

Investigations as a central focus for a

Mathematics curriculum, Charles

Lovitt pointed out “the unfortunate

perception that one aspect of the problem

solving picture is delivered through

games and puzzles and therefore is

relegated to the periphery or margins of

mathematics”.

He suggests that the term ‘Problem

Solving’ might better be replaced by the

term ‘Investigation Process’, as “The

word (problem solving) has become so

blurred that we have no common shared

agreement on what it means”.

Lovitt cites Derek Holton’s article What

Mathematicians Do — and why it is

important in the classroom (Item 6: Best

of Set, ACER, Melb. 1994), where he

listed the investigative process by which

mathematicians create knowledge and

solve problems.

In his presentation Lovitt paraphrases the

stages of this process (see also diagram

on previous page), which could also provide easy to follow guidelines for both

constructing and assessing curriculum:

Edward de Bono

Two types of

thinking

According to de

Bono, thinking can

be divided into two

types:

Vertical thinking:

the traditional-

historical method

that uses the

processes of logic

Lateral (or

‘creative’) thinking:

a relatively new

type of thinking

that complements

analytical and

critical thinking

Lateral thinking

seeks to solve

problems by

apparently illogical

means; it is a

learnable set of

skills (a ‘tool’) that

uses a process

and willingness to

look at things in a

different way,

using insight and

creativity.

He uses this

analogy to explain:

“Lateral thinking is

like the reverse gear

in a car. One would

never try to drive along in reverse gear

the whole time. On the other hand one

needs to have it and to know how to use

it for manoeuvrability and to get out of a

blind alley.”

De Bono stresses that lateral thinking

isn’t something that occurs by chance

and that the technique needs to be

taught, preferably as a defined lesson,

rather than along with another subject:

“The best way to acquire skill in lateral

thinking is to acquire skill in the use of a

collection of tools which are all used to

bring about the same effect.”

After discussing attitudes towards lateral

thinking and its use, de Bono describes in

detail in the book, section by section, the

different processes, including background

material, theory and nature of the process

being discussed in that section, followed

by practical formats for trying out and

using the process under discussion –

“actual involvement” is vital, he says.

However, de Bono stresses that “what is

supplied is supplied more as an example

than as anything else. Anyone who is

teaching lateral thinking…must

supplement the material offered here with

his own material.” He suggests different

types of materials that could be collected

for this purpose.

De Bono stated that: “Some people with

high IQs turn out to be relatively

ineffective thinkers and others with much

more humble IQs are more effective.”

Edward de Bono

Edward de Bono was

born in Malta in 1933. He

developed the concept

and tools of lateral

thinking, making his work

practical and available to

everyone, from five years olds to adults.

In his Problem Solving article on the

thinking-approaches website, de Bono

suggests that the traditional method of

problem solving “is that you analyse the

problem, identify the cause and then

proceed to remove the cause. The cause

of the problem is removed so the problem

is solved.” However, this does not work in

all cases, for example, where there is

more than one cause for the problem, or

where the cause cannot be found or

cannot be removed. He says that analysis

and argument are not enough; we need

to develop the habits of constructive

thinking: “The whole thrust of education is

towards analysis…Everything should

yield to analysis in our traditional methods

of thinking. Very little emphasis is given to

creativity.”

In his 1970 book Lateral Thinking, de

Bono says of the two types of thinking:

“Lateral thinking is not a substitute for

vertical thinking. Both are required. They

are complementary. Lateral thinking is

generative. Vertical thinking is selective.”

Six Thinking Hats

Using Six

Thinking Hats in

the classroom

In this nine-minute

video Kim Wells,

one of three de

Bono master

trainers in

education,

describes what de

Bono's thinking

hats are and how

they can be used

as a learning and

thinking strategy.

In her article Dr

Edward de

Bono’s six

thinking hats and

numeracy

(Australian

Mathematics

Primary Classroom

(3) 2006), Anne

Patterson, a

teacher and

lecturer in Victoria,

applies the

teaching approach

of “thinking hats”

to mathematics

education.

This thinking tool is designed to help

people think clearly and thoroughly by

directing their thinking attention in one

direction at a time. Each metaphorical

'Thinking Hat' is a different colour that

represents a different style of thinking.

If you look at a problem with the 'Six

Thinking Hats' technique, then you will

solve it using all approaches.

This four-minute video

explains the Six Thinking

Hats method further.

De Bono defined thinking as: The

operating skill with which intelligence

acts upon experience.” He felt that

there is enough individuality in thinking

styles and sufficient difference between

individuals to suggest that thinking may

be a skill that can be developed. With this

in mind, de Bono designed the CoRT

Thinking Lessons for schools; these

lessons have been in use since 1970.

De Bono summarises the CoRT Cognitive

Research Trust) aims as follows:

1. To acknowledge thinking as a skill.

2. To develop the skill of practical

thinking.

3. To encourage students to look

objectively at their own thinking and the

thinking of others.

Edward de Bono explains

the importance of and need

for thinking tools in a ten-minute video

introduction to a lecture that has been

uploaded in six parts. This includes a

demonstration of the process of addition,

and how we can rearrange things in our

mind so that we deal with them more

simply and more effectively. His ‘Six

Thinking Hats’ method is also mentioned.

The ‘Six Thinking Hats’ framework is

widely used across the world from

primary school to board level for any

sort of discussion or debate, as an

alternative to traditional argument. The

‘parallel’ nature of this method, where

everyone is thinking in the same

direction, from the same perspective, at

the same time, “enables each person's

unique point of view to be included and

considered.”

The White Hat calls for

information known or

needed.

The Red Hat signifies

feelings, hunches and

intuition.

The Black Hat is judgment --

the devil's advocate or why

something may not work.

The Yellow Hat symbolizes

brightness and optimism.

The Green Hat focuses on

creativity: the possibilities,

alternatives and new ideas.

The Blue Hat is used to

manage the thinking

process.

Marilyn vos Savant

The problem

The teaser was based on one of the games in the US show Let's Make a Deal! hosted 1963-1991 by Monty Hall.

Let's Make a Deal unexpectedly spawned a mathematical conundrum dubbed the Three

Door Puzzle.

Yet this wasn’t a new problem to some mathematicians, who called it the Monty Hall Problem. An earlier version of the problem, the Three Prisoner Problem, was

analysed in 1959 by Martin Gardner in his Mathematical Games column in the journal Scientific American, noting that "in no other branch of mathematics is it so easy for experts to blunder as in

probability theory."

In a follow-up column vos Savant called

on school teachers to show the problem

to classes, and published the results of

more than 1,000 school experiments.

Nearly 100% found it pays to switch.

Despite this, the Monty Hall Problem

continues to be a much-debated topic –

we have found a few links that we hope

will help to model the problem:

The Dr Math forum illustrates the debate

well, with several people posting their

explanations and solutions to the

problem. The Virtual Laboratories in

Probability and Statistics project

provides free, high quality, interactive,

web-based resources for students and

teachers of probability and statistics. The

Games of Chance section includes the

Monty Hall Problem.

Alan Davies and Oxford

Mathematics Professor

Marcus Du Sautoy test out the Monty Hall

problem in this YouTube video in

response to the comments in this video.

The Monty Hall problem was

also featured in Mark

Haddon’s novel The Curious

Incident of the Dog in the

Night-time; the SparkNotes

literature study guide to the

book (see Analysis:

Chapters 97-101) sums it up thus:

“In essence, Christopher shows that

intuition, which says is what people use in

life to make decisions, can lead a person

to the wrong answer. A problem that

appears straightforward turns out to be

not straightforward at all.”

Marilyn vos Savant

A good example of using

thinking hats to use different

perspectives to consider a

problem is that of Marilyn

vos Savant and the Monty Hall Problem. In September 1990, vos Savant, puzzle

columnist for the U.S. magazine Parade,

was sent a probability teaser by a reader.

Its publication in her "Ask Marilyn"

column together with her solution has

produced much debate amongst

mathematicians and laymen ever since.

“Suppose you're on a game show, and

you're given the choice of three doors:

Behind one door is a car; behind the

others, goats. You pick a door, say No. 1,

and the host, who knows what's behind

the doors, opens another door, say No. 3,

which has a goat. He then says to you,

‘Do you want to pick door No. 2?’ Is it to

your

advantage to

switch your

choice?”

Marilyn vos Savant replied:

“Yes; you should switch. The first door

has a 1/3 chance of winning, but the

second door has a 2/3 chance. Here’s a

good way to visualize what happened.

Suppose there are a million doors, and

you pick door #1. Then the host, who

knows what’s behind the doors and will

always avoid the one with the prize,

opens them all except door #777,777.

You’d switch to that door pretty fast,

wouldn’t you?” She came under

vehement criticism from

mathematicians for her reply.

Further Reading

Teachers’ resources and

ideas shared

The NCETM weekly Twitter #mathscpdchat on 1 July 2014 discussed different ways to promote problem solving in maths and useful resources. The chronological account of the chat is available on the NCETM website, with some of the ideas highlighted. Several links to relevant books are

also provided.

A search in TES Connect reveals a wide array of problem solving resources shared by teachers for use in the secondary mathematics

classroom.

In her article Mathematics Through Problem Solving, Margaret Taplin (Institute of Sathya Sai Education, Hong Kong) asks What Is A 'Problem-Solving Approach'?, and examines The Role of Problem Solving in Teaching

Mathematics as a Process.

MATHEMATICAL PROBLEM SOLVING by James W. Wilson, Maria L. Fernandez, and Nelda Hadaway (Wilson, P. S. (Ed.)(1993). Research Ideas for the Classroom: High School Mathematics.

New York: MacMillan. Chapter 4). The authors review and discuss the research on how students in secondary schools can develop the ability to solve a wide

variety of complex problems:

PISA 2012 results: Creative Problem Solving (OECD Publishing). This volume presents an assessment of student performance in creative problem solving, which measures students’ capacity to respond to non-routine situations in order to achieve their potential as constructive

and reflective citizens.

LeMaPS: Lessons for Mathematical Problem Solving (University of Nottingham School of Education; Centre for Research in Mathematics Education. Principal Investigators: Geoff Wake; Co-Investigators: Malcolm Swan, Colin Foster). “This Nuffield funded project seeks proof-of-concept of new and sustainable models of partnerships that support professional learning in secondary school mathematics with the involvement of Higher Education. The focus is on improving students’ problem-solving capabilities in mathematics. “The project will build on the outcomes of a Bowland Maths funded pilot that explored the use of Japanese lesson study principles to consider the teaching

of mathematical problem solving.”

Further Reading

Laura E. Hardin’s 2002 paper Problem Solving Concepts and Theories provides an overview of educational research on problem solving. Hardin considers problem solving in the context of behavioural, cognitive, and information-processing pedagogy, concluding that “both content knowledge and general problem-solving skill are necessary for

expert problem solving to occur.”

Why Is Teaching With Problem Solving Important to Student Learning? In their NCTM research brief authors Jinfa Cai and Frank Lester provide some directions and useful suggestions, for both teachers and curriculum writers, on teaching with problem solving based on research . (Judith Reed Quander, Series Editor. National Council of Teachers of

Mathematics Research Brief. 2010.)

Fostering Mathematical Thinking and Problem Solving: The Teacher’s Role Nicole R. Rigelman’s article includes considerations for teachers who want to foster their students’ mathematical thinking and problem solving. (Teaching Children Mathematics. February 2007. The National Council of

Teachers of Mathematics, Inc.)

Kevin Niall Dunbar (Professor of Human

Development and Quantitative Methodology at the University of Maryland College Park) has written several papers on scientific thinking heuristics, including PROBLEM SOLVING (1998. In W. Bechtel, & G. Graham (Eds.). A companion to Cognitive Science. London, England: Blackwell, pp

289-298.); Problem Solving and Reasoning (2006. Kevin Dunbar & Jonathan Fugelsang. To appear in E.E. Smith & S. Kosslyn. An introduction to

cognitive psychology. Chapter 10).

MEI and FMSP Problem Solving

Resources Links

GCSE Problem Solving

Resources

Problem Solving

Resources

Problem Solving - CPD for teaching problem solving in GCSE Maths or in

the Sixth Form

STEP/AEA/MAT support - The FMSP will be running several national programmes of CPD to support teachers helping students to develop problem solving skills and to prepare for examinations, also Student and Teacher Problem Solving Conferences where students and teachers will look at various aspects of problem

solving skills.

STEP/AEA/MAT

Year 12 Problem Solving Summer School - A series

of five live online workshops to help year 12 students develop their problem solving skills in pure

mathematics.

delivered by Stella Dudzic:

IQM: Modelling and estimation

IQM: Probability and risk

IQM: Financial problem solving

IQM: Statistical problem solving

London Schools’ Excellence Fund CPD:

Introducing problem-solving into the

Key Stage 4 curriculum

Developing mathematical thinking in

post 16 students

LSEF Problem Solving Conference Resources - Resources from the first London Schools’ Excellence Fund Mathematical Problem Solving Conference that took place on Wednesday 9th July 2014 at Birkbeck, University of London.

STEP and AEA support - MEI provides

real-time online tutorials and teaching sessions in STEP and AEA Mathematics. Students can access live interactive tuition at a time and location to suit them

through an online learning platform.

Problem Solving and STEP – An MEI Conference 2014 session delivered by FMSP Area Coordinators Martin Bamber

& Abi Bown

The Further Mathematics Support Programme (FMSP) supports the

development of problem solving in mathematics at both GCSE and A level with professional development courses for teachers, events and activities for students and resources for use in schools

and colleges. See left column for links.

MEI curriculum development,

resources and professional

development

Links to web pages about problem

solving in mathematics education, with

links to publications and resources.

Realistic Mathematics Education

(RME)

Integrating Mathematical Problem

Solving (IMPs)

Integrating Mathematical Problem Solving (IMPS) resources - Free of charge, designed to help teachers of mathematics and teachers of other subjects at A level to teach relevant aspects of mathematics and statistics, showing how they are used in solving real

problems.

Critical Maths: a mathematics-based

thinking curriculum for Level 3

Critical Maths resources - Designed for

post-16 students at level 3; especially useful for Core Maths classes. The resources enable students to think about real problems using mathematics. Many start by engaging the students in giving an initial opinion and then encourage them to think more deeply and to

evaluate their initial thoughts.

Quantitative Methods

OCR Level 3 Certificate in Quantitative Methods resources - Can be subscribed to free of charge by centres, thanks to

sponsorship from OCR.

Teaching Introduction to Quantitative

Methods CPD

A series of MEI Conference 2014 session

Spot the Pattern On the next slide is a grid and on each subsequent slide there are 4 pieces of information. Can you work out how the grid should be coloured in?

Spot the Pattern There are 4 red squares (arranged in a square) in the middle of the design.

There are 7 red squares in the bottom right hand quarter of the design.

There is one square of each colour in the top row of the design (the rest are blank).

No blue square is directly next to a yellow square.

Spot the Pattern There are 9 blank squares (arranged in a square) in the bottom right hand corner of the design.

There is one square of each colour in the first column of the design (the rest are blank).

The top left corner to the bottom right corner is a line of reflection symmetry.

The blue square in the top row has two blank squares between it and the yellow square in the top right hand corner.

Spot the Pattern The yellow squares are only on the top right to bottom left diagonal.

There are 5 more red squares than blue squares.

The design has one line of reflection symmetry.

There are 6 blue squares in the top left hand quarter of the design

Spot the Pattern There are 6 blue squares in the top left hand quarter of the design

There are 13 red squares in the design.

Some of the squares are not coloured in.

The design does not have rotation symmetry.

Spot the Pattern The top left corner to the bottom right corner diagonal has 5 red squares on it (the rest are blank)

The design uses 3 different colours.

The top right corner to bottom left corner diagonal has red and yellow squares only in the ratio 1:3.

The ratio of yellow squares to blue squares is 3:4.

Algebra with cards and paper

Three rectangular business cards are shown. What is the perimeter of each?

7cm

5 cm

7cm

a

b

a

Algebra with cards and paper

5 cm+7cm+5 cm+7cm = 24 cm 7cm

5 cm

7cm

a

b

a

a cm+7cm+a cm+7cm = 14+2a cm

a cm+ b cm+ a cm+ b cm=2a + 2b cm

Algebra with cards and paper Using this business card, three arrangements of 2 cards are shown below. What is the perimeter of each?

Algebra with cards and paper

Which is smaller: 2a +4b or 4a+2b?

2a+4b

4a+2b 2a+4b

Algebra with cards and paper Putting 2 cards together ‘edge to edge’, what other perimeters can you find? How might you write an expression for the perimeter of these arrangements?

Algebra with cards and paper Can you describe how to arrange the two cards to obtain: • the maximum perimeter? • the minimum perimeter? Can you explain how you know these are the maximum and minimum values? How many different arrangements can you find for 3 cards?

Algebra with cards and paper Explore the maximum and minimum perimeters for: • 3 cards • 4 cards • 5 cards • … • n cards Can you come up with general algebraic expressions for the maximum and minimum perimeters for n cards?

Algebra with cards and paper Does using a different sized rectangular card affect the arrangements that give the minimum perimeter? Does it affect the algebraic value of the minimum perimeter?

Perimeter of Rectangular Rings Putting 4 cards together, it is possible to make a ring as shown. Write down an expression for:

• the perimeter of the outer rectangle of the ring; • the perimeter of the inner rectangle of the ring; • the total perimeter.

Can you simplify your expressions?

Perimeter of Rectangular Rings outer rectangle of the ring: 4a + 4b inner rectangle of the ring: 4b – 4a the total perimeter: 8b

Perimeter of Rectangular Rings Putting 6 cards together there are 2 possible rings. Find them and write expressions for: • the perimeter of the outer rectangle of the ring; • the perimeter of the inner rectangle of the ring; • the total perimeter.

Simplify the expressions where possible

Perimeter of Rectangular Rings outer rectangle of the ring: 4a + 6b inner rectangle of the ring: 6b – 4a the total perimeter: 12b

Perimeter of Rectangular Rings outer rectangle of the ring: 4a + 6b inner rectangle of the ring: 6b – 4a the total perimeter: 12b

Perimeter of Rectangular Rings Explore for different numbers of cards. What do you notice each time? Can you explain why?

Monty Hall Problem Monty Hall was a U.S. game show host in the 1970s. His show provides us with a probability problem. Contestants on the show would either win a car… …or a goat.

Monty Hall Problem Monty presents the contestant with a choice of 3 doors. Behind one of them is a car, behind the other two are goats.

Green!

Monty Hall Problem Having chosen a door, Monty shows her what is behind one of the other doors – he knows where the car is and always shows her a goat.

Monty Hall Problem He now asks her whether she wants to stick with the green door or switch to the pink one.

Monty Hall Problem Should she stick with the door she chose first or switch? What’s your initial instinct? Try it out several times with a partner to see what happens. Do you win more times if you stick or switch?

Monty Hall Problem Let’s look at the problem mathematically. Supposing the car is behind the green door. Fill in the table on the following slide and decide whether it’s generally better to stick or switch.

Monty Hall Problem

Door chosen by you:

Behind it is a..

Door Monty would then show you:

Stick, and you win a…

Switch and you win a…

Monty Hall Problem

Door chosen by you:

Behind it is a..

Door Monty would then show you:

Stick, and you win a…

Switch and you win a…

or

Monty Hall Problem So if you switch, you can expect to win a car 2 out of 3 times, whereas if you stick you would only win the car 1 out of 3 times. This problem is famous for puzzling mathematicians during the last century and illustrates that although probability questions can seem confusing and even counter-intuitive sometimes, using a logical approach helps to unravel them.

Teacher notes

In this edition, 4 short activities from the MEI conference are used. All four activities could be used with a wide range of students, although they are in approximate order of ‘age appropriateness’. The full session PowerPoint and materials can be downloaded from the conference page ‘Spot the Pattern’ problem by Phil Chaffé was in Session B ‘Algebra with cards and paper’ by Kevin Lord was in Session C ‘Perimeter of Rectangular Rings’ is from the same session. ‘Introduction to Probability (S1)’ by Clare Parsons was in session I

Teacher notes: Problem Solving, Phil Chaffé

During his session, Phil led teachers to consider what is meant by ‘problem solving’, why it is important, how we might enable students to improve their problem solving skills and he also looked at a range of resources and their sources. One of his problems is presented here which requires students to discern a pattern detailed by a series of snippets of information. There are 20 information cards and a blank grid for students to work on, which also appear on the PPT slides. This activity can be tackled in small groups with rules for collaborative work imposed to prevent some students from dominating and others from not participating, or it could be tackled in pairs or individually.

Teacher notes: Problem Solving, Phil Chaffé

Either present the whole class with the slides one at a time so that they have 4 bits of information to work with at a time (some to-ing and fro-ing may be necessary to check wording) or print the slides for groups to use. Extension ideas: • Ask if there are any pieces of information that are unnecessary • Ask students to come up with their own designs (on a 4x4 grid,

perhaps) and describe them with only 8 pieces of information.

Teacher notes: Algebra with cards and paper, Kevin Lord

Kevin’s session began very simply and built up to more complex use of algebra. The activities can be used to support students with reasoning, justification and proof as well as use of algebra. Just two of his activities are used here; several more are available on the conference page. His activities all used business cards as a starting point, but identical rectangles of paper or card would work just as effectively. The first activity looks at the maximum and minimum areas for two or more cards put together, the second at perimeters of rings of cards.

Teacher notes: Algebra with cards and paper, Kevin Lord

Slide 13 Since b is the long edge, 4a + 2b will be smaller in value than 2a + 4b Slide 14 The maximum perimeter will be when the two cards are almost separate (as shown) so the limit is 4a+4b The minimum occurs when the cards have the long edges together:

Teacher notes: Algebra with cards and paper, Kevin Lord

Slide 16 When working with n cards, the maximum perimeter will be n(2a + 2b) ‘Taking out’ the longest edges by putting them together will always minimise the perimeter, reducing n(2a + 2b) by 2b each time, thus the minimum perimeter is n(2a + 2b) – (n-1)2b = 2na + 2b Explanation: Placing one card initially, the perimeter is 2a + 2b; another card makes the perimeter 2(2a+2b), but by placing a long edge of the second card against the first one this is reduced by a maximum of 2b, there being a ‘b’ ‘taken out’ on each side of the join. Adding a third card, the maximum perimeter is 3(2a +2b),this is reduced by a maximum of 2(2b) by ensuring that joins are made at 2 long edges.

Teacher notes: Algebra with cards and paper, Kevin Lord

Slide 17 The ratio of the sides of the rectangle affects the arrangements that are possible to obtain the minimum perimeter, but the algebraic value of the minimum perimeter is unchanged; being 6a+2b for 3 cards . Similar arrangements for 3 different rectangles are shown.

Arrangement for minimum perimeter

Arrangement for minimum perimeter

Not an arrangement for minimum perimeter

Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs:

Length b from each rectangle on the outside.

Length a from 4 of the rectangles on the outside at the ‘corners’.

Total outside perimeter is 4a + nb

Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs:

Length b-a from 4 rectangles on the inside.

Length b from n-4 of the rectangles on the inside’.

Total inside perimeter is (n-4)b + 4(b-a) = nb - 4a

Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs:

Total perimeter is inside + outside: nb - 4a + 4a + nb =2nb

Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs:

An alternative way of getting to this result is to consider the perimeter of all the cards used: 2n(a+b) = 2na + 2nb At each ‘join’, 2a is lost from the perimeter. There are n such joins. Perimeter = 2na + 2nb – 2na Perimeter = 2nb

Teacher notes: Introducing Probability (S1), Clare Parsons To begin her session, Clare used a short activity which highlighted the challenge that understanding probability presents. Not wishing to spoil the session for her with future TAM teachers, I have instead used the ‘Monty Hall’ problem, of which it was reminiscent. During the session Clare used several approaches to teaching probability which made solving problems much more straight-forward, retaining understanding and insight whilst giving a very helpful structure.