Monthly
Maths
I s s u e
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.
4 1
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
In his 1998 paper well as those posed by others. Because
Problem Solving, the whole point of learning maths is to be
Kevin Dunbar
able solve problems. Learning those rules
gives focuses on a and facts is of course important, but they
are the tools with which we learn to do
number of
important issues in maths fluently, they aren’t maths itself.”
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.”
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.”
Click here for the MEI
Maths Item of the Month
www.mei.org.uk
O c t o b e r
2 0 1 4
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.
Disclaimer: This newsletter provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these
external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites.
George Pólya
Teaching
resources
Carol Knights has
adapted resources
from three problem
-solving sessions
delivered by MEI
staff at the MEI
Conference 2014:
Clare Parsons
Kevin Lord
Phil Chaffé
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.
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.
2.
3.
4.
Understand the problem
Devise a plan
Carry out the plan
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
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.
Derek Holton
Problem solving
Derek Holton
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.
British-born and Australianeducated 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, peopleside 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
The Problem
mathematics, at least at the macro level,
Solving section of
they are problems, people and proof.”
the nzmaths
website provides
The Mathematics
problem-solving
Association publishes the
lessons that cover
magazine Mathematics in
Levels 1 to 6 (ages
School; in the March 2006
5-15) of the New
issue there is an article
Zealand
by Holton where he describes and
Curriculum.
discusses ‘The 400 Problem’: “This
problem really does contain
experimenting, conjecturing, proving,
extending and generalizing. All these are
the mathematician’s tools in trade”.
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.
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 selfcontradictory or
'paradoxical' logic.
Give your brain a
workout by
thinking about
these scenarios”.
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:
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.
Edward de Bono
Two types of
thinking
According to de
Bono, thinking can
be divided into two
types:
Vertical thinking:
the traditionalhistorical 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.
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.”
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.”
Six Thinking Hats
Using Six
De Bono defined thinking as: The
Thinking Hats in operating skill with which intelligence
acts upon experience.” He felt that
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.
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.”
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.
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.
This four-minute video
explains the Six Thinking
Hats method further.
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."
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.
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.”
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.
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 informationprocessing 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).
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; CoInvestigators: 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’ problemsolving 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.”
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.
MEI curriculum development,
resources and professional
development
Links to web pages about problem
solving in mathematics education, with
links to publications and resources.
delivered by Stella Dudzic:
IQM: Modelling and estimation
IQM: Probability and risk
IQM: Financial problem solving
IQM: Statistical problem solving
Realistic Mathematics Education
(RME)
London Schools’ Excellence Fund CPD:
Integrating Mathematical Problem
Solving (IMPs)
Introducing problem-solving into the
Key Stage 4 curriculum
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
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.
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
7cm
b
7cm
5 cm
a
Three rectangular business cards are shown.
What is the perimeter of each?
a
Algebra with cards and paper
7cm
5 cm
5 cm+7cm+5 cm+7cm = 24 cm
a
a cm+7cm+a cm+7cm = 14+2a cm
7cm
b
a
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
2a+4b
4a+2b
2a+4b
Which is smaller: 2a +4b or 4a+2b?
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 Switch and
you win
you win
a…
a…
Monty Hall Problem
Door
chosen by
you:
Behind it is
a..
Door Monty
would then
show you:
or
Stick, and
you win
a…
Switch and
you win
a…
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 a from 4 of the rectangles
on the outside at the ‘corners’.
Length b from each rectangle
on the outside.
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 from n-4 of the rectangles
on the inside’.
Length b-a from 4 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.