Transcript Problem Solving - St Joseph's Boorowa

Establishing a culture of thinking in your class.
Problem Solving & Modelling
‘Solving problems is a practical art like swimming, or
skiing, or playing the piano: you can learn it only by
imitation and practice... if you wish to learn
swimming you have to go into the water, and if you
wish to become a problem solver you have to solve
problems.’ (Polya 1962)
Problem Solving
In mathematics education, problem solving has been
emphasised since Polya’s work in the 1940s. His
four step problem solving model, which has been
used as the basis for many subsequent models, is
linear in design:
1. Understand the problem
2. Devise a plan
3. Carry out the plan, and
4. Look back.
National Numeracy Review Report
May 2008
Commissioned by the Human Capital Working Group,
Council of Australian Government
That from the earliest years, greater emphasis be
given to providing students with frequent exposure to
higher-level mathematical problems rather than
routine procedural tasks, in contexts of relevance to
them, with increased opportunities for students to
discuss alternative solutions and explain their
thinking.
Sample Problem
Paper Fold
Take a strip of paper and fold it in half. Now fold what
you get in half. If you fold the strip like this 10 times
and then undo it, how many creases will there be?
Taken from ‘Teacher Tactics for Problem Solving’ by
K Stacey and Beth Southwell.
Working Mathematically
Investigation Processes:
Questioning
Problem solving
Communicating
Verifying
Reflecting
Using Technology
What is problem solving?
‘Solving a problem is finding the unknown means to
a distinctly conceived end ... To find a way where no
way is known off hand. For a question to be a
problem, it must present a challenge that cannot be
resolved by some routine procedure. Problem
solving is a process of accepting a challenge and
striving to resolve it.’ (Polya)
What is problem solving?
‘For any student, a mathematical problem is a task:
 in which the student is interested and engaged and
for which they wish to gain a resolution; and
 for which the student does not have readily
accessible mathematical means by which to achieve
that resolution.’
(Schoenfeld, 1989)
Types of Problems
Taken from ‘Modelling with mathematics in Primary and Secondary Schools’ by Mason and Davis.
Action Problems
Believable problems
Curious problems
Dubious problems
Educational problems
where the results affect the
pupil’s lives;
which could plausibly be
Action problems at some time;
which intrigue and stimulate;
which are really covers for
exercising mathematical
techniques; and
which are constructed to make
some important point but which
are not directly related to pupil
experience
Types of Problems
Taken from ‘Modelling with mathematics in Primary and Secondary Schools’ by Mason and Davis.
Burkhardt claimed that textbooks rarely rise above
the curious, and are almost always,
“Frankly dubious.”
Sample Problem
Diagonals
How many diagonals does an octagon have?
How many diagonals does a decagon have?
How many diagonals does a polygon with 100 sides have?
(A diagonal is a straight line that joins two vertices in a
polygon or polyhedron.)
Taken from ‘Teacher Tactics for Problem Solving’ by K Stacey and Beth
Southwell.
Primary Maths Challenge
The problem you are about to attempt comes from
the Primary Maths Challenge organised by The
Australian Maths Trust. I use it for my top maths
students.
They work in a small group for approx 3 weeks
without teacher assistance on 4 demanding
problems. Each problem consists of several parts
which gets progressively harder.
Ramped Problems
A ramped problem is one that can be adapted to a
variety of abilities or levels (age/class). It may start
as an easy one which is made more difficult or it may
be a difficult one which is simplified.
Ramped Problems
Can you work out how many squares altogether in
the square below?
My friend gave me a chessboard (8 x 8) and wanted
to know how many squares were on it altogether.
How many do you get?
Hint: try 3 x 3 squares, then 4 x 4 etc
Work Sample –Year 4
Ramped Problems
With 64 squares – our chessboard
Size
2x2
3x3
4x4
5x5
6x6
7x7
8x8
1x1
4
9
16
25
36
49
64
2x2
1
4
9
16
25
36
49
1
4
9
16
25
36
1
4
9
16
25
1
4
9
16
1
4
9
1
4
3x3
4x4
5x5
6x6
7x7
8x8
Total
1
5
14
30
55
91
140
204
Sample problem
Handshake
You entered a room in which there were six other
people standing. If everyone was to shake hands
with every other person in the room once and only
once, how many handshakes would take place?
Problem Solving Strategies
 Guess and check
 Look for a pattern
 Draw a table
 Reduce to a simpler case
 Act it out
 Work backwards
 Draw a sketch
 Divide into subtasks
 Substitute simple values
Problem Solving Enhances Children’s Numeracy Learning
Ann Gervasoni
Australian Primary Maths Classroom Vol 5 Number 4 2000 AAMT
Problem solving activities, including those that have no immediately obvious solution, can quite
significantly change the nature and power of the mathematical thinking in which children engage at
school. This, I believe, is the greatest value of problem solving.
Below are her six reasons why she believes problem solving enhances mathematical thinking and
numeracy development.

Problem solving resembles the work of mathematicians.
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Problem solving enables students to become the knowers and creators of mathematics.
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Problem solving requires the proving and justifying of solutions.
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Problem solving encourages mathematical conversations.
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Problem solving requires students to play significant communication roles.
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Problem solving provides a meaningful purpose to write about mathematics.
Limitations of using problem solving.
 Unless the problems are motivating students may see
them as busy work and react to them accordingly.
 Unless students are interested and believe they have a
chance of solving the problem, they may be reluctant to
try.
 Appropriate problems/investigations take time to develop
since each problem needs to be carefully structured to
produce specific student learning outcomes.
 Unless your students understand why they are
attempting to solve a particular problem, they may not
want to learn what you want them to learn.
Taken from ‘Using Problem Solving As A Teaching Strategy’ R Killen, 1996
What are open-ended questions?
Sullivan and Clarke (1991) suggest that ‘good’
questions have three main features:
1. They require more than remembering a fact or
reproducing a skill;
2. Pupils can learn by answering the questions, and
the teacher learns about each pupil from their
attempt;
3. There may be several acceptable answers.
Open ended questions
Good or ‘fat’ questions
1. What three numbers add up to 18?
2. In the bag in front of me I have a number of regular
shapes whose sides when added together equal
forty-three? What shapes might I have in my bag?
Are there any other possibilities?
3. If the area of a plot of land is 15m², what are its
dimensions?
Caution
In a study examining the effectiveness of using
open-ended questions (Zevenbergen, Sullivan &
Mousley, 2001) researchers found that students from
poor socio-economic and ESL backgrounds may find
these questions more difficult due to the richness of
the language.
Open-ended tasks and barriers to learning: teacher’s perspectives – APMC,
Vol 6, Number 1, 2001. AAMT
Investigations – A Central Focus For Mathematics
Charles Lovitt
Australian Primary Maths Classroom Vol 5 Number 4 2000 AAMT
Lovitt asserts that despite the central focus given to problem solving in
curriculum documents both here and overseas from the 1980’s onwards, it
has failed to eventuate.
‘I do accept that the problem solving push has contributed much to the
vitality of many classrooms and significantly influenced the thinking of many
teachers. But it has not become the ‘central’ theme it was supposed to be.
Two major reasons I believe are:
 Lack of clear and widely accepted criteria. All sorts of things, some
diametrically opposite to each other are all dressed up as problem solving.
The word has become so blurred that we have no common shared
agreement on what it means.
 Another reason is the unfortunate perception that one aspect of the problem
solving is delivered through games and puzzles and therefore is relegated to
the periphery or margins of mathematics. ‘I do these really interesting things
on Friday afternoons,’ say many teachers to me. I am not sure if they are
conscious that the act of doing so is to send a message to students that it is
not really important – merely a bit of fun to be done after the ‘real stuff’.
Maths Investigation
A mathematical investigation;
Has multidimensional content;
Is open-ended, with several acceptable solutions;
Is an exploration requiring a full period or longer to
complete; is centred on a theme or an event; and
Is often embedded in a focus question
Maths Investigation
In addition, a mathematical investigation involves
processes that include:
Researching outside sources,
Collecting data, collaborating with peers, and
Using multiple strategies to reach conclusions.
STAGES OF INVESTIGATIVE PROCESSES
 Find an interesting (meaningful/worthwhile) problem.
If he fails this he doesn’t go on.
 Informally explore, unstructured ‘play’ which generates data.
From this comes some data from which theories form.
 From patterns in the data, create hypotheses, conjectures, and theories.
 Invoke problem solving strategies to prove or disprove any theories.
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Using problem solving strategies
Apply any basic skills I know as part of this proof process.
Calls on acquired skills – algorithms, graphing etc
Extend and generalise the problem – what else can I learn from it.
How can I stretch this problem?
Publish.
Go back to step 1.
Investigations – A Central Focus For Mathematics byCharles Lovitt
Australian Primary Maths Classroom Vol 5 Number 4 2000 AAMT
MANSW Maths Investigations
Worth Your Weight In Gold
Gold is currently selling for
approx $990 per ounce.
If you were made of gold
how much would you be
worth?
How much would the whole
class be worth?
A Truckload of Money
You have seen the Lotto
advertisement where a tip truck
is loaded with money and the
tyres blow out.
How much money do you think
this truck would hold if it was
that full?
Would different denominations
change the amount?
Reflection
 Is an integral part of any maths lesson.
 The student reflects on what is being asked.
 On various strategies to solve the problem.
 On how they solved the problem. Is there another
way they could have solved it.
 Reflection can be done individually or within a group.
 It allows for the social construction of knowledge.
 It allows for the development of a common
mathematical language.
Posing Problems and Solving Problems
Tom Lowrie in Australian Primary Maths Classroom Vol 4 Number 4 1999 AAMT
 Problem posing is an important companion to problem solving and
lies at the heart of mathematical activity (Kilpatrick, 1987). Silver
(1995) identified three types of problem-posing experiences that
provide opportunities for children to engage in mathematical activity.
He argued that problem posing could occur prior to problem solving
when problems were being generated from a particular situation,
during problem solving when the individual intentionally changes the
problem’s goals or conditions, or after solving a problem when
experiences from the problem-solving context are modified or
applied to new situations.
 How much do young children learn from posing problems for one
another to solve?
Whose Questions Are Being Answered In The Maths Classroom?
Jennie Bickmore-Brand
Australian Primary Maths Classroom Vol 3, Number 1. 1998 AAMT
 Do students have a sense of the bigger picture and where this problem fits
in?
 Do students see the relevance of doing this activity?
 What prior knowledge would students need to know before doing this
activity? This could be literacy skills as well as mathematical concepts and
strategies.
 Do students have a vested interest in the outcome of the activity?
 Do the students have to accept any responsibility for their answers in any
tangible way?
 Are there any real consequences to the students’ calculations?
 Can students apply common sense to this problem?
 Have students seen you or one of your peers solve a problem like this?
 Can the activity be done by a group who have different contributions to make
to the overall result?
 Does the activity allow for a variety of learning styles?
Suggestions For Classroom Development
 ‘An environment that encouraged the children to pose problems for friends to solve
increased the likelihood of the students developing mathematical power (Lowrie,
1999). In order to create teaching/learning situations that provide positive problemsolving situations, the classroom teacher should:
 Encourage students to pose problems for friends whom are at or near their own
‘standard’ until they become more competent in generating problems;
 Ensure that students work cooperatively in solving the problems so that the problem
generator gains feedback on the appropriateness of the problems they have designed;
 Ask individuals to indicate the type of understandings and strategies the problem
solver will need to use in order to solve the problem successfully before a friend
generates a solution;
 Encourage problem-solving teams to discuss, with one another, the extent to which
they found problems to be difficult, confusing, motivating or challenging;
 Provide opportunities for less able students to work cooperatively with a peer who
challenges the individual to engage in mathematics at a higher level than they were
usually accustomed; and
 Challenge students to move beyond traditional ‘word problems’ by designing problems
that are open ended and associated with real-life experiences.’