Why teach problem solving?
• It enables children to utilise and develop their deep conceptual understanding.
• It can provide a social context for their learning.• Solving a problem can motivate students to develop new ways of
thinking• It allows for differentiation of the task.• It promotes cognitive development.• Promotes mathematical discussion.• It provides opportunities for teaching at the point of need• Remember that you have to ask the harder question to get the
children thinking – enter the confusion zone!• It allows children to have FUN with maths!• By teaching children a range of strategies they are empowered to
become real life problem solvers.• It takes around 200 repetitions before the synapses in your brain
make the connection. Stress is a major inhibitor of the growth of synapses
George Polya – the grandfather of problem solving
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 in
the water, and if you wish to become a problem solver you have to solve
problems.
What is your definition of a problem?
A problem is merely a question to which you have not found the
answer……yet.
A problem is an opportunity for all students to explore something that they did not instantly know the answer to or how to go about solving it.
The Problem Solving Proficiency Strand
The Problem Solving Proficiency Strand can be seen as the ability to formulate,
represent and solve maths problems, and communicate solutions effectively.
We understand something if we see how it is related or connected to other things
we know.How might we translate this into
practice?Discuss what problem solving currently looks like, feels like and sounds like in
your classroom.
Mathematical problem solving skills are critical to successfully
function in today’s technologically advanced society. Yet, improving
students’ problem solving has proved to be a significant
challenge – solving problems requires understanding the
relations and goals in the problem and connecting the different
meanings, interpretations, and relationships to the mathematical
operations.
Jitendra 2008
George Polya’s Four Principles
• A Hungarian born Mathematician who, in his later life worked on trying to characterize the methods that people use to solve problems, and to describe how problem-solving should be taught and learned.
• He is often considered to be ‘the father of problem solving’• He believed there were four steps to problem solving.
SEE – understand the problem.PLAN – devise a plan.DO – carry out the plan.CHECK – check the answer.
1. SEE Understand the problem
• Do you understand all the words stating the problem?
• What are you asked to do or show?• Can you restate the problem in your own words?• Can you think of a picture or diagram that might
help you to understand the problem?• Is there enough information to enable you to
solve the problem? • Is there too much information?• Do you need to ask a question to get the
answer?
2. PLAN Devise a plan
There are many reasonable ways to solve problems but children need to have
experience with a wide range of problems so they can develop a repertoire of strategies from which to choose. The more problems they solve the easier they should find it to
choose an appropriate strategy for the particular problem they are dealing with.
3. DO – Carry out the plan
• Implement a particular plan of attack.• Look into your Mathematician’s Tool Box.• Persist with the plan you have devised;
you may need to modify or revise it.• If your plan continues not to work then
discard it and choose another. • This is how mathematicians work!
4. CHECK - review your answer
• Make sure you have taken all of the important information into account.
• Decide whether or not the answer makes sense. Is it reasonable?
• Make sure the answer meets all of the conditions of the problem.
• Put your answer into a complete sentence.• Take time to reflect on your answer and on what
did and did not work, and on what you would try again.
• Consider how you will present your answer.
It is better to solve one
problem five different ways than to solve five different
problems.George Polya
What is open questioning?Requires
more than YES or NO answers
Encourages Mathematical
Dialogue
Requires ‘wait time’
to think about
responses
Requires the teacher to respond in ways to direct students to think mathematically
Show me how to…
Would that work with
other numbers?
Could you work that
out another way?
May require abandoning planned program
Where does Problem Solving fit?
DATA
Children’s development
Thinking strategie
s
Learning behaviours
Interactions
Communication skills
Knowledge
Use of mathematica
l language
Children’s content knowledge :
Number and Algebra, Measurement and
Geography, Statistics and Probability
Children’s fluency, problem solving,
reasoning, understandin
g – The Proficiencies
Choose appropriate and worthwhile tasks
Choose tasks which :- Are challenging Provide students with opportunities to reinforce
and extend their knowledge Are intriguing and invite speculation and
exploration Provoke for robust discussion Provide opportunities for the learning of
significant mathematicsIf you can’t identify the significant maths in
the activity then why do it?
Metacognition‘pupils whose teachers made effective use of plenaries to evaluate approaches, summarize key issues and encourage collective reflection were not only the most successful in practical
problem solving situations but also showed the greatest improvement in the content
areas of mathematics.’Howard Tanner and Sonia Jones
Becoming a Successful Teacher of Mathematics
Something to think aboutStudents are not taught the problem solving process. The thinking part of problem solving is typically suppressed. They are primarily exposed to the result of the process. In the typical textbook the thought processes, the planning the problem solver used to solve the problem are omitted. Only the results of that planning are displayed. Thus the very thing most needed, training in thinking, is omitted from the examples of problem solutions presented to the student. This can be verified by examining any textbook and associated teaching material. It is simple to see that the solutions presented do not start at the point where the writer started in solving the problem. The thinking that forms the basis for the solution is not shown. Students then feel that there is something different about problem solving that requires special aptitudes. This induced misconception leads to failure to develop facility in mathematical methods and failure to use mathematical methods effectively in subject matter courses such as physics, chemistry, business and other subjects.
http://www.hawaii.edu/suremath/why1.html
Some of my favourite resources: Brainstrains by Chris Kunz (available from Hawker Brownlow
Education)
Problem-solving in Mathematics by George Booker & Denise Bond (RIC) (Books A to G)
Math-e-magic by Anne Joshua (Books A to E)
Nrich.maths.org fantastic website at Cambridge University, UK
Maths Problem Solving by Peter Maher Macmillan Boxes 1- 6
Thinking through mathematics – engaging students with inquiry-based learning. Books 1, 2 & 3 by Sue Allmond, Jill Wells & Katie Makar (Education Services Australia)
Get It Together – Maths Problems for Groups Grades 4 - 12
Working Mathematically with Puzzles and Problems by Thelma Perso & Gillian Neale (MAWA)
Sometimes we make it much more difficult than it needs to be!``
Top Related