The problem with problem solving! Dianne Rundus Triple M Consulting rundus@bigpond.com

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? Children’s development Interactions Knowledge Children’s fluency, problem solving, reasoning, understanding – The Proficiencies DATA Communication skills Learning behaviours Children’s content knowledge : Number and Algebra, Measurement and Geography, Statistics and Probability Thinking strategies Use of mathematical language

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 mathematics If 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 about Students 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!``