Problem Solving

1. Problem Solving Establishing a culture of thinking in your class.

2. 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)

3. 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: • Understand the problem • Devise a plan • Carry out the plan, and • Look back.

4. National Numeracy Review ReportMay 2008Commissioned 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.

5. 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.

6. Working Mathematically Investigation Processes: • Questioning • Problem solving • Communicating • Verifying • Reflecting • Using Technology

7. 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)

8. 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)

9. Types of ProblemsTaken from ‘Modelling with mathematics in Primary and Secondary Schools’ by Mason and Davis. Action Problems where the results affect the pupil’s lives; Believable problems which could plausibly be Action problems at some time; Curious problems which intrigue and stimulate; Dubious problems which are really covers for exercising mathematical techniques; and Educational problems which are constructed to make some important point but which are not directly related to pupil experience

10. Types of ProblemsTaken 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.”

11. 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.

12. 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.

13. 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.

14. 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

15. Work Sample –Year 4

16. Ramped Problems With 64 squares – our chessboard

17. 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?

18. 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

19. Problem Solving Enhances Children’s Numeracy LearningAnn 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. • Problem solving enables students to become the knowers and creators of mathematics. • Problem solving requires the proving and justifying of solutions. • Problem solving encourages mathematical conversations. • Problem solving requires students to play significant communication roles. • Problem solving provides a meaningful purpose to write about mathematics.

20. 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

21. What are open-ended questions? Sullivan and Clarke (1991) suggest that ‘good’ questions have three main features: • They require more than remembering a fact or reproducing a skill; • Pupils can learn by answering the questions, and the teacher learns about each pupil from their attempt; • There may be several acceptable answers.

22. Open ended questions Good or ‘fat’ questions • What three numbers add up to 18? • 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? • If the area of a plot of land is 15m², what are its dimensions?

24. 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

25. Investigations – A Central Focus For MathematicsCharles LovittAustralian 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’.

26. 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

27. 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.

28. 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. 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 byCharlesLovittAustralian Primary Maths Classroom Vol 5 Number 4 2000 AAMT

29. MANSW Maths Investigations Worth Your Weight In Gold A Truckload of Money 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? 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?

30. 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.

31. Posing Problems and Solving ProblemsTom 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?

32. Whose Questions Are Being Answered In The Maths Classroom?Jennie Bickmore-BrandAustralian 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?

33. 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 problem-solving 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.’