1. MAKING PROBLEM SOLVING LESS PROBLEMATIC
Prepared by SER Literacy & Numeracy Lead Coaches

2. What is problem solving?
The term ‘problem’ and ‘problem solving’ occur in many subject areas, however, is most commonly associated with mathematics. ‘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 represent a challenge that cannot be resolved by some routine procedure. Problem solving is a process of accepting a challenge and striving to resolve it.’ (George Polya – ‘the father of problem solving’ 1945)
Schoenfeld (1989) states that for any student, a mathematical problem is a task:
-in which the student is interested and engaged and for which they wish to obtain a resolution AND
-for which the student does not have readily accessible mathematical means by which to achieve that resolution.
Siemon & Booker (1990) have a similar definition which states a problem is a task or situation:
-that you want to or need to solve
-that you believe you have some reasonable chance of solving, either individually or in a group BUT
-for which you or the group have no immediately available solution strategy.
In summary – a problem is a task for which there is no immediate or obvious solution and that problem solving is the process students undertake when engaging in this task.

3. Why teach problem solving?
An interesting and enjoyable way to learn mathematics.
An opportunity to learn mathematics with greater understanding.
Produces positive attitudes towards mathematics.
Teaches varied ways of thinking, flexibility and creativity.
Teaches general problem solving skills generally applicable in other KLAs and life.
Encourages cooperative skills.
Initialise a think pair share session before displaying the dot points.
Why teach problem solving:
Firstly it is one of the proficiency strands of the Australian Curriculum - …... the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify their answers are reasonable.
Problem solving seems to employ problems that are implicitly interesting to children. This is partly because problem solving does not involve a sequence of very similar questions that are designed to practice the same skill. The novelty of the problems seems to add to their interest.
In the process of struggling with a problem, children can often obtain a fairly deep understanding of the mathematics surrounding the problem. This understanding is often enhanced, when, in a whole class setting, teachers draw together the various threads from all of the children in the class.
While the children seem to enjoy the problems, and get quite involved with them it helps them to gain a positive attitude towards the subject. Some of them have even expressed the opinion that what they were doing was not mathematics and have asked to do more problem solving instead of mathematics! But we want them to see that problem solving is mathematics and that it is an enjoyable subject.
Problem solving provides an opportunity for children to explore ideas and so gives them the chance to extend their creativity. Children are continually coming up with ways of tackling problems that we hadn’t thought of before. The interesting thing is that the children who are producing these ideas are not always the ones who we generally think of as being good at mathematics.
many of the strategies and techniques that are used in mathematics are used in any type of problem. The four stages of problem solving due to Pólya (which we will look at later) are quite general steps that can be applied to any problem whether mathematical or not!
Working in cooperative groups does seem to have advantages. It promotes peer collaborative thinking Somehow, talking mathematics out loud (like ‘think-alouds’) appears to help learning and understanding, and it also seems to help many children produce original ideas. Cooperative learning is crucial in promoting the students’ ability to communicate and reason mathematically (Booker et al – 1997)

4. The Importance of Problem Solving
‘With exposure, experience, and shared learning, children will develop a repertoire of problem-solving strategies that they can use flexibly when faced with new problem-solving situations.’
Burris, Anita (2005). Understanding the Math You Teach. Merrill Prentice Hall, ISBN

5. Australian Curriculum
Problem solving is one of the four proficiency strands of the Australia Curriculum.
“The proficiency strands are Understanding, Fluency, Problem Solving, and Reasoning. They describe how content is explored or developed, that is, the thinking and doing of mathematics. They provide the language to build in the developmental aspects of the learning of mathematics and have been incorporated into the content descriptions of the three content strands described above.” (

6. Why do children have difficulties with PROBLEM SOLVING?
When solving problems students will need to know general strategies and techniques that will guide the choice of which skills or knowledge to use at each stage in problem solving.
When a problem has ‘a new twist’ students cannot recall how to go about it – this is when general strategies are useful in providing possible approaches that may lead to a solution.
Some children find it difficult to think of ideas and strategies. A brain-storming session might help students reflect on problems they have previously solved.
The misconceptions that students have, are a problem with the understanding proficiency strand rather than with fluency.  When students don't understand something we can't fix it by just doing more of the same. 
The most common struggle students report when attempting to solve problems is not understanding what is being asked of them. That is, the context of the problem does not make sense and/or is not clearly translatable to a number sentence.
Student insecurity may occur because the children have never met open-ended problems before. Some teachers in mathematics have traditionally given children algorithms to practice and copy. It is not surprising that in more open problem solving situations, some children will feel insecure. However, by careful handling and by introducing things gradually, children should be able to overcome their initial insecurity

7. Approaches to teaching problem solving …
The approaches teachers use:
Teaching for problem solving - knowledge, skills and understanding (the mathematics)
Teaching about problem solving - experience-based techniques for problem solving, learning, and discovery that give a solution which is not guaranteed to be optimal and behaviours (the strategies and processes)
Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet)

8. Procedural ( Closed) and Open-ended Problems
Procedural problems:
One- or two-step simple word problems
Open-ended
problems:
Problems that require mathematical
analysis and reasoning;
Open-ended problems
can be solved in more than one way,
and can have more than one solution.

9. Procedural or Open-ended?
A typical conventional classroom task is: a Find the perimeter of this rectangle cm cm a A corresponding open-ended task is: a The perimeter of a rectangle is 28cm. What might be the length and width of the rectangle?
The latter task is open ended in that many answers are possible – some students might find only one or two while others will use experimentation and patterns to identify many correct possibilities.
This task (in either form) is content specific in that children need to come to grips with the generalised meaning of ‘perimeter’ and ‘rectangle’. However, with the latter format it is not possible to follow a rule or apply a procedure and to complete it successfully, children need to understand the general idea – that any rectangle has four sides with opposite pairs being equal. These powerful, general ideas that are not learned by mimicking a process or using a formula. (Making Classroom Organisation Explicit)

10. Newman’s analysis of children’s problem solving errors
1. Reading
Can students read the words of the problem?
2. Comprehension
Can students understand the meaning?
3. Transformation
Can students determine a way to solve the problem?
4. Process Skills
Can students do the mathematics?
5. Encoding
Can students record and interpret their answer?
The Australian educator Anne Newman (1977) suggested five significant prompts to help determine where errors may occur in students attempts to solve written problems. She asked students the following questions as they attempted problems.
1.       Please read the question to me. If you don't know a word, leave it out.
2.       Tell me what the question is asking you to do.
3.       Tell me how you are going to find the answer.
4.       Show me what to do to get the answer. "Talk aloud" as you do it, so that I can understand how you are thinking.
5.       Now, write down your answer to the question.
These five questions can be used to determine why students make mistakes with written mathematics questions.
If when reworking a question using the Newman analysis the student is able to correctly answer the question, the original error is classified as a careless error.
Research using Newman's error analysis has shown that over 50% of errors occur before students get to use their process skills. Yet many attempts at remediation in mathematics have in the past over-emphasised the revision of standard algorithms and basic facts.
Expanded information Source: cs/numeracy/newman/index.htm
(cited in NAPLAN Numeracy 2009 by Bob Wellham K-12 Mathematics Consultant, Swansea)

11. So what do we do? Help students to ……… - read the questions
- comprehend what they read
provide strategies that aid understanding
teach students to check their answers are reasonable.
In multiple choice questions – many students guess or are mislead.

12. How can we do this? Teach a problem solving model explicitly
Increase the mathematical metalanguage used
Train students to explain how they get their answers
Scaffold the teaching of the problem solving strategies
Assess for differentiation of learning
Explicit teaching of concepts
Creating a problem solving environment:
Integrate problem solving into the school’s mathematics program across all strands
Value the construction of meaning and the justifications provide by the students
Develop a culture in which students are expected to share, elaborate and even defend their ideas.
Adopt an enquiry based approach
Recognise that problem solving is an experience to be shared and discussed.
Model problem solving behaviour, engage in think-alouds and ask the students to check your reasoning.
(Dianne Siemon – Problem Solving: Addressing the Myths.)

13. Polya’s Model – Problem Sovling
George Polya has had an important influence on problem solving in mathematics education. He stated that good problem solvers tend to forget the details and focus on the structure of the problem, while poor problem solvers do the opposite.
Four-Step Process:
1. Understand the problem (See)
2. Devise a plan (Plan)
3. Carry out the plan (Do)
4. Look back (Check)

14. Four Step Process
Polya outlined four basic steps to solving problems, understanding the problem, devising a plan, carrying out the plan, and looking back. In the first step, understanding the problem, the student has to determine if they comprehend what it is they are to solve or find. When they understand what it is they are looking for or trying to solve, then they can choose a plan to reach that goal. Several strategies are available depending on the type of problem the student is dealing with. When a plan is selected that the student is confident with they can carry out or implement the strategy to find the unknown or arrive at a solution to the problem. When the student is finished, they can look back and prove or justify their answer or solution to the problem. Their answer should be reasonable and justifiable. When these steps are used, this process can apply to any problem that needs solving.

15. Newman’s Analysis aligned with Polya’s modelDo
Check
Plan
See
1.Reading the problem
2.Comprehending what is read
3. Carrying out a transformation from the words of the problem: the selection of an appropriate mathematical strategy
4. Applying the process skills demanded by the selected strategy
5. Encoding the answer in an acceptable written form
Newman’s analysis, displayed in the centre of Polya’s model, identifies where the five basic steps fit into Polya’s problem solving model.

16. Step 1 – SEE: Understand the problem
Students can be overwhelmed just by reading a problem. At this point group discussion is beneficial.
Questions that may help you lead students to an understanding of the problem:
What are you asked to find or show?
What information is given? (valuable/useless?)
Has anyone seen a problem like this before?
Can you restate the problem in your own words?
What conditions/operations apply?
What type of answer do you expect/Can you give an estimate?
What units will be used in the answer?
The SEE step is identified as READ and UNDERSTAND the problem.

17. Visualisation Is an important way to read and understand the problem.
Do I see pictures in my mind?
How do they help me understand the situation?
Imagine the situation
What’s going on here?
Is an important way to read and understand the problem.
Do I see pictures in my mind?
How do they help me understand the situation?
Imagine the situation
What’s going on here?

18. Sally had 1034 stickers in the first place.
Drawing pictures/models: Using models to visualise/understand the problem
Problem:
Sally had some stickers. After she gave her best friend 280 stickers she had 754 stickers left. How many stickers did Sally have in the first place?
Model
drawing
Sally had 1034 stickers in the first place.
"A picture is worth a thousand words."
Although ‘drawing a model’ is a strategies used in the ‘Plan Step’ which is the next step of Polya’s model, using modelled drawing helps students ‘SEE’ and understand the problem better through the use of a visual model.
Model drawing means translating a word problem into a diagram or ‘model’.
Most younger children find equations and abstract calculations difficult to understand. Model drawing helps to convert the data provided in a problem into a concrete visual image.
This enables students to comprehend and convert problem situations into relevant mathematical expressions or ‘number sentences’ and to solve them.
Animation can be used with younger students to enhance, explain and convey, both mathematical concepts and solutions. Using visual models allows teachers to follow these basic principles:
Teach from the known to the unknown.
Teach from the simple to the complex.
Teach from the concrete to the abstract.
The ‘Visual Model' method may appear to be simple and quite obvious when only two or three quantities are involved. However, its real benefit will be realised when dealing with difficult and challenging problems, especially those involving decimals, fractions and ratio.

19. Step 2- PLAN: Devise a Plan
Is this problem similar to a problem you’ve done before? (Can you use the same method or part of the previous plan?)
What strategy would help you solve it? (Students can best learn skills at choosing an appropriate strategy by exposure to many different problems.)
Are the units consistent?
Units – examples km and cm or L and ml

20. Problem-solving strategies include:
Draw a picture or diagram
Act it out
Use concrete materials/make a model
Look for a pattern
Make an orderly list or table
Work backwards
Use direct/logical reasoning
Solve a simpler problem
Predict and test (guess & check)
Write an equation/use a formula
Eliminate possibilities
First 5 strategies are primary strategies – need to be explicitly taught from Prep – Year 7
Remaining strategies are more suitable for years 3 – 7 but may be introduced earlier. All strategies need to be consolidated in upper grades (from year 3 upwards).
Teachers need a plan for introducing strategies; it is not feasible to focus on them all at once.
Points to consider:
While “act it out” may be appropriate for lower elementary, it may not be for middle grades.
It is rare that a problem can be solved only with one strategy. For this reason, a repertoire of strategies is useful

21. Make a Drawing or Diagram
Stress that there is no need to draw detailed pictures.
Encourage children to draw only what is
essential to tell about the problem.
As mentioned earlier this strategy may be used to assist students to understand the question. The diagram represents the problem in a way which allows the student to “see” it, understand it, and think about it while looking for the next step. 

22. Act It Out
Stress that other objects may be used in the place of the real thing.
The value of acting it out becomes clearer when the problems are more challenging.
This strategy assists students to visualise what the problem is and what is involved. They go through the actions of what the problem says and at the end come to a conclusion. The physical action makes the problem components and their relationship clearer in the students minds.
When teaching this strategy explain to the students that they can use different objects to represent the real thing. Children are very good at pretending so they will probably suggest substituting objects anyway. Teachers need to make sure the attention is on the actions and not the actual objects.
An easy problem to solve is, 'Six children are standing at the teacher's desk. Four more join them. How many are at the teacher's desk now?' The value of acting out the problem becomes clearer. If the problem is harder then it may be more important to act it out and show relationships.

23. Use concrete materials/ make a model
Students use materials to help them discover the relationship they need to see, to lead them to a solution.
Using concrete materials may make the problem easier to see.
Similarly to acting it out, students can use objects to represent the real thing to assist them to solve the problem.

24. Look for a Pattern This involves identifying a pattern and
predicting what will come next.
Often students will construct a table, then use it to look for a pattern.
Explain to the students that:
They can look at a series of shapes, colours or numbers to see if you can find a pattern
The pattern should repeat
The pattern is not always obvious
In early mathematics learning children are taught what patterns are and how to construct them in pictures, objects and numbers. It involves a more active search when recognising a pattern in problem solving. Students' can use tables of results then look for a pattern.
Once a student can see a pattern they can make a prediction, which is the essence of the problem solving strategy: see the pattern, make a prediction.
One of the most famous patterns in mathematics is known as the Fibonacci series, named for a mathematician who lived in Italy in the 13th Century.

25. Example: Look for a pattern using a table
PROBLEM: Restaurants often use small square tables to seat customers. One chair is placed on each side of the table. Four chairs fit around one square table.
Restaurants handle larger groups of customers by pushing together tables.
Two tables pushed together will seat six customers. How many people will 4 tables pushed together seat?
Number of tables
Number of people 1 4 2 6 3 8 ?

26. Construct a Table or Organised List
The letters ABCD, can be put into a different order: DCBA or BADC. How many different combinations of the letters ABCD can you make? To answer this question students may choose to make a list. By making a SYSTEMATIC list, students will be able to see every possible combination.
This is an efficient way to classify or order a large amount of data.
An organised list provides a systematic way to record computations.
There is a common saying, "Don't just stand there, do something!" When solving a problem, don't just think, write something! Draw a diagram, construct a table or make a list!
It is often obvious when you should draw a diagram and it is often obvious when you should make a list. Take a very simple example: putting things in order.
The letters ABCD, can be put into a different order: DCBA or BADC. How many different combinations of the letters ABCD can you make? To answer this question students may choose to make a list. By making a SYSTEMATIC list, students will be able to see every possible combination.

27. Work Backwards
This strategy seems to have limited application opportunities, however, is a powerful tool when it can be used.
Some problems are posed in such a way that students are given the final conditions of an action and are asked about something that occurred earlier.
Some times problems are worded in a way that give the final outcome and ask about something that happened before that. Students need to work backwards in solving these problems.

28. Example of Working Backwards

29. Direct /Logical Reasoning
Explicitly teach students:
To tackle the problem step by step
That each piece of information is a piece of the puzzle, put all pieces together to find the solution
To read each clue thoroughly and work by a process of elimination
When the first plan (strategy) is unsuccessful, to try another one
Students need to use the process of deduction to decide on the answer to the problem. Sometimes students will have an answer that does not make sense. It is important for students to arrive at a solution that is feasible. For example, a student solving how far a person walked in a day must decide that 1 million miles is not logical and if that is the answer they calculated there was an error in the calculation or the numbers used.

30. Solve a Simpler or Similar Problem
Some solutions are difficult because the problem contains large numbers or complicated patterns.
Sometimes a simpler representation will show a pattern which can help solve a problem.
Teach the students to:
· Set aside the original problem and work through a simpler related problem
· Replace larger numbers with smaller numbers to make calculations easier, then apply same method of solving it to the original problem
· Look for a pattern that may be emerging
Sometimes a problem is too complex to solve in one step. When this happens, it is often useful to simplify the problem by dividing it into cases and solving each one separately.
Teach the students to:
· Set aside the original problem and work through a simpler related problem
· Replace larger numbers with smaller numbers to make calculations easier, then apply same method of solving it to the original problem
· Look for a pattern that may be emerging

31. Predict & Test (Guess and Check)
This strategy does not include “wild” or “blind” guesses.
Students should be encouraged to incorporate what they know into their guesses - a logical guess.
The “Check” portion of this strategy must be stressed.
When repeated guesses are necessary, using what has been learned from earlier guesses should help make each subsequent guess better and better.
Guessing often produces the wrong answer. But the strategy called "Guess and Check" often produces the right answer. It should probably be called "Guess and Check and Guess Again," because the process of checking the accuracy of each guess and then making another, more informed guess is an essential part of the strategy.
An educated guess is based on careful attention to pertinent aspects of the problem, plus knowledge from previous related experiences. Guessing is a viable strategy if students are encouraged to include what they know into their guess instead of taking "wild" guesses. The student then must check the answer to make sure it is correct.

32. Write an equation/ use a formula
Students can sometimes make sense of a problem by changing the written problem to a number sentence.
To use this strategy students must choose the appropriate formula and substitute data in the correct places of the formula. It can be used for problems that involve converting units or measuring geometric objects. Also, real-world problems such as tipping in a restaurant, finding the price of a sale item, and buying enough paint for a room all involve using formulas.

33. Eliminate possibilities
Eliminating possibilities is a strategy where students use a process of elimination until they find the correct answer.
This is a problem-solving strategy that can be used in basic math problems or to help solve logic problems.
Students may find this problem solving strategy fun because it makes them feel like a detective.  For this strategy they are using deductive reasoning skills. It can sometimes be used in conjunction with "look for a pattern" and "construct a table."
The students do not need to examine every possibility. They just need to account for all possibilities in a logical and systematical way with proof. Students may be able to sort the possibilities into categories and dismiss some categories before checking the rest of them.
Sometimes it is important to look at all possibilities and it may be a good idea to write out all the possibilities of the problem and then cross out each one as they go.
Eliminating Possibilities is a strategy in which students remove possible answers until the correct answer remains.

34. Step 3 – DO: Carry out the plan
At this stage students need to follow the relevant steps to solve the problem.
They need to learn how to check each step of the solution and ensure the accuracy of the computation as they implement their chosen problem solving strategy.  They need to be explicitly taught how to keep an accurate record of each step and how to avoid making careless mistakes and computational errors.
At this stage students need to follow the relevant steps to solve the problem.
They need to learn how to check each step of the solution and ensure the accuracy of the computation as they implement their chosen problem solving strategy.  They need to be explicitly taught how to keep an accurate record of each step and how to avoid making careless mistakes and computational errors.

35. Step 4 – CHECK: Look Back Does my answer make sense?
Check the mathematics by working backwards, estimating, showing that the answer is reasonable, or doing the problem another way.
Have I answered the question?
Have I learned anything new from solving this problem?
Students need to be explicitly taught to check the results of their solution, determine whether the final answer is reasonable, and whether it answers the question of the problem. They also learn to generalise and compare this to other related problems. This is the point where they reflect on each step so far.
Students reflect on the effectiveness of the strategy used, if they are satisfied with the solution and what other problems might be solved in the same way .

36. Polya’s Model in action
Maddison, Bella, Tia and Talia all exchanged Valentine’s Day cards. How many Valentine’s Day cards were exchanged?

37. See
Understanding the Problem

38. Do I understand all of the words?
Exchange = swap
Valentine = loved one
Many = more than one

39. What am I asked to do?
I am asked to find out how many cards were exchanged between the 4 friends.

40. Can I restate the problem in my own words?
Four friends want to give each other a card. How many cards would be swapped if they were each to get a card from one another?

41. Can I think of a picture or diagram that might help me understand the problem?
Four Friends

42. Is there enough information to enable me to solve the problem? Yes!

43. Plan
Devise a plan

44. List Maddison Bella-Tia-Talia Bella Tia-Talia-Maddison
Tia Talia-Maddison-Bella
Talia Maddison-Bella-Tia

45. Diagram
Maddison
Tia
Talia
Bella

46. Act it out
Three friends and I could stand in a circle. Each of us would hold enough pieces of paper to give one another one each. Then I could count how many pieces of paper we held altogether.

47. Look for a Pattern Maddison would need *** Bella would need ***
Tia would need ***
Talia would need ***
SO… ***+***+***+*** = 12

48. Solve a Simpler Problem
If Maddison was to give a card to each of her three friends, She would need 3 cards.
SO…
There are four friends, each needing 4 cards.
4 X 3 = 12

49. 4 friends each need to give out 3 cards.
Write an Equation
4 friends each need to give out 3 cards.
SO…
4 X 3 = 12
OR…
= 12

50. Problem Solving Think board
What is the question asking?
What information is important?
What strategy will you use to solve the problem?
Think board – collaborative tool to promote deep quality conversation around the posed problem. DO
Write your answer as a complete
sentence.
Attack the problem!
Do the maths!