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In this section some well-known definitions of ‘mathematics problem' will be presented. The aim is to provide background information of what a mathematical problem.
Frobisher (1994, p. 154) describes a problem as a ‘situation that has interest and appeal to a child, who therefore wishes to explore the situation more fully in order to gain understanding of it. Goals arise naturally during the exploration and are determined not by the setter of the problem but by the child. The child in turn surveys the problem situation before exploring avenues of interest, following paths which may or may not lead to a satisfactory conclusion. As Ernest(1991) so succinctly puts it, ‘the emphasis is on the exploration of the unknown land rather than a journey to a specified goal' '. Kilpatrick(1985, p. 2) defines a problem as ‘ a situation in which a goal is to be obtained and a direct route to the role is blocked'. R. Mayer (1985, p. 123) states that ‘ A problem occurs when you are confronted with a given situation- let's call that the given state - and you want another situation - let's call that the goal state - but there is no obvious way of accomplishing your Goal.' Furthermore Mayer(1985) gives the example of finding the volume of a frustrum a right pyramid where the value of the sides of the two bases and the height is given. ‘ If you did not know a formula for the volumes of frustrums, this would be a problem for you (Polya, 1965)' (Mayer, 1985, p. 123).
In Frobisher's definition he mentions that the goal is not determined by the setter ( teacher) but by the student. This statement is not valid for all problem types. In some problems such as class room word problems the student does not determine the ‘goal' (answer).In some questions different paths can be taken to obtain the desired ‘goal' (answer ) which is determined by the teacher. Furthermore in problem types like investigation problems the children could obtain different answers (goals).
Orton & Frobisher (2005, p. 25) says ‘ a mathematics problem for one learner may be an exercise for another' because if a student has had a similar situation before student may consider the problem to be an exercise, where another who has never come across with a situation may see the activity as a problem. He continues describing ‘a mathematical problem can be said to be a situation in which an individual student :
a. Recognizes or believes that there exists a mathematical goal to be achieved, usually an answer of some kind;
b. Accepts the challenges to perform some mathematical task in order to reach the goal;
c. Has no readily known or recallable mathematical procedure available to enable the goal to be attained directly' (Orton & Frobisher, 2005, p. 25)
It can be concluded by the above given definitions that a mathematical problem can be a situation in which a student recognizes the mathematical ‘goals', performs some mathematics to achieve the goal although there is no known or recalled mathematical procedure for getting to the final ‘goal' which is usually the answer.
Types Of Mathematical Problems
According to LeBlanc, Proudfit, & Putt (1980, p. 104) standard textbook problems( word problems) and process problems are the two main problems types that are widely used in school mathematics. The characteristics of the two methods will be explained broadly.
Word (Standard Text Book ) Problems
Word problems are commonly used in elementary school mathematics textbooks and mathematics books. Frobisher (1994, p. 152) states that in a word problem a task is presented in words or symbols, and a goal is set by asking a question. The main characteristic of a word problem is that it can be solved ( achieving of the goal) by using previously mastered algorithms or operations ( addition, subtraction , multiplications or divisions)LeBlanc, et al (1980, p. 105). LeBlanc continues saying that the purposes of the word problems are improve the ability of recalling factors, strengthening of the skills with operations and algorithms and building the relationship between operations and their relationship between real world situations. The main factors that affect the difficulty of these kind of problems are the level of mathematics, the complexity of the algorithm and the number of steps involved in solving. (LeBlanc, et al., 1980)
Orton & Frobisher (2005, p. 27) describes ‘Routine problems' which is a category of word problems that ‘uses knowledge and techniques already acquired by a student in a narrow and synthetic context'. The example given below is a routine problem.
‘How many more than 432 is 635?'
In these type of questions a student is expected to understand the ‘linguistic complexity' and change it to a model with mathematics symbols or operations. Story problems are also a type of word problems that are set in a ‘ real context' which often needs an understanding of the real world situation. An example of a story problem is given below:
‘A postman has ninety four letters in his bag.
Twenty five of them are first class. How many are second class?' (Orton & Frobisher, 2005, p. 27)
The understanding of the different classes of letters is essential for a student to solve this problem.
As word problems use the facts and techniques taught recently, it ‘ not an appropriate method for development of new knowledge and its contribution to mathematics knowledge is minimal'(Orton & Frobisher, 2005). Furthermore Orton & Frobisher (2005) argues that word problems are mathematical exercises rather than questions since, they do not satisfy the criterion that there is ‘no readily known or recallable mathematical procedure available to enable the goal to be attained directly' nevertheless I feel that the answer ( ‘goal' ) is also predetermined by the question setter .
Process problems are also type of problems that appear in mathematics textbooks , but not available to elementary school students. Unlike in word problems this type requires ‘strategies and non algorithmic approaches'(LeBlanc, et al., 1980, p. 105) and often has more than one answer.
According to LeBlanc (1980) more emphasis is given to the process of obtaining the solution rather than the final solution moreover the success of solving the problem depends on the use of one or more strategies and not on the ‘ application of specific mathematical concept, formulas or algorithms'(LeBlanc, et al., 1980, p. 105). Orton & Frobisher (2005) takes a similar view and says that children who reflects on different processes develop the ability to solve other problems.
Nevertheless process problems ‘encourage the development and the practice of problem solving strategies……., provides an opportunity for students to devise creative methods of solution, to share their method with other students , to build confidence in solving problems ….. and to enjoy mathematical problem solving' (LeBlanc, et al., 1980)
According to LeBlanc the difficulty of the process problem depends on the number of conditions that must be satisfied, the complexity of the conditions and the type of strategy used by the solver .
Butts (1980) in Orton & Frobisher (2005, p. 158) describes ‘open search type of problems' as ‘one that does not contain a strategy for solving the problem in its statement. Thus the openness refers to the method of solution, not to the solution'. He gives the following example:
How many different triangles with integer sides can be drawn having a longest side(or sides) of length 5cm? 6 cm? n cm? In each case, how many of the triangles are isosceles?
As it does not have one specific path to the solution, I feel ‘open search problems' are similar to process problems.
At this juncture, I would like to express my ideas about these two problem types. Firstly in the elementary school level where student more on the basic operations and algorithms it is appropriate to use word problem as it helps the children to familiarize the mathematic concept. With my former experience as a mathematics teacher, I have observed that when students do not master in their basic mathematical skills in lower level classes, they are unable to solve complex problems or think innovatively when they come to higher classes. More seriously students come to a conclusion that ‘Only geniuses are capable of discovering or creating mathematics'(Schoenfeld, 1997). Whereas actually the problem is the lacking of the basic mathematics. On the other hand I believe that as the students come to higher grades it is essential to engage in process problems as it broadens the students mathematical knowledge and the reasoning abilities.
Research On Mathematical Problem Solving
According to Suydam (1980, p. 35) early research on mathematics problem solving has focused mainly on word (textbook) problems . Their main emphasis was on how children solved problems. In doing so they thought they could find a way as problem solving could be taught. It was during this period that Polya produces How to Solve It (1954) ‘a charming exposition of the problem-solving introspection' (Schoenfeld, 1987, p. 17) . Every researcher since then has based their research on mathematical Problem Solving on Polya's work.
According to English, Lesh, & Fennewald (2008) Most research has tried to investigate the questions: ‘(a) Can Polya-style heuristics be taught? (b) Do learned heuristics/strategies have positive impacts on students' competencies?'. But no response was made.
Begel's(1979) and Silver (1985) in (English, et al., 2008) has concluded after review of literature in mathematics education that there is little evidence that transfer of learning has been successive .Though it has been reported in some studies successful learning has occurred, Silver suggests that it is due to students mastering the ‘mathematics concept , rather than from problem solving strategies , heuristics, or problem solving process'
Schoenfeld (1992, p. 53) states ‘Pólya's characterizations did not provide the amount of detail that would enable people who were not already familiar with the strategies to be able to implement them' he also says that the reason for the lack of success was due to Polya's heuristics being ‘descriptive rather than prescriptive'. According to Schoenfeld in English, et al (2008) problem solving research should help students to develop larger number of ‘specific problem stratogies', learn metacognitive strategies and to ‘improve beliefs about nature of mathematics , problem solving and their personal compitancies'.
Lester Koehle(2003) in (English, et al., 2008) states that even ten years after Schoenfeld proposal, research on problem solving has failed. According to him ‘schoenfeld's proposal simply moved the basci heuristics to a higher level' not achieving ‘prescriptive power' .However as a response to this Schoenfeld at the 2007 NCTN Research Pre-session proposed that reaserchers should forcus on ‘ meta-meta-cognitive processes'(English, et al., 2008) . But then again as in the earlier cases this had the same shortcomings and was remarked as a ‘short list of descriptive rules lack of perscriptive power and a longer list of perscriptive rules involve knowing when and why to use them'(English, et al., 2008).
Finally as the field of research has failed more than 50 years, Lesh and Zawojewski(2007) conclues that it is time to ‘re-examine the fundemental level of assumption' and suggests that an alternative is to use theorotical perspectives and use methodologies suchas ‘models & modeling perspective (MMP) on mathematics problem solving.
Polya's Four Phases
Polya (1973) grouped his observations into four phases (understanding the problem, planning, carrying out the plan and looking back) describing the four stages the person passes through during the problem-solving process. These four phases would be a guideline as to how to solve a problem. Let us examine them separately by using a 5th grade mathematical problem.
Susan wants a candy bar that costs 25cents. The machine would take pennies, nickels and dimes in any combination. List out the different coins she could pay. (Krulik, Reys, & National Council of Teachers of Mathematics., 1980)
Understanding the problem
The student should be able to point out the unknown, the given data and the conditions. The teacher's role at this point is to ask questions and guide the student. ‘Some probable questions the teacher can ask are : What is the unknown? What are the data? What is the condition?' (Polya, 1973). Depending on the question sometimes it is easier to understand if the student draws a figure and use notations.
Some questions that a teacher or 5th grade student would ask.
How much does a candy bar cost? Which coins does the machine take? Can all the coins be the same? Can Susan pay with quarters? Do y think there are more than one answer to this question? Can you tell in your own words what the problem is asking you to find?(Krulik, et al., 1980)
Devising a plan
It needs to be planned which calculation or construction should be done to obtain the unknown answer. According to Polya (1973) it is often easy to look at a formally solved problem and try to relate to it or think of the acquired mathematical knowledge or even present a problem which has been solved before and ask them to use it.
Some suggested they would make a table like below
Others suggested to write down the denominations 2 dimes and 1 nickel, 25 pennies. The teacher labeled these strategies making a table and making a list and wrote these strategies on the board. (Krulik, et al., 1980)
Carrying out the plan
When it comes to this step because the most critical step in planning is over it is just a matter of carrying out the plan patiently. If the student is convinced of the plan he may carry out the plan smoothly, but in an instance where he has reached the plan by an outside source he might forget the plan. Therefore, it is important that the student understands the steps of the plan. The teacher too has to insist that the student ‘check each step' (Polya, 1973).
Most 5th grade children can get some entries in the list they end up with incomplete list because the entries are not organized. If the children work in groups, the sharing and discussion of answers may lead to a complete solution (Krulik, et al., 1980).
It is important to re-examine and reconsider the way they solved the question and the answer. It allows them to rethink about the question and the method of solving and sometimes to come up with a different, easier method. By doing so, the children learn to believe in themselves. Shumway (1982, p. 134) states
one could argue problem solving ends and the concept learning begins when one begins looking back, identifying similar problems, and engaging in other post-solution activities.
Example : After solutions to the problems are obtained, the class should focus on having the children analyze their own strategies and consider alternative strategies.
Teacher: John would you show the class how your group solved the problem?
John : we wrote the answers we could find like this:
Teacher: can you name the strategy your group used, john?
John : we called it “listing”
Teacher : did another group use a different strategy? Becky would u show the class how your group solved the problem?
Becky: we also made the list by writing the names of the types of coins at the top of the three columns.
Teacher: are you sure you have all the answers to this problem?
Becky : I think so
Teacher: Is there some way we can write down all the answers that Beckys' group worked out without missing any of them.
Stephen : we could find out the largest dime we have , then start with them and work down to the smallest number of dime.
Teacher: Would you draw it on the board so that everyone can see what you mean.
Stephen: I would make columns like beckys group did, but with dimes first then nickels and then pennies. Like this:
Teacher: that's very good. When we draw up a list like Stephens' and put the numbers in some order , we call it an organized list
The teacher can continue the discussion by asking the children what if a candy bar cost 30 cents show the ways Susan could put coins into the same machine.
How does the number of ways alter if the machine will also take quarters?
(Krulik, et al., 1980)
Burtons' three phases
According to Burton three phases of activity namely Entry, Attack and Review(L Burton, 1984) can be observed in the process of problem solving. Let us examine each phase separately using the same question used above.
This is the first step where the solver tries to understand the questions as to what it is about and what needs to be done. Mason, Burton, & Stacey (1982, p. 29) suggests to answer the three questions: What do I KNOW?
What do I WANT?
What can I INTRODUCE?
In this stage the question needs to be read carefully and the relevant facts need to be extracted from the question. Solver also can restate the question in his own words so that he understands what need to be discovered. Introducing diagrams, symbols, images and introducing notations not only makes it easier to obtain a clear picture but also makes it easier for the next phase.
Some ways that a 5th grade student would think :
I WANT to find the different coins Susan could use to pay for her candy.
I KNOW how much a candy bar costs
I KNOW which coins the machine takes
I Know Susan cannot pay with quarters because the machine does not accept quarters.
I know there are more than one answer to this question so I WANT to find different ways of paying.
Can I INTRODUCE a table , draw the different coins or write down the denominations?
It is at this stage that the question starts to resolve. In the attempt to solve the question the solver might take several approaches or different plans may be executed. It is quite normal for the solver to get ‘stuck' (Mason, et al., 1982) and therefore re-think of a different plan or even go back to the entry phase . ‘Stuck' is always accompanied by ‘aha!' (Mason, et al., 1982) which means a different approach or a way out.
When the solver is ‘stuck' TRY….., MAY BE……, BUT WHY……. (Mason, et al., 1982) may help him to overcome the problem. However, this phase is complete when the problem is resolved.
Children can get some entries in the list but try different combinations: Let us TRY 4 nickels and 5 pennies. MAYBE we should TRY all possible ways with 2 Dimes…… BUT WHY can't we use 3 quarters?
As the name suggests, it is the step where you look back at what you have done, try to extend it to a higher level or improve it. The three words that help to structure this phase:
CHECK… the calculation, arguments to verify the computations are appropriate, that the resolution is correct
REFLECT…on the key points, arguments, resolution
EXTEND…the results , by seeking a new path , by altering some of the constraints
(Mason, et al., 1982)
This is where they will CHECK the answers, where the teacher compares Becky's and Steplen's answers.
REFLECT on the key point like where Stephen says “ but with dimes first, then nickels and then pennies.” REFLECT on putting the numbers in some order.
They can REFLECT on different results of other groups.
To EXTEND the problem the teacher or the students could ask:
If a candy bar cost 30 cents show the ways Susan could put coins into the same machine.
How does the number of ways alter if the machine will also take quarters?
Schoenfeld (1997) has parsed protocols into episodes for the purpose of understanding the process. These are periods of time where the solver engages in similar actions. Episodes include reading, analyzing , exploration, planning , implementation and verification. We shall try to explore the episodes using the previous question.
This episode starts when the solver starts to read the problem. This includes the time he spends to understand the problem and the rereading of the problem.
This is where the children would read the problem, may be read it again and try to understand the question.
Can you tell in your own words what the problem is asking you to find?
In this stage the solver tries to fully understand the problem by selecting an appropriate perception and re organizing the question accordingly and introducing the mechanisms that might be suitable for the problem. In cases where the solver knows the relevant perspective this episode may be bypassed. Some analysis questions are:
Example: At this point the children will try to understand and find out how much a candy bar costs? Which coins does the machine take?
This is where the solver explores for relevant information that can be used in other phases. This is less structured than analysis and loosely related to the conditions and the goals of the problem.
Example: The solver would explore for more relevant information like can all the coins be the same? Can Susan pay with quarters? Can there be more than one answer to this question?
Planning - implementation
In these stages issues such as whether the plan is well structured, whether it can be orderly implemented, whether its progress is being observed and reported to the solver is addressed.
This is where the solver revives the solution. Verification questions are:
Does the problem solver review the solution?
Is the solution tested in anyway? If so, how?
Is there any assessment for the solution?
Going further Schoenfeld presents three categories of knowledge and behaviors that should be considered to obtain an accurate idea of the actual problem solving performance. The levels are resources , control and belief system .
Relating Polya's Phases And Scheonfeld's Episodes.
According to the protocol analysis using the two models Polya's Phase's(1973) and the Scheonfeld's Episodes(1997) the following observations can be made.
Line 1- 4, shows the student reads the problem and tries to understand the question therefore is can be categorize as Polya's understanding phase. In Sheonfelds episodes line 1 falls under reading and lines 3-4 falls under analyzing. In Lines 3-4 the student is tries to understand the question by summarizing the given information. The basic information he obtains by reading is not sufficient to solve the problem. Thus, although lines 1-4 is listed under understanding no proper understanding will be acquired until the student ‘plays around' with the given information. But reading and then analysing is a better way of attempting to solve the question.
It is difficult to categorize Lines 5-8 in one of the categories under Polya's Phases. Polya (1973, p. 34)says to make a plan (obtaining different ideas) it is necessary to ‘ emphasize different parts, examine different details, examine the same detail repeatedly but in different ways , combine the details differently, approach them from different sides' . Moreover the student tries to identify details and to ‘contact with formally acquired knowledge' (Polya, 1973, p. 34) (‘It's Just a fixed triangle' (line 4)and ‘those triangles are similar(line7)'). By doing so the student tries to gain understanding of the problem as he is unable plan the method of solving. Therefore these lines could be categorized under understanding as well as planning.
Item 9-19 planning and implementing, It is difficult to separate the planning and implementation as the child does calculations while devising the plan mentally ( thinking loud) . It is illustrated in the statements ‘so if I construct the√2…..then I should be able to draw this line' , ‘ so I just got to remember how to make this construction' and ‘the best way to do that is to construct A' that there is planning going on his mind. Whereas in ‘ 1/√2 - let me see here - ummm. That's ½ plus ½ is 1' and the hand written workings of the student illustrates the implementation of the plan.
However I cannot categorize lines 22-38 and 40-48 under ‘understanding or planning'. The solver ‘works for better understanding'(understanding) and ‘examines the same detail repeatedly but in different ways' (planning)(Polya, 1973, p. 34) and acquires a better understanding of the problem ulitmately. But this cannot be listed under any of the above two phases as this makes the student engage in much complicated analysis than in a direct word problem (where the student is able to understand the problem by examine the given data). Further more in theses lines the student is ‘playing around'(Frobisher, 1994, p. 164) with the ideas. Hence it is difficult to identify into which category these lines would fall into.
Lines 3- 8 , 22- 38 and 40-48, are categorized as the ‘analyzing episode' according to Schoenfeld episodes. Analyzing is ‘an attempt made to fully understand the problem, to select an appropriate perspective and reformulate the problem in those terms and to introduce for consideration whatever principles or mechanisms might be appropriate' (Schoenfeld, 1997, p. 298). In these lines the student tries mechanisms to approach the problem. So, it is appropriate to list it in ‘ analysis episode'
Item 49-53, shows that the student explores better ways of constructing the triangle by ‘playing around' with the information. Frobisher (1994, p. 164) states ‘ understanding of a problem only emerges slowly as it is explores' and that ‘it is difficult for a pupil to develop an understanding of the problem without first attempting to explore whatever appears appropriate'. In these lines the student ‘explore' possibilities. But Polya does not define this type of student behavior thus, this does not fall under Ploya's Phases. These lines are listed under Schoenfeld's exploring episode.
Relating Burtons's Phases And Scheonfeld's Episodes.
One common observation that is visible throughout this protocol analysis was that when using Burtons Phases, ‘chunks' of lines into could not be categorized into one particular Phase. Let us examine the analysis in detail.
Lines 1- 8 , can be recognized as the entry phase. The student reads the problem( line1) tries to understand the question by ‘organising the information'(L Burton, 1984, p. 26) (line 3) . Furthermore in lines 4, 7 and 8 the student uses the rubric ‘I want to' (Leone Burton, 1984) which clearly defines this is the entry phase. On the other hand Schoenfeld categorizes line 1 as reading and lines 3-8 as analysing. Student ‘organizing information' and ‘exploring the problem' which is listed under entry, can be categorized under ‘analysis episode' as well. Therefore entry phase can be interpreted as a collection of different processes.
Lines 9-19 can be categorized as the ‘attack' . In this section it can be seen that the student gets ‘stuck' while tries to find resolutions to the problem and overcomes the situation by getting an answer ( Ah huh! Ah huh! - in line 13) . Schoenfeld categories these lines under ‘planning and implementing'.
Lines 20-24 is the ‘review phase' according to Burton and ‘verification' under Schoenfelds episodes.
In Line 25-39 and lines 40 -48, the student analyses the information trying to figure out what should be done to solve the question. Burton (1984, p. 38) says in ‘attack ' , ‘ several approaches may be taken and several plans may be formulated and tried out' therefore one could think this is ‘attack' phase. But a proper plan has not yet been formulated as the student is still trying to understand the information by analyzing. The student uses the rubric ‘ I know' and ‘now it seems more possible' (line 39) makes it visible that at the end of exploring he gets an understanding of what should be done . But after closely observing , I feel these sections fall in to a phase in between ‘entry' and ‘attack' which is not separately defined by Burton. Although the most activates the student is engaged in are defined by Burton I feel that there is a undefined gap which is very important as the student cannot proceed further without analyzing and understanding the information.
Lines 49-53, the student tries to explore the possibilities of ‘constructing √15' by recalling his past experiences (“trying to remember my algebra”) . Someone could think that these lines are ‘entry phase' as Burton(1984) lists ‘ explore the problem' or even attack because he is taking action to overcome being ‘stuck' . nevertheless it is difficult to list these sections under one category. Therefore the necessity of a more broader and well defines phases arises.
In the light of the protocol analysis I like to raise these ideas.
Schoenfeld (1987, p. 17) states that ‘Polya's characterization of problem solving strategies were in essence accurate summary description' and that they are larger sections of processes . He continues to say that Polya's phases are more descriptive but not prescriptive therefore the phase's do not provide sufficient information for students to use it as a problem solving strategy. Similarly when examining burtons phases I felt that the details are sufficient to recognize the procedure but not precise enough to use as a guideline for the implication of the strategy.
Polya's and Burtons phase's fits perfectly for the word problem where the student can think of a method of solving (plan) directly. But in some problems the student needs to ‘play around' with the given information before coming up with a plan. Schoenfeld (1997, p. 298) says ‘analysis leads directly into plan development' this shows the importance of the analysis episode. When a student try to solve a problem he sometimes ‘search for relevant information so that they can incorporated to the analysis-plan-implementation sequence' (Schoenfeld, 1997, p. 298) this behavior is also not recognized by Polya or Burton.
The planning and implementation is combined in Scheoldfelds model. Though they are combined implementation does not always follow the plan.
‘verification' ‘look back' and ‘check' are similar processses in all three modals, except for the fact that the ‘reflection' and ‘extending' process do not have corresponding processes in the Schoenfeld's episodes as these two processes ‘intends to develop problem solving expertise rather than help in solving a particular problem'(Goos & Galbraith, 1996, p. 242).
Therefore when closely examining to all three processes , I feel that Schoenfeld's episodes are well defined processes when compared to the other two phases.
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