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An Investigation of Secondary Teachers’ Understanding and Belief on Mathematical Problem Solving

Weaknesses on problem solving of Indonesian students as reported by recent international surveys give rise to questions on how Indonesian teachers bring out idea of problem solving in mathematics lesson. An explorative study was undertaken to investigate how secondary teachers who teach mathematics at junior high school level understand and show belief toward mathematical problem solving. Participants were teachers from four cities in East Java province comprising 45 state teachers and 25 private teachers. Data was obtained through questionnaires and written test. The results of this study point out that the teachers understand pedagogical problem solving knowledge well as indicated by high score of observed teachers‘ responses showing understanding on problem solving as instruction as well as implementation of problem solving in teaching practice. However, they less understand on problem solving content knowledge such as problem solving strategies and meaning of problem itself. Regarding teacher's difficulties, teachers admitted to most frequently fail in (1) determining a precise mathematical model or strategies when carrying out problem solving steps which is supported by data of test result that revealed transformation error as the most frequently observed errors in teachers’ work and (2) choosing suitable real situation when designing context-based problem solving task. Meanwhile, analysis of teacher's beliefs on problem solving shows that teachers tend to view both mathematics and how students should learn mathematics as body static perspective, while they tend to believe to apply idea of problem solving as dynamic approach when teaching mathematics.


Introduction
It has already been agreed that problem solving is an essential issue deeply discussed in mathematics education in recent decades since its practical role to the individual and society. Problem-solving as one of five standard competences in mathematics mentioned by NCTM (National Council of Teachers of Mathematics) [1] not only develops individuals' conception about aspects of mathematics, but also it helps to adapt to various problems in many aspects of their lives. NCTM [1] also recommended that problem solving be the focus of mathematics teaching because it encompasses skills and functions which are an important part of everyday life.
This issue brings a variety of responds which regards to both story of success and difficulties in applying it within practical mathematics teaching and learning in many countries. It is generally known that mathematics curriculum in some countries can be ascertained put problem solving as a and learning of mathematics [20]. Teachers' beliefs about students' ability and learning greatly influence their teaching practices [21]. The study of Stipek, Givvin, Salmon, MacGyvers [16] show that there is a substantial coherence among teachers' beliefs and consistent associations between their beliefs and their practice. Thus, it is important to know what beliefs that teachers in Indonesia typically have, particularly, in order to understand their influence toward teaching practice.
Hence, this issue gives rise to the questions of how Indonesian teachers bring out idea of problem solving in their understanding related to both content and pedagogical knowledge as well as how they experience difficulties, and how are their beliefs on this issue. As an initial step to address these questions, this present study aims to investigate secondary teachers' understanding and belief on mathematical problem solving.Therefore, the following research questions were addressed: 1. How did secondary teachers understand mathematical problem solving regarding problem, problem solving as instruction, problem solving steps, problem solving strategies, and instructional practice of problem solving as well as level of their performance on problem solving task?
2. What are secondary teachers' difficulties which are regarded to understand mathematical problem solving?
3. How are secondary teachers' beliefs on mathematical problem solving?

Teachers' understanding on the nature of problem solving
Teaching problem solving needs some understandings which are related to some points of knowledge. Chapman [22] mentioned three types of knowledge for teaching problem solving: problem solving content knowledge, pedagogical problem solving knowledge, and affective factors and beliefs. (see table 1). Structuring some knowledge within this table was then confirmed as follows. First, figuring out what it means by problem. Chapman [22] argued that teachers should understand problems based on their structure and purpose in order to make sense of how to guide students' solutions including understanding on types of tasks, such as cognitively demanding tasks; multiple-solution tasks; tasks with potential to occasion/promote mathematical creativity in problem solving; demanding problems that generally allowed for a variety of problem-solving strategies; rich mathematical tasks, and particularly open-ended problems. Second, understanding problem solving in instruction which means teachers are encouraged to foster their students in completing problem solving steps precisely.
Third, coming up with idea of instructional practices for problem solving, teachers need to consider a series of activities which give students opportunity to solve problems which needs challenging complex thinking and logical reasoning. The activity depends on how the teacher's ability to prepare a problem. Crespo & Sinclair [in 23] explained that teachers who are able to create questions in the initial situation will be more successful learning rather than the teacher who asked the problems spontaneously.Thus, teaching for problem solving, teachers should be proficient in as well as understand the nature of it in order to teach it effectively. In general, this categorization appears to give insight on emerging teachers' conception of problem solving theoretically to teachers' actual teaching practically. In addition to the specific issue related to problem solving content knowledge, teachers should also be proficient to deal with a variety of problem solving task, such as completing problem solving steps as well as applying problem solving strategies on various types of mathematical task. Level of understanding regarding this issue was discussed by some frameworks, such as Polya's problem solving step [24], and some error analysis guidelines on performing mathematical tasks developed by, for instance, Wijaya et al [13] who adapted from three main frameworks, namely Newman's error [25], Blum and Less' modelling stages [as cited in 26], and PISA's mathematization stages [27], and Kohar & Zulkardi [12] who adapted from Valley, Murray, & Brown [28] and PISA's mathematical literacy [29]. To figure out the possibility of those frameworks are applied to investigate level of teachers' proficiency in solving a variety of problem solving task, the following table shows characteristics of some of those. Understanding what a student knows, can do, and is disposed to do (e.g., students' difficulties with problem solving; characteristics of good problem solvers; students' problem solving thinking) Instructional practices for problem solving Understanding how and what it means to help students to become better problem solvers (e.g., instructional techniques for heuristics/strategies, metacognition, use of technology, and assessment of students' problem solving progress; when and how to intervene during students' problem solving).

Affective factors and beliefs
Understanding nature and impact of productive and unproductive affective factors and beliefs on learning and teaching problem solving and teaching In general, from table 2 we can notice that element of steps among those three frameworks corresponds each other. Specifically, steps of understanding problem and devising strategies, simultaneously, has likely similar idea with steps of reading, comprehension, and transformation in Newman analysis, while this idea also appear on mathematical literacy, i.e. formulate. As early stages in solving mathematical task, they end up with determining precise mathematical model or strategies before performing further steps of solving problem. Likewise, each idea of carrying out step in Polya's process, process skill in Newman, and employ in PISA's mathematical literacy deals with undertaking mathematical procedure to find mathematical results, such as performing arithmetic computations, solving equations, making logical deductions from mathematical assumptions, performing symbolic manipulations, or extracting mathematical information from tables and graph. Furthermore, the last step of Polya's, i.e. looking back, corresponds to final stage of Newman analysis, i.e. encoding and PISA's mathematical literacy, i.e. interpretation. The idea of this stage is interpreting the mathematical result to the initial problem such as checking the reasonableness of the answer or considering other strategies and solution of the problem. The difference, obviously, only appear on the type of the tasks examined where PISA's mathematical literacy specifies on contextual task [29], while Polya and Newman respectively deals with general mathematical problem [24] and written mathematical task [25]. Comparing those three frameworks, it is known that Polya's problem solving steps, which was introduced before the other two frameworks, have an agreement with both Newman analysis and PISA's mathematical literacy. Thus, Newman's error categories can be used to analyze teachers' level of performance in solving context-based mathematical problem solving tasks, which were used in this study.
The typical teaching of problem solving should also need to be considered as important parts of understanding instructional practices for problem solving pedagogical knowledge. As examples, typical learning in Japan [30] consists of (1) discuss the previous lesson, (2) presents a problem, (3) ask students work individually or in groups, (4) discussion of problem-solving method, and (5) provides a summary and discussion of an important point. Shimizu [30] explains that underline and summarizes the activities or "Matome" has the function (1) underline and summarize the main points, (2) to encourage reflection of what we have done, (3) determine the context to recognize new concepts or terms based on previous experience, and (4) make the connection (connection) between the new and the previous topic. Similarly, teachers in Finland as reported by Koponen [31] in his study carry out problem solving lesson by firstly introducing the problem, then working on the problem in pairs or in small groups, instructing the individual students on their solutions, and the final whole-classroom discussion. Based on the above discussion it is necessary to know actually how teachers understand the nature of problem solving that includes problem itself, problem-solving, problem-solving strategies, problem solving approach brought into classroom practical teaching, and posing problem solving task. Besides, it is also necessary to look into teachers' performance in solving problem solving task.

Teachers' beliefs on mathematical problem solving
Mathematical beliefs, as Raymond said, are regarded as "personal judgments about mathematics formulated from experiences in mathematics [32]. They play role as prerequisite development of problem solving itself [33]. Regarding their categorization related to other interrelated fields, Weldeana & Abraham [34] summarized frameworks of teachers' mathematical belief systems into smaller subsystems, including beliefs about the following: (a) the nature of mathematics, (b) the actual context of mathematics teaching and learning, and (c) the ideal context of mathematics teaching and learning. These subsystems are wide-ranging, including, for examples, teachers' views on mathematical knowledge; the role of learners and learning; the role of teachers and teaching; and nature of mathematics activities. This categorization was also conceptualized by Ernest [35] who described three views of nature of mathematics: the instrumentalist view, the platonist view, and the problem solving view. The instrumentalist believes mathematics is useful and collects unrelated facts, rules and skills. The platonist views mathematics as a consistent, connected and objective structure, which means mathematics is a unified body of knowledge that is discovered, not created. The problem solving view sees mathematics as a dynamically organized structure located in a social and cultural context.
In attempt to simplify these views, Beswick [30] has tried to make connections among the nature of mathematics, mathematics learning, and mathematics teaching as follows. The relationship between teachers' beliefs about mathematics and its teaching and learning practice, has been investigated in many studies. Ernest [in 31] claimed that a teacher's personal view of mathematics underpins beliefs on the teaching and learning of mathematics. In line with this claim, Schoenfeld [37] stated that the teacher's sense of the mathematical enterprise determines the nature of the classroom environment that the teacher creates. Thus, beliefs influence teaching and learning practice. Other study, Ruthven's [38] study, for instance, argued that teachers need to broaden their perspective about ability and quality in mathematical learning which is probably more easily changed by changing practical teaching in classroom first, as the teachers' understanding of mathematics teaching and learning. Thus, teaching and learning practice influence teacher's belief.
Studies on how deep mathematics teacher develop beliefs on mathematics and its practice in teaching and learning were then investigated by some researchers. A study of Zhang and Sze [39] comparing preservice teachers' belief on mathematics in China and Thailand, for instance, revealed that generally teachers participating in this study believe that mathematics is about thinking, logic, and usefulness, rather than a subject of calculableness and preciseness. Regarding their belief on mathematics teaching and learning, the results showed that Chinese preservice teacher's beliefs are more like constructivist. Beswik [40] in his study on two mathematics teacher's view on the nature of mathematics and mathematics as school subject regarding on the three views: platonist, instrumentalist, and problem solving, suggested that more attention needs to be paid to the beliefs about the nature of mathematics that the teachers have constructed as a result of the cumulative experience of learning mathematics. Thus, we get insight on how important these views offer opportunities for teachers to rethink their own beliefs and get to know more about teaching practices.
These beliefs, as Schoenfeld [37] argued, give impact on students' belief in learning mathematics which then obviously influence their mathematics performances. The cause, for instance, is that teachers rely on established beliefs to choose pedagogical content and curriculum guidelines [e.g. 17]. If teachers tend to believe mathematics as a set of tools that contain facts, rules, and skills, the lesson will likely to be centered on teachers instead of students [41]. Furthermore, The importance of students 'and teachers' beliefs about the role of problem solving in mathematics is a prerequisite development of problem solving itself [41]. Romberg in [42] shows the relationship of elements in the teaching of mathematics as follows. only teacher's mathematical content, but also teachers' beliefs influence students' performance. This view illustrates the importance of types of teacher beliefs which are needed to be investigated as attempt to improve teachers' proficiency dealing with problem solving.

Methods
This is a descriptive explorative research which aims to explore teachers' understanding on mathematical problem solving and their belief.

Participants
Participants were secondary teachers who has minimum a bachelor degree, have taught more than 5 years, from four cities: Surabaya, Sidoarjo, Gresik, and Mojokerto. There were 25 private teachers and 45 state involved in this study.  practice of problem solving (3 items), and designing problem solving task (3 items). Other three items are categorized to identify the difficulties of teachers in problem solving while the other three items are categorized to explore teacher beliefs, i.e regarding mathematics, how to teach mathematics, and how students should learn mathematics.

Data collection
Meanwhile, the problem solving tasks were designed to explore teachers' understanding regarding contextual tasks which do not likely need any higher prerequisite formal mathematical knowledge. See the tasks at appendix. The following is the description on the tasks' demands. The difference between those two task, particularly, appears on the most dominant stages which are likely more needed to be performed when solving the task. Here, task 1 needs more performance on final stage of problem solving, i.e. interpreting mathematical result to initial problem, while task 2, in opposite, demands more likely performance on early stages, i.e. from understanding the task to devising strategies or mathematical model of the task.

Data analysis
Descriptive analysis on investigating teachers' understanding and belief was carried out by using score given on each group of questions. Each option on a question has score either 1 (not understanding), 2 (partial understanding), or 3(full understanding). As an example, we give one question contained in group of question: problem solving content knowledge including its options and its score as follows.
An open-ended mathematics task is the task which...
A. contains open sentences (score 1) B. produces a variety of strategies to find out the solutions (score 3) C. gives an open opportunity to persons who want to solve (score 1) D. contains higher level of difficulty so that needs higher mathematical skill as well (score 2) E. has more than one solutions (score 3) F. has opportunity to be developed into other type of tasks by changing information or requirements from the solved task (score 2) The score varies to show level of understanding from 1.00 (do not understand), to 3.00 (fully understand), while the other scores varies to show level of beliefs on mathematical problem solving from 1.00 (platonist view/as tool) to 3.00 (problem solving view). The score is given to each participant on each question based on the following formula. In detail, we have developed a guideline to categorize these levels as shown by the following table. Regarding teachers' performance on problem solving task, we used framework on investigating individual's performance based on stages of mathematical modelling adapted from Wijaya [13], and Kohar [12] since the task is in the category of context-based problem solving task. This is shown as follows.

Teachers' understanding in questionnaire results
There were seven categories of questions which were tested to measure teachers' understanding on problem solving. Each category could contain more than one questions. For instance, category of problem solving strategies contained two questions, while category of experience in designing problem solving tasks contained three questions. Table 7 shows teachers' average score on the questionnaire.  Table 7 shows that the lowest score of understanding problem solving appears on category problem solving strategy (1.83) which means teachers did not really.understand toward this group of questions. In giving response on questions related to problem solving strategy, data show that most teachers chose wrong options related to type of problem solving strategy should be applied on a given information on the question. Moreover, teacher did not really show good understanding on the meaning of problem as indicated by its score, i.e., 2.36, which could be interpreted as low score of full understanding based on scoring category on table 5. However, higher score appears on category problem solving as instruction (2.91) and designing problem solving task (2.64), both of which are related to practical knowledge of problem solving. It shows even though teachers are aware of the importance of problem solving as the focus of learning but there are still weaknesses in selecting a task question as a problem and solution strategies. Thus, regarding Chapman's category of knowledge needed to understand problem solving, we then could note that the teachers had relatively better understanding on pedagogical problem solving knowledge rather than problem solving content knowledge.

Teachers' understanding in performing problem solving task
In total, we had 140 possible responses (number of tasks done by all teachers in total) which included 35 correct responses (25%), 80 incorrect responses (57.15%), i.e., no credit or partial credit, and 25 missing responses (17.85%). Each incorrect response had an opportunity to be coded by more than one sub-type code since its different errors found from this response. For instance, a response could be coded as mathematical processing error subtype algebraic error (P-1) and interpretation simultaneously (I). Thus, the total number of responses was no longer 140 items, instead we found 176 coded responses. Then, the percentage of each category of responses is given as follows.   Table 8 shows that transformation/devising strategies error constitute to the most frequently found from teachers' work (23.30%), while mathematical process error, conversely, become the least frequently observed (4.55%). Morover, they also performed comprehension error highly, i.e., 21.03%. This point out that teachers found some difficulties in early stages of problem solving steps.
As examples on how teachers perform those errors, the following figures show comprehension error and transformation-interpretation error, respectively on task 1.

Figure 1. Examples of errors on task 1
On comprehension error, teacher at figure 1a was unable to distinguish between relevant and irrelevant information given from the table. He only considered information in column won, lost, and drawn without giving more attention to the column goals for and goals against to convey a calculation. Thus, we coded it as C-2. Meanwhile, transformation error was performed by teacher at figure 1b who tended to use directly a mathematical procedure, i.e. addition and subtraction without analyzing whether or not it was needed and did not provide precise mathematical argumentation supporting the procedure he used. Consequently, he got the score in negative number, which is not possible to happen in real world setting. Thus, we also coded it as I error.
Regarding transformation error and comprehension error on task 2, we give these following examples.

Figure 2. Examples of errors on task 2
Figure 2a was coded as transformation error because the teacher provided wrong mathematical model to find the best time chosen for holding the competition. She seemed trying to find LCM of the differences of hours from the three cities which is not suitable to find the solution. Hence, this error was coded as T-3. Meanwhile, figure 2b actually shows a unique strategy, i.e. using algebraic approach, which was not found from other teachers' work. However, the teacher did not perform carefully in selecting information related to model of inequality for Ankara time, i.e. writing 12≤x+14≤23, instead of 12≤x+2≤23. Thus, we coded it as P-1 since it contains algebraic error of finding solution of inequality.
Teachers' complete performance in solving the tasks is also interested to be discussed. Here, we found two approaches, i.e. arithmetical and algebraic approach.
Here are examples of these approaches from task 1. a. T-3 b. P-1

Translation:
Finding least common multiples from differences of hours among those three countries, i.e. LCM from 4,15, and 5 = 60 hours So, it should start from 12.00 a.m. to 12.00 p.m.

Translation
: let x is time for Greenwich, then allowed time for other cities to hold the competition is given as follows.
Since the duration of competition is 1 hour, then range of time that can be used is from 5 to 8 a.m (Greenwich time) shows that teacher used symbols representing number of goals successfully shot to another team, then carrying out a well-operated algebraic form by seeing about information of goals for and goals against from the table in task 1 to find out each of goals produced by each of teams. Interestingly, the teacher started to symbolize score of Mentari vs Surya by a : a, which means he know that number of goals produced between Mentari and Surya is same based on information of number of drawn match given in the table. Similarly, teacher at figure 3b also applied this information to find each of score between two teams of each match, but he applied arithmetical approach by listing some possibilities of score and then carrying out simple operation (addition) of number to find out the score of Mentari vs Surya FC.

Teachers' difficulties on understanding problem solving
We categorized teachers' difficulties into three type of questions, i.e. problem solving steps, designing problem solving task, and causes of difficulties in designing problem solving task. Table 7 show teachers' responses on this issue.

Teachers' beliefs on mathematical problem solving
Belief in mathematics can be classified into three parts, namely mathematics as a tool, mathematics as body static and dynamic mathematics as human creations. Table 7 shows teachers' average score of beliefs on mathematics, teaching mathematics, and learning mathematics.

Discussion and Conclusion
Results of studying teachers' understanding on problem solving show that most teachers were likely found difficulties on problem solving content knowledge, especially in identifying problem solving strategy. Analysis of their performance in solving problem solving task also reveal that errors mostly happened when they started to devise strategies or precise mathematical model. These findings were then supported by their opinion on questionnaire that the most difficult step of carrying out problem solving step is devising strategies. Moreover, understanding this knowledge needs likely more attention to deeply comprehend interrelated content knowledge regarding certain topic. For instance, when investigating prospective elementary teachers' common thinking about open-ended, Chapman [17] found that they actually know that open ended problem meant more than one answer but they were doubt about what this meant mathematically such as inability to provide example of other answers on a given open-ended task.
Regarding teachers' belief, the fact that teachers tend to view both mathematics and how students should learn mathematics as body static perspective, while they tend to believe to apply idea of problem solving as dynamic approach when teaching mathematics shows that belief in mathematics is not the only factor affecting the practice of teaching and the views of how students should learn. Raymond [in 4] describes other factors besides belief in mathematics such as teacher education programs, teaching social norms, teachers' life outside the classroom, characteristics of teacher's personality, the situation in the classroom, and student life outside the classroom. Teachers also may have a tendency toward a view on teaching and learning mathematics which includes encouraging students to be actively involved in solving problems in various contexts. However, our findings about teachers' view on teaching problem solving seemed contrast with the study of Wijaya, et al [20] and Maulana [43] which point out that teachers' view of teaching mathematics task in their study admit to mostly gave context-based tasks which explicitly provide the needed procedures and contain only the information that is relevant for solving the tasks. This teachers' view does not support the idea of teaching mathematics as problem solving as our finding on teachers which agreed to teach mathematics as problem solving. Furthermore, teachers in their study prefer to teach using directive teaching approach by mostly explaining a topic while students write, listen, and answer closed questions. Thus, there is an inconsistency between teachers' actual practical teaching and their view on teaching problem solving. A conjecture of this issue could be related with teacher's understanding on problem solving knowledge, particularly on the lack of teachers' problem solving content knowledge as we found in this study since there is a significant association among teachers' understanding, beliefs, and teaching practice on problem solving [16,44,45].
To sum up, we argue that: (1) teachers' understandings on problem solving content knowledge was less than those on pedagogical problem solving knowledge, (2) teachers more believed on mathematics and mathematics learning as body static, while in practice, they tend to believe to the views that they should teach mathematics as a dynamic/problem solving activity. The implication of this study recommends the need to develop a program of teacher professional development in understanding and applying problem solving in teaching practices taking into consideration the difficulties experienced by teachers primarily on problem solving content knowledge. Thus there will be a balance between teachers' view and teachers' actual knowledge and practice toward mathematical problem solving.