Problem-Solving Approach in Secondary School Students - Research Paper Example

 The Impact of applying Problem Solving Approach in the Development of Secondary School Students’ Performance in Saudi Arabia The Effect of Applying Problem Solving approach in the Development of Secondary School Students performance in Saudi Arabia Abstract Investigating the teaching methods that used by teachers at the present time, most find that teachers focuses on memorization and then applied to the familiar routine of teaching, teacher centred (traditional Methods(1)), leading the stagnation of the analytical skills of students. Perhaps that is due to inability of most teachers to develop good methods of teaching different thinking skills in students. However, these capabilities can be developed through training. This has emerged as a necessity at the present for most if not all professions, occupations and sectors. Moreover, Problem Solving is one of the most important methods of teaching mathematical thinking skills in many countries (National Council of Teachers of Mathematics, 1989, 1998). The purpose of this study is to examine the effects of applied problem solving approach on the development of secondary school students’ performance in Saudi Arabia. The study will be focus particularly on three teachers as a case study to observe the change of their teaching skills and their students’ performance when they undertake a particular training program of problem solving. The participant teachers’ will receive a 2 hours weekly training program of problem solving activities. However, the problems will be used in teaching mathematics problem solving are a textbook story problems. Data will be collected through interviews, classroom observations and consideration of students’ results in examinations which will be analysed in the context of an interpretive qualitative study to understand the meanings of the participants’ classroom development. (1)The traditional Methods: A way of teaching that relies on memorization and presentation. It does not include any independent discovery by the learner. The focuses are on information contained in the textbook and have a negative role of the learner in the educational process while the teacher is the centre of activity and effectiveness. Some teachers may teach according to the way they were taught mathematics or they may just do the same thing each year without reflecting on their practices. Objective Teachers can use different methods to teach mathematics to include varying proportions of the display and presentation, dialogue, discussion, review and raise questions that require logical thinking. However, some of these methods may have a greater impact than others. Method of solving problems is one of the modern teaching methods that help students to raise questions that require logical thinking. The importance of the method of problem solving through literature, articles and many research, which revolve around this vital issue which has become the target of many conferences and served a lot of educators and specialists in curriculum development (National Council of Teachers of Mathematics, 1989, 1998). It shows the method of solving problems as one of the most important methods of teaching mathematics, and this stems from the importance that the solution to problems gives students the opportunity to heighten their logical thinking, analysis and synthesis of information in the discovery of new facts. In the mathematics curriculum in Saudi Arabia at secondary grades there is a significant opportunity for problem solving practice, but this is not availed of by most teachers. The focus of this study is to assist teachers to move from routine and algorithmic teaching to integrating into their classes. As part of the project, participant teachers will be encouraged and supported to use problem solving strategies in mathematics in order to improve the students’ mathematical skills. The Objective of this research is to lay the basis for the creation of a program for teachers of mathematics at the secondary level in the field of teaching methods of problems solving. This will focus on a particular unit in the curriculum. Moreover, there would be an assessment of the effectiveness of the program in the development of knowledge and skills of teachers. In addition, to know the impact of this proposed program at the level of student achievement in secondary education in the same particular unit of the curriculum. The study will dwell, according to the sound judgement of the researcher, on the following questions regarding mathematics teaching at secondary schools in Saudi Arabia: 1) What is the advanced program will be required for Saudi teachers to enhance the use of problem solving approach in their mathematics classroom? 2) How much the effectiveness of the proposed program in the development of knowledge and skills of participant teachers in this research? 3) What is the effect of the program in academic development for secondary school students in the particular unit of the curriculum? Background Approximations of the population holding Saudi citizenship have varied widely. Official figures published by the Saudi government indicate a population of 14,870,000 in 1990. Based on this figure, at the 1990 rate of growth, a population of 20 million was projected by the year 2000. According to the Saudi government, the country's population was more than 27 million in 2009. The education system in Saudi Arabia consists of four levels, Primary, Secondary, High school and Tertiary (colleges or universities) and an important aspect of the education system in Saudi Arabia is that the schooling for all four different levels is free of charge. The schools in Saudi Arabia are all single gender schools which are separating boys and girls. Both genders follow the same curriculum and take the same annual examinations. In starting this literature review it is important, particularly as the word problem, regularly means different things to different people (Blum & Niss, 1991; Nesher, Hershkovitz & Novotne, 2003; Wilson, Fernandez & Hadaway, 1993; Goos, Galbraith & Renshaw, 2000), to consider how it is defined. Most researchers would agree that the defining of a problem situation is that there must be some blockage of the potential problem solver. That is, a mathematical task is a problem only if the problem solver reaches a point where he or she does not know how to proceed. Krulik & Rudnick (1980) define a problem as a situation that requires resolution and for which the individual sees no apparent or obvious means or bath to obtaining the solution. A common definition is that a mathematical problem presents an objective or goal with no immediate or obvious solution or solution process (Blum and Niss, 1991; Schrock, 2000, Polya, 1981; Nunokawa, 2005). A mathematical problem is like every other problem; it is a situation that requires a solution. This solution can either be approached through a simple procedure or certain strategies have to be employed. Problem solving is the procedure during which the problem solver understands the problem, separates the given information from what he/she is asked to achieve and uses all the elements of the problem in order to reach a solution. According to Chapman (1997) problem solving means different things to different people, having been viewed as a goal, process, basic skill, mode of inquiry, mathematical thinking and teaching approach. However, most research in the area seems to regard problem solving as the process of achieving a solution (Chapman, 1997; Blum & Niss, 1991; Franke & Carey, 1997; Hart, 1993). Famously, Polya (1981) described it as a means of “finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable”. This research aims to be a building block in the area of teaching methods in mathematics, and to contribute to the definition of problem solving strategies, which may result in the development of ways of teaching are currently used, and the formation of attitudes towards this method, so that they can use some strategies that may give rise to the thinking of students and encourage them to solve problems facing them. In the reality, teacher plays a vital role in the development of education system. The success of education system mainly depends on the quality of the teachers. In Saudi Arabia as well as many Arab countries, education focuses on the teacher as a pivotal person in the educational process who provides the material and knowledge for the students. However, in many countries, problem solving has become one of its main focuses and teachers are recommended to give appropriate emphasis to problem solving in their planning and teaching (Kroll, & Miller, 1993). They need to be as (Anderson, 2003) suggested: “Actually, teachers generally report that they endorse a focus on problem solving in school curriculum and agree that problem solving is an important life skill for students to develop”. George Polya (1945) said that we should have steps for solving problem which he thought it is legal topic in teaching mathematics .This was George Polya opinion and it was not until the publication of the National Council of Teachers of Mathematics (NCTM) 1980 Yearbook which considered solving problems is the focus topic of reforming teaching mathematics. Problem solving simply refers to the set of processes that are carried out in order to come up with solutions for a given problem. The problem is first defined, analysed and the possible solutions to the problem brought forward. Problem solving is considered as an area of cognitive psychology (Anderson & white, 2004). Kepner-Tregoe (1958) suggests in their Problem Solving and Decision Making (PSDM), that problem solving starts with defining the problem, describing it, establishing possible causes, testing the most probable cause and verifying the most probable cause, a process they called System Analysis. In the first step, defining the problem, the management team cannot fail to define the problem as failing to understand the problem results in wasting valuable time. Inexperienced troubleshooters see this step as time a waste and this leads to poor judgment and therefore incompetence. A good definition should have sufficient information in order to bring forward a good focus, viewpoint, purpose and object. In the second step the problem should be described on the criteria of where and when. The problems that could be put are not highlighted in this step. After the problem is described the next step is establishing its possible causes after which the most probable of them is identified and the last step is verifying the most probable cause. Problem solving refers to the steps used by people in order to adapt to both external and internal problems that need decision making (Heppner & Krauskopf, 1987). Heppner explains that people’s view and attitudes as problem solvers is largely related to their problem-solving techniques, how they perceive challenges and their degree of anxiety. According to him problem-solving skills can be enhanced through training. Heppner even came up with The PSI (Problem Solving Inventory) as a way of quantifying how people associate with their problem solving capability and this model has three determinants; Confidence, Individual control and Approach style. Problem solving according to D’zurilla & Goldfried (1971) refers to a cognitive process that avails various response alternatives for dealing with problematic situations and increases the chances of choosing the most efficient alternative. He identified a process for solving problems that consists of the set, definition of the problem, available alternatives, decision-making and verification as the main steps. According to him good problem-solvers have a variety of skills which include: Identifying problematic situations, coming up with many possible courses of action, planning and coming up with the future challenges and advantages of the chosen alternative. A motivational fact on a certain alternative should also be identified. The terms “problem solving” and “problem-solving teaching approach” are variously defined in the literature as above. In this research problem solving is used to refer to the process of students exploring exercises about same situation in the life from textbook example; using a range of strategies to solve the task, and developing the processes of analysing, reasoning, generalising and abstracting. Textbook story problems actually provide little practice in “problem solving” because no blockage exists; if student can recognize what strategy or operations to apply, the situation is not a problem but merely an exercise. A problem-solving teaching approach refers to the practices teachers choose to use to engage students in problem solving experiences. Role and importance of problem solving in mathematics education Problem-solving is very essential in mathematics education. It is the basic component to achieving the school level. A problem that has been presented can only be solved if skills on have been developed and this is more rewarding than teaching the skills without context. However, for this an approach to solving the problem must first be put forward. This kind of teaching is motivating especially to the learning of new mathematical concepts (Stanbic & Kilpatrick, 1989). Using problem-solving forms a simulation of real life and this makes it easier for students to relate to. It also helps in adapting to unforeseen changes and problems in students. Using problem-solving to teach mathematics gives students a more bright view on mathematics’ power in the surrounding world. Resnick (1987), contributes that use of problem-solving as an approach in teaching mathematics, enhances the practical use and helps them to adapt in the instance of technology breakdown, for example. Resnick believed that modelling adaptive learners who can work in changing environments. Problem-solving contribute in a major way to mathematics as it helps students to think in a logical way, be responsible for their personal learning and also to create their own ideas. It gives them necessary knowledge to deal with new situations. Teachers are therefore faced with the challenge to develop this kind of thinking in their students. However, the teachers of mathematics find it difficult to inspire in their students problem solving skills, because students generally have difficulty in solving the problems facing them, and students needs to solve the problem by analysis of the essential elements of the problem and to find relationships and linkages between those elements, and these are not easy teaching them to students only after the teacher is becoming familiar with this type of teaching methods for a long time. There were many problems which related to the problems themselves .Such as, first, ambiguity of the problems as Branca (1980) explained saying that solving problems is subjective which means that a mathematical problem is relative to the solver. To explain more, some problems can be related to the solvers understanding, thoughts and the way the problem can be solved from their views. Moreover, some problems are hard for people to solve it, while it is not a problem for others. The second problem is the teacher himself is the one who cannot help his students to do exercises or solve problems accurately. Furthermore, how teachers shape the curriculum to meet the needs of their students (e.g., Rickard, 1998) it is important to examine the process of how a teacher might integrate her or his own understanding and teaching of problem solving. Finally, there is no specific definition for the term solving problem. To better support reform efforts, the term solving problem should be determined as Anderson (2003) said:" Problem solving is the process of applying previously acquired knowledge to new and unfamiliar situations". This definition enables the differentiation between problems and exercises, which typically involve practicing a procedure. (e.g., an algorithm) or rehearsing specific facts or concepts (e.g., multiplication facts or definitions), to build proficiency and quickly obtain a correct answer (NCTM, 2000). This difference helped to advance the reform goal .To make further reform, the NCTM Principles and Standards for School Mathematics (2000) provides detailed examples and methods of teaching and learning problem solving in context of the grades in this study. However, Branca (1980), for example, argued that problem solving in school mathematics can be viewed as a goal, a process, or as a basic skill, with each interpretation having different implications for teaching and learning problem solving. Moreover, many considerations to the method of problem solving in mathematics, one way is considering it as a teaching method, and the other direction as an area of mathematics in general education. However, in the literature, there is a long-standing distinction between “teaching for problem solving, teaching about problem solving, and teaching through problem solving,” as set out by Schroeder and Lester (1989). Moreover, as Killen (2009) give explanation for the three different ways, teaching for problem solving means when teacher concentrate on helping students to acquire the knowledge, understandings and skills that are useful for solving problems. Simply put; the teacher provides the students with the foundations for later problem solving. When teachers teach about problem solving they concentrate on the processes of problem solving and teach students how to solve problems. While problem solving has always had a role in learning mathematics, its role has evolved over the years. The oldest role that problem solving has and continues to have in learning mathematics is that of a context for practicing and applying concepts and skills. This role has been referred to as “teaching for problem solving.” In this role, concepts and skills are developed and then real-world problems, usually called “applications,” are presented where students must choose and apply appropriate concepts and skills to find solutions (Charles, 2009). However, teaching through problem solving is using problem solving as a technique for helping students to learn other things (Killen, 2009). Moreover, teaching via problem solving is a method of direct approach of a mathematical concept and is a way of connecting a mathematical concept with daily life situations. Furthermore, teaching via problem solving can be used as a way for introducing new concepts and the journey towards the solution of a problem is much more important than the solution itself. The solver has to find her/his own way which will eventually lead her/him to the solution. Only then has the problem solver really understood the problem and its solution and can explain her/his journey to others. It is a bit late that mathematics curriculum in Saudi Arabia has just started to pay attention to problem solving as an educational concept that improves students' mathematical cognition. Whereas in the developed countries: Over 60 years ago, the importance of problem solving was recognised (Brownell, 1942, cited in Suydam, 1980) and the importance was emphasised strongly throughout the 1980s (Suydam, 1980). NCTM (1989) asserted that problem solving “should be the central focus of the mathematics curriculum” (NCTM, 1989, p. 23) and Polya (1949) and others (e.g., Branca, 1980) maintain that problem solving is the goal of mathematics learning. More recently the NCTM reiterated its call for problem solving to form an integral part of the mathematics curriculum (NCTM, 2003). There is a little evidence, however, to indicate that this focus on problem solving is occurring in today’s classrooms (Muir, Beswick, & Williamson, 2008; Anderson, 2003; Lovitt, 2000; Schoenfeld, 1992). This focus addresses the concern raised by Anderson and White (2004) that research needs to provide teachers with a clearer picture of their role in problem solving classrooms in order for them to adopt more contemporary practices that embrace problem solving teaching approaches. A framework in my study is the teaching problem solving which is exactly teaching students how to solve problems; for example, teaching student how to solve word problems in mathematics. Actually, teachers generally report that they support the focus on problem solving in syllabus documents and agree that problem solving is an important life skill for students to develop (Anderson, 2003). Moreover, the rate of success and nature of implementation vary depending on the teachers’ understanding and conceptions about problem solving. As the term “Problem Solving” has just recently used in Saudi mathematics curriculum, Word (Story) problem comes first as a type of problem solving (Jonassen, 2000) and seems to be a good start for researchers in this field to make effective studies in Saudi mathematics education in which the problem solving approaches can be promoted and applied smoothly. Moreover, a successful problem solver must incorporate a range of heuristic approaches when solving problems (Schoenfeld, 1980). The heuristic plan which is most widely advocated by researchers (e.g., Schoenfeld, 1980; Suydam, 1980) as facilitating problem solving is derived from Polya (1957) and requires the solver to understand the problem, devise a plan, carry out the plan and look back and examine the solution obtained (Polya’s four stage heuristic model). This heuristic framework will be used as a basis for applying problem solving in this research. When solving mathematical problems a procedure is very important. A procedure is basically composed of four steps which are planning, understanding, trying and looking back. The first step understands, and here the problem is read and re-read to determine what is to be done. The condition, data and the variables should be well understood. The second step after understanding is planning which is done with respect to past experiences regarding problems like the one being solved. The information provided as understood is then considered for strategies to be used. This is then followed by attempting the question after a plan is put forward. It is then looked at to see whether correctness can be determined but proving can also be done. After this has been done and the answer found, the problem is revisited to determine if the problem has been solved or something was not put into consideration. If something has been missed the plan is revisited and the problem attempted again. Methodology Choosing the method on research depends on the purpose of the research and the conceptual framework assumed (Anderson, 1998). The methodology for the study will be frame in the context of a case study. This research project will focus on the effect of the problem solving approach on developing teacher skills in mathematics’ class over a short period of time in a secondary setting. The study will be conducted in Saudi Arabia at governmental schools for boys (because males are not allowed to enter female’s schools in Saudi) at grade 7 (because it has new mathematics curriculum). This study will focus on three teachers and their way of making sense of teaching problem solving. However, intensive workshops on problem solving strategies and related processes will be organised for them. Moreover, the work shop can be done in the beginning of the research for teachers who will be involved in this project, Two hours per week or by immediate appointment. This step will be synchronised with the observation which is important to see how is it going with the three teachers and their mathematics classroom when applied problem solving. This will include interviewing them as to their experiences. The analysis of these outcomes will be left to the researcher as the supervisor will verify or validate it. One unit from mathematics curriculum of grade 7 will be taken and developed with the teachers in which well organised activities will be designed for the students' groups to apply problem solving using the story problems stated in this unit of the textbook. The choice as to the unit will be left to the researcher and the sound advisement of the supervisor. An important teaching approach that seems to aid problem solving in classroom is the use of collaborative practices. Cooperative groups of three or four students to share solving a problem will be a structure used in the classroom. The researcher favours structuring activities for students to discuss problem solving with one another and to describe their own attempts at solving a problem. This project will involve three secondary teachers who are teaching all their mathematics, including computation, through small group problem solving and whole-class discussion. Typically the children first work on problem-centered mathematical activities in pairs or occasionally in groups of three or four. During this phase of the lesson, the teacher moves from group to group, observing and interacting with the students while they do mathematics. After fifteen or twenty minutes, the teacher asks the children to stop working and begins a whole-class discussion of their solutions to the problems. However, when solving mathematical problems a procedure is very important. A procedure is basically composed of four steps which are planning, understanding, trying and looking back. The first step understands, and here the problem is read and re-read to determine what is to be done. The condition, data and the unknown should be well understood. The second step after understanding is planning which is done with respect to past experiences regarding problems like the one being solved. The information provided as understood is then considered for strategies to be used. This is then followed by attempting to formulate questions about a plan put forward. It is then assessed as to its correctness but validating it can also be undertaken. After this has been done and the answer found, the problem is revisited to determine if the problem has been solved or something was not put into consideration. If something has been missed the plan is revisited and the problem attempted again. Some research terms Training program: Dictionary of Education realizes that the training program: A set of activities organized and planned, which aims to develop knowledge and attitudes of the trainees and help them refine their skills and increase their efficiency and guide their thinking and improve their performance in their work. . Training programs can manifest itself as a series of seminar – workshops, modules, forums, and other activities that inculcate varied approaches as to mathematical education with regards to problem solving. The traditional Methods: A way of teaching that relies on memorization and recitation. It does not include any independent discovery by the learner. The focuses are on information contained in the textbook and have a negative role of the learner in the educational process while the teacher is the centre of activity and effectiveness. Some teachers may teach according to the way they were taught mathematics or they may just do the same thing each year without reflecting on their practices. Problem solving: Problem solving can be defined as attempting to find a solution in situations where the series of actions which need to be performed to reach that solution are not immediately known (Cooper, 1986). Significance In Saudi Arabia, it is clear that the examinations are the only way to evaluate students' mathematics skills and examine their attitudes in mathematics classes and therefore there is no link between teachers and their students. Through doing this study it will give the teachers a good point of reference in the formulation of their teaching methods in mathematics and help them to change their individual approaches to better gain students' attention in mathematics lessons. The second reason is that the researcher holds the conviction that the curriculum has a great impact on students’ learning and perceptions. In the typical mathematics curriculum, students were not free to express themselves by sharing their own ideas or opinions in a mathematics classroom. By using Problem Solving Strategies through new curriculum, teachers and students will get benefits and be connected to what they are working on together. I will organise some activities for students to discuss problem solving with each other as group activity and to describe their own attempts of solving a problem with their teachers, in which I think it is an important part of good problem solving instruction. Moreover, the outcomes of this research project will benefit early adolescence educators as it outlines the benefits of implementing problem solving approach and highlights the short term effects on young children’s attitudes toward mathematics. By doing this research project, the researcher will enable educators in Saudi Arabia to assess the effectiveness of the problem solving approach within their own classrooms. It is expected that the results of this research to guide the attention of interested people and officials in the mathematics curriculum in Saudi Arabia to the need to focus on teaching problem-solving strategies and to include it in school curricula, and training teachers to apply it in their classrooms. Finally, it helps curriculum planners in Saudi Arabia to organise the academic content of mathematics in more efficient and clearer way, so, teachers, learners and their parents will exist mathematics. Moreover, it is expected that this research opens the way for research and other studies in the preparation of training programs for teachers in general and in the method of problem solving in particular. Ethics Any research that involves the participation of human subjects requires consideration of the potential impact of that research on those involved (Elliot, 2005). There are many issues should be addressed and to consider during the research. Ethics are required to protect the participant from any harm or abuse of power by the researcher. In fact, ethics parts should be considered in many different times during the research; before commencing the fieldwork, during and following the fieldwork. In my research, I will use action research methodology, interviews and observations are going to be used. There are many ethics issues in these types of methods which I will cover in my research. Facilities and Resources The major facilities and resources required for this proposed research include library facilities, access to a computer and printer, online access to email and library facilities, photocopying and printing facilities, and a desk for the researcher to work. These items will be provided on Curtin University campus, through the Science and Maths Education Centre. Also, an authorisation to do the research at the chosen schools with the chosen teachers is needed. Data Storage Data will be stored on campus at Curtin University in paper format (as the original questionnaires, surveys and logs), and in electronic form. All electronic work will also be stored in an archived disk. The data files will be maintained electronically for five years after which they will be destroyed. Access to the data will be limited to the researcher and the Thesis Committee. As a precautionary measure a copy of the paper as well as pertinent documents regarding it will be stored online via means. The particular selection of means will be left to the sound discretion of the researcher. Time line 1. Consultancy with Supervisor will be every 2 weeks or when the need arises by immediately by appointment. 2. The candidacy’s target date of completion and submission will be December 2010. 3. Literature of the research will be p to the end of the year 2010. But this will be ongoing in the sense that new material may become available. 4. Data collection after the completion of the proposal for this study with the approval of the supervisor. The target start will be on March 2011 5. Data analysis will follow after the completion of the Data Collection. 6. Writing report and editing will occur after the data analysis as well as any other changes to be made to the paper upon advisement of the supervisor. 7. Submission of research to the supervisor by approximation will be from December 2010 to July 2013. However this period may change as the study progresses. Factors that appear to impact on implementation of problem solving: 1) Teachers’ beliefs. 2) Students’ abilities. 3) Grade level of the class. 4) Assessment requirements. 5) Level of professional. Different type of problems: teaching problem solving to year 7 1) Logical reasoning problems. 2) Problem with extra information. 3) Spatial reasoning problems. 4) Real life application problems. 5) Puzzle problems. 6) Problems that are on students’ interest or events in their environment or that can be personalized by adding their names. References: Alfred S. Posamentier, Beverly S. Smith, Jay Stepelman, (2010). Teaching secondary mathematics: techniques and enrichment units (8th ed.). Allyn & Bacon. Anderson, G. (1998). Fundamentals of educational research (2nd.). Bristol, PA: Falmer Press. Anderson, J. (2003). Teachers’ choice of tasks: A window into beliefs about the role of problem solving in learning mathematics. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.), Mathematics education research: Innovation, networking, opportunity—Proceedings of the 26th annual conference of the mathematics education research group of Australasia, 72–79. Melbourne: MERGA. 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Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Tea Anderson, G. (1998). Fundamentals of educational research (2nd.). Bristol, PA: Falmer Press. Muir, T., Beswick, K., & Williamson, J. (2008). “I’m not very good at solving problems”: An exploration of students’ problem solving behaviours. The Journal of Mathematical Behavior 27, 228–241. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning, 334–370. New York: Macmillan. Schroeder, T. L., & Lester, F. K., Jr. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New directions for elementary school mathematics. Yearbook of the National Council of Teachers of Mathematics, 31–42. Reston, VA: National Council of Teachers of Mathematics. Sigurdson, S. A., Olson, A. T., & Mason, R. (1994). Problem solving and mathematics learning. Journal of Mathematical Behavior 13, 361-388. Suydam, M. N. (1980). Untangling clues from research on problem solving. In S. Krulik & R. E. Reys (Eds.), Problem solving in school mathematics 1980 yearbook, 34–50. Reston, VA: National Council of Teachers of Mathematics. Read More