A primary goal of mathematics teaching and learning is to develop the ability to solve problems (Wilson, Fernandez & Hadaway, 1993; Lester, 1980; National Council of Supervisors of Mathematics, 1978). We solve problems every time we achieve something without knowing beforehand how to do so. Problem solving involves the process of confronting a novel situation formulating connections between given facts, identifying the goal, and exploring possible strategies for researching the goal (Szetula, 1992). Mathematics teachers have voiced concern about the inability of students to solve any problems but the most routine problems even though they have mastered all the required computational skills (Lester, Garofalo & Kroll, 1989). Until recently researchers have attributed problem solving difficulties almost exclusively to cognitive aspects. One aspect getting considerably more attention of late is metacognition. Metacognition refers to knowledge and control one has of one’s cognitive functioning. With respect to mathematical performance this includes knowing one’s strengths and weaknesses together with an awareness of one’s repertoire of strategies and how these can enhance performance (Lester et al., 1989). Lester et al (1989) contend that for students struggling to become competent problem solvers, the complexity of problem solving is compounded by the fact that most of them do not receive adequate instruction, either in quality or quantity. Many mathematics teachers have received little or no systematic training in problem solving. They suggest two reasons why problem solving competence is so difficult for students to develop: i) problem solving is a complex cognitive activity which requires much more than just direct application of some mathematical content knowledge. Further, these cognitive activities are influenced by a number of non-cognitive factors. ii) students need to be given appropriate opportunities to become proficient problem solvers. They need to be given carefully designed problem solving instruction and they need to be given extensive problem solving experience. The purpose of this proposed study is to develop an instructional intervention to promote the development of cognitive and metacognitive problem solving skills.

Research has been carried out both by cognitive psychologists and mathematics educators in an attempt to understand the cognitive aspects of mathematical problem solving. George Polya (1957) an eminent mathematician has written extensively on the topic of mathematical problem solving. He developed a model of problem solving, which includes the following processes: understanding the problem, devising a plan, carrying out the plan and looking back. Much research on problem solving has been based on his work (Schoenfeld, 1987).

A current approach to studying cognition is the information processing model. This model has three major components: problem task environment, working memory and long term memory (Silver, 1987, Stiff et al., 1993). The task environment is the structure of facts, concepts and interrelationships that make up the problem. Working memory or short term memory is where most of the cognitive action is occurring. Working memory maintains an internal representation of the current state of cognitive activity. It is also where information obtained from the external problem task environment interacts with knowledge retrieved from long term memory. Mathematical knowledge, such as basic facts, processes, generalized problem types, schemas, algorithms as well as metacognitive knowledge is stored in long term memory. Information processing involves controlling information into and out of working memory, recognizing, comparing and manipulating symbols in working memory, and storing information in long term memory (Silver, 1987). Information can be stored in long term memory only after it is has been processed in working memory and it can be used in thinking only after being retrieved from long term memory and placed in working memory.

Most researchers taking a cognitive science approach to learning agree that individuals construct their own knowledge bases (Silver and Marshall, 1990). The constructivist approach basically states that students construct knowledge to fit what they already know or believe about the world. In order to solve a problem one must construct an understanding of the problem that connects his or her present knowledge with the task requirements of the problem. These problem representations are viewed as central to the problem solving process (Silver and Marshall, 1990). Problem representation is linked to Polya’s first stage of problem solving, understanding the problem. Failure to solve problems can often be traced back to failing to understand the problem and consequently failing to construct adequate problem representations (Silver & Marshall, 1990; Mayer & Hegarty, 1996). "The major creative work in solving word problems rests in understanding what the problem means." (Mayer & Hegarty, 1996).

Metacognition has been described as ‘reflections on cognition’ or ‘thinking about one’s own thinking’ (Schoenfeld, 1987). It includes awareness of how one learns; awareness of what one does and does not understand; knowledge of how to use available information to achieve a goal; ability to judge the cognitive demands of a particular task; knowledge of what strategies to use for what purposes; and assessment of one’s progress both during and after performance (Flavell, 1979). It is these skills that enable one to plan, monitor and evaluate performance throughout the execution of a task. Simply stated, cognition can be described as those skills involved in doing, whereas metacognition involves those skills required to choose and plan what to do, as well as, to monitor what is being done (Garofalo & Lester, 1985).

While teaching basic skills may seem to be the most straightforward way to improve problem solving performance, research indicates that knowledge of basic skills alone is not sufficient (Mayer, 1998). Problem solvers in addition to requiring specific math skills, also require the ability to mange these skills. Schoenfeld (1985) in videotaping students solving problems aloud in pairs, observed that even when the students demonstrated mastery of the course material, they still failed to solve familiar problems correctly. He would observe students solving a geometry problem where they immediately chose a solution method without discussing if or why the method was appropriate. When they encountered difficulties they did not stop to consider whether they were on the right track. Further, when students got stuck, they often moved on to another strategy without reflecting on what had gone wrong. As a result they would abandon good ideas along with the bad ones. Schoenfeld (1985) describes good ’control’ and not necessarily always making the right decisions, but being able to realize that a strategy in not working, and to consider alternatives. In the above case, the students had the resources to solve the problems, but they were unable to apply them successfully because they lacked the knowledge of how to regulate their thinking.

Schoenfeld (1987) investigated the metacognitive skills of novice and experts in problem solving. His study found that novice students quickly choose a solution strategy and then spent their remaining time in the execution of the strategy. In contrast, Schoenfeld found that the expert problem solvers spent most of their time analyzing the problem making sure that they understood it. The approach of the experts was to constantly ask themselves if the chosen strategy was working and if the answer was ‘no’ to search for a new strategy.

Swanson (1990) investigated whether high levels of metacognitive knowledge about problem solving could compensate for low overall aptitude. He looked at students in grades 4 and 5 and based on their scores on the Cognitive Abilities Test (CAT) categorized them as high-aptitude (scores above 120) and low-aptitude (scores below 105) students. The children were also administered a metacognitive questionnaire to assess their level of metacognition in the general area of problem solving. Children were categorized as high and low metacognitive performers. The problem solving tasks the children were given were specific to mathematics specific but were general tasks frequently used in Piagetian studies. One task involved a pendulum where students were given string of 3 lengths and 3 sets of weights and they were asked to identify the factor that determined the frequency of oscillations. The second, combinatorial task, required the children to discover a combination of 3 clear, colourless liquids that yielded a particular colour when an indicator was added. Swanson found that the high-metacognitive individuals outperformed the lower metacognitive individuals in problem solving regardless of their overall aptitude level. Based on these findings the authors suggest that high metacognitive skills can compensate, to some degree, for overall ability.

Cognitive and Metacognitive Framework

Garofalo and Lester (1985) presented a cognitive and metacognitive framework for studying mathematics performance. The framework (see Appendix A) comprises four categories of activities involved in performing a mathematical task such as problem solving: orientation, organization, execution and verification. The framework does not list all possible cognitive and metacognitive behaviors that might occur , rather it specifies key points where metacognitive decisions are likely to influence cognitive actions. The orientation category includes strategic behaviors to assess and understand the problem, as well as, strategies to obtain an appropriate representation of the problem situation. The organization category includes the planning of one’s behavior and choice of actions. The execution phase involves the problem solver regulating behavior to conform to the plans and monitoring the implementation of the strategy. The verification category includes evaluating the decisions made as well as the outcomes of the executed plan. Included in this category would be evaluating the adequacy and reasonableness of the solution. These phases are related to Polya’s four phases of problem solving. The cognitive-metacognitive framework is intended to serve as a tool for analyzing metacognitive aspects of mathematical performance (Garofalo & Lester,1985). In addition, they believe that the mathematics education community needs to adapt more of a metacognitive perspective in regard to the development of instructional treatments. If the critical role of metacognition is made clearer, educators will be in a better position to incorporate metacognitive aspects into mathematics education.

Instructional Implications

Teaching mathematics is a complex task. How a student learns or acquires new skills and information needs to impact on how teachers should teach (Stiff et al., 1993). Based on the constructivist approach to learning, teachers cannot expect transfer of mathematical concepts and relationships to occur by simply telling their students what they know. At one time the best advice from the research literature to teachers given the task of developing student’s problem solving abilities was that problem solving should be taught by giving students lots of problems to solve (Silver & Marshall, 1990). Recent research suggests a more refined approach. While extensive practice in solving appropriate problems is a necessary requirement for the development of expertise, it is not sufficient.

Gourgey (1998) asks the question how are students able to achieve good grades in mathematics courses and then demonstrate such poor ability to solve the same problems? It is common practice, she notes, for similar problem types to be tested together within a particular lesson. Students, as a result, can apply these procedures mechanically without having to understand them. In addition, it could also be that instruction which focuses on performing techniques could be neglecting the ‘when’ and ‘why’ of strategy use. If this is the case, students may not be internalizing the skills that would enable them to analyze a problem and to draw upon their knowledge to solve it (Gourgey, 1998).

The learning of problem solving has been shown to be facilitated by student discussion. Hart (1993) undertook a study involving seventh grade students who were placed into cooperative groups. The students were asked to solve a four part applied problem. The students had been exposed to the mathematical concepts necessary for solving the problem, but had received no formal instruction on problem solving strategies. One of the factors shown to enhance problem solving performance was group collaboration. The collective experience of the group frequently supplied background information that individual students did not possess. In addition, a second benefit of group work was that the challenge and disbelief of peers acted as a form of external monitoring when self monitoring was not apparent. Although students seldom questioned their own strategies, they occasionally did challenge each other. Such encounters seemed to force them to examine their own knowledge, strategies and beliefs more closely (Hart, 1993). She found that one of the factors that impeded problem solving performance was lack of individual monitoring or regulation of cognitive activity.

Mayer (1998) suggests that a metacognitive approach to the learning of problem solving skills should involve the modeling of how and when to use strategies in realistic academic tasks. If problem solving performance is to improve, students need to learn how to represent problems within the context of actually trying to solve it. Mayer believes that the most successful instructional technique for teaching students how to control their mathematical problem solving strategies is having a competent problem solver describe his/her thinking processes as he/she solves a real problem in an academic setting.

Incorporating the above instructional strategies, student discussion and modeling among others, Lester et al.(1989) undertook a study to look at the role of metacognition in seventh grader’s problem solving and explore the extent to which these students can be taught to be more strategic and self-aware of their problem solving behaviors. Prior to instructional intervention all students in two grade 7 classes were administered a pre-test and a parallel version of this test was given to all students within a week after the end of the instructional phase. The primary assessment was conducted analyzing videotapes of individual students and pairs of students working on multi-step problems. The students were also interviewed on their mathematical knowledge, strategies, decisions, beliefs and affects. The instruction was presented by one of the investigators 3 days per week for a period of 12 weeks. The instruction consisted of 3 concurrent components: the teacher as external monitor, the teacher as facilitator of students’ metacognitive development, and the teacher as a model of a metacognitively aware problem solver.

The teacher as external monitor, emphasized the 10 teaching actions for the teacher to engage in (see Appendix B). The teacher directs whole class discussions about a problem that is to be solved; observes, questions and guides students as they work either individually or in small groups to solve the problem and leads a whole-class discussion about students’ solution efforts. The teacher as facilitor of students’ metacognitive development, assumes this role when he/she asks, questions and devises assignments that require students to analyze their mathematical performance, points out aspects of mathematics and mathematical activity that have bearing on performance, and helps students build a repertoire of heuristics and control strategies along with knowledge of their usefulness.

The teacher as a model, involves teacher explicitly demonstrating regulatory decisions and actions while solving problems for students in the classroom. The intent is to give students the opportunity to observe the monitoring strategies used by an ‘expert’.

Lester et al.(1989) found that the more successful problem solvers were better able to monitor and regulate their problem solving activity than were the poorer problem solvers, with the difference being most apparent during the orientation phase. They also found that effective monitoring requires students knowing not only what and when to monitor, but also how to do so. The authors believe that students can be taught what and when to monitor relatively easily but helping them acquire the skills needed to monitor effectively is more difficult. Metacognitive training is likely to be most effective when it takes place in the context of learning specific mathematical concepts and skills.

Adibnia and Putt (1998) designed a study to investigate the effects of an instructional intervention based on Garofalo and Lester ‘s(1985) cognitive-metacognitive framework. Specifically, they investigated the effect of the teaching intervention on the mathematical problem solving performance of grade 6 students of different ability levels. The instructional intervention consisted of fourteen 90 minute lessons involving a variety of process problems. Lessons plans were designed using the Garofalo and Lester framework. The focus was on developing students’ metacognitive behavior primarily, to think about their own thinking and to become aware of the regulation and monitoring of their activities. The instruction had three components: the teacher as external monitor, the teacher as facilitator of students’ metacognitive development, and the teacher as a model of a metacognitively aware problem solver similar to that of Lester et al (1989). In addition, the daily lesson plans consisted of four phases – Orientation, Organization, Execution and Verification – which were followed by the teacher.

To measure the effectiveness of the instructional intervention, the students were administered a process problem test before and after the instruction. The tests were scored using an analytic scoring scheme (see Appendix C). The scoring assigned a maximum of two points to each of the cognitive and metacognitive phases, namely Orientation, Organization, Execution and Verification.

The authors found that the instructional intervention was more effective for above-average than for below-average students. Above-average students were found to make greater use of the cognitive-metacogntive strategies following the instructional intervention. The study did not address, however, how the use of these strategies impacted their final grade in the course.

In 1999, The Ontario Curriculum, Grades 9 and 10: Mathematics, was introduced. The new curriculum embeds the learning of mathematics in the solving of problems based on real-life situations. It assumes that students will be called upon to explain their reasoning in writing, or orally to the teacher, to the class, or to a few other students. An important part of every course in the mathematics program is the process of inquiry, where students develop a systematic method for exploring new problems or unfamiliar situations. The inquiry process is a major strategy that underlies teaching and learning of mathematics at all levels and all grades.

The new curriculum is based on two essential elements: expectations and achievement levels. Expectations are the knowledge and skills students are expected to demonstrate. These expectations are subject, grade and level (academic/applied) specific. The achievement levels describe four possible levels of student achievement. In the area of mathematics these levels focus on four knowledge/skill categories: knowledge/understanding, thinking/inquiry/problem solving, communication and application. Levels 1 and 2 identify achievement that falls below the expectations specified for the grade but are passable levels of achievement. For example, a student is not passing the course would be below Level 1. Level 3 identifies achievement that meets the expectations and is considered the provincial standard. Level 4 achievement is that which surpasses the expectations (Ministry of Education, 1999).

To measure the extent to which the curriculum expectations are being met across the province of Ontario, standardized testing was introduced. This testing is administered by The Education Quality and Accountability Office (E.Q.A.O.) whose mandate is to provide reliable data about student achievement in Ontario’s publicly funded schools. In the most recent Ontario Provincial Report on Achievement, 200-2001: English-Language Secondary Schools, are the results of the Grade 9 Assessment of Mathematics, administered for the first time in the province. The assessment is based on the expectations for the entire Grade 9 curriculum for applied and academic courses.

On this first test, in terms of overall achievement 89% of the students in the province taking mathematics at the academic level achieved at a ‘passable level of achievement’ (Level 1 to 4) and 49% of these students achieved at or above the provincial standard. The results for the students taking mathematics at the applied level are an area of concern. Sixty two percent of the students at the applied level achieved at a ‘passable level of achievement’ and only 13% of all the students at the applied level achieved at or above the provincial standard. Specifically with respect to the category of Thinking/Inquiry/Problem Solving 80% students studying at the academic level achieved a passable level of achievement and 40% achieved at or above the provincial standard. At the applied level of study, 52% achieved at a passable level of achievement and 10% of the students achieved at or above the provincial standard in the area of problem solving. Problem solving at the applied level is a particular area of concern in the province.

The aims of the proposed study are as follows: