Pedagogical and cognitive aspects of problem solving

As modern civilization requires relentless quantification and critical evaluation of information in daily transactions, it becomes necessary to develop newer ways of thinking and reasoning that can be used to learn and do mathematical activities. Through problem solving for instance, we acquire a functional understanding of mathematics needed to cope with the demands of society.

School mathematics of the twenty first century is viewed by educators to be that which should engage a learner in problem solving and reasoning. It should also foster deep understanding and develop the learner’s critical and analytical thinking. Instruction should not be limited to plain mastery of algorithms or the development of certain mathematical skills. It should involve learners in investigation through “exploring, conjecturing, examining and testing” (NCTM, 1990, p.95). It should be tailored to promote reflective thinking among students. A wealth of research on mathematics education and cognitive science in the last decade has dealt with the pedagogical and cognitive aspects of problem solving. Rivera and Nebres (1998) note specifically “the numerous published research studies of Fennema and Carpenter on Cognitively Guided

Instruction (CGI) in the last quarter of this century [which] point to the pernicious effects of status quo ways of thinking about mathematics and problem solving (i.e. existing mathematics culture)”. CGI recognizes the “acculturation of school children to an algorithmic approach to learning basic arithmetical facts” which pervade the current school mathematics culture and which have been proven to be “detrimental to children’s own ways of thinking about problem solving and computations”.

Bishop (1999) adds that “research has shown the importance of the idea of situated cognition which describes the fact that when you learn anything you learn it in a certain situation” . Thus for learning to become meaningful, the learner has to actively participate in the formation of mathematical concepts. She should not passively receive knowledge from an authority but should be involved in the construction of knowledge. The theory of active construction of knowledge influenced many learning theories formulated by staunch contemporary mathematics educators like Von Glasersfeld, Cobb, Bauersfeld, Vygotsky and numerous others (Rivera, 1999). In fact, “problem solving and mathematical investigations based on a constructivist theory of learning, have been the main innovations or revivals for the last decade” according to Southwell (1999, p.331).

Willoughby (1990) believes that the abundant books, pamphlets and courses on critical thinking and problem solving that have been propagated in the 1980s cannot be of help unless certain pedagogical misconceptions are clarified. This includes prescribed rules such as finding key words in a problem to decide the appropriate operations on the values given in the problem, or applying arithmetic algorithm to any word problem. Developing critical and analytical thinking through problem solving takes time and a lot of teacher’s commitment and dedication. (Willoughby, 1990; Barb and Quinn, 1997). Developing critical and analytical thinking involves pedagogical conceptions with a philosophical basis. This paper adheres to the constructivist theory of learning and promotes the belief that problem solving processes rest on basic thinking skills which are best developed within a constructivist framework. Another challenge of the new millenium is the proper use of the ever advancing technology in education. Researchers have to look into the quality of instruction and curriculum which utilize technology. Educational technology should be guided by pedagogical principles that guarantee effective  learning, and not subordinated to technological ends. Thus, “technology should be used to advance educational programs, [and] should not determine programs” (Witt, 1968, p. 145). How to empower students further in learning with the use of technology should be the concern of curriculum designers. In the light of existing literature base on mathematics instruction and flourishing research studies on mathematics teaching and learning, this paper explores issues and finds ways of fostering critical and analytical thinking through problem solving. Then it draws implications regarding the design of a techno mathematics curriculum for algebra at the collegiate level that establishes problem type schema. This design is supported by a philosophical basis of the role of technology in the acquisition of mathematical knowledge. The design is not instrument specific, since it is intended to be adaptable to whatever technology is available to both teachers and students be it in progressive countries or in the third world countries.

Recent research studies on mathematics education have placed its focus on the learners and their processes of learning. They have posited theories on how learners build tools that enable them to deal with problem situations in mathematics. Blais reveals that the philosophical and theoretical view of knowledge and learning embodied in constructivism offers hope that educational processes will be discovered that enable students to acquire deep understanding rather than superficial skills. (Blais, 1988, p.631)

As learners experience their power to construct their own knowledge, they achieve the satisfaction that mathematical expertise brings. They acquire the ability to engage in critical and analytical context of reflective thinking. They develop systematic and accurate thought in any mathematical process. O’Daffer and Thorquist (1993) define critical thinking as “a process of effectively using skills to help one make, evaluate and apply decisions about what to believe or do”(p.40). They cited the observations of Facett(1938) on a student using critical thinking as one who

1. Selects the significant words and phrases in any statement that is important and asks that they be carefully defined;

2. Requires evidence supporting conclusions she is pressed to accept;

3. Analyzes that evidence and distinguishes fact from assumption;

4. Recognizes stated and unstated assumptions essential to the conclusion;

5. Evaluates these assumptions, accepting some and rejecting others;

6. Evaluates the argument, accepting or rejecting the conclusion;

7. Constantly reexamines the assumptions which are behind her beliefs and actions.

Critical thinking abilities can only be developed in a setting which the learner has ample knowledge and experience. Thus, fostering critical thinking in a certain domain entails developing deep and meaningful learning within the domain. Learners can acquire critical thinking strategies by using what cognitive and developmental psychologists call a cognitive schema. Smith, Knudsvig and Walter (1998, p.50) describe a cognitive schema to be “a scheme, method, process by which (one) can see, organize and structure information” for better comprehension and recall. Through the schema learners interpret, analyze, organize and make sense of every information given in a problem situation through a constructive process called reflective abstraction.

Through reflective abstraction, critical thinkers are able to assimilate information into their mathematical network and build from their prior knowledge. They can accommodate new ideas including those that conflict with what they know or believe and negotiate these ideas. They are willing to adjust their belief systems after reexamining information. They are also able to generate new ideas based on novel ideas that are available to them. They are expert problem solvers who can handle abstract problem information and make sense of different problem situations.

On the other hand, novice problem solvers are not able to handle abstract mathematical concepts. They have difficulty recognizing underlying abstract structures and often need to make detailed comparisons between current and earlier problems before they can recognize the abstract information in the solution of the current problem ( Reed ,1987; Reed, Dempster, Ettinger, 1985; Anderson, 1984; Ross, 1987, as cited by Bernardo, 1994). They usually resort to algorithmic activity and not to the perception of essence. Blais (1988) observed that “they resist learning anything that is not part of the algorithms they depend on for success”(p.627). They tend to be very shallow in dealing with problem situations because of the lack of depth in their experiences while engaging in mathematical activities.

All problem solvers, whether experts or novices, develop a cognitive schema which cognitive scientists call problem-type schemata when confronted with a mathematical problem. According to Bernardo (1994), “[k]nowledge about the problem categories include information about the relevant underlying principles, concepts, relations, procedures, rules, operations and so on”(p.379). Further, he adds, “problem-type schemata are acquired through some inductive or generalization process involving comparisons among similar or analogous problems of one type”(p.379). Learners represent, categorize and associate problems to be able to determine the appropriate solution. The expert’s schematic processing leads to an accurate analysis of the problem which the novice hardly achieves. Bernardo (1994) claims that “the novices’ schemata (expectedly) include[s] mainly typical surface-level information associated with a problem type, whereas experts’ schemata include[s] mainly statements of abstract principles that [are] relevant to the problem type”(p.380). One example of the difference in the processing of experts and novices given by Blais (1988) is on their reading process of a mathematical material. Blais (1988) observes that, [w]hen novices read, the process almost always appears to be directed toward the acquisition of specific information that will be needed for algorithmic activity, (whereas) the reading process used by experts is directed toward the perception of essence. (p.624)

Experts seem to readily categorize the mathematical information in the material being read, thus facilitating the processing of information that lead to the correct solution. They are able to attain some sort of a visual form of say an algebraic expression and are able to communicate this before they perform the algorithmic activity. Besides, they can determine errors and attain a deep understanding of the underlying structure of the mathematical concept. Experts rely not only on concepts and procedures when confronted with a mathematical problem. They also have access to metacognition which is the knowledge used by experts in “planning, monitoring, controlling, selecting and evaluating cognitive activities” (Wong, 1989, Herrington, 1990, English, 1992 as cited by English-Halford, 1992; Bernardo, 1997). With this higher order thinking skill, problem solvers are assured of the success of every mathematical strategy they employ.

It is therefore the goal of education to help novices gain expertise in mathematical activities such as problem solving. In the next section, we deal with a few different views of studies conducted on didactics of problem solving.

Smith, Knudsvig and Walter (1998) advocate a cognitive schema which learners can use to acquire critical thinking strategies. They call it the TCDR for TOPIC-CLASS-DESCRIPTION-RELEVANCE. Thus, when given a learning material, students should ask the following questions:

· What TOPIC I must understand?

· What overall CLASS does this topic belong?

· What is the DESCRIPTION of the topic?

· What is the RELEVANCE of the topic?

These questions help learners interpret, analyze, organize and make sense of the information that are given in the material for better processing of learning. Once this becomes the framework of the learners, they gain strength and clarity of thinking. Several schemes have been offered by mathematics educators for solving word problems. The most versatile and widely used scheme for problem solving is the one formulated by George Polya (1957). These include working simpler problems, restating a problem, decomposing or recombining a problem, drawing figures, making charts or organized lists, exploring related problems, using logical deduction, using successive approximations, using guess-and-check methods, and working backwards. 

   

                      

Polya (1957) also developed a framework for problem solving in terms of such general phases as “understanding the problem, divising a plan, carrying out the plan and looking back” (cited by Barb and Quinn, 1997, p. 537). If carried out effectively, then the problem solver becomes successful in handling a problem situation. But the process involved in traversing these steps is quite complex. The learner has to use her prior knowledge, apply acquired mathematical skills, understand the context of the problem situation, and choose the appropriate strategy in solving the problem. This requires formal abstraction, a higher order thinking skill that is available to experts alone. What, then, can be done to help novices gain intellectual power?

By their success in working with simpler problems, novices gain confidence and are motivated to work with more difficult ones. Their analogical thinking can be best harnessed by using very concrete prior experiences. They are able to build their mathematical ideas from simple tasks and are able to acquire mathematical skills. Bernardo (1997) emphasizes the importance of the use of context problems that are familiar to the students which “provides students with a concrete (possibly, real) grounding on the problem, and which allows students to more easily draw from their existing knowledge about similar situations”(p. 11). Hopefully, students become more involved in the difficult task of making learning meaningful.

Mathematics educators recommend the use of mental models to guide learning. These mental models (aids) come in the form of diagrams or drawings used to represent the structure of the concept. The development of strategies and mental modeling fall under the theory of analogies. The effectivity of the analogy lies in a learner’s ability to recognize the “correspondence between the structure of the aid and the structure of the concept to be understood” (English-Halford, 1992, p. 121). In this case, the learner is able to map the essence of the model into the essence of the concept, and match or transfer specific conceptual aspects of one domain into another. This cognitive process promotes reflective abstraction. It is unfortunate, though, that certain popular pedagogical practices are counterproductive.

In the process of streamlining the problem solving task, teachers are sometimes tempted to use artificial and fabricated ways of building skills which Blais (1988) refers to as remedial processing. One good example is the prescription of finding key words in a problem which may work for experts, but not necessarily for novices. Some novices use these key words to decide on the algorithm to apply, with complete disregard of the essence of the problem. Key words prompt novices to add when they see the word increase, or subtract when they see the word decrease in a problem. Worse, some apply an arithmetic operation on any two numbers that they see depending upon the key words that they find in the problem. In fact, even their use of formal symbolic expressions in the solutions of the problems may not even communicate the essence of the given problem.

Blais (1988) laments that “[c]onventional instruction permits, allows, and sometimes blatantly encourages algorithmic activity that is separate and isolated from the perception of essence”(p. 627). This may be due to the focus of instruction on the product and not the process of the mathematical activity. In  fact, explanations sometimes send the wrong signal that problem solving processes are neat, well organized and easy as the teacher’s presentations on the board. Consequently, novices are tempted to resort to rote memorization of the algorithms, rules and formulas presented by the teacher. They do not realize that proficiency in problem solving is best achieved in recognizing the essence of a given problem and the application of the proper problem solving heuristics. Understanding the structural relations in a mathematical problem ushers the learners to reflective abstraction and gives them a sense of direction and feeling of certainty.

Barb and Quinn (1997) advocate the use of multiple methods of problem solving including such intuitively based methods as the guess-and-check method approximation. Problem solvers can use arithmetic computation with figures and charts and logical reasoning, and not necessarily algebraic equations in finding solutions. They believe that this strategy is more meaningful to a learner who is beginning to use some form of reflective abstraction, than rote application of algorithms usually found in textbooks. Teachers who usually look for algebraic solutions should be convinced of the value of developing the students’ problem-solving skills and refining their strategies using intuition and logic. It should be noted that the ultimate goal of this instructional method is to help learners build a good knowledge base in solving word problems so they can achieve reflective abstraction in the process. This belief was expressed by Owen and Sweller (1989) when they challenged the emphasis placed on problem solving and heuristics in the 1980s and pointed out that “superior problem solving performance does not derive from superior heuristics but from domain specific skills” (cited by Puut and Isaacs, 1992, p.215). They claim that the use of general cognitive strategies such as the means-end strategy impose heavy cognitive load and hamper schema acquisition and rule automation. It is because “a means - end tactic involves comparing the initial conditions of a task against the goal set for that task, then searching for a tactic that will transform either the goal or the initial conditions to be a bit more like one another” (Wine & Stockley, 1998, p. 124). This becomes very difficult especially when solving multistep problems. The solver has to analyze and break down the problem to subgoals, successfully transform each initial condition and subgoal into the desired condition, repeat the tactic until the final goal of the problem is achieved. The learner has to see the overall structure of information, concepts, operations, rules, and all other elements  that make up the whole schema of the problem. It is preferred that problems be freed of a single goal.

When the problem becomes goal-free, solvers are able to work forward from givens of the problem that they are able to generate. According to Wine & Stockley, “each iteration is a self-contained step that uses whichever problem-solving technique is easiest for the student, [in which case] the drain on working memory’s resources is minimized”(1998, p.125). In fact, Sweller (1989) claims that “research shows that freeing problems of singular goals can help students acquire schemas for solving problems “(cited by Wine & Stockley, p. 125). The development of domain-specific skills of learners may facilitate the development of schemas that underlie genuine understanding and meaningful learning.

Another issue that is worth considering is the question of when students should engage in word problems. Word problems are usually treated as application problems since they are given after certain mathematical concepts are introduced, with the aim of using the concepts in solving the problems. On the other hand, word problems may be taught in context, i.e. they may be used to teach a mathematical idea or process. According to Laughbaum (1999) “[t]eaching in context also uses problems or situations, but they are used at the beginning of a math topic for the purpose of helping students understand the mathematics to be taught, or to create a motivating experience of the mathematics to follow” (p.1). Certain groups looked into the effects of application problems to the development of the skills of the learners. One such group called the Cognition and Technology Group of Vanderbilt (CTGV) identified the shortcomings of the application problems and came up with efficient ways of teaching word problems in context. The CTGV has these to say about application problems:

1. Instead of bringing real world standards to the work, students seem to treat word problems mechanically and often fail to think about constraints imposed by real-world experiences.

2. Single correct answers to application problems lead to misconceptions about the nature of problem solving and inadvertently teaches students for a single answer rather than seek multiple answers.

3. The goal of one’s search for a solution is to retrieve previously presented information rather than rely on one’s own intuition. This may limit the development of people’s abilities to think for themselves.

4. They explicitly define the problems to be solved rather than help students to learn to generate and pose their own problems. Mathematical thinkers tend to generate their own problems.

5. The use of application problems lead to inert knowledge. Inert knowledge is that which is accessed only in a restricted set of contexts even though it is applicable to a wide variety of domain. (1997, p. 40)

These application problems are traditionally presented using general problem solving strategies which Polya prescribed or the means-end strategy. While some educators and researchers express the above mentioned concerns, many mathematics educators still adhere to the conventional practices of teaching problem solving. Lawson (1990), in defense of conventional methods, explained that when done properly, “general problem solving strategies play an important role in learning and transfer” (cited by English-Halford, 1992, p. 120). He described the three different types of general problem-solving strategies to include:

Task orientation strategies (which) influence the dispositional state of the student and include the broad affective, attitudinal, and attributional expectations held by the student about a particular task. Executive strategies are concerned with the planning and monitoring of cognitive activity, while domain-specific strategies include heuristics such as means-ends analysis and other procedures developed by the problem solver for organizing and transforming knowledge (e.g., constructing a table or drawing a diagram). (p. 120).

Lawson insisted that these strategies “have a general sphere of influence on cognitive activity during problem solving and should be seen as distinct from strategies specific to a particular task” (p. 404, 1990, cited by English-Halford, 1992, p.120).

Bernardo (1997) recommends the use of variable problem contexts to promote abstraction. He claims that “[b]y presenting concepts in variable problem contexts, students will come to appreciate the meaning and use of a particular concept or procedure in a variety of contexts”(p.12). Problem solvers cannot possibly recognize problem structure of single problems, thus the need for use of a wide range of diverse problems to facilitate the abstraction of specific concepts and transfer of knowledge to various problem contexts. He believes that a “deeper engagement of the problem information should lead to better conceptual understanding of the problem, and hopefully, result to higher level of abstract thinking about the problems”(p. 13). He proposes teaching strategies that promote analogical transfer. It should be noted that “many theorists argue that specific experiences are represented in memory as cases that are indexed and searched so that they can be applied analogically to new problems that occur”(Kolodner, 1991, Riesbeck and Schank, 1989; Schank, 1990 cited by CTGV, 1997, p.37). It is therefore the task of mathematics educators to determine ways of facilitating analogical transfer among learners.

One such instructional strategy that promotes analogical transfer involves presenting students with a context problem and then asking them to make their own problem using a different context. The effectiveness of this strategy according to Bernardo seems to be due to the deeper level of understanding of the problem structure achieved by the problem solver…[as she] explores the problem structure while attempting to create an analog, …[and] as a result of correctly mapping the problem structural information to create a true analog of the original problem. (Bernardo, 1998, p. 7)

Through this problem posing strategy the learners are able to recognize the essence of a problem and construct similar problems with the same essence. Mathematical problem posing, according to Silver (1994, cited by Ban-Har and Kaur, 1999) “is the generation of new problems or the re-formulating of existing ones ”(p. 77). It is recognized as “a valuable process that is motivating, challenging and allows students to exercise their creativity and independent learning skills” (Southwell, 1999; Silver, 1994, Kilpatrick, 1987 as cited by Ban-Har and Kaur, 1999). There are variety of ways to pose problems as a mathematical activity. These include writing questions based on given set of facts, on a given calculation, or on certain information. The benefits of the activity are the same whichever form is used. While results of recent studies give no clear correlation between quality of problem posing responses and problem solving ability (Ban-Har and Kaur, 1999), there are indications that, when performed in the context of analogical problem construction, analogical transfer is facilitated (Bernardo, 1998, p. 7).

There are other ways of facilitating recognition of problem structures, one of which is the use of text editing skills. In this activity, problem solvers are asked to identify missing information from problems or point out information that are irrelevant to the problems. Low and Over (1989) showed the significantly high correlation between students’ ability to edit the text of algebraic story problems and their ability to solve these problems; as well as between students’ ability to edit the text and categorize problems as being similar or different from each other (cited by Putt and Isaacs, 1992, p. 215). This activity enhances the problem solvers’ awareness of their own thinking processes. Such awareness helps learners identify their points of strengths and weaknesses and regulate their own ways of knowing. Garofalo and Lester (1985) claimed that “most problem solvers do not develop the appropriate metacognitive knowledge that should accompany the execution of computational procedures for doing  problems”(cited by Bernardo, 1997, p. 8). Wong (1989) and Herrington(1990) showed otherwise in their studies (cited by English-Halford, 1992). According to Wong (1989), “most students indicated that they were conscious of metacognitive processes and used strategies for monitoring and regulating the processes necessary for problem solving” (cited by English-Halford, 1992, p.118). Herrington (1990) also observed that “upper primary school children had well formed views on the process of learning mathematics and were able to confidently express them”(cited by English-Halford, 1992, p.119). Inspite of these varying opinions, Wong and Bernardo both agree about the need to use guided instruction in the use of metacognitive strategies for problem solving especially among lower ability students. Bernardo (1997) echoed Schoenfeld’s suggestion (1987, cited by Bernardo, 1997) that teachers model the metacognitive processes in problem solving when they present solutions to their students. A teacher thinks aloud and exhibits the process of planning, organizing, analyzing and carrying out the solution. The teacher articulates questions, makes mistakes, traces and corrects mistakes, deals with incorrect approach, backtracks, evaluates her progress, and struggles to arrive at the correct solution. This teaching strategy demonstrates the complexity of the process involved in solving problems and the reality that there are many possible ways of arriving at the correct answer.

In the light of all the issues and conflicts on various aspects of problem solving, particularly on developing cognitive strategies among students, and with the assumption that teachers hold wholesome beliefs and attitudes towards mathematics teaching, this paper attempts to offer suggestions on effective ways of fostering critical and analytical thinking through problem solving at different school levels.

At this point, we all agree that an expert problem solver is a critical and analytical thinker. When a learner gains expertise, she has acquired all the qualities of strong and smart thinking. She becomes insightful, and logical. The expert is also a constructive learner. She participates actively in the learning process and is able to build from her prior knowledge while assimilating and accommodating new knowledge. She appreciates the variety of ways of solving mathematical problems and recognizes a good solution. She is not afraid to use intuition and logic in her solutions. She makes good models of the problems and recognizes the essence and structure of a given problem. She employs a cognitive schema that helps her organize and plan her strategies. Her metacognitive skills help her monitor and evaluate her progress.

Expertise can be attained at an early age. Blais (1988) cites indicators of a schooler’s expertise once a teacher expresses doubt in her work. According to Blais, [I]f the child does not erase, if she or he refuses to accept the hint from an outside authority and tries to ponder whether the answer is correct, that student is an expert. Being willing and able to think and act independently, she or he will decide what is sensible and reasonable based on informal concepts already acquired (Mills, 1859). A child accustomed to accepting rules and procedures on faith has subordinated his or her own reasoning to outside authority and would have

yielded to it once again; the child would have erased. (Blais, 1988, p. 626) 

This suggests that teachers should allow their students to experience the joy of working independently by simply guiding and facilitating their learning and by not doing all the thinking and solving for them.