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Center for Teaching and Learning •University of North Carolina at Chapel Hill CTL Number 20 June 1998 Teaching ProblemSolving Skills Based on a workshop developed by Robert Keller (Mathematics), Graduate Teaching Consultant, Center for Teaching and Learning, and Thomas Concannon (Physics & Astronomy), Graduate Teaching Assistant Liaison to the Center. The ability to solve problems is a basic life skill and is essential to understanding technical subjects. Problemsolving is a subset of critical thinking and employs the same strategies. Although the line between the two is fuzzy, in general, the goal of problemsolving is to adduce correct solutions to wellstructured problems, whereas the goal of critical thinking is to construct and defend reasonable solutions to illstructured problems. Basically, problemsolving is the process of reasoning to solutions using more than simple application of previously learned procedures. There are several reasons that college students often fail to reach a satisfactory level of proficiency in problemsolving. They frequently suffer from fears and anxieties, especially fear of failure, that hamper their efforts to solve problems. Particular learning styles may make it harder to learn to solve problems. Also, general thinking patterns may inhibit students’ problemsolving ability. Below, we examine these problems and explore several strategies that students and instructors can use to address them. Barriers to Problem Solving Even if you are a practiced problemsolver, holding your emotions in check so that you can think clearly may be difficult at times. For many of our students, however, falling prey to these anxieties is the rule rather than the exception. Many of them seriously question their ability to problemsolve effectively, and in fact may avoid courses that involve problemsolving altogether because of these concerns. Overcoming these fears and anxieties is the first step in learning to problemsolve effectively, but a step that instructors frequently ignore. Student learning styles also affect how students learn to solve problems. Some people learn primarily visually, others aurally; some learn stepbystep, others employ an allornothing process; some cogitate on a problem introspectively, while others find they work problems best when they can discuss them. There is also evidence that some thinking styles that affect the ability to solve problems are genderlinked (Kimura, 1992). For example, a marked discrepancy exists between males and females in visualizing the structure of chemical molecules because males are better able to manipulate 3dimensional objects in space. However, females organize and relate data more efficiently than males. In addition to the emotional and psychological issues outlined above, there are numerous cognitive barriers to mastering problemsolving. The primary difficulty for many students is the inability to identify and use concepts and procedures in analogous but novel situations. The lack of transfer of structure between problems is a significant cognitive difficulty, not only for inexperienced problemsolvers but also for experts. Successful transfer rests on the ability to recognize analogies, but even when given an analogy, students often fail to see how to employ it. In order to understand this phenomenon more concretely, consider the following problem: A patient has a cancerous tumor. Beams of radiation will destroy the tumor, but in high doses will also destroy healthy tissue surrounding the tumor. How can you use radiation to safely eradicate the tumor? This structure of this problem follows the general pattern common to all problems. It has a set of facts (tumor, radiation, tissue) and unknowns (ways to administer radiation), together with relationships between them (radiation destroys tumor and tissue). Gick and Holyoak (1983) gave volunteers the story below and then asked them to solve the tumor problem. A fortress surrounded by a moat is connected to land by numerous narrow bridges. An attacking army successfully captures the fortress by sending only a few soldiers across each bridge, converging upon it simultaneously. The story and the problem have exactly the same logical structure, but only a small percentage of subjects were able to solve the tumor problem after being told the story. The solution is to bombard the tumor from different directions with lowintensity radiation so as not to harm healthy tissue. The convergence of the beams at the tumor provides sufficient intensity to destroy it. Only when the subjects were overtly prompted to use the story as an analogy to help them solve the problem were most of them able to solve it. The inability to transfer in the absence of prompting may be one of the greatest hurdles for student and instructor. A lack of transfer skills is frequently marked by functional fixedness, the perception that a particular object or concept has only one use. For example, in the tumor problem students might interpret the word “dose” as implying oral medication. Or they may believe, erroneously, that the word “beam” implies that there can be only one direction from which radiation can be applied. Since successful transfer may require seeing a familiar concept or procedure in a new way, functional fixedness handicaps the transfer process. Another handicap for students is superficial transference, where students identify and link words or variables between problems instead of linking deeper, more meaningful structures. For example, physics experts represent problems in terms of the laws or principles needed to solve them, e.g., energy equations or Newton’s laws of motion. Novices, on the other hand, categorize problems on the basis of superficial features such as whether they involve pulleys, inclined planes or other objects (Kurfiss, 1988). Teaching and Learning Strategies that Enhance ProblemSolving Skills There are two types of strategies that can overcome difficulties in problemsolving: pedagogical strategies, which are teachercentered methods, and methodological strategies, which tend to be learnercentered. Pedagogical Strategies Some pedagogical strategies allow the teacher to address the emotional, psychological, and cognitive barriers to problemsolving simultaneously. For example, on the first day of class, an instructor could conduct an open discussion about the nature of the course material, encouraging students to voice their fears and concerns about it. This approach helps create a comfortable learning environment—a classroom in which students are encouraged to question and take risks without penalties. Continuing this kind of open dialogue in the classroom throughout the semester will strengthen rapport between teacher and student and provide many opportunities for students to discuss different ideas and approaches to solving problems. Class discussion also reinforces success and transfer of learned skills. Studies suggest that active involvement is critical in developing problemsolving skills, so using student learning groups to promote active experimentation with problems is a sound pedagogical strategy. Other effective strategies include accepting multiple attempts of solutions for an assignment, assigning personal journals in which students describe their problemsolving strategies, and allowing students to rework homework and exams for credit. These strategies both dissipate anxiety because they reduce the sole emphasis on “getting the right answer” and encourage reflection on the problemsolving process. Different learning styles as well as genderspecific differences in thinking can be addressed by employing a variety of activities and approaches in teaching. The traditional instructional mode of lecturing and explaining is effective for only one learning style. To address other learning styles, you might use graphics to illustrate concepts, provide opportunities for practice in class, ask for student interpretations of data, and require students to work on problems in groups. Making students aware of their learning styles and preferences can also be helpful. The following Web site, developed by Prof. Richard Felder of North Carolina State University, provides an explanation as well as a selftest for learning styles: http://www2.ncsu.edu/unity/lockers/users/f/ felder/public/ILSpage.html Methodological Strategies Methodological strategies provide a series of steps to assist students in addressing and solving a new problem, and work handinhand with the pedagogical techniques discussed above. There are two basic types: algorithmic and heuristic methods. An algorithmic procedure is a “stepbystep prescription for achieving a goal” (Woolfolk, 1993). The mnemonic PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) is an algorithm that math students use to remember the order of operations used in simplifying algebraic expressions. Students appreciate algorithms because they are easily applied. However, students may “algorithmize” methods they have observed others using and bring them to bear in a given situation whether applicable or not. Algorithmic methods are limited to lowlevel tasks and tend to be domainspecific. Heuristic methods, general schemes used to derive solutions to problems, are more useful than algorithms. There are a variety of heuristics that can be useful to students. Bransford and Stein (1984) use the acronym IDEAL to represent the five steps usually contained in many solution strategies. Identify the problem. Define and represent the problem. Explore possible solution strategies. Act on the strategies. Look back and evaluate. This scheme is beneficial in a large number of disciplines. Students like the IDEAL heuristic because it is easy to remember and widely applicable. For example, in a composition class, students might follow these steps in developing a response against the argument that Twain’s Adventures of Huckleberry Finn should be removed from libraries because it is a racist book. Of course the problem is that banning this book would essentially mean removing a great historical and cultural artifact from circulation. In defining the problem, a student would need to determine which passages are offensive to particular audiences. One possible solution strategy might include claiming that these passages can be used to stimulate discussion on racism, its role in late 19th century America, and how our culture has changed over the years. The student might then compose a persuasive response based on these ideas. Using their prepared responses students could also debate the two positions in order to evaluate their persuasiveness. Promoting Transfer Other strategies assist students in transferring problemsolving techniques from one problem to very similar or analogous problems. For successful transfer to occur, it is essential for students to identify the central theme that is common to a set of problems so they can readily recognize and apply it in more abstract settings. Through the conscious use of analogy, students can explore situations which are similar, transferring structure to the problem at hand. A powerful aspect of this technique is that it appeals to past experience and common sense. For example, the problem of eradicating the tumor may appear to be too specialized to some students, but the analogous problem of the fortress and its attacking army may seem more familiar and therefore easier to understand. In fact, in the Gick and Holyoak study only 10% of the volunteers were able to solve the tumor problem without being given the analogy. After hearing the fortress story, 30% could solve the problem, and if prompted to relate the two, this number jumped to 75%. This outcome suggests that instructors need to use a wide variety of analogies in their repertoire of examples. Also, instructors should encourage students to develop appropriate analogies of their own. They also need to show explicitly, perhaps through diagrams, how elements of one problem map onto elements of an analogous problem, and discuss the underlying relationships between them. Dialogue can also be useful in promoting transfer by highlighting the differences between the problemsolving techniques used by experts and novices. In order to solve a problem, both experts and novices follow the same pattern: they read and analyze, plan a strategy, act on that strategy to produce a solution, and then try to verify it. But experts work harder on the initial stage than do inexperienced problemsolvers, and inappropriate or superficial transfer frequently characterizes the novice. Getting students to talk through the differences between problems that have similar superficial structures but different deep structures decreases the risk of incorrect transfer. Center for Teaching and Learning CB# 3470, 316 Wilson Library Chapel Hill, NC 275993470 9199661289 5,000 copies of this document were printed at a cost of $453.10, or $.09 per copy. References Gage, N.L. and Berliner, D.C., Educational Psychology, 5th ed., Houghton Mifflin Co., 1992. Gick S. and Holyoak G., “Schema Induction and Analogical Transfer,” Cognitive Psychology, 15, 1 38, 1983. Kimura, D., “The Mind and the Brain,” Scientific American, September 1992. Kurfiss J., Critical Thinking: Theory, Research, Practice, and Possibilities, ASHEERIC Higher Education Reports #2, 1988. Wason, P., “Realism and Rationality in the Selection Task,” Thinking and Reasoning: Psychological Approaches, Evans, J. ed., Routledge and Kegan Paul, 1983. Woolfolk, A.E., Educational Psychology, 5th ed., Allyn and Bacon, 1993. Having students work on numerous problems individually and in groups also facilitates transfer. The traditional method of giving two examples in class and assigning twenty exercises for homework fails to give students a sufficient base from which to work. Once students have mastered problems of a particular type, they can begin to tackle problems of a more general nature. Choosing problems which evolve from simple and welldefined to complex and illdefined will help them develop transfer skills. Using realworld data in sample problems will also help facilitate the transfer process, since students can more easily identify with the context of a given situation. Using these strategies, students will learn the relevance of course material to daily life and will begin to transfer concepts between disciplines, moving toward a more cohesive understanding of the real world. Summary To develop better problemsolvers, instructors must help students overcome both emotional and cognitive barriers to learning effective problemsolving skills. By first creating a comfortable classroom environment and helping students overcome their fears and anxieties related to problemsolving, teachers lay the necessary foundation for successful learning. Then using an array of pedagogical and methodological strategies, instructors can promote student reflection on the problemsolving process itself and provide them critical tools for and practice in productive problemsolving. As a result students will become increasingly effective problem solvers, able to solve more and more complex problems with greater and greater independence.
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Title  For your consideration : suggestions and reflections on teaching and learning 
Date  199806 
Description  no. 20 (June 1998) 
Digital CharacteristicsA  288 KB; 4 p. 
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Full Text  Center for Teaching and Learning •University of North Carolina at Chapel Hill CTL Number 20 June 1998 Teaching ProblemSolving Skills Based on a workshop developed by Robert Keller (Mathematics), Graduate Teaching Consultant, Center for Teaching and Learning, and Thomas Concannon (Physics & Astronomy), Graduate Teaching Assistant Liaison to the Center. The ability to solve problems is a basic life skill and is essential to understanding technical subjects. Problemsolving is a subset of critical thinking and employs the same strategies. Although the line between the two is fuzzy, in general, the goal of problemsolving is to adduce correct solutions to wellstructured problems, whereas the goal of critical thinking is to construct and defend reasonable solutions to illstructured problems. Basically, problemsolving is the process of reasoning to solutions using more than simple application of previously learned procedures. There are several reasons that college students often fail to reach a satisfactory level of proficiency in problemsolving. They frequently suffer from fears and anxieties, especially fear of failure, that hamper their efforts to solve problems. Particular learning styles may make it harder to learn to solve problems. Also, general thinking patterns may inhibit students’ problemsolving ability. Below, we examine these problems and explore several strategies that students and instructors can use to address them. Barriers to Problem Solving Even if you are a practiced problemsolver, holding your emotions in check so that you can think clearly may be difficult at times. For many of our students, however, falling prey to these anxieties is the rule rather than the exception. Many of them seriously question their ability to problemsolve effectively, and in fact may avoid courses that involve problemsolving altogether because of these concerns. Overcoming these fears and anxieties is the first step in learning to problemsolve effectively, but a step that instructors frequently ignore. Student learning styles also affect how students learn to solve problems. Some people learn primarily visually, others aurally; some learn stepbystep, others employ an allornothing process; some cogitate on a problem introspectively, while others find they work problems best when they can discuss them. There is also evidence that some thinking styles that affect the ability to solve problems are genderlinked (Kimura, 1992). For example, a marked discrepancy exists between males and females in visualizing the structure of chemical molecules because males are better able to manipulate 3dimensional objects in space. However, females organize and relate data more efficiently than males. In addition to the emotional and psychological issues outlined above, there are numerous cognitive barriers to mastering problemsolving. The primary difficulty for many students is the inability to identify and use concepts and procedures in analogous but novel situations. The lack of transfer of structure between problems is a significant cognitive difficulty, not only for inexperienced problemsolvers but also for experts. Successful transfer rests on the ability to recognize analogies, but even when given an analogy, students often fail to see how to employ it. In order to understand this phenomenon more concretely, consider the following problem: A patient has a cancerous tumor. Beams of radiation will destroy the tumor, but in high doses will also destroy healthy tissue surrounding the tumor. How can you use radiation to safely eradicate the tumor? This structure of this problem follows the general pattern common to all problems. It has a set of facts (tumor, radiation, tissue) and unknowns (ways to administer radiation), together with relationships between them (radiation destroys tumor and tissue). Gick and Holyoak (1983) gave volunteers the story below and then asked them to solve the tumor problem. A fortress surrounded by a moat is connected to land by numerous narrow bridges. An attacking army successfully captures the fortress by sending only a few soldiers across each bridge, converging upon it simultaneously. The story and the problem have exactly the same logical structure, but only a small percentage of subjects were able to solve the tumor problem after being told the story. The solution is to bombard the tumor from different directions with lowintensity radiation so as not to harm healthy tissue. The convergence of the beams at the tumor provides sufficient intensity to destroy it. Only when the subjects were overtly prompted to use the story as an analogy to help them solve the problem were most of them able to solve it. The inability to transfer in the absence of prompting may be one of the greatest hurdles for student and instructor. A lack of transfer skills is frequently marked by functional fixedness, the perception that a particular object or concept has only one use. For example, in the tumor problem students might interpret the word “dose” as implying oral medication. Or they may believe, erroneously, that the word “beam” implies that there can be only one direction from which radiation can be applied. Since successful transfer may require seeing a familiar concept or procedure in a new way, functional fixedness handicaps the transfer process. Another handicap for students is superficial transference, where students identify and link words or variables between problems instead of linking deeper, more meaningful structures. For example, physics experts represent problems in terms of the laws or principles needed to solve them, e.g., energy equations or Newton’s laws of motion. Novices, on the other hand, categorize problems on the basis of superficial features such as whether they involve pulleys, inclined planes or other objects (Kurfiss, 1988). Teaching and Learning Strategies that Enhance ProblemSolving Skills There are two types of strategies that can overcome difficulties in problemsolving: pedagogical strategies, which are teachercentered methods, and methodological strategies, which tend to be learnercentered. Pedagogical Strategies Some pedagogical strategies allow the teacher to address the emotional, psychological, and cognitive barriers to problemsolving simultaneously. For example, on the first day of class, an instructor could conduct an open discussion about the nature of the course material, encouraging students to voice their fears and concerns about it. This approach helps create a comfortable learning environment—a classroom in which students are encouraged to question and take risks without penalties. Continuing this kind of open dialogue in the classroom throughout the semester will strengthen rapport between teacher and student and provide many opportunities for students to discuss different ideas and approaches to solving problems. Class discussion also reinforces success and transfer of learned skills. Studies suggest that active involvement is critical in developing problemsolving skills, so using student learning groups to promote active experimentation with problems is a sound pedagogical strategy. Other effective strategies include accepting multiple attempts of solutions for an assignment, assigning personal journals in which students describe their problemsolving strategies, and allowing students to rework homework and exams for credit. These strategies both dissipate anxiety because they reduce the sole emphasis on “getting the right answer” and encourage reflection on the problemsolving process. Different learning styles as well as genderspecific differences in thinking can be addressed by employing a variety of activities and approaches in teaching. The traditional instructional mode of lecturing and explaining is effective for only one learning style. To address other learning styles, you might use graphics to illustrate concepts, provide opportunities for practice in class, ask for student interpretations of data, and require students to work on problems in groups. Making students aware of their learning styles and preferences can also be helpful. The following Web site, developed by Prof. Richard Felder of North Carolina State University, provides an explanation as well as a selftest for learning styles: http://www2.ncsu.edu/unity/lockers/users/f/ felder/public/ILSpage.html Methodological Strategies Methodological strategies provide a series of steps to assist students in addressing and solving a new problem, and work handinhand with the pedagogical techniques discussed above. There are two basic types: algorithmic and heuristic methods. An algorithmic procedure is a “stepbystep prescription for achieving a goal” (Woolfolk, 1993). The mnemonic PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) is an algorithm that math students use to remember the order of operations used in simplifying algebraic expressions. Students appreciate algorithms because they are easily applied. However, students may “algorithmize” methods they have observed others using and bring them to bear in a given situation whether applicable or not. Algorithmic methods are limited to lowlevel tasks and tend to be domainspecific. Heuristic methods, general schemes used to derive solutions to problems, are more useful than algorithms. There are a variety of heuristics that can be useful to students. Bransford and Stein (1984) use the acronym IDEAL to represent the five steps usually contained in many solution strategies. Identify the problem. Define and represent the problem. Explore possible solution strategies. Act on the strategies. Look back and evaluate. This scheme is beneficial in a large number of disciplines. Students like the IDEAL heuristic because it is easy to remember and widely applicable. For example, in a composition class, students might follow these steps in developing a response against the argument that Twain’s Adventures of Huckleberry Finn should be removed from libraries because it is a racist book. Of course the problem is that banning this book would essentially mean removing a great historical and cultural artifact from circulation. In defining the problem, a student would need to determine which passages are offensive to particular audiences. One possible solution strategy might include claiming that these passages can be used to stimulate discussion on racism, its role in late 19th century America, and how our culture has changed over the years. The student might then compose a persuasive response based on these ideas. Using their prepared responses students could also debate the two positions in order to evaluate their persuasiveness. Promoting Transfer Other strategies assist students in transferring problemsolving techniques from one problem to very similar or analogous problems. For successful transfer to occur, it is essential for students to identify the central theme that is common to a set of problems so they can readily recognize and apply it in more abstract settings. Through the conscious use of analogy, students can explore situations which are similar, transferring structure to the problem at hand. A powerful aspect of this technique is that it appeals to past experience and common sense. For example, the problem of eradicating the tumor may appear to be too specialized to some students, but the analogous problem of the fortress and its attacking army may seem more familiar and therefore easier to understand. In fact, in the Gick and Holyoak study only 10% of the volunteers were able to solve the tumor problem without being given the analogy. After hearing the fortress story, 30% could solve the problem, and if prompted to relate the two, this number jumped to 75%. This outcome suggests that instructors need to use a wide variety of analogies in their repertoire of examples. Also, instructors should encourage students to develop appropriate analogies of their own. They also need to show explicitly, perhaps through diagrams, how elements of one problem map onto elements of an analogous problem, and discuss the underlying relationships between them. Dialogue can also be useful in promoting transfer by highlighting the differences between the problemsolving techniques used by experts and novices. In order to solve a problem, both experts and novices follow the same pattern: they read and analyze, plan a strategy, act on that strategy to produce a solution, and then try to verify it. But experts work harder on the initial stage than do inexperienced problemsolvers, and inappropriate or superficial transfer frequently characterizes the novice. Getting students to talk through the differences between problems that have similar superficial structures but different deep structures decreases the risk of incorrect transfer. Center for Teaching and Learning CB# 3470, 316 Wilson Library Chapel Hill, NC 275993470 9199661289 5,000 copies of this document were printed at a cost of $453.10, or $.09 per copy. References Gage, N.L. and Berliner, D.C., Educational Psychology, 5th ed., Houghton Mifflin Co., 1992. Gick S. and Holyoak G., “Schema Induction and Analogical Transfer,” Cognitive Psychology, 15, 1 38, 1983. Kimura, D., “The Mind and the Brain,” Scientific American, September 1992. Kurfiss J., Critical Thinking: Theory, Research, Practice, and Possibilities, ASHEERIC Higher Education Reports #2, 1988. Wason, P., “Realism and Rationality in the Selection Task,” Thinking and Reasoning: Psychological Approaches, Evans, J. ed., Routledge and Kegan Paul, 1983. Woolfolk, A.E., Educational Psychology, 5th ed., Allyn and Bacon, 1993. Having students work on numerous problems individually and in groups also facilitates transfer. The traditional method of giving two examples in class and assigning twenty exercises for homework fails to give students a sufficient base from which to work. Once students have mastered problems of a particular type, they can begin to tackle problems of a more general nature. Choosing problems which evolve from simple and welldefined to complex and illdefined will help them develop transfer skills. Using realworld data in sample problems will also help facilitate the transfer process, since students can more easily identify with the context of a given situation. Using these strategies, students will learn the relevance of course material to daily life and will begin to transfer concepts between disciplines, moving toward a more cohesive understanding of the real world. Summary To develop better problemsolvers, instructors must help students overcome both emotional and cognitive barriers to learning effective problemsolving skills. By first creating a comfortable classroom environment and helping students overcome their fears and anxieties related to problemsolving, teachers lay the necessary foundation for successful learning. Then using an array of pedagogical and methodological strategies, instructors can promote student reflection on the problemsolving process itself and provide them critical tools for and practice in productive problemsolving. As a result students will become increasingly effective problem solvers, able to solve more and more complex problems with greater and greater independence. 
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