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Research on Reasoning Much fruitful research on reasoning has been generated by the contradistinction between general reasoning skills and knowledge of specific subject matter as instructional goals. Some investigators have concentrated on identify- ing the substantive knowledge needed for problem solving in a particular area, others on the strategies used by good problem solvers, and still others on impediments to the use of reasoning skills. An understanding of how a particular reasoning skill operates in a specific substantive context can lead to improved ability to teach the skill. For example, Larkin and Reif (1976) analyzed what information a learner should acquire while reading a description of a scientific relation and used their findings to design special training. In this training, students reading scientific text practiced finding particular kinds of information, e.g., units and symbols for quantities, typical values, scaling properties, and features dis- tinguishing them from other quantities. After they had practiced answering such questions for a variety of passages over a period of six weeks in a physics course, the number of students who were able to learn how to use a scientific relation from its written descriptions increased from 40 to 80 percent. As another example, Anderson (1981) has developed a computer model that solves problems in geometry. Anderson's system is based on a model of how experts solve geometry problems. This model also can ~understand" a student's proposed solution and give flexible advice when the student executes steps that are either wrong or unproductive. Because the model itself has an understanding of high school geometry, it does not rigidly constrain the learner to particular solution paths, but gives advice only when the student 5
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6 does something entirely incorrect or unproductive Although it has not yet been proven, Anderson estimates that an instructional system based on this model may improve the efficiency of teaching geometry by a factor Of two. Knowledge required to solve arithmetic word problems has been studied in detail. . Being skilled at such prob- tems appears to be associated with being skilled not merely in the basic arithmetical operations, but also in categor iz ing problems according to the relationships they involve. Riley et al. (1983) have categorized word prob- lems according to the implicit problem structure. In "change problems," a quantity (Jane's three marbles) is increased or decreased (by Tom giving Jane two more or taking two away). In "equalizing problems,. two quan- tities must be considered and made the same (three of Jane's seven marbles have to be taken away and added to Tom's one marble). In "combine problems, n two separate quantities have to be considered in combination (Jane's marbles and Tom's marbles). In "compare problems, n again, the quantities remain the same but have to be compared (how many more marbles than Tom does Jane have?). To solve such problems, students must recognize distinct patterns that involve the ways in which quantities are related. This research has been used to provide explana- tions of the different levels of problem-solving skill that are observed among elementary school children. Schoenfeld (1979) has developed a theory of problem- solving competence in mathematics that describes the "executive" knowledge good problem solvers use to make efficient use of their resources. The theory also deals with attitudes of students about problem-solving tech- niques that prevent them from using methods they have mastered. Schoenfeld presented general heuristic strategies (shown in Figure 1) that could facilitate problem solving in mathematics to a group of college science and mathematics majors and a control group. He concluded that the likelihood of students' picking up such strategies from experience is small and that problem-solving strategies must be taught explicitly as are other mathematical techniques. He also found that, even when students master problem-solving techniques, there is no guarantee that they will use them. Although experts find these strategies easy to use, students must be taught not only how to use them, but also when. When students do use them, the impact on their problem solving is substantial. However, much more research is required
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(1) Draw a diagram if at all Possible. Even if you finally solve the problem by algebraic or other means, a diagram can help give you a "feel. for the problem. It may suggest ideas or plausible answers. You may even solve a problem graphically. (2) If there is an integer parameter, look for an inductive argument. Is there an "na or other parameter in the problem that takes on integer values? If you need to find a formula for f(n), you might try one of these: (A) (B) (3) Calculate f(l), f(2), f(3), f(4), f(5); list them in order, and see if there's a pattern. If there is, you might verify it by induction. See what happens as you pass from n objects to n + 1. If you can tell how to pass from f(n) to f(n + 1), you may build up f(n) inductively. Consider arguing by contradiction or contrapositive. Contrapositive: instead of proving the statement waif X is true then Y is true,. you can prove the equivalent statement waif Y is false then X must be false.U Contradiction: assume, for the sake of argument, that the statement you would like to prove is false. Using this assumption, go on to prove either that one of the given conditions in the problem is false, that something you know to be true is false, or that what you wish to prove is true. If you can do any of these, you have proved what you want. Both of these techniques are especially useful when you find it difficult to begin a direct argument because you have little to work with. If negating a statement gives you something solid to manipulate, this may be the technique to use. (4) Consider a similar problem with fewer variables. If the problem has a large number of variables and is too confusing to deal with comfortably, construct and solve a similar problem with fewer variables. You may then be able to tA) Adapt the method of solution to the more complex problem. (B) Take the result of the simpler problem and build up from there. (I) Try to establish subgoals. Can you obtain part of the answer, and perhaps go on from there? Can you decompose the problem so that a number of easier results can be combined to give you the total result you want? FIGURE 1 The five problem-solving strategies. Source: Schoenfeld (1979)
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8 to identify truly useful strategies, how experts learn how and when to apply them, and efficient means of passing this knowledge on to students. A nether of studies have identified obstacles to learning reasoning that lie in the preconceptions students bring to school. For example, Champagne et al. (1980), Clement (1982), and McCloskey (in press) have shown that college students have preconceived notions about common physical situations that run counter to and can interfere with learning the principles of physics taught in the classroom. Even after instruction, naive pre-Newtonian beliefs about basic mechanics interfere with students' understanding of the physical world. Various strong convictions, which are often reinforced by unaided common- sense perception (as in naive accounts of forces acting on the movement of objects on an inclined plane), become intertwined with new learning and inhibit its progress. Alternative learning experiences can connect these tendencies. By presenting computer simulations of events (for example, objects falling in accordance both with Newtonian principles and commonsense perceptions), exper imental instruction in this area has been shown to influence strongly held beliefs and deepen understanding of the power of scientific accounts (Champagne et al. , 1980; diSessa, 1982; White, 1984). Such results are examples only, but they portend a deeper understanding of the mechanisms underlying reasoning skills in science, mathematics, and technology, and the development of effective instructional strategies to teach these skills. DEVELOPING COMPETENCE To understand the mechanisms of reasoning skill, it has proven useful to compare and contrast the performance of more and less skilled individuals. Much more work needs to be done, but good examples of such research include that of Chase and Simon (1973), Egan and Schwartz (1979), Chi et al. (1981), Jeffries et al. (1981), Clement (1982), Lesgold (1983), and Voss et al. (1983). These studies are concerned both with how a reasoning skill operates and with the difficulties and limitations of students who have not acquired this skill. The studies indicate that problem solving proceeds on the basis of the solver's representation of the problem. Students with less skill tend to represent problems through recognition of literal surface features and not by
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9 inferences from abstracted principles in the domain of knowledge pertinent to a problem. Yet investigations in radiology, architecture, electronics, chess playing, and physics show that experts categorize problems according to principles--this is a Newton 's-second-law problem, a conservation-of-energy problem, etc.--rather than according to the specific set of factors and conditions presented by the problem. The relations between a structured body of knowledge about the pertinent domain and the problem-solving process is mediated through the quality of the solver's representation. Both the scope and depth of the solvers domain-related knowledge and its organization--its completeness and its coherence-- determine the efficiency of progress toward the solution. In addition, experts' knowledge includes information about conditions of applicability for various procedures. Average performers in a field often know enough about the domain to construct an efficient initial representation of a problem, but they lack knowledge of the conditions for applying certain procedures. A major research challenge is to understand better the dynamic process through which reasoning skills are acquired, to develop a rich theory of learning particu- larly targeted toward reasoning skills that are difficult to acquire. Such a theory would facilitate the develop- ment of more effective interventions to help learners acquire these skills more efficiently and reliably. Productive work in this area includes that of Greeno (1980), Anderson (1981), Larkin et al. (1983), and Riley et al. (1983). Anderson's instructional system for geometry, mentioned earlier, not only includes sufficient knowledge to solve a wide range of geometry problems, but it can also adjust the nature of this knowledge to match the state of the learner. For example, for a beginner, the system might break up the recognition of congruent angles into several substeps. For a more advanced learner, the system might expect the learner to recognize congruent angles in a single step. Larkin has noted that experts and novices solving textbook physics problems use a very different set of subgoals. A typical novice, however, exhibits a strategy that shows a few expert features as well as a predominantly novice structure. Thus it seems that learners gradually acquire the system of subgoals that experts use productively. As another example, Riley et al. have developed a model of how children's knowledge for solving simple word problems changes over time. Primitive versions of the model are
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10 based on extremely simple mental structures, while more powerful versions can take advantage of more elaborate structures. In studies contrasting more and less skilled indi- viduals, the more skilled individuals are often much easier to understand. They have well-formed processes and models of knowledge that correspond accurately to the discipline. In contrast, less skilled individuals often have processes and organizations of knowledge that are rich and complex, unstable, and do not correspond accurately to the discipline. Current research (McCloskey et al., 1980; Clement, 1982; McCloskey and Kohl, 1982) suggests that learners do not simply discard these processes when they are instructed. Instead, their initial reasoning processes interact with instruction in complex ways. Research efforts should be targeted toward understanding those initial processes and showing how they can be addressed through instruction (Glaser, 1984). One approach might be through confrontation so that new, more effective processes can replace old. Another approach might be through incorporation, where useful aspects of initial processes can be incorporated into more accurate and effective processes. Cognitive studies use the notion of prototypical knowledge structures or schemata to account for various phenomena in memory, comprehension, problem solving, and understanding. Schema theory attempts to describe how acquired knowledge is organized and represented and how such cognitive structures facilitate the use of knowledge in particular ways. This theoretical construct has particular utility for devising approaches to instruction as individuals attempt to interpret new information on the basis of prior knowledge. Modes of instruction that demand interrogation of the learner's knowledge and thinking and that demand confrontation with new knowledge are being investigated by a number of researchers. For example, Collins and Stevens (1982) have studied effective teachers' procedures for teaching students domain-specific rules and theories. The procedures involve shaping a line of inquiry that helps students articulate their naive initial conception of facts and principles and then accept, modify, or reject them in the light of their predictive power, congruence with new facts, and the like. This inquiry approach both enables the student to assimi- late new information efficiently and provides practice in deriving rules or theories for related knowledge. An important feature of the approach is the selection of
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11 cases and questions that enable students to use extant knowledge as a framework for new learning. Research toward better understanding of children's developing capabilities for scientific reasoning is needed, including more detail on the capabilities of children at varying ages and on the kinds of scientific experiences that aid the development of those capabil- ities. Work on understanding scientific reasoning in young children has illustrated that they are capable of using quite intricate thought processes (Carey, 1985). The nature of mathematical concepts on which young children build their learning is beginning to be des- cribed. Pervasive changes in children's reasoning and learning abilities appear as they gain knowledge in various domains. Researchers need to study what children can do as well as what they cannot do. As Gelman and Gallistel (1978:242) note: "the discoveries we have made about development would not have been possible if we had followed the trend of considering preschoolers merely as beings who lacked the capacity of their older siblings. Our hypothesis of more capacity than meets the eye has served us well. We expect that researchers who keep their eyes open will find still more unexpected ability in young children. The committee recommends research on how competence in reasoning skills is acquired, including the mechanisms of reasoning skills, particularly as evidenced in the differences between novice and experienced learners; the dynamic processes through which reasoning skills are acquired in the context of specific domains of knowledge; and the scientific reasoning skills of children. THE SEARCH EOR GE~I~ IN CONING SKILLS Despite many efforts to understand and teach "generals reasoning skills, success has been unclear. For example, Polya (1957) outlined some general help for solving problems. He divided problem solving into four phases--understanding, planning, executing, and looking back--and formulated heuristics such as thinking of a simpler problem that is similar to the difficult problem at hand. While almost everyone finds these formulations appealing, success in improving problem solving behavior based on such general advice remains to be demonstrated. Early computer models of problem solving were also quite general . Ernst And Newell (1969), for example, developed
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12 a general problem solver" with computer codes for solving problems that could be applied to a variety of specific task domains, including algebra, geometry, several puzzles, and logic. Again the example is appealing, but this line of work has not developed into truly powerful general problem solvers. One explanation may be that effective reasoning requires a large amount of domain- specific knowledge, as illustrated by the work cited in the previous section, and relatively little general- purpose knowledge. This conclusion is further supported by considerable recent work on modeling and simulating experts in various fields. The models generally do require a large amount of domain-specific knowledge and relatively little general-purpose knowledge (Davis et al., 1977; Duda et al., 1978). Studies of problem solving by cognitive scientists have indicated a strong influence of specific knowledge structures on effective performance in a discipline (Lesgold, 1983; Voss et al., 1983). Studies in the social sciences, particularly proto- col analyses of expert and novice political scientists, indicate that problem solving in these fields generally proceeds through an analysis of the historical background of the problem, the positing of a solution, and its con- sideration in the light of the suburoblems it would oen- erate. For example, experts who were asked to consider the Soviet agricultural situation laid out the problem first either in terms of its actual history or in terms of long-standing ideological factors influencing agri- cultural policy. Their solutions consisted of lines of intersecting argumentation on subgroblems such as tech- _ - nological capacity, transportation, supply and demand, and agricultural education. In this instance, experts were differentiated from novices not only by the range of subproblems they uncovered in examining the implications of their solutions, but also in the quality and depth of the knowledge they incorporated into their lines of argument (doss et al., 19531. The latter is important in the social sciences, in which the test of a solution is the strength of the argument since hypotheses can seldom be experimentally validated. It appears that problem solving and comprehension are based on knowledge, and that people continually Try to understand and think about the new in terms of what they already know. If this is indeed the case, then it seems best to teach reasoning skills--e.g., skills needed for solving problems and for correcting errors of
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13 understanding--in terms of knowledge domains in which individuals are attempting to become competent. To further such teaching, the committee recommends research on reasoning in particular disciplines, aimed at understanding how abilities to make inferences, to reason, and to generate new information can be fostered by ensur- ing contact with prior knowledge that can be restructured and further developed as learning takes place. There is, however, research that suggests how par- ticular general-purpose reasoning skills may operate (Larkin and Reif, 1976; Schoenfeld, 1980; Larkin et al., 1983; Palincsar and Brown, 1984). In these studies, researchers have specified the skills to be taught in detail and applied them to a focused set of related domains. For example, in an integrated set of training studies, Palincsar and Brown have shown that instruction based on a system of inquiry can greatly increase children's skills in understanding what they read--from 20 percent to 60 percent correct scores on reading comprehension tests. The training sessions they designed focus on encouraging children to internalize skills that foster comprehension. These skills are modeled bythe instructor, who leads dialogues involving paraphrasing main ideas, questioning ambiguities, predicting questions that are implicit in a given passage, and hypothesizing the array of themes of a reading passage. The aim of the sessions is to enable the children to lead such dialogues and eventually develop techniques of self-criticism. An important aspect of the instruction is keeping students fully aware of the purpose of the activity, its utilities for comprehension, how it improves their deficiencies, how and when to use the various interrogative techniques, and the expectation that they will eventually internalize the techniques in their own performance. Currently, significant effort is being devoted to thinking and reasoning courses in secondary and post- secondary education; the effectiveness of such instruc- tion requires careful analysis and evaluation (Chipman et al., in press; Segal et al., in press). Of particular interest are the self-regulatory or metacognitive capabil- ities present in mature learners. Examples of these abilities include knowing what one does and does not know, predicting the outcomes of one's performance, planning ahead, efficiently apportioning time and cog- nitive resources, and monitoring and tailoring one's efforts to solve a problem or to learn (Brown, 1978). These skills vary widely. Although individuals can be
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14 taught knowledge of subject-matter rules and procedures or appropriate theories, if transfer of learning to new situations is a criterion, then they need to know how to monitor the use of this knowledge. Self-regulatory activities thus become important candidates for instruc- tion, and their presence may predict student abilities to solve problems and to learn successfully. Hence, the committee recommends focused research on self-regulatory or metacognitive capabilities what they are, how they develop, and how learners can be helped to acquire them. We also recommend systematic tracking of outcomes resulting from efforts to teach generalized thinking and reasoning skills. . .
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