Chapter 3 Understanding Elementary Mathematics

The Nature and Origins of Mathematical Skills J.I.D. Campbell (Editor) 0 1992 Elsevier Science Publishers B.V. All rights reserved.

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Chapter 3 UNDERSTANDING ELEMENTARY MATHEMATICS Jeffrey Bisanz University of Alberta Jo-Anne LeFevre Carleton University

Summary Understanding often is defined inconsistently, ambiguously, or narrowly, and consequently the relation between understanding and cognitive processing on mathematical tasks is not very clear. Many forms of behavior are related to understanding and a framework is needed to describe these various forms in an integrated way. A "contextual space" is proposed for classifying different types of perjormance related to understanding elementary mathematical concepts. The two dimensions of this space are the type of activity involved (applying justifying, and evaluatingsolution procedures) and the degree of generality with which these activities are exercised. These two aspects can be used to construct a 'Iprofile" that reflects the contexts in which an individual shows various forms of understanding. Acquisition of understanding can be described in terms of the sequences of profiles that emetge, and these sequences have implicationsfor characterizing the mechanisms underlying changes in knowledge. Introduction Over the past 25 years, cognitive, developmental, and instructional psychologists have made a great deal of progress in determining how children and adults compute answers to simple arithmetic problems. Research focused on the processes that underlie arithmetic computation has been important for generating insights about remembering, attention, problem solving, and development, and these insights have led to hypotheses and conclusions that extend far beyond the domain of arithmetic (e.g., Ashcraft, 1987; Campbell & Clark, 1989; Kail, 1988;

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Siegler, 1986; Siegler & Jenkins, 1989). This research also has considerable promise for improving instructional practice by increasing the sensitivity of assessment and by contributing to the development of better teaching methods (e.g., Grouws, in press; Leinhardt, Putnam & Hattrup, in press; Romberg & Carpenter, 1986). Despite the success of research on identifying computational processes, we still lack an integrated and detailed account of the relation between computational processes and the common, if ambiguous, notion of understanding Certainly the degree or level of understanding must be reflected in the solution processes used by children and adults. What remains unclear, however, are a number of questions about the precise relation between understanding and solution processes. How, for example, should understanding be inferred from behaviors, how might it be represented internally, how is it involved in the construction, selection, or modification of solution procedures, and how might it be modified by the use of solution procedures? Questions such as these are of critical importance for psychologists who seek to provide a full account of remembering and problem solving, and of how these processes develop. Answers to these questions are also of interest to teachers, who seek to help students achieve levels of understanding that transcend the low levels of performance often required in classrooms (Cum'culum and Evaluation Standards for School Mathematics, 1989). Details about the relation between understanding and solution procedures are also very relevant for evaluation specialists, who are beginning to recognize the shortcomings of current achievement tests (Kulm, 1990). This confluence of interests between research psychologists, educational practitioners, and evaluation specialists has begun to provide a basis for a mutually beneficial interaction, and the opportunity for enhancing this interaction should not be missed. In our view, research on the relation between understanding and solution procedures has progressed slowly because understanding is often defined poorly or inconsistently. In this chapter we therefore focus on the preliminary and important issue of how understanding in elementary mathematics might be conceived and operationalized optimally for developmental, cognitive, and instructional purposes. We begin by describing general difficulties in defining what it means to understand simple arithmetic, and we note the kinds of problems that can arise when understanding is defined operationally in terms of performance on a single type of task. We suggest, instead, that different forms of understanding must be recognized. Next we describe a general, two-dimensional framework for classifying the contexts in which forms of understanding can be evaluated, and we highlight the advantages of focusing on relations among tasks as opposed to single tasks. Finally we explore some implications of this classification scheme for the

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development of understanding. Our intent is to provide a multifaceted view of understanding that facilitates research on the relation between understanding and solution processes and that contributes to improved assessment and instructional practices. Problems in defining understanding What does it mean to understand simple mathematics? Teachers, parents, researchers, and sometimes even students emphasize the importance of understanding in learning mathematics. Despite this consensus, a clear and comprehensive definition of understanding remains elusive. Consider, for example, the case of students who are presented with two problems: 85 + 29 and 29 + 85. Suppose that students answer both problems correctly, and then they perform with continued success on several similar pairs of problems. Do these students understand simple multidigit addition? If we focus on correctness of the answers, as is often the case in conventional methods of educational assessment, then we might conclude that these students indeed understand addition of two-digit numbers. If, however, we focus on the way in which students solve the problems or respond to verbal questions about the problems, our judgment might be quite different. For example, if some students solve the problems correctly but fail to realize that the 8 and 2 represent 80 and 20, respectively, then we might conclude that these students have only memorized a solution algorithm and do not understand the role of place value in addition (Fuson, 1990). If the students are incapable of decomposing the problem into a simpler form (e.g., 85 + 29 = 85 t 30 - l), then we might conclude that they do not completely grasp part-whole relations (Putnam, DeBettencourt & Leinhardt, 1990). If the same left-to-right procedure is used on every single problem, then we might be concerned that the students do not understand commutativity; that is, the students may fail to grasp that 85 + 29 and 29 + 85 must yield the same answer, and so calculation on the second problem is unnecessary (Baroody, Ginsburg & Waxman, 1983). Finally, if some students solve two-digit addition problems but fail to solve three-digit problems or word problems with two-digit values, then we probably would conclude that their understanding of multidigit addition is illusory or, at best, quite limited. Clearly different forms of understanding can be identified, and this example highlights two problems in defining what it means to understand in simple mathematics. First, our judgment about whether a person understands simple addition depends on the elements entailed in our definition of simple addition.

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Depending on our d e f ~ t i o n these , elements can include various concepts and principles (e.g., place value, part-whole relations, commutativity, inversion, associativity, compensation). Moreover, determining whether our hypothetical students understand simple addition depends not only on the products of their solution processes (i.e., the answers provided by the students), but also on the processes that underlie performance (e.g., fact retrieval, counting, decomposition, and use of heuristics). Thus the first problem consists in defining the specific domain of what is to be understood, and it is addressed by developing theories about processes and/or concepts involved in performing tasks related to that domain. The vast bulk of research on mathematical cognition is consistent with this agenda, including research ranging from the identification of memory retrieval processes (Campbell & Oliphant, this volume) to classification of the knowledge structures necessary for solving arithmetic word problems (Kintsch & Greeno, 1985).

Also implicated in the example is a second, somewhat more subtle problem, and one on which we focus in this chapter. After researchers or teachers have identified the domain of understanding, the next step typically is to select or develop a task to elicit relevant behavioral evidence on whether the student understands the domain. In the example above, our judgment about a student’s understanding could well depend on whether we examine solution processes or pose questions about the properties of addition. Similarly, our conclusion may depend greatly on whether we present symbolic arithmetic problems or word problems to the student. Thus this second problem amounts to defining the context in which understanding is being assessed. For present purposes, context refers broadly to task demands and materials that are used to evaluate understanding; we do not use the term to refer only to certain environments (e.g., academic versus nonacademic situations). Effects of context have been amply demonstrated, but there has been little effort to systematically examine the role of context and to explore and integrate the cognitive and developmental implications of these effects. We now turn to this issue. Contexts of understanding As illustrated in the example we have described, an individual may show evidence for some forms of understanding but not others, depending on context. We assume that no single context is definitive for assessing understanding. Moreover, we assume that discrepancies as a function of context reflect potentially important differences in underlying processes and/or representations. A narrow definition or criterion would tend to make understanding an all-or-none

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phenomenon (Greeno, 1983) and would mask relations among the varieties of understanding that may emerge during the course of acquisition. We propose that evidence for different forms of understanding can be organized, at least initially, in terms of a few orthogonal aspects or dimensions, and that understanding and its development can be characterized in terms of the contextual space defined by these dimensions. Such an organizational scheme should be useful for determining a profile of understanding that reflects the contexts in which an individual evinces performance related to understanding. These profiles should be useful for identifying common patterns that reflect underlying individual differences or sequences in acquisition, as well as for providing insights for instructional intervention. As an initial step toward identifying different forms of understanding in simple mathematics, we propose the classification scheme in Table 1. The scheme consists of two aspects, activity and generality, that jointly define a contextual space for assessing understanding.

Table I. Contexts for Assessing Understanding

Generality

.....................

Activity

Narrow

Application of procedures

Using an appropriate procedure spontaneously on one particular task.

Using a similar procedure on a variety of related tasks.

Justification of procedures

Describing the principle or concept that makes a procedure appropriate for one particular task.

Generating a similar explanation on a variety of related tasks.

Evaluation of procedures

Recognizing the validity of a procedure on one particular task.

Making a similar judgment on a variety of related tasks.

>

Broad

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Activity

Tasks used to assess understanding vary considerably in terms of the demands placed on an individual. Children and adults respond to these demands with a wide variety of problem-solving activities. Despite their diversity, the activities required to perform many tasks can be grouped into three general classes: application of procedures to solve problems; explicit justification of procedures; and evaluation of the procedures. To illustrate these activities, consider the problem of determining whether a person understands the principle of inversion as it applies to addition and subtraction (Starkey & Gelman, 1982). When presented with a problem of the form a +. b - b, a person who understands inversion presumably knows that the answer to this problem must be equal to a and that successive addition and subtraction is unnecessary. The critical issue is how this Understanding, or lack of understanding, can be assessed. Application of procedures. One approach is to observe whether an individual uses solution procedures that are consistent with the principle of inversion. When presented with a problem such as 4 + 9 - 9, some children engage in a laborious, left-to-right solution procedure that involves sequential addition and subtraction (i.e., 4 t 9 = 13, and then 13 - 9 = 4), whereas others answer quickly without adding or subtracting (Bisanz & LeFevre, 1990). Children in the latter group appear to use a shortcut based on the principle of inversion. Application of procedures simply refers to the use of a solution procedure that reflects, or is at least consistent with, a concept or principle appropriate for that problem. Knowledge of appropriate concepts or principles often enables children to use alternative solution procedures that are easier, more efficient, or more accurate than the standard algorithms that typically are taught. For example, a child who solves 5 + 6 by transforming the problem to (5 t 5 ) t 1 may be using a procedure based on the concept of part-whole relations (Putnam et al., 1990). Whether the child obtains the right answer is largely irrelevant; the important point is that the child activates a procedure that is consistent with the underlying concept. Use of an appropriate procedure may appear to be compelling evidence that a child understands the underlying concept or principle. This type of evidence alone often can be insufficient, however, for two reasons. First, children may use a conceptually appropriate procedure for reasons unrelated to the underlying concept. In the inversion problem, for example, a child may simply respond with the first number because that type of response had been rewarded on previous problems, or because the child has been trained in a rote fashion to respond in

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this way on problems having the form of inversion problems. In this case, attributing understanding to the child would be overestimating the child's competence. Second, children may be fully capable of using a conceptually appropriate procedure but simply may do not do so, perhaps because they think that adult observers expect a more prosaic algorithm. In this case, attributing a lack of understanding to the child would be underestimating the child's competence. Thus conclusions based only on the application of a conceptually appropriate procedure can be misleading, and researchers often seek other forms of evidence (such as justification and evaluation of procedures) to confiim or refute their assessments. Justification of procedures. Another way to assess children's knowledge of inversion is to ask them to provide a rationale for the shortcut procedure on inversion problems. For example, when a child uses a shortcut to solve 4 + 9 9, or when a shortcut is presented to the child, can he or she explain why the answer must be 4 and why adding and subtracting are not necessary? An explanation in which the child focuses on the logical necessity of the answer might be taken as evidence that the child understands inversion. Tasks requiring justification can be used to assess whether an individual can explicitly explain or describe the principles, concepts, or rules that account for the appropriateness or validity of a particular procedure. If the principle or concept in question is featured in a reasonable way, the child may be credited with having "explicit knowledge" of the underlying concepts (Green0 & Riley, 1987), and the possibility that the procedure was applied in a rote fashion may be regarded as minimal or negligible. Piaget, of course, relied heavily on the justifications of children and adolescents in determining whether their reasoning was characteristic of preoperational, concrete operational, or formal operational thought (Miller, 1982; Piaget, 1970). In research on mathematical cognition, justifications are often used to assess understanding. Putnam et al. (1990), for example, sought to determine whether elementary school children were able to justify the use of derived-fact procedures to solve certain addition and subtraction problems. A derived-fact procedure involves the transformation of a problem so that known number facts can be used to solve the problem. For instance, a child might solve 4 + 5 by transforming the problem to (4 + 4) t 1, thus making it possible to use his or her knowledge that 4 + 4 = 8 to solve the more difficult problem. In this study, children watched as puppets used derived-fact procedures to solve arithmetic problems. The children were asked, by one of the puppets, to explain why the derived-fact procedure was valid. Putnam et al. evaluated the explanations as a means of assessing whether children understand part-whole relations (Resnick, 1983). Understanding part-

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whole relations, according to their analysis, consists in the knowledge that the combined value of the parts (the augend and addend) must equal the value of the whole (sum) and that transformations of a part must be compensated by transformations to another part or to the whole, Children were credited with understanding part-whole relations if, for example, they recognized that a change in the value of one of the addends or parts (i.e., changing the 5 to a 4 in the example above) must be compensated by a change in the sum or whole (i.e., changing the 8 to 9). Similarly, Greeno (1983) analyzed verbal protocols to assess the extent to which high school students used a part-whole schema when solving geometry problems. In each case, the justifications provided by students were crucial for determining students’ understanding of the conceptual basis for problem-solving activities. The conditions under which children and adults acquire explicit knowledge of problem-solving procedures, as indexed by their verbal justifications, are of considerable interest. For example, Schliemann and Acioly (1989) studied the performance and knowledge of bookies for a lottery in Brazil and found that justifications can vary as a function of schooling. Although all the bookies showed similarly high performance, only those with some formal schooling were able to satisfactorily explain why their solutions worked. In contrast, young children’s justifications on conservation problems appear to improve as a function of age, not schooling (Bisanz, Dunn & Morrison, 1991). Clearly the relation between schooling and the quality of justification is variable, and the boundary conditions on this relation require clarification. When individuals provide an adequate justification for a procedure, it often is reasonable to assume that they have explicit, accessible knowledge about the underlying principles. Two points of caution should be noted, however. First, failure to provide an adequate justification does not imply that the person lacks the knowledge in question; he or she simply may have difficulty verbalizing that knowledge (e.g., Brainerd, 1973). Second, being able to provide adequate explanations or rationales does not necessarily imply that the person can use the corresponding procedure spontaneously. The procedure simply may be too dificult to implement. For example, a student may be able to justify the use of a multiplication procedure to solve a particular problem but be incapable of executing that procedure mentally because of the memory load involved. Alternatively, a person may be able to justify a solution algorithm but may fail to recognize situations in which that algorithm might be applied. Thus the criterion of justification can be important, but it need not be the sine qua non for evaluating understanding.

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Evaluation of procedures. To assess a child‘s understanding of inversion, a third approach is to describe or present a shortcut procedure based on inversion and to ask the child simply to judge, without performing or justifying the solution procedure, whether the procedure is valid. In the case of inversion, a child who observes a puppet solving the problem without successive addition and subtraction could be asked whether an answer really can be found without adding and subtracting. An argument could be made that children who do not understand inversion would judge the shortcut to be invalid, whereas a child who understands inversion would find the procedure to be perfectly acceptable. Evaluation of procedures refers to a person’s decision about the applicability and correctness of a particular solution to a problem (Greeno, Riley & Gelman, 1984). It reflects a potentially different type of knowledge than application of procedures, in much the same way that one’s judgments of grammatical correctness are different than the production of grammatical speech. Judgment tasks have been used to assess understanding of mathematical cognition, although less frequently than application and evaluation tasks. In a study of counting, for example, Gelman and Meck (1983) discovered that children could detect that a puppet had counted incorrectly even though they could not count accurately themselves (but see Briars & Siegler, 1984, and Wynn, 1990). In a study of estimation, Sowder and Wheeler (1989) found that more children could choose a best-case solution from among alternatives than could apply their knowledge to open-response problems. Appropriate evaluations about the validity of solution procedures reflect knowledge about the conditions and/or constraints under which a concept applies. In contrast with application of procedures, evaluation does not require execution of the solution procedure and so processing demands presumably are minimized. In contrast with justification, evaluation does not require explicit verbalization about a principle and so assessment is not confounded with verbal skill. Moreover, evaluation tasks are relatively simple to perform because of the minimal demand placed on the subject. The simplicity of the task does not necessarily beget simplicity of interpretation, however, because the knowledge used to make the evaluation may be difficult to identify. If, for example, a child indicates that a puppet’s count is inaccurate, it may be difficult to discern whether this judgment reflects sensitivity to a counting principle (e.g., double counting is invalid) or whether the child decided on the basis of some irrelevant feature of the observed procedure (e.g., a dislike of the object on which the count ended). This difficulty can be minimized by combining evaluation and justification activities, so that a child must justify his or her evaluation, but again the verbal demands of justification may lead to an

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underestimation of the child’s competence. Moreover, an evaluation situation may be so ambiguous or difficult that a child could fail to decide correctly, even if the child is fully capable of justification, application, and evaluation in a simpler situation. Generality Each of the activities in Table 1 can be used independently to assess understanding, but each has shortcomings. One way of minimizing this difficulty is to combine two or more activities, but another is to examine the generality of performance by varying the degree of similarity among the problems used. In the case of assessing knowledge of inversion, for example, a researcher could test a child’s understanding of inversion by studying performance on a variety of problems in which an inversion-based shortcut could be used. Problems could vary in form (a + b - b, b t a - 6 , orb + c t a - c - b), size (small or large numbers), or format (symbolic equations or word problems). Children who solve a broader range of problems using shortcut procedures have, arguably, a better Understanding of inversion. The importance of generality for defining the contextual space in Table 1 for assessments of understanding can be illustrated with two examples. Carraher, Schliemann, and Carraher (1988) examined the use of ratio conversion processes in Brazilian foremen who were experienced in converting measurements from the scale of blueprints to the scale of the construction projects on which they worked. Carraher et al. found that all of the foremen were successful when they used blueprints drawn in familiar scales, but only 34% were successful when an unfamiliar scale was used. The fact that many of the men could not adapt the ratio-conversion process to unfamiliar scales was taken to imply that their understanding of ratio conversions was strikingly limited. Whereas Carraher et al. (1988) focused on individual differences, Lawler (1981) examined changes in the generality of knowledge with development. Lawler carefully observed how his 6- year-old daughter’s knowledge about arithmetic changed over several months. Initially his daughter’s knowledge was highly particularistic. For example, her use of solution procedures for numerically identical problems depended on the specific domain or microwodd of the problems: An addition problem presented verbally elicited a combination of derived-fact and counting procedures; a problem involving money elicited addition in coin-based units; and a problem in vertical, paper-and-pencil form elicited the usual right-to-left algorithm with carries. These three procedures, and the domains in which they were applied, were viewed by her as being entirely independent. The

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generality of each procedure continued to be tightly constrained until, according to Lawler, these separate microworlds became integrated into a structure that enabled the use of any of these specific procedures in any of the formerly separate domains. In the case of the 6-year-old, as with the Brazilian foremen, greater generality was interpreted to imply a greater understanding of the underlying mathematical principles. Indeed, generality in problem solving is often taken to be the primary characteristic of true understanding. Wertheimer, for example, distinguished between rote learning and meaningful learning (see Greeno, 1983). According to this view, rote learning results in adequate performance only in situations identical to, or very similar to, the original context of learning. Meaningful learning, in contrast, involves the acquisition of a general principle that enables successful performance in a variety of situations where the principle is relevant. The flexible and general understanding that results from meaningful learning is characterized by demonstrations of transfer from original learning situations to new and different problems. The importance of generality is that successful performance is unlikely to be due to rote learning or to spurious factors if it can be observed on a variety of new or different problems. As Greeno et al. (1984) noted, "the case for understanding of principles is strongest if a child is required to generate a new procedure or a modification of a known procedure, and the procedure that is formed is consistent with the principles" (p. 105). Generality often is minimal in learning and development. As children develop, for example, they often fail to use mnemonic strategies even though they are fully capable of executing the strategies and have used them successfully in other situations (e.g., Brown, Bransford, Ferrara & Campione, 1983). In adults, knowledge obtained in one problem-solving situation often does not transfer to a second, analogous situation (e.g., Gick & Holyoak, 1980). Hatano (1988) found it useful to distinguish between two types of experts: Adaptive experts use their knowledge flexibly, whereas routine experts are proficient only in the specific domain in which they have perfected their performance. At a minimum, the implication of these findings is that thinking and problem-solving skills can be heavily influenced by the nature and organization of specific knowledge. Bransford, Vye, Kinzer, and Risko (1990) refer to the commonly observed lack of generality as one aspect of the "problem of inert knowledge" and clearly illustrate the pervasive importance of this problem for the study of cognition, development, and instruction. Some researchers (e.g., Greeno, 1983) have tried to identify the properties of knowledge structures that enable generality, as opposed to those that enable only very restricted application. Other researchers have focused on factors

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that seem to decontextualie learning and enable increased generality (e.g., Bransford et al., 1990; Lehman, Lempert & Nisbett, 1988; Perfetto, Bransford & Franks, 1983), and these investigationsshould provide explicit clues to the question of how knowledge structures influence generality. We do not attempt here to solve the problem of inert knowledge, but we want to underscore the importance of generality in assessing understanding. Profiles. The principle contribution of the scheme in Table 1 is that it helps to outline the kinds of research that need to be conducted to fully evaluate relations among different assessments of "understanding". No single definition or operationalization of understanding is sufficient for capturing the range of contexts in which understanding, in some form, may be demonstrated. Instead, we conceive of Understanding in terms of aprofile in the contextual space defined by activity and generality. Thus the issue for assessment of individuals is not one of deciding whether an individual has or does not have understanding, but rather one of determining the pattern of successes and failures in the contextual space. Similarly, the issue for studying acquisition is not how one group performs on a particular task as opposed to another group, but rather how understanding, or evidence for some form of understanding, "spreads"in this contextual space during the course of development or instruction. In the next section we speculate on some possible developmental sequences and the implications of each for mechanisms of knowledge acquisition. Before proceeding to implications for development, we should note several qualifications that pertain to the contextual space represented in Table 1. First, the activities described in this organizational scheme are illustrative but not necessarily exhaustive. Many researchers have used one or more of the three types of activity to assess understanding (e.g., Greeno et al., 1984), and no doubt other activities (or variations on the three activities) could be used. Second, our characterization of generality is incomplete. A full account of this dimension would require a psychological analysis of the gradients of similarity among problems, a description that necessarily will vary with the particular types of problem solving being studied. Moreover, generality of performance can be assessed in a number of ways (see Greeno & Riley, 1987, for an illustrative list). Third, we have focused on aspects of task contexts that should apply across many domains, rather than on the knowledge structures that might be required for performance in these contexts. We highlight these three limitations because each represents an issue that must be addressed in research on understanding in a specific domain. That is, the scheme in Table 1 serves as a broad framework for investigating understanding in any domain, but the agenda for research in a particular domain consists largely of identifying and studying (a) the most

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important activities relevant to that domain, (b) the dimension(s) of generality for problems in that domain, and (c) the knowledge structures required for performance. In summary, we propose that the study of understanding is best conceived in terms of a space of contexts defined, at least initially, by activity and generality. One implication of this approach is that researchers need to focus not on single tasks but rather on relations among tasks. Given this orientation, the empirical goal becomes one of describing how profiles of understanding vary as a function of domain, development, and individual differences. Development of understanding When research is designed to assess profiles of understanding in a contextual space, then development can be represented as a sequence of profiles or a spread of understanding from an initial point to other parts of the space. This formulation leads to a number of pertinent questions. For example, are application, justification, and evaluation tasks always mastered in the same order? Is some generalization (horizontal spread) typically observed before a new activity is mastered (vertical spread)? How general are changes in profiles across different domains of problems? Answers to these questions should provide valuable insights into characteristics of the knowledge acquisition process that underlies cognitive development (see Klahr, 1976). In this section we briefly describe two plausible, partial sequences to illustrate how different sequences implicate different developmental mechanisms.

Evaluation before application One plausible sequential relation is that evaluation precedes application. Suppose, for example, that children are able to recognize that a shortcut on inversion problems is legitimate before they generally use the shortcut spontaneously. In such a case, evaluative judgment precedes spontaneous application in the course of acquisition. This sequence seems fairly prevalent in a variety of domains. Children often can judge the grammatical appropriateness of a sentence well before they spontaneously incorporate the underlying grammatical rule in their own speech, much as researchers and students often can evaluate the validity of using certain statistical procedures without being able to implement those procedures. Similarly, children and adults typically are better at recognizing features of a "good experiment than they are at designing experiments (Bullock, 1991; Sodian, Zaitchik & Carey, 1991). In mathematical cognition,

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Gelman and Gallistel (1978) have proposed that knowledge about the principles of counting precedes and guides acquisition of counting procedures, and Byrnes and Wasik (1991) have argued that children acquire conceptual knowledge about fractions before they add fractions accurately. The common theme is that knowledge about the underlying principles or concepts is acquired in some form before the corresponding procedures are used. An evaluation-before-application sequence would be expected if knowledge is represented initially in terms of general, relatively abstract principles, which in turn can be used to guide or constrain the construction of more specific procedures (Wynn, 1990). The assumption is that possession of abstract principles is sufficient for evaluating the validity of an observed procedure, even if the child has not yet used that procedure. If this hypothesis about the genesis of knowledge structures is correct, then two critical questions about development need to be addressed. First, how do abstract knowledge structures influence the construction or selection, and subsequently the implementation, of task-specific procedures? Second, what accounts for the lag between the acquisition of abstract principles and the use of procedures reflecting those principles?' A survey of proposed answers to the first two questions is beyond the scope of this chapter, but a few examples serve to highlight the issues. Bisanz and LeFevre (1990) speculated about how knowledge about the principle of inversion might be adapted to enable children and adults to use shortcuts on problems of the form a t b - b. They proposed that conceptual and procedural knowledge about mental arithmetic could be represented in terms of productions, which are condition-action statements that can vary greatly in complexity and specificity (cf. Neches, Langley & Klahr, 1987; Newell & Simon, 1972). A production system can contain not only abstract, conceptual productions and task-specific procedural productions, but also knowledge-acquisition productions that, under specific conditions, can modify old productions or create new productions (Kail & Bisanz, 1982; Klahr, 1984; Neches et al., 1987). Bisanz and LeFevre proposed that conceptual productions combine with these knowledge-acquisition productions to create task-specific procedures that enable children to use shortcuts on certain

'

A third and more basic question concerns the origins of abstract principles. This question is of central importance to any developmental theory about understanding, but it is not particularly tractable, theoretically and methodologically, and hence it is rarely addressed thoroughly in research or theory. Basic concepts or capacities are often described as natural or innate. This proposal has practical value because it functionally defines the state of the cognitive system at the point of development where analysis begins. Ultimately, however, simple proposals about innateness have little explanatory value from a developmental-systems view (e.g., Cottlieb, 1991).

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kinds of arithmetic problems. Similar theoretical issues have been addressed with comprehensive models by a number of researchers (e.g., Anderson, 1983; Greeno, 1983; Ohlsson, 1987; Ohlsson & Rees, 1991), and in each case the focus is on the developmental mechanisms that account for the acquisition of task-specific representations of knowledge (procedures) from preceding, abstract representations of knowledge (principles and concepts). The lag between knowledge of principles and use of procedures raises not only issues about representational changes but also about processing factors that are associated with the task or the problem solver and that may interfere with the expression of abstract knowledge (e.g., Flavell & Wohlwill, 1%9; Pascual-Leone, 1970). For example, spontaneous application of a procedure may require more resources (e.g., short-term memory capacity) or processing speed than an individual can manage, whereas the evaluation process may be less demanding. Thus hypotheses about developmental changes in processing efficiency or speed (e.g., Case, 1985; Kail, 1991) also may be pertinent for explaining the transition from the availability of abstract principles to the spontaneous use of procedures based on those principles.

Application before evaluation A second plausible sequence is the opposite of the first. Suppose, for example, that children are able to use shortcuts spontaneously before they are able to evaluate the validity of the procedure. This application-before-evaluationsequence might imply that an individual can activate a procedure initially without having knowledge of the underlying principles or concepts that make the procedure valid. This particular profile may not be flattering, but it appears to capture the capabilities of many school children who are capable of executing arithmetic procedures but have little sense for where those procedures are appropriate (e.g., Carpenter & Lindquist, 1989). Assuming that many children progress beyond this profile, it is possible that, after repeated applications, individuals may construct or infer the principles that underlie these procedures, thus enabling performance on evaluation and justification tasks. Such a sequence, ideal or not, also seems to characterize the way in which some adults learn to program computers or to analyze data statistically. Application of procedures without an appreciation for why the procedures are relevant is often referred to as rote or mechanical performance (e.g., Baroody & Ginsburg, 1986), and it is often viewed as the nemesis of true understanding. To the extent that this sequence represents the use of procedures before acquisition of principles, however, it may be important for acquiring some kinds of knowledge.

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For example, several researchers (Briars & Siegler, 1984, Fuson, 1988; Fuson & Hall, 1983; Wynn, 1990) have proposed that children learn to count, at least with small numbers, before they learn at least some of the underlying principles of counting. Upon examining relations between preschoolers’ use of counting procedures and their understandingof cardinality,Wynn (1990) found that children younger than approximately 3.5 years of age showed little evidence of comprehending cardinality, even for numerosities within their counting range. Older children, in contrast, often showed a generalized understanding of cardinality. Wynn concluded that counting is, initially, a relatively meaningless, recitation-like activity, and that children later infer principles such as cardinality on the basis of this activity. A similar scenario seems plausible for forms of arithmetic. If a child is able to execute, however mechanically, the appropriate procedures for adding 2 + 3 and 3 + 2, then he or she will have the opportunity to observe relations among answers and thus to infer underlying principles (e.g., commutativity). The concern, of course, is that children might learn procedures rotely but never gain the insights that are afforded (Baroody & Ginsburg, 1986). An application-before-evaluationsequence would be expected if knowledge is represented first in terms of task-specific procedures, which subsequently enable inferences about more general and abstract principles. Observation of this sequence leads directly to two questions. First, how do task-specific procedures influence the process by which principles are inferred? Second, what accounts for the lag between use of procedures and acquisition of the corresponding principles? One possible approach toward answering these two questions involves hypotheses about resource limitations and automatization. A common set of assumptions in developmental and cognitive research is that (a) the amount of mental resources that can be allocated to processing at any one time is limited, (b) execution of procedures requires resources, and (c) procedures can become increasingly resource efticient with repeated practice, even to the extent that they become automatized or completely independent of resources (see Bjorklund & Harnishfeger, 1990; Case, 1985; Kail & Bisanz, in press). It can be argued that individuals must construct or infer the concepts necessary for evaluation and justification, and that the process of construction or inference is extremely resource dependent. That is, an individual cannot construct or infer the concept necessary for evaluation or justification unless sufficient resources are available. By repeatedly applying the procedure on appropriate problems, the procedure may become increasingly efficient, that is, less dependent on resources. Consequently, greater procedural efficiency may result in the liberation of resources that can be allocated to the process of constructing or inferring the underlying concepts. Consequently, repeated applications and the resulting increase in efficiency could

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be a necessary precursor for acquisition of the concept or principle that is, in turn, necessary for performance on evaluation or justification tasks. (For similar arguments in the domain of elementary arithmetic, see Kaye, 1986; Kaye, dewinstanley, Chen & Bonnefil, 1989; Kaye, Post, Hall & Dineen, 1986). This sort of resource-efficiency hypothesis has several appealing characteristics for explaining developmental phenomena, but many problems with this approach have been in encountered developmental research (Howe & Rabinowitz, 1990). One problem, in the present context, is that the hypothesis includes a description about necessary, but not sufficient, conditions for inference to occur. Specifically, the increasing efficiency of procedures is not sufficient, theoretically, for inferences of a concept or principle; some unspecified process would be required. One possibility is that productions exist that monitor and modify procedures to make them more efficient and more general (Kail & Bisanz, in press). A thorough description of methods by which this type of self-modification could be implemented is beyond the scope of this chapter (see Anderson, 1983; Klahr, Langley & Neches, 1987). The point is that a lag between application and evaluation highlights the need to understand the processes by which inferences about concepts and principles occurs. Other developmental sequences The number of ways in which evidence for different forms of understanding might "spread through the contextual space in Table 1 certainly is not limited to the two cases we have described. For example, the underlying relation between evaluation and application could be reciprocal in that primitive concepts lead to acquisition of simple procedures, use of these procedures could lead to acquisition of more complex or differentiated concepts, which in turn could guide the development of more advanced procedures, and so on (Case, 1985; Fuson, 1988, Chapter 10). This type of account is particularly appealing because it implies that mechanisms for constructing procedures and for inferring concepts are interactive and equally important in understanding cognitive development. Methodologically, repeated observations over time with a variety of evaluation and application tasks might be the most appropriate way to identify and assess reciprocal interactions of this type (cf. Siegler & Crowley, 1991). To this point we have not considered how performance on justification tasks might fit into various sequences, but several possibilities seem plausible. If justification requires verbal skills that lag behind other means of performing (e.g., Brainerd, 1973; Miller, 1982), then performance on justification might occur last in acquisition. For some kinds of expertise, however, a different sequence might

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be expected. An excellent figure skating coach, for example, may be quite capable of explaining the principles of performing a routine without being able to execute the routine, and a researcher who described how an idiot savant performs calendrical calculations may not be able to perform those calculations. Thus the relation between justification and application need not be unidirectional. To summarize, different sequences of development in the contextual space lead to different questions about the underlying nature of the knowledge-acquisition process. The evaluation-before-application sequence requires hypotheses about how a cognitive system is able to use general, abstract principles to generate specificprocedures, the application-before-evaluationsequence requires hypotheses about how the system might progress from use of specific procedures to inferences about general principles, and more complex sequences may require hypotheses about reciprocal interactions among procedures and principles.

Conclusions

As we pondered what our contribution to this volume might be, we considered describing our work on memory and problem-solving processes in mental arithmetic (e.g., Bisanz & LeFevre, 1990; LeFevre, Kulak & Bisanz, 1991). We were struck, however, by the enormous gulf between two solitudes in the study of mathematical cognition. One solitude consists of cognitive and developmental psychologists who conduct experimental work on the ways in which children and adults solve arithmetic problems. For this group, the focus is on careful identification and measurement of the processes and representations that underlie performance, and the goal is to generate a model of cognition and its development that accounts for observed performance on carefully defined tasks. The other solitude consists of teachers and some educational researchers who are less interested in memory or problem-solving processes and more interested in the degree to which students understand mathematics. To the experimental psychologist, the educator’s concept of understanding seems soft, fuzzy, and bereft of ties to rigorous theory or methodology. To educators, the psychologist’s preoccupation with retrieval and decisions processes seems misplaced. In our view, crossing the gulf between these two solitudes is critical for addressing fundamental questions about the acquisition of elementary mathematical skills. With some notable exceptions (e.g., Fuson, 1988, 1990; Greeno & Riley, 1987; Resnick, 1983), attempts at connecting research on procedures with research on various forms of understanding have been few and far between, and an integrative framework for this line of research is needed.

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Our approach in this chapter has been to outline such a framework by exploring the kinds of methods that are useful for addressing questions about the development of understanding. This framework is based on three central propositions. The first proposition is that many forms of behavior are related to understanding and that defining understanding in terms of performance on a single task or measure is counterproductive. Narrow definitions promote an all-or- none view of understanding that simply is not very informative for developmental and instructional concerns, where the focus is on the sequence in which different behaviors and skills are acquired. The second proposition is that a contextual space can be defined to identify different types of performance related to understanding a particular concept or principle. We suggest that activity, as exemplified by three general classes of tasks, and generality are particularly appropriate for identifying apmfile of performance in this contextual space, and that this profile provides the empirical basis for constructing theories about the relations among skills required for different tasks. The third proposition is that the acquisition of understanding be studied in terms of sequences of profiles that emerge during the course of development or instruction, as opposed to studying changes in performance on a single task. As we have illustrated, this approach should provide valuable insights into mechanisms underlying the acquisition processes, Given this framework, the empirical goal is to describe relations among tasks, or profiles, that emerge during development and instruction. The theoretical goal, then, is to describe knowledge acquisition systems that can account for the observed changes in profiles. We expect that these theories will provide useful insights for both the assessment and instruction of mathematical understanding and, moreover, that the process of attempting to convert these insights into educational practice will contribute to richer and more revealing theories and methods of research. ACKNOWLEDGEMENTS Preparation of this chapter was supported with grants from the Natural Sciences and Engineering Research Council of Canada to both authors. We are grateful to Jamie Campbell, Karen Fuson, Alison Kulak, Don Mabbott, and Laura Novick for helpful comments on a previous draft. Correspondence may be addressed to J. Bisanz, Department of Psychology, University of Alberta, Edmonton, AB, Canada T6G 2E9, or to J. LeFevre, Department of Psychology, Carleton University, Ottawa, Ontario, Canada K l S 5B6.

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