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The development of mathematical cognition
Cognitive
Development,
1, 157-170
(1986)
The Development
of Mathematical Cognition Daniel
University
of California,
B. Kaye Los
Angeles
Whereas descriptions of the developmental transition from slow, implicit counting to more rapid, effortless memory retrieval of simple addition problem solutions are available in the literature, it remains unclear what are the procesGng advantages of the shift from counting to retrieval. Studies within the psychometric tradition have pointed to the importance of the efficiency of basic computational processes in the development of mathematical ability, and the development of automatic routines for solving problems or retrieving declarative knowledge from memory are richly described in a variety of information processing theories. A theoretical account of the developmental transition from counting to retrieval based upon principles of attention theory is offered. Research using dual-task methods is described and a preliminary developmental model is outlined.
A central theme in much of the work on numerical cognition is the nature of the developmental transition from reliance on slow-acting procedural knowledge to fast-acting procedural knowledge or rapid access of declarative knowledge. Although there is debate (Ashcraft, 1983; Ashcraft, Fierman, & Bartolotta, 1984; Baroody, 1983, 1984) as to the relative importance of these two basic types of knowledge, I would concur with the essence of Ashcraft’s (1983) description that younger children are more reliant on slower implicit counting strategies, and with development, more children retrieve a declarative representation of numerical facts from memory. This proposed transition to increased reliance on memory retrieval is correlated with the increase in speed of numerical problem solving, at least in the context of simple addition problems (see also Ashcraft, 1985; Rabinowitz, 1985; Siegler, 1984). In this article, I discuss the implications of these process and speed changes on the problem-solving abilities of children and adults. Reviews of research on the transition from counting to memory retrieval are available elsewhere (Ashcraft, 1982, 1985; Kaye, Post, Hall, & Dineen, in press). Portions of this article were presented at the Biennial Meetings of the Society for Research in Development, Toronto, April, 1985; and at the Eighth Biennial Meetings of the I.S.S.B.D., Tours, France, July, 1985. Contributions by Virginia Bonnefil and Chen Qi are greatly appreciated. Research reported herein was supported in part by a Spencer Foundation seed grant (through the University of California). Constructive comments from Wendell E. Jeffrey are greatly appreciated. Child
Manuscript
received
October
28, 1985;
revision
accepted
November
29, 1985
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It is one thing to describe these process changes and to note the increases in simple problem-solving speed as children grow more experienced and older, but it is another matter to understand the advantages of these changes. What is the nature of the gain, if any, on .information processing or problem solving that accompanies the change from reconstructive processing to reproductive processing? Is there an advantage to relying on knowledge structures in memory when solving simple addition problems? Does extended reliance on counting strategies inhibit successful performance in mathematical problem solving? These are questions that should be answered for a better understanding of the acquisition of mathematical skill. LIMITED
CAPACITY
INFORMATION
PROCESSING
If we assume that complex problem solving is constrained by the limited capacity of the information processing system, then it perhaps is required that some component processes be executed automatically (i.e., without attention) because other processes will be capacity demanding. I have pointed out elsewhere (Kaye, Bonnefil, & Chen, 1983), as has Ashcraft (1982), that one important aspect of the development of numerical skills between middle-elementary school and adulthood is the efficiency of simple numerical computation processes. In the domain of arithmetic processing, it may be that the simple numerical computation processes not only become faster, but probably require less and less attention. Perhaps these computations are calculated automatically or the declarative knowledge is accessed automatically, i.e., in an effortless fashion, with increasing experience. This notion of the role of automaticity in information processing has been the focus of much theory in the recent past. In the domain of mathematical problem solving, the roots of this idea can be traced at least to some of the early psychometricians. Spearman (1927) isolated a numerical factor over and above the g factor, and this factor captured the fundamental computation processes. Thurstone (1938) described the “N” factor, or numerical ability that was tested pith simple and speeded numerical computations, as in the addition or multiplication of pairs of single-digit numbers-essentially the task used by those who study the chronometry of simple arithmetic. But it was the Swedish researcher Werdelin (1958) who described a numerical factor (“N”) in terms more similar to modem concepts of the automaticity of overlearned routines. Coombs (1941) had concluded that numerical ability represents agility in handling well-established associations and Werdelin emphasized the importance of automatizing reasoning, especially with regard to numerical operations, for the purpose of economizing mathematical problem solving. He saw the numerical factor as capturing the ability to automatize mathematical reasoning through extensive practice. In this way, little if any conscious thought or mental effort would be required to apply mathematical rules or principles in
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numerical problem solving. Automaticity would be possible for any level of problem complexity, requiring more practice for greater complexity. All in all, the psychometric approach to the study of mathematical ability and intelligence has provided a broad picture of a distinct general intellectual ability complemented by two other factors: (a) a numerical factor concerned mostly with basic computational skills, and (b) a mathematical factor that incorporates advanced ability in mathematics relevant to school achievement. If we accept this type of evidence that numerical proficiency is an important and basic component of intelligence, we might further assume that at least part of the achievement in higher order mathematical problem solving would be contingent upon some level of efficiency of basic numerical skills. An interesting question is not only what the consequences of experience or numerical fact representation might be for simple addition problems, but what are the processing consequences on tasks in which computation is but a component task? When doing algebra or calculus or some other type of mathematics, what is the effect of computational processing efficiency? If computation is an effortful, time-consuming process, then will further acquisition of procedures and concepts in the mathematical domain be inhibited? Or, even if more complex procedures or concepts are within the knowledge schemata of an individual, will effortful computation diminish performance in higher level matehmatical tasks in which computation is a component process? It is conceivable that experts in mathematics need not be the most proficient “calculators.” Indeed, I’m sure we all have a friend or acquaintance who appears otherwise intelligent or proficient in quantitative tasks but is not the best at adding, subtracting, multiplying, or dividing even the smallest of numbers. One question for us to consider in the future is to what extent the strategy and efficiency with which we attack simple computational problems constrain performance in related, but perhaps more complex, mathematical tasks. With regard to development, if a child is less than an automatic calculator of simple numerical problems or retriever of stored number facts, will attempts to acquire other skills, e.g., word problem-solving skills, algebra, calculus, or even geometry, be roadblocked? My position is that the development of some critical level of computational proficiency will be predictive of success at higher levels of mathematical complexity. Such a position is consistent with that of Toyama (cited in Hatano, 1982b), who argued that computational ability is an important component of analytical ability, so that early competence in calculation ability may facilitate analytic problem solving ability in general.
Knowledge Structures The question of the relationship between knowledge and the processes that operate upon that knowledge is one of the basic issues in modem-day cognitive science: If we assume that there are at least two types of knowledge-procedural and declarative (cf. Anderson, 1983)-then we may ask what is the interre-
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lationship between these types of knowledge, developmentally speaking as well as in terms of the acquisition of any particular skill. For example, in learning that “3 + 2 = 5,” a child could (a) learn the basic numerical principles (declarative knowledge) that are requisite for performing this addition problem (Gelman, 1982); (b) memorize this numerical fact (declarative knowledge); (c) learn that in order to compute this sum, one must begin at “3” and increment this number by “2” more, with the result of “5” (procedural knowledge); or (d) learn to retrieve the sum of 3 + 2 from declarative memory, where the numerical fact is stored (procedural knowledge). Although it is conceivable that procedures and declarative facts may be acquired independently, it is more likely that procedures and declarative knowledge develop in synchrony: As the declarative networks of numerical knowledge become more complex, more sophisticated procedures are developed and optimized, in turn leading to the acquisition of yet more declarative knowledge.
Modes of Information
Processing
One coherent theoretical system within which it is feasible to place Baroody’s (1983, 1984) emphasis on specialized procedures as well as Ashcraft’s (1982, 1983) emphasis on access of declarative knowledge from memory is Anderson’s (1983) ACT* theory. At early levels of skill acquisition, knowledge is represented in declarative form. General procedures (productions) use declarative knowledge “interpretively”: This process is not automatic and is a critical step in the acquisition of skill since the learner has some cognitive control over the process by acknowledging errors and correcting procedures. Through the process of knowledge compilation, which includes composition (or the combination of several productions into a single production) and proceduralization (which results in productions performing behaviors directly), relatively automatic production systems evolve at high levels of skill. The later processes of strengthening and tuning of production systems result in the raw increases in performance speed seen at later stages of skill development. Thus, the problem “3 + 2 = ?” may first be dealt with by primitive counting procedures operating on simple arithmetic facts stored in declarative knowledge. Later, more specific counting procedures may be developed, resulting in more efficient counting performance. But when highly organized declarative networks of numerical knowledge develop, then a shift to memory retrieval procedures may replace the less efficient counting procedures. Finally, the problem may be executed rapidly and correctly, because the productions responsible for encoding the addends-retrieving the sum from memory and responding-will be executed without attention. Essentially, this sequence of stages from goal-directed, to data-driven, processing corresponds rather well to other transitions in cognitive processing, among which are the processing dichotomies of controlled and automatic-processing (Schneider & Shiffrin, 1977), conscious attention and automatic activation
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(Posner, 1978), and various conceptualizations of top-down and bottom-up processing (e.g., Laberge & Samuels, 1974). Let’s presume that the procedural knowledge and the declarative knowledge upon which the procedures act form production systems. These production systems fall along a continuum of attentional capacity demanded. At early levels of experience, we would expect the production systems to be invoked only with effortful processing; later, after extensive task practice, the systems could be invoked automatically. With development, specific mathematical knowledge becomes packed into local processing systems that act autonomously, as production systems, and free attention to execute other, more global, aspects of problem solution (Stemberg, 1984a). Increased problem-solving efficiency is the result, with more cognitive acts performed correctly per unit time. A similar view of development is given by Siegler (1984). In his model of the “distribution of associations” between particular addition problems and their potential answers, strength of associations in the declarative network is what increases with development. As the strength of association increases, the likelihood that an addition problem is solved via rapid, error-free retrieval, rather than some other slower overt strategy such as counting, increases as well. Computer simulations (Siegler, 1984; Siegler & Shrager, 1984) based on the associations model accou$ed for performance of 4- and 5-year-olds studied by Siegler and Robinson (198 1) and Siegler and Shrager (1984) (also see simulations by Ashcraft, 1985, and Rabinowitz, 1985, for influence of speed). The developmental mechanism of increasing strength of association was attributed to several sources, among them, the frequency with which different problems are presented to children. Certain problems were found to be more frequently presented by parents; hence, the strength of association between these problems and their answers was higher, leading to more frequent executions of memory-retrieval procedures and hence faster and more accurate responding. The question of economy of information processing with retrieval versus overt strategies was not the focus of these studies, but the model was proposed to account for development of skill in other cognitive domains as well (e.g., spelling). In this spirit, Siegler (1984) is proposing that the sort of mechanism of development that accounts for arithmetic achievement in particular may be a fundamental process of intellectual development in general. It may be possible to understand how the psychometricians have found numerical computation efficiency to comprise a basic factor of intelligence if what underlies such a factor is a general ability to organize basic knowledge in a declarative network and then to maximize access to and from the network with experience. Cognitive Research on Simple Arithmetic The general approach that I am describing fits well with much of the literature on cognitive research on addition and subtraction. There have been many studies
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describing the counting strategies of very young children (see Carpenter, Moser, & Romberg, 1982). Elaborate descriptions of the operations that need to be applied to problems representing different quantitative relationships presented in different formats have led some researchers to formulate models of the child’s internal representations of computational procedures (procedural schemata) and of conceptual understandings (conceptual schemata). Riley, Greeno, and Heller (1982), Resnick (1982), and Vergnaud (1982) have alluded to the interaction of these cognitive structures in accounting for children’s performance in simple arithmetic computation and word problems. Romberg (1982) claimed that there is a general paradigm evolving in which researchers are constructing cognitive models of children’s mathematical problem solving appropriate for different developmental levels. From this perspective, development is viewed as contingent upon successful induction both of proper computational strategies and of concepts about numbers and numerical relationships, which are stored in declarative knowledge networks. Development of adequate declarative schemata will allow for the acquisition of more sophisticated strategies (procedures), and these in turn will allow for the increase in complexity of conceptual understanding. If the procedural knowledge is emphasized at the expense of the conceptual, then highly skilled computational experts may develop. But, as Hatano (1982a) warned, these may be only “routine experts”-experts in terms of speed, accuracy, and automaticity of simple calculations. Without the concomitant training in conceptual knowledge, further achievement may not occur. One would presume that increases in declarative knowledge can enable the generation of new knowledge through general inferential processes. One of my contentions is that this automaticity of procedural knowledge, or access of the declarative knowledge instead, may be a necessary prerequisite for successful achievement in conceptual domains. This may turn out to be too strong a hypothesis; instead, it may be that some critical level of efficiency of simple computation, short of complete automaticity, may allow a reasonable level of conceptual understanding to evolve.
Mathematics and Processing Efficiency So how would we go about testing some of these notions of processing efficiency from a cognitive information-processing perspective? In the most general sense, we would have to relate, at each “stage” of mathematical development, the level of efficiency of simple computation with the acquisition of declarative knowledge. Structural modeling techniques in a longitudinal design would be useful for determining the causal links between the development of automatic computational skills and the acquisition of conceptual nodes in the declarative network. But what is the essence of this iink between computational automaticity and uonceptual acquisition? As I suggested earlier, as mathematical problems become more complex, their solution may depend on more than just adequate procedural and declarative knowledge: It is possible that efficient allocation of
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attention to various components of a task will be essential for optimal application of the knowledge schemata. Inefficient allocation could inhibit to varying degrees the application of available knowledge-hence, competence/performance disparities would be understood in terms of processing inefficiency. For example, Siegler and Robinson (1981) describe the inability of preschoolers to draw connections between separate mathematical abilities. With increasing proficiency in these skills that “develop in relative isolation from one another . . . then greater use [may be made] of the potential interconnections among them” (Siegler & Robinson, 198 1, p. 70). This perspective has been a popular one in the recent literature but the processes by which independent domains of knowledge become interdependent are as yet unspecified. In part, I suspect that capacity limitations during problem solving often prevent interconnections from being realized. If simple addition is a difficult, capacity-demanding task for a preschooler, then spare capacity may not be available to incorporate knowledge of number comparisons in the addition process. The proficiency Siegler and Robinson refer to may be best understood by reference to the notions of automatic processing resulting from extended practice (Schneider & Shiffrin, 1977), with which cues in stimulus information would result in more rapid and effortless access of information in memory. In this way, available processing capacity could be reallocated to more capacity-consuming aspects of the problem-solving situation. Alternatively, proficiency could refer to increases in the number or total capacity of resources available for problem solving. Case (1982, 1985) has argued that parallel developments across different content domains may be accounted for by general growth of central processing capacity. Shavelson (1981) has drawn attention to Case’s views and suggested that automatization of arithmetic operations, through extensive practice and overlearning, will reduce demands on memory and thereby allow for additional storage space for new incoming information. That is, the development of processing efficiency may be seen as an enabling condition for the acquisition of new information, whether procedural or declarative. Such notions are consistent with the views of Coombs (1941), Werdelin (1958), Hatano (1982a, 1982b), and Siegler (1984), all mentioned earlier. As an example of research concerning the role of efficient processing in higher level mathematical problem solving, Muth (1984) studied some of the component abilities contributing to arithmetic word problem solving in sixth graders. She focused on subjects’ computational ability as well as their attention to extraneous information in the problem. Once reading ability was partialed, it was found that computational ability and the effect of extraneous information in the problem both predicted the number of correct word-problem answers and correct set-ups (defined by correct analysis but incorrect answer). Extraneous information, but not computational ability, predicted solution latency over and above reading ability. The results were consistent with the view that the extraneous information placed “formidable demands on the student’s limited process-
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ing capacities” (p. 210) and that better computational skills were related to higher level functioning in math. It would have been interesting to see what the effect of extraneous information on performance was as a function of level of computational skill, but this analysis was not conducted. Brainerd (1983) argues that the emphases on chronometric analysis of arithmetic as well as the more traditional Piagetian and neo-Piagetian views of number development fail to differentiate between development of processing skill and short-term memory effects. In his view, memory failures play a large role in the development of numerical expertise, independently of changes in processing operations, which have been studied predominantly in recent research. In an elegant mathematical analysis of addition and subtraction performance of 4-, 5, and 6-year-olds, Brainerd demonstrates that it was short-term memory (encoding and retrieval together) that developed more rapidly during this age span than arithemtical processing. I agree with Brainerd’s assertion that emphasis on the nature of arithemtic processing operations (e.g., counting vs. retrieval) generally has led to failures to acknowledge the important performance-limiting role played by short-term memory factors. But it does not follow that the chronometric and rule-assessment paradigms cannot be used to study the influence of short-term memory processes on arithmetic performance in young children. On the contrary, we have found it possible to measure the efficiency of arithmetic processing at different stages of processing corresponding to encoding, storage, retrieval, and decision phases of information processing (Kaye, Bonnefil, & Chen, 1983). Our technique is described later in this paper. Although the chronometric studies, along with interview studies, have been helpful in describing the strategy shifts through elementary school, these descriptions are insufficient to account for development of hierarchically more complex mathematical skills. Again, Siegler and Robinson (198 1) suggest that models of children’s knowledge can be built upon “upward and outward” to account for development of skill in more complex mathematical domains. But what allows for the actual transition from one level of skill to another? Surely, instruction, formal or informal, must play a role; extended practice is important as well, for reasons mentioned above. We must continue to specify the nature of the missing link: What limits the interconnections between separate sources of knowledge present in one’s memory systems (cf. Stemberg, 1984b)?
Addition and Theories of Attention It would seem that we must study tasks in which more than one source of knowledge must be used. Many modem theories of attention address the task of solving one problem while simultaneously attempting memory retrieval of taskrelevant information (see Eysenck, 1982). Whereas some theorists maintain that there is always a fixed pool of resources that may be distributed to different tasks performed simultaneously (Norman & Bobrow, 1975), others have argued that the total amount of processing capacity changes both within individuals and
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across tasks as a function of task demands (Kahneman, 1973). Although the differences in empirical predictions derived from such different theories are necessary to consider, most modem attention theorists believe that we can learn a great deal about people’s levels of skill from their performance on tasks that demand divided attention. Kerr (1973), in her review of limited-capacity attention theories, remarked that we can observe the limits of processing capacity when we observe how performance decreases when a person is asked to do more than one thing at a time. Although we may be able to attend to but one thing at a time, we can process several things simultaneously if the processes become habitual or automatic. Either highly practiced skills require less attentional capacity and hence the total pool of resources is not exceeded by multiple task processing, or greater levels of skill derived from extended practice allow the subject to divide attention more efficiently among relevant tasks. There is no known limit to human mental-processing capacity; it has been shown that continued practice with multiple, high-level cognitive tasks can result in quite amazing levels of performance (Hirst, Spelke, Reaves, Caharack, & Neisser, 1980). How can such theoretical notions be applied to the development of mathematical ability? There is little doubt that even smiple arithmetic problem solving (e.g., addition or subtraction) involves multiple-task components (encoding, retrieval or computation, response). To the extent that any of these operations can be made automatic, problem solution will be derived in less time and perhaps with less effort expended. Recent research by Kaye et al. (1983) involves measurement of attentional allocation during simple arithmetic problem solving using dual-task techniques. We wanted to show that strategy changes are of functional importance since they result in additional processing capacity that can be used to accomplish more complex problem solving, as in proficient word-problem solving. The work of Carpenter and Moser (1982) suggests that basic computational strategies are related to word problem-solving performance; we would like to show that this in part is due to processing efficiency. If a child or adult is attempting to solve a word problem involving addition, subtraction, or some other operation, then the extent to which the computational aspects of the problem demand conscious attention may interfere with the subject’s ability to process the syntactic, logical, or semantic aspects of the sentences. Perhaps the child has the requisite skills to add two numbers together or to read a sentence and understand its meaning. Unless these separate skills are proficient, then, when their combined effort is required, the processing resources available may not be sufficient for adequate solution. The bottleneck of processing efficiency could be almost anywhere in the problem solution. It is our belief that the systematic, hierarchical developments in mathematical ability alluded to by so many researchers may be understood better in terms of mechanisms of efficient capacity allocation. Using a dual-task paradigm, Kaye et al. (1983) were able to measure the attentional capacity left free during different stages of information processing of
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a two-term addition verification problem, i.e., the encoding of the addends and computation or retrieval of the actual sum, encoding of the stated answer and comparison with the computed or retrieved answer, and finally the decision and response stages. The addition problem comprised the primary task, and a simple reaction time to detection of an auditory probe interpolated at different intervals during the visual presentation of the addition problem served as the secondary task-dependent measure. We tested children in grades 2, 4, and 6, as well as college students and derived samples of higher and lower achieving subjects in each grade based on scores on standardized tests of computation and conceptual understanding. By so doing, we were able to pinpoint stages of problem performance that were attention demanding and those that were not, and then we were able to relate performance at these different stages to arithmetic ability. In brief, we found that ability differences in middle-elementary school were correlated with attentional efficiency during the encoding and computation phases, whereas the ability differences in college were correlated with attentional and response efficiency in comparison and decision stages. This pattern of results is consistent with much of the literature to which I have referred. In middle-elementary school, more subjects are making the transition from counting to retrieval strategies, and development after this point is largely a matter of increasing speed of processing. Anderson’s (1983) theory gives a coherent structure for these results: There is a transition from early interpretive productions to later knowledge compilation and finally the strengthening and tuning of productions. In the larger view of development across many years of practice, both procedural and declarative knowledge would be come integrated in production systems that are invoked involuntarily and executed accurately and quickly when the appropriate problem solving situation presents itself. The work of Siegler (1984), Ashcraft (1985), and Rabinowitz (1985) strengthens my belief that such development centers about the availability and the accessibility of the computational facts of arithemtic in memory. MODEL
OF THE DEVELOPMENT
OF NUMERICAL
COGNITION
The theory and research that I have reviewed may be summarized best by presenting a general, preliminary model of how mathematical proficiency develops. The following propositions form the crux of this model. 1. In the most general form, I maintain in this model that the development of arithmetic skill is generally a function of increased efficiency of information retrieval (to replace overt computation). Not only does development involve a shift from counting to fact retrieval, but this shift produces cognitive savings due to reduced attentional demands of computation. These cognitive savings are realized in time in the acquisition of higher order concepts and procedures. 2. Once the basic number concepts have been acquired in the preschool years (Gelman, 1982; Gelman & Gallistel, 1978), practice with counting objects and
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numbers results in arithmetic skill development and increasing strength of associations between sums (in addition) and numbers to be added (Ashcraft, 1985; Siegler, 1984). 3. As in Brainerd’s (1983) model, short-term memory interfaces between problems that are presented and the information to be accessed in long-term memory. If encoding is inaccurate, then performance will be impaired even if access of information from long-term memory is proficient. If access of information is difficult, then subjects may resort to overt strategizing, e.g., counting (Siegler, 1984). In this event, processing will take longer, allowing for other short-term memory failures, such as forgetting the value of one or more of the addends in an addition problem. Thus, even if the processing skill is applied correctly, incorrect solution may be attained. 4. If the child is required to use arithmetic computation to solve a more complex mathematical problem, then performance will be a function of a number of component skills, among them, computational proficiency. To the extent that computation is attention demanding, then other aspects of problem solution will be impaired. If attentional load is high due to inefficient computation, then conceptual linkages will fail to be realized, and thus development of mathematical expertise will be delayed. 5. Proposition 4 above predicts that some criteria1 level of computational efficiency will be requisite for acquisitions of higher level mathematical skill, both declarative/conceptual and procedural. This assertion is of course empirically testable (cf. Muth, 1984). 6. If computational efficiency is a necessary condition for the development of expertise in the mathematical domain, then “experts” at different levels of development should have developed efficient, and even automatic, computation skills. 7. At higher levels of mathematical complexity, once other knowledge domains become relevant to mathematical problem solution, then continuing expertise development will be a matter of developing automatic routines to access and use the relevant knowledge structures. 8. Propositions 6 and 7 imply that the critical aspect of performance at any level of complexity of mathematics is the degree to which the access and application of knowledge necessary to solve a problem is efficient and the extent to which processing capacity is left free to execute relevant strategies that may not depend on automatic information retrieval. To avoid circularity, it would be imperative to conduct task analyses at each level of mathematical problem so as to be able to differentiate aspects of a problem that are directly relevant to task solution. One obvious criticism is that computation itself is often irrelevant to problem solution in various mathematical domains (e.g., some types of geometry, calculus). But it is my contention that, developmentally, there will be a fairly consistent sequence of knowledge structures that are “compiled, strengthened, and tuned” (Anderson, 1983). Basic number and computation facts comprise the first, or one of the first-such, declarative networks in the mathematical domain.
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So, it is predicted that efficient use of this network in early elementary school will be the strongest predictor of later mathematical achievement. ‘CONCLLJSION In sum, 1 think that one new direction for chronometric studies of numerical cognition is to determine the information-processing consequences of different types of computational and retrieval strategies. If an accurate characterization of the development of addition skill is one of a transition from reliance on computation to reliance on retrieval from a declarative-knowledge network, then we should determine the cognitive savings of retrieval strategies over computation. Once retrieval strategies, or perhaps also advanced procedures (of the type Baroody studies), become depended upon, then the extent to which increased practice produces economical problem solving when computation is but a component of more complex mathematical problems may predict the development course of expertise. As I have suggested, the richness of theories of attention affords the cognitive researcher a variety of methods with which to study mathematical cognition. REFERENCES Anderson,
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Psychology.
Resnick,
Development Toronto. L.B. (1982). Syntax
Riley,
& T.A. Romberg (Eds.), Erlbaum. M.S., Greeno, J.G., &
and semantics
Addirion Heller,
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and subtraction: J.I.
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Carpenter,
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Moser,
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ability. In H.P. Ginsburg (Ed.), The development of ma~hemarical rhinking. New York: Academic Press. Romberg, T.A. (1982). An emerging paradigm for research on addition and subtraction skills. In T.P.
J.M. Moser, & T.A. Romberg (Eds.), Addition and subfraction: Hillsdale: NJ: Erlbaum. & Shiffrin, R.M. (1977). Controlled and automatic human information
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R.S. (1984, October). Symposium on Child Siegler, R.S., & Robinson,
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Reese & L.P. Lipsitt (Eds.). York: Academic Press. Siegler. R.S., & Shrager, J. (1984).
The
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development
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know what to do? In C. Sophian (Ed.), Origins Spearman. C. (1927). The abilities of man. New York:
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R.J. (1984a). A triarchic theory of intelligence. Behavioral a&Bruin Sciences, 7, 269287. Stemberg, R.J. (Ed.). (1984b). Mechanisms ofcognirive developmenr. New York: Freeman. Thurstone, L.L. (1938). Primary mental abilities. Psychometric Monogrphs, No. I. Vergnaud, G. (1982). A classification~of cognitive tasks and operations of thought involved in addition and subtraction problems. In T.P. Carpenter, J.M. Moser, & T.A. Romberg (Eds.), Addirion and subrracrion: A cognirive perspecrive. Hillsdale, NJ: Erlbaum. Werdelin, L. (19.58). The mdremuricul nbiliry. Lund, Sweden: Ohlsson.
Development,
1, 157-170
(1986)
The Development
of Mathematical Cognition Daniel
University
of California,
B. Kaye Los
Angeles
Whereas descriptions of the developmental transition from slow, implicit counting to more rapid, effortless memory retrieval of simple addition problem solutions are available in the literature, it remains unclear what are the procesGng advantages of the shift from counting to retrieval. Studies within the psychometric tradition have pointed to the importance of the efficiency of basic computational processes in the development of mathematical ability, and the development of automatic routines for solving problems or retrieving declarative knowledge from memory are richly described in a variety of information processing theories. A theoretical account of the developmental transition from counting to retrieval based upon principles of attention theory is offered. Research using dual-task methods is described and a preliminary developmental model is outlined.
A central theme in much of the work on numerical cognition is the nature of the developmental transition from reliance on slow-acting procedural knowledge to fast-acting procedural knowledge or rapid access of declarative knowledge. Although there is debate (Ashcraft, 1983; Ashcraft, Fierman, & Bartolotta, 1984; Baroody, 1983, 1984) as to the relative importance of these two basic types of knowledge, I would concur with the essence of Ashcraft’s (1983) description that younger children are more reliant on slower implicit counting strategies, and with development, more children retrieve a declarative representation of numerical facts from memory. This proposed transition to increased reliance on memory retrieval is correlated with the increase in speed of numerical problem solving, at least in the context of simple addition problems (see also Ashcraft, 1985; Rabinowitz, 1985; Siegler, 1984). In this article, I discuss the implications of these process and speed changes on the problem-solving abilities of children and adults. Reviews of research on the transition from counting to memory retrieval are available elsewhere (Ashcraft, 1982, 1985; Kaye, Post, Hall, & Dineen, in press). Portions of this article were presented at the Biennial Meetings of the Society for Research in Development, Toronto, April, 1985; and at the Eighth Biennial Meetings of the I.S.S.B.D., Tours, France, July, 1985. Contributions by Virginia Bonnefil and Chen Qi are greatly appreciated. Research reported herein was supported in part by a Spencer Foundation seed grant (through the University of California). Constructive comments from Wendell E. Jeffrey are greatly appreciated. Child
Manuscript
received
October
28, 1985;
revision
accepted
November
29, 1985
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It is one thing to describe these process changes and to note the increases in simple problem-solving speed as children grow more experienced and older, but it is another matter to understand the advantages of these changes. What is the nature of the gain, if any, on .information processing or problem solving that accompanies the change from reconstructive processing to reproductive processing? Is there an advantage to relying on knowledge structures in memory when solving simple addition problems? Does extended reliance on counting strategies inhibit successful performance in mathematical problem solving? These are questions that should be answered for a better understanding of the acquisition of mathematical skill. LIMITED
CAPACITY
INFORMATION
PROCESSING
If we assume that complex problem solving is constrained by the limited capacity of the information processing system, then it perhaps is required that some component processes be executed automatically (i.e., without attention) because other processes will be capacity demanding. I have pointed out elsewhere (Kaye, Bonnefil, & Chen, 1983), as has Ashcraft (1982), that one important aspect of the development of numerical skills between middle-elementary school and adulthood is the efficiency of simple numerical computation processes. In the domain of arithmetic processing, it may be that the simple numerical computation processes not only become faster, but probably require less and less attention. Perhaps these computations are calculated automatically or the declarative knowledge is accessed automatically, i.e., in an effortless fashion, with increasing experience. This notion of the role of automaticity in information processing has been the focus of much theory in the recent past. In the domain of mathematical problem solving, the roots of this idea can be traced at least to some of the early psychometricians. Spearman (1927) isolated a numerical factor over and above the g factor, and this factor captured the fundamental computation processes. Thurstone (1938) described the “N” factor, or numerical ability that was tested pith simple and speeded numerical computations, as in the addition or multiplication of pairs of single-digit numbers-essentially the task used by those who study the chronometry of simple arithmetic. But it was the Swedish researcher Werdelin (1958) who described a numerical factor (“N”) in terms more similar to modem concepts of the automaticity of overlearned routines. Coombs (1941) had concluded that numerical ability represents agility in handling well-established associations and Werdelin emphasized the importance of automatizing reasoning, especially with regard to numerical operations, for the purpose of economizing mathematical problem solving. He saw the numerical factor as capturing the ability to automatize mathematical reasoning through extensive practice. In this way, little if any conscious thought or mental effort would be required to apply mathematical rules or principles in
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numerical problem solving. Automaticity would be possible for any level of problem complexity, requiring more practice for greater complexity. All in all, the psychometric approach to the study of mathematical ability and intelligence has provided a broad picture of a distinct general intellectual ability complemented by two other factors: (a) a numerical factor concerned mostly with basic computational skills, and (b) a mathematical factor that incorporates advanced ability in mathematics relevant to school achievement. If we accept this type of evidence that numerical proficiency is an important and basic component of intelligence, we might further assume that at least part of the achievement in higher order mathematical problem solving would be contingent upon some level of efficiency of basic numerical skills. An interesting question is not only what the consequences of experience or numerical fact representation might be for simple addition problems, but what are the processing consequences on tasks in which computation is but a component task? When doing algebra or calculus or some other type of mathematics, what is the effect of computational processing efficiency? If computation is an effortful, time-consuming process, then will further acquisition of procedures and concepts in the mathematical domain be inhibited? Or, even if more complex procedures or concepts are within the knowledge schemata of an individual, will effortful computation diminish performance in higher level matehmatical tasks in which computation is a component process? It is conceivable that experts in mathematics need not be the most proficient “calculators.” Indeed, I’m sure we all have a friend or acquaintance who appears otherwise intelligent or proficient in quantitative tasks but is not the best at adding, subtracting, multiplying, or dividing even the smallest of numbers. One question for us to consider in the future is to what extent the strategy and efficiency with which we attack simple computational problems constrain performance in related, but perhaps more complex, mathematical tasks. With regard to development, if a child is less than an automatic calculator of simple numerical problems or retriever of stored number facts, will attempts to acquire other skills, e.g., word problem-solving skills, algebra, calculus, or even geometry, be roadblocked? My position is that the development of some critical level of computational proficiency will be predictive of success at higher levels of mathematical complexity. Such a position is consistent with that of Toyama (cited in Hatano, 1982b), who argued that computational ability is an important component of analytical ability, so that early competence in calculation ability may facilitate analytic problem solving ability in general.
Knowledge Structures The question of the relationship between knowledge and the processes that operate upon that knowledge is one of the basic issues in modem-day cognitive science: If we assume that there are at least two types of knowledge-procedural and declarative (cf. Anderson, 1983)-then we may ask what is the interre-
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lationship between these types of knowledge, developmentally speaking as well as in terms of the acquisition of any particular skill. For example, in learning that “3 + 2 = 5,” a child could (a) learn the basic numerical principles (declarative knowledge) that are requisite for performing this addition problem (Gelman, 1982); (b) memorize this numerical fact (declarative knowledge); (c) learn that in order to compute this sum, one must begin at “3” and increment this number by “2” more, with the result of “5” (procedural knowledge); or (d) learn to retrieve the sum of 3 + 2 from declarative memory, where the numerical fact is stored (procedural knowledge). Although it is conceivable that procedures and declarative facts may be acquired independently, it is more likely that procedures and declarative knowledge develop in synchrony: As the declarative networks of numerical knowledge become more complex, more sophisticated procedures are developed and optimized, in turn leading to the acquisition of yet more declarative knowledge.
Modes of Information
Processing
One coherent theoretical system within which it is feasible to place Baroody’s (1983, 1984) emphasis on specialized procedures as well as Ashcraft’s (1982, 1983) emphasis on access of declarative knowledge from memory is Anderson’s (1983) ACT* theory. At early levels of skill acquisition, knowledge is represented in declarative form. General procedures (productions) use declarative knowledge “interpretively”: This process is not automatic and is a critical step in the acquisition of skill since the learner has some cognitive control over the process by acknowledging errors and correcting procedures. Through the process of knowledge compilation, which includes composition (or the combination of several productions into a single production) and proceduralization (which results in productions performing behaviors directly), relatively automatic production systems evolve at high levels of skill. The later processes of strengthening and tuning of production systems result in the raw increases in performance speed seen at later stages of skill development. Thus, the problem “3 + 2 = ?” may first be dealt with by primitive counting procedures operating on simple arithmetic facts stored in declarative knowledge. Later, more specific counting procedures may be developed, resulting in more efficient counting performance. But when highly organized declarative networks of numerical knowledge develop, then a shift to memory retrieval procedures may replace the less efficient counting procedures. Finally, the problem may be executed rapidly and correctly, because the productions responsible for encoding the addends-retrieving the sum from memory and responding-will be executed without attention. Essentially, this sequence of stages from goal-directed, to data-driven, processing corresponds rather well to other transitions in cognitive processing, among which are the processing dichotomies of controlled and automatic-processing (Schneider & Shiffrin, 1977), conscious attention and automatic activation
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(Posner, 1978), and various conceptualizations of top-down and bottom-up processing (e.g., Laberge & Samuels, 1974). Let’s presume that the procedural knowledge and the declarative knowledge upon which the procedures act form production systems. These production systems fall along a continuum of attentional capacity demanded. At early levels of experience, we would expect the production systems to be invoked only with effortful processing; later, after extensive task practice, the systems could be invoked automatically. With development, specific mathematical knowledge becomes packed into local processing systems that act autonomously, as production systems, and free attention to execute other, more global, aspects of problem solution (Stemberg, 1984a). Increased problem-solving efficiency is the result, with more cognitive acts performed correctly per unit time. A similar view of development is given by Siegler (1984). In his model of the “distribution of associations” between particular addition problems and their potential answers, strength of associations in the declarative network is what increases with development. As the strength of association increases, the likelihood that an addition problem is solved via rapid, error-free retrieval, rather than some other slower overt strategy such as counting, increases as well. Computer simulations (Siegler, 1984; Siegler & Shrager, 1984) based on the associations model accou$ed for performance of 4- and 5-year-olds studied by Siegler and Robinson (198 1) and Siegler and Shrager (1984) (also see simulations by Ashcraft, 1985, and Rabinowitz, 1985, for influence of speed). The developmental mechanism of increasing strength of association was attributed to several sources, among them, the frequency with which different problems are presented to children. Certain problems were found to be more frequently presented by parents; hence, the strength of association between these problems and their answers was higher, leading to more frequent executions of memory-retrieval procedures and hence faster and more accurate responding. The question of economy of information processing with retrieval versus overt strategies was not the focus of these studies, but the model was proposed to account for development of skill in other cognitive domains as well (e.g., spelling). In this spirit, Siegler (1984) is proposing that the sort of mechanism of development that accounts for arithmetic achievement in particular may be a fundamental process of intellectual development in general. It may be possible to understand how the psychometricians have found numerical computation efficiency to comprise a basic factor of intelligence if what underlies such a factor is a general ability to organize basic knowledge in a declarative network and then to maximize access to and from the network with experience. Cognitive Research on Simple Arithmetic The general approach that I am describing fits well with much of the literature on cognitive research on addition and subtraction. There have been many studies
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describing the counting strategies of very young children (see Carpenter, Moser, & Romberg, 1982). Elaborate descriptions of the operations that need to be applied to problems representing different quantitative relationships presented in different formats have led some researchers to formulate models of the child’s internal representations of computational procedures (procedural schemata) and of conceptual understandings (conceptual schemata). Riley, Greeno, and Heller (1982), Resnick (1982), and Vergnaud (1982) have alluded to the interaction of these cognitive structures in accounting for children’s performance in simple arithmetic computation and word problems. Romberg (1982) claimed that there is a general paradigm evolving in which researchers are constructing cognitive models of children’s mathematical problem solving appropriate for different developmental levels. From this perspective, development is viewed as contingent upon successful induction both of proper computational strategies and of concepts about numbers and numerical relationships, which are stored in declarative knowledge networks. Development of adequate declarative schemata will allow for the acquisition of more sophisticated strategies (procedures), and these in turn will allow for the increase in complexity of conceptual understanding. If the procedural knowledge is emphasized at the expense of the conceptual, then highly skilled computational experts may develop. But, as Hatano (1982a) warned, these may be only “routine experts”-experts in terms of speed, accuracy, and automaticity of simple calculations. Without the concomitant training in conceptual knowledge, further achievement may not occur. One would presume that increases in declarative knowledge can enable the generation of new knowledge through general inferential processes. One of my contentions is that this automaticity of procedural knowledge, or access of the declarative knowledge instead, may be a necessary prerequisite for successful achievement in conceptual domains. This may turn out to be too strong a hypothesis; instead, it may be that some critical level of efficiency of simple computation, short of complete automaticity, may allow a reasonable level of conceptual understanding to evolve.
Mathematics and Processing Efficiency So how would we go about testing some of these notions of processing efficiency from a cognitive information-processing perspective? In the most general sense, we would have to relate, at each “stage” of mathematical development, the level of efficiency of simple computation with the acquisition of declarative knowledge. Structural modeling techniques in a longitudinal design would be useful for determining the causal links between the development of automatic computational skills and the acquisition of conceptual nodes in the declarative network. But what is the essence of this iink between computational automaticity and uonceptual acquisition? As I suggested earlier, as mathematical problems become more complex, their solution may depend on more than just adequate procedural and declarative knowledge: It is possible that efficient allocation of
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attention to various components of a task will be essential for optimal application of the knowledge schemata. Inefficient allocation could inhibit to varying degrees the application of available knowledge-hence, competence/performance disparities would be understood in terms of processing inefficiency. For example, Siegler and Robinson (1981) describe the inability of preschoolers to draw connections between separate mathematical abilities. With increasing proficiency in these skills that “develop in relative isolation from one another . . . then greater use [may be made] of the potential interconnections among them” (Siegler & Robinson, 198 1, p. 70). This perspective has been a popular one in the recent literature but the processes by which independent domains of knowledge become interdependent are as yet unspecified. In part, I suspect that capacity limitations during problem solving often prevent interconnections from being realized. If simple addition is a difficult, capacity-demanding task for a preschooler, then spare capacity may not be available to incorporate knowledge of number comparisons in the addition process. The proficiency Siegler and Robinson refer to may be best understood by reference to the notions of automatic processing resulting from extended practice (Schneider & Shiffrin, 1977), with which cues in stimulus information would result in more rapid and effortless access of information in memory. In this way, available processing capacity could be reallocated to more capacity-consuming aspects of the problem-solving situation. Alternatively, proficiency could refer to increases in the number or total capacity of resources available for problem solving. Case (1982, 1985) has argued that parallel developments across different content domains may be accounted for by general growth of central processing capacity. Shavelson (1981) has drawn attention to Case’s views and suggested that automatization of arithmetic operations, through extensive practice and overlearning, will reduce demands on memory and thereby allow for additional storage space for new incoming information. That is, the development of processing efficiency may be seen as an enabling condition for the acquisition of new information, whether procedural or declarative. Such notions are consistent with the views of Coombs (1941), Werdelin (1958), Hatano (1982a, 1982b), and Siegler (1984), all mentioned earlier. As an example of research concerning the role of efficient processing in higher level mathematical problem solving, Muth (1984) studied some of the component abilities contributing to arithmetic word problem solving in sixth graders. She focused on subjects’ computational ability as well as their attention to extraneous information in the problem. Once reading ability was partialed, it was found that computational ability and the effect of extraneous information in the problem both predicted the number of correct word-problem answers and correct set-ups (defined by correct analysis but incorrect answer). Extraneous information, but not computational ability, predicted solution latency over and above reading ability. The results were consistent with the view that the extraneous information placed “formidable demands on the student’s limited process-
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ing capacities” (p. 210) and that better computational skills were related to higher level functioning in math. It would have been interesting to see what the effect of extraneous information on performance was as a function of level of computational skill, but this analysis was not conducted. Brainerd (1983) argues that the emphases on chronometric analysis of arithmetic as well as the more traditional Piagetian and neo-Piagetian views of number development fail to differentiate between development of processing skill and short-term memory effects. In his view, memory failures play a large role in the development of numerical expertise, independently of changes in processing operations, which have been studied predominantly in recent research. In an elegant mathematical analysis of addition and subtraction performance of 4-, 5, and 6-year-olds, Brainerd demonstrates that it was short-term memory (encoding and retrieval together) that developed more rapidly during this age span than arithemtical processing. I agree with Brainerd’s assertion that emphasis on the nature of arithemtic processing operations (e.g., counting vs. retrieval) generally has led to failures to acknowledge the important performance-limiting role played by short-term memory factors. But it does not follow that the chronometric and rule-assessment paradigms cannot be used to study the influence of short-term memory processes on arithmetic performance in young children. On the contrary, we have found it possible to measure the efficiency of arithmetic processing at different stages of processing corresponding to encoding, storage, retrieval, and decision phases of information processing (Kaye, Bonnefil, & Chen, 1983). Our technique is described later in this paper. Although the chronometric studies, along with interview studies, have been helpful in describing the strategy shifts through elementary school, these descriptions are insufficient to account for development of hierarchically more complex mathematical skills. Again, Siegler and Robinson (198 1) suggest that models of children’s knowledge can be built upon “upward and outward” to account for development of skill in more complex mathematical domains. But what allows for the actual transition from one level of skill to another? Surely, instruction, formal or informal, must play a role; extended practice is important as well, for reasons mentioned above. We must continue to specify the nature of the missing link: What limits the interconnections between separate sources of knowledge present in one’s memory systems (cf. Stemberg, 1984b)?
Addition and Theories of Attention It would seem that we must study tasks in which more than one source of knowledge must be used. Many modem theories of attention address the task of solving one problem while simultaneously attempting memory retrieval of taskrelevant information (see Eysenck, 1982). Whereas some theorists maintain that there is always a fixed pool of resources that may be distributed to different tasks performed simultaneously (Norman & Bobrow, 1975), others have argued that the total amount of processing capacity changes both within individuals and
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across tasks as a function of task demands (Kahneman, 1973). Although the differences in empirical predictions derived from such different theories are necessary to consider, most modem attention theorists believe that we can learn a great deal about people’s levels of skill from their performance on tasks that demand divided attention. Kerr (1973), in her review of limited-capacity attention theories, remarked that we can observe the limits of processing capacity when we observe how performance decreases when a person is asked to do more than one thing at a time. Although we may be able to attend to but one thing at a time, we can process several things simultaneously if the processes become habitual or automatic. Either highly practiced skills require less attentional capacity and hence the total pool of resources is not exceeded by multiple task processing, or greater levels of skill derived from extended practice allow the subject to divide attention more efficiently among relevant tasks. There is no known limit to human mental-processing capacity; it has been shown that continued practice with multiple, high-level cognitive tasks can result in quite amazing levels of performance (Hirst, Spelke, Reaves, Caharack, & Neisser, 1980). How can such theoretical notions be applied to the development of mathematical ability? There is little doubt that even smiple arithmetic problem solving (e.g., addition or subtraction) involves multiple-task components (encoding, retrieval or computation, response). To the extent that any of these operations can be made automatic, problem solution will be derived in less time and perhaps with less effort expended. Recent research by Kaye et al. (1983) involves measurement of attentional allocation during simple arithmetic problem solving using dual-task techniques. We wanted to show that strategy changes are of functional importance since they result in additional processing capacity that can be used to accomplish more complex problem solving, as in proficient word-problem solving. The work of Carpenter and Moser (1982) suggests that basic computational strategies are related to word problem-solving performance; we would like to show that this in part is due to processing efficiency. If a child or adult is attempting to solve a word problem involving addition, subtraction, or some other operation, then the extent to which the computational aspects of the problem demand conscious attention may interfere with the subject’s ability to process the syntactic, logical, or semantic aspects of the sentences. Perhaps the child has the requisite skills to add two numbers together or to read a sentence and understand its meaning. Unless these separate skills are proficient, then, when their combined effort is required, the processing resources available may not be sufficient for adequate solution. The bottleneck of processing efficiency could be almost anywhere in the problem solution. It is our belief that the systematic, hierarchical developments in mathematical ability alluded to by so many researchers may be understood better in terms of mechanisms of efficient capacity allocation. Using a dual-task paradigm, Kaye et al. (1983) were able to measure the attentional capacity left free during different stages of information processing of
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a two-term addition verification problem, i.e., the encoding of the addends and computation or retrieval of the actual sum, encoding of the stated answer and comparison with the computed or retrieved answer, and finally the decision and response stages. The addition problem comprised the primary task, and a simple reaction time to detection of an auditory probe interpolated at different intervals during the visual presentation of the addition problem served as the secondary task-dependent measure. We tested children in grades 2, 4, and 6, as well as college students and derived samples of higher and lower achieving subjects in each grade based on scores on standardized tests of computation and conceptual understanding. By so doing, we were able to pinpoint stages of problem performance that were attention demanding and those that were not, and then we were able to relate performance at these different stages to arithmetic ability. In brief, we found that ability differences in middle-elementary school were correlated with attentional efficiency during the encoding and computation phases, whereas the ability differences in college were correlated with attentional and response efficiency in comparison and decision stages. This pattern of results is consistent with much of the literature to which I have referred. In middle-elementary school, more subjects are making the transition from counting to retrieval strategies, and development after this point is largely a matter of increasing speed of processing. Anderson’s (1983) theory gives a coherent structure for these results: There is a transition from early interpretive productions to later knowledge compilation and finally the strengthening and tuning of productions. In the larger view of development across many years of practice, both procedural and declarative knowledge would be come integrated in production systems that are invoked involuntarily and executed accurately and quickly when the appropriate problem solving situation presents itself. The work of Siegler (1984), Ashcraft (1985), and Rabinowitz (1985) strengthens my belief that such development centers about the availability and the accessibility of the computational facts of arithemtic in memory. MODEL
OF THE DEVELOPMENT
OF NUMERICAL
COGNITION
The theory and research that I have reviewed may be summarized best by presenting a general, preliminary model of how mathematical proficiency develops. The following propositions form the crux of this model. 1. In the most general form, I maintain in this model that the development of arithmetic skill is generally a function of increased efficiency of information retrieval (to replace overt computation). Not only does development involve a shift from counting to fact retrieval, but this shift produces cognitive savings due to reduced attentional demands of computation. These cognitive savings are realized in time in the acquisition of higher order concepts and procedures. 2. Once the basic number concepts have been acquired in the preschool years (Gelman, 1982; Gelman & Gallistel, 1978), practice with counting objects and
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numbers results in arithmetic skill development and increasing strength of associations between sums (in addition) and numbers to be added (Ashcraft, 1985; Siegler, 1984). 3. As in Brainerd’s (1983) model, short-term memory interfaces between problems that are presented and the information to be accessed in long-term memory. If encoding is inaccurate, then performance will be impaired even if access of information from long-term memory is proficient. If access of information is difficult, then subjects may resort to overt strategizing, e.g., counting (Siegler, 1984). In this event, processing will take longer, allowing for other short-term memory failures, such as forgetting the value of one or more of the addends in an addition problem. Thus, even if the processing skill is applied correctly, incorrect solution may be attained. 4. If the child is required to use arithmetic computation to solve a more complex mathematical problem, then performance will be a function of a number of component skills, among them, computational proficiency. To the extent that computation is attention demanding, then other aspects of problem solution will be impaired. If attentional load is high due to inefficient computation, then conceptual linkages will fail to be realized, and thus development of mathematical expertise will be delayed. 5. Proposition 4 above predicts that some criteria1 level of computational efficiency will be requisite for acquisitions of higher level mathematical skill, both declarative/conceptual and procedural. This assertion is of course empirically testable (cf. Muth, 1984). 6. If computational efficiency is a necessary condition for the development of expertise in the mathematical domain, then “experts” at different levels of development should have developed efficient, and even automatic, computation skills. 7. At higher levels of mathematical complexity, once other knowledge domains become relevant to mathematical problem solution, then continuing expertise development will be a matter of developing automatic routines to access and use the relevant knowledge structures. 8. Propositions 6 and 7 imply that the critical aspect of performance at any level of complexity of mathematics is the degree to which the access and application of knowledge necessary to solve a problem is efficient and the extent to which processing capacity is left free to execute relevant strategies that may not depend on automatic information retrieval. To avoid circularity, it would be imperative to conduct task analyses at each level of mathematical problem so as to be able to differentiate aspects of a problem that are directly relevant to task solution. One obvious criticism is that computation itself is often irrelevant to problem solution in various mathematical domains (e.g., some types of geometry, calculus). But it is my contention that, developmentally, there will be a fairly consistent sequence of knowledge structures that are “compiled, strengthened, and tuned” (Anderson, 1983). Basic number and computation facts comprise the first, or one of the first-such, declarative networks in the mathematical domain.
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So, it is predicted that efficient use of this network in early elementary school will be the strongest predictor of later mathematical achievement. ‘CONCLLJSION In sum, 1 think that one new direction for chronometric studies of numerical cognition is to determine the information-processing consequences of different types of computational and retrieval strategies. If an accurate characterization of the development of addition skill is one of a transition from reliance on computation to reliance on retrieval from a declarative-knowledge network, then we should determine the cognitive savings of retrieval strategies over computation. Once retrieval strategies, or perhaps also advanced procedures (of the type Baroody studies), become depended upon, then the extent to which increased practice produces economical problem solving when computation is but a component of more complex mathematical problems may predict the development course of expertise. As I have suggested, the richness of theories of attention affords the cognitive researcher a variety of methods with which to study mathematical cognition. REFERENCES Anderson,
J.A.
(1983).
Ashcraft,
M.H. (1982). mental Review, Ashcraft. M.H. (1983).
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A.J. (1983). The development of procedural knowledge: An alternative explanation for , chronometric trends of mental arithmetic. Developmenfal Review, 3. 225-230. Baroody, A.J. (1984). A reexamination of mental arithmetic models and data: A reply to Ashcraft. Brainerd.
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algorithms in arithmetic. In T.P. Carpenter, J.M. Moser, & T.A. and sub/rat/ion: A cognitive perspective. Hillsdale, NJ: Erlbaum. R. ( 1985). Inre/lectual developmen/: Birth IO adulthood. Orlando,
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