How Children Discover New Strategies

Robert Siegler, Eric A. Jenkins

• Publisher: Psychology Press
• Year: 2014
• Print ISBN: 9780805804720
• eBook ISBN: 9781317784456
• Topic: Psychology
• Subtopic: Developmental Psychology
• Index: Psychology

This well-documented book divides the process of constructing new problem-solving strategies into two parts: discovery of the new strategy, and its generalization to new contexts. By using a trial-by-trial analysis, the authors are able to identify the exact trial on which the new strategy is first used, the circumstances that lead to the discovery, and the generalization of the strategy beyond its initial use. These observations disconfirm popular stereotypes of the discovery process and provide important insights into the nature of long-term learning and strategy discovery.

1

Strategy Discovery and Strategy Generalization

A child’s mind is like a workshop. This workshop contains a remarkable collection of materials (knowledge) and tools (learning processes) that can be used to make new products (rules, strategies, hypotheses, schema, causal networks, etc). Some of the tools and materials are useful for a great many tasks. Many others are specialized for a particular purpose, but are invaluable when they are needed.

Orders constantly arrive at the workshop for products that need to be made. Most of the requested products are familiar, and the child already knows how to make them. Others are new, though. Choosing the tools and materials for building these new products can be frustrating, but it also challenges the workshop to produce its finest wares.

Many products that are fabricated in the process of meeting one order themselves become materials and tools for making additional products. For this reason, the broader the range of products the workshop has produced in the past, the greater its potential for meeting future demands. Realizing this potential, however, requires not only that diverse tools and materials be on hand but that they be efficiently organized. Otherwise, they will simply represent clutter.

The life of the workshop is not limited to its internal workings. It has daily contact with other workshops. These other workshops tend to be friendly, experienced, and eager to demonstrate new techniques. Still, there are limits to how helpful they can be. Often, they do not say how a particular job can be done. Even when they attempt to provide instructions, the ones they give are often confusing because they call for tools that the workshop has not yet acquired. Many of the greatest challenges must be met by discovering how to use available resources for novel purposes.

The workshop is a proud and industrious place. Its workers invent tools and materials even without outside requests for them. In part, this is due to knowing that the new discoveries may someday be useful. In part, it is due to a desire to find easier ways of getting jobs done. Not all of the efforts, however, can be attributed to foresight or laziness. Some are due to simple enjoyment of the work.

The purpose of this book is to explore how children’s minds construct one of their most important products, new problem-solving strategies. The strategy-construction process is divided into two parts: strategy discovery and strategy generalization. Among the central issues that are addressed are the experiences that promote discovery and generalization of strategies, the qualities of children that predict who will make the discoveries and who will generalize them most widely, and how strategies differ from related cognitive activities such as procedures and plans.

THE ISSUE OF STRATEGY CONSTRUCTION

We accomplish most of our goals in unremarkable ways. Sometimes, however, we must abandon the routine. Highway repairs, forgotten wallets, malfunctioning slide projectors, lost keys, and many other minor tragedies challenge us daily. Often, we already know alternate approaches. Taking a different route to work, borrowing money for lunch, checking whether all of the proper connections have been made on the slide projector, and finding a janitor with a key provide straightforward ways around these momentary problems. Occasionally, however, we need to go beyond previously discovered or easily generated alternatives. That is, we need to construct a new strategy.

Although all of us must construct new strategies, the process may be especially prominent in the lives of children. As Brown and DeLoache (1978) noted, children are in a sense universal novices. They constantly need to discover strategies to cope with situations that adults find quite routine. They also need to find out where newly discovered strategies apply. Thus, they seem a particularly appropriate group for studying the process of strategy construction.

HISTORICAL BACKGROUND

The Role of Learning in Cognition

Construction of new strategies is one facet of the larger topic of learning. However, it is a facet that has historically received relatively little attention. The reasons for this are instructive.

Learning has been a central topic within psychology from the field’s earliest days as a science. The grand theories of Thorndike, Guthrie, Skinner, and Hull established learning as the central topic in psychology in the 1930s, 1940s, and 1950s. However, construction of new strategies never became a major topic within these learning theories. Part of the reason was that the idea of “strategies” was uncommon; the standard unit of analysis was the stimulus-response connection. Another reason was that most of the tasks that were studied were extremely simple. To the extent that any strategy needed to be constructed, the strategy usually could be assembled very quickly.

The cognitive revolution of the late 1950s brought with it the strategy as a basic unit of analysis. This centrality was evident even in some of the earliest cognitive research: that of Newell and Simon (1956); Bruner, Goodnow, and Austin (1956); Brown, (1956); and Miller, Galanter, and Pribram (1960). Along with this attention to strategies and other higher order cognitive units, however, came a de-emphasis on learning. This de-emphasis represented an explicit decision concerning the approach that seemed most likely to advance rapidly understanding of cognition. As Newell and Simon (1972) commented:

Turning to the performance-learning-development dimension, our emphasis on performance again represents a scientific bet.… The study of learning, if carried out with theoretical precision, must start with a model of a performing organism, so that one can represent, as learning, the changes in the model.… It is our judgment that in the present state of the art, the study of performance must be given precedence, even if the strategy is not costless. (pp. 7–8)

This research approach did allow rapid progress in our understanding of strategies and other types of cognitive activity. However, as Newell and Simon anticipated, it was not without cost. By the late 1970s, numerous investigators were lamenting the lack of data and the paucity of cognitive theories of learning, especially complex, extra-laboratory learning (e.g., Anderson, 1976; Bransford, 1979; Glaser, 1976; Stevenson, 1983; Voss, 1978). For example, Voss (1978) wrote, “Although the concept of learning may be found in cognitive psychology, it also must be conceded that the cognitive view of learning is vague, is abstract, and most important is lacking a substantive data base” (p. 13). Similarly, Anderson (1976) commented, “Interest in the mechanisms of procedural learning seems to have died in cognitive psychology with the demise of the stimulus-response theories.… It is important that cognitive psychology concern itself more with the issue of learning, both because of its important practical implications and because it serves to place important constraints on cognitive theories” (p. 20).

In the 1980s, research on learning of complex, extra-laboratory skills assumed a more prominent place. Major theories of adult cognition, such as those of Anderson (1983); Holland, Holyoak, Nisbett, and Thagard (1986); McClelland and Rumelhart (1986); and Newell (1989) have focused on problems of learning and induction. A similar trend has been evident in cognitive-developmental theories, such as those of Case (1985), Klahr (1984), Sternberg (1985), and Siegler and Shrager (1984). There is considerable consensus now that a model that does not account for knowledge acquisition is a seriously incomplete model of cognition.

Even with the resurgence of interest in learning, large gaps in our understanding remain. One of the most serious of these involves our lack of knowledge about how new strategies are constructed. A number of current models deal quite well with the strengthening and weakening of rules and strategies that already exist. However, this is not sufficient. As Holland et al. (1986) noted, “Although it is possible to understand much of learning in terms of revision of rule strength, it is not possible to understand all of learning in that way. Somehow, at some point, those rules that are to undergo strength revision have to be generated” (p. 344).

The generation of strategies over long and variable periods of time is a special mystery. We usually think of learning as involving either a gradual build up of associative strength of some behavior or as involving the testing of hypotheses until one is found to be correct. It is difficult to explain long-term strategy construction comfortably within either of these frameworks, however. If all that was needed for a strategy to appear was sufficient asociative strength, why would some novel strategies gain the required strength within 10 minutes, yet others require 10 weeks or months? If hypotheses were being tested, why would it often take so long for the new strategy to be hypothesized (or if the hypothesis were not in the set being sampled, how would it eventually get in there)?

Two barriers must be overcome for progress to be made in understanding this type of learning. One is an adequate empirical base. Because of the time and effort inherent to studying long-term strategy construction, few studies of the process have been conducted. The other barrier concerns ideas about mechanisms that could produce such discoveries. Conceiving of mechanisms that, like people, would form new strategies after not doing so on many previous occasions, is difficult. Yet we know from people’s behavior that such mechanisms must exist.

Some Personal History

Any research program reflects the personal history that led the investigators to study the problem, and study it in a particular way, as well as the history of the field. For one of us (RS), studying the process of strategy construction represents a coming together of two main research programs pursued since the mid-1970s. The first project (the rule-assessment studies) focused on identifying the strategy a given child used to solve a set of problems, the sequence of strategies through which children’s thinking progressed with age, and the ways in which existing rules and encodings influenced acquisition of new knowledge. The more recent project (the strategy-choice studies) has focused on identifying the diverse strategies that a given child uses to solve certain types of problems, the changes over time in the relative frequency of the strategies used by each child, and the mechanisms that enable children to choose adaptively among the multiple strategies they know and use for solving these problems.

The two sets of studies differ in numerous ways. The first group of studies examined children’s performance on variants of Piaget’s scientific reasoning tasks: balance scale, projection of shadows, and probability problems (Siegler, 1976, 1978); conservation of liquid quantity, solid quantity, and number problems (Siegler, 1981); time, speed, and distance problems (Siegler & Richards, 1979); and so on. The latter group of studies focused on academic tasks such as addition, subtraction, and multiplication (Siegler, 1988a; Siegler & Shrager, 1984); spelling and reading (Siegler, 1986, 1988b); and time-telling (Siegler & McGilly, 1989). The first group emphasized the rule as the basic unit of analysis; the second group emphasized associations among strategies, problems, and answers. The earlier studies depicted development as a sharp qualitative change from use of one rule to use of another; the more recent ones depict development as a gradual change in the distribution of strategy use, with new strategies coexisting with, rather than replacing, previous ones.

In my own mind, at least, the two sets of studies form a natural progression. The initial studies focused on situations in which individual children consistently use a single rule (strategy). As I reflected on this property of the situations, I became increasingly convinced that although such situations are an interesting part of cognitive development, they are far from the whole story. Often, children (and adults) use a variety of strategies. Understanding both how they come to use a single consistent rule in some situations and how they come to use multiple approaches in others seemed important for understanding cognition in general. This was part of my motivation for focusing on problems on which children use multiple strategies rather than a single rule.

Another reason for moving from the one emphasis to the other concerned the complexity of the issues raised by the situations. At first glance, understanding children’s problem-solving strategies on balance scale, projection of shadows, and conservation tasks sounds like it would be inherently more challenging than understanding how they solve arithmetic and time-telling problems. Intuitive estimates of the difficulty of understanding cognitive processes are often wildly misleading, however. The first computer simulations of cognitive activity (Newell & Simon, 1956) modeled quite well how people perform such high-level cognitive activities as solving logic problems. Other profound human accomplishments, such as playing chess, solving physics problems, and proving geometry theorems, quickly proved to be similarly tractable. Yet we remain unable to model such seemingly mundane activities as seeing, hearing, and understanding language. In the same way, understanding how children choose among multiple strategies on such “simple” tasks as adding and subtracting small numbers has proven to be more challenging than understanding their use of a single strategy on scientific reasoning and other “high-level” tasks. Thus, although it may sound ironic, part of the reason that I moved from studying balance scales, projection of shadows, and conservation to studying addition and subtraction was that understanding the issues raised by the latter posed the greater challenge for me.¹

One difference between the earlier and later projects was critical to the present investigation. Although both sets of studies have focused on learning processes, the nature of the processes differs. The learning processes of central interest in the earlier studies involved acquisition of new problem-solving strategies. Those in the later studies involved changing frequencies of use of existing strategies.

The experiments that are reported in this book represent a synthesis of the two types of change processes. Like the earlier studies, they focus on acquisition of new strategies. Like the more recent studies, they focus on situations in which any given strategy, new or old, will be only one among several approaches that children know and use. In such situations, new strategies do not replace previous ones (at least not for a long time), but rather find a niche among them. This means that the research in this monograph is aimed at explaining not only how new strategies are first generated but also how they find their niches among previously acquired approaches.

Two particular themes that emerged in the earlier studies of learning also emerged as important in the present ones. One was the influence of existing knowledge on learning. New strategies do not emerge in a vacuum. Instead, they seem to be constructed from the materials of previous strategies. At least in this domain, the strategy-construction process seems to involve combining parts of existing strategies in novel ways, or grafting on new segments to existing procedures, rather than creating the new strategy out of whole cloth.

The second common theme was the importance of encoding in strategy construction. In the earlier studies, young children often failed to learn from relevant experience on balance scale and temporal duration problems because they did not encode relevant dimensions such as the distance of the weights on each side from the fulcrum (Siegler, 1976, 1983). In the present study, much of the challenge of constructing new strategies appears to lie in encoding which goals were met by different parts of existing strategies and in selecting the right segments of existing strategies for inclusion in the new strategy.

Methods for Studying Learning

Studying complex extra-laboratory learning poses not only conceptual difficulties but also methodological ones. At least three distinct research strategies have been developed for dealing with these methodological challenges. Each approach has distinct strengths and weaknesses. However, one of them, the microgenetic approach, seems particularly well suited for studying the construction of strategies over long periods of time. Here, we review the three methods for investigating complex learning and discuss why we concluded that the microgenetic approach was best suited for the goals of the present investigation.

One approach to studying learning of complex real-world skills has been to focus on quantitative changes in performance with practice (Fitts & Posner, 1967; Mazur & Hastie, 1978). Here, individuals are followed over a very large number of trials as they became increasingly competent at executing a skill. Often, the emphasis is on the shape of the mathematical function that best describes the reduction in solution times with practice. Newell and Rosenbloom (1981) summarized much of this literature in their article on the ubiquity of the power law as a description of practice effects in such diverse areas as reading inverted letters, solving geometric proof problems, rolling cigars, and learning text-editing commands. This approach has yielded striking quantitative regularities in the rate of improvement of performance with practice. By its focus on purely quantitative measures, however, the approach precludes obtaining qualitative information about knowledge and processes underlying construction of new strategies. Thus, it is not well suited to the issues of central interest in the present investigation (though see Rosenbloom & Newell, 1986, for ways in which a rich theoretical framework can be used to explain how such quantitative regularities can arise.

A second strategy for studying learning of complex, real-world skills has been to examine how students learn from explicit instruction. Such studies typically include identification of a concept or skill that students have difficulty acquiring, analysis of why the learners’ current capabilities or the usual instruction make the new understanding difficult to obtain, and formulation of a new type of instruction that will overcome the difficulty. Experiments test whether the new instruction is effective in producing the hoped-for progress. Not surprisingly, this approach has been particularly prominent in educational areas such as reading (Bradley & Bryant, 1983; Palincsar & Brown, 1984), writing (Bereiter & Scardamalia, 1987; Gray, 1977), mathematics (Case, 1985; Schoenfeld, 1987), and computer programming (Klahr & Carver, 1988). It has proven useful both theoretically and practically; much learning is in response to instruction, and what is learned from instruction often differs from what the teacher intended. Because teaching is such a complex activity, however, it often is impossible to identify the point at which a given child acquired a new strategy or the specific factors of the teaching strategy that led to the discovery. Again, the approach does not seem ideal for studying strategy construction.

A third approach to learning of complex real-world knowledge and skills, the microgenetic approach, combines a number of advantages of the other approaches. The term microgenetic approach has been used in different ways by different investigators (e.g., Kuhn & Phelps, 1982; Metz, 1985; Paris, Newman, & McVey, 1982). In our view, microgenetic methods have two key properties: (a) Subjects are observed over an extended period of time; (b) Their learning is subjected to intensive trial-by-trial analysis, with the goal being to infer the underlying representations and processes that ga...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• CHAPTER 1 Strategy Discovery and Strategy Generalization
• CHAPTER 2 Strategy Choices and the Development of Arithmetic Skills
• CHAPTER 3 A Study of Strategy Construction
• CHAPTER 4 Case Studies
• CHAPTER 5 Summary, Conclusions, and Ideas
• References
• Author Index
• Subject Index