Perceptual Learning Modules in Mathematics: Enhancing Students’ Pattern Recognition, Structure Extraction, and Fluency

1. Introduction

What aspects of scientific research on learning have relevance for education? In the educational literature, we find discussions of fact learning, conceptual understanding, procedure learning, constructing explanations, analogical reasoning, and problem-solving strategies. Except for an occasional mention of pattern recognition, little hints at any role for perceptual learning (PL). Likewise, studies of PL involving sensory discriminations among a small set of fixed stimuli appear to have little connection to real-world learning tasks, least of all to high-level, explicit and symbolic domains such as mathematics. Despite this apparent irrelevance or neglect, PL—improvements in information extraction as a result of practice—is one of the most important components of learning and expertise in almost any domain, including mathematics. It is also the component least addressed by conventional instruction; thus, problems of information selection and fluent extraction of structure pose major obstacles for many learners.

In this article, we describe the relevance and promise of applications of PL to high-level cognitive domains. We focus on mathematics learning, but we hope the relevance to other areas of learning will be obvious. By providing several brief examples of PL interventions in mathematics, we hope to give a useful overview of the scope, character, and possibilities of PL in instructional settings.

Applying PL to rich instructional domains is a startling and exciting endeavor. It is startling because some current conceptions of PL would preclude this kind of connection, and it is exciting because considerable evidence suggests that PL technology can address crucial and neglected dimensions of learning, offering the possibility of major advances in the effectiveness of learning in many domains.

We first summarize some characteristic improvements in human information extraction that derive from PL. Next we explain the connection between these changes in high- and low-level PL tasks, focusing on the notions of discovery and selection as unifying concepts. We then describe three examples of research on perceptual learning modules (PLMs) that apply these ideas to mathematics learning, each illustrating a separate strength of PL interventions. The current treatment is brief, and we refer the reader to other sources for more expansive discussion of some issues (Garrigan & Kellman, 2008; Kellman, 2002; Kellman & Garrigan, 2009; Kellman et al., 2008; Massey et al., in press).

2. Characteristics of expertise generated by PL

In contrast to the literature on learning, scientific descriptions of expertise are dominated by PL effects. In describing their studies of master level performance in chess, Chase and Simon (1973) wrote: “It is no mistake of language for the chess master to say that he ‘sees’ the right move; and it is for good reason that students of complex problem solving are interested in perceptual processes” (p. 387). Table 1 summarizes some of the well-known information processing changes that derive from PL. Kellman (2002) suggested that these can be divided into discovery and fluency effects. Discovery pertains to finding the features or relations relevant to learning some classification, whereas fluency refers to extracting information more quickly and automatically with practice. Both discovery and fluency differences between experts and novices have since been found to be crucial to expertise in a variety of domains, such as science problem solving (Bransford, Brown, & Cocking, 1999;Chi, Feltovich, & Glaser, 1981; Simon, 2001), radiology (Kundel & Nodine, 1975; Lesgold, Rubinson, Feltovich, Glaser, & Klopfer, 1988), electronics (Egan & Schwartz, 1979), and mathematics (Robinson & Hayes, 1978).

3. Scope of PL

Eleanor Gibson, who pioneered the field of PL (for a review, see Gibson, 1969), defined it as “an increase in the ability to extract information from the environment, as a result of experience…” (p. 3). She described a number of particular ways in which information extraction improves, including both the discovery and fluency effects noted above. Of particular interest to Gibson was “…discovery of invariant properties which are in correspondence with physical variables” (Gibson, 1969, p. 81). This view of PL applies directly to many real-world learning problems; although Gibson did not mention mathematics learning, her examples included chick sexing, wine tasting, map reading, X-ray interpretation, sonar interpretation, and landing an aircraft.

Much contemporary PL research differs from these earlier ideas, both in the types of learning problems studied and in theoretical context. It is common to consider PL as described by Fahle and Poggio (2002): “…parts of the learning process that are independent from conscious forms of learning and involve structural and/or functional changes in primary sensory cortices.” This emphasis derives from several related ideas. PL effects often show specificity to stimulus variables (such as orientation or motion direction) or observer variables (such as the eye or part of the visual field used in training). Specificity in learning (lack of transfer) has been assumed to indicate a low-level locus of learning effects, implying that mechanisms of PL are modifications, such as receptive field changes, in early sensory areas.

Confining PL to changes early in sensory systems would separate it from most real-world learning tasks involving higher-order relations and extraction of invariant patterns from varying instances. If the supporting assumptions are correct, then the type of PL being studied recently has little to do with mathematics learning. Perhaps the discovery of invariance in complex tasks and modification of receptive fields in simple discriminations have little do with each other, and the best course would be to adopt different names for these different kinds of learning.

Such a division would be unfortunate, for two reasons. First, the assumptions supporting confinement of PL to early sensory cortices and simple sensory phenomena are not sustainable. Those that are logical assumptions, such as the idea that specificity of learning implies a low-level locus, are fallacies (Mollon & Danilova, 1996). Those that are empirical hypotheses appear to be false (for more extended discussions, see Garrigan & Kellman, 2008; Kellman & Garrigan, 2009). For example, data about transfer from PL experiments have been inconsistent, and small task variations can lead to large differences in generality of learning (e.g., Liu, 1999; Sireteanu & Rettenback, 2000); single-cell recording data offer little support for receptive field modification as the explanation for PL, at least in vision (e.g., Ghose, Yang, & Maunsell, 2002). Furthermore, substantial evidence suggests that PL, even when oriented toward a low-level sensory discrimination, is guided by higher-level tasks (Ahissar & Hochstein, 2004) and attention (Seitz, Lefebvre, Watanabe, & Jolicoeur, 2005; Seitz & Watanabe, 2005), and it may in fact be required to work through constancy-based representations (Garrigan & Kellman, 2008).

Perhaps the most important rationale for considering “high-” and “low-” level tasks together in PL is that they may share common principles and mechanisms. For Gibson, a key to PL was selection. Recent experimental results suggest that selection is a better explanation of many low-level PL phenomena than receptive field modification (Ahissar, Laiwand, Kozminsky, & Hochstein, 1998; Petrov, Dosher, & Lu, 2005). Higher levels of processing receive inputs from below; learning processes discover which inputs are most relevant, and these inputs become more heavily weighted (Mollon & Danilova, 1996; Petrov et al., 2005). Processes of discovery and selective weighting apply to both laboratory studies of sensory discrimination and discovery of high-level invariance in ecologically important tasks.

Does specificity of learning, then, indicate a gulf between low and high-level PL? In real-world tasks, detecting structural invariance across otherwise variable cases is crucial to useful learning (e.g., detecting a wrist fracture in an unfamiliar X-ray or seeing that an algebraic transformation involves the distributive property). Yet simple laboratory PL tasks often show strong specificity of learning. We believe there is really no conflict between these observations. Specificity in learning reflects the task posed to the learner (Mollon & Danilova, 1996). If thousands of trials using only two stimuli are used, PL processes of discovery and selection may exploit local analyzers specifically relevant to those two stimuli. If accurate classification in a task requires extracting invariance and decoupling it from irrelevant, variable attributes, PL will lead to selection of relevant features or patterns that are not tied to irrelevant attributes of specific cases. For this reason, PL based on practice in extraction of structure across changing cases offers a powerful method for producing transfer of learning to novel cases. The idea that specificity in learning relates to the design and requirements of the learning task is consistent with the variability of results regarding transfer in the PL literature (e.g., Liu, 1999); with the evidence favoring selection processes over receptive field changes in PL; and with evidence and models suggesting that PL can involve information at multiple levels in a flexible manner (e.g., Ahissar & Hochstein, 2004).

4. Applications of PL to high-level learning domains

Assuming the relevance of PL to complex tasks, one might still wonder about symbolic domains such as mathematics. Even Gibson’s examples—wine tasting, recognizing aircraft, or reading X-rays—seem much less “cognitive.” Mathematics might be thought to involve only declarative knowledge and procedures. There are inherently symbolic aspects of mathematical representations that cannot be apprehended via information “in the stimulus.” This is true, but it is also true that mathematical representations pose important information extraction requirements and challenges. Characteristic difficulties in mathematics learning may directly involve issues of discovery and fluency aspects of PL. A number of studies indicate the role of PL in complex cognitive domains, such as mathematics (Goldstone, Landy, & Son, 2008; Silva & Kellman, 1999), language or language-like domains (Gomez & Gerken, 1999; Reber, 1993; Reber & Allen, 1978; Saffran, Aslin, & Newport, 1996), chess (Chase & Simon, 1973), and reading (Baron, 1978; Reicher, 1969; Wheeler, 1970). Some have asserted that in general, abstract concepts have crucial perceptual foundations (Barsalou, 1999; Kosslyn & Thompson, 2000; Landy & Goldstone, 2008). Moreover, in complex cognition it is important to realize that conceptual and procedural knowledge must work together with structure extraction. Both declarative and procedural knowledge depend on pattern recognition furnished by PL. Which facts and concepts apply to a given problem? Which procedures are relevant? How do we appropriately map parts of the given information into schemas or procedures? These are fundamentally information selection and pattern recognition problems.

5. PL technology

The lack of PL techniques in instructional contexts owes not only to its neglect in learning research but also to the lack of suitable methods. The expert’s pattern extraction and fluency are thought to develop separately from formal instruction, as a result of experience. Yet recent efforts suggest that there are systematic ways to accelerate the growth of perceptual expertise, in areas as diverse as aviation training (Kellman & Kaiser, 1994), medical learning (Guerlain et al., 2004), language difficulties (Merzenich et al., 1996; Tallal, Merzenich, Miller, & Jenkins, 1998), and mathematics (Kellman et al., 2008; Silva & Kellman, 1999).

In our work, we implement PL principles in PLMs. Although a full description is beyond our scope here, we mention some elements of PL interventions. Although we lack complete models of PL in complex tasks, it appears that information extraction abilities advance when the learner makes classifications and (in most cases) receives feedback. Digital technology makes possible many short trials and appropriate variation in short periods of time, allowing the potential to accelerate PL relative to less frequent or systematic exposure to structures in a domain. Unlike conventional practice in solving problems, learners in PLMs typically discriminate patterns, compare structures, make classifications, or map structure across representations.

6. Specificity of PL interventions

How do we know that a learning intervention targets PL rather than other aspects of learning? This is a complex question, and one which, in realistic instructional settings, has no absolute answer. In the work reported here, for example, a basic commitment is to use PL interventions to address core domains and known problem areas in mathematics. In doing so, less stimulus and task control are available relative to artificial materials or laboratory tasks. We have little doubt that there are some unsystematic opportunities for PL present in ordinary instruction, and, conversely, that students’ declarative and procedural knowledge may interact with our PL interventions. A more global problem in making crisp distinctions is that it is likely that improved information extraction obtained through PL normally interacts strongly with other cognitive processes. Such synergy may account for the general tendency found by Fine and Jacobs (2002) for PL effects to be larger in higher-order tasks. Supporting thought and action are, after all, the functions of perception and PL. It is likely that research communities focusing on one or another of our cognitive faculties are more clearer partitioned than our use of these faculties in complex tasks.

Despite such complexities, we believe several characteristics distinguish PL interventions from conventional instruction. These characteristics involve both design of an intervention and outcomes related to characteristics of expert information extraction (Table 1).

At least three general properties are common to PL interventions:

That an intervention impacts PL is a function of its design but also its outcomes. Some potential signatures of PL effects include the following:

7. Experiment 1—Mapping across multiple representations: The MultiRep PLM

Mathematical representations are aimed at making concepts and relations accurate and efficient, but they pose complex decoding challenges for learners. Each representational type (e.g., a graph or an equation) has its own structural features and depicts information in particular ways. Perceptual extraction of structure from individual representations and mapping across representations present learning hurdles that are not well addressed by ordinary instruction. We developed the Multi-Rep PLM to help middle and high school students develop pattern recognition and structure mapping with representations of linear functions, in graphs, equations, and word problems. As in many PLMs, rather than having students solve problems for a numerical answer, we presented them with short, interactive classification tasks that facilitated fluent extraction of important features and patterns.

On each trial of the PLM, either an equation, graph, or word problem expressing a particular linear function was presented. The learner was asked to select an equivalent function among three possible choices in a different representation (e.g., if an equation was presented, the choices could be three possible graphs).¹ An example is shown in Fig. 1. There were six types of mapping trials, comprising all possible pairs of word, equation, and graphical representations given as targets and choices. Incorrect choices usually shared some values with the target display (e.g., having a common slope) but differed in some other respect (e.g., having a different y-intercept). Across problems, a variety of contexts and numerical values were used.

The rationale for the PLM was that fluent use of each representational type requires the ability to extract particular structural attributes (e.g., knowing where to look in an equation to obtain the slope). Practice in mapping across representations requires accurate selection of information in each representational type and may also lead to intuitions about the way equivalent structures relate across representational types (e.g., learning the graphical consequences of slopes <1 or negative intercepts). All of these notions have been explicitly instructed earlier in the mathematics curriculum; moreover, the PLM contained no additional explicit instruction (other than feedback indicating the correct answer on each trial). It was predicted that improvements in selective information extraction and mapping in the PLM would transfer to important core mathematical tasks, such as generating a correct equation from a word problem or a correct graph from an equation.

7.2. Results and discussion

Primary results for translation problems for the PLM and control conditions are shown in Fig. 2. There were no significant differences in pretest accuracy between the control and PLM conditions, t(67) = 1.11, p = .27. There was a robust interaction of test by condition, F(1,66) = 21.17, indicating that the PLM group improved from pretest to posttest more than the control group. Word problem solving (not shown) averaged about 85% in the pretest in both groups and did not vary between groups.

These results, from two short sessions of PLM use, indicate that practice in mapping problems across multiple representations led to strong improvements on a transfer task—generating the correct equation or graph from a word problem, graph, or equation. In contrast, for the Control group, the translation task in the posttest was not one of transfer; it was the same task practiced during training. The remarkable fact that the PLM group performed better on this transfer task than a control group that practiced the actual task suggests that the PLM produced improvements in structure extraction that are useful to other mathematical skills, such as representation generation. We have also studied this PLM with other age groups. For comparison, we include here a 12th grade sample (Fig. 1, rightmost data). Pretest scores indicate that even in grade 12, the initial ability to generate correct equations and graphs is poor.

The current results do not provide insight regarding every variable that differed between the experimental and control groups. In some sense, both manipulations encouraged mapping structure across representations; it appears that the organized trials of the PLM were more effective. One important difference may have been the fact that the PLM group received feedback after each trial. Not only might this have facilitated PL, but immediate feedback can be valuable in various learning contexts (e.g., Mathan & Koedinger, 2003). We have carried out further studies attempting to isolate effective ingredients of the PLM; these raise a number of interesting issues and will be reported elsewhere (J. Son, J. Zucker, N. Chang, & P. J. Kellman, unpublished data).

The MultiRep PLM included the three design properties described earlier. Students performed discriminations and mappings involving key structures in the PLM task; they did not receive explicit procedural or declarative instruction, and they did not solve equations or word problems in the PL task. The results also reflect outcomes that we suggested are consistent with PL effects. The core notions in this module (e.g., the structure of the equations given in slope-intercept form, the depiction of linear functions in Cartesian coordinates, and the interpretation of word problems) are heavily instructed topics in middle and high school mathematics curricula. The improvement produced by a short PLM intervention at both grade levels is generally consistent with the idea that the PLM addressed dimensions of learning that are not well-addressed by conventional instruction. The results also indicated generative use of structure. Achieving learning criteria within the PLM required accurate and fluent processing of novel exemplars. Moreover, evidence of remote transfer was found, in learners’ markedly enhanced abilities to generate correct equations and graphs from word problems, a correct equation from a graph, and so on.

8. Experiment 2—From knowledge to fluency: Algebraic transformations

One prediction of a PL approach is that it should be possible for a student to have relevant declarative and procedural knowledge in some domain and yet lack fluent information extraction skills. We tested this idea in work in algebra learning with students who had been instructed for half of a school year on the basic concepts and procedures for solving equations. The hypothesis was that despite reasonable student success in declarative and procedural learning, the “seeing” part of algebra is poorly addressed by ordinary methods and might be accelerated by a PL intervention focused on structures and transformations.

The task we chose was mapping algebraic transformations. On each trial, a target equation appeared, and below it was given four other equations. One equation was a legal algebraic transformation of the target; the others were not. The learner was instructed to choose the legal transform as accurately and quickly as possible. The task was constructed to require comparison of structure between the target equation and possible choices. Targets and choices were novel on each trial, and pretest and posttest problems were not used in the learning phase. The PLM design incorporated the design criteria described above. In particular, learners did not practice solving equations in the PLM. We hypothesized, however, that this PL task, by inducing attention to structure and transformation, would improve the seeing of patterns and relations in algebra, and perhaps transfer to improved fluency in actual problem solving. Additional discussion may be found in Kellman et al. (2008).

8.1. Method

8.1.1. Participants

Participants were 30 eighth and ninth grade students at an independent philanthropic school system in Santa Monica, California, tested after mid-year of a year-long Algebra I course.

8.1.2. Apparatus and materials

The PLM was tested on standard PCs using the Windows operating system in computer-equipped classrooms. All assessments and the PLM were presented on computer, with participants’ data being sent to a central server.

8.1.3. Design and procedure

The experiment was set up to assess the effects of PL techniques on learners’ speed and accuracy in recognizing algebraic transformations and the transfer of PL improvements in information extraction to algebra problem solving. A pretest was given on one day, followed by 2 days in which students worked on the PLM for 40 min/day. A posttest was administered the next day. For a subset of subjects, a delayed posttest was administered 2 weeks later. In the Algebraic Transformations PLM, participants on each trial selected from several choices the equation that could be obtained by a legal algebraic transformation of a target equation. An example is shown in Fig. 3. Problems involved shifts of constants, variables, or expressions. Accuracy and speed were measured, and feedback was given. Parallel versions of assessments were constructed such that corresponding problems on separate versions varied in the specific constants, variables, or expressions appearing in each equation. Each participant saw a different version in pretest, posttest, and delayed posttest, with order counterbalanced across participants. Each version of the assessment contained recognition problems similar to those in the PLM and solve problems, used as a transfer test. Solve problems were basic Algebra I equations in a single variable, ranging from simple items (such as x– 5 = 2) to more complex “two-step” problems (such as −6 = 3t/5).

8.2. Results and discussion

Fig. 4 shows the data from this study on the transfer task of equation solving, for students who completed the pretest, learning phase, posttest, and delayed posttest. A key insight from this study comes from the pretest data. The accuracy of algebra problem solving was quite high for learners at the beginning of the study, averaging almost 80%. This level of competence indicates the success of instructional efforts in conveying concepts and procedures for solving equations. Yet students’ explicit knowledge contrasts with an obvious difficulty in fluent processing of structure: Students take about 28 s per problem to solve simple algebra problems! This aspect of their problem solving was dramatically improved by PLM use. After two sessions, speed of solving had dropped to about 12 s per problem. These gains were fully preserved after a 2-week delay. Note that these learners never practiced solving equations in the learning phase; the PLM activity focused on recognizing structure and transformations. These outcomes are consistent with several of the characteristic PL effects we noted earlier: increased fluency, generative use of structure, and persistence over a delay. Perhaps most striking, the attainment of large and lasting gains in fluency from a short intervention suggests that PL methods can produce rapid advancement on dimensions of learning that are not well addressed by conventional instruction.

9. Experiment 3—Fostering structural insight: Linear measurement

U.S. students perform poorly on measurement problems on national and international standardized tests. Even basic skills, such as linear measurement with rulers, show significant deficits. For example, National Assessment of Educational Progress (NAEP) results indicate that many elementary and middle school students are unable to use a ruler that has been broken to measure a 2½ inch toothpick whose left end is aligned with 8 rather than 0 (National Center for Education Statistics, 2008). Students’ incorrect responses suggest that they do not conceive of units of linear measurement as having extent. Further, they do not make a clear distinction between position and distance, and they have great difficulty using fractions to represent subdivisions of units.

In a current project, we are applying PL principles to concepts of measurement and fractions. One PLM addressed learning difficulties related to linear measurement. Explanations and demonstrations that students normally receive may be insufficient for them to extract relevant features and relations in measurement. As a result, they learn blind procedures that involve misunderstandings of measurement. We applied PL principles using interactive trials that emphasized students’ discrimination of position and distance and fostered their structural intuitions about units, including fractional units, in measurement problems.

9.2. Results and discussion

As can be seen in Fig. 5, prior to instruction, the sixth graders and the seventh and eighth grade control groups scored similarly. This result suggests that the substantial focus on these topics in normal curricula for these grades produces little improvement through the middle school years. PLM use produced significant improvement in the sixth grade intervention group, confirmed by a one-way anova comparing the sixth, seventh, and eighth grade groups [F(2,138) = 19.687, p < .001]. The sixth graders achieved nearly identical scores on a delayed posttest administered 4.5 months later, indicating that their learning gains were fully maintained. Among the subscales, students’ performance improved in reading and constructing lengths with conventional and broken rulers, and they also made strong gains in problems involving fractions. As assessment problems were transfer items of varied kinds, it appears that the PLM guided students to see the relevant structures underlying units, measurement, and fractions, replacing blind procedures used initially by many students.

10. General discussion

The study of PL interventions in education and training has barely begun, yet the promise is already clear. PL techniques have the potential to address crucial, neglected dimensions of learning. These include selectivity and fluency in extracting information, discovering important relations, and mapping structure across representations. Each PLM described here addressed an area of mathematics learning known to be problematic for many students. In each case, a relatively short intervention produced major and lasting learning gains, and in each case the learning transferred to key mathematical tasks that differed from the training task.

How do we know that students’ learning gains involved PL? As noted earlier, it is difficult in realistic learning domains to exclude all but one type of learning. Both the design and results of the interventions reported here, however, implicate PL as the primary driver of learning. All of the PLMs described here incorporate the general design criteria for PL interventions that we noted. The primary learning tasks required learners to classify or distinguish key features and relations that carry important mathematical information in each domain. Learning occurred over numerous short classification trials with varied instances and involved minimal explicit instruction. Moreover, the results of these interventions show important signatures of PL effects. All of the PLMs showed generativity in structure use, as evidenced by transfer to tasks that differed from the training task. The three PLMs described here also have complementary characteristics with regard to PL effects. The Algebraic Transformations PLM particularly highlights fluency effects in extracting pattern information and the importance of PL manipulations in attaining it. Both the Algebraic Transformations and Linear Measurement PLMs showed no decrement in performance in a delayed posttest. Full preservation of learning gains after a 4.5-month delay in the measurement PLM is an especially striking result. Although persistence of learning may not exclusively implicate PL effects, it is consistent with them.

Perceptual learning methods bear interesting relations to other work applying cognitive principles to improve learning. Some researchers have found that learning of concepts, relations, or problem-solving strategies can be facilitated by comparisons. A number of studies have shown that learning can be enhanced when learners consider two or more cases that involve a common structure but differ in superficial respects (Gick & Holyoak, 1983; Loewenstein, Thompson, & Gentner, 2003); others indicate that comparison of contrasting cases can produce better understanding of relevant structure (Bransford & Schwartz, 2001; Gick & Paterson, 1992; Rittle-Johnson & Star, 2007; Star & Rittle-Johnson, 2009).

Although sometimes described as a cognitive or learning mechanism, “comparison” denotes a procedure. It remains to be determined what information processing effects are triggered by comparison, that is, what learning mechanisms are engaged. One answer to this question was suggested by Schwartz and Bransford (2004; see also Bransford & Schwartz, 2001). They refer to:

This interpretation of contrasting cases is highly consistent with views of PL and E—Gibson’s view in particular. Gibson (1969) emphasized differentiation, specifically, the learning of “distinguishing features,” in PL. Likewise, the converse idea of comparing varied instances that share some structure describes conditions Gibson noted were relevant to discovery of invariance.

Both of these effects can be modeled in rigorous ways in some PL tasks (e.g., Petrov et al., 2005), although usually in tasks much simpler than those studied here. Still, it has been argued that PL tasks of varying levels and complexity share an underlying commonality in terms of processes of discovery and selection (Kellman & Garrigan, 2009). In neural network approaches to PL, for example, discrimination or classification experience, along with feedback, can lead to the strengthening of the weights of analyzers that detect certain information and downweight other analyzers. Such a mechanism concurrently handles both discovery of distinguishing features between categories and invariance within categories. This description is not meant to oversimplify the modeling task in PL; indeed, high-level PL probably involves generation of candidate structures that are not initially on some fixed list of properties or analyzers, an aspect of discovery that remains mysterious (Kellman & Garrigan, 2009). However, PL models do suggest some promising avenues for understanding the learning mechanisms underlying comparison.

More could be said about comparison procedures and PL interventions; we mention a few interesting issues here. First, whether comparison of a couple or a few cases suffices or whether more extensive classification of examples is needed may depend on the learning task. Second, comparison tasks, more so than simple PL interventions, often involve some explicit instruction. The value of such input, however, may not indicate non-PL factors but simply indicate that language may help guide search and discovery processes in PL (Kellman & Garrigan, 2009). Not much evidence suggests that language alone can accomplish effective discovery; most experiments using comparison appear to rely heavily on representations provided to learners. Furthermore, even when an invariant or distinguishing feature has been explicitly taught, such a manipulation likely does little to enhance fluency of classification.

We note one final issue. Although “differentiation learning” has been used as a synonym for PL (Gibson, 1969), it is conceivable that there are domains in which differentiation learning can occur in the complete absence of perceptual representations. Such an effect might allow some uses of comparison to be clearly distinguished from PL; however, as noted above, most comparison experiments make extensive use of representations that permit PL. For example, in a recent study of comparison in numerical estimation procedures (Star & Rittle-Johnson, 2009), learners who compared two estimation strategies viewed 32 worked examples. Clearly, these comparisons provided ample opportunities for PL. Whether there are cases of comparison that do not rely on PL, whether PL and comparison have different effects on fluency, and how explicit inputs may assist discovery processes in PL all pose interesting issues for further research.

Taken together, the results reported here suggest that PL components play a strong role even in complex tasks that involve symbolic processing. They further indicate that appropriately designed PL technology can produce rapid advances in learning. Few learning interventions produce large learning improvements and transfer from short interventions as occurred in each experiment reported here. Further research will undoubtedly reveal even more about how to optimize discovery of structure and fluency in complex domains. Moreover, as we have briefly considered, the synergy in complex tasks among perceptual, declarative, and procedural learning poses important questions and opportunities regarding both the detailed nature of the interactions and how they may be optimally combined in instruction.