On Learning Natural-Science Categories That Violate the Family-Resemblance Principle

Abstract

The general view in psychological science is that natural categories obey a coherent, family-resemblance principle. In this investigation, we documented an example of an important exception to this principle: Results of a multidimensional-scaling study of igneous, metamorphic, and sedimentary rocks (Experiment 1) suggested that the structure of these categories is disorganized and dispersed. This finding motivated us to explore what might be the optimal procedures for teaching dispersed categories, a goal that is likely critical to science education in general. Subjects in Experiment 2 learned to classify pictures of rocks into compact or dispersed high-level categories. One group learned the categories through focused high-level training, whereas a second group was required to simultaneously learn classifications at a subtype level. Although high-level training led to enhanced performance when the categories were compact, subtype training was better when the categories were dispersed. We provide an interpretation of the results in terms of an exemplar-memory model of category learning.

Keywords

category learning similarity computational modeling open data

Classic work in psychological science suggests that categories in the natural world are structured in accord with a family-resemblance principle (Rosch & Mervis, 1975; Smith & Medin, 1981). According to this principle, members of the same category share bundles of characteristic features that are not shared by members of contrasting categories. This principle produces structures in which there tends to be high within-category similarity and low between-category similarity.

In this article, we are concerned both with the universality of this principle, especially with regard to science categories that students must learn in educational contexts, and with effective learning methods for categories that violate this venerable principle. We focus on the geoscience domain and, in particular, rock categories. Rock categories are the foundation for gaining expertise in geological sciences and must be learned by students in introductory college-level geoscience courses. Further, learning rock categories is incorporated into middle-school science curricula (e.g., Daniel, Hacket, Moyer, & Vasquez, 2005) and is a task that also challenges hobbyists. Thus, rock categories constitute a ubiquitous natural-science domain that is encountered by many people.

As is true of all natural categories, rock categories have a hierarchical structure (e.g., Collins & Quillian, 1969); each of the broad categories—igneous, metamorphic, and sedimentary rocks—is composed of many subtypes. For example, common subtypes of igneous rocks include granite, basalt, and obsidian. Consider two possible structures that might characterize how the three main rock categories and their various subtypes are situated in a psychological similarity space. The left panel of Figure 1 illustrates a case in which three subtypes of each broad category form a compact, coherent cluster in the space. Within each high-level category, the subtypes are similar to one another, whereas between the high-level categories, similarity is low. This structure reflects the classic theoretical assumption that these natural categories obey a family-resemblance principle. By contrast, the right panel of Figure 1 illustrates an extreme case in which the category structure is highly disorganized and dispersed. The three subtypes of igneous rocks are located in separate clusters, and likewise for the subtypes of metamorphic and sedimentary rocks. In this dispersed structure, there are many cases in which between-category similarity is high, but within-category similarity is low.

Fig. 1.

Schematic illustration of compact and dispersed category structures, using the example of subtypes of igneous (I), metamorphic (M), and sedimentary (S) rocks.

To the extent that rock categories are characterized by the dispersed structure just illustrated, this would provide an important example of a natural-category structure that violates the family-resemblance principle. Indeed, the strong intuition of our fourth author and other geology-education collaborators was that the structure of the broad rock categories is disorganized and dispersed. To formally investigate this possibility, in Experiment 1 we used multidimensional-scaling (MDS) methods to illuminate the salient dimensions along which rocks are discriminated and to determine whether the structure of the rock categories within this dimensional space is relatively compact or dispersed. Were rock categories to reflect a relatively dispersed structure, then learning these categories would presumably present significant challenges. Experiment 2 was then conducted to begin the investigation of what techniques may facilitate the learning of dispersed category structures. We obtained evidence that, in certain respects, techniques that are effective for learning dispersed categories may be the polar opposite of those that facilitate learning the compact category structures that are more usually investigated in psychological science.

Experiment 1

Using major college-level textbooks in the geosciences as a guide (Marshak, 2013; Tarbuck & Lutgens, 2015), we compiled a list of 10 of the most common subtypes of each of the three major categories of rocks: igneous, metamorphic, and sedimentary. The 30 subtypes are listed in Table 1. In consultation with an expert in geoscience education (our fourth author), we conducted Web searches for a representative token of each subtype. Subjects provided pairwise similarity judgments among all subtypes, and the averaged similarity judgments were submitted to an MDS analysis (Shepard, 1974, 1980). This analysis allowed us to observe the overall proximity relations among the rocks and also provided information about some of the salient dimensions that influenced the participants’ similarity judgments.

Table 1.

Subtypes of Igneous, Metamorphic, and Sedimentary Rocks Used in Experiment 1

Igneous	Metamorphic	Sedimentary
Andesite	Amphibolite	Bituminous coal
Basalt	Anthracite	Breccia
Diorite	Gneiss	Chert
Gabbro	Hornfels	Conglomerate
Granite	Marble	Dolomite
Obsidian	Migmatite	Micrite
Pegmatite	Phyllite	Rock gypsum
Peridotite	Quartzite	Rock salt
Pumice	Schist	Sandstone
Rhyolite	Slate	Shale

Method

Subjects

The subjects were 82 undergraduates from Indiana University Bloomington who participated in partial fulfillment of a requirement for an introductory psychology course. We used a large sample size to ensure that any evidence for a dispersed category structure in the MDS solution could not be attributed to noise in the similarity judgments.

Stimuli

The stimuli were photographs of rocks obtained from Web searches. The set included one representative token of each of the 30 subtypes listed in Table 1. Each rock picture was square in shape and subtended a visual angle of approximately 7° × 7°. The stimuli were displayed on a gray background on a CRT monitor. The experiment was conducted on PCs running MATLAB (The MathWorks, Natick, MA) and the Psychophysics Toolbox (Brainard, 1997).

Procedure

On each trial, participants were shown a pair of pictures of two different rock subtypes from Table 1. The pictures were presented side by side. Each of the 435 unique pairs was shown once in a unique random order for each individual subject, with the left/right placement of the members of each pair also randomized. Subjects rated the similarity of each pair using the keys from “1” (extremely dissimilar) through “9” (extremely similar) on the keyboard.

Results

The data for the MDS analysis were the averaged similarity ratings for each of the 435 unique pairs. (We deleted from the analysis the data of 14 subjects whose individual similarity ratings showed low correlations with the group average.) The averaged similarity ratings were submitted to a standard nonmetric MDS analysis (the MDSCALE function of MATLAB). This analysis locates each object as a point in an M-dimensional space such that the computed distances among the points optimize the rank-order-based criterion known as stress (Kruskal & Wish, 1978). The number of dimensions required for a complete account of the similarity structure of natural real-world categories is undoubtedly quite large (e.g., Storms, De Boek, & Ruts, 2000; Storms, Navarro, & Lee, 2010; Swets et al., 1991). However, for present purposes, our goal is simply to provide prima facie evidence of the highly dispersed nature of the similarity structure of the rock categories, as well as to provide some sense of the primary dimensions that influenced subjects’ similarity judgments. Therefore, we present outputs obtained only from three- and four-dimensional solutions (see the Supplemental Material available online for more detailed statistical results).

For purposes of easy visualization, the three-dimensional solution is pictured in Figure 2. The spheres, cubes, and diamonds correspond to individual subtypes of igneous, metamorphic, and sedimentary rocks, respectively. It is obvious from inspection that the category structures are highly dispersed. There are numerous cases in which the similarity between subtypes of different high-level categories is quite high, as well as numerous cases in which the similarity between subtypes of the same high-level category is quite low.

Fig. 2.

Three-dimensional scaling solution for the 30 rock subtypes from Table 1. Spheres represent igneous rocks, cubes represent metamorphic rocks, and diamonds represent sedimentary rocks.

To provide greater insight into these results, in Figure 3 we show the pictures of the rock tokens on two-dimensional projections of the four-dimensional solution. Figure 3a provides a plot of the pictures on Dimensions 1 and 2 of the space, whereas Figure 3b provides a plot on Dimensions 3 and 4. (As detailed in the Supplemental Material, we conducted a separate individual-dimensions ratings study to rotate the derived MDS solution onto the interpreted dimensions shown in the figure.) As is apparent from inspection, Dimension 1 corresponds reasonably well to the attribute of lightness of color, with dark rocks located to the left of the space and light rocks located to the right. Dimension 2 corresponds reasonably well to the attribute of average grain size: Rocks with no visible grain are located toward the bottom of the space, rocks with fine and coarse grains are located in the middle, and rocks with highly fragmented grain are located toward the top. Dimensions 3 and 4 are more difficult to interpret; however, our individual-dimensions rating study revealed significant correlations between the Dimension 3 coordinates and an attribute that we refer to as “saturation” or “coloration”; white and black rocks are to the left, and rocks with coloration are to the right. In addition, Dimension 4 corresponds roughly to the attribute of “organization”: Rocks toward the top of the space have organized layers or bands, whereas rocks toward the bottom are composed of disorganized fragments that seem glued together in haphazard fashion.

Fig. 3.

Results of the four-dimensional solution from the multidimensional-scaling analysis of the 30 rock subtypes in Experiment 1. The graph in (a) plots the locations of the rocks on Dimensions 1 and 2, and the graph in (b) plots the locations of the rocks on Dimensions 3 and 4. Membership in the high-level categories is indicated by the letters preceding the subtype names: I = igneous, M = metamorphic, and S = sedimentary.

Inspection of these graphs also allows deeper insight into the highly dispersed category structure depicted in Figure 2. For example, obsidian (igneous), anthracite (metamorphic), and bituminous coal (sedimentary) are the dark black rocks with no visible grain that lie clustered together to the lower left of the plot for Dimensions 1 and 2. By comparison, granite (igneous) and breccia (sedimentary) are lighter colored, coarse-grained rocks that are clustered together at the opposite, top right corner of the space. Numerous other examples of the dispersed structure of the rock categories are easily visible in the plots.

Discussion

The scientific classifications igneous, metamorphic, and sedimentary are based on the manner in which rocks were originally formed. However, these formation processes occurred eons ago and are not directly accessible. An observer who needs to classify rocks into these categories must therefore infer their membership by integrating information across numerous other component dimensions to which there is access. Here, we have provided prima facie evidence that the organization of the rock categories along these component dimensions is highly disorganized and dispersed.

The disorganized structure of the rock categories violates the general family-resemblance principle of natural categories. It is true that Rosch, Mervis, Gray, Johnson, and Byes-Braem (1976) argued that family-resemblance structure is strongest at a privileged “basic” level in which within-category similarities are large relative to between-category similarities (e.g., birds vs. insects). As one moves to higher, superordinate levels (e.g., animals vs. plants), within-category similarities grow smaller, so family resemblance may weaken. However, it is clearly not the case that, for example, there are multitudes of subtypes of animals that are highly similar to multitudes of subtypes of plants—yet we found high between-category similarity for numerous subtypes of the disorganized rock categories. Alternatively, as one moves to lower, subordinate levels (e.g., crows vs. ravens), between-category similarities grow larger. However, it is clearly not the case that, for example, there are multitudes of subtypes of crows that are highly dissimilar to one another—yet we found high within-category dissimilarity for numerous subtypes in the disorganized rock categories. In a nutshell, whatever the category level at which igneous, metamorphic, and sedimentary rocks are presumed to exist, our results provide an example of significant violations of the universality of a family-resemblance principle for natural categories.

Admittedly, there are other dimensions that provide information about rock-category membership that are nonvisual and that were not tapped by our similarity-ratings study. For example, various sources of tactile, mechanical, and chemical information contribute to geologists’ ability to identify and classify rocks. In addition, expert geologists may have learned to attend to alternative dimensions of rock stimuli that are more diagnostic of category membership than the ones that influenced our naive subjects. (It is important to note, however, that in describing the characteristics of rocks, college-level geology textbooks make reference to the same dimensions revealed by our MDS study. For example, the dimension of lightness/darkness of color reveals much about the chemical content of igneous rocks, and the dimension of average grain size reveals whether igneous rocks solidified deep underground, nearer the surface, or in the air—e.g., Tarbuck & Lutgens, 2015, chap. 3.) However, although future research on the topic is needed, it seems extremely unlikely that building higher-dimensional similarity representations that include these alternative sources of information would change the extremely dispersed category structures into compact ones. At the least, the results of our initial scaling study provided strong motivation for pursuing the question of what might be effective procedures for teaching dispersed category structures.

Experiment 2

An initial common goal for most introductory science courses is to teach students to classify at a high level of categorization. For example, in introductory geosciences courses, a goal is to teach students to classify rocks into the three broad categories: igneous, metamorphic, and sedimentary. To meet this objective, an intuitive approach would be to focus training at this high level. It is also possible, however, for training to occur at a lower, subtype level. Specifically, rather than engaging solely in focused, high-level training, a student might be required to simultaneously learn to classify at the subtype level.

Theoretically, general learning principles based on transfer-appropriate processing (Blaxton, 1989; see also Thomas & McDaniel, 2007) suggest that if the goal is to learn categories at a high level, then it might be best for training to focus on the categories at that high level. Indeed, recent work reported by Noh, Yan, Vendetti, Castel, and Bjork (2014) confirms this intuition. In their study, subjects learned to classify a set of pictures of snakes into high-level categories, such as venomous versus nonvenomous, or into lower-level categories, such as distinct subtypes of venomous snakes. Noh et al. found that learning at the subtype level interfered with subjects’ ability to learn the high-level categories. However, there are also findings that point in the opposite direction. Using one specific category structure, Lassaline, Wisniewski, and Medin (1992) found that learners displayed more accurate classification when they were trained and tested at a lower level of a hierarchy than when they were trained and tested at a more general level. Palmeri (1999) and Verheyen, Ameel, Rogers, and Storms (2008) tested the same category structure in variants of the original paradigm and obtained the same results.

We propose a more nuanced hypothesis—that the relative merits of focused high-level category training and simultaneous subtype training will depend on whether the categories form compact or dispersed clusters in psychological similarity space. On the basis of the workings of a well-known exemplar-memory model of categorization (Nosofsky, 1986, 2011), we suggest that focused high-level training will tend to be most effective when the category structure is compact, but that simultaneous high-level and subtype training will be most effective when the category structure is dispersed. We describe the exemplar-model machinery in a theoretical discussion that follows our presentation of Experiment 2 and its results.

Design

To test our predictions, we conducted a classification-learning study in which the category structures closely approximated the schematic compact and dispersed ones illustrated in Figure 1. Subjects learned to classify pictures of rocks into the three high-level categories (i.e., igneous, metamorphic, or sedimentary). Each high-level category was composed of three subtypes. As illustrated in Figure 1, in one condition we selected the three subtypes such that they combined to form compact high-level categories, whereas in a second condition the three subtypes formed dispersed high-level categories. We conducted an MDS study separate from Experiment 1 to confirm that our choice of subtypes produced the desired category structures (see the Supplemental Material for details).

One group of subjects learned the high-level categories through direct high-level training, whereas a second group simultaneously learned to classify at the subtype level. Following training, subjects’ ability to generalize to novel instances was tested in a transfer phase.

Method

Subjects

The subjects were 133 undergraduates from Indiana University who participated in partial fulfillment of a requirement for an introductory psychology course. All subjects had normal or corrected-to-normal vision. There were 33, 34, 32, and 34 subjects in the compact-structure/high-level-training, compact-structure/subtype-training, dispersed-structure/high-level-training, and dispersed-structure/subtype-training conditions, respectively. Our sample sizes per condition were based on Noh et al. (2014; initial data collection from slightly more than 20 subjects in each of four conditions); to be conservative, we exceeded their sample sizes to obtain power of approximately .80 to identify a medium-size effect.

Stimuli

The stimuli were photographs of rocks obtained from Web searches. In both the compact-structure and dispersed-structure conditions, there were three subtypes of each of the high-level categories (i.e., igneous, metamorphic, and sedimentary rocks) and six tokens of each subtype. The subtypes used in each condition are listed in Table 2. (The subtypes granite, quartzite, and breccia were common to the compact- and dispersed-structure conditions, whereas all the other subtypes were unique to one of these conditions. Note also that the coquina subtype had not been used in Experiment 1. Because of its resemblance to breccia and conglomerate—all three are sedimentary rocks composed of large fragments that seem glued together—it was a convenient subtype to use to create the compact-structure condition.) Other aspects of the stimuli were the same as described in Experiment 1.

Table 2.

Rock Subtypes Used in Experiment 2

Compact-structure condition		Dispersed-structure condition
Subtype	Rock name	Subtype	Rock name
I1	Gabbro	I1	Pegmatite
I2	Granite	I2	Granite
I3	Diorite	I3	Obsidian
M4	Slate	M4	Amphibolite
M5	Quartzite	M5	Quartzite
M6	Hornfels	M6	Anthracite
S7	Conglomerate	S7	Dolomite
S8	Coquina	S8	Bituminous coal
S9	Breccia	S9	Breccia

Note: I = igneous; M = metamorphic; S = sedimentary.

Procedure

In all conditions, for each individual subject, half the rocks in each subtype were randomly chosen to serve as training items, and the other half served as novel transfer items. The experiment was organized into three training blocks followed by a transfer block. During each training block, each of the 27 training items was presented twice. The order of presentation of the 54 total items was randomized for each subject. During the transfer block, each of the 27 original training items and each of the 27 novel transfer items was presented twice, again in a random order.

On each trial, a rock was presented at the center of the computer screen, and the subject classified it into a category by pressing a key on the computer keyboard. In the high-level-training conditions, the categories were “igneous,” “metamorphic,” and “sedimentary.” In the subtype-training conditions, the categories were the nine individual subtypes. However, rather than using the subtype names (e.g., “granite”), we used a nomenclature that combined each rock’s high-level category with a subtype number: “igneous 1” through “igneous 3,” “metamorphic 4” through “metamorphic 6,” and “sedimentary 7” through “sedimentary 9.” The letter keys “I”, “M,” and “S” were used for responses in the high-level-training condition. The number keys with attached labels “I1,” “I2,” “I3,” “M4,” “M5,” “M6,” “S7,” “S8,” and “S9” were used for responses in the subtype-training condition.

In the training blocks, corrective feedback was provided for 3 s following each classification response. The nature of the feedback was appropriate to the training level. For example, in the high-level-training condition, the feedback might be “Correct! Igneous” or “Incorrect: Metamorphic,” whereas in the subtype-training condition, the feedback might be “Correct! Igneous-1” or “Incorrect: Igneous-2.” Corrective feedback was withheld in the transfer block: Following each response, the computer simply displayed “Okay.” At the end of each block, subjects were informed of their overall percentage of correct responses.

Results

Figure 4 summarizes performance during the transfer phase as a function of category structure (compact vs. dispersed), training condition (high-level vs. subtype), and item type (original training vs. novel transfer). The measure of performance is the mean proportion correct with respect to the high level of categorization. For example, if a subject in the subtype-training condition classified a rock from subtype Igneous 1 as Igneous 2, this response was scored as correct. This method of scoring is sensible because our primary goal was to assess how well subjects were able to classify at the high level of categorization.

Fig. 4.

Mean proportion correct in Experiment 2 as a function of category structure (compact vs. dispersed), training condition (high-level vs. subtype), and item type (original training vs. novel transfer). Accuracy was scored with respect to the high level of categorization. Error bars represent ±1 SEM, calculated between subjects.

Inspection of Figure 4 reveals several clear effects, which were confirmed by a mixed-model 2 (category structure; between groups) × 2 (training condition; between groups) × 2 (item type; within groups) analysis of variance on the proportion-correct data. First, performance was better in the compact-structure condition than in the dispersed-structure condition, F(1, 129) = 207.0, MSE = 0.018, p < .001, η_p² = .616. This result is not surprising given that the subtypes formed coherent clusters with respect to the high level of categorization in the compact-structure condition, but not in the dispersed-structure condition. Second, performance was better for the original training items than for the novel transfer items, F(1, 129) = 442.3, MSE = 0.005, p < .001, η_p² = .774. This result indicates that there was significant within-subtype variability among the rock tokens.

Third, and of greatest interest, the hypothesized interaction between category structure and training condition was confirmed, F(1, 129) = 21.0, MSE = 0.018, p < .001, η_p² = .140. Performance was better in the compact-structure condition (collapsed across the original training and novel transfer items) when training focused on only the high level of categorization. This result was confirmed by an a priori test for simple effects, F(1, 129) = 6.03, MSE = 0.018, p = .015. By contrast, performance was better in the dispersed-structure condition (collapsed across the original training and novel transfer items) when subjects were required to simultaneously learn the subtypes, F(1, 129) = 15.05, MSE = 0.018, p < .001.

We conducted a conceptual replication of the dispersed-structure condition of Experiment 2 (see the Supplemental Material for details). The procedure was the same except that the subtype labels were the actual rock names rather than numbers (e.g., “Igneous: Granite,” “Metamorphic: Quartzite”). The pattern of results was identical to the one reported for Experiment 2: Performance was significantly better when subjects were required to simultaneously learn the subtypes rather than the broad categories only, F(1, 71) = 4.75, MSE = 0.032, p = .033, η_p² = .063. In short, the advantage of subtype training observed for the present dispersed category structure is robust.

Toward a formal account of the subtyping effect

Our hypothesis that subtype training would lead to enhanced high-level categorization in the dispersed-structure condition was motivated by conceptual predictions derived from Nosofsky’s (1986, 2011) generalized context model (GCM), an exemplar-memory model of category learning (for closely related clustering models of categorization, see, e.g., Anderson, 1991; Love, Medin, & Gureckis, 2004; Sanborn, Griffiths, & Navarro, 2010). According to the GCM, people learn categories by storing individual exemplars in memory and classify items on the basis of their summed similarity to the exemplars of the alternative categories. Exemplars are represented as points in a multidimensional space, and similarity is a decreasing function of the distance between objects in the space. Observers are presumed to attend selectively to dimensions of the similarity space that are highly diagnostic of category membership. This process acts to increase within-category similarities among exemplars and to decrease between-category similarities in the psychological similarity space.

As explained in greater detail in the Supplemental Material, in the formal model, the summed similarities are “magnified” by a response-scaling parameter, γ (Ashby & Maddox, 1993; McKinley & Nosofsky, 1995). This parameter can be interpreted in terms of a process-based implementation of the GCM in which retrieved exemplars drive an evidence-accumulation process (Nosofsky & Palmeri, 1997; see also Giguère & Love, 2013). When the magnitude of γ is low, the observer retrieves few exemplars and tends to make noisy, inconsistent decisions, whereas when γ grows larger, the observer retrieves many exemplars and responds more deterministically with the category that yields the greatest overall summed similarity to the test item.

The model naturally predicts our finding that overall performance was better in the compact-structure condition than in the dispersed-structure condition: In the compact-structure condition, within-category exemplar similarity was high, and between-category exemplar similarity was low, whereas the reverse tended to be true in the dispersed-structure condition. The model also naturally predicts that the original training items were classified with higher accuracy than the novel transfer items: An old item provides a perfect match to its exemplar trace in memory, and this boosts its summed similarity to the target category.

In the case of the compact-structure condition, the model predicts more accurate performance when subjects received focused high-level training rather than simultaneous training in discriminating among subtypes. The reason is that highly diagnostic dimensions tend to be available for separating the categories in compact structures. If observers are allowed to focus on the high level, then they can learn to attend selectively to these diagnostic dimensions. By contrast, if observers must simultaneously learn the subtypes, then they are forced to split their attention across numerous idiosyncratic dimensions. Numerous studies involving tests of the GCM (and closely related models) have documented this key role of selective attention in making high-level category learning more efficient than low-level category learning (e.g., Kruschke, 1992; Nosofsky, 1986, 1987; Shepard, Hovland, & Jenkins, 1961). We provide an illustration of these ideas in our Supplemental Material.

Of greatest interest, however, is that the model also predicts our finding that high-level classification of objects in the dispersed category structure was substantially better when subjects were required to simultaneously identify the rocks at the subtype level. A formal statement of the reason for this prediction is provided in the Supplemental Material. Here, we try to convey the key intuitions.

For the dispersed structure, there were no highly diagnostic dimensions for separating objects into the high-level categories, so the role of selective attention to particular dimensions was minimized. In the absence of the selective-attention process, the interexemplar similarities were nearly invariant across the high-level-training and subtype-training conditions. Figure 5 illustrates the resulting scenarios in the two training conditions. In the case of the subtype-training condition, test item S1_i, assumed to belong to subtype S1, yielded a very strong summed-similarity signal with respect to that subtype, because it was highly similar to the exemplars within that subtype (top panel of Fig. 5). The strong summed-similarity signal was further magnified by the response-scaling parameter (γ). The magnified summed-similarity signal for the target subtype was therefore very large relative to the summed-similarity signals for all other subtypes (because the exemplars of other subtypes were dissimilar to the test item). Thus, the model predicts a high proportion of correct classifications when subjects learned to classify at the subtype level. But the model assumes that when subjects were trained solely at the high level, they summed the similarity of the test item to all exemplars of all subtypes in that high-level target category, including those subtypes in the dispersed structure to which the test item was very dissimilar (bottom panel of Fig. 5). When the exemplars from the dissimilar subtypes were included, the magnitude of the target-category summed-similarity signal was weakened relative to the summed-similarity signals of the contrasting categories, so the model predicts a lower proportion of correct classification at the high level.

Fig. 5.

Schematic illustration of the relative strength of the summed-similarity signal of the target category in the subtype-training condition (top panel) and high-level-training condition (bottom panel) of the dispersed-structure condition. The letters and shapes represent the three broad categories of rocks: igneous (I; circles), metamorphic (M; squares), and sedimentary (S; diamonds). Each cluster of three shapes represents a group of similar subtypes. (Recall that in the dispersed-structure condition, the high-level categories were composed of subtypes that belonged to different similarity clusters.) Test item S1_i (dashed diamond) is assumed to be a member of subtype S1, at the far left. In the subtype-training condition, because S1_i was highly similar to all exemplars within its target subtype, it gave rise to a strong summed-similarity signal with respect to its target subtype. In the high-level-training condition, although S1_i was highly similar to the exemplars in subtype S1, it was very dissimilar to the exemplars of the two other subtypes that composed its high-level category. Summing across the exemplars of all subtypes of the high-level category led to a target-category summed-similarity signal that was weakened relative to the summed-similarity signals of the alternative categories.

Beyond the conceptual predictions discussed here, an ambitious goal for future research is to achieve true quantitative predictions from the model of the complete set of rock-classification data. Achieving this goal will require the prior derivation of a precise, high-dimensional scaling representation for the very large number of rock tokens that our subjects learned to classify. This high-dimensional scaling representation would then be used in combination with the GCM for generating quantitative predictions of the probability with which each individual rock token will be classified into the alternative categories in the different conditions (for a review of the procedure in simpler perceptual domains, see, e.g., Nosofsky, 1992).

Summary and Conclusions

Although most categories in the natural world appear to be structured in accord with a family-resemblance principle, our research provides prima facie evidence that the fundamental categories of igneous, metamorphic, and sedimentary rocks strongly violate this principle. Whereas one might expect that these rock categories form coherent clusters in psychological similarity space, our study reveals instead that the category structures are highly disorganized and dispersed. Although the generality of our finding remains to be investigated, it seems likely that many other natural-science categories that need to be learned also have dispersed structures.

Our finding therefore motivates the question of what methods might enhance the learning of such categories. Intuition, as well as classic and recent studies (Noh et al., 2014; Shepard et al., 1961), suggests that if the goal is to learn categories at a high level, then it might be best to receive training focused solely at the high level rather than to also receive training on more fine-grained distinctions at lower subtype levels. In the present research, however, we found support for the hypothesis that the preferred method depends on the detailed structure of the to-be-learned categories. Focused high-level training is indeed the preferred method if the high-level categories comprise subtypes that form compact clusters in psychological similarity space; however, simultaneous subtype-level training appears to be the preferred method if the high-level categories comprise subtypes that are dispersed. It is noteworthy that this result was predicted a priori by a successful exemplar-memory model of categorization and that we obtained the effect using materials from the natural, real-world categories of rocks. Thus, the present results could have significant practical bearing on optimal strategies for teaching rock classification in the geological sciences, and perhaps for teaching classifications in other scientific domains as well.

Footnotes

Action Editor

Kathleen McDermott served as action editor for this article.

Declaration of Conflicting Interests

The authors declared that they had no conflicts of interest with respect to their authorship or the publication of this article.

Funding

This work was supported by National Science Foundation Grant No. 1534014 (EHR Core Research) to R. M. Nosofsky, B. J. Douglas, and M. A. McDaniel.

Supplemental Material

Additional supporting information can be found at

Open Practices

All data have been made publicly available via the Open Science Framework and can be accessed at https://osf.io/. The complete Open Practices Disclosure for this article can be found at http://pss.sagepub.com/content/by/supplemental-data. This article has received the badge for Open Data. More information about the Open Practices badges can be found at https://osf.io/tvyxz/wiki/1.%20View%20the%20Badges/ and .

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