Appendix D
The Role of Spatial Representations in Learning, Problem Solving, and Transfer

Spatial representations are powerful cognitive tools that can enhance learning and thinking. This position is based on three claims, each of which has significant implications for teaching and learning about spatial thinking. First, creating spatial representations is a powerful way to encode new information that one wishes to recall at a later time. Second, generating images of “old” information that has already been learned and of the situations in which it was learned can powerfully aid in recalling the information at later times. Third, some problems are more readily solved using spatial representations, whereas in other cases, trying to use spatial representations can interfere with problem solving.

LEARNING AND ENCODING NEW INFORMATION

The oldest, best known and most effective mnemonic techniques are based on the power of interactive imagery (Bower, 1973). Possibly the oldest encoding technique using interactive images is the method of loci, which can be traced back to Greek oratory. Like any memory strategy, the method of loci involves two phases: encoding and recall. The key element of the encoding phase begins with imaging a well-known area, generating a perceptually rich spatial representation of a path connecting memorable places through the area, and then encoding each to-be-remembered item by imagining it linked with an easily recalled place along the path. The key element of the recall phase involves generating an image of the familiar path, and then recalling (“stopping at”) each place along the way in order to recall the to-be-remembered object associated with that location.

From the standpoint of encoding, retaining, and retrieving information, imagery-based mnemonic systems take advantage of the unique characteristics of interactive images. Consider, for example, the peg word system, where a series of items to be remembered are associated with a series of target objects (denoted by “peg” words), which are embedded in and ordered by a rhyming scheme.

For the method of loci, instead of reciting a rhyme to prompt recall, one calls to mind the details of a route of travel and recalls items that were associated with different places along the



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Learning To Think Spatially Appendix D The Role of Spatial Representations in Learning, Problem Solving, and Transfer Spatial representations are powerful cognitive tools that can enhance learning and thinking. This position is based on three claims, each of which has significant implications for teaching and learning about spatial thinking. First, creating spatial representations is a powerful way to encode new information that one wishes to recall at a later time. Second, generating images of “old” information that has already been learned and of the situations in which it was learned can powerfully aid in recalling the information at later times. Third, some problems are more readily solved using spatial representations, whereas in other cases, trying to use spatial representations can interfere with problem solving. LEARNING AND ENCODING NEW INFORMATION The oldest, best known and most effective mnemonic techniques are based on the power of interactive imagery (Bower, 1973). Possibly the oldest encoding technique using interactive images is the method of loci, which can be traced back to Greek oratory. Like any memory strategy, the method of loci involves two phases: encoding and recall. The key element of the encoding phase begins with imaging a well-known area, generating a perceptually rich spatial representation of a path connecting memorable places through the area, and then encoding each to-be-remembered item by imagining it linked with an easily recalled place along the path. The key element of the recall phase involves generating an image of the familiar path, and then recalling (“stopping at”) each place along the way in order to recall the to-be-remembered object associated with that location. From the standpoint of encoding, retaining, and retrieving information, imagery-based mnemonic systems take advantage of the unique characteristics of interactive images. Consider, for example, the peg word system, where a series of items to be remembered are associated with a series of target objects (denoted by “peg” words), which are embedded in and ordered by a rhyming scheme. For the method of loci, instead of reciting a rhyme to prompt recall, one calls to mind the details of a route of travel and recalls items that were associated with different places along the

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Learning To Think Spatially route. The peg word system, the method of loci, and other imagery-based systems produce dramatically better memory performance than do the techniques people tend to use in everyday contexts such as verbal rehearsal. Interactive images are remarkably long-lived in terms of their resistance to being forgotten over time, as illustrated in Luria’s (1968) book about a famous memory performer who used the method of loci on stage. Imagery-based systems entail a structure—a rhyme in the case of the peg word system, a route in the case of the method of loci—that provides a systematic way to interrogate one’s memory and to retain the ordering of items. The spatial version of the system has been used and taught for thousands of years. It exemplifies the power of a spatial representation to encode any type of information for later recall. IMPROVING RECALL THROUGH SPATIAL REPRESENTATIONS Bower (1973) not only has demonstrated the power of imagery and representation in the construction of memories, but also has shown that the perspective we adopt in constructing such visualizations can influence heavily what we are later able to recall and the properties of the imaginable representations we construct. Physically reinstating the environmental context in which information was learned (that is, revisiting the physical context) can improve even an infant’s ability to recognize information weeks or months later (Rovee-Collier et al., 1999). For older children and adults, however, a physical visit is not necessary: simply imagining the environmental context in which things were learned can improve recall (Bruck and Ceci, 1999; Smith and Vela, 2001). Significantly, mental reinstatement of the original context has about the same benefits as physically reinstating the context itself (Bjork and Richardson-Klavehn, 1989; Smith et al., 1978). Such findings may explain reports from test takers that recalling where, exactly, a needed fact was on a page of text helps them recall the fact. The fact that mental reinstatement can aid in retrieval of episodic information has been incorporated in the design of cognitive interviews used by police officers (Fisher et al., 1989). The interview procedure depends in part on generating images of the environmental context as a method of prompting memory by eyewitnesses and has been used successfully with children as well as adults (McCauley and Fisher, 1995). Witnesses are asked to replay the event in their minds, first from one perspective, such as the event unfolding as they witnessed it, and then from another perspective, such as a bird’s eye view of the event. Details that witnesses are unable to recall from one perspective can sometimes be recalled from another perspective. Recall is an active process. When people are asked to remember a set of words, recall is better if they are given a task that leads them to generate the list on their own than if they are asked simply to study the list (Slamecka and Graf, 1978). The benefits of generating one’s own knowledge are general and include the memory benefits of elaborating the to-be-learned information with one’s own experiences (Hirshman and Bjork, 1988; Stein and Bransford, 1979). Indeed, learners are often better off when they think partially through a topic before being told about it than when they are told about the topic before thinking (Schwartz and Bransford, 1998). The benefits of generating one’s own knowledge apply to pictures, graphics, and many other types of spatial representations. For example Wills et al. (2000) asked undergraduate students to study complex sentences and either to use a picture that was provided to help them understand the sentences or to generate their own picture. Students showed better comprehension and later recall when they generated their own pictures. The production of spatial representations—diagrams, flow charts, and concept maps—can create spatial schemas that link related items of information and can provide a way to interrogate one’s memory. Reinstatement and generation techniques can be used in teaching spatial thinking. Reinstatement techniques can facilitate classroom learning, especially in the recall of science experiments, laboratory demonstrations, and other classroom or field experiences in perceptually rich contexts.

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Learning To Think Spatially Effective teachers understand, at least implicitly, the importance of repetition and creating exercises that lead learners to reinstate the context of what they have learned. Given the effectiveness of regenerating spatial representations of situations in aiding memory, it makes sense for science students to call to mind the perceptual details of laboratory demonstrations and experiences as one method of remembering them when they are needed. Generation techniques show that learners are better at remembering and understanding when they have generated the knowledge or elaborated it on their own, without being “told” to do so. Spatial representations, especially pictures and diagrams, are particularly helpful in remembering and understanding information. SPATIAL REPRESENTATIONS AND PROBLEM SOLVING Generating an image of a physical system can help adults reason about the properties of the system. Clement (2000) showed this when he asked adults to reason about why the force exerted by a spring increases as a function of its length. Hegarty and Just (1993) showed the power of images when they asked adults to reason about and explain mechanical advantage in pulley systems. They demonstrated that people generate spatial representations in order to reason about word problems that describe pulley systems. Schwartz and Black (1996) showed that spatial representations aid people in reasoning about systems of interconnected gears. Schwartz (1999) went on to show that physical actions can be integrated with spatial representations, such that physical actions can facilitate how readily people can imagine dynamic changes in imagined physical systems. How a problem is represented can affect how easy that problem is to solve or whether it can be solved at all. The history of science is rife with examples of where scientists believe they solved difficult problems at least in part because they thought about them in spatial terms (see Chapters 1, 2, and 3). However, using a spatial representation is not a panacea. Spatial representations can interfere when trying to solve problems—the particular representation must fit the problem’s structure. The classic “Buddhist monk” problem, originated by Duncker (1945), is a good example of where a spatial representation can lead to a simple solution: One morning a Buddhist monk sets out at sunrise to climb a path up the mountain to reach the temple at the summit. He arrives at the temple just before sunset. A few days later, he leaves the temple at sunrise to descend the mountain, traveling somewhat faster since it is downhill. Show that there is a spot along the path that the monk will occupy at precisely the same time of day on both trips. The problem is difficult when people try to solve it mathematically by focusing on issues such as relative speeds of travel. A spatial representation can make the answer transparent. If one imagines two monks, one ascending the mountain starting at sunrise, the other descending the mountain starting at sunrise, it is obvious that they must meet on the path, and when they do they will be at the same point at the same time of day. One can do this imagining in different ways, ranging from generating perceptually rich images of monks and mountains to creating a simple schematic diagram to represent the problem’s key spatial and temporal features. The fit of the problem representation to the problem structure is crucial. Ill-chosen spatial representations can lead one down the “garden path” to problem-solving methods that are doomed. Consider a “knock-out” tournament: Suppose that 130 children enter a single elimination tennis tournament, where all of the children are originally paired, the winners of the first round are paired, the winners of the second round are paired, and so forth, until the tournament champion wins the final round. How many matches will be played altogether in the tournament. Now consider the general case—how many matches will it take to determine the winner when n children enter the tournament?

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Learning To Think Spatially To solve this problem, people often represent it by a simple tree structure, reasoning that the first round will include 65 matches, the second round 32 matches plus one “bye,” and so forth. However, people using this tree structure approach stumble badly and fail when trying to figure out the general case of n entrants. The problem is that a spatial diagram like a tree structure does not lead to a simple solution; simple solutions arise from considering the verbal fact that in order for there to be a single winner, every other entrant must lose. Given that one match determines one and only one loser, there must be a total of n - 1 matches to determine a champion. In general, problem solving by college students shows that translating a problem into terms that fit it, often spatial terms, aids greatly in solving the problem. The type of representation that leads to the fastest and best solutions depends on what the learner already knows and on the structure of the problem—that is, on what type of solution can, in principle, be used to solve the problem. Novick and Morse (2000) demonstrated that spatially diagrammed representations can help to solve problems, and Novick and Hurley (2001) showed that some problems are most efficiently solved with tree hierarchy schematics, whereas others are best approached via matrices or networks. Many problems are more readily solved using spatial representations, but in some cases, trying to use spatial representations can interfere with problem solving. Perceptually rich images can enhance reasoning. Barsalou (1999) showed, however, that perceptually rich images do not necessarily help us to think about things in order to recall and reason about their features. For example, in one experiment, participants were asked to think about a situation and then to list all of the characteristics of the main concept. In one condition, people were asked to think about a grassy field and then to generate a list of all of the features of their concept of “grass.” In the other condition, people were asked to think about sod being rolled and transported on a truck, and then to generate the same type of list. The results were revealing—in the former condition, people generated relatively few features, and very few listed “roots”; in the latter condition, people generated many more features and everyone listed roots, many listed root hairs, and so forth. The difference between the images is the degree of perceptual richness. Indeed, rich perceptual images involve some of the same neural activation patterns as those apparent when someone is actually looking at such a situation instead of just remembering it.