How can we improve students’ reasoning about large magnitudes?
Skills in reasoning about size and scale are central to performance across STEM disciplines (for example, Hawkins, 1978; Tretter, Jones, Andre, Negishi, & Minogue, 2006). Many fundamental science, technology, engineering, and mathematic (STEM) phenomena occur at extreme scales that cannot be directly perceived. For example, a core geologic concept is that geologic events can last billions of years (for example, the Earth formed approximately 4.6 billion years ago). Reasoning about geologic time allows geologists to reconstruct the surface conditions of ancient Earth, produce an accurate time line of Earth’s history, and understand the imperceptibly slow processes that have led to the current environment. Practically, understanding geologic time allows people to reason about the sustainability of non-renewable resources and the consequences of anthropogenic climate change. Given the importance of reasoning about scale, it should be no surprise that the National Research Council in A Framework for K-12 Science Education (National Research Council, 2011) and the American Association for the Advancement of Science in Benchmarks for Science Literacy (American Association for the Advancement of Science, 1993) identified “size and scale” as fundamental scientific concepts, and suggested them as a unifying theme in science education.
Unfortunately, novices have trouble reasoning about phenomena outside human perception. Although novices are sometimes reasonably accurate at ranking phenomena in a correct sequence, they have difficultly comparing the magnitude between phenomena at extreme scales (for example, Delgado, Stevens, Shin, Yunker, & Krajcik, 2007; Jones, Tretter, Taylor, & Oppewal, 2008; Libarkin, Anderson, Dahl, Beilfuss, & Boone, 2005). For example, although students are fairly accurate in identifying a correct sequence of events in Earth’s geologic history (Trend, 2001), they fail to understand the magnitude of time between events (Tretter et al., 2006).
Analogy is potentially valuable for learning and reasoning about phenomena outside human perception because such phenomena cannot be directly experienced (Jones, Taylor, & Broadwell, 2009). Analogy refers to a process of aligning structural similarities between a base concept and a target concept (Gentner, 1983). In fact, analogy is frequently used to teach phenomena at extreme scales (for example, Clary & Wandersee, 2009; Petcovic & Ruhf, 2008; Semken et al., 2009; Sibley, 2009). Unfortunately, even with a wide range of analogies, students struggle to comprehend phenomena outside human perception (for example, Delgado et al., 2007; Jones et al., 2008; Libarkin et al., 2005). Moreover, analogies can mislead students (Brown & Salter, 2010; Duit, 1991). For example, geologic time is often represented using a spatial analogy that compresses the time before life on Earth becomes relatively more complex. Although functional, learning from this nonlinear representation can mislead students into thinking that biologic events occurred very early in the Earth’s history (Resnick, Davatzes, Newcombe, & Shipley, 2016; Resnick, Newcombe, & Shipley, 2016).
Different kinds of spatial analogies may elicit different kinds of cognitive barriers to aligning extreme scales with human scales (for review see Resnick, Davatzes, et al., 2016). For example, a common analogy is to map an extreme scale (for example, Earth’s history) onto a spatial structure, such as the Eiffel Tower (Clary & Wandersee, 2009). However, without knowledge of the base concept (How tall is the Eiffel Tower?), it is difficult to identify corresponding relations between the base concept and target concept (for example, Gentner, 1983; Kotovsky & Gentner, 1996). It can also be difficult to identify the relevant relations to align if the base concept and target concept are different in many ways (Gentner, 1983; Gentner, 2001; Markman & Gentner, 1996, 1997; Kokinov & French, 2003). For example, Earth’s history is also commonly mapped onto a 24-hour clock. However, this analogy contains at least two differences in addition to differences in magnitude (24 hours versus billions of years): clocks are cyclical whereas Earth’s history is linear, and clocks are composed of equal temporal divisions whereas geologic time comprises unequal temporal divisions based on major geologic events. Thus, students may not be able to identify the appropriate analogy to make and, subsequently, draw incorrect conclusions (Brown & Salter, 2010; Gentner, 1983). In this instance, students may erroneously believe that, just like the 24-hour clock, periods of Earth’s history are also evenly spaced, and fail to make the appropriate analogy between relative magnitudes of time between events.
A review of analogical reasoning literature offers three techniques that may be useful in the development of effective analogies for phenomena at extreme scales. First, the base concept and target concept should be structurally aligned; that is, as similar as possible with just one “alignable difference” (Goldstone, 1994; Markman & Gentner, 1993a, 1993b; Medin, Goldstone, & Gentner, 1993). An alignable difference is a common relation shared by the base concept and target concept which differs along one dimension. In the case of aligning an extreme scale with a human scale, both scales should be constructed in the same format (for example, a linear number line), with the only difference being magnitude.
However, because the difference in magnitude between extreme and human scales is so vast, it may not be possible for the base concept and target concept to be structurally aligned. Thus, a second technique to align very different concepts is called progressive alignment (Kotovsky & Gentner, 1996; Thompson & Opfer, 2010). Using progressive alignment, an analogy may consist of more than one analogical step, beginning with a comparison of a base concept and a highly similar intermediate concept. Comparing two very similar concepts as an intermediate analogical step will help extend the analogy to the subsequent alignment of increasingly unfamiliar concepts (Gentner & Namy, 2006). For example, instead of mapping Earth’s history directly onto a spatial time line, the analogy may first map a human lifespan, and increase the amount of time the spatial time line represents in separate analogies (for example, American history, human evolution, and so on) until all of Earth’s history is included in the spatial time line. In learning about scale information, progressive alignment may alleviate conceptual dissimilarity by providing structural alignment across smaller increases of scale (Resnick, Davatzes, et al., 2016).
Finally, a third technique is called hierarchical alignment (Resnick, Newcombe, et al., 2016). Hierarchical alignment advocates the hierarchical organization of all analogical steps within each new analogy to highlight common relational structures between the base, intermediate, and target concepts. In the Earth’s history example, the learner would identify where each previous division in time (for example, human lifespan) would be located relative to the new spatial time line (for example, American history). This process of hierarchical alignment provides salient internal anchor points, which emphasize corresponding relations between each analogical step (Resnick, Newcombe, et al., 2016). Hierarchical alignment specifically supports learning about phenomena outside human perception by highlighting the proportional relation between magnitudes across multiple scales.
In addition to the principles of analogical reasoning discussed above, immediate (Coulter & Grossen, 1997) and corrective (Hao, 1991; Sharpe, Lounsbery, & Bahls, 1997) feedback has also been found effective in increasing student learning. In particular, corrective feedback has been found to be effective for learning about unfamiliar magnitudes in young children (Thompson & Opfer, 2010). Even a single trial of feedback can increase estimation accuracy by providing learners with a salient anchor (Opfer & Siegler, 2007; Opfer & Thompson, 2008; Opfer, Thompson, & Kim, 2016; Thompson & Opfer, 2008).
The current studies
In the current studies we ask whether analogies can foster accurate reasoning about phenomena at scales outside human perception and, if so, what the most efficient and effective techniques in teaching scale information are. While the instructional analogies developed here could be designed for use in teaching any magnitude-based context, we examine analogical reasoning in the context of a large temporal magnitude: geologic time.
Over two experiments we develop two instructional analogies using a combination of structural alignment, progressive alignment, and hierarchical alignment. Of importance, all three techniques provide multiple opportunities to practice making relevant analogies. Thus, both Experiments 1 and 2 examine the efficacy of providing multiple opportunities to align geologic time to a spatial linear representation using structural alignment to improve understanding and reasoning about geologic time. Across both experiments, students are also provided with corrective feedback. A key difference between Experiments 1 and 2 is that Experiment 1 also assesses the benefit of hierarchical and progressive alignment, whereas Experiment 2 assesses structural alignment and corrective feedback alone, without progressive or hierarchical alignment, in an effort to devise a more time-efficient means of intervention.