E-Letter responses to:

ed-forum: Jennifer A. Kaminski, Vladimir M. Sloutsky, and Andrew F. Heckler LEARNING THEORY: The Advantage of Abstract Examples in Learning Math Science 2008; 320: 454-455 [Summary] [Full text] [PDF]

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Published E-Letter responses:

Response to N. S. Podolefsky and N. Finkelstein’s E-Letter

Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler,   (2 February 2009)

Counterexamples and Concerns Regarding the Use of Abstract Examples in Learning Math

N. S. Podolefsky, N. D. Finkelstein   (2 February 2009)

Modulo 3 Means Something

Anne Watson   (28 October 2008)

Response to W. McCallum's E-Letter

Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler   (11 August 2008)

Another Interpretation for Learning Math

William McCallum   (11 August 2008)

Response to N. S. Podolefsky and N. Finkelstein’s E-Letter 2 February 2009
Jennifer A. Kaminski Center for Cognitive Science, The Ohio State University, Columbus, OH 43210, USA, Vladimir M. Sloutsky, Andrew F. Heckler, Respond to this E-Letter: Re: Response to N. S. Podolefsky and N. Finkelstein’s E-Letter	N. S. Podolefsky and N. Finkelstein reference their studies investigating what types of representations best facilitate learning physics concepts (e.g. electromagnetic waves) (1, 2). They have found that students who learned by scaffolding with concrete analogues outperformed students who learned with only abstract diagrammatic representations. We find these results to be both interesting and practical. However, these studies differ from ours in both goal and method. The goal of the Podolefsky and Finkelstein studies was to consider conditions that improve learning of one physics domain. The tutorials taught students about electromagnetic waves with concrete analogies to string waves and sound waves. Across condition, only the physical appearance of the representations was varied; potential prior knowledge of strings, sound, and graphs was not systematically controlled. Correct responses to questions about electromagnetic waves were compared across learning conditions. In sum, their research considered how to improve learning. In contrast, our research considered how to improve transfer. Our participants received no instruction in the novel transfer domain. The analogy between the learning instantiation(s) and the transfer instantiation was not told to participants. They were only told to apply what they have learned. This is the challenge of applying mathematics. Mathematical structure underlies phenomena in many fields such as physics, biology, and economics; and often there is not assistance in aligning the mathematics and the application. Podolefsky and Finkelstein offer an interesting hypothesis that in the generic condition participants learn (i) rules of the system and (ii) the meta-knowledge that the salient information in the representations is irrelevant to the rules of the system, whereas in the concrete condition they learn only the rules. To test this possibility, they suggest an additional condition which would use concrete instantiations where the concreteness is irrelevant to the abstract rules to be learned. In fact, this experiment was conducted in our previous research (3). The results do not support their hypothesis—irrelevant concreteness markedly hindered transfer. Podolefsky and Finkelstein suggest that students attempt to apply to a novel domain “what is readily apparent,” and this is what differed across condition and lead to the observed pattern of transfer performance. However, this is exactly the point of our argument: the extraneous information of concrete instantiations diverts the learner’s attention from the relevant mathematical structure making it difficult to recognize common structure in a novel, superficially dissimilar situation leading to transfer failure. Furthermore, it is well-documented that transfer is more likely to occur when the novel domain shares superficial features with the learned domain (4–7). However, mathematical concepts have an unlimited number of instantiations. Since we certainly do not know a priori the superficial features of each, we argue that future transfer will be best facilitated by minimizing the superficial features of the learning instantiation. Jennifer A. Kaminski Center for Cognitive Science, The Ohio State University, Columbus, OH 43210, USA. Vladimir M. Sloutsky Center for Cognitive Science and Department of Psychology, The Ohio State University, Columbus, OH 43210, USA. Andrew F. Heckler Department of Physics, The Ohio State University, Columbus, OH 43210, USA. References 1. N. S. Podolefsky, N. D. Finkelstein, Phys. Rev. ST - Phys. Educ. Res. 3, 010109 (2007). 2. N. S. Podolefsky, N. D. Finkelstein, Phys. Rev. ST - Phys. Educ. Res. 3, 020104 (2007). 3. V. M. Sloutsky, J. A. Kaminski, A. F. Heckler, Psychonom. Bull. Rev. 12, 508 (2005). 4. K. J. Holyoak, K. Koh, Mem. Cognition 15, 332 (1987). 5. K. J. Holyoak, P. Thagard, Am. Psychol. 52, 35 (1997). 6. B. H. Ross, J. Exp. Psychol. Learn. Mem. Cogn. 13, 629 (1987). 7. B. H. Ross, J. Exp. Psychol. Learn. Mem. Cogn. 15, 456 (1989).
Counterexamples and Concerns Regarding the Use of Abstract Examples in Learning Math 2 February 2009
N. S. Podolefsky Department of Physics, University of Colorado at Boulder, Boulder, CO 80309, USA, N. D. Finkelstein Respond to this E-Letter: Re: Counterexamples and Concerns Regarding the Use of Abstract Examples in Learning Math	J. Kaminski et al. (Education Forum, "The advantage of abstract examples in learning math," 25 April 2008, p. 454) examine student learning with abstract vs. concrete representations. They claim that students were better able to transfer understanding of modulus-3 addition when taught with abstract, information-sparse representation compared to students who learned the same mathematical ideas using concrete, information-rich representations. While we do not dispute their particular findings, the generalizations drawn by Kaminski et al. are not substantiated. Moreover, we question certain assumptions about the nature of students’ knowledge upon which their general conclusions are based. Kaminski et al. claim that abstract representations lead to better transfer and this may in fact be a generalizable finding; our own findings on students’ use of representations lead to a different conclusion about student learning of abstract ideas (2, 3). In a similar population of college students, we examined whether students learning about electromagnetic (EM) waves were better prepared to transfer their understanding of plane wave characteristics when they learned these properties using either concrete or abstracted representations. In many regards the experiment is similar to Kaminski et al; students learn properties of a system in one domain (in our case string and sound waves) and apply these ideas (the transverse propagation of three-dimensional waves) to another domain. Students learned these ideas using concrete, abstract, or blend (depicting both concrete and abstract simultaneously) representational formats. We find that students taught EM waves using both abstract and concrete representations simultaneously (the blend condition) outperformed students taught the same ideas, but using only abstract representations, by a factor of three on a pre-post assessment [74% vs. 24% correct, p = 0.002 (3)]. In the same experiment, students taught using only concrete representations outperformed the students taught with only abstract representations by a factor of two (57% vs. 24% correct, p = 0.002). We therefore suggest that the statement "Instantiating an abstract concept in a concrete, contextualized manner appears to constrain that knowledge and to hinder the ability to recognize the same concept elsewhere; this, in turn, obstructs knowledge transfer." (1), is not generalizable. In fact, our findings lead to quite the opposite conclusion—concrete representations can facilitate the learning of ideas and transfer of those ideas to new domains. We note that in both the initial learning tasks and the transfer task, students in Kaminski’s experiments appear to learn the rules of each domain—modulus-3 addition, at least from the expert point of view. We question whether it is solely or necessarily the rules of modulus-3 that are transferred. Central to our point of view is that, in educational environments, tacit knowledge that students learn and transfer can encompass much more than the content that is taught explicitly (4). Therefore, instruments meant to measure the explicitly taught content may, alternatively, measure students' understanding of the nature and utility of this knowledge (5). While the explicitly taught content included rules of mathematics, the tacitly learned knowledge might include rules about the activity itself, such as what is to be learned, what resources students should bring to bear, and which resources students should suppress or ignore (6, 7). In the concrete domains used by Kaminski et al., students may have not only learned the mathematical rules of those domains, but students may have also learned epistemic rules (8). That is, rules about the activity such as whether students should use their salient knowledge and rely on associations within the symbol systems that are readily apparent (9). For instance, we may safely assume that the idea "1/3 cup added to 1/3 cup results in 2/3 cups" is highly salient to students. Students learn to rely on their prior held conceptions in the concrete domains. Conversely, we may also assume that the idea "diamond and diamond results in circle" is not highly salient to students. Importantly, other associations within this system may be salient, such as the number of sides of each shape. In this case, students learn to ignore such salient associations. Our claim is that rules such as "what is readily apparent to me is relevant to the task" in the concrete domain vs. "what is readily apparent to me is irrelevant to the task" in the abstract domain may be what transfer. If this is so, students in the abstract domain would be better prepared to learn the abstract rules in the transfer task since these students would know to ignore salient information. Note that according to this hypothesis, we do not require students who learned in the abstract domain to transfer the rules of modulus-3 addition in order to perform better on the transfer task. Only the tacitly learned meta-knowledge of what knowledge to use and what to ignore (or suppress) is required to predict the same outcomes (6). This may be the "recognition" that Kaminski et al. speak of in the quote above; however, such recognition has less to do with the content per se, and more to do with students learning about how to use and value their prior knowledge—that which is both "concrete" and "salient" to students. This distinction in interpretation is testable—one can conduct an experiment similar to that of Kaminski et al., but an additional learning domain should be introduced. This domain would use concrete representations where the concreteness is irrelevant to the abstract rules to be learned. This learning domain would be similar to the transfer task in that students would be asked to learn two key ideas: modulus-3 arithmetic and the meta-knowledge that the salient information in the representations is irrelevant to the rules of the system. Our hypothesis as stated above would predict that students in this group would outperform students taught with concrete representations where the salient information is relevant to the abstract rules of the system. If this is the case, the important finding in the results of Kaminski et al. is not one of abstract vs. concrete, but of whether students learn to expect activities to draw on their salient knowledge or not. Such findings would impact educational choices, insofar as how and when to value student prior knowledge, and mechanisms, such as choice of representation, that inform these educational decisions. N. S. Podolefsky and N. D. Finkelstein Department of Physics, University of Colorado at Boulder, Boulder, CO 80309, USA. References 1. J. A. Kaminski, V. M. Sloutsky, A .F. Heckler, Science, 320, 454 (2008). 2. N. S. Podolefsky, N. D. Finkelstein, Phys. Rev. ST - Phys. Educ. Res. 3, 010109 (2007). 3. N. S. Podolefsky, N. D. Finkelstein, Phys. Rev. ST - Phys. Educ. Res. 3, 020104 (2007). 4. E. F. Redish, Teaching Physics with the Physics Suite, (Wiley, 2003). 5. D. Hammer, A. Elby, R. F. Scherr, E. F. Redish, in Transfer of Learning from a Modern Multidisciplinary Perspective J. Mestre, Ed. (Information Age Publishing, Charlotte, NC, 2005) pp. 89–120. 6. A. H. Schoenfeld, Cognitive Science and Mathematics Education (Lawrence Erlbaum Associates, Mahwah, NJ, 1987). 7. S. Scribner, M. Cole, Science, 182, 553 (1973). 8. J. Tuminaro, E. F. Redish, Phys. Rev. ST - Phys. Educ. Res. 3, 020101 (2007). 9. N. S. Podolefsky, N. D. Finkelstein, Proceedings of the 2007 Physics Education Research Conference 951, (AIP Press, Melville, NY, 2007).
Modulo 3 Means Something 28 October 2008
Anne Watson, Professor of Mathematics Education Department of Education, University of Oxford, Oxford OX2 6PY, UK Respond to this E-Letter: Re: Modulo 3 Means Something	I, too, am concerned about the results reported from the study by Kaminski and colleagues (Education Forum, "Learning theory: The advantage of abstract examples in learning math," 25 April 2008, p. 454). It is true that if you want arithmetic modulo 3 to be a meaningless game with symbols, then combining ladybugs and vases (as I think was suggested in the post-test) to make something else would map neatly onto the kind of meaningless manipulation of circles and other shapes offered in the intervention. I would rather, however, that students understood the residue nature of modulo 3 arithmetic, and this is more easily understood from the jug model—which I think is a useful model—even if it is harder to learn and harder to apply elsewhere. It is mathematically meaningful and cognitively challenging, but learning mathematical ideas is cognitively challenging; otherwise, why do we teach it? I have to admit that if I had been taught the jug model and then asked to combine ladybugs and jugs I would have been mystified, and I am no slouch at mathematics, or at teaching it. Their approach also confuses situativity with artificiality. I understand jugs—they are in my ordinary life. The swapping game is not, but given time I could learn it. Anne Watson Department of Education, University of Oxford, Oxford OX2 6PY, UK.
Response to W. McCallum's E-Letter 11 August 2008
Jennifer A. Kaminski Center for Congitive Science, The Ohio State University, Columbus, OH 43210, USA, Vladimir M. Sloutsky, Andrew F. Heckler Respond to this E-Letter: Re: Response to W. McCallum's E-Letter	W. McCallum brings up an interesting aspect of some mathematical groups, namely, that they are modular addition. However, we disagree with his interpretation of our results. First, across the different experimental conditions, the mathematical principles were presented analogously through text and examples. It is not that one group saw a table, while another did not. More importantly, the underlying mathematics is identical across conditions; this is elementary abstract algebra (1). To elaborate, up to isomorphism, there is one commutative group of order three, isomorphic to addition modulo 3. We can instantiate this as cyclic addition involving 0, 1, and 2, but we can also instantiate it through scenarios such as the children's game. Mathematically speaking, "1" can generate the group or "bug" can generate the group. Most importantly, McCallum's construal of our results is wrong because it leads to wrong empirical predictions. Based on his construal of the locus of difficulty for the learner, McCallum makes the following prediction: "It would be interesting to know what happens when students are given pure instruction in addition modulo 3, with statements such as 1 + 1 = 2, 2 + 2 = 1, 1 + 3 = 3, 2 + 3 = 2. I hypothesize they would do just as poorly as the students given concrete examples." However, the prediction (and subsequently the construal that gives rise to the prediction) is wrong. We have already conducted this study, and our data falsify that hypothesis. Those students who were taught modular addition with numbers were able to successfully transfer knowledge to the children’s game (2). We therefore emphatically disagree with the statement that "there is a purely abstract mathematical hurdle for students" who learned the concrete instantiations. No, the hurdle is not a mathematical one; it is a cognitive one. Jennifer A. Kaminski Center for Cognitive Science, The Ohio State University, Columbus, OH 43210, USA. Vladimir M. Sloutsky Center for Cognitive Science and Department of Psychology, The Ohio State University, Columbus, OH 43210, USA. Andrew F. Heckler Department of Physics, The Ohio State University, Columbus, OH 43210, USA. References 1. I. N. Herstein, Topics in Algebra (Xerox, 1975). 2. J. A. Kaminski, V. M. Sloutsky, A. F. Heckler, manuscript in preparation.
Another Interpretation for Learning Math 11 August 2008
William McCallum, Mathematician University of Arizona, Tucson, AZ 85721, USA Respond to this E-Letter: Re: Another Interpretation for Learning Math	The experiment by J. A. Kaminski et al. (Education Forum, "The advantage of abstract examples in learning math," 25 April 2008, p. 454) is interesting, but it seems to me that you have ignored a rather large elephant. Both the examples presented in experiment 1 and the children's game have the same underlying mathematical object, namely, the commutative group of order 3 given by an abstract symbolic multiplication table. The instruction in experiments 2, 3, and 4, involves an isomorphic but different mathematical object, the group of integers modulo 3. In experiments 2, 3, and 4 the symbols have obvious numerical interpretations as 1, 2, or 3, and the binary operation matches addition modulo 3 (at least, that's the way the measuring cups work, and I assume the tennis balls and pizza worked the same way). Thus there is a purely abstract mathematical hurdle for the students in experiments 2, 3, and 4, that has nothing to do with the concrete representations they studied. Namely, when observing the abstract multiplication table represented in the children's game, they have to understand that it is isomorphic to addition modulo 3. It would be interesting to know what happens when students are given pure instruction in addition modulo 3, with statements such as 1 + 1 = 2, 2 + 2 = 1, 1 + 3 = 3, 2 + 3 = 2. I hypothesize they would do just as poorly as the students given concrete examples. If this were the case, another possible interpretation of your work would be that when teaching with concrete examples you must make sure that the examples match the underlying mathematics. William McCallum University of Arizona, Tucson, AZ 85721, USA.