> a 0jbjb11 $[[-ZZZ^&&& |9|9|9|9D9T x :6:6:6:6:6:.d:x:wwwwwww,H{R}z'x&:6:6:::'xA6:6:t,w$Rx0x!uAXwA& 8"D 8"Student Awareness of Analogy: Mathematical & Ethical Implications
(Martina Metz)
Introduction
I.
The storied leaves fall through the stories
of air. Their call is to return. The ground
gathers them all into its plot, as it gathers
fallen hands. Autumnal as stories begin,
they end when their patterns rhyme, in beauty, vernal.
(from A Grace, written by Wendell Berry and
dedicated to Gregory Bateson; Berry, 1980, p. 36)
In recent years, the fundamental role that analogical reasoning plays in mathematical thinking (cf. English, 1999, 2004; Lakoff & Nez, 2000) and cognition in general (cf. Lakoff & Johnson, 1980) has been increasingly recognized. As English (2004) points out, however, studies of childrens use of analogical reasoning have typically focused on classical analogies (A:B::C:D), problem analogies (transferring solution strategies between problems recognized as similar), and pedagogical analogies (where the teacher selects analogies deemed useful and is careful to emphasize where they break down). In each of these contexts, students are expected to develop understandings that conform to predetermined expectations. In this paper, I consider the generative power of complexly interacting multiple metaphors (see also Pimm, 1987; Sfard, 1999) as well as the development of student awareness of their own analogical faculties. To do so, I summarize a framework developed in the context of a science classroom that describes various levels of student awareness of analogical reasoning (Schmidt, 1999), consider its relation to mathematical intution (cf. Burton, 1999; Fischbein, 1982; 1987; Stavy & Tirosh, 2000), and further consider how students interactions with complex mathematical problem spaces might interact with their evolving awareness of their use of analogy. In particular, I draw from Watson and Masons (2005) work with learner-generated examples and Lesh and Doerrs (2003) work with mathematical modeling, both of which attend explicitly to the structure of mathematical understanding and both of which allow for the emergence and development of new mathematical ideas. Finally, I further suggest that awareness of analogical reasoning might itself be abstracted from experiences with metaphor in diverse contexts within and beyond mathematics, and that such awareness could have important ethical implications.
Student Awareness of Analogy
The student learns more if he sees an analogy which is not formalized in the language of isomorphism, than if isomorphisms which he cannot grasp are forced upon him. The analogy is such an effective means of building inside and outside mathematical relations because it is the most natural and primitive means of all with which the organization of the world is attempted, a means which nevertheless remains vigorous at higher levels too. The student knows it even before he learns mathematics. Its overestimation is dangerous, in particular in mathematics, but to weigh it up correctly one has to learn its application consciously.
(Freudenthal, 1975, p. 78)
To begin with, I would like to clarify my use of the terms metaphor and analogy. For purposes of this discussion, analogical reasoning will be used to describe those aspects of metaphorical transfer that are consciously applied. I intentionally collapse distinctions sometimes drawn between analogies, examples, generalizations, models, and representations as well as those between within and between-domain analogies. While these distinctions are relevant in certain contexts, I wish to draw attention more generally to the nature of meaning transfer that might occur in any of these cases. In particular, the importance of closely attending to whether perception of similarity justifies transfer of meaning is a theme that I return to time and again. With Sfard (1999), I observe that transfer is a recursive process, impacting understanding of what are often referred to as source and target analogs. Here I include the recursive development of examples and definitions, models and that which they describe. Furthermore, multiple metaphors may interact in complex ways. I do not distinguish metaphor from analogy by the formers creation of a new conceptual system (Sfard, 1999, p. 345) such as the invention of the negative integers by projection of our understanding of positive integers. In a broad sense, every situation that involves analogical reasoning involves the creation of new understanding. Pimms (1987) description of metaphor is helpful: i.e. metaphor is a connection between different areas of understanding that allows one thing to be understood in the terms of another (p. 30).
Children already, necessarily, and often sub-consciously think with their own analogs, with or without attention to their appropriateness. It is important that they develop the habit of considering the strengths, weaknesses, and applicability of these analogs:
In short, students adapt their own ways of thinking rather than adopting the authors (or teachers) ways of thinking, and the adaptation (modification, extension, and revision) of existing conceptual systems is given as much attention as the construction (or assembly) of conceptual systems that are assumed to be completely new to the student(s). (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003, p. 57)
Elsewhere, I have reported four levels of student awareness of their use of analogy as evident the context of an elementary science classroom (cf. Metz, forthcoming; Schmidt, 1999). In summary, at Level 1, students explored new situations and enacted new understandings through subconscious reference to more familiar experience. By Level 2, they were explicitly aware of their use of analogy in formulating ideas but explained away the parts that dont work with the rationale that the idea is just an analogy. At Level 3, students evaluated the applicability of selected analogs and explicitly recognized that just because you can relate two things doesnt necessarily mean that conclusions based on the relationship make sense. While this may at first seem obvious, the distinction between recognition of similarity and applicability of transfer can be a profound insight that makes it easier to interrupt entrenched ways of thinking. In some cases, evaluation of an analogical relationship involves the creation of further analogs that vary the features in question by taking them to extremes or by creating more familiar cases that (arguably) fall within the scope of the analogs explanation. Clement (1981) provides a fascinating description of physicists use of extreme cases and what he terms bridging analogies, which share features of two or more situations perceived as similar and which may make the transfer of meaning between them seem more acceptable (or not). Level 4 was initially based on my interpretation of Clements expert studies, although I have since observed students working at this level: These students seek awareness of unconscious constraints on their thinking and make conscious attempts to understand gut feelings that may be convincing but wrong. By identifying the sources of certainty that can underlie gut feelings, they are better able to evaluate their validity.
It is important to emphasize that the same student may exhibit different levels in different contexts. However, as students learn (a) to seek out potential analogs, (b) to consider what features are allowed to vary between analogs in order to justify conclusions based on their relationship and (c) to attend to ways in which implicit metaphors might be brought to consciousness and dealt with analogically, it is possible to see more generalized growth in their use of analogical reasoning. In the next section, I consider the relevance of awareness of analogical reasoning for mathematics education. Following that, I further suggest possible ethical implications of a broader awareness of analogical reasoning - awareness that is likely best developed simultaneously in both mathematical and non-mathematical contexts that themselves may recursively interact.
Mathematical Implications
Our brains are not like computers, working systematically and logically. They are metaphor machines that leap to creative conclusions and belatedly shore them up with logical narratives.
(Stewart, 2006b, pp. 22-23)
Mathematics is particularly well-suited to the development of (as well as dependent on) analogical reasoning. Mathematical levels of abstraction enable the perception of many connections that might otherwise be overlooked. Of course, the higher the level of abstraction, the greater the caution that needs to be exercised in determining the applicability of transfer:
[We] emphasize the role of mathematization in modeling. We are not arguing that all models are or should be mathematical, but as experience becomes progressively mathematized, it becomes both more mobile and more extensible more model-like. By mobility, we mean that modeling processes can easily be transferred from one place or one process to another, and extensible means that the implications can be worked out in the model-world and then tested against the real world. Mathematization not only helps students see things that they otherwise might not; it also makes disconfirmation a real possibility that must be reckoned with when a solution is proposed. (Lehrer & Schauble, 2003, p. 61)
Learner-generated examples (Watson & Mason, 2005) and mathematical modelling (Lesh & Doerr, 2003) may both be seen as particular instances of the broader class of analogical reasoning. In focusing attention in this manner, my aim is not to summarize the incredible richness that either entails; rather, I hope to foreground the manner in which each attends to (1) what comes to mind in response to a particular problem or prompt, (2) how what comes to mind is appropriate / inappropriate to the situation at hand, and (3) whether conclusions based on what comes to mind are justifiable. In so doing, explicit attention is directed to the structure of mathematical understanding. Does my understanding of this case justify conclusion(s) about a larger class? What else belongs in such a class? Should I reject an example, alter a definition, or both? How do extreme and special cases inform this understanding? Does my understanding of a particular model justify conclusion(s) about the situation being modeled? If not, should I alter my mathematical model, alter my understanding of the situation that the model is designed to represent, or both?
While Watson and Mason (2005) do not make explicit reference to analogical reasoning in their discussion of learner-generated examples, their description of deliberate exploration of dimensions of possible variation and consideration of range of permissible change is inherently analogical. Perceiving examples as examples of something is an analogical process what makes them qualify? When students are asked to generate their own example spaces and to consider what features must remain constant to justify their creation of classes and examples, they are given important opportunities to intentionalize their use of analogical reasoning. Attending to dimensions of possible variation also encourages the expansion of example spaces beyond what first comes to mind: Watson and Mason repeatedly ask, What else? in order to push understanding beyond entrenched ways of thinking.
Watson and Mason emphasize that the development of learner-generated example spaces is possible within mathematics itself, in a world of abstract structures, rather than merely in real applications (p. 31). While this is certainly possible, I also see great value in using real-world situations (by which I mean situations that students might encounter and find relevant outside of the classroom) to prompt the exploration and development of structured mathematical understandings. Lesh and Doerrs (2003) work with mathematical modeling encourages students to define, explore, and re-define experience in mathematical terms. This goes far beyond using mathematical concepts and procedures in utilitarian contexts: The situations themselves are an integral part of the mathematics, embodying concepts pertaining to number, geometry, measurement, algebra, and data analysis:
[M]odel-eliciting activities usually involve mathematizing by quantifying, dimensionalizing, coordinatizing, categorizing, algebratizing, and systematizing relevant objects, relationships, actions, patterns, and regularities. (Lesh & Doerr, 2003, p. 5)
As students develop and refine their own models, they gain access to the power of mathematics as a tool for recognizing similarity among seemingly disparate ideas: The best mathematics has a curious kind of universality, so that ideas derived from some simple problem turn out to illuminate a lot of others (Stewart, 2006a, p. x; also see Wigner, 1960). Like Watson and Masons work with learner-generated examples, Lesh and Doerrs description of mathematical modeling emphasizes the importance of attending to those aspects of a relationship that must be preserved to allow justifiable conclusions. Once a situation has been mathematized, the mathematized situation must be treated as an analog to the original, and conclusions based on the model cannot be assumed to apply to the situation from which the model was abstracted. The map is not the territory (Bateson, 1979/2002). Furthermore, symbolic and / or geometric methods developed in response to a modeling problem may be extended and modified in ways that are not immediately relevant to the real-world problem that prompted them, and this may be facilitated by attending to the dimensions of possible variation and range of possible change emphasized in Watson and Masons work with learner-generated sample spaces.
When students are actively involved in structuring their own mathematical understandings, they may structure their understandings in ways very different (but equally defensible) than those presented by a teacher or textbook. For example, in a discussion with a colleague about exponents, it occurred to me that exponentiation might be understood as a particular instance of repeated multiplication. While this may seem obvious, I have not seen the broader notion of repeated multiplication (i.e. where the multiplicands need not be alike) treated as a concept in any of the teaching resources or curricula that I have encountered. Yet it is conceivable that students might transfer significant understandings about the nature of multiplicative growth to their understanding of exponentiation (and vice versa). Repeated multiplication and division produce rapid change whether or not the bases remain the same, but this change of course could not be represented with traditional base / exponent notation. Further exploring how exponentiation and repeated multiplication are alike / different might help address common misconceptions such as, If you multiply by 2 and then by 3, its the same thing as multiplying by 5, If you divide by 2 then by 4, its the same things as dividing by 6, or 3 X 4 X 5 = 4 X 4 X 4, because the 5 balances the 3. I am not suggesting that math curricula should explicitly acknowledge this connection. I am offering an example to support the importance of remaining open to alternative ways of connecting ideas by analogy to emphasize that students ways of transferring meaning between their own understandings might look quite different from those we are accustomed to seeing. That something is true says nothing of how we came to believe it or of how else a situation might be understood (or metaphorically, what else it might be understood as), both of which have great relevance for the mathematics classroom:
That is, we easily come to believe that the way we dissected the real world in order to make our description was the best and most correct way to dissect it. (Bateson & Bateson, 1987/1988, p. 153)
As Davis and Simmt (2006) emphasize in their discussion of mathematics-for-teaching,
[The] approach is necessarily recursive in the complexity science sense: it always results in elaborations of understandings as teachers confront, analyze, and blend represented ideas, concepts and beliefs is not simply aimed at identifying what is, but contributes to the production of new interpretive possibilities (p. 299).
Much has been made of expert-novice differences in categorizing math problems: experts have been found to attend to structural features, while novices attend to surface details (English, 1999 gives several examples of such studies; also see Schoenfeld & Herrmann, 1982). Of course, there is nothing inherently wrong with grouping two problems about, say, jellybeans together instead of two problems about the probability of selecting a certain color from a particular sample: This is only a problem if a student assumes that since both problems are about jellybeans, solution strategies are therefore transferable from one problem to the other. Regularly asking children to consider whether the ways that problems are similar / different justifies transfer could prompt them to consider deeper structural features while at the same time honoring their own ways of structuring defensible mathematical understandings. Lobato and Siebert (2002) similarly argue that such actor-oriented transfer involves the personal construction of relations of similarity between activities (p. 89).
What comes to mind in immediate response to a problem or prompt is often in a form that can be termed intuition, i.e. the fast processing of information, so that the person is able to make a judgement before knowing the reasons for their choice (Graham, 2006, p. 29). If handled with care, intuition can be a powerful source of ideas (cf. Fischbein, 1982; Burton, 1999). By attending more closely to cues such as It seems like., I think it has something to do with., or Thats sort of like., ideas that might otherwise influence thinking on a mostly subconscious level are brought to fuller awareness where they may be evaluated for their accuracy as well as their applicability (a distinction not obvious to many children): How is it like? How is it different? Of particular interest to the discussion here, do the likenesses justify transfer of meaning to the situation under consideration (or inclusion in a set of examples)? Often, prior experience (including of problems situations perceived in some way as similar) influences our thinking without us being fully (or even partially) aware that this is happening, making certain strategies or solutions seem plausible (or not). Exploring the feeling that something seems plausible can bring these connections to fuller consciousness.
Consider the following problem:
In a warehouse you obtain 20% discount but you must pay a 15% sales tax. Which would you prefer to have calculated first, discount or tax? (from Mason, Burton, & Stacey, 1982, p. 1)
After spending some time working through this problem with a group of colleagues, one member commented that although he agreed with both the deductive (mostly algebraic) arguments and the large example spaces developed by others, "it just felt wrong" that the solutions should come out the same either way. Rather than simply (a) assuming the feeling was accurate or (b) dismissing it when it contradicted the evidence, he was further able to locate the source of his discomfort in a perceived connection to another familiar situation: "It seems like those problems where you add 20% then subtract 20% and you don't come back to the original price." This resonated with my own initial doubts, even though I, too, was very confident that my solution to the problem was correct.
My own first reaction to the discovery that order did not seem to matter was that my choice of a starting number of $100 might somehow have produced an anomalous solution. Surprisingly (at least to me), despite many subsequent experiences with special cases that produce unusual results, this suspicion remains most strongly rooted in an analogy to the apparent similarity between 2 + 2 and 2 X 2! As I attended more closely to the operations I had performed on my chosen starting price of $100, it became clear that I had subtracted a larger percentage of a smaller amount and added a smaller percentage of a larger amount, but it wasnt immediately obvious why what was gained and what was lost should perfectly balance. In fact, I was immediately suspicious of this potential symmetry, and I was able to locate my suspicion in an analogy to work my students had done in developing ways to compare fractions: In comparing 2/3 and 3/4, it is not sufficient to say that since 3/4 has more but smaller pieces that the fractions are equivalent.
I decided to condense what I had done in each case: First, I calculated 80% of the total amount, then 115% of what was left: .8 * $100 * 1.15 = $92. Then I calculated 115% of the original amount and took 80% of what was left: 1.15 * $100 * .8 = $92. It now seemed obvious that the only difference was order of multiplication a difference that didnt matter and that this would be true regardless of what number I chose for a starting number, for tax, or for the discount. I was convinced, but something remained unresolved in my broader understanding. I decided to explore the connection between this problem and the +20%, -20% problem that my colleague had identified. As I expected, when I added 20% then subtracted 20%, I did not end up where I had started. But why did such a seemingly trivial result seem problematic? It was only after working out the details of what was essentially a routine problem and considering them alongside the warehouse problem that I realized that a fair comparison of the two problems would not ask, Did I end up where I started?, but rather Does it matter in which order I add and subtract? In the warehouse problem, I did not end up where I started (i.e. $100), either. Exposing the source of my faulty intuition did little to bolster my confidence in my original solution (in which I was already very confident), but it did foster greater consistency to my broader understanding. In other words, intentional use of analogy prompted me to consider more deeply the connections between these problems and resulted in a clarified understanding of both.
Interestingly, the notion that adding 10% to a particular value and then subtracting 10% from the result does not effect a return to the starting point is itself counter-intuitive to many children. Stavy and Tirosh (2000) make a strong case for the notion that this is due to an intuitive rule that they deem Same A Same B, which they claim leads to the assumption that both 10%s are equivalent (and further affects thinking in a wide variety of problem contexts). Children who have not worked through this problem (or one like it) would probably not be surprised to find that it makes no difference in which order tax and discount are calculated in the warehouse problem.
While the solution to the warehouse problem described above deals with a solution that is correct but counter-intuitive, it is perhaps more common for students to develop solutions that are convincing but flawed. This is evident in common overgeneralizations that children make. If students learned to question the feeling of sameness and to consider what specifically is the same and what is different (i.e. to treat the items as analogous rather than identical), they might avoid errors like the following:
What I did was I wanted to make 36 times 17 easier, so I added 4 to 36 to make it easier, and I added 3 to 17 to make it 20, so I timesed 40 times 20 I knew it wasnt the answer, so I minused 4 because I added 4 to get up to 40 and that brought me to 796, and I minused 3 because I had to add 3 to get to 20, and I got the answer 793. (from Russell, 1999, p. 11)
Russell further notes that even though other students had correctly calculated 36 X 17, there seemed to be some logic in this way of reasoning; by further analyzing how Thomas result differed from theirs, they, too, developed a deeper understanding of the situation. Analyzing both the similarities and differences between these approaches prompted deeper understanding of multiplication. While counter-intuitive problems provide an excellent way to prompt childrens encounters with their own assumptions, it is important to remember that no problem is inherently counter-intuitive any more than two problems are inherently analogous. However, there are many problems that challenge commonly held intuitions (see Stavy & Tirosh, 2000). By identifying the sources of associated intuitions, students may more consciously attend to the analogic that underlies them.
Ethical Implications
II.
In rhyme of hand with leaf, time with time,
we recall what grief forgets, what joy has never
seen: the chief beauty of the world,
pattern of patterns. Though deaf, we dance. The music
that moves us, deaf and blind, is our relief.
(Berry, 1980, p. 36)
Berry (1993) warns:
[A]bstraction, of course, is what is wrong [with our society]. The evil of the industrial economy (whether capitalist or communist) is the abstractness inherent in its procedures its inability to distinguish one place or person or creature from another (p. 23).
Perhaps generalization inappropriately universalizes only if we ignore (a) what Mason, Burton, & Stacey (1982) call specializing - the recursive testing of particularities against evolving generalities that are not assumed as universals and (b) the collection of relationships associated with a particular claim, which can be highly particular. Both Weaver (1948) and Bateson (1979/2002) further caution against confusing the general with the specific and the predictable with the unpredictable. Mathematics that attends to the role of analogical reasoning is much more than an oppressive and universalizing quantification and abstraction of experience: It enables the perception of deep symmetries that emphasize connectedness rather than separation and attends directly to the danger of inappropriate universalization.
By nurturing a habit of mind that continuously asks, Does my perception of similarity justify transfer of meaning? mathematics may also offer insight into other metaphorical constructs. For example, stereotypes often form when we confuse our experience of a particular person or situation with a type of person or situation, thereby inappropriately transferring meaning from the specific to the general. Misunderstandings can occur when we assume that our own experience of a particular situation is the same as that of a friend who has chosen to confide in us: I know exactly what you mean. The notion of precedent, whether pertaining to informal argument, institutional policy, or courts of law, is inherently metaphorical: If theyre going to ban smoking, why dont they ban fast food? But thats different! is a common protest in arguments such as these. One of the key points I have attempted to develop in this paper is the need to pursue this protest more deeply: How is it different? and Is it a difference that makes a difference? (Bateson, 1979/2002). If all students developed the habit of probing more deeply into assumptions such as these, the impact on society would be profound.
Finally, mathematics may be uniquely positioned to interrupt entrenched habits of mind. Fawcett (1938) argues that demonstrable geometry provides a context devoid of strong emotional content (p. 12) that allows students to become more aware of how conclusions are determined by their own definitions and assumptions. Although Maturana (1988) argues that we only stop asking questions when the emotion or mood of the observer shifts from doubt to contentment (p. 28), [M]athematics more often than not presents its counterpoint to false suppositions in the barest of guises. The truths (results) of mathematics cannot as easily be denied (Handa, 2006, p. 60). Consider the following reflection offered by a Grade Six student after working on a number problem with a provable solution:
[W]hen we have an idea we usually are almost attached to it and we believe so much that our idea is true we often become in a way blind to proof that disproves your idea. I like the feeling when you know only you can solve the problem not someone else for you. You have a nice feeling of control.
The fact that the content of the problem was in itself not a source of deep emotion for this student made it easier for him to locate the source of his emotions in his desire to be right and in control. While considerable and important work has been done on error analysis as a way of deepening mathematical understanding (cf. Borasi, 1996), students awareness of error in mathematics might interact with their broader awareness of what and why they implicitly and explicitly believe. With Jardine (2000), then, we might ask: What witness do such things bear on us and our doings? Not what do we have to say about them but what do they say about us (p. 214).
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