MATHEMATICAL THINKING AND LEARNING, 3(2&3), 93–124 Copyright © 2001, Lawrence Erlbaum Associates, Inc. Using Metaphors to Understand and Solve Arithmetic Problems: Novices and Experts Working With Negative Numbers Ming Ming Chiu Department of Educational Psychology The Chinese University of Hong Kong In this study, novices and experts used the same metaphors to understand and solve problems with negative numbers. However, they used them differently. Twenty-four participants (12 middle school children and 12 postsecondary adults) computed arith- metic expressions during the problem-solving task. During this task, children used metaphors more often than adults did to compute, detect and correct errors, and justify their answers. Metaphorical computations were more accurate but slower than other methods. The participants explained 6 arithmetic expressions during the understand- ing task. During this task, the adults used more metaphors (with fewer details) and used them more often than the children did. Compared to the median child, the median adult showed a more integrated understanding of arithmetic through multiple meta- phors, mathematical rules, and transformations. These results suggest that the meta- phors used by both the children and the adults are central to understanding arithmetic. Thus, these metaphors are likely candidates for theory-constitutive metaphors. Metaphors were once viewed as simply literary devices. Now many researchers con- sider metaphors to be important mechanisms for understanding concepts (Black, 1979; Davis, 1984; Gentner, 1989; Kuhn, 1979; Lakoff & Johnson, 1980; Petrie & Oschlag, 1993; Pimm, 1987; Presmeg, 1992; Reddy, 1979; Silva & Moses, 1990; see Ortony, 1993, for a collection of articles on metaphors). Students can use their Supplementary materials to this article are available on the World Wide Web at http://tcct.soe. purdue.edu/books_and_journals/mtl_journal/ Requests for reprints should be sent to Ming Ming Chiu, Department of Educational Psychology, The Chinese University of Hong Kong, 314 Ho Tim Building, Shatin, N.T., Hong Kong. E-mail: mingming@cuhk.edu.hk 94 CHIU prior knowledge of familiar phenomena to understand difficult concepts metaphori- cally. For example, they can use their knowledge of motion to make sense of static polygons, viewing them as paths through the POLYGONS ARE PATHS metaphor. (In this article, all metaphor names appear in small capitals; see Appendix A for a list of met- aphors.) By doing so, students can organize their computation of the polygon’s per- imeter by summing the number of steps needed to walk each consecutive portion (side) of the path (polygon). People reason metaphorically by projecting a source sit- uation’s (e.g., motion) goals, entities, relations, and actions onto the target situation (e.g., polygon) to create new target goals, entities, relations, and actions. However, metaphors also have limitations, such as inherent differences in the source and target. When students acquire expertise, do they stop using their earlier metaphors because of their limitations? Or do experts adapt to their limitations? Boyd (1993) argued that metaphors are necessary to understand some theoretical ideas at the heart of a discipline. These are theory-constitutive metaphors. If experts continue using the same metaphors that novices use, these metaphors are candidates for theory-constitutive metaphors. In this study, I compare novices and experts using mathematical metaphors to solve problems involving negative numbers. (See Chiu, 2000, and English, 1997, for more articles on metaphors in mathematics.) THEORETICAL PERSPECTIVE In this section I discuss the benefits and limitations of metaphorical reasoning and possible adaptations to these limitations. Benefits Researchers have argued that students can benefit from metaphorical reasoning in several ways. These include understanding concepts, interpreting representations, connecting concepts, improving recall, computing solutions, and detecting and correcting errors. Building on their prior knowledge, students can understand diffi- cult concepts metaphorically (Lakoff & Johnson, 1980; Lakoff & Nunez, 1997). For example, they can use their understanding of space and containers to make sense of sets (SETS ARE CONTAINERS). Many mathematical representations such as Venn diagrams, number lines, and graphs rely on metaphorical inferences from a student’s source understanding of space. (In this article, a representation is any per- ceptually accessible stimulus that a student interprets, e.g., drawings, gestures, talk, etc.) Lakoff (1987) showed how Venn diagrams can be interpreted through SETS ARE CONTAINERS. Elements of Set A are understood as objects inside the container la- beled A. Likewise, elements of Set B are objects inside the container labeled B. Through metaphorical reasoning, students can also connect different concepts. For  USING METAPHORS 95 example, they can use SETS ARE CONTAINERS to connect sets and logic (Lakoff, 1987). Consider a Venn diagram with one oval (A) completely inside another oval (B). Be- cause A is inside B, an element in Set A is also in Set B (A implies B or A → B). Likewise, an element outside Set B (~B) is also outside Set A (~B → ~A). These ex- tra connections between concepts provide more ways for students to remember the ideas. As a result, metaphors can improve recall, as shown in Reynolds and Schwartz’s (1983) study. In Chiu (1994), students used metaphors to compute and to detect and correct errors. Using ARITHMETIC IS MANIPULATING OBJECTS (also known as OBJECTS meta- phor), a student viewed the problem “–5 + 2” as combining five holes and two ob- jects (e.g., marbles). Matching each hole with a marble neutralized the paired entities. Then, counting the remaining entities (in this case, holes) yielded the an- swer (–3). Likewise, a student can use a metaphor to detect and correct errors. In the problem “–5 + 2,” a student who mistakenly computes “3” recognized that there are more holes (–5) than marbles (2). After the pairings, only holes can re- main. Thus, the answer must be negative. Novices uncertain about their mathematical knowledge while solving an ap- plied mathematics problem can create a chain of metaphors. For example, a stu- dent faced with a target problem situation (e.g., stock market change) can use a recently learned source (e.g., arithmetic) to understand it metaphorically (STOCK MARKET CHANGES ARE ARITHMETIC COMPUTATIONS). If the student’s arithmetic is rela- tively weak, he or she can use a metaphor to do the arithmetic (e.g., ARITHMETIC IS MOTION ALONG A PATH, also known as MOTION metaphor). That student has then cre- ated a chain of metaphors: STOCK MARKET IS ARITHMETIC, ARITHMETIC IS MOTION (see Table 1). Limitations Metaphorical reasoning, however, also has several potential disadvantages. These include invalid inferences, unreliable justifications, and inefficient procedures. A metaphor’s source and target are inherently different phenomena. Thus, there must be omissions or invalid metaphorical inferences in the target. Consider a student us- ing PRIME NUMBERS ARE PRIMARY COLORS (Nolder, 1991). That student knows that the number of primary colors is finite. Therefore, the student infers that there must be a finite number of primes. However, this metaphorical inference is false; there are an infinite number of primes. Metaphorical inference is therefore not a generally reli- able method, so experts do not use metaphors to justify their results in formal dis- course (e.g., journals such as Acta Mathematica). Finally, metaphorical reasoning is slower and less efficient than facts or algorithms (Chiu, 1994). Consider alternate methods for solving “–5 + 2.” A student can simply recall the memorized fact “–5 + 2 = –3.” Or, a student can compute the difference (5 – 2 = 3) and attach the sign of  96 CHIU TABLE 1 A Chain of Metaphors: Using Arithmetic to Compute Stock Market Transactions and Motion to Understand Arithmetic Stock Market Arithmetic Motion (Target 2) Is (Source 2/Target 1) Is (Source 1) Value ← Number ← Location No net gain or loss at the start of today’s ← Zero ← Origin transactions Profit ← Positive number ← Location in front of the origin Loss ← Negative number ← Location behind the origin Net gain or loss ← Sum all numbers ← Complete journey of all motions Determine effect of Transaction A ← Add A ← Move A steps Determine effect of a profit ← Add a positive number ← Move A steps forward Determine effect of a loss ← Add a negative number ← Move A steps backward And so forth And so forth And so forth the larger number (–5 > 2, so –, and thus, –3). Both of these methods are faster than the metaphorical computation described previously. Adapting to a Metaphor’s Limitations In this section, I briefly note the conflicting views regarding experts’ continuing use of earlier metaphors. Then I consider possible adaptations involving knowl- edge compilation, greater source comprehension, multiple metaphors, and condi- tions of use. Expert use of earlier metaphors? Some researchers have claimed that metaphors are only temporary scaffolds for novices. However, others have argued that experts continue using them. According to Searle (1979), metaphorical expres- sions that people understand quickly must be dead metaphors, in which the source is lost and unrecoverable. Likewise, Post, Wachsmuth, Lesh, and Behr (1985) ar- gued that abstraction during developing expertise entails target understanding without any concrete metaphorical source. In contrast, Boyd (1993) and Lakoff and Nunez (1997) argued that there are theory-constitutive metaphors in which people must use their human experiences to understand central mathematical ideas. Knowledge compilation. One possible resolution of this dispute involves knowledge compilation (Anderson, 1987). Anderson argued that people condense their chains of reasoning during knowledge compilation. So, A → B → C → D be- comes A → D. Likewise, students can compile their metaphorical reasoning and eventually bypass it to create autonomous reasoning in the target. Instead of pro-  USING METAPHORS 97 jecting all source properties onto the target mathematics, experts do not invoke un- needed parts of the metaphor. They do not need the scaffolding that was useful in building their initial understanding (Vygotsky, 1935/1978). Consider solving the problem “–2 + –1” with ARITHMETIC IS MOTION ALONG A PATH. Initially, a novice may draw a line, label it from –5 to 5, start at the origin (0), walk backward two steps (–2) to –2 and walk one more step backwards (–1) to –3. As that student develops exper- tise, he or she recognizes that some source-to-target projections are unnecessary. So, the student omits them to increase efficiency. For example, he or she labels fewer locations on a line, starts at –2 rather than walking there from the origin (0), and so on. Eventually, experts can recognize the target problem “–2 + –1” and simply recall the target result, –3. By doing so, they have bypassed metaphorical reasoning through the source (motion). Thus, experts may reference fewer source details compared to novices. Knowledge compilation explains why experts may not refer to a metaphorical source at all while solving a problem, consistent with Searle (1979) and Post et al. (1985). Knowledge compilation does not preclude the re-creation of the metaphorical inferences. Students can easily do so for grounded metaphors (Lakoff & Nunez, 1997; Reyna’s, 1986, mundane metaphors) in which the source consists of basic intuitions (Chiu, 1996). In short, knowledge compila- tion explains how experts can both have a metaphor and use it less often. Expert’s greater source comprehension. With more experiences to use as sources for metaphors, experts can create both more metaphors and more infer- ences per metaphor. Goswami’s (1991) review of developmental research showed that knowledge predicted metaphorical reasoning better than age did. Gibbs’s (1990, 1992) studies showed that the source of a metaphor provided the raw mate- rial for potential target inferences. People also develop the ability to reason meta- phorically at an early age. Specifically, Chen, Sanchez, and Campbell (1997) showed that 13-month-old infants could reason metaphorically. Inadequate understanding of a metaphor’s source limits a student’s capacity to use the metaphor. Consider ROOTS OF A NUMBER ARE COMPLEX PLANE ROTATIONS OF ONE ANOTHER. A mathematician would view a root of a number as a vector and the other roots as rotations of it. In contrast, a lay person who does not understand the com- plex plane source cannot. Source comprehension enables and limits reasoning through specific meta- phors. Therefore, experts can potentially use both more metaphors and more infer- ences per metaphor. Multiple metaphors. Experts can use more metaphors, so they can use two or more metaphors with different sources more often to understand a target concept (Chiu, 2000). For example, a person can use A VARIABLE IS A PLACEHOLDER FOR A NUM- BER (PLACEHOLDER metaphor) to understand that a variable refers to a specific num- ber. However, using this metaphor to understand that a variable can have many val- 98 CHIU ues can be difficult (e.g., y = 2x). Students can use VARIABLES ARE TRAVELERS to understand that a variable may assume different values. However, it is less suited than the PLACEHOLDER metaphor for making sense of equations such as 3/x + 2 = 5. By using both metaphors, experts can overcome each metaphor’s limitations. Conditions of use. Experts also learn when to use their metaphors. As stu- dents develop expertise, they are told which metaphorical inferences are wrong, so they learn to avoid them. As metaphorical inferences are not always reliable, ex- perts find other ways to justify their results (e.g., proofs). Schunn and Dunbar’s (1996) study showed that experts do so even if they initially used a metaphor. Ex- perts also use metaphors less often because they learn or create more efficient algo- rithms. In short, experts use their metaphors less often due to invalid inferences, un- reliable justifications and inefficient procedures. A comparison of metaphorical reasoning with other types of reasoning can be found at http://tcct.soe.purdue.edu/books_and_journals/mtl_journal/, together with an example of the use of metaphors in the classroom. Returning to this study’s central questions: Do experts continue using the same metaphors as novices? If yes, do they use these metaphors in the same way? Spe- cifically, do they both use metaphors to (a) understand mathematical ideas, (b) connect different ideas, (c) compute, or (d) detect and correct errors? Are experts more likely to (a) compile their metaphors using less detail, (b) have more metaphors, (c) have more inferences per metaphor, (d) use multiple metaphors to understand a single concept, (e) justify their answers without meta- phors, or (f) use metaphors less often in favor of more efficient methods? METHOD Twelve children and 12 adults solved problems involving negative numbers and showed their understanding of arithmetic expressions. The specific problems and subproblems can be solved with different metaphors that the participants likely learned in school. Data included audiotaped semistructured interviews of each par- ticipant, written work, and field notes. Participants The 12 children were 12 to 13 years old (7 boys and 5 girls). The 12 adults were 18 to 25 years old (7 men and 5 women). Six weeks before this study, the children’s seventh-grade teacher used the Keedy and Bittinger (1987) textbook to teach nega- tive numbers. She also discussed temperatures and debts as examples of negative  USING METAPHORS 99 numbers. The 12 adults were studying mathematics or engineering at an elite uni- versity at the graduate or undergraduate level. Tasks An interviewer presented each participant with a description of the stock market, a two-part problem-solving task, and an understanding task (see Appendix B for de- tails). The problem contexts encourage use of nonmathematical information, both from the problem situation and from other sources. The contexts also clarify the participants’ metaphor use and their situation-based reasoning (or distributed rea- soning; see http://tcct.soe.purdue.edu/books_and_journals/mtl_journal/; e.g., Gib- son, 1950; Lave, 1983; Scribner, 1986). Also, the novel problem context is more likely to engage the participants. The two tasks help clarify the differences between their uses of metaphors for problem solving and for understanding. Each participant read a written description of the problem. Then, in the first half of this problem, the interviewer asked the participants to calculate the day’s earn- ings or losses. A report listed four transactions and the changes in price. The solu- tion entailed multiplying the ounces by the change in price per ounce for each transaction and summing the results: (10 oz. × $8/oz.) + (30 oz. × –$5/oz.) + (–40 oz. × –$1/oz.) + (10 oz. × –$4/oz.) = $80 + –$150 + $40 + –$40 = –$70. The overall result was a loss (–$70). After each participant produced a result, the in- terviewer asked him or her to explain it. In the second half of the problem, each participant could change a single num- ber by 5 (+5 or –5) to make a profit. For example, subtracting 5 from the change in the price of platinum (–$1/oz.) yields –$6/oz. (–1 – 5 = –6). So, the new profit is –40 oz. × –$6/oz. = $240. The additional $200 ($240 – $40 = $200) changes the net loss, –$70, to a net gain of $130 (–$70 + $200 = $130). In the understanding task, the interviewer asked each participant for his or her understanding of six arithmetic expressions. Scoring Each participant’s computation correctness and computation time was calculated. Each arithmetic operation between two numbers was a computation. So, “30 + 50 – 20 is 80, 60” was two computations. Next, a colleague and I blind coded the tran- scripts (see Appendix C for details). The participants’ written work and the inter- viewer’s field notes provided supporting evidence. For both tasks, we coded each 100 CHIU computation for solution method. For instances of metaphorical reasoning, we also coded for metaphor type and level of detail. In the problem-solving task data we also coded for type of metaphor function. We used a decision tree to classify the participants’ solution methods into the several categories. (Appendix C includes the complete decision tree and detailed examples.) These categories were answer only, mathematical rule, mathematical transformation, situation reasoning, metaphorical reasoning, and other. Analyses Differences between children and adults were tested with respect to computational accuracy, speed, frequency of metaphor use, and gender effects. Regressions and analyses of variance in this type of study are isomorphic to t tests. Therefore, with two exceptions, the results are presented simply via t tests. First, a regression exam- ined whether the child or adult distinction and the number of computations both predict the frequency of metaphor use during the problem-solving part. Second, when an event occurs less than five times, Fischer exact tests were used to provide more precise estimates (Tabachnick & Fidell, 1989). Next, case studies examined the work of the median accuracy child and the me- dian accuracy adult in detail. Cohen’s kappa was used to test for intercoder reli- ability. When the coders assigned different codes, they resolved them through consensus. RESULTS The participants used a metaphor 155 times during their 582 computations between two numbers (Cohen’s κ = 0.872, z = 11.1, p < .001). See Tables 2 and 3 for a sum- mary of solution methods. None of the following results show any significant gen- der differences (all p > .10). The adults and children used similar metaphors but their uses and their accu- racy differed. Children used metaphors more often to solve problems. In contrast, adults showed their understanding with more metaphors. While solving the invest- ment problem, children used metaphors more often (M = 3.67 times, SD = 2.99) than adults did (M = 0.25, SD = 0.62), t = 2.97, p < .02, two-tailed. Indeed, only two adults used metaphors during the investment problem. Predicting metaphor uses during problem solving with both expertise (adult vs. child) and number of compu- tations yielded similar results (adult b = –1.82, SE = 0.77; number of computations b = 0.16, SE = 0.06; R2 = .71). However, adults showed their understanding of the six arithmetic expressions with metaphors more often (M = 7.00 times, SD = 2.70) than children did (M = 2.25, SD = 0.35), t = 3.40, p < .01, two-tailed.  USING METAPHORS 101 TABLE 2 Mean Uses and Standard Deviations of Each Computational Method by Participants for Each Task Mathematical Mathematical Answer Only Metaphor Rule Transformation Situation Task and Participants M SD M SD M SD M SD M SD Problem solving Children 16.25 6.34 3.67 2.99 0.58 0.47 0.42 0.75 0.67 0.54 Adults 10.17 1.67 0.25 0.62 0.33 0.51 0.17 0.39 0.67 0.29 Understanding Children 1.92 0.62 2.25 0.35 1.17 0.55 1.17 0.62 NA NA Adults 0.67 0.78 7.00 2.70 1.58 1.38 1.67 0.90 NA NA Note. Participants may use multiple methods for a single computation. NA = not applicable. TABLE 3 Mean Uses and Standard Deviations of Each Type of Metaphor by Participants for Each Task Opposing Social Motion Objects Transactions Others Task and Participants M SD M SD M SD M SD Problem solving Children 2.67 1.11 0.92 0.86 0.00 0.00 0.09 0.23 Adults 0.08 0.29 0.00 0.00 0.08 0.29 0.00 0.00 Understanding Children 1.75 1.02 0.34 0.40 0.09 0.23 0.09 0.23 Adults 2.08 0.90 3.83 1.67 0.75 0.50 0.33 0.51 Accuracy was higher for adults and was higher during metaphor use. As ex- pected, adults computed more accurately (98.8%, 248 of 251) than the children did (83.1%, 275 of 331), χ2(1, N = 582) = 38.74, p < .001. When children used a meta- phor to compute, they were more accurate (93%, 66 of 71) than when they used other methods (81%, 211 of 260), χ2(1, N = 331) = 5.48, p < .02. Adult accuracy also improved during metaphor use, but not significantly so (metaphor = 100%, 86 of 86; other = 98.2%, 163 of 166; Fisher’s exact test φ < 0.08). However, children were slower when using a metaphor (M = 5.93 sec, SD = 2.73) than when they used other methods (M =3.14, SD = 1.27), z = 6.28, p < .001. Thus, using a metaphor to compute tends to improve accuracy but reduces efficiency. Next I discuss two participants, a child (Nina) and an adult (Eva), both with me- dian computational accuracy and median instances of metaphor use in their re- spective groups. (Both names are pseudonyms. They need not reflect gender or ethnicity.) 102 CHIU Median Child (Novice): Nina Nina eventually solved the investment problem after a few errors and showed few ways of understanding the arithmetic expressions. Her computational accuracy was 87%, and she used metaphors eight times. Problem solving. Most students (10 of 12) solved the accounting part of the problem correctly. However, half of the students (6 of 12, including Nina) made at least one mistake. Consider Nina’s solution. In the transcript, numbers in parenthe- ses, (7), indicate the duration of pauses in seconds. Numbers and words within sin- gle quotes inside brackets, [‘lost 70’], indicate the student’s writing. Vertical bars, ||, indicate simultaneous speech. Int represents the interviewer. Int: [Reading problem question] How much money did you win or lose? Nina: (2) So, I just multiply. 10 times 8 is 80 [‘80’], I guess 30 times negative 5 is negative 150 [‘–150’]. Negative 40 times negative 1. That’d be lower, negative 40, right? Int: How can you check? Nina: Can I look at the sheet? Int: Uh-huh. Nina: [Looks at instruction sheet] (20) 40 [‘40’]. Uh, 10 times negative 4 is negative 40 [‘–40’]. Won 120 [‘120’ (from 80 + 40)] lost 190 [‘–190’ (from –150 + –40)]. I lost. I lost more than I won. 190 minus 120 is, um, lost 70 [‘–70’]. Nina immediately converted the problem into computations (“I just multiply”) and computed the first two expressions correctly. However, she multiplied –40 × –1 in- correctly, “that’d be lower, negative 40, right?” After the interviewer asked for an evaluation strategy, Nina looked at the instruction sheet and corrected the error. Then, she computed the final expression correctly. She then combined the gains and losses (“won 120, lost 190”), noted that she lost money (“overall I lost”), and computed the exact result (“lost 70”). When asked to justify her answer, she used different computations to get the same answer. Nina: … lost 70 [‘–70’]. Int: How can you tell if that’s right? Nina: See it was positive 80 and I add 150, negative 150, so it’d just be 70 [‘–70’] and then I add positive 40, so that’d be positive 30 [‘30’], no, I haven’t gone all the way up yet so it’d be negative 30 [‘–’ to make ‘–30’]. Int: You didn’t go all the way up?  USING METAPHORS 103 Nina: I hadn’t gone all the way up to 1 or 0 [raises her horizontally flat right hand to eye level] when you change this [left finger points below right hand] into the positives [left finger moves above right hand]. Um, nega- tive 30 and negative 40 is negative 70. Starting with the results of each metal, Nina summed them in a different order. After adding 80 + –150 = –70, she mistakenly added –70 + 40 = 30. She then detected and corrected this error via the MOTION metaphor (“no, I haven’t gone all the way up yet, so it’d be negative 30”). When asked to elaborate, Nina di- vided the space in front of her into an upper region and a lower region (“[raises her horizontal flat right hand to eye level]”). She argued that she had not crossed over into the upper region (“I hadn’t gone all the way up to 1 or 0”). Finally, she added –30 + –40 = –70. In short, Nina quickly converted the problem into computations and solved it primarily with arithmetic facts, correcting an error en route. When asked to justify her answer, she did a different set of computations, using a metaphor to detect and correct an error. During the computer part of the investment problem, Nina erred more often (as did 7 of 12 children) and used metaphors more often (as did 7 of 12 children). Nina: Which one should I change? Int: Any one you want. Nina: (6) Uh, I’m going to do 40 [platinum ounces] ’cause, um, it’s the biggest. [Softly] It doesn’t go past it, so it’s minus 35 [–40 + 5]. Int: Doesn’t what? I’m sorry, I didn’t hear you. Nina: Yeah, it doesn’t goes past 0, so it’s negative 35. Minus 35 times minus 1 is minus 35 [‘–35’]. So, minus 40 plus minus 35 is 75, minus [‘–75’], so minus 80 is minus 5, uh, 5? (3) Oh, it doesn’t matter because the minus 150 is bigger. Int: Bigger? Nina: Yeah, it’s bigger, so it’ll wipe out the 5 even if it’s positive. Int: OK. Nina tried to analyze the problem. However, her heuristic of choosing the largest number required more computations than the “test and check” approach used by eight children. To compute –40 + 5, she used a metaphorical constraint (MOTION metaphor) to recognize that the result was still in the negative region (“it doesn’t go past zero, so it’s negative 35”). After multiplying two negative numbers incorrectly again (–35 × –1 = –35), Nina correctly added –40 + –35 = –75. She then had diffi- culty deciding whether –75 + 80 was 5 or negative 5. However, she decided that the question was moot because –150 was bigger and would “wipe out the 5.” Viewing 104 CHIU the computation as a conflict between two numbers (OBJECTS metaphor), Nina rec- ognized that the one with the greater magnitude dominated. Nina continued adding and subtracting 5 from the largest remaining investment numbers without success until she subtracted 5 from the silver price change. Nina: OK, uh, 5 minus 5 is 0, so that’s just 0. Oh, that works because 40 and mi- nus 40 is 0 and we have positive 80 left. So I win $80. Int: How can you tell if that’s right? Nina: Well, 5 minus 5 is 0 because anything take away itself leaves nothing, and 0 times anything is 0, and the 40 and negative 40 wipe each other out, so all that’s left is 80. After calculating the correct answer (“I win $80”), Nina justified her answer with metaphors and arithmetic rules. Using her OBJECTS metaphor, she argued that given the presence of an object, removing it left nothing behind (“anything take away it- self leaves nothing”). Then she applied an arithmetic rule (“0 times anything is 0”) to justify the new silver result (0 × –30 = 0). Applying a conflict (OBJECTS) metaphor again, Nina argued that positive and negative numbers of equal magnitudes, specif- ically 40 and –40, would neutralize each other (“40 and negative 40 wipe each other out”). In short, Nina’s analysis of the computer part of the problem increased the num- ber of computations, and she made several mistakes. In particular, she consistently and incorrectly computed a negative product from two negative numbers. When asked to justify her answer, she used metaphors and arithmetic rules. Understanding task. All of the middle school students in this study used metaphors to explain some arithmetic expressions during the understanding task. Nine of the 12 children (including Nina) made at least one mistake. Int: How do you make sense of negative 5 plus 8? Nina: I think of like, numbers, like the negative numbers, like a line and they have negatives, can I write it down? Int: Sure. Nina: It’s easier for me [writes the following in a column from the top with a line in the place of 0 as a boundary: ‘10, 9, 8, 7, 6, 5, 4, 3, 2, 1, –, –1, –2, –3, –4, –5, –6, –7, –8, –9, –10’ (see Figure 1)]. These are the positives [points toward the top] and these are the negatives [points below the boundary line]. Int: So what’s the answer to this problem, –5 + 8? Nina: Oh, um [circles –5], plus 8; 1, 2, 3, 4, 5, 6, 7, 8 [finger bounces up and ends at 4, skipping 0 [puts ‘–’ after 4, ‘4 –’], that’d be right here. 4.  USING METAPHORS 105 Int: 4? Nina: Yeah. 1, 2, 3, 4, 5, 6, 7, 8 [finger bounces up starting from –5 and skips 0 again]. 4. Int: Are there other ways of making sense of it? Nina: (3) I don’t know. Nina depicted signed numbers vertically and used a boundary line to separate them. Using this detailed drawing, she used her MOTION metaphor to compute the expres- sion “–5 + 8.” Starting at –5, she moved her finger up eight steps to the final destina- tion of 4 (“plus 8; 1, 2, 3, 4, 5, 6, 7, 8 [finger bounces up and ends at 4, skipping 0]”). Nina omitted zero, suggesting that she viewed zero only as a boundary between positive and negative numbers. She did not view zero as a metaphorical location on par with the other numbers, so she did not include zero in her computation. Her in- correct metaphorical projection of motion onto arithmetic resulted in the wrong an- swer. Additional prompts did not elicit any changes or further explanation. Nina also explained the second expression “–4 – 6” with the same MOTION metaphor. Int: How about negative 4 minus 6? Nina: Start down, right here [points to –4 on her vertical line] and you go down 6 more [her finger bounces from –4 to –10] to end up at negative 10. FIGURE 1 Nina’s depiction of her vertical spatial metaphor with a posi- tive–negative boundary and no zero. 106 CHIU After locating –4 (“start down, right here [points to –4 on her vertical line]”), Nina computed the result metaphorically (“you go down 6 more [her finger bounces from –4 to –10] to end up at negative 10”). She did not use this metaphor in her next computation. Int: How about 7 minus negative 2? How do you make sense of this one? Nina: Um, negative 5, cause 7 minus 2 is 5 and you add the negative. Nina used an incorrect procedure of subtracting the numbers and adding a negative sign. Unlike her earlier computations, she did not check her answer with a meta- phor. Nina correctly computed the remaining three expressions. Int: How do you make sense of negative 2 times 3? Nina: Negative 2 times 3 is negative 6 because a negative times a positive is a negative. Using an arithmetic rule, she correctly justified the sign of her answer. Int: How about negative 7 times negative 4? Nina: Negative 7 times negative 4 is 28, negative times a negative is a positive. Although Nina consistently multiplied negative numbers incorrectly for the earlier expressions, she invoked a general rule to compute “–7 × –4” correctly (perhaps cued by the previous problem). Int: And negative 8 divided by negative 4? How do you make sense of that? Nina: Same as before, negative divided by a negative is a positive, so negative 8 divided by negative 4 is positive 2. Likewise, she noted that division of two negative numbers yielded a positive number. In short, Nina showed correct understanding of most of the expressions, giving one explanation for each expression (via a metaphor or an arithmetic rule). How- ever, she computed an incorrect answer using her faulty motion metaphor, and she did not use a metaphor to detect or correct another error. She also used an arithme- tic rule to compute the product of two negative numbers correctly although she had incorrectly computed three such expressions earlier. Nina summary. Nina computed most of the expressions correctly, but made several mistakes. She used detailed metaphors several times to compute results, de- tect and correct errors, and justify her results. Nina showed her understanding of  USING METAPHORS 107 arithmetic expressions using either a metaphor or an arithmetic rule. However, she did not use them consistently. Adult (Expert): Eva Eva solved the investment problem correctly with few computations. She also showed many ways of understanding the arithmetic expressions. She made no un- detected errors and reasoned metaphorically eight times. All of the adults solved the accounting part of the problem correctly without any errors or metaphors. Consider Eva’s solution for example. Int: [Reads] How much money did you win or lose? Eva: [Looks at instruction sheet] (20) OK, we multiply, so that’s 80 [‘80’], minus 150 [‘–150’], 40 [‘40’], and minus 40 [‘–40’]. 150 and 40 is 190 [‘–190’], 80 and 40 is 120 [‘120’], lost 70 [‘70’]. Like Nina, Eva converted the problem into arithmetic computations (“OK, we mul- tiply”). After computing the results for each metal, Eva summed the negative re- sults (“150 and 40 is 190 [‘–190’]”). Then, she summed the positive results (“80 and 40 is 120 [‘120’]”). Finally, she added the negative and positive partial sums to conclude “lost 70.” Next, Eva justified her result. Int: How can you tell if that’s right? Eva: I looked at the examples. We have to figure out the result for each one of these by multiplying the ounces by the change per ounce to get the profit and then we sum up all the profits for all the metals. Unlike Nina, Eva did not recompute arithmetic expressions to show that her answer was correct. Eva justified her answer by referring to the instruction sheet and the problem situation, suggesting that she did not believe that she needed to justify her computations. First, she stated a set of subgoals (“figure out the result for each [metal]”). Then, she described computations to achieve them (“multiplying the ounces by the change per ounce to get the profit”). Last, she explained how to com- pute the final answer (“sum up all the profits for all the metals”). As with the first part of the problem, all of the adults also correctly solved the second part. Furthermore, they all analyzed the problem to reduce the number of necessary computations, unlike the test and check approach used by many of the children. Three adults (including Eva) made mistakes, and only Eva corrected her error. Consider Eva’s solution. 108 CHIU Eva: [Reads instructions] Which number should you change? (5) I think I can figure out each little effect and that’ll be the overall difference [draws a table, frequently looking back at the previous problem (see Figure 2)]. Eva stated a goal of computing the local effect of each change (±5) and using it to find the overall effect (“I can figure out each little effect and that’ll be the overall difference”). Then, she organized her computations with two tables, the first for changes in the ounces bought and sold and the second for price changes. After she systematically added and subtracted 5 from each quantity, she computed the new results for each metal. Eva: 80, 40, 120 [writes in the fourth column ‘120’], 40 [‘40’], 150, 25, 175 [‘–175’], 125 [‘–125’], 45 [‘45’], 35 [‘35’] minus 60 [‘–60’], minus 20 [‘–20’], 130 [‘130’], 30 [‘30’], minus 300 [‘–300’], 0 [‘0’], minus 160 [‘–160’], 240 [‘240’], 10 [‘10’], minus 90 [‘–90’]. After systematically computing the new results for each metal (all correct), she sub- tracted the original results from them. FIGURE 2 Eva’s diagram explaining that 10 – (–40) = 50, not 30. Eva’s table of computations for the second (computer) part of the problem-solving task.  USING METAPHORS 109 Eva: [120 – 80] 40, no [writes in the fifth column ‘X’], no [‘X’], no [‘X’], [125 – –150] 25, no, [‘X’], [45 – 40] 5, no [‘X’], no [‘X’], no [‘X’], nope [‘X’], [130 – 80] 50, no [‘X’], no [‘X’], no [‘X’], [0 – –150] 150, won that one [‘!’], down [‘X’], [240 – 40] 200, win [‘!’], [10 – –40] 30, wait, I’m passing the 0 line. Int: |You’re passing | Eva: |I’m going forward | Yeah, ’cause I’m going from a positive to a negative value, so the actual difference is not just 30 right? It’s 10 plus 40, so the actual difference is 50. Int: What’s the 0 line? Eva: Like I’m imagining 10 being on the line, and minus 40 would be on the other side of the 0. Int: Is there a picture you’re thinking of? Eva: Yeah, the number line, like this [draws line, writes under the line ‘–40,’ ‘0,’ and ‘10,’ appropriately spaced, and draws a brace between –40 and 10]. Right? Int: Oh, so this is the 0 line? Eva: Yeah, actually I shouldn’t say line, the 0 point, so 50 [10 – –40], no [‘X’], no [‘X’]. Nearing the end of numerous subtractions, Eva incorrectly subtracted 10 – (–40) = 30. However, she used her motion metaphor to detect her mistake (“wait, I’m pass- ing the 0 line”). Imagining the numbers to be locations (“I’m imagining 10 being on the line, and minus 40 would be on the other side of the 0.”), Eva argued that “I’m going from a positive to a negative value.” So, the distance traveled was the sum of the distances from 10 to 0 and from 0 to –40. As a result, Eva added the numbers (“it’s 20 plus 40”) instead of subtracting them. She then drew a picture to support her explanation (see Figure 2). The interviewer then asked Eva to justify her answer. Int: How can you tell if that’s right? Eva: It’s 50 because subtraction of a negative number is the same as adding a positive number, like the inverse of an inverse is the original number. Eva used a metaphor to compute her answer but not to justify it. Instead, she re- ferred to two arithmetic rules (“subtraction of a negative number is the same as add- ing a positive number, like the inverse of an inverse is the original number”). The interviewer then asked for Eva’s final answer and her justification of it. Int: What’s your final answer? Eva: Subtracting 5 from the price in silver and platinum wins money overall. Int: How can you tell if that’s right? 110 CHIU Eva: I added and subtracted 5 from every possible number, and only subtract- ing 5 from silver and platinum gets you an additional $70 or more. Be- fore, I lost 70, so I had to make up for that. Eva used a metaphor (ARITHMETIC IS SOCIAL TRANSACTION or SOCIAL metaphor) to iden- tify her goal (“I lost 70, so I had to make up for that”). Also, she briefly described her method (“I added and subtracted 5 from every possible number”). Then, she summarized her final results (“only subtracting 5 from silver and platinum gets you an additional $70 or more”). In short, Eva systematically computed each possible change. However, she did fewer computations by comparing the local change to the original overall result (unlike the children’s typical test and check approach). Like Nina and unlike most other adults, Eva metaphorically detected and corrected an error. She also used metaphors to identify goals. Unlike Nina, Eva never used a metaphor to begin a computation or to justify an answer. Understanding task. The adults computed each arithmetic expression in the understanding task correctly and explained them with several metaphors and arith- metic rules. For example, Eva explained the first expression, –5 + 8, with three met- aphors. Int: How do you make sense of negative 5 plus 8? Eva: Initially I flip to 8 minus 5 and I get 3. How do I make sense of it though? It’s like I have to give 5 gifts let’s say, and I have 8 toys and I have to give 5 toys as gifts and so I only have 3 left. Beginning with a procedural solution, Eva mathematically transformed the prob- lem into a familiar subtraction expression with positive numbers. Through her SOCIAL metaphor, she explained the –5 as a social obligation (“I have to give 5 toys as gifts”) and the 8 as property (“I have 8 toys”). To fulfill her social obli- gation, she used her property (“I have to give 5 toys as gifts and so I only have 3 left”). Eva: Another way I think about it is a number line with a 0 point [draws hori- zontal line and marks ‘0,’ ‘–5,’ and ‘8’ on it] and a 8 and a minus 5 and some sort of [draws bars from –5 to 0 and from 0 to 8]. Int: What are those things? Eva: I don’t know, they’re sort of like bars representing like 8 right, and this represents minus 5, and it has nothing to do with how I actually solve the problem, but that’s how I think about it. Int: OK.  USING METAPHORS 111 Eva: They’re both in conflict, right? And this sort of bar is pulling toward mi- nus 5. It wants to go toward minus 5 as a goal and this bar wants to go to- ward 8 as a goal. Int: Uh-huh. Eva: And in order to resolve the conflict those two have to fight it out and they’re going to get 3. Int: What happens when they fight it out? Eva: There’s tension, going in opposite directions, and they start canceling each other, so the 5 cancels out 5 parts of the 8, and so the 8 can only go to 3. Eva delineated a clear separation between her problem solution and her under- standing (“it has nothing to do with how I actually solve the problem, but that’s how I think about it”). Combining MOTION and OBJECTS metaphors, she gave a composite explanation. She identified each number with particular locations and desired des- tinations (“It [–5 bar] wants to go toward minus 5 as a goal and this bar [points to 8 bar] wants to go toward 8 as a goal”). Consistent with her MOTION metaphor, Eva en- dowed the –5 and 8 objects with sufficient size to occupy the distances from the ori- gin to their metaphorical locations (“[draws bars from –5 to 0 and from 0 to 8]”). However, Eva explained the computation not as motion, but as opposition between the objects (“those two have to fight it out”). By neutralizing paired components (“the 5 cancels out 5 parts of the 8”), only three remained (“they’re going to get 3”). Eva used her MOTION metaphor again when she elaborated her explanation (“the 8 can only go to 3”). Unlike Nina, who showed only one way of understanding “–5 + 8,” Eva explained it in several ways, via a transformation to a subtraction problem and via two explanations with three metaphors. Next, the interviewer asked her about –4 – 6. Int: OK. And negative 4 minus 6? Eva: Minus 10. I’m at minus 4 and I want to go down 6, so I go down to nega- tive 10. This time, Eva began with a MOTION metaphor explanation. She identified the initial location (“I’m at minus 4”), the motion (“go down 6 more”), and the destination (“I go down to negative 10”). Then, she explained with her SOCIAL metaphor: Eva: Also, extending the analogy I started out using earlier and say, and let’s say, I owed someone 4 toys and then I borrowed 6 toys and then so I owe a lot more toys now. It might make more sense not with toys but with like money or something. Like you go borrow money from someone to go buy candy and then eat it and then you borrow more money but then you have to pay them back more. 112 CHIU Eva explained –4 as a social obligation (“I owed someone 4 toys”) and –6 as incur- ring another obligation (“I borrowed 6 toys”). These result in a greater net obliga- tion (“so I owe a lot more toys now”). Replacing toys with money, she provided a similar explanation. Then, Eva explained 7 – (–2) using some dynamic imagery. Int: How do you make sense of, um, 7 minus negative 2? Eva: I’m thinking that the two negative signs mush together and make a posi- tive sign. Exploiting the symbols, Eva transformed the subtraction problem into an addition problem. She combined the subtraction symbol and the negative number’s sign into an addition symbol (“the two negative signs mush together and make a positive sign”). Next, the interviewer asked Eva to explain –2 × 3. Int: How about negative 2 times 3? Eva: It’s like I owe two people some money and I owe each of them $3, and then so I owe $6. Using her SOCIAL metaphor, Eva explained that the first number (–2) was the num- ber of people to which she owed money (“I owe two people some money”). Mean- while, the second number was the quantity of each obligation (“I owe each of them $3”). Eva similarly explained her understanding of –7 × –4: Int: And how about negative 7 times negative 4? Eva: Same as before, 28 because two negatives cancel each other out, and a negative times a negative is a positive. Int: Same as before? Eva: Uh-huh, I mean it’s like 7 minus negative 2, the minus signs cancel. Eva explained her solution with an arithmetic rule (“a negative times a negative is a positive”). She also argued that the solution to this multiplication problem was the “same as” the earlier “7 – –2” problem (“it’s like 7 minus negative 2”) because “two negatives cancel each other.” Lastly, Eva explained –8 ÷ –4. Int: Finally, how do you make sense of negative 8 divided by negative 4? Eva: Same as the multiplication, negative divided by negative is also posi- tive, too.  USING METAPHORS 113 Noting the similarity between these two problems, Eva used an analogous division rule (“negative divided by negative is also positive”). Unlike Nina, Eva computed all the expressions correctly and provided multiple explanations. Eva used several metaphors, arithmetic transformations, and rules to show her understanding of each arithmetic expression and the relations among them. Eva also used fewer metaphor details than Nina did. Eva summary. Eva solved the investment problem quickly and efficiently primarily through direct computations. She reasoned metaphorically twice during the problem-solving task, to create a goal and to detect and correct an error. During the understanding task, she used multiple metaphors, arithmetic transformations, and rules to show her integrated understanding of arithmetic expressions. DISCUSSION These statistical analyses and case studies of median participants suggest that both novices and experts have the same arithmetic metaphors but use them differently. These novices and experts use substantially the same set of metaphors, therefore they are candidate theory-constitutive metaphors. The results also support Boyd’s (1993) and Lakoff and Nunez’s (1997) claim that understanding mathematics in- volves theory-constitutive metaphors. Both novices and experts used metaphors to understand arithmetic ideas, to connect different ideas, to compute arithmetic ex- pressions, and to detect and correct errors. These benefits of metaphorical reason- ing are consistent with past research results (e.g., Petrie & Oschlag, 1993; Silva & Moses, 1990). However, novice and expert metaphor use and understanding differed in sev- eral ways. Experts used metaphors less often in favor of more efficient methods. Both used metaphors when they faced difficulties. However, novices had more difficulties and used metaphors more often. These behaviors are consistent with the view that metaphors serve as scaffolds (Hutchins, 1995; Rogoff & Gardner, 1984; Vygotsky, 1935/1978; Wood, Bruner, & Ross, 1976). In the comparison of the median expert and the median novice, the expert showed more metaphors, had more inferences per metaphor, used metaphors to connect different ideas, used metaphors with less detail, and did not use metaphors to justify her answers. In the problem-solving task, both novices and experts used metaphors to detect and correct errors. However, only the novice used them to start computations and to justify results. In contrast, the expert used metaphors to create broader goals (Carbonell, 1983; Hall, 1989). 114 CHIU Both the median novice and the median expert used metaphors to show their understanding of arithmetic. However, the expert showed a more consistent, inte- grated understanding. The novice only provided single explanations of each arith- metic expression and failed to use her understanding in the earlier problem-solving task. In contrast, the expert gave multiple explanations for an arithmetic expres- sion and used a metaphor to explain multiple arithmetic expressions. This expert’s behaviors support Dowker’s (1992) and Moschkovich, Schoenfeld, and Arcavi’s (1993) claim that mathematical expertise requires integrated multiple understand- ings of mathematical ideas. Finally, the expert used less detail in her metaphorical problem solving and ex- planations compared to the novice. This result is consistent with Anderson’s (1987) knowledge compilation. Many questions still remain. In particular, do adults use these same metaphors while solving a more challenging problem? How and where do students learn these metaphors—from teachers, textbooks, or sources outside of the classroom? How do they put these metaphors together? Are there intermediate phases of metaphori- cal reasoning? Do people’s uses of standard mathematical metaphors continue changing as they develop expertise (e.g., mathematicians)? 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Source entities, properties, and relations in the columns on the right pro- vide the bases for metaphorical inferences in the target in the columns on the left. Target is Source. Target ← Source  USING METAPHORS 117 Arithmetic is manipulation of objects Numbers are groups of blocks Arithmetic is social transaction Arithmetic is manipulation of lengths Primary numbers are primary colors Roots of numbers are complex plane rotations of one another Sets are containers TABLE A1 Partial Metaphorical Mappings Arithmetic Is Manipulating Objects 0 ← No objects Unit (1) ← An object Add A + B ← Put B objects into the current pile of A objects Subtract A – B ← Remove B objects from the current pile of A objects Multiply A × B ← Replace the original A pieces with B copies OR Multiply A × B ← Cut each of the current pile of A objects into B pieces (Confrey, 1994) OR Divide A ÷ B ← Put the A pieces evenly into B boxes and count the number of pieces in each box Negative numbers ← Antimatter objects (pairs of matter and antimatter objects mutually annihilate one another) Add a negative number A + (–B) ← Put B antimatter objects into the current pile of A objects (or antimatter objects) Subtract a negative number A – (–B) ← Remove B antimatter objects from the current pile of A objects (if there are less than B antimatter objects in the current pile, it gets very messy—need to create pairs of matter and antimatter objects out of nothing) Multiply by a negative number A × (–B) ← Replace the existing pile of A pieces with B replications of A pieces; change the objects into their mirror images (matter becomes antimatter and vice versa) OR Multiply by a negative number A × (–B) ← Cut each of the current pile of A objects into B pieces; change the objects into their mirror images Divide A ÷ (–B) ← Put the A pieces evenly into B boxes and remove all but one of the B boxes; change the objects into their mirror images OR Divide A ÷ (–B) ← Merge every B of the A objects into new objects, and change the objects into their mirror images (continued)  TABLE A1 (Continued) Numbers Are Groups of Blocks Each power of 10 (1, 10, 100, …) ← An object of different size Each power of 10 is equal to 10 of the next ← An object is equal in size to 10 of the next smaller size lesser power of 10 object Each digit of a number ← A group of blocks of the same size A number consists of digits ← A collection of blocks consists of groups of different size blocks Arithmetic Is A Social Transaction 0 ← Possession of no objects (cf. POSSESSION IS HOLDINGa) Unit (1) ← Possession of 1 object Add A + B ← Receive B objects to combine with current A possessions Subtract A – B ← Give away B objects from current A possessions Multiply C × D ← C people each give me D objects Divide C ÷ D ← C objects shared among D people (including self) Negative number ← Bill (debt) Add a negative number A + (–B) ← Receive a bill for B objects Subtract a negative number A – (–B) ← Give someone a bill for B objects Multiply a positive by a negative C × –D ← C people each give me a bill for D objects Multiply a negative number by a positive ← Give C objects to D people number –C × D Multiply a negative by a negative –C × –D ← Give each of C people a bill for D objects Divide a positive by a negative C ÷ –D ← C objects will be given away by your group of D people Divide a negative by a positive –C ÷ D ← A bill for C objects has been received by your group of D people Divide a negative by a negative –C ÷ –D ← A bill for C objects will be given by your group of D people Arithmetic Is Motion Along a Path Number ← Location relative to the origin 0 ← Origin or starting point Positive number A ← Location A steps to the right of the origin Negative number (–A) ← Location A steps to the left of the origin Absolute value |A| ← Distance from the origin (A steps away) A>B ← Location A is to the right of location B A<B ← Location A is to the left of location B Operations The default operation is to face right and move forward in that direction; (–A) indicates A steps backward Add A ← Move A steps; if A = 0, then hop and land in the same place A+B=B+A ← The order of steps does not change the destination (continued) 118  TABLE A1 (Continued) Arithmetic Is Motion Along a Path Subtract B ← Turn around and move B steps If A ≠ B, ← If the number of steps differ (A and B), A–B≠B–A Walking some (A) steps, turning around, and walking some more (B) steps ends in a different destination than walking B steps, turning around, and walking A steps A + (–B) = A – B ← Walking backward is the same as turning around and walking forward Addition is the inverse of subtraction ← Moving forward is the opposite of moving backward Additive inverse ← Direction and steps needed to return to the origin Multiplication M × N (e.g., –7 × 4) M is the number of steps (e.g., –7 is turn around and move 7 times) N is the step size (e.g., for 4, each step is 4 units long) Execute the “–” sign on M (if any) by turning around Repeat M times: Move step size N M×N=N×M ← Switching the number of steps and the step size does not change the destination M M × N = ΣN ← Multiply by repeating a particular step (N) several times (M) Division P ÷ Q (e.g., –20 ÷ 2) P is the final destination, Q is the step size How many steps of size Q should a student take to go to P? Do you need to turn around “–”? –20 ÷ 2 = –10 ← How do you reach location –20 using a step size of 2? Turn around (–) and take 10 steps, so the answer is –10 4 ÷ 0 =? ← How many steps of size 0 are needed to go to 4? Impossible 0÷0=? ← How many steps of size 0 are needed to go to 0? Any number of steps … 0, 1, 2 … Division is the inverse of ← Ask how to get there is the opposite of ask where are we multiplication going? Lines Are Paths Line segment ← Path of a traveler Polygon ← Round-trip path in which a traveler begins and ends at the same location Side of a polygon ← A straight portion of a round-trip path Adjacent sides of a polygon ← Straight portions of a round-trip path that are traversed one after the other (continued) 119  TABLE A1 (Continued) Lines Are Paths Length of a polygon side ← Number of steps needed to traverse a particular straight portion of a round-trip path Perimeter ← Total steps needed to traverse a round-trip path Prime Numbers Are Primary Colors Natural numbers ← Colors Composite numbers ← Composite colors 1 ← Transparency Product of any number and 1 is that ← Composition of any color and a transparency is that color number, A × 1 = A Product of two different prime numbers ← Composition of two different primary colors yields a yields a composite number secondary color Product of two natural numbers other than ← Composition of two different colors yields a composite 1 yields a composite number color Product of two identical prime numbers ← X Composition of two identical colors yields the same color yields a composite number Infinite number of prime numbers ← X Finite number of primary colors Roots of Numbers Are Complex Plane Rotations of One Another Number ← Vector in complex plane starts at the origin Nth root of a number ← Rotation of rescaled vector around origin Remaining N – 1 roots ← Current vector + rotation of (i/N × 360) degrees around origin, for i = 1, 2, … N – 1 Sets Are Containers Members ← Objects x is a member of Set S, x ∈ S ← Object x is inside Container S Intersection A ∩ B ← Container of objects inside both containers Union A ∪ B ← Container of objects inside either container Negation ~ A ← Container of all objects outside Container A Implication A → B ← Container A is inside Container B, thus the objects inside Container A are inside Container B A → B implies ~ B → ~ A ← An object inside Container A must be inside Container B likewise implies that an object outside Container B must be outside Container A A∨B ← Object is inside Container A or Container B or both A∧B ← Object is inside both Container A and Container B ~A ← Object is outside Container A aIndicates a conventional metaphor from Lakoff, Espenson, and Goldberg (1991). 120  USING METAPHORS 121 APPENDIX B Problem Solving and Understanding Tasks Written Description The stock market is a place for gambling, like a casino. In the stock market, you can buy and sell things like gold and silver. Suppose you buy an ounce of gold for $100. If the price increases by $1 the next day, you have $101 and you win. If the price decreases (– $1), then you lose. On the other hand, you sell some of your silver for $50 an ounce. If the price increases ($5) the next day, then you lose because you should have kept your silver, which is now worth $55 instead of $50. If the price decreases (– $3), you win by selling early and getting more money for it, $50 instead of $47. Problem-Solving Task, Part 1 How much money did you win or lose? (See Table B1.) [How can you tell if that’s right?] TABLE B1 Stock Market Summary Metal Bought/Sold(–) Ounces Gain/Loss(–) Change per Ounce Total Gold 10 $8 Silver –30 $5 Platinum –40 –$1 Copper 10 –$4 Problem-Solving Task, Part 2 It looks pretty bad, but fortunately, you’re a computer expert. You can break into the computer account and change any one of the numbers by 5 (either +5 or –5). Which number should you change? [How can you tell if that’s right?] Understanding Task How do you make sense of –5 + 8? [The interviewer follows with similar questions about –4 – 6, 7 – (–2), –2 × 3, –7 × –4, –8 ÷ –4] 122 CHIU APPENDIX C Coding the Data: Details and Examples Does the student give an answer immediately without further explanation? Yes, code as result only No, Does the student invoke a mathematical rule? Yes, Does the student create another mathematical expression using the rule? Yes, code as mathematical transformation No, code as mathematical rule No, Does the student invoke a situation different from the problem? Yes, Does the student apply an inference from the invoked situation to the problem? Yes, code as metaphorical reasoning No, code as result only No, code as problem situation When solving a problem such as “Find the net gain or net loss of the following transactions: –$4 and –$2,” people could give an answer without any visible work, saying “minus 6” or writing “–6.” The interviewer prompted the participants to elaborate on these answers, but if they did not add anything further, their methods were coded as answer only. The participants explained many of their solution meth- ods as they solved the problem, so many of these answer only solutions may have been arithmetic facts. They could also apply a procedure (mathematical rule), “negative and negative is negative, so negative 6,” or change the problem into a dif- ferent problem (mathematical transformation), “that’s like the negative of 4 plus 2 [–(4 + 2)], so negative 6.” They could also use the constraint information from the problem (situation reasoning), “I lost $4 and then I lost 2 more dollars, so alto- gether I lost $6.” Finally, they could (a) invoke a new situation different from the problem situation (and different from the gambling explanation) and (b) apply an inference pattern from the invoked situation to the problem (metaphorical reason- ing); for example, “[draws horizontal line with hashmark and writes ‘–4’ under- neath] we’re going left, so 1 [pen bounces left and hashmarks line], 2 [pen bounces left and hashmarks line; writes ‘–5’ under previous hashmark and ‘–6’ under the last hashmark], minus 6.” For instances of metaphorical reasoning, we classified them by type into the following categories: motion, opposing object, social transaction, and other. All of these categories are from Lakoff, Espenson, and Goldberg (1991). Briefly, motion is a basic category hardwired into our brain—we have groups of nerves that func- tion as motion detectors. Objects are also somewhat hardwired through edge de- tectors—objects are configurations of edges. Physical opposition is a basic  USING METAPHORS 123 experience of contact with another object and its resistance to motion. Social trans- actions also stem from basic experiences of gifts and exchanges. Consider for example, the analyses of the following segments (see Table C1), drawn from an earlier pilot study. TABLE C1 Metaphorical Reasoning Transcripta Analysis Use: Computing a Solution Metaphor: Arithmetic Is Motion Along a Path AZ: Minus 40 minus 50 [–40 – 50]; do AZ does not give an immediate answer, and asks for help you add or subtract? choosing between two arithmetic operations IN: How can you figure it out? Interviewer prompts for a strategy AZ: Umm (3) [draws vertical line, labels AZ invokes an new situation, space, and represents numbers as it from “–5” at the bottom,“ –4, –3, –2, locations along a line; AZ interprets “minus 50” as going –1, 0, 1, 2, 3, 4, 5” at the top]. So down, so he begins counting and moving down minus 40, minus 50, so down. 1 [pen bounces from –4 to –5]. Wait [extends line further down and However, AZ recognizes that the line does not include enough writes “–6, –7, –8, –9, –10”] 1, 2, 3, 4, numbered locations for him to compute his answer and adds 5 [pen bounces down from –4 to –9 as more; AZ metaphorically computes the answer with a moving she counts], minus 90 [writes “–90”]. count along the numbered locations OK, let’s see, 10 minus 80 … AZ begins the next computation Use: Detecting and Correcting an Error Metaphor: Arithmetic Is Manipulating Objects EL: Minus 5 plus 8 is minus 3. No, EL initially states a result, but changes her mind. She invokes a wait, the pluses wipes out the minuses. new situation of objects, characterizing each number as Then, there are only pluses left over. It entities (pluses and minuses).The pluses eliminate the should be plus, positive 3. minuses, and by applying the inference metaphorically, the answer must be “plus” or positive 3. EL does not explicitly state the mechanism by which “the pluses wipes out the minuses” (e.g., by pairwise mutual annihilation of one plus for each minus) Use: Justifying a Result Metaphor: Arithmetic Is Social Transactions AZ: Minus 5 plus 8 is 3. AZ gives an immediate answer IN: How do you know that’s right? Interviewer asks for a justification AZ: Because it’s like I owe $5 and then I AZ invokes the situation of a debt; after earning money, he earned $8, so when I pay back the $5, I metaphorically repays the debt from his earnings, thus, the have $3 left. Does that make sense? result is 3 Note. AZ = Latino male seventh-grade student; IN = interviewer (author); EL = Latina female seventh-grade student. aNumber in parentheses indicates the duration of pause in seconds. 124 CHIU The coders took a conservative approach toward coding for metaphors. For the purpose of this article, using the broader problem context to understand the mathe- matical problem is coded as problem situation, not metaphorical. Also, the mathe- matical problem was part of the broader problem situation, so a student need not invoke another situation. I have coded problem situation separately to allow for both types of analysis. Similarly, the instruction sheet explained the problem through gambling, so we coded participants’ use of gambling inferences as prob- lem situation rather than metaphorical because participants may view the gam- bling scenario as part of the problem context. In the same conservative spirit, referencing another situation does not necessar- ily imply metaphorical reasoning. A student may use metaphors for non- mathematical, non-problem-solving reasons. Consider the following example (drawn from an earlier pilot study). AZ: [Computes –4 + –3] Negative 7. Int: Did you say negative 7? AZ: Yeah, because negative 4 plus 3 is negative 7. So, we end up at nega- tive 7. AZ may have understood the problem metaphorically through vertical space to compute the result of negative seven (“we end up at negative 7”). Or, he could have computed the answer first then metaphorically marked this part of the prob- lem-solving discourse as completed (reaching its destination). Additional evidence for the latter interpretation includes both the reference to motion after the result of the computation and the use of the temporal and effect marker so (Halliday & Hasan, 1976). AZ may have reasoned metaphorically, but he did not provide clear evidence of a metaphorical computational strategy. If the computational means was not visible despite interviewer prompts for explanations, the computation was not coded as an instance of metaphorical reasoning.