informing practice research matters for teachers

Developing Symbol Sense for the Minus Sign Lisa L. Lamb, Jessica Pierson Bishop, Randolph A. Philipp, Bonnie P. Schappelle, Ian Whitacre, and Mindy Lewis

Imagine that you have just asked your sense regarding the minus sign. In our students to read this equation: research, we have learned that this limited view of the minus sign inter- -x – 9 = 12 feres with students’ abilities to—

What would they say? If you expect, 1. truly understand the process of “Negative x minus 9 equals 12,” then solving equations, and that statement would be consistent 2. make sense of variables. with how most students would re- spond. However, students could read Three Meanings this equation in a way that conveys a of the Minus Sign different, even deeper, understanding. The minus sign is used in three We highlight three meanings of the common ways. The three problems iminus sign and explore specifc dif- in table 1 use the symbol “–” that we fculties related to the minus sign that refer to, in general, as the minus sign. many students experience. We also However, each problem may elicit a suggest ways that you as a teacher can different meaning for students. support your students in developing The frst meaning is shown in a robust understanding of the minus problem 1, in which the minus sign sign. indicates subtraction, the original use Research has shown that how of the symbol that young children en- students interpret and use the minus counter. In problem 2, the minus sign sign are facets of symbol sense, which is part of the symbolic representation Arcavi (1994) described as “making for a negative number, in this case, friends with symbols” (p. 25). “Mak- “negative 2.” In problem 3, however, ing friends” with symbols includes an the frst minus sign may be viewed as understanding of and feel for symbols the opposite of so that one could read Edited by Debra I. Johanning, debra and how to use them and read them. - -4 as “the opposite of negative 4” [email protected]. Readers are Research indicates that many students rather than students’ more common encouraged to submit manuscripts in middle school and even high school reading of “negative negative 4” (see through http://mtms.msubmit.net. do not have fully developed symbol Bofferding 2010; Vlassis 2008).

Vol. 18, No. 1, August 2012 ● Mathematics Teaching in the Middle School 5 Copyright © 2012 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Table 1 Exploring the three meanings of the minus sign will allow students to equations (Vlassis 2008). For example, differentiate among them. in the equation

Problem Meaning of the Minus Sign 5 – x = 12, 1. 5 – 8 = Subtraction as a binary operation the initial meaning of the minus sign is 2. + 5 = –2 A symbolic representation for a negative subtraction. When one solves for x by number subtracting 5 from both sides, writing 3. Which is larger, – – 4 or – 4? The opposite of, a unary operation -x = 7,

Fig. 1 Three explanations offer justifcation for the equivalence of two expressions. that action necessitates a change in the meaning of the minus sign from 3 – (x + 5) = 3 – x – 5 because— a binary operator (subtraction) to a unary operator (the opposite of ). Stu- • 3 – 1(x + 5) = 3 + -1(x + 5). Subtracting x is equivalent to dents who recognize the minus sign adding - x. as sometimes meaning the opposite of could reason that because the opposite • 3 + -1(x + 5) = 3 + -x + -5. Using the distributive property of of x is 7, then x must be equal to -7. multiplication over addition For students to truly understand the removes the parentheses. process of solving equations, they need to be able to fexibly move • 3 + -x + -5 = 3 - x - 5. Adding - x is equivalent to subtracting x. among the different interpretations of the minus sign and be aware of when and why this sign can change In problem 1, the minus sign func- Treating One Meaning as if It interpretations. tions as a binary operator in that two Were Another inputs are used to produce one output. We often treat one meaning of the Meaning Opposite Of Addition, subtraction, multiplication, symbol as if it were another. For The notion of the minus sign as the and division are examples of binary example, mathematically profcient unary operator opposite of is critical operators. For example, subtraction is students and adults will often rewrite in supporting a sophisticated un- a binary operator because the inputs 3 – (x + 5) as 3 – x – 5 without pause. derstanding of variables, algebraic of 5 and 8 result in one output, -3. They implicitly invoke two mathemat- expressions, and symbolically repre- In problem 3, in contrast, the minus ical principles: sented defnitions. However, research sign is used as a unary operator in that it indicates that middle school and high involves only one input and one output. a. The equivalence of subtraction and school students have diffculty with When one thinks of –(-4) as “the op- addition of the subtrahend’s inverse problems that require them to (at posite of negative 4,” then one is view- b. The distributive property of multi- least implicitly) conceive of the minus ing the frst minus sign as the unary plication over addition (see fg. 1) sign as the opposite of. For example, operator, the opposite of. However, in in our research, we found that only problem 2, some people may view the Those who reason in this way treat the about one-fourth of middle school minus sign in -2 as a unary operator, subtraction symbol as if it were a nega- students who were interviewed cor- not as part of the number but instead as tive sign, although they rarely identify it rectly identifed - -4 as larger than the opposite of 2. One who can view -2 as such. This ability to treat one mean- -4. Further, they did not recognize in both ways can be said to fexibly hold ing of the sign as if it were another is that they had insuffcient informa- both meanings of the minus sign. effcient for making calculations. tion to determine the larger of -x or Although students may initially face x. Having a conception of the minus diffculties because three meanings are Changing the Meaning When sign representing a negative number assigned to the same symbol, having Solving Equations or representing subtraction is not par- the same symbol represent several ideas The meaning of the minus sign can ticularly helpful when users are faced is important. change during the process of solving with these types of problems.

6 Mathematics Teaching in the Middle School ● Vol. 18, No. 1, August 2012 Table 2 These questions highlight the different meanings of the minus sign.

Student Work Teacher Questions A student remarks that “–3 – 5 has “I see one negative [points to –3] and a subtraction sign [points to the two negatives so my answer is –8.” subtraction sign]. Can you explain how you see two negatives?” A student has written “–5” and “–x” After a teacher sees this student’s work, he or she asks: on the board. 1. “What would you call this symbol [points to –5]? Can you place this on the number line?” 2. “What would you call this symbol [points to –x]? Can you place this on the number line?” 3. “Is there another place we could locate –5? Is there another place we could locate –x? In what ways do these two symbols [points to the minus sign in each expression] have the same meaning? In what ways do they have different meanings?” 4. “Could we place the opposite of x [–x] on the positive side of the number line? Why, or why not?”

We view the responses to these the defnition of absolute value) and your classroom and how you might problems as evidence that students appropriately make sense of algebraic leverage those problems to explicitly not only struggle to respond cor- expressions (such as -x). support students’ understanding and rectly but also, by and large, have not awareness of the multiple meanings of engaged with the meaning of the Supporting Minus-Sign the minus sign. A student might state minus sign as opposite of. Students Symbol Sense that “-3 – 5 has two negatives” (see who conceive of the expression -x We believe that supporting students’ the frst row), even though this prob- as the opposite of x may be poised to sense making of the three meanings lem as stated contains one negative understand that -x may represent of the minus sign and the students’ number and subtraction of a positive a positive number. But students ability to identify when each meaning number. Pressing students to explain who lack a strong conception of -x might be appropriately invoked are their strategies will draw out both and opposite of can, understandably, important. Time and attention need interpretations for discussion. struggle. to be paid for students to— Similarly, when -x appears in a When students use the language problem or as part of a solution (see “negative x” to read or say “-x,” they • learn the different meanings of the the second row of table 2), stop and may think that “-x” should represent minus sign; ask students to read the expression a negative number. Negative is used • recognize the appropriate meaning out loud. Many students will likely in the term, so why would one expect in a problem; refer to it as “negative x.” You might that “negative x” could represent a • understand when the meaning positive number? shifts during problem solving; and Although we have found that stu- • fexibly move among the meanings. dents can move between the meanings of the minus sign as subtraction and In the following sections, we share as the sign of the number, students three ideas to help students (1) better often implicitly make these shifts understand the three meanings of the and struggle to explain their think- minus sign and (2) become aware of ing. Further, very few students invoke the shifting meanings. o t the meaning of the symbol as opposite ho p k c of. This limited view of the minus sign Discuss and Make Explicit the o t S i hinders students’ abilities to under- Meanings of the Minus Sign o/ li fo stand expressions related to symboli- Table 2 presents two examples of icta

P cally represented defnitions (such as problems that may arise naturally in

Vol. 18, No. 1, August 2012 ● Mathematics Teaching in the Middle School 7 Table 3 Various examples can help students become aware of language use in relation to the minus sign.

Equation Teacher Actions Points to the minus sign and reminds students that the minus sign means subtraction. 6 – x = 9

Points to the minus sign and states, “The opposite of x is equal to 3.” – x = 3

Points to the minus sign and reminds students that the minus sign can also mean the x = – 3 sign of the number.

Table 4 The meaning of the minus sign as an opposite can be illustrated with several permutations.

Tasks Directions for Each Task a. – (–4) 4 Ask students to circle the larger quantity or write an equal sign if the values are equal. b. – 6 – (–6) Have students write a question mark if too little information is given to determine the larger quantity. After each task, ask students how they determined the larger quantity c. –4 x and which meaning of the minus sign they used and why. d. x + x x e. x –x f. 5 = – Ask, “Is there anything that you can put in the blank to make this equation true?”

then ask students how they would equation-solving process. Table 3 on the value of x, either expression read the expression “­­-5.” We expect provides suggestions for explicitly may be larger. After students have that students will respond, “nega- helping students become aware of reasoned through several problems tive 5.” If students think of and read language use in relation to the minus similar to (a)–(d), pose a task such as -x as “negative x,” we suspect that sign. In 6 – x = 9, the initial meaning (e) to give students opportunities to when asked to place -x and -5 on the of – is subtraction. If one subtracts expand their ideas about the symbol number line, they will place -x on the 6 from both sides, the resulting equa- by combining ideas about possible same side of the number line as -5. tion is –x = 3, necessitating a change values for x with the meaning of the Using the line of questioning sug- in meaning from subtraction to minus sign as the opposite of. gested in the second row of table 2, opposite of. In the fnal answer, x = –3, the-opposite-of interpretation can the minus sign represents the sign of CONCLUSION be discussed in conjunction with the a number. In closing, we believe that students, negative-number interpretation. given opportunities to engage in dis- Comparison Tasks That Support cussions like those suggested in tables Use Opposite of, Subtraction, the Opposite-of Interpretation 2-4, can develop a sophisticated and Negative Language The tasks in table 4 can be used to understanding of the three meanings For students to truly understand support students in making meaning of the minus sign. Explicitly discuss- the process of solving equations, of the minus sign as the opposite of. ing each meaning of the symbol with they need to fexibly move among In discussing problems (a) and (b), a particular emphasis on the opposite the different interpretations of the students can reason about the minus of concept will enable students to minus sign and be aware of when and sign as the opposite of by comparing robustly reason about operations and why they can change. Being aware numerical expressions. By using prob- algebraic expressions. We also believe of the language you use can support lems (c) and (d), students can expand that these experiences will support students’ understanding and help their conceptions so that the possible students in developing symbol sense them recognize that the meaning of values for x include negative numbers. in relation to the minus sign that will the symbol can change during the Students can discuss how, depending foster their future learning as they

8 Mathematics Teaching in the Middle School ● Vol. 18, No. 1, August 2012 move from middle school into high Mathematics.” For the Learning of Lisa L. Lamb, Lisa school and beyond. Mathematics 14:24–35. [email protected]; Bofferding, Laura. 2010. “Addition and Jessica Pierson Bishop, This material is based on work Subtraction with Negatives: Acknowl- [email protected]; supported by the National Science edging the Multiple Meanings of the Randolph A. Philipp, Foundation (NSF) under grant num- Minus Sign.” In Proceedings of the [email protected]; ber DRL-0918780. Any opinions, 32nd Annual Conference of the North Bonnie P. Schappelle, fndings, conclusions, and recommen- American Chapter of the Psychology of [email protected] dations expressed in this material are Mathematics Education, edited by .edu; Ian Whitacre, those of the authors and do not neces- Patricia Brosnan, Diana B. Erchick, [email protected]; sarily refect the views of NSF. The and Lucia Flevares, pp. 703–10. and Mindy Lewis, authors thank Debra Johanning for Columbus, OH: The Ohio State mindydanenhauer@ her thoughtful comments on drafts of University. http://pmena.org/2010/ hotmail.com, work together this article. Vlassis, Joelle. 2008. “The Role of Math- at San Diego State ematical Symbols in the Development University to investigate REFERENCES of Number Conceptualization: The grades K-12 students’ Arcavi, Abraham. 1994. “Symbol Sense: Case of the Minus Sign.” Philosophical conceptions of negative Informal Sense-Making in Formal Psychology 21:555–57. numbers.

(Ed. note. For more on this topic, read “A Proposed Instructional Theory for Integer Addition and Subtraction,” by Michelle Stephan and Didem Akyuz, in the July 2012 issue of the Journal for Research in Mathematics Education.)

NCTM Honors Research Articles Linking research and practice has long been an NCTM strategic directive. NCTM’s Research Committee has now advanced that goal with the Linking Research and Practice Outstanding Publication Award. Based on criteria ranging from timeliness to applicability, the annual award is given to a research-based article in one of the NCTM school journals. The 2011–2012 volume-year recipients are the following:

connecting research Mark (empirical argument) Jared (empirical argument) informing practice Teaching young children to to teaching research matters for teachers mathematics (Stylianou 2011). Their idea to combine tables, considering Using their representation, Gareth Despina a. stylianou and Maria L. Blanton representations become models of that some seats would disappear? How and Ellen determined an initial create equal-size groups is their thinking. many seats would be eliminated? solution to the Party problem. Their your treasure map for building Let us join a sixth-grade classroom teacher listened and examined the Fair Shares, Matey, in which students were beginning to Gareth: But if you said that it elimi- representation that students produced students’ fl exible, connected The Process of Abstracting study algebraic concepts and were nates one, then it’s gonna… as they described their process. The offered a “cognitively demanding task” Ellen: Exactly, ’cause, because if it fgure and their thinking were strongly understanding of and reasoning (Smith and Stein 1998). This task eliminates . . . grounded in the context. The teacher (a) (b) about ratios, fractions, and in Students’ Representations required that students explore and Gareth: No, it actually eliminates two, eventually wanted them to generalize Alex (moving toward a deductive argument) Pat (moving toward a deductive argument) understand the nature of mathematical doesn’t it, because you got one, two, their fndings, so she encouraged them multiplicative operations. Despina A. Stylianou concepts, without being offered repre- three, four, fve, six, and there’s sup- to pursue their investigation through sentations, algorithms, or worked-out posed to be seven, and eight. the use of their representation. “I see sentations typically varies among stu- examples up front. Such tasks natural- Ellen: So you draw one table; you put what you are saying: You eliminate or Walk Developing students’ dents, representation also varies within ly allow for a variety of approaches to each person on each side. two guests. But could this drawing the work of each student. Representa- emerge. The Party problem was used Gareth: Yeah. have more to say? I wonder if there’s tion, then, becomes a personally mean- to start this unit: Ellen: Then, you, you, uh . . . . more here for us to fnd out. We might capacity for constructing ingful activity for each student, central Gareth: You add another table. want to have more tables.” After to his or her problem solving and We plan on having a party to cel- Ellen: Yeah, you add another table, and asking these questions and making construction of meaning. Researchers ebrate the upcoming elections. We then you put each person on each these comments, the teacher left the Proofs through Discourse have highlighted how students’ rep- want to sit with our friends and watch side and . . . students so they could negotiate their theBy P. Holt Wilson, Marrielle (c) (d) resentations evolve from concrete to the results as they are announced Gareth: But, you know, you have to next steps. Plank hether sharing a collection of Curriculum Focal Points (NCTM 2006) pri- Myers, Cyndi Edgington, abstract during mathematical activity on TV. A party rental company will eliminate that because he’s going to toys among friends or a pie oritized the development of rational number (see Gravemeijer et al. 2002). Students bring several small square tables (each be squished. iterative cycles in the and Jere Confrey for dessert, children as young reasoning (RNR) during the elementary school Fig. 1 students’ proofs indicate a variety of approaches. draw from their earlier applications of side fts one person) that we will put Ruby: So you’re saying there’s going Process of rePresenting W as kindergarten age are keen years. on making sure that everyone gets their “fair Although proficiency in RNR is funda- ematics, and we share some of our strategies for stUDent-constrUcteD ProoFs representations by using them as tools together to form one long table. How to be four people at each table, or The representation that Gareth and share.” In the classroom, fair-sharing activities mental for success in higher mathematics, teaching proof and proving. The work reported here was taken from a year- for thinking. For teachers, attention to many people can we invite to our three? Ellen produced was only the frst step call for creating equal-size groups from a col- students have diffi culties in coming to under- t was early in the fall, and we asked students in What is the roLe oF ProoF in long mathematics class that had as one of its the evolution of representation in each party if we rent several such tables? Ellen: Three. Three at each table. in their mathematical activity. Grave-

JAmES PAulS/iSTockPhoTo.com lection of objects or creating equal-size parts stand and use rational numbers (Lamon our mathematics class to explore and prove a MatheMatics teaching anD Learning? What is ProoF? objectives helping students understand and build student’s work opens a window to all Gareth: No, no, look, you eliminate meijer (1999) described four levels of a whole and are generally used by teach- 2007). Teachers are familiar with these strug- iproposition: “The sum of an even number and Proof has been described as the “soul of mathemat- A proof is a “mathematical argument, a connected proofs. The class was taught by one of the two Representation lies at the heart of students developing problem-solving After reading this problem’s statement: two . . . . of mathematical activity: (1) activity ers to support students in formulating ideas gles, like the belief that “multiplication makes an odd number is odd.” ics” (Schoenfeld 2009, p. 12). NCTM (2000) has sequence of assertions for or against a mathematical authors, and all classes were videotaped and sub- doing mathematics. It is the activity processes and mathematical reasoning. within the task, (2) activity that refers of unit fractions. Yet most of us will readily bigger, division makes smaller” or choosing Students’ arguments varied substantially (see recognized the importance of proof by elevating it claim” (Stylianides 2007). As this definition suggests, sequently analyzed to trace students’ development of creating and using mathematical • How would the students in your As the students debated, they created back to work in the task, (3) general acknowledge that as many students progress to multiply or divide on the basis of the num- fig. 1 for some types of arguments that students cre- to one of five Process Standards for thinking math- proofs are built on statements that are accepted by a with respect to proof. symbols, signs, and diagrams. NCTM MatheMatizing and class approach this task? drawings to represent their thinking. activity, and (4) formal activity. The through the elementary school mathematics bers in a problem rather than the problem ated). Mark and Jared used a few examples to show ematically, emphasizing that proof should be part given community—that is, the audience determines Let’s revisit the student-constructed proofs (2000) advocates that students be fu- rePresenting • What types of representations Figure 1a shows this frst drawing. goal is for students to gradually move curriculum toward rational numbers, they context. However, young children are success- that the proposition is true. Alex and Pat attempted of students’ experiences across all grades. More what counts as “proof.” To prove that the sum of whose types are provided in figure 1. Follow- ent users of representations and that Fosnot and Dolk (2002) describe the would you expect them to build? Each table is clearly represented as a from the activity within the task—the have great diffi culties in learning to reason ful at creating equal-size groups or parts of to build logical arguments using properties of odd and recently, the Common Core Standards (2010) have two odd numbers is even, a third-grade class may ing Sowder and Harel (1998), we call the student instruction should support students process of mathematizing by contrast- • How would you support them in square with four guests, one on each counting of guests at the tables—to about fractions, ratios, and multiplicative collections and wholes, an idea that Confrey even numbers; they, too, were convinced that the sum reiterated the importance of justification as a math- use an argument different from an argument used by proofs by Mark and Jared (see figs. 1a and 1b) operations—all of which require reasoning and colleagues refer to as equipartitioning of an even and an odd number is an odd number. ematical practice. a ninth-grade class. Similarly, when constructing a empirical arguments—that is, arguments that use in learning to navigate mathematical ing a “traditional approach to teach- their efforts to use these represen- side. But as tables are moved together, a general activity that allows them to about equal-size groups or parts. As teachers, (Confrey et al. 2009). In contrast to breaking a Before we proceed, think about the students in Despite its central role in mathematical thinking, geometry proof, high school students may use criteria numerical examples as their basis. Although not concepts and problem solving through ing mathematics” with the work of tations as tools to solve this task? some of the seats for the guests are see a growth pattern and generalize it how can we assist students in using their tal- collection or whole into unequal-size groups your class. How would they prove this proposition? however, proof is challenging. Research has docu- different from those that professional mathemati- considered a proof by mathematicians, empirical the use of a variety of representations. mathematicians. Of mathematicians, eliminated. Figure 1a was, in fact, the to other situations. This move is rarely ent for and interest in fair sharing to overcome or parts, equipartitioning describes children’s How would their arguments look? And how would mented that students of all ages face difficulties in con- cians use. Thus, proof is a socially constructed con- arguments are commonly used in high school and Representation is often understood they write that “at the heart of math- The teacher in this class introduced the frst step that Gareth and Ellen took linear. As students move back and the challenges of learning rational numbers? cognitive ability to partition a set of objects or you support them in their efforts to construct proofs? structing rich mathematical arguments. Recent cur- cept whose validity is based on criteria established even in early college (see, e.g., Healy and Hoyles rto be a product—a static picture or set ematics is the process of setting up task and allowed students to work with in mathematizing this situation. forth using their representations, they One of a child’s greatest mathematical a whole into groups or parts of the same size. ricula in middle school and high school have attempted by a community—a community that is accountable [2000]). of symbols. These static representa- relationships mathematically in order their peers. Some attempted to act it Note that the work in this class move toward abstraction. This is a accomplishments is reasoning about rational In our work with children from prekinder- to bring proof to the forefront of instruction by provid- to mathematics. Ultimately, however, we want stu- Mark’s explanation, following his attempt to connecting research to teaching appears in alternate issues of Mathematics tions or products are often used to aid to communicate them to others” out, some used square tiles, and others had been grounded in the belief that dynamic process of gaining and losing numbers. A rich understanding of fractions, garten through the middle grades, we sought ing more opportunities for students to build appropri- dents’ understanding of proofs and proving to evolve write a proof, illustrates the strategies used by Teacher and brings research insights and fndings to the journal’s readers. Manu- instruction or illustrate a mathematical (p. 8). Central to this is the process of started drawing fgures. In all cases, mathematical reasoning can best be information in their representations: ratios, and multiplicative operations (mul- to understand the ways that the children scripts for the department should be submitted via http://mt.msubmit.net. For more ate mathematical arguments. Still, teachers have few toward an understanding held by mathematicians. most students before they develop a formal notion idea. However, representation is also representing. Mathematicians repre- students began their investigation developed when students are given the Students lose contextual detail and tiplication, division, and scaling) requires learned equipartitioning through fair-sharing information on the department and guidelines for submitting a manuscript, please tools at their disposal to help students overcome their If proof is indeed a process of argumentation, of proof: a process—the path that one follows sent problem situations and use these grounded in the context of the party. opportunity to engage in contextual gain abstracted information. gradual development and integration over the activities and to chart these ways as they visit http://www.nctm.org/publications/content.aspx?id=10440#connecting. struggles with constructing proof. of questioning statements and using evidence while developing mathematical un- representations as devices in advancing Two students, Gareth and Ellen, situations that the students themselves Ellen’s group resumed its work on elementary and middle grades. Recognizing developed across the grades. Here, we present What makes proof so difficult? Why does it appropriately, then it is reasonable to look for ways I started with some basic things like 2 or 3, like a Edited by Margaret Kinzel, [email protected] derstanding. Diagrams and symbolism their own understanding and commu- debated the fate of seated party guests mathematize—exploring situations the Party problem and produced a new this, NCTM recommends in its Number and the different strategies, justifi cations, names, remain so elusive for the majority of students to strengthen students’ argumentation skills in real easy number, and then I used like a more …, Boise State University, Boise, ID Operations Standard that “beyond under- and mathematical relationships that students across the grades, particularly in high school? And mathematics. One important approach is to look at like a number that often disproves things, like 0 or Edited by debra i. Johanning, evolve dynamically during problem nicating their thinking to their peers. if the tables were pushed together. mathematically, looking for relations, representation (see fg. 1b). The square standing whole numbers, young children can used as they engaged in fair-sharing activities what can be done to improve students’ ability to how classroom discourse can support students in 1. And then I used ridiculously large numbers and [email protected]. Readers solving, assuming different roles and Similarly, as students explore problem Would the guests be “squished”? and building explanations and conjec- tables were now moved together, Laurie Cavey, [email protected] be encouraged to understand and represent (Confrey et al. 2010). From these observations, Boise State University, Boise, ID prove their mathematical claims? In this article, we engaging in productive conversations for proving going all over the spectrum to all sorts of different are encouraged to submit manuscripts providing insights into this process. contexts, they create representations Would they be asked to move to the tures as they do so (Fosnot and Dolk indicating that the issue of displacing commonly used fractions in context … and we offer an outline of the ways children use discuss proving as it applies to high school math- mathematical claims. things. through http://mtms.msubmit.net. Although the use of various repre- as part of processing and exploring next available spot? Would it be a good 2002; Freudental 1968). guests had been resolved. The guests, to see fractions as part of a unit whole or of a this early understanding to build successively

PIRATE: GLENDA POWERS/ISTOCKPHOTO.COM; TREASURE BORIS YANKOV/ISTOCKPHOTO.COM; X MARKS THE SPOT: ISTOCKPHOTO.COM X MARKS THE SPOT: TREASURE BORIS YANKOV/ISTOCKPHOTO.COM; GLENDA POWERS/ISTOCKPHOTO.COM; PIRATE: collection” (NCTM 2000, p. 33). More recently, more sophisticated ideas of rational number. 140 MatheMatics teacher | Vol. 105, No. 2 • september 2011 Vol. 105, No. 2 • september 2011 | MatheMatics teacher 141 8 MatheMatics teaching in the Middle school ● Vol. 17, No. 1, August 2011 Vol. 17, No. 1, August 2011 ● MatheMatics teaching in the Middle school 9

www.nctm.org teaching children mathematics • April 2012 483 MATHEMATICS TEACHER MATHEMATICS TEACHING TEACHING CHILDREN MATHEMATICS Despina A. Stylianou and Maria IN THE MIDDLE SCHOOL P. Holt Wilson, Marrielle Myers, L. Blanton,“Connecting Research Despina A. Stylianou, “Informing Cyndi Edgington, and Jere Confrey, to Teaching: Developing Students’ Practice: The Process of Abstracting “Fair Shares, Matey, or Walk the Capacity for Constructing Proofs in Students’ Representations” Plank” (April 2012, pp. 482–89) through Discourse” (September (August 2011, pp. 8–12) 2011, pp. 140–45)

For research award criteria, down- NCTM congratulates the recipients of this award, load one of the free apps for your who will be acknowledged at the 2013 NCTM smartphone. Then scan this tag to Research Presession and at the Annual Meeting access http://www.nctm.org/news/ content.aspx?id=31248. and Exposition in Denver.

Vol. 18, No. 1, August 2012 ● Mathematics Teaching in the Middle School 9