Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero

Journal of Mathematical Behavior 26 (2007) 83–95

Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero Esther Levenson ∗ , Pessia Tsamir, Dina Tirosh Tel-Aviv University, Israel Available online 22 June 2007

Abstract This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically based explanations relying on everyday contexts. By using mathematically based explanations that rely solely on mathematical notions, students were able to correctly conclude that zero is an even number. Extending the natural number system in elementary school to include zero can be used as springboard to encourage the use of mathematically based explanations. © 2007 Elsevier Inc. All rights reserved. Keywords: Elementary school; Explanations; Mathematically based and practically based explanations; Parity; Zero

1. Introduction Children learn to count well before they begin formal schooling (Siegler, 1998). As such, their world of numbers consists of the counting numbers or natural numbers. Eventually, this number system is extended to include zero, rational numbers, and integers. As children extend their world of numbers, they must consider how the new numbers differ from previously recognized numbers as well as how these numbers fit into existing conceptual schemas. Students’ tacit models, as well as the more explicit models introduced by their teachers, are most often based on natural numbers. These models often cause complications and need to be rethought when rational numbers are introduced (Fischbein, Deri, Nello, & Marino, 1985; Lubinsky & Fox, 1998; Mack, 1995) as well as when integers are introduced (Linchevksi & Williams, 1999; Schwartz, Kohn, & Resnick, 1993). It is also widely known that students of all ages consistently find the incorporation of zero to cause tremors in their mathematical foundation (Anthony & Walshaw, 2004; Henry, 1969; Pogliani, Randic, & Trinajstic, 1998; Wheeler & Feghali, 1983). Extending the number system is not a trivial matter. Concepts and operations defined in one domain may need to be redefined in an extended domain. Definitions and number properties that exist for the set of natural numbers may not exist for all elements in an extended number system. Yet, even when a certain definition is extended or a property is conserved, it may not be a trivial matter for the student (Ball & Bass, 2000; Borasi, 1992). What stumbling blocks lie in the way of students as they edge forward to an extended system? What are the dilemmas that students raise when exploring these extensions? What is the relationship between different explanations for a concept and the adaptability of that concept to include new numbers? ∗

Corresponding author at: Bar Ilan 50, Ra’annana 43701, Israel. E-mail address: [email protected] (E. Levenson).

0732-3123/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2007.05.004

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Different types of explanations are considered appropriate for students of different ages. Prominent psychological and developmental perspectives (e.g., Piaget & Inhelder, 1969) suggest that in elementary schools practically based (PB) explanations should prevail. These explanations use everyday contexts and/or manipulatives to “give meaning” to mathematical expressions. Yet, the very accessibility of PB explanations may render them less appropriate when students begin to expand their world of numbers (Fischbein, 1987; Wu, 1999). Perhaps, extending the world of numbers may be used as a springboard by which children may be encouraged to appreciate the more formal nature of mathematics. Although elementary school students may be too young for rigorous and formal explanations, they may be ready for mathematically based (MB) explanations that are less formal but nevertheless rely solely on mathematical notions. In this study we investigate the extension of the natural numbers to include zero and the use of MB and PB explanations when integrating parity, the property of being even or odd, into this extended system. We focus on two sixth grade students, both of whom claimed at the outset of this study that zero was neither even nor odd. After considering various MB and PB explanations, both students reached the same conclusion, namely that zero is an even number. Yet the paths they took varied greatly. Specifically, this paper will (1) investigate the difficulties these students exhibited when classifying zero as an even number, (2) describe the students’ preferences for MB and PB explanations when regarding the parity of a natural number, and (3) explore the role of MB and PB explanations in correctly classifying zero as an even number. 2. Related background and research In this section we first review studies related to the expansions of number systems in the elementary school, focusing on the expansion of the natural numbers to include zero. We then discuss the property of parity, its place in the curriculum and some related research. Finally, we review the different types of explanations used by students and teachers in the elementary school mathematics classroom. 2.1. Expanding number systems Several different aspects may be considered when expanding number systems in the elementary school. In this study we focus on two separate but intertwined issues. The first is students’ conceptions of the new numbers and the second is extending known definitions, properties, and operations to the expanded system. Young children are most familiar with the natural numbers encountered in their day-to-day lives. Negative numbers do not appear in the students’ physical world and are therefore more difficult to conceptualize. Recognizing the gap between the positive and negative numbers, many studies examine ways to introduce negative numbers in relation to concrete models or real-life situations (Ball, 1993; De Souza, Mometti, Scavazza, & Baldino, 1995; Linchevski & Williams, 1999; Lytle, 1994). Fischbein (1987), on the other hand, used the negative numbers as an example of when a teacher should forgo the use of any artificial models and treat the subject formally from the beginning. Other studies focused on operating with negative numbers. For example, Schwarz et al. (1993) proposed a computerized environment that would encompass both the sense of negatives as a new sort of quantity as well as extending addition and subtraction to negatives. Sfard (2001) used teaching of negative numbers as an example of when mathematical discourse should be separated from everyday discourse. Introducing rational numbers in the elementary school has a long research history. As with the integers, many studies focused on various ways of representing the rational numbers by relating them to students’ informal knowledge and everyday experiences (Ball, 1993; Freudenthal, 1983; Mack, 1990; Streefland, 1993). Others focused on extending addition and subtraction to rational numbers (Cramer & Henry, 2002) as well as multiplication (Mick & Sinicrope, 1989) and division (Sharp & Adams, 2002). It was found that students’ reasoning with fractions was often based on their reasoning with whole numbers (Fischbein et al., 1985; Mack, 1995; Newstead & Murry, 1998; Ni & Zhou, 2005). This study focuses on the extension of the natural numbers to include zero. Students’ conceptual development of the number zero seems to parallel the historical development of zero (e.g., Blake & Verhille, 1985; Pogliani et al., 1998; Seife, 2000; Wilson, 2001). When interviewing fourth and eighth graders, Reys and Grouws (1975) found that many did not consider zero to be a number. Part of the confusion over zero may have been caused by students equating zero with nothing. For example, if you have eight cookies and you eat all eight of them you are left with nothing. Simply put, eight take away eight leaves nothing. Translating the English sentence into a mathematical sentence, we write: 8 − 8 = 0. The term “nothing” is translated as zero. Other students focus on the notion that zero “doesn’t do anything . . . you can add

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zero, or take away zero, because nothing happens” (Anthony & Walshaw, 2004, p. 40). Similarly, Levenson, Tsamir, and Tirosh (2007) found that first and second grade students claimed that 3 × 0 = 3 because you start with 3 and “do nothing.” Watanabe (2003) reviewed the teaching of multiplication in Japan and found that multiplication by zero is introduced only after third grade when the multiplicands from one through nine have been taught. A possible reason for this is that although students may easily learn to multiply by zero, they might not see this as being a multiplicative situation. Blake and Verhille (1985) agreed that the “zero is nothing” analogy “effectively prevents the teaching of the deep, complex structure of zero” (p. 37). This analogy seems to persist among high school students, albeit mostly among lower achieving students (Tsamir, Sheffer, & Tirosh, 2000). Even pre-service elementary school teachers were found to frequently use “zero” and “nothing” interchangeably or synonymously (Wheeler & Feghali, 1983). In high school, students may be further confused by the concept of a horizontal line that has slope zero (Pogliani et al., 1998). For most students, this means that there is no slope, just as a flat road without hills may be referred to as not having any slope. However, in mathematics, having no slope refers to a vertical line for which slope is undefined. The use of common language creates a superficial structure of zero, which effectively prevents a deeper understanding of the concept of zero (Blake & Verhille, 1985). Students’ difficulties with division by zero have been widely documented (Blake & Verhille, 1985; Reys & Grouws, 1975; Tsamir et al., 2000). As they try to grapple with dividing “nothing” into “something” and “something” into “nothing” they find once again that equating zero with nothing causes much confusion. The latter is allowed but the former is not. Other students claimed that division of a non-zero number by zero results in a number (either zero or the dividend) and that zero divided by zero results in zero. It was also found that prospective and practicing teachers were not always clear on the results of division by zero (Ball, 1990; Wheeler & Feghali, 1983) and even when teachers knew that this division is undefined they could not supply an appropriate explanation (Even & Tirosh, 1995). Moving beyond the natural numbers is not a simple task. Although including zero is one of the first extensions encountered by students, it continues to cause confusion and dilemmas throughout the elementary school years and beyond. This study focuses on extending the property of parity to zero. In the following section we review in general the concept of even and odd numbers, its place in the curriculum and some related research. We then focus on the parity of zero. 2.2. The property of parity The parity of a number is its attribute of being even or odd. Several common definitions of even numbers are (1) numbers that can be expressed as the sum of two equal whole numbers; (2) numbers that can be expressed as a multiple of two; (3) numbers that are divisible by two without a remainder; (4) numbers that give a whole number quotient when divided by two. Definitions for odd numbers are similar. For example, an odd number is a number that leaves a remainder of one when divided by two. In the decimal system, even numbers are recognized by the ones’ digit belonging to the set {0, 2, 4, 6, 8}. In the binary system, even numbers end with a zero. Introducing the concept of even and odd numbers is usually relegated to the early elementary school grades and is often combined with learning other concepts. By the third grade, students in most parts of the United States, Canada, and Great Britain are expected to demonstrate if a number from 1 to 100 is even or odd. In Australia, recognizing even and odd numbers by grouping objects into two rows is considered part of the early development of multiplication and division skills (Board of Studies NSW, 2006). The Principles and Standards for School Mathematics (NCTM, 2000) explicitly mentions even and odd numbers several times, yet within different contexts. In the chapter discussing representations several examples are given of third graders’ exploration of even and odd numbers in the multiplication table. In the chapter discussing reasoning and proof, examples are given of first and third graders arguing and reasoning about even and odd numbers. Within the context of algebra, the Standards encourage the beginning of recursive thinking by having young students explore number patterns such as the even numbers. According to the Israel National Mathematics Curriculum (INMC, 2005) the topic of even and odd numbers is initially introduced in the first grade. At this point the mandatory curriculum suggests using physical items such as beads to introduce the concept. It then suggests incorporating even numbers into the study of addition as numbers that can be written as the sum of two equal whole numbers. In the second grade the INMC specifically mentions that at this point an even number may be defined as a multiple of two. By the end of third grade students should be able to recognize numbers that are divisible by two. From the fourth grade on, the property of parity is mentioned in the curriculum in relation to other topics such as number sense and inquiry based activities.

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Research in the past has used the concept of even and odd numbers to explore general pedagogical issues. Ball and Bass (2000) used the context of even and odd numbers to investigate the construction of public mathematical knowledge in a third grade classroom. They describe a student who proposes to his class that six “can be even and it can be odd.” When another student says to the first, “Prove it,” Ball asks the class to restate the “working definition of an even number.” Restating the definition emphasizes that conjectures and justifications must be grounded in agreed-upon definitions. Zazkis and Campbell’s study (1996) on preservice elementary school teachers’ understanding of divisibility led to Zazkis’ report (1998) on students’ understanding of the parity of whole numbers. Results indicated that many students do not perceive evenness as being equivalent to divisibility by two. Rather, the parity of a number was often perceived as a function of its last digit. Regarding the parity of zero, the INMC does not state anywhere that the parity of zero should be made explicit to students at a certain age. Instead, when setting the guidelines for the first grade curriculum a note to the teacher is added, reminding the teacher that zero is an even number. However, the handbook also adds that it is not necessary to mention this in the first grade unless it comes up in class. The explicit mention of the parity of zero in not mentioned elsewhere in the handbook. In contrast, the NCTM Standards use an example of a first grader discussing the parity of zero to illustrate young children’s ability to reason mathematically when justifying a conjecture. The first grader used an informal proof by contradiction to argue that the number zero is even: “If zero were odd, then zero and one would be two odd numbers in a row. But even and odd numbers alternate. So zero must be even.” (p. 59). In a similar manner, Ball and Bass (2000) gave an example of third grade students raising conjectures regarding the parity of zero and justifying them as well. One student claimed that it must be even because “the ones on each side is odd”. Another student challenges this reasoning, “What two things can you put together to make it?” Both children are basing their arguments on publicly shared mathematical knowledge. In this section we reviewed various explanations given for the concept of parity as well as using the concept of parity to explain conjectures. Explanations play a significant role in the conceptualization of mathematical ideas. In the next section we examine types of explanations and their uses in the classroom. 2.3. Explanations In the literature we find various ways of classifying explanations used in mathematics classrooms. In this section we briefly review different types of explanations including MB and PB explanations. For a more extensive review see Levenson, Tirosh, and Tsamir (2006). Several studies (e.g., Ginsburg & Seo, 1999; Hiebert & Carpenter, 1992; Mack, 1995; Raman, 2002) differentiate between informal and formal explanations. Informal explanations are based on the informal knowledge that students bring to the classroom from their real-world experiences. Formal explanations consist of more symbolic representations. Tsamir and Sheffer (2000), in their study of high school students’ explanations for division with zero, differentiate between concrete and formal explanations. While a concrete explanation uses “real world experience to give meaning to the mathematical expression, a formal explanation uses only mathematical definitions and theorems” (p. 93). Other studies (Bowers & Doerr, 2001; Cobb, McLain, & Gravemeijer, 2003; Thompson, Philipp, Thompson, & Boyd, 1994) differentiate between calculational explanations and conceptual explanations. Calculational explanations describe a process, procedure, or the steps taken to solve a problem. Conceptual explanations describe the reasons for the steps, which link procedures to the conceptual knowledge of the student. Basing explanations on real-life contexts is another category much discussed in the literature (see for example Koirala, 1999; Nyabanyaba, 1999; Streefland, 1991; Van den Heuvel-Panhuizen, 2003). In this study we focus on MB and PB explanations. MB explanations are based on mathematical definitions or previously studied mathematical properties, and often use mathematical reasoning. However, they are not necessarily rigorous and complete formal explanations. Formal explanations are usually used at the high school and undergraduate level. MB explanations may be more appropriate for elementary school students than formal explanations. PB explanations include any explanation that does not rely solely on mathematical notions. They include those explanations that use visual aids or manipulatives, explanations based on real-life contexts, and informal explanations. The research and recommendations concerning PB explanations are varied. Wu (1999), in a discussion of using visual aids when teaching division of fractions claims that students need also to understand problems that cannot be visualized. Wu is not against the use of visual aids but claims that teachers need to know the limitations of these aids. He suggests increasing the use of abstraction in fifth through seventh grades.

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Much debate has risen around PB explanations that use real-life contexts. Mack (1990) showed that it is possible to use children’s informal knowledge of fractions to build meaningful formal symbols and procedures. Others are more skeptical. Koirala (1999) and Nyabanyaba (1999) argued that using everyday contexts to teach mathematics does not necessarily enhance the understanding of academic mathematics. In discussing the coordination of informal and formal aspects of mathematics, Raman (2002) found that many high school and university students do not coordinate the formal definitions with informal characterizations. As a result, some students suppress helpful intuitions while others are so convinced by the informal explanations that they may not recognize the need for formalism. Raman concludes that both aspects are needed and useful. Many researchers consider it necessary that the development of formal mathematics among students be an outgrowth of their PB experiences and activities (Bonotto, 2005; Freudenthal, 1983; Linchevski & Williams, 1999). Major theorists (Piaget, 1952; Bruner, 1966; Skemp, 1987) support this view and claim that the interaction with physical activities allows children to form abstract concepts. It is therefore not surprising that so few studies specifically investigated the use of MB explanations in elementary school classes. When studying their own classes, Ball and Bass (2000) and Lampert (1990) described examples of young children using MB explanations. Levenson, Tirosh, and Tsamir (2004) also showed that many elementary school students use MB explanations when solving multiplication tasks. In their study of the sociomathematical norms related to MB and PB explanations, Levenson et al. (2006) found that fifth grade students are capable of understanding MB explanations and some might even prefer them. They also found that although a teacher might personally prefer MB explanations, this preference might be set aside for didactical considerations. Levenson et al. (2007) examined the use of MB and PB explanations among first and second graders solving multiplication tasks, a context not yet introduced formally in school. This study explores the use of MB and PB explanations among fifth and sixth graders within a familiar context, that of even numbers. It focuses on the use of MB and PB explanations when extending a known concept to include zero. 3. Method 3.1. Subjects Two sixth grade students are the subjects of this study. Both students learned in the same class in an elementary school located in a middle socio-economic neighborhood. Both students have been learning with the same mathematics teacher since the beginning of fifth grade and were considered by their mathematics teacher to be of high ability. This study took place in the fall term of the sixth grade. 3.2. Tools Questionnaires were handed out to the students in which they were asked to identify several numbers as being even or odd and to explain their reasoning. Students’ responses to the parity of fourteen and zero were then used as a springboard for the interviews which followed the questionnaires. A semi-structured interview was used that began with the researcher asking the student to evaluate the parity of fourteen and explain the reasoning behind the evaluation. Although students were asked a similar question on the questionnaires, interviews allow for a more in-depth and elaborate discussion of the issues at hand. A semi-structured interview also leaves room for additional questions as well as clarifications. Each student was then presented, one at a time, with five different explanations for why fourteen is an even number (see Figs. 1 and 2). The order of presentation was the same for each subject and was as follows: PB1 (pairs of students), MB1 (divisibility by 2), PB2 (motorcycle story), PB3 (marbles), and MB2 (sum of two equal whole numbers). The discrepancy between the number of MB and PB explanations was caused by the need to differentiate between PB explanations presented with accompanying pictures and those presented without accompanying pictures. As previously discussed, several equivalent definitions exist for even numbers. The explanations used in this study were based on the following definitions: (1) An even number is divisible by 2 without a remainder. (2) An even number is the sum of two equal whole numbers. Students were requested to evaluate each explanation by stating (1) if the explanation was valid; (2) if the explanation could by useful when explaining the parity of fourteen to a fellow student and (3) which of the explanations were convincing.

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Fig. 1. Mathematically based explanations for why 14 is an even number.

Students were then asked to evaluate the parity of zero and explain the reasoning behind their evaluation. They were then asked if they could use the explanations presented to them for the parity of fourteen to help them evaluate and clarify their understanding as to the parity of zero. All interviews were audio taped and fully transcribed. Interviews lasted between 60 and 80 minutes.

Fig. 2. Practically based explanations for why 14 is an even number.

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4. Results In this section we review the results of each student individually as they clarify for themselves the parity of zero. We begin with the student’s initial responses on the questionnaires as to the parity of fourteen and zero. We then review their evaluations of the explanations shown to them during the interview for the parity of fourteen. We conclude with each student’s realization that zero is an even number. Throughout this section the problems, dilemmas, and resolutions of each student are highlighted. 4.1. Johnny On his questionnaire, Johnny wrote “fourteen is an even number because it’s divisible by two.” Regarding the parity of zero he wrote, “zero is neither an even nor an odd number. The number zero is not divisible by anything so it can’t be even.” There are two interesting points to be noted here. First, Johnny is consistent. Even numbers are divisible by two. He uses this definition to explain both why fourteen is an even number and why zero is not. In other words, Johnny is trying to incorporate zero into his previous knowledge of numbers, treating zero as he would a natural number. Second, Johnny’s first stumbling block is the well-documented common error claiming that zero is not divisible by anything. From the questionnaire alone we do not know where Johnny’s error stems from. During the interview Johnny was presented with the five explanations shown in Figs. 1 and 2 for the parity of fourteen. In general, he considered all of the explanations to be valid and convincing. However, Johnny claimed that MB1 (divisibility by 2) was the most convincing because “it shows that it’s divisible by two and also there is no remainder.” When asked if he would use PB1 (pairs of students) to explain to a friend in class why fourteen is an even number he answered, “No because most [students] are smart enough to understand mathematical language. I would tell them using mathematical language because they are smart.” Instead he would use MB1 (divisibility by 2) for all of the students claiming, “Some children would just understand faster and others slower.” In general, Johnny has a strong preference for MB explanations and believes that all students are capable of understanding them. Returning to the parity of zero, Johnny restates that zero is not divisible by anything. At this point, the interviewer, along with Johnny, goes through a series of division exercises with non-zero numbers checking each one by multiplication. For example, 10:5 = 2 because 2 × 5 = 10. At the end, Johnny agrees that zero divided by five equals zero and writes this down (see Fig. 3). Although at this point we are tempted to conclude that Johnny’s problem is solved, the discussion which followed the above exercises showed that this was not the case. I: J: I: J: I: J: I:

You said that zero divided by five is . . . Zero divided by five is zero. So . . . but like it can’t even divide by five because it has nothing to divide. How do I check it? Zero times five. It equals zero. But, you don’t really have anything to divide from. I’m dividing from zero. (quiet) So, you just don’t divide anything. You’re the one that told me that you like mathematical language. So, if we look at it just from a mathematical point of view, then zero divided by five is zero because when I check it and multiply it we get zero.

Fig. 3. Johnny’s division exercises.

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J: I: J: I: J: I: J: I: J:

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Yeah. You can divide [0] by two, but you don’t get any answers. I get an answer. What’s the answer that I get? Zero. Is zero an answer? Yeah. So, what would zero divided by fifteen equal to? Zero. So, now I want to check if zero is an even number or not. How would I check to see if it’s an even number? If you do zero plus zero it equals zero so it could be an even number

There are two interesting points to note in this discussion. The first is that by the end of the above discussion, Johnny can write division sentences with zero. We might have thought from his questionnaire that he was merely confused between zero divided by two and two divided by zero but this does not seem to be the underlying dilemma. Instead, his confusion appears to be related more to his conception of the number zero and possibly his conception of division. For him, in order to divide there must be “something” to divide and with zero there is “nothing to divide.” That he can write a division sentence using zero is not enough and he finds it difficult to accept zero as an answer. The second point of interest is that although Johnny agrees, albeit reluctantly so, that zero divided by two is indeed zero, he still does not use this to explain why zero is an even number. Instead, rather surprisingly, he refers to an altogether different explanation presented to him during the interview, that of even numbers being the sum of two equal whole numbers (MB2). In other words, Johnny seeks out a different MB explanation in order to justify to himself why zero is indeed an even number. 4.2. Miri On her questionnaire, Miri stated that “Fourteen is an even number because it’s a multiple of 2. Fourteen is divisible by 2. 7 × 2 = 14 and there is no remainder at the end.” Regarding the parity of zero she wrote, “Zero is neither even nor odd because zero is the smallest number so it can’t be divided or multiplied (by 2).” First, it is interesting to note that Miri gives two different explanations for the parity of fourteen. Second, like Johnny, she attempts to use her working definition of even numbers to assess the parity of zero. Although Miri claims, as Johnny did, that zero cannot be divided, she adds that it also cannot be multiplied. Her reasoning is that zero is the smallest number. In other words, for Miri, zero is not some mysterious number that behaves differently because of unexplained reasons. Rather, zero is the smallest number (according to Miri) and therefore, dividing it is problematic. At this point, although it is clear that Miri has erred regarding zero being the smallest number, it seems likely also that Miri’s conceptions regarding division and multiplication may also be problematic. When considering the explanations shown to her during the interview, the only explanation that she does not consider to be valid and useful is MB2 (sum of two whole numbers). This is because she finds it difficult to understanding this explanation exclaiming, “When I read the other ones, I immediately understood how they were trying to explain it. And this one I didn’t really get it.” After the interviewer reviews this explanation again and it is understood by Miri, she agrees that it is valid but still would not use it to explain to a friend in class why fourteen is an even number. On the other hand, she is very enthusiastic regarding the PB explanations: Explanation one (PB1 – pairs of students) is very good. It explains all the details. The kids are on a trip and the kids can be divided equally and there’s no remainder and that is why it’s an even number. . . I really got that one and I thought it was really convincing . . . I also think that this one (PB3 – marbles) is a good one. Unlike Johnny, Miri clearly has a preference for PB explanations, especially those with illustrations. Returning to the parity of zero, Miri claims that she wasn’t sure about her answer. Referring to what she wrote on her questionnaire regarding multiplying by zero, she now clarifies M: Like, you can’t . . . you can do zero times two but. . . I wasn’t exactly sure. Or I thought that [it is neither even nor odd] or I thought that it is an even number. I: OK. So, it’s either an even number or it isn’t an even number. Could it be an odd number? M: No. I don’t think it would be.

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I: You don’t think it would be. M: Because it’s the lowest number you can’t really do zero times two is zero. I: But, you just did it. Didn’t you just tell me that zero times two is equal to zero? You say you can’t do it but what is zero times two? M: Zero. But like also, even numbers are made up from two. I: Oh. What does that mean, made up from two? M: Like six is built from two. Like two and two and two. That’s why it could be divided by two. From her questionnaire, it seemed that Miri had a problem operating with zero. That is, her problem was with multiplying and dividing by zero. However, from the interview we learn something new. Her conception of even numbers is that of a number being “made up” of twos or being “built” from twos. In other words, her dilemma with the parity of zero is related more to her conception of even numbers than with her conception of the number zero. Being “made up of twos” requires there to be at least one “two” and zero doesn’t have any “twos.” The term “built from twos” is somewhat ambiguous. Miri does not explicitly mention that she is adding twos or multiplying twos. In fact, she had mentioned this concept previously when discussing PB1 (pairs of students), “If I think an even number is made up of twos and when you divide it you won’t have a remainder and all that, then that’s why this (pointing to PB1) is a good explanation. Because I see that here there isn’t a remainder.” It could be that PB1, with its illustration of pairs of students, reinforces Miri’s conception of even numbers being “built from twos,” which in turn corresponds to her preference for this explanation. When reflecting upon Miri’s conception of even numbers it becomes apparent that her definition of the term “multiple” is also questionable. On the one hand, she wrote on her questionnaire that 14 is a multiple of two. She also wrote that 7 × 2 = 14. This led us to believe that Miri’s definition of the term “a is a multiple of b” was synonymous with being able to express a by writing the mathematical sentence “a = b × n, n ∈ N.” However, although Miri agrees that zero times two equals zero, she does not equate this with zero being a multiple of two. Apparently, her concept image of the term “multiple” includes an image of “twos” and not just the ability to write a multiplication sentence. After restating her problem with the parity of zero, Miri is invited by the interviewer to review each explanation for the parity of 14 and try to use these explanations when clarifying the parity of zero. The following discussion ensues after reading PB1 (pairs of students) while substituting the number 0 for 14: I: What do you think? Does this help explain why zero might be an even number? M: Yeah. Cause it’s showing that there’s no remainder. Like, kind of. It’s showing that there’s no remainder but it’s not exactly showing that it’s built up from twos. Cause there’s zero students at the end. I: So, what does this make you think? Does this make you think that zero is an even number or that zero isn’t an even number? M: It would make me think that it’s not even and not odd. At this point, Miri has a dilemma. On the one hand she argues that since there is no remainder visible, zero could be an even number. On the other hand, Miri cannot see how zero is built from twos. The interviewer then turns to the second explanation (MB1 – divisibility by 2): I: Ok. Let’s look at the next explanation (MB1 – divisibility by 2). You didn’t like it as much as explanation one (PB1) but you said that it was a good explanation and it’s correct. So, let’s see if we could put zero here. You read. M: Zero is divisible by two without a remainder. I: So, what would you write here (meaning a mathematical sentence)? M: Zero divided by two equals zero without a remainder. I: So, what does that make you think? M: It would make me feel more like it’s even. Because it more makes you feel that it could be divided by two. Like . . . this one (PB1) makes you feel that there won’t be a remainder but that one (MB1) also makes you feel both because it also shows that there won’t be remainder and it also shows that it could be divided by two. Now I think that zero is even. This one (MB1) would be a good one to show zero.

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The PB explanation only partially convinces Miri that zero might be an even number, leaving her unsure as to the parity of zero. In fact, when reviewing the other PB explanations, Miri claims that all of them leave her undecided about the parity of zero. When referring to MB2 (sum of two equal whole numbers), Miri claims, “it’s showing that zero plus zero equals zero and here it doesn’t have halves or anything.” For Miri, the MB explanations convince her that zero is indeed an even number. 5. Discussion This paper investigates the problems and dilemmas faced by students when extending the property of parity to include zero. It also examines the types of explanations, MB and PB, which students use to resolve these problems. In this section we discuss these findings as well as the educational implications. 5.1. Dilemmas and resolutions When extending the number system we take into account both students’ conceptions of the new numbers as well as extending known operations and properties to the new numbers. We begin with the students’ conceptions of zero. Both Johnny and Miri classified zero as being neither even nor odd. This classification points to the students’ willingness to set zero apart from the natural numbers. This is not necessarily a bad thing. Zero does behave differently from the natural numbers in many mathematical situations and students are wise to proceed cautiously. However, there is a difference between proceeding cautiously and assigning zero to a totally different plateau. Johnny’s first problem with evaluating the parity of zero was that he related zero to nothing. This relationship complicated his conception of division, which in turn caused part of his confusion regarding the parity of zero. On the other hand, Miri did not apparently have a specific problem with zero. In fact, on her questionnaire and during her interview, Miri considered zero very carefully as to how it could be incorporated into the various schemas she had acquired previously. When extending operations and properties to the new numbers we consider students’ conceptions of operations and properties within the more familiar system that is being extended. Both students classified fourteen as an even number and both students used MB explanations to explain their reasoning. However, during his interview, it became apparent that Johnny’s conception of division included the image of “something” being divided. Miri’s conception of even numbers included being “built from twos.” These images caused a dilemma for both Johnny and Miri when extending the property of parity to zero. Johnny agreed that zero is divisible by two and knew that if zero was divisible by two it must be an even number. His dilemma was how to reconcile zero being divisible by two with the concept images of both zero and division that were so ingrained within him. Miri too had a dilemma. She agreed that if zero times two equals zero, then zero must be an even number. On the other hand, she could not see that zero is built from twos. She was, furthermore, drawn by the image of pairs of students in PB1, reinforcing her concept image of even numbers. Both students eventually resolved their dilemmas. Johnny, who preferred MB explanations, naturally turned to a different MB explanation to explain why zero is indeed an even number. He did not consider using any of the PB explanations for the parity of fourteen and did not turn towards these explanations when clarifying the parity of zero. Miri, on the other hand, started off by using MB explanations but preferred the PB explanations that were presented to her during the interview. If she had relied solely on the PB explanations to help her clarify the parity of zero, she would have concluded incorrectly that zero is neither even nor odd. Miri resolved her dilemma by accepting the fact that zero is divisible by two and must therefore be an even number. After struggling with her concept image of even numbers and setting aside her preference for the PB explanations, Miri ultimately was convinced by the MB explanations that zero is an even number. 5.2. Educational implications The interplay between concept definition and concept image is part of the process of concept formation (Vinner, 1991). Although both students used formal concept definitions when explaining the parity of fourteen, their concept images were revealed when discussing the parity of zero. These images are most often formed by models based on the natural number system. Fischbein (1987) claimed that the intuitive interpretations created by the concrete instructional materials and models used during the elementary school years often become rigid and it may be difficult at a later stage

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to undo this rigidity. Although a certain model might be very useful initially because of its concreteness, the primacy effect of that first model may make it impossible later on for the child to move on to more general and more abstract interpretations of the same concept. The Israel National Mathematical Curriculum (INMC, 2005) suggests that in the first grade, physical items should be used to introduce the “intuitive definition of even numbers: when arranging pairs of items no item is left without a partner” (p. 4). Miri, currently in the sixth grade, still retained this concept image of even numbers. Presenting her with PB1 (pairs of students) reinforced this image. Johnny also retained conceptions based on concrete models, such as his notion that division requires dividing concrete items. He also retained the intuitive notion of zero being related to nothing. However, Johnny did not receive any reinforcement for these conceptions since he did not give a second glance to the PB explanations presented to him. At the beginning of this paper we raised the issue of the relationship between different explanations for a concept and the adaptability of that concept to include new numbers. For Miri, the PB explanations hindered her acceptance of zero as an even number. On the other hand, MB explanations were used by both students to correctly classify zero as an even number. Indeed, when presented with MB explanations, Miri was able to set aside her intuitive conception of even numbers and correctly classify zero as an even number. One may regard extending the number system as an end in itself. Students should become familiar with negative numbers, rational numbers, as well as zero. They should also learn to operate within these systems. However, extending the number system may also be used as a means to a different end. Fischbein (1987) advocates introducing activities that help the child assimilate concepts of higher complexity and abstraction during the concrete operational stage. “One has to start, as early as possible, preparing the child for understanding the formal meaning and the formal content of the concepts taught” (Fischbein, 1987, p. 208). Expanding the set of even numbers to include zero is just such an activity. As students grapple with the parity of zero, they begin to feel the necessity for relying on a systemic structure as opposed to concrete examples. Vinner (1991) indicates that in order to encourage students to use mathematical definitions, “one should point at the conflicts between the concept image and the formal definition” (p. 80). We believe that this may be done at the elementary school level by pointing out the conflicts and inconsistencies that come up when extending known properties to include zero and by encouraging the use of MB explanations when extending those properties. According to Freudenthal (1983), the process of extending the number system should be made explicit in the classroom. However one proceeds in extending the number concept, it is a necessity that the fact and the mental process of extending are made conscious, that it is made conscious why one has the extension take place in this, and in no other way, and that as a background of the compulsory, the arbitrary in the definitions becomes clear. (Freudenthal, 1983, p. 460) Extending the number system is not a trivial matter. Nor should it be trivialized in the classroom. Instead, by discussing with students issues such as the parity of zero, we bring to light many of the conceptions students harbor that otherwise go unrecognized. A simple statement such as “zero is neither even nor odd” can also be used as a springboard by which children may be encouraged to appreciate the more formal nature of mathematics. We have shown that MB explanations can be very useful when extending the number system. Knowing that the move to more formal mathematics may be difficult, we should take advantage of such opportunities to introduce more MB explanations to elementary school students.

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