# First and second graders use of mathematically-based and practically-based explanations for multiplication with zero.

Abstract

In this study, we use the context of multiplication by zero, to reexamine the premise that elementary school students prefer practically-based explanations that rely on tangible items or real life stories to that of mathematically-based explanations that rely solely on mathematical notions. Two major aims of this investigation were: (1) to explore students' preconceptions of multiplication with zero, and (2) to investigate the types explanations, mathematically-based and/or practically-based, that students use. Thirty-one first and second grade students were interviewed and asked to solve given multiplication tasks that included multiplication without zero and with zero. Results show that (1) not all students, who correctly solved 3x2 before being introduced to multiplication in class, knew that multiplication with zero always results in zero, and (2) most of the explanations given by students were mathematically-based explanations.

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"Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well" (NCTM, 2000). Understanding what students know includes knowing the preconceptions and misconceptions students of different age levels and backgrounds bring to the mathematical classroom (Shulman, 1986). A teacher must also understand the reasoning behind these conceptions and anticipate common mistakes (Even & Tirosh, 1995). However, according to the NCTM, teachers must be aware of what students need to learn in the future. Long-term goals, such as helping our students make the transition to the formal use of definitions in mathematics, must be kept in mind. This paper does both; it looks at the preconceptions that students bring to the classroom with an eye towards what they need to know in the future.

In this study we explore students' explanations of multiplication with zero before they are introduced to multiplication in school. Although explanations have been classified in many different ways throughout the years, this study investigates mathematically-based (MB) explanations and practically-based (PB) explanations. MB explanations employ only mathematical notions. PB explanations use daily contexts and/or manipulatives to "give meaning" to mathematical expressions (Koren, 2004). As children extend their world of numbers to include zero, they must consider how this new number is different from previously recognized numbers as well as how known operations must be extended to include the new number. This study not only allows us to investigate students' preconceptions of multiplication with zero, but also allows us to investigate students' conceptions of the number zero as well as the types of explanations, PB and/or MB explanations, they use to incorporate a new concept into their mathematical world. By focusing on the types of explanations students' use we may begin to explore the types of explanations appropriate for elementary school students with an eye towards the types of explanations we would like to encourage in the future.

Background

Two main issues are at the heart of this investigation: expanding the natural number system to include zero and the types of explanations students use to accommodate this expansion. In this section we review studies related to these issues.

Students' conceptions of zero

One of the first mathematical ideas children are exposed to is counting. As such, children's first experiences with the world of numbers are with natural numbers. This number system is eventually extended to include the number zero. In this section we first discuss children's development of number concepts, specifically the concept of zero. We then discuss the educational implications of zero's behavior under the basic operations of mathematics.

It is commonly believed that for students to understand a number concept they need to understand two basic properties of numbers: cardinality and ordinality (Siegler, 1998). A cardinal number refers to the number of elements in a set. In the case of zero, zero represents the number of elements in the empty set. An ordinal number refers to the position of an element within a set. The status of zero as an ordinal number may be cause for confusion among children. If the number three represents the third tallest child in the class, then what does zero represent? Although the ground floor of a building is clearly the "zeroth" floor, it is not commonly denoted as such (Pogliani, Randic, & Trinajstic, 1998).

Piaget (1952) hypothesized that the construction of number goes hand in hand with the development of logic and that classification serves as a basis for mathematical concepts. Children classify objects based on physical properties or certain patterns of behavior. By the time they reach the age of 8-9 years old they have developed an understanding of the inclusion relation as well as the concept of the singular class. However, the empty or null class still poses a problem. According to Inhelder and Piaget (1969), the null set poses special difficulty "because a class without any elements is incompatible with the logic of concrete operations, i.e., operations in which form is inseparably bound up with content" (p. 149).

Many studies have been conducted regarding students' conceptions of zero. Reys and Grouws (1975) interviewed fourth and eighth graders and found that many did not consider zero to be a number. This seems to parallel the historical development of zero (e.g., Blake & Verhille, 1985; Pogliani et al, 1998; Seife, 2000; Wilson, 2001). Reys and Grouws found that part of the confusion over zero may have been caused by students equating zero with nothing. 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).

The analogy of "zero as nothing" finds its way into many mathematical operations. Zero is the additive identity for the set of whole numbers. This may be explained by the following: if you have any number of items, and add nothing, then you remain with the same amount of items you had to begin with. When explaining subtraction: If you have eight cookies and you eat all eight of them you are left with nothing. This can be written: 8 - 8 = 0. As stated above, students of all ages find this analogy very satisfying and use it for several years. It is easy to learn and retrieve and seems to work--until division. As students try to grapple with dividing "nothing" into "something" and "something" into nothing: they find that this analogy causes much confusion. The first is allowed but the latter is not. In high school, students may be 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 having no hills may be referred to as not having any slope. However, in mathematics, having no slope refers to a vertical line, a line for which no slope exists. 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, Sheffer, & Tirosh, 2000). Many 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 teachers and 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).

In light of these studies, it is not surprising that much has been written regarding pedagogical and instructional approaches to explaining the concept and use of zero in different mathematical contexts. Pogliani et al. (1998) suggest the following example to illustrate the distinction between 'empty', 'nothing', and 'zero': "When zero, as an element is added to an empty set {}, the set {0} is no longer empty; it has an element, which is zero" (p. 730). In the elementary school, Leeb-Lundberg (1997) advocates relating zero to a place of origin, such as on the number line.

Regarding multiplication and zero, most instructional suggestions relate to the mathematical statement "if the product of two numbers is zero, then one of the factors must be zero" (Allinger, 1980; Forringer, 1994; Newman, 1967). One study (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.

Regarding division by zero, some have suggested that teachers must first emphasize the relationship between division and multiplication as inverse operations and then use this relationship when explaining division by zero (Henry, 1969; Reys & Grouws, 1975; Sundar, 1990). Others offer explanations that arise from the common models used to teach whole number division (Knifong & Burton, 1980). Tsamir & Sheffer (2000) offer two possible recommendations for teaching division by zero: using concrete explanations in the elementary school, or, alternatively, postponing the introduction of division by zero until secondary school. Newman (1967) argued that explaining the behavior of zero under the basic operations of mathematics offers the elementary school teacher an opportunity to introduce her students to the importance of definitions in mathematics and the logical use of these definitions in discovering new mathematical properties.

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.

Explanations

The Standards for School Mathematics (NCTM, 2000) attribute different types of explanations to different age students. According to the Standards, "young children will express their conjectures and describe their thinking in their own words and often explore them using concrete materials and examples" (p. 56). However, by the "middle and high grades, explanations should become more mathematically rigorous" (p. 61). According to the Standards, students in middle school and high school should understand the role of mathematical definitions and use them, as well as previously learned mathematical properties, in their explanations.

The NCTM Standards initimate that there are basically two different categories or levels of explanations corresponding to younger and older students. This is somewhat similar to Piagetian theory, which differentiates between the concrete operational stage and the formal operational stage. Children in the concrete operational stage demonstrate their intelligence through logical and systematic manipulation of symbols related to concrete objects. In this stage, "the logical organization of judgments and arguments is inseparable from their content" (Inhelder & Piaget, 1969, p. 132). Children in the formal operational stage demonstrate their intelligence through logical use of symbols related to abstract concepts. There is a disconnection of thought from objects that "liberates relations and classifications from their concrete or intuitive ties" (Inhelder & Piaget, 1969, p. 132). Elementary and early adolescents may be said to be in the concrete operational stage. Adolescents and adults may be said to be in the formal operational stage.

This study focuses on mathematically-based (MB) explanations and practically-based (PB) explanations. MB explanations employ only mathematical notions. PB explanations use daily contexts and/or manipulatives to "give meaning" to mathematical expressions (Koren, 2004). This classification distinguishes between explanations that are based solely on mathematical notions but are not necessarily rigorous, and complete, formal explanations. Formal explanations are usually referred to at the high school and undergraduate level. The term PB explanation was coined to include any explanation that does not rely solely on mathematical notions. According to Piagetian theory it would seem that PB explanations are appropriate for younger children in the concrete operational stage and MB explanations are more appropriate for adolescents. Yet, Ball and Bass (2000) and Lampert (1990) describe classrooms where third and fourth graders use MB explanations as well as PB explanations. The NCTM Standards (2000) also provide examples of third grade students using MB explanations.

Much research has been done relating to the use of PB explanations in the elementary school mathematics classroom. Mack (1990) showed that it is possible to use children's informal knowledge of fractions, based on their real-life experiences, to building meaningful formal symbols and procedures. However, many studies (e.g., Koirala, 1999; Nyabanyaba, 1999; Szendrei, 1996; Wu, 1999) found that each type of PB explanation has its own set of pitfalls that need to be avoided or remedied by the teacher.

Fischbein (1987) claimed that until the age of 11-12 the child is in the concrete operational period and one cannot force upon the child concepts that he is not intellectually mature enough to understand. Yet the intuitive interpretations created by the concrete instructional materials and models used during this period often become rigid and it may be difficult at a later stage 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. Therefore, Fischbein 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" (p. 208). In other words, according to Fischbein, MB explanations should be used alongside PB explanations in the elementary school.

Is it possible to introduce elementary school students to formal mathematics if they are so reliant on concrete examples? Perhaps elementary school students are too young for rigorous explanations but not too young for explanations that are less formal but nevertheless rely solely on mathematical notions. This study focuses on the types of explanations, MB and PB, which students use for a mathematical concept before they are introduced to this concept in class. By focusing on the types of explanations used we reexamine the premise that elementary school students need explanations that rely on tangible items or real life stories and investigate the possibility of introducing explanations that rely solely on mathematical notions in these grades.

Method

First and second graders from three different elementary schools located in middle-class neighborhoods were asked to solve 3x2. Only students who could correctly solve this task were asked to participate in the study. As a result, ten first graders (six girls and four boys) and twenty-one second graders (nine girls and twelve boys) participated in this investigation. None of the students had been introduced to multiplication in school. Being that the first and second graders had only recently learned to read and write in school, we suspected that they may have trouble expressing their ideas to the fullest in writing. It was therefore decided to interview all subjects. Each student was interviewed individually allowing for ample time to think and respond to the questions. Each interview was audio taped.

During the interview, students were asked to solve and give their own explanations for multiplication problems without and with zero. Specifically, the following multiplication examples were given to each student:

3x2 =

2x3 =

3x0 =

0x3 =

0x0 =

The first two problems sought to establish how multiplication without zero was solved and explained and if the subject used the commutative property of multiplication as an explanation. The second two problems allowed us to investigate how subjects solved and explained multiplication of a non-zero number by zero. Specifically, the second two questions investigated if the types of explanations used by the subject for multiplications by non-zero numbers differed from the explanations used for multiplication with zero. Furthermore, these two questions allowed us to investigate if the subject differentiated between 3x0 and 0x3. The problem 3x0 fits well into the definition of multiplication as repeated addition when the multiplier is a positive integer and indicates the number of times 0 is to be added to itself. However, when the multiplier is non-positive, as in the case of 0x3, difficulties may arise. Therefore, it was of particular interest to investigate how subjects would explain this problem and how prevalent the use of the commutative property would be in this case. The last problem allowed us to investigate multiplication that did not involve any non-zero numbers and sought to examine if the explanations given would differ from those already offered.

According to the Israeli National Mathematics Curriulum (ISNM), the number zero is first introduced in kindergarten. At this age, the children are taught to associate zero with emptiness, or the absence of matter. Teachers are encouraged to use stories to introduce the concept of zero. This is illustrated by the following example taken from the ISNM handbook, "There are 6 nuts on one plate. On the second plate there aren't any nuts. We can say that the second plate has zero nuts" (Ministry of Education, 1988, p. 12). This example illustrates the cardinal property of zero. Zero is used to express the number of elements in the null set. The curriculum booklet offers a second example: "If the temperature outside is zero degrees then it is colder than when it is one degree" (Ministry of Education, 1988, p. 12). It is important to note that the handbook suggests using this second example only for "advanced or high ability" students. This example illustrates the ordinal property of numbers in that zero comes before one when ordering the numbers. However, the primary example is the one that relates zero to "nothing".

First graders, according to Israel's National Mathematics Curriculum, are reintroduced to zero, but only after they become familiar with the numbers one through twenty. The curriculum handbook recognizes that many children do not accept zero as a number and suggests including zero in addition and subtraction tasks. By second grade students should know that zero is the additive identity and that when subtracting any number from itself the result will always be zero. Multiplication is not introduced until the end of second grade. Students were interviewed in the middle of the school year, during the months of January and February.

Results

This section discusses the results of the interviews. First, we discuss the various explanations students gave for the multiplication tasks without zero and how they were categorized into MB and PB explanations. We then discuss students' responses to the tasks of 3x0 and 0x3 and the explanations given by students for these tasks. Finally, we compare the types of explanations students used for the multiplication of 0x0 with those already given for previous tasks.

Multiplication without zero

Most students were consistent in the types of explanations given for both tasks. In other words, students who used a MB explanation for 3x2 used a MB explanation for 2x3 and likewise for PB explanations. One first grader gave both a MB and PB explanation for 3x2 but only a MB explanation for 2x3. One second grade gave both a MB explanation and a PB explanation for both tasks. Results are summed up in Table 1. Percentages are based on the number of students in each grade.

Results show that both first and second graders are more likely to use MB explanations than PB explanations. One might have expected that such young children would be more likely to base their explanations on practical life experiences. However, this was not the case in this study.

MB explanations

As stated above, MB explanations employ only mathematical notions. In this category we included explanations that did not rely on the use of pictures, concrete objects, or stories. Many explanations were based on the definition of multiplication as repeated addition. As one first grader said, "Instead of doing a lot of addition, you do multiplication." This type of explanation usually involved representing the multiplication (the second number) and then successively adding these numbers. An example of this is the following explanation for 3x2 given by a second grader, "2 and another 2 is 4 and another 2 is 6." Also in this category were explanations that relied on sequencing, such as one second grader's explanation on the word "times," such as, "This is like saying 3 two times, which would be six." When asked to elaborate and explain the word times, many students replied with an exercise based on repeated addition such is "twice 2 ... 3 plus 3." Other students could not elaborate. Finally, explanations that were based on the commutative property of multiplication were considered MB explanations. As none of the students had learned multiplication in class, it was not expected that they would refer to this property by its proper name. Instead, this category included explanations that stated that the order of the factors is irrelevant to the solution. When one first grader replied that 2x3 is the same as 3x2 he explained, "It's the same numbers. It's easy." One second grader was even more explicit. When responding to the problem of 2x3 the following dialogue ensued:

Interviewer: Now tell me what 2x3 is.

Student: 6

Interviewer: 6? Again? You have to explain it to me.

Student: It's the same thing. If you do 2 time 3 then it's the same as 3 times 2.

Interviewer: And then you do the same thing as you told me (for 3x2)?

Student: Yes. They just say it backwards. 2 times 3.

The most common MB explanation given by both first and second graders was repeated addition (56% and 76% respectively of all MB explanations). Surprisingly, although none of the students had experience with multiplication in class, three students understood the commutative property of multiplication and used it as an explanation.

PB explanations

PB explanations were defined above as explanations that use daily con texts and/or manipulatives to "give meaning" to mathematical expressions. Although paper and pencils were brought to each interview and students were free to use any object at hand, students primarily chose to use their fingers as manipulatives. Such was the case with one first grader who held up three fingers and said, "Here's 3," and then held up three fingers on the other hand and said, "and another 3 makes 6." Only first graders used their fingers. Second graders who gave PB explanations drew pictures. Figure 1 is an illustration of a second grader's PB explanation of why 2x3=6 and 3x2=6.

[FIGURE 1 OMITTED]

This student originally answered that 2x3 equals 12. When explaining her solution, she drew 2 sets with 3 pencils in each. The student realized her mistake and then wanted to figure out how many sets of 3 pencils she would need in order to have 12 pencils. This led her to draw 4 sets of 3 pencils and write "4x3=12." Finally, she drew 3 sets of 2 pencils to illustrate why 3x2=6.

Multiplication with zero: 3x0 and 0x3

Two new issues arose in students' explanations for 3x0 and 0x3 that were not present in the examples without zero. First, not all students knew the correct results of multiplication of a non-zero number with zero. Second, some students could not explain their answers while other students offered explanations that could not be clearly categorized as MB or PB. In this section we will first discuss the solutions given by the students and then present the types of explanations used.

Students' solutions for 3x0 and 0x3

Although all of the students interviewed knew multiplication without zero, 60% of the first graders and 14% of the second graders incorrectly solved 3x0, and 30% of the first graders and 33% of the second graders incorrectly solved 0x3 (see Table 2). It should be noted that many children changed their minds several times and only their final answers are considered. It is very interesting to note that all (except for one) of the students who answered incorrectly, claimed that 3 times 0 (or 0 times 3) equals 3 and not all students who answered one question correctly answered the second correctly. These results show that although students may know multiplication without zero, it does not necessarily follow that they will know multiplication with zero. Possibly, this is because children's first experiences are with the set of natural numbers. As mentioned in the background, students, especially younger ones, often relate zero to "nothing" or "emptiness." At times, this can lead to an incorrect solution.

From Table 2 we see that there are similarities and differences in the results of 3x0 and 0x3. MB explanations are used more often than PB explanations for both tasks. However, second graders' use of MB explanations far exceeded that of the first graders. This is not surprising considering that second graders have had more experience with arithmetic operations in the classroom. In Table 2 we encounter explanations that could not be categorized as either MB or PB. Among first graders who correctly solved both multiplication tasks with zero, one student could not explain the reason why 3x0 and 0x3 are equal to zero. Among second graders, there were two students whose explanations for 3x0 and 0x3 could not be categorized as either MB or PB. However, there were an additional 3 second graders who explained 3x0 but could not explain 0x3. This added difficulty may be a result of the multiplier being a non-positive integer.

MB explanations

As discussed previously some MB explanations were based on the meaning of the word "times." Regarding 3x0 and 0x3, this type of explanation was usually given for a correct solution, such as the second grader who claimed, "3 times 0 is 0, 3 times." When asked to clarify, he continued, "3 zeros would just equal zero." Another second grader explained 3x0=0 because, "0 is nothing. 3 times 0 is also nothing. The first time 0 is 0. The second time 0 is 0." When explaining 0x3, one first grader explained that 0x3=0 "because you don't even do the 3. So it's 0." A second grader explained that 0x3=0 because "0 means that there are none of 3's." The following second grader was even more explicit:

Student: If it's 1 times it will be 3.

Interviewer: Aha. Why is that?

Student: Cause it is, like, you say 1, you have to write once 3. That's why it's 3.

Interviewer: I see. And 0 times 3?

Student: 0 times 3? It says 0 times 3. It means you don't have to write any times the 3.

One student knew the correct answer because he had been taught by his older brother that every number multiplied by zero results in zero. When asked to think of an explanation for this rule, the student reverted back to the meaning of times which he had used successfully for multiplication without zero:

Interviewer: Why do you think 3 times 0 is 0?

Student: (thinks) OK. Because, let's say, 3 times or 4 times 2 that would be 4, 2 times. And 0 is 0, 0 times.

Interviewer: It's 0, 0 times, for 3 times 0?

Student: Yeah. So, it must makes 0. You get it?

Interviewer: I think I get it, but I'm not sure where the 3 comes in then.

Student: (thinks) OK. (thinks)

Interviewer: Would it be different if I asked you what 0 times 3 is?

Student: Uhm. (thinks) I don't think so.

Interviewer: You're stuck.

Student: Yeah.

Not all students who used this type of MB explanation gave a correct solution. When asked to solve 0x3, one student responded, "0 time 3? It's 3 ... because you do the 3 zero times and you get 3." It is possible tha this student was confused by what it means to do something zero times.

Zero caused much confusion among the many students who based their explanation on the definition of multiplication as repeated addition. Only two students explicitly said that 3x0 is 0 because "0 plus 0 plus 0 equals 0." Many students were confused as to the appropriate addition problem for 3x0 and concluded that 3x0 is 3 because "you don't add anything so it stays the same number." This response implies that the student is seeking to add to 3 another number, as he did for 3x2, but cannot do so because of the zero. Other students held the intuitive belief that multiplication always makes bigger (Fischbein, Deri, Nell, & Marino, 1985) and were therefore confused as to how a number may be multiplied by zero.

Student: 3 times 0 is 3 and you can't do such a thing that it will be bigger than 3.

Interviewer: Why can't you make it bigger?

Student: Because 0 doesn't add anything.

This student expects that when multiplying the 3 by some number the 3 will "get bigger." She reasons that the zero stops her from adding on to the 3 and therefore the result must be the same 3 she started with. Another student gives a similar explanation when explaining the difference between 3x0 and 0x3, "0x3 is 3 because ... you can't raise the 3. So it stays 3. But 3 times 0 you can't even raise the 0 so it's 0." This misconception is most probably due to students' first experiences with multiplication within the natural numbers.

Many students who answered correctly 3x0=0 explained that multiplication by zero cannot be done and therefore the result is 0. One student explained that 3x0=0 "because you can't multiply by zero. 3 times 1, that could be 3 ... like, if you multiply by zero then it stays 0." When asked to explain why multiplication by zero could not be done, this student responded that 3 times 0 "is like you take it down to 0 and then add 0." Somehow, the 3 turns magically into 0 and then, if you must add something, add zero.

Other students also answered correctly that 3x0=0 but were still confused as to whether zero should be considered a number or not. This is illustrated by the following exchange with a second grader who had clearly used repeated addition when multiplying 3 by 2 but found multiplying by 0 quite different:

Interviewer: And what is 3 times 0?

Student: 0.

Interviewer: Why?

Student: Because ... it doesn't have a number. If you had one, then it could be different. Because you can't do 3 times 0. It's still 0.

Interviewer: Why can you do 3x2 but you can't do 3x0?

Student: Because 0 is a number but it's ... it's nothing. It's nothing.

As mentioned in the beginning, the problem 3x0 fits well into the definition of multiplication as repeated addition because the multiplier is a positive integer and indicates the number of times 0 is to be added to itself. However, when the multiplier is non-positive, as in the case of 0x3, difficulties may arise. One student was deliberating between 0x3=0 and 0x3=3. He changed his mind frequently and finally explained that 0x3 is 0 "because ... you can't do it." The student then explained why 0x3 can't be done, "because it's 0 plus 3 ... uh ... no ... 3." Another student also deliberated with himself and finally concluded that 3x0=3 "because it's like 0 plus 0 plus 3. Like, you lift up another 3 to 0." Again we see a student who wants to add, is not sure how to do so, and intuitively wants to "lift" or "make bigger" the number he started with.

PB explanations

As with multiplication without zero, PB explanations for multiplication with zero consisted of finger manipulation or drawing pictures. One first grader explained her response that 3x0=3 by raising 3 fingers on one hand and making a fist with her other hand. When asked to explain 0x3 this same student started by holding up a closed fist. She then unfolded 3 fingers and proclaimed that 0x3=3. Using a similar explanation, a different first grader exclaimed that 3x0=0 by holding up a closed fist and then thrusting it into the air 3 times. When asked to explain 0x3 this student simply said, "If I do 3, 0 times ... then you do nothing."

One second grader drew 3 tally marks and claimed that 3x0=3. When asked to solve 0x3 he drew a big circle and claimed, "It's just nothing." This student had previously illustrated with tally marks why 3x2 must be the same as 2x3 but was seemingly not bothered that 3x0 would not be the same as 0x3. Another second grader, who had drawn groups of pencils to illustrate multiplication without zero, was not sure how to draw 3x0 or 0x3:

Interviewer: Tell me what you think about 3x0.

Student: 0?

Interviewer: Why?

Student: 3.

Interviewer: 3. Why?

Student: Cause then you have 3 ... 3 pencils?

Interviewer: You decide.

Student: (very long pause) 0.

Interviewer: 0. We're back to 0. So tell me why you think it's 0.

Student: Because it can't be ... If we take 3 minus 0 it's 3. But if We do ... but you can't do it ... because pencils is like ... If you take ... So it's 0 because it's 3 groups of nothing.

Interviewer: Ok. What about 0x3?

Student: Also 0.

Interviewer: Why?

Student: Because no groups of 3. Because 0 is like nothing so you have nothing and let's say there's nothing, like no house, and then, times, so I go, 0, so I go nothing is nothing.

This student only came to a correct solution when she was able to part from drawing pictures. This release from the concrete picture allowed her to reason about sets, even sets that she could not draw. Instead of drawing pictures, this student uses the "zero is nothing" analogy. As discussed previously, this analogy often leaves students confused as to whether zero is a number or not. In fact, this second grader stated later in the interview that zero is not a number.

Multiplication of zero by zero

All of the students knew that 0x0=0. However, the explanations were quite varied and sometimes difficult to categorize (see Table 3). Most of the students still used MB explanations. None of the students gave both a MB and PB explanation for 0x0.

MB and PB explanations

MB explanations followed the examples already seen for 3x0 and 0x3. Some students still relied on th meaning of "times" such as "0, 0 times, so it's 0." Some student still tried adding:

Interviewer: Can you explain to me why 0x0 is 0?

Student: (thinks) Is it like 0 plus 0?

Interviewer: Is 3 times 2 like 3 plus 2?

Student: (Shakes his head no.)

Interviewer: So? Why is 0 times 0 like 0 plus 0?

Student: Because both are zero.

Some students simply stated that 0x0=0 because "you don't add anything to the 0". Others intimated that there is no definitive amount of addends: "0x0=0 because 0 plus 0 plus 0 plus 0 plus 0 ... As much zeros as there would be, it equals 0." Other students still inferred that multiplication makes bigger, "0 times 0 ... you can't do 0 times 0. It's 0. And the second 0 can't make me bigger. It's 0."

From Table 3 we see that few students used PB explanations. One student pointed to the big circle he drew to illustrate 0x3 as if to say that the two examples are the same. A different student explained 0x0 as follows, "cause 0 groups and there's 0 thingies inside. It makes 0 cause there's no number that makes a group." Finally, those who had used their fingers continued to show closed fists to illustrate the meaning for them of 0x0.

Uncategorized explanations

Although we have already shown examples where students used the "zero is nothing" analogy when explaining 3x0 and 0x3, many uncategorized explanations for 0x0 were related to this analogy. One first grader explained that 0x0=0 "because 0 is nothing and 0 times 0 is also nothing. And you know that it's also 0." Another student said, "Nothing times nothing equals nothing." When asked to explain how one does nothing times nothing, he answered, I don't know. But it's 0." A different student claimed, "0 times 0 is nothing because the 0 makes it nothing."

Student's confusion over the status of zero as a number became more apparent in this task. One second grader claimed that the product is 0 because "when you say 0 times 0 there is no number." A different second grader explained, "0 times 0 always will be 0 because both of them are 0. 0 isn't a number. That means there are none of them. 0 plus 3 equals 3 because that isn't times. 0 plus 0 will also equal 0 because 0 isn't a number. And 0 in times also isn't a number." Another student referred to zero as a "non-number."

Discussion

Two major aims of this investigation were: 1) to explore students' preconceptions of multiplication with zero and 2) to investigate the types of explanations, MB and/or PB that students use. We now discuss these findings and the educational implications of each.

Students' preconceptions of multiplication with zero

The findings of this study show that before they are introduced in school to the concept of multiplication with zero, not all students know that multiplication with zero will always result in zero. Many studies have pointed out students' difficulties understanding the concept of zero in general, and specifically, their difficulties with division by zero. The result has been a plethora of suggestions regarding how to introduce elementary school students to division by zero, and to a lesser extent, how to introduce the number zero in the early grades. Perhaps few studies have dwelled on other operations with zero because of the seemingly few difficulties that arise in class. Unlike division, there are no exceptions to the rule that every number times zero is zero. The commutative property of multiplication is preserved. Yet the findings of this study showed that this rule was not obvious to young students.

We feel that there could be two major consequences to ignoring students' preconceptions of multiplication by zero. First, we have shown that students' misconceptions of the number zero prevail during multiplication. It would be a mistake not to give sufficient attention to operations with zero before introducing division. By ignoring students' misconceptions at an early age, we allow these misconceptions to take root and grow therefore making it more difficult to undo later on.

Furthermore, many students believe that mathematics is all about learning rules (Schoenfeld, 1989). According to these students, succeeding in mathematics means knowing the rules and when to apply them. When does this belief develop? One of the first rules taught to young students is that every number times zero must equal zero. By having students memorize this rule, and ignoring their preconceptions, we encourage their belief that mathematics is indeed all about rules, some of which make sense and others that do not. Is this a belief that we want to encourage?

Students' use of MB and PB explanations

We begin our discussion by highlighting the fact that over 80% of the students used MB explanations for multiplication without zero. This is somewhat surprising considering that the students interviewed had not yet been introduced formally to multiplication in class. Despite the lack of formal learning, these students felt comfortable using explanations that relied solely on mathematical notions.

All students interviewed knew multiplication without zero. Introducing the question of multiplication with zero allowed us to investigate how students might redefine their knowledge of operations to include an expanded number system. It is a long held belief that when elementary school children seek to describe their mathematical thinking or explore mathematical concepts they will use tangible items to manipulate or relate these concepts to real life contexts (e.g., Cramer & Henry, 2002; Fischbein, 1987; National Council of Teachers of Mathematics [NCTM], 1989; NCTM, 2000). Results of this study showed otherwise. Over two-thirds of the students continued to use MB explanations for 3x0. In other words, more students looked within the mathematical system for an explanation to this new problem and did not rely on stories and concrete objects taken from the "real" world.

Although the use of MB explanations declined for the task of 0x3 (a little more than half of the students tried basing their explanations solely on mathematical notions), the use of PB explanations did not increase. Instead, there was an increase in uncategorized explanations. The task of 0x0 was even more difficult to explain leading students to rely on the analogy of "zero is nothing" and raising doubt in the students' minds as to the status of zero as a number.

One of our goals as mathematics educators is to help our students move from PB explanations to MB explanations. In the beginning of this paper we asked if it is possible to introduce more formal mathematics to young children. Knowing that the move to formal mathematics may be difficult, we should examine the possibility of introducing more MB explanations to elementary school students.

This study shows that even young students are capable of using explanations that rely solely on mathematical notions. Is this true only for multiplication tasks? We need to examine students' use of MB explanations in other mathematical contexts as well. We also need to investigate how these findings may be used in practice by teachers in the classroom and, in line with Fischbein's (1987) recommendation, investigate how MB explanations may be used to prepare students for the formal content of mathematics.

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Esther Levenson, Pessia Tsamir, and Dina Tirosh

Tel-Aviv University

In this study, we use the context of multiplication by zero, to reexamine the premise that elementary school students prefer practically-based explanations that rely on tangible items or real life stories to that of mathematically-based explanations that rely solely on mathematical notions. Two major aims of this investigation were: (1) to explore students' preconceptions of multiplication with zero, and (2) to investigate the types explanations, mathematically-based and/or practically-based, that students use. Thirty-one first and second grade students were interviewed and asked to solve given multiplication tasks that included multiplication without zero and with zero. Results show that (1) not all students, who correctly solved 3x2 before being introduced to multiplication in class, knew that multiplication with zero always results in zero, and (2) most of the explanations given by students were mathematically-based explanations.

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"Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well" (NCTM, 2000). Understanding what students know includes knowing the preconceptions and misconceptions students of different age levels and backgrounds bring to the mathematical classroom (Shulman, 1986). A teacher must also understand the reasoning behind these conceptions and anticipate common mistakes (Even & Tirosh, 1995). However, according to the NCTM, teachers must be aware of what students need to learn in the future. Long-term goals, such as helping our students make the transition to the formal use of definitions in mathematics, must be kept in mind. This paper does both; it looks at the preconceptions that students bring to the classroom with an eye towards what they need to know in the future.

In this study we explore students' explanations of multiplication with zero before they are introduced to multiplication in school. Although explanations have been classified in many different ways throughout the years, this study investigates mathematically-based (MB) explanations and practically-based (PB) explanations. MB explanations employ only mathematical notions. PB explanations use daily contexts and/or manipulatives to "give meaning" to mathematical expressions (Koren, 2004). As children extend their world of numbers to include zero, they must consider how this new number is different from previously recognized numbers as well as how known operations must be extended to include the new number. This study not only allows us to investigate students' preconceptions of multiplication with zero, but also allows us to investigate students' conceptions of the number zero as well as the types of explanations, PB and/or MB explanations, they use to incorporate a new concept into their mathematical world. By focusing on the types of explanations students' use we may begin to explore the types of explanations appropriate for elementary school students with an eye towards the types of explanations we would like to encourage in the future.

Background

Two main issues are at the heart of this investigation: expanding the natural number system to include zero and the types of explanations students use to accommodate this expansion. In this section we review studies related to these issues.

Students' conceptions of zero

One of the first mathematical ideas children are exposed to is counting. As such, children's first experiences with the world of numbers are with natural numbers. This number system is eventually extended to include the number zero. In this section we first discuss children's development of number concepts, specifically the concept of zero. We then discuss the educational implications of zero's behavior under the basic operations of mathematics.

It is commonly believed that for students to understand a number concept they need to understand two basic properties of numbers: cardinality and ordinality (Siegler, 1998). A cardinal number refers to the number of elements in a set. In the case of zero, zero represents the number of elements in the empty set. An ordinal number refers to the position of an element within a set. The status of zero as an ordinal number may be cause for confusion among children. If the number three represents the third tallest child in the class, then what does zero represent? Although the ground floor of a building is clearly the "zeroth" floor, it is not commonly denoted as such (Pogliani, Randic, & Trinajstic, 1998).

Piaget (1952) hypothesized that the construction of number goes hand in hand with the development of logic and that classification serves as a basis for mathematical concepts. Children classify objects based on physical properties or certain patterns of behavior. By the time they reach the age of 8-9 years old they have developed an understanding of the inclusion relation as well as the concept of the singular class. However, the empty or null class still poses a problem. According to Inhelder and Piaget (1969), the null set poses special difficulty "because a class without any elements is incompatible with the logic of concrete operations, i.e., operations in which form is inseparably bound up with content" (p. 149).

Many studies have been conducted regarding students' conceptions of zero. Reys and Grouws (1975) interviewed fourth and eighth graders and found that many did not consider zero to be a number. This seems to parallel the historical development of zero (e.g., Blake & Verhille, 1985; Pogliani et al, 1998; Seife, 2000; Wilson, 2001). Reys and Grouws found that part of the confusion over zero may have been caused by students equating zero with nothing. 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).

The analogy of "zero as nothing" finds its way into many mathematical operations. Zero is the additive identity for the set of whole numbers. This may be explained by the following: if you have any number of items, and add nothing, then you remain with the same amount of items you had to begin with. When explaining subtraction: If you have eight cookies and you eat all eight of them you are left with nothing. This can be written: 8 - 8 = 0. As stated above, students of all ages find this analogy very satisfying and use it for several years. It is easy to learn and retrieve and seems to work--until division. As students try to grapple with dividing "nothing" into "something" and "something" into nothing: they find that this analogy causes much confusion. The first is allowed but the latter is not. In high school, students may be 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 having no hills may be referred to as not having any slope. However, in mathematics, having no slope refers to a vertical line, a line for which no slope exists. 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, Sheffer, & Tirosh, 2000). Many 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 teachers and 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).

In light of these studies, it is not surprising that much has been written regarding pedagogical and instructional approaches to explaining the concept and use of zero in different mathematical contexts. Pogliani et al. (1998) suggest the following example to illustrate the distinction between 'empty', 'nothing', and 'zero': "When zero, as an element is added to an empty set {}, the set {0} is no longer empty; it has an element, which is zero" (p. 730). In the elementary school, Leeb-Lundberg (1997) advocates relating zero to a place of origin, such as on the number line.

Regarding multiplication and zero, most instructional suggestions relate to the mathematical statement "if the product of two numbers is zero, then one of the factors must be zero" (Allinger, 1980; Forringer, 1994; Newman, 1967). One study (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.

Regarding division by zero, some have suggested that teachers must first emphasize the relationship between division and multiplication as inverse operations and then use this relationship when explaining division by zero (Henry, 1969; Reys & Grouws, 1975; Sundar, 1990). Others offer explanations that arise from the common models used to teach whole number division (Knifong & Burton, 1980). Tsamir & Sheffer (2000) offer two possible recommendations for teaching division by zero: using concrete explanations in the elementary school, or, alternatively, postponing the introduction of division by zero until secondary school. Newman (1967) argued that explaining the behavior of zero under the basic operations of mathematics offers the elementary school teacher an opportunity to introduce her students to the importance of definitions in mathematics and the logical use of these definitions in discovering new mathematical properties.

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.

Explanations

The Standards for School Mathematics (NCTM, 2000) attribute different types of explanations to different age students. According to the Standards, "young children will express their conjectures and describe their thinking in their own words and often explore them using concrete materials and examples" (p. 56). However, by the "middle and high grades, explanations should become more mathematically rigorous" (p. 61). According to the Standards, students in middle school and high school should understand the role of mathematical definitions and use them, as well as previously learned mathematical properties, in their explanations.

The NCTM Standards initimate that there are basically two different categories or levels of explanations corresponding to younger and older students. This is somewhat similar to Piagetian theory, which differentiates between the concrete operational stage and the formal operational stage. Children in the concrete operational stage demonstrate their intelligence through logical and systematic manipulation of symbols related to concrete objects. In this stage, "the logical organization of judgments and arguments is inseparable from their content" (Inhelder & Piaget, 1969, p. 132). Children in the formal operational stage demonstrate their intelligence through logical use of symbols related to abstract concepts. There is a disconnection of thought from objects that "liberates relations and classifications from their concrete or intuitive ties" (Inhelder & Piaget, 1969, p. 132). Elementary and early adolescents may be said to be in the concrete operational stage. Adolescents and adults may be said to be in the formal operational stage.

This study focuses on mathematically-based (MB) explanations and practically-based (PB) explanations. MB explanations employ only mathematical notions. PB explanations use daily contexts and/or manipulatives to "give meaning" to mathematical expressions (Koren, 2004). This classification distinguishes between explanations that are based solely on mathematical notions but are not necessarily rigorous, and complete, formal explanations. Formal explanations are usually referred to at the high school and undergraduate level. The term PB explanation was coined to include any explanation that does not rely solely on mathematical notions. According to Piagetian theory it would seem that PB explanations are appropriate for younger children in the concrete operational stage and MB explanations are more appropriate for adolescents. Yet, Ball and Bass (2000) and Lampert (1990) describe classrooms where third and fourth graders use MB explanations as well as PB explanations. The NCTM Standards (2000) also provide examples of third grade students using MB explanations.

Much research has been done relating to the use of PB explanations in the elementary school mathematics classroom. Mack (1990) showed that it is possible to use children's informal knowledge of fractions, based on their real-life experiences, to building meaningful formal symbols and procedures. However, many studies (e.g., Koirala, 1999; Nyabanyaba, 1999; Szendrei, 1996; Wu, 1999) found that each type of PB explanation has its own set of pitfalls that need to be avoided or remedied by the teacher.

Fischbein (1987) claimed that until the age of 11-12 the child is in the concrete operational period and one cannot force upon the child concepts that he is not intellectually mature enough to understand. Yet the intuitive interpretations created by the concrete instructional materials and models used during this period often become rigid and it may be difficult at a later stage 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. Therefore, Fischbein 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" (p. 208). In other words, according to Fischbein, MB explanations should be used alongside PB explanations in the elementary school.

Is it possible to introduce elementary school students to formal mathematics if they are so reliant on concrete examples? Perhaps elementary school students are too young for rigorous explanations but not too young for explanations that are less formal but nevertheless rely solely on mathematical notions. This study focuses on the types of explanations, MB and PB, which students use for a mathematical concept before they are introduced to this concept in class. By focusing on the types of explanations used we reexamine the premise that elementary school students need explanations that rely on tangible items or real life stories and investigate the possibility of introducing explanations that rely solely on mathematical notions in these grades.

Method

First and second graders from three different elementary schools located in middle-class neighborhoods were asked to solve 3x2. Only students who could correctly solve this task were asked to participate in the study. As a result, ten first graders (six girls and four boys) and twenty-one second graders (nine girls and twelve boys) participated in this investigation. None of the students had been introduced to multiplication in school. Being that the first and second graders had only recently learned to read and write in school, we suspected that they may have trouble expressing their ideas to the fullest in writing. It was therefore decided to interview all subjects. Each student was interviewed individually allowing for ample time to think and respond to the questions. Each interview was audio taped.

During the interview, students were asked to solve and give their own explanations for multiplication problems without and with zero. Specifically, the following multiplication examples were given to each student:

3x2 =

2x3 =

3x0 =

0x3 =

0x0 =

The first two problems sought to establish how multiplication without zero was solved and explained and if the subject used the commutative property of multiplication as an explanation. The second two problems allowed us to investigate how subjects solved and explained multiplication of a non-zero number by zero. Specifically, the second two questions investigated if the types of explanations used by the subject for multiplications by non-zero numbers differed from the explanations used for multiplication with zero. Furthermore, these two questions allowed us to investigate if the subject differentiated between 3x0 and 0x3. The problem 3x0 fits well into the definition of multiplication as repeated addition when the multiplier is a positive integer and indicates the number of times 0 is to be added to itself. However, when the multiplier is non-positive, as in the case of 0x3, difficulties may arise. Therefore, it was of particular interest to investigate how subjects would explain this problem and how prevalent the use of the commutative property would be in this case. The last problem allowed us to investigate multiplication that did not involve any non-zero numbers and sought to examine if the explanations given would differ from those already offered.

According to the Israeli National Mathematics Curriulum (ISNM), the number zero is first introduced in kindergarten. At this age, the children are taught to associate zero with emptiness, or the absence of matter. Teachers are encouraged to use stories to introduce the concept of zero. This is illustrated by the following example taken from the ISNM handbook, "There are 6 nuts on one plate. On the second plate there aren't any nuts. We can say that the second plate has zero nuts" (Ministry of Education, 1988, p. 12). This example illustrates the cardinal property of zero. Zero is used to express the number of elements in the null set. The curriculum booklet offers a second example: "If the temperature outside is zero degrees then it is colder than when it is one degree" (Ministry of Education, 1988, p. 12). It is important to note that the handbook suggests using this second example only for "advanced or high ability" students. This example illustrates the ordinal property of numbers in that zero comes before one when ordering the numbers. However, the primary example is the one that relates zero to "nothing".

First graders, according to Israel's National Mathematics Curriculum, are reintroduced to zero, but only after they become familiar with the numbers one through twenty. The curriculum handbook recognizes that many children do not accept zero as a number and suggests including zero in addition and subtraction tasks. By second grade students should know that zero is the additive identity and that when subtracting any number from itself the result will always be zero. Multiplication is not introduced until the end of second grade. Students were interviewed in the middle of the school year, during the months of January and February.

Results

This section discusses the results of the interviews. First, we discuss the various explanations students gave for the multiplication tasks without zero and how they were categorized into MB and PB explanations. We then discuss students' responses to the tasks of 3x0 and 0x3 and the explanations given by students for these tasks. Finally, we compare the types of explanations students used for the multiplication of 0x0 with those already given for previous tasks.

Multiplication without zero

Most students were consistent in the types of explanations given for both tasks. In other words, students who used a MB explanation for 3x2 used a MB explanation for 2x3 and likewise for PB explanations. One first grader gave both a MB and PB explanation for 3x2 but only a MB explanation for 2x3. One second grade gave both a MB explanation and a PB explanation for both tasks. Results are summed up in Table 1. Percentages are based on the number of students in each grade.

Results show that both first and second graders are more likely to use MB explanations than PB explanations. One might have expected that such young children would be more likely to base their explanations on practical life experiences. However, this was not the case in this study.

MB explanations

As stated above, MB explanations employ only mathematical notions. In this category we included explanations that did not rely on the use of pictures, concrete objects, or stories. Many explanations were based on the definition of multiplication as repeated addition. As one first grader said, "Instead of doing a lot of addition, you do multiplication." This type of explanation usually involved representing the multiplication (the second number) and then successively adding these numbers. An example of this is the following explanation for 3x2 given by a second grader, "2 and another 2 is 4 and another 2 is 6." Also in this category were explanations that relied on sequencing, such as one second grader's explanation on the word "times," such as, "This is like saying 3 two times, which would be six." When asked to elaborate and explain the word times, many students replied with an exercise based on repeated addition such is "twice 2 ... 3 plus 3." Other students could not elaborate. Finally, explanations that were based on the commutative property of multiplication were considered MB explanations. As none of the students had learned multiplication in class, it was not expected that they would refer to this property by its proper name. Instead, this category included explanations that stated that the order of the factors is irrelevant to the solution. When one first grader replied that 2x3 is the same as 3x2 he explained, "It's the same numbers. It's easy." One second grader was even more explicit. When responding to the problem of 2x3 the following dialogue ensued:

Interviewer: Now tell me what 2x3 is.

Student: 6

Interviewer: 6? Again? You have to explain it to me.

Student: It's the same thing. If you do 2 time 3 then it's the same as 3 times 2.

Interviewer: And then you do the same thing as you told me (for 3x2)?

Student: Yes. They just say it backwards. 2 times 3.

The most common MB explanation given by both first and second graders was repeated addition (56% and 76% respectively of all MB explanations). Surprisingly, although none of the students had experience with multiplication in class, three students understood the commutative property of multiplication and used it as an explanation.

PB explanations

PB explanations were defined above as explanations that use daily con texts and/or manipulatives to "give meaning" to mathematical expressions. Although paper and pencils were brought to each interview and students were free to use any object at hand, students primarily chose to use their fingers as manipulatives. Such was the case with one first grader who held up three fingers and said, "Here's 3," and then held up three fingers on the other hand and said, "and another 3 makes 6." Only first graders used their fingers. Second graders who gave PB explanations drew pictures. Figure 1 is an illustration of a second grader's PB explanation of why 2x3=6 and 3x2=6.

[FIGURE 1 OMITTED]

This student originally answered that 2x3 equals 12. When explaining her solution, she drew 2 sets with 3 pencils in each. The student realized her mistake and then wanted to figure out how many sets of 3 pencils she would need in order to have 12 pencils. This led her to draw 4 sets of 3 pencils and write "4x3=12." Finally, she drew 3 sets of 2 pencils to illustrate why 3x2=6.

Multiplication with zero: 3x0 and 0x3

Two new issues arose in students' explanations for 3x0 and 0x3 that were not present in the examples without zero. First, not all students knew the correct results of multiplication of a non-zero number with zero. Second, some students could not explain their answers while other students offered explanations that could not be clearly categorized as MB or PB. In this section we will first discuss the solutions given by the students and then present the types of explanations used.

Students' solutions for 3x0 and 0x3

Although all of the students interviewed knew multiplication without zero, 60% of the first graders and 14% of the second graders incorrectly solved 3x0, and 30% of the first graders and 33% of the second graders incorrectly solved 0x3 (see Table 2). It should be noted that many children changed their minds several times and only their final answers are considered. It is very interesting to note that all (except for one) of the students who answered incorrectly, claimed that 3 times 0 (or 0 times 3) equals 3 and not all students who answered one question correctly answered the second correctly. These results show that although students may know multiplication without zero, it does not necessarily follow that they will know multiplication with zero. Possibly, this is because children's first experiences are with the set of natural numbers. As mentioned in the background, students, especially younger ones, often relate zero to "nothing" or "emptiness." At times, this can lead to an incorrect solution.

From Table 2 we see that there are similarities and differences in the results of 3x0 and 0x3. MB explanations are used more often than PB explanations for both tasks. However, second graders' use of MB explanations far exceeded that of the first graders. This is not surprising considering that second graders have had more experience with arithmetic operations in the classroom. In Table 2 we encounter explanations that could not be categorized as either MB or PB. Among first graders who correctly solved both multiplication tasks with zero, one student could not explain the reason why 3x0 and 0x3 are equal to zero. Among second graders, there were two students whose explanations for 3x0 and 0x3 could not be categorized as either MB or PB. However, there were an additional 3 second graders who explained 3x0 but could not explain 0x3. This added difficulty may be a result of the multiplier being a non-positive integer.

MB explanations

As discussed previously some MB explanations were based on the meaning of the word "times." Regarding 3x0 and 0x3, this type of explanation was usually given for a correct solution, such as the second grader who claimed, "3 times 0 is 0, 3 times." When asked to clarify, he continued, "3 zeros would just equal zero." Another second grader explained 3x0=0 because, "0 is nothing. 3 times 0 is also nothing. The first time 0 is 0. The second time 0 is 0." When explaining 0x3, one first grader explained that 0x3=0 "because you don't even do the 3. So it's 0." A second grader explained that 0x3=0 because "0 means that there are none of 3's." The following second grader was even more explicit:

Student: If it's 1 times it will be 3.

Interviewer: Aha. Why is that?

Student: Cause it is, like, you say 1, you have to write once 3. That's why it's 3.

Interviewer: I see. And 0 times 3?

Student: 0 times 3? It says 0 times 3. It means you don't have to write any times the 3.

One student knew the correct answer because he had been taught by his older brother that every number multiplied by zero results in zero. When asked to think of an explanation for this rule, the student reverted back to the meaning of times which he had used successfully for multiplication without zero:

Interviewer: Why do you think 3 times 0 is 0?

Student: (thinks) OK. Because, let's say, 3 times or 4 times 2 that would be 4, 2 times. And 0 is 0, 0 times.

Interviewer: It's 0, 0 times, for 3 times 0?

Student: Yeah. So, it must makes 0. You get it?

Interviewer: I think I get it, but I'm not sure where the 3 comes in then.

Student: (thinks) OK. (thinks)

Interviewer: Would it be different if I asked you what 0 times 3 is?

Student: Uhm. (thinks) I don't think so.

Interviewer: You're stuck.

Student: Yeah.

Not all students who used this type of MB explanation gave a correct solution. When asked to solve 0x3, one student responded, "0 time 3? It's 3 ... because you do the 3 zero times and you get 3." It is possible tha this student was confused by what it means to do something zero times.

Zero caused much confusion among the many students who based their explanation on the definition of multiplication as repeated addition. Only two students explicitly said that 3x0 is 0 because "0 plus 0 plus 0 equals 0." Many students were confused as to the appropriate addition problem for 3x0 and concluded that 3x0 is 3 because "you don't add anything so it stays the same number." This response implies that the student is seeking to add to 3 another number, as he did for 3x2, but cannot do so because of the zero. Other students held the intuitive belief that multiplication always makes bigger (Fischbein, Deri, Nell, & Marino, 1985) and were therefore confused as to how a number may be multiplied by zero.

Student: 3 times 0 is 3 and you can't do such a thing that it will be bigger than 3.

Interviewer: Why can't you make it bigger?

Student: Because 0 doesn't add anything.

This student expects that when multiplying the 3 by some number the 3 will "get bigger." She reasons that the zero stops her from adding on to the 3 and therefore the result must be the same 3 she started with. Another student gives a similar explanation when explaining the difference between 3x0 and 0x3, "0x3 is 3 because ... you can't raise the 3. So it stays 3. But 3 times 0 you can't even raise the 0 so it's 0." This misconception is most probably due to students' first experiences with multiplication within the natural numbers.

Many students who answered correctly 3x0=0 explained that multiplication by zero cannot be done and therefore the result is 0. One student explained that 3x0=0 "because you can't multiply by zero. 3 times 1, that could be 3 ... like, if you multiply by zero then it stays 0." When asked to explain why multiplication by zero could not be done, this student responded that 3 times 0 "is like you take it down to 0 and then add 0." Somehow, the 3 turns magically into 0 and then, if you must add something, add zero.

Other students also answered correctly that 3x0=0 but were still confused as to whether zero should be considered a number or not. This is illustrated by the following exchange with a second grader who had clearly used repeated addition when multiplying 3 by 2 but found multiplying by 0 quite different:

Interviewer: And what is 3 times 0?

Student: 0.

Interviewer: Why?

Student: Because ... it doesn't have a number. If you had one, then it could be different. Because you can't do 3 times 0. It's still 0.

Interviewer: Why can you do 3x2 but you can't do 3x0?

Student: Because 0 is a number but it's ... it's nothing. It's nothing.

As mentioned in the beginning, the problem 3x0 fits well into the definition of multiplication as repeated addition because the multiplier is a positive integer and indicates the number of times 0 is to be added to itself. However, when the multiplier is non-positive, as in the case of 0x3, difficulties may arise. One student was deliberating between 0x3=0 and 0x3=3. He changed his mind frequently and finally explained that 0x3 is 0 "because ... you can't do it." The student then explained why 0x3 can't be done, "because it's 0 plus 3 ... uh ... no ... 3." Another student also deliberated with himself and finally concluded that 3x0=3 "because it's like 0 plus 0 plus 3. Like, you lift up another 3 to 0." Again we see a student who wants to add, is not sure how to do so, and intuitively wants to "lift" or "make bigger" the number he started with.

PB explanations

As with multiplication without zero, PB explanations for multiplication with zero consisted of finger manipulation or drawing pictures. One first grader explained her response that 3x0=3 by raising 3 fingers on one hand and making a fist with her other hand. When asked to explain 0x3 this same student started by holding up a closed fist. She then unfolded 3 fingers and proclaimed that 0x3=3. Using a similar explanation, a different first grader exclaimed that 3x0=0 by holding up a closed fist and then thrusting it into the air 3 times. When asked to explain 0x3 this student simply said, "If I do 3, 0 times ... then you do nothing."

One second grader drew 3 tally marks and claimed that 3x0=3. When asked to solve 0x3 he drew a big circle and claimed, "It's just nothing." This student had previously illustrated with tally marks why 3x2 must be the same as 2x3 but was seemingly not bothered that 3x0 would not be the same as 0x3. Another second grader, who had drawn groups of pencils to illustrate multiplication without zero, was not sure how to draw 3x0 or 0x3:

Interviewer: Tell me what you think about 3x0.

Student: 0?

Interviewer: Why?

Student: 3.

Interviewer: 3. Why?

Student: Cause then you have 3 ... 3 pencils?

Interviewer: You decide.

Student: (very long pause) 0.

Interviewer: 0. We're back to 0. So tell me why you think it's 0.

Student: Because it can't be ... If we take 3 minus 0 it's 3. But if We do ... but you can't do it ... because pencils is like ... If you take ... So it's 0 because it's 3 groups of nothing.

Interviewer: Ok. What about 0x3?

Student: Also 0.

Interviewer: Why?

Student: Because no groups of 3. Because 0 is like nothing so you have nothing and let's say there's nothing, like no house, and then, times, so I go, 0, so I go nothing is nothing.

This student only came to a correct solution when she was able to part from drawing pictures. This release from the concrete picture allowed her to reason about sets, even sets that she could not draw. Instead of drawing pictures, this student uses the "zero is nothing" analogy. As discussed previously, this analogy often leaves students confused as to whether zero is a number or not. In fact, this second grader stated later in the interview that zero is not a number.

Multiplication of zero by zero

All of the students knew that 0x0=0. However, the explanations were quite varied and sometimes difficult to categorize (see Table 3). Most of the students still used MB explanations. None of the students gave both a MB and PB explanation for 0x0.

MB and PB explanations

MB explanations followed the examples already seen for 3x0 and 0x3. Some students still relied on th meaning of "times" such as "0, 0 times, so it's 0." Some student still tried adding:

Interviewer: Can you explain to me why 0x0 is 0?

Student: (thinks) Is it like 0 plus 0?

Interviewer: Is 3 times 2 like 3 plus 2?

Student: (Shakes his head no.)

Interviewer: So? Why is 0 times 0 like 0 plus 0?

Student: Because both are zero.

Some students simply stated that 0x0=0 because "you don't add anything to the 0". Others intimated that there is no definitive amount of addends: "0x0=0 because 0 plus 0 plus 0 plus 0 plus 0 ... As much zeros as there would be, it equals 0." Other students still inferred that multiplication makes bigger, "0 times 0 ... you can't do 0 times 0. It's 0. And the second 0 can't make me bigger. It's 0."

From Table 3 we see that few students used PB explanations. One student pointed to the big circle he drew to illustrate 0x3 as if to say that the two examples are the same. A different student explained 0x0 as follows, "cause 0 groups and there's 0 thingies inside. It makes 0 cause there's no number that makes a group." Finally, those who had used their fingers continued to show closed fists to illustrate the meaning for them of 0x0.

Uncategorized explanations

Although we have already shown examples where students used the "zero is nothing" analogy when explaining 3x0 and 0x3, many uncategorized explanations for 0x0 were related to this analogy. One first grader explained that 0x0=0 "because 0 is nothing and 0 times 0 is also nothing. And you know that it's also 0." Another student said, "Nothing times nothing equals nothing." When asked to explain how one does nothing times nothing, he answered, I don't know. But it's 0." A different student claimed, "0 times 0 is nothing because the 0 makes it nothing."

Student's confusion over the status of zero as a number became more apparent in this task. One second grader claimed that the product is 0 because "when you say 0 times 0 there is no number." A different second grader explained, "0 times 0 always will be 0 because both of them are 0. 0 isn't a number. That means there are none of them. 0 plus 3 equals 3 because that isn't times. 0 plus 0 will also equal 0 because 0 isn't a number. And 0 in times also isn't a number." Another student referred to zero as a "non-number."

Discussion

Two major aims of this investigation were: 1) to explore students' preconceptions of multiplication with zero and 2) to investigate the types of explanations, MB and/or PB that students use. We now discuss these findings and the educational implications of each.

Students' preconceptions of multiplication with zero

The findings of this study show that before they are introduced in school to the concept of multiplication with zero, not all students know that multiplication with zero will always result in zero. Many studies have pointed out students' difficulties understanding the concept of zero in general, and specifically, their difficulties with division by zero. The result has been a plethora of suggestions regarding how to introduce elementary school students to division by zero, and to a lesser extent, how to introduce the number zero in the early grades. Perhaps few studies have dwelled on other operations with zero because of the seemingly few difficulties that arise in class. Unlike division, there are no exceptions to the rule that every number times zero is zero. The commutative property of multiplication is preserved. Yet the findings of this study showed that this rule was not obvious to young students.

We feel that there could be two major consequences to ignoring students' preconceptions of multiplication by zero. First, we have shown that students' misconceptions of the number zero prevail during multiplication. It would be a mistake not to give sufficient attention to operations with zero before introducing division. By ignoring students' misconceptions at an early age, we allow these misconceptions to take root and grow therefore making it more difficult to undo later on.

Furthermore, many students believe that mathematics is all about learning rules (Schoenfeld, 1989). According to these students, succeeding in mathematics means knowing the rules and when to apply them. When does this belief develop? One of the first rules taught to young students is that every number times zero must equal zero. By having students memorize this rule, and ignoring their preconceptions, we encourage their belief that mathematics is indeed all about rules, some of which make sense and others that do not. Is this a belief that we want to encourage?

Students' use of MB and PB explanations

We begin our discussion by highlighting the fact that over 80% of the students used MB explanations for multiplication without zero. This is somewhat surprising considering that the students interviewed had not yet been introduced formally to multiplication in class. Despite the lack of formal learning, these students felt comfortable using explanations that relied solely on mathematical notions.

All students interviewed knew multiplication without zero. Introducing the question of multiplication with zero allowed us to investigate how students might redefine their knowledge of operations to include an expanded number system. It is a long held belief that when elementary school children seek to describe their mathematical thinking or explore mathematical concepts they will use tangible items to manipulate or relate these concepts to real life contexts (e.g., Cramer & Henry, 2002; Fischbein, 1987; National Council of Teachers of Mathematics [NCTM], 1989; NCTM, 2000). Results of this study showed otherwise. Over two-thirds of the students continued to use MB explanations for 3x0. In other words, more students looked within the mathematical system for an explanation to this new problem and did not rely on stories and concrete objects taken from the "real" world.

Although the use of MB explanations declined for the task of 0x3 (a little more than half of the students tried basing their explanations solely on mathematical notions), the use of PB explanations did not increase. Instead, there was an increase in uncategorized explanations. The task of 0x0 was even more difficult to explain leading students to rely on the analogy of "zero is nothing" and raising doubt in the students' minds as to the status of zero as a number.

One of our goals as mathematics educators is to help our students move from PB explanations to MB explanations. In the beginning of this paper we asked if it is possible to introduce more formal mathematics to young children. Knowing that the move to formal mathematics may be difficult, we should examine the possibility of introducing more MB explanations to elementary school students.

This study shows that even young students are capable of using explanations that rely solely on mathematical notions. Is this true only for multiplication tasks? We need to examine students' use of MB explanations in other mathematical contexts as well. We also need to investigate how these findings may be used in practice by teachers in the classroom and, in line with Fischbein's (1987) recommendation, investigate how MB explanations may be used to prepare students for the formal content of mathematics.

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Esther Levenson, Pessia Tsamir, and Dina Tirosh

Tel-Aviv University

Table 1. Distribution (in %) of types of explanations per grade for multiplication without zero. 3x2 2x3 Task 1 2 Total 1 2 Total Grade n=10 n=21 n=31 n=10 n=21 n=31 MB 70 86 81 80 86 84 PB 20 9 13 20 9 13 MB & PB 10 5 6 - 6 3 Note. MB = mathematically-based explanation; PB = practically-based explanation. Table 2. Distribution (in %) of types of explanations per grade for multiplication without zero. 3x0 0x3 Task 1 2 Total 1 2 Total Grade n=10 n=21 n=31 n=10 n=21 n=31 MB 40 (10) 81 (67) 67 (45) 40 (20) 66 (38) 58 (29) PB 30 (10) 5 (5) 13 (6) 30 (30) 5 (5) 13 (6) MB & PB 20 (10) 5 (5) 10 (3) 20 (10) 5 (5) 10 (6) Other 10 (10) 9 (9) 10 (10) 10 (10) 24 (19) 19 (13) Note. Percentages of correct solutions are given in parenthesis. MB = mathematically-based explanation; PB = practically-based explanation. Table 3. Distribution (in %) of types of explanations per grade for zero times zero. Task 0x0 Grade 1 (n = 10) 2 (n = 21) Total (n = 31) MB 50 48 48 PB 20 4 10 Other 30 48 42 Note. MB = mathematically-based explanation; PB = practically-based explanation.

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Author: | Tirosh, Dina |
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Publication: | Focus on Learning Problems in Mathematics |

Date: | Mar 22, 2007 |

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