Students’ perceived sociomathematical norms: The missing paradigm

Journal of Mathematical Behavior 28 (2009) 171–187

Contents lists available at ScienceDirect

The Journal of Mathematical Behavior

journa l homepage: www.e lsev ier .com/ locate / jmathb

Students’ perceived sociomathematical norms: The missing paradigm

Esther Levenson ∗, Dina Tirosh, Pessia TsamirTel Aviv University, Israel

a r t i c l e i n f o

Keywords:Sociomathematical normsPerceived normsMathematically based explanationsPractically based explanationElementary school

a b s t r a c t

This study proposes a framework for research which takes into account three aspectsof sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enactednorms, and students’ perceived norms. We investigate these aspects of sociomathematicalnorms in two elementary school classrooms in relation to mathematically based and prac-tically based explanations. Results indicate that even when the observed enacted normsare in agreement with the teachers’ endorsed norms, the students may not perceive thesesame norms. These results highlight the need to consider the students’ perspective wheninvestigating sociomathematical norms.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

In their landmark article regarding the interpretation of mathematics classrooms, Yackel and Cobb (1996) introduced thenotion of sociomathematical norms to describe “normative aspects of mathematical discussions that are specific to students’mathematical activity” (p. 458). They were interested in how normative aspects of mathematics discussion are developed,such as what counts as mathematically different, mathematically efficient, and mathematically elegant. Since then, manystudies have focused on the role of the teacher in establishing and maintaining classroom norms (Kazemi & Stipek, 2001;Martin, McCrone, Bower, & Dindyal, 2005; McClain & Cobb, 2001; Yackel, 2002). These studies viewed sociomathematicalnorms mainly from the teacher’s perspective. Other studies focused on the continuous negotiation of sociomathematicalnorms occurring in the classroom (Ju & Kwon, 2007; Yackel & Cobb, 1996; Yackel, Rasmussen, & King, 2000). These studiesviewed sociomathematical norms from the perspective of observable actions in the classroom. Borrowing from the theory ofmeta-discursive rules, we may say that the above mentioned studies investigated both the endorsed norms that are declaredby teachers and the enacted norms of both teachers and students (Ben-Yehuda, Lavy, Linchevski, & Sfard, 2005). Accordingto this theory, “. . . enacted meta-rules describe the discourse as it actually is, whereas the endorsed meta-rules say how itshould be according to the actors themselves” (p. 183). Although meta-rules are not necessarily norms, they may eventuallyfunction as norms in that they may become “value laden and count as preferred ways of behavior” (Sfard, 2000, p. 170).

Certainly, previous studies have contributed much to our knowledge of the mathematics classroom as a community ofparticipants and related sociomathematical norms. The classroom is a complex environment consisting of many individ-uals coming together to form a community and the notion of sociomathematical norms may be even more complex thanthus far described. In a study concerning sociomathematical norms and individual preferences, it was found that some stu-dents retain their personal preferences for different types of explanations despite the sociomathematical norms observedin their classroom (Levenson, Tirosh, & Tsamir, 2006). This raises to the forefront the issue of the students’ perspective onsociomathematical norms. What do the students perceive as being normative in their classroom?

∗ Corresponding author at: Tel Aviv University, POB 39040, Ramat Aviv 69978, Israel. Tel.: +972 9 742 6681.E-mail address: [email protected] (E. Levenson).

0732-3123/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2009.09.001

http://www.sciencedirect.com/science/journal/07323123

http://www.elsevier.com/locate/jmathb

mailto:[email protected]

dx.doi.org/10.1016/j.jmathb.2009.09.001

172 E. Levenson et al. / Journal of Mathematical Behavior 28 (2009) 171–187

To the notions of endorsed and enacted norms we add the notion of perceived norms. These are the norms which studentsperceive to be in place in their classroom. Although a few studies investigated the relationship between students’ conceptualunderstanding of mathematics and sociomathematical norms (Kazemi & Stipek, 2001; McClain & Cobb, 2001; Stylianou &Blanton, 2002; Yackel, Cobb, & Wood, 1993), these studies did not investigate sociomathematical norms from the students’perspective. Are individual students aware of the sociomathematical norms enacted in the classroom? And if so, to whatextent are the norms perceived by students in agreement with the teacher’s endorsed norms as well as the teacher’s andstudents’ enacted norms in the classroom? This study takes into consideration part of the complexity of the classroomcommunity by investigating three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’enacted norms, and students’ perceived norms. By adding the voice of the students, we enhance the original picture portrayedby many studies and emphasize the need to further investigate the students’ perspective if we are to faithfully portray thesociomathematical norms in a classroom.

2. Theoretical background

A major issue of this paper is the complexity involved when investigating sociomathematical norms. This section offers ageneral overview of various studies related to sociomathematical norms. These studies review specific types of sociomathe-matical norms observed in mathematics classrooms, the relationship between sociomathematical norms and mathematicalcontexts, the development of sociomathematical norms, and the role of the teacher in developing these norms. Sociomathe-matical norms guide the explanations given and evaluated in the mathematics classroom. Although not a focus of this study,the issue of explanations is related to this study and is thus reviewed at the end of this section.

2.1. Sociomathematical norms

Specific sociomathematical norms may vary from classroom to classroom. Kazemi and Stipek (2001) analyzed two teach-ers who established similar social norms in their classes, but different sociomathematical norms. For example, both teachersencouraged their students to offer multiple strategies when solving a problem. This is a social norm. However, only oneteacher established the sociomathematical norm of exploring the mathematical relationships between the strategies. Intheir study of prospective elementary school teachers learning in a mathematics content course, Szydlik, Szydlik, and Benson(2003) found that social norms identified by the instructor included the following: solutions and arguments come primar-ily from the students and not the instructor, students share ideas with each other, the role of the instructor is to provideguidance and encouragement. Some of the sociomathematical norms that were observed included: an argument or solutionis different if it uses a different strategy or reveals a different aspect of the mathematical structure, complete deductiveargument is required for all mathematical claims, inquiry does not end until a problem is solved and understood, reveal-ing the underlying structure. Sociomathematical norms may also guide the types of explanations used and accepted in theclassroom and may vary according to the specific environments or contexts being discussed in the classroom. For example,in an undergraduate differential equations class, acceptable explanations needed to be grounded in an interpretation of therates of change (Yackel et al., 2000). In a middle-school class studying data analysis, acceptable explanations were thosegrounded in an investigation of the data, rather than calculational procedures (Cobb, McLain, & Gravemeijer, 2003).

Several studies investigated the construction and development of sociomathematical norms. Yackel and Cobb (1996)described a second grade classroom where the students differentiated between explanations that described procedures andexplanations that described actions on mathematical objects such as quantities. Through classroom discussion, sociomath-ematical norms were negotiated and it was agreed that only explanations that describe actions on mathematical objectswould be acceptable. Hershkowitz and Schwarz (1999) studied the sociomathematical norms related to rich learning envi-ronments that use computerized tools. They found that in such classes, “judging mathematical hypotheses with tools inorder to confirm or refute them, becomes normative in the class” (p. 155). As pairs of students worked together at thecomputer they were negotiating sociomathematical norms without guidance from the teacher. Bowers, Cobb, and McClain(1999) described a third grade class where the participation of individual students in classroom mathematical practicescontributed to the development of sociomathematical norms.

Many studies were interested in the role of the teacher in developing normative aspects of mathematical discussions.Lampert (1990) claimed that the role of the teacher is to follow and engage in mathematical arguments with the studentsas well as regulate classroom discourse. In order to accomplish this, the teacher must know more than the answer or therule used to find the solution. The teacher must know why a solution’s strategy is legitimate or useful and raise the issuesof legitimacy and usefulness in classroom discussions. The teacher demonstrates what is acceptable and thus demonstratesthe sociomathematical norms she wishes to establish in her classroom. According to Yackel and Cobb (1996), one of theteacher’s roles in developing sociomathematical norms in the classroom is to encourage the students to compare solutions.“Such reflective activity has the potential to contribute significantly to children’s mathematical thinking” (p. 464). Theteacher may then indicate which solution is desirable and which solution needs more explaining. McClain and Cobb (2001)explicitly investigated the teacher’s role in the development of sociomathematical norms claiming that the teacher supportsstudents’ development of a “mathematical disposition” and of intellectual autonomy. By taking a proactive role in guidingclass discussions and by symbolizing students’ way of reasoning, it was shown that the teacher supports the emergence ofsociomathematical norms related to sophisticated and efficient solutions. Recently, Rivera (2007) described the teacher’s

E. Levenson et al. / Journal of Mathematical Behavior 28 (2009) 171–187 173

role in developing sociomathematical norms related to mathematical justifications associated with the use of instruments,such as graphic calculators.

2.2. Explanations

As indicated in the above section, the framework of sociomathematical norms has been used to investigate the typesof explanations used and accepted in various classroom environments. In the mathematics classroom, explanations maytake on different forms depending on the classroom tradition. According to Yackel, Cobb, and Wood (1998), in the schoolmathematics tradition, where students engage in practices of carrying out prescribed procedures or following instructions,an explanation may consist of describing the steps of the procedure used. In the inquiry mathematics tradition, studentsare expected to give explanations which communicate their interpretations and mathematical activity to others in order toconvince others that their solutions are legitimate. Thus an explanation may describe how to do something as well as whyto do something (Perry, 2000). In addition, one may provide warrants which legitimize the relevance of the explanation andbackings which indicate why the warrant should be accepted as having authority (Toulmin, 1969).

In this study, we investigate the sociomathematical norms related to mathematically based (MB) and practically based(PB) explanations. MB explanations are based on mathematical definitions or previously learned mathematical properties,and often use mathematical reasoning. They are not necessarily formal and rigorous as those explanations which are typicallya focus at the high school and undergraduate level. PB explanations use daily contexts and/or manipulatives to “give mean-ing” to mathematical expressions. They include those explanations that use manipulatives or visual aids and explanationsbased on real-life contexts. It is unlikely that MB and PB explanations form a dichotomy. More likely, there is a continuumof explanations, beginning with PB explanations that use every day concrete objects, proceeding to semi-structured manip-ulatives, models, and generalized visual arguments, continuing to MB explanations, and ending with formal explanations.Previous studies found that some elementary school students prefer PB explanations while others prefer MB explanationsand that both types of explanations are used by students in various mathematical contexts (Levenson et al., 2006; Levenson,Tsamir, & Tirosh, 2007a). In this study, we use the context of MB and PB explanations to investigate sociomatheamticalnorms.

3. Research aims

The major aim of this study is to present a framework for research which takes into account three aspects of sociomathe-matical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. It is not theaim of this study to investigate the types of explanations used and accepted in the classroom. Nor is it the aim of this studyto investigate the relative merits of using MB and PB explanations in the elementary school (for a fuller discussion regardingMB and PB explanations see e.g. Levenson et al., 2006; Levenson, Tsamir, & Tirosh, 2007b). Rather, in this study we focuson the sociomathematical norms. Sociomathematical norms guide different aspects of mathematical activity such as whichexplanations are considered elegant and which are considered different. We investigate sociomathematical norms within thecontext of MB and PB explanations. Specifically we ask: (1) Are individual students aware of the sociomathematical normsregarding MB and PB explanations which are endorsed by their teachers? (2) Are students aware of the sociomathematicalnorms regarding MB and PB explanations which are enacted in their classroom and if so, is there a relationship between thisawareness and the degree of agreement between a teacher’s endorsed norms and those enacted in the classroom? (3) Towhat extent are the sociomathematical norms regarding MB and PB explanations perceived by students in agreement withthe teacher’s endorsed norms as well as the teacher’s and students’ enacted norms in the classroom?

4. Research design

4.1. Participants

Two fifth grade classes from two elementary schools participated in this study. These classes were chosen based on initialresults of questionnaires and classroom observations (described in the following sections) which showed that in one classthere was a consistency between the teacher’s endorsed norms and the teacher’s and students’ enacted norms whereas inthe other class there was a discrepancy between the teacher’s endorsed norms and those enacted by the teacher and studentsin class. We were interested in investigating students’ perceived norms in both of these situations. Teachers and studentswere not aware that the focus of the research was on sociomathematical norms regarding MB and PB explanations. Bothschools were located in the same middle-socioeconomic town. The curriculum used in both classes was the Israel NationalMathematics Curriculum (INMC, 2006), mandatory in all Israeli elementary schools.

The first teacher, whom we shall call Rose, had 31 years experience teaching fifth and sixth grade classes. She was agraduate of a teachers’ college and she had also completed a three-year national program certifying specialized mathematicsteachers in elementary schools. There were 25 students in Rose’s fifth grade classroom. According to her evaluations, 18 ofthe students were of high mathematical ability, four were of average ability, and three were of low ability.

The second teacher, whom we shall call Hailey, had 14 years experience teaching mathematics in the elementary schoolsystem, mostly teaching fifth and sixth grade classes. She had an undergraduate degree and a graduate degree in education


Fig. 1. MB explanations for the parity of 14.

as well as an additional graduate degree in Russian language and literature. There were 25 students in Hailey’s fifth gradeclassroom. According to her evaluations, 13 students were of high mathematical ability, five were of average ability, andseven were of low ability. Both teachers based their evaluations of mathematical ability on a combination of classroommathematics test scores, national mathematics achievement test scores, and personal observations of the students’ classroomwork.

4.2. Tools and procedure

4.2.1. BackgroundIn order to gain an overall picture of the teachers’ and students’ individual use and preferences for MB and PB explanations,

all participants were asked to fill out two questionnaires, each with the same format but using two mathematical contexts:parity (a familiar concept first introduced in first grade) and equivalent fractions (a concept recently introduced in the fifthgrade). The parity questionnaire was filled out during the first quarter of the school year, as this context was familiar tostudents.1 The fractions questionnaire was filled out during the third quarter of the year after students had been introducedto the concept of equivalent fractions.

Each questionnaire began by investigating the types of explanations given spontaneously by each participant. On theparity questionnaire, teachers and students were asked to state whether 14 is an even number or an odd number and toexplain their reasoning. On the fractions questionnaire, teachers and students were asked to assess the equivalence of two-fourths and six-twelfths, and to explain their reasoning. Participants were then presented with several explanations for why14 is an even number (see Figs. 1 and 2) and for why two-fourths is equal to six-twelfths (see Figs. 3, 4a, and 4b). Teachersand students were asked to read each explanation carefully and to comment on which explanation was most convincingto them. Teachers were also asked which explanations they would use in class and why. As will be discussed in the nextsections, interviews were conducted later on with both teachers and students which allowed for a deeper understanding ofthe results of the questionnaires.

The explanations presented to participants on these questionnaires did not move beyond the specific example to moregeneral cases. On the one hand, this may have skewed participants’ viewpoints of MB and PB explanations. On the otherhand, they were more closely related to the explanations given in the classroom and therefore served as a basis for discussingthe sociomathematical norms related to these types of explanations.

4.2.2. Teachers’ endorsed sociomathematical normsA first glimpse into teachers’ endorsed norms regarding MB and PB explanations was gained from the questionnaires

described above. In addition, each teacher was interviewed separately at the end of the school year. Both the parity andfraction questionnaires served as a context for discussion. The first part of the interview focused on the teachers’ responsesto the questionnaires, clarifying the reasons for their responses. The second part of the interview focused on the criteriathese teachers used when evaluating explanations in their mathematics classrooms. Both teachers were asked to differentiatebetween explanations they would use in class when addressing the whole class as opposed to explanations they would usefor individual students of different mathematical abilities. They were also asked to comment on which explanation theythought their students would find most convincing and which explanation they thought their students should give out loudwhen addressing the whole class.

1 The school term begins September 1st and ends June 30th.


Fig. 2. PB explanations for the parity of 14.

4.2.3. Teachers’ and students’ enacted sociomathematical normsClassroom observations took place over a six-month period, from mid-January till the end of June. It was thought that

by mid-year, sociomathematical norms would already be established in the classes. Each class was observed nine timesand was recorded with both a video camera and a tape recorder with the first author present at all times. Observationstook place randomly based on the teachers’ and schools’ schedule. The purpose of the observations was to record if theenacted norms regarding MB and PB explanations were in agreement with the endorsed norms proposed by the teacherand the norms perceived by students. The focus of the observations was on the use of different types of explanations givenby the teacher and the students in different circumstances and how these explanations were evaluated by members ofthe classroom community. Analysis of the observations was guided by the following questions: (1) What types of expla-nations are given by the teacher and under what circumstances? (2) Are students expected to explain their answers andif so, what types of explanations, MB, PB, or others, do students give? (3) Who is responsible for evaluating the expla-nations and on what basis? (4) How did the teacher respond to different types of explanations given by students in theclass?

4.2.4. Students’ perceived sociomathematical normsTen students from each class were interviewed individually in order to assess the sociomathematical norms they per-

ceived to be in place in their classes. Students were chosen in order to include those of mixed mathematical abilities but alsoin consideration of their being highly capable of verbal expression. Interviews took place between April and June of the schoolyear. Students were presented with the same MB and PB explanations used on the parity and fractions questionnaires. Theseexplanations were then used as a context with which to discuss the students’ perceived sociomathematical norms regardingMB and PB explanations. Each student was asked the following questions: (1) Which explanation would the teacher use inclass? (2) Which explanation would you use if a classmate asked you to explain this concept? (This question was repeatedusing another classmate’s name each time. Names of classmates were chosen based on their different mathematical abilitiesas evaluated by the teacher.) (3) Which explanation would you use if the teacher in class asked you to explain the concept outloud? (4) Would the teacher accept a different explanation from the one you chose? Taken together, these questions allowedus to investigate the degree of consensus among students regarding the use and acceptance of MB and PB explanations intheir classroom.

5. Results

This section examines the teachers’ endorsed sociomathematical norms, the teachers’ and students’ enacted sociomath-ematical norms, and the students’ perceived sociomathematical norms in both Rose’s and Hailey’s mathematics classrooms.Each classroom community is described separately beginning with background information regarding individual teachers’and students’ spontaneous use of explanations and the types of explanations considered most convincing. As will be shown,participants related to the term “convincing” in different ways. The section continues by reviewing the teacher’s endorsed


Fig. 3. MB explanations for why 2/4 = 6/12.

sociomathematical norms regarding MB and PB explanations and proceeds by describing teachers’ and students’ enactedsociomathematical norms observed during lessons. Finally, the students’ perceived sociomathematical norms relating to MBand PB explanations are reviewed.

5.1. Rose’s Classroom

5.1.1. BackgroundOn both questionnaires, Rose spontaneously gave MB explanations and considered the MB explanations presented to her

as most convincing. When considering which explanation was most convincing on the parity questionnaire, she commented:

First, I remove all the pictures and stories. It’s more comfortable for me to look at mathematical formulas. That leavesme with these (selects MB1, MB2, and MB3). This (she chooses MB1—divisibility by two). It’s the most basic. All of theothers are derived from this.


Fig. 4. (a and b) PB explanations for why 2/4 = 6/12.

Although Rose states that the MB explanations are most convincing, she adds that these explanations are more “com-fortable.” In other words, it may not be that she regards PB as unconvincing or that PB explanations do not convey certainty.Rather, she prefers explanations without pictures or stories. On the other hand, Rose felt that the PB explanations would bemost convincing to her students for both the parity and equivalent fractions concepts and clarifies this later on.

Nearly all explanations given spontaneously by students, for both mathematical contexts, were MB explanations (seeTable 1). On the other hand, not all students considered the MB explanations presented to them as most convincing. Morestudents chose PB explanations as most convincing for the concept of parity. On the fractions questionnaire, there seemedto be no consensus as to which type of explanation was most convincing. Students’ conceptions of a convincing explanationranged from an explanation being easily understood to one where the student could “see” that the statement was true.For example, on the fractions questionnaire, one student chose PB2 (chocolate explanation with illustration) as the mostconvincing and claimed “I saw that each one ate a part like the fraction.”

5.1.2. Rose’s endorsed normsDuring her interview, Rose was asked which of the seven explanations would be the most convincing to her students.

She responded:

I think the first explanation (PB1—pairs of students). Because I would take children and ask them to stand up and Iwould show them. You call a group of children and say to them you are on the A team and you are on the B team. Youare on the basket-ball team and you are on the soccer team. And you show them concretely that there is the half andthe half. Or . . . No. You know what I would do? Yes. I would say to one [child], you are the captain of a team and toanother [child] you are the captain of a team. And then, you pick. And now you pick. And then again. First you, thenyou. And then it’s clear. Until there are no children left. And then they would count seven and seven.

Table 1Frequency of types of spontaneous and most convincing explanations by Rose’s Students.

Context Spontaneous Most convincing

MB PB MB PB

Parity 26 1 6 14Equivalent fractions 17 1 7 11


Rose’s reply to this question is quite lengthy. She carefully considers how this explanation could be implemented andused in her classroom. Even though Rose was only asked to state which of the seven explanations would most convince herstudents, we understand by her reply that she would not only present this explanation to her class but she would physicallydemonstrate this explanation using the actual students in her fifth grade class. Rose responded to the question of whichexplanation would most convince her students by telling the interviewer which explanation she would use in class. Sheimplicitly related the two questions.

When explicitly asked which explanation she would use for the entire class, Rose chooses PB explanations for both theparity and fractions contexts. On the fractions questionnaires, she specifically chooses PB2 (chocolate explanation withillustration) and elaborates:

To begin with, I need to appeal to the common denominator. I can’t start with . . . In order to convince a heterogeneousclass, I would start with this (points to PB2—chocolate explanation with illustration). Reducing and expanding fractionsis a difficult topic to understand. For the good students and the not so good students.

We learn from this statement two endorsed norms which are important to Rose. First, the teacher ought to “appeal to thecommon denominator.” This implies that, according to Rose, in a whole class situation, explanations should be understoodby everyone and that, according to Rose, PB explanations are best suited to this norm. Second, for the topic of equivalentfractions, which Rose claims is a difficult topic, PB explanations are to be endorsed. Later in the interview Rose adds, “Mygoal is to choose the simplest [explanation] and what I think will be quickly grasped.”

As Rose proceeds down the class list choosing different explanations for different students, she comments, “I wouldchoose [to use] . . . the one that I think would convince the student most. I would want him to understand the first timearound and not the second.” In other words, a convincing explanation is one which would be understood quickly. Again,Rose’s choice of which explanation to use is based on which explanation she feels would most convince that student or beunderstood in the least amount of time. After several students are considered, it becomes apparent that she separates thestudents as well as the explanations into two groups which are hardly ever mixed. Rose chooses to use MB explanationsfor the high-ability students and PB explanations for those students she considers of average ability. There are only twolow-ability students. For these she would use all of the PB explanations and, regarding the parity explanations, adds, “Iwould just take the money and show them . . . Money coins work very well with students who have learning disabilities . . .Money is tangible and it helps them.”

Rose was also asked to choose from the explanations presented to her, which explanations she would expect her studentsto use in class. Interestingly, Rose expected each one of her students to use only MB explanations, regardless of the contextbeing discussed. Although Rose claimed she would give PB explanations to low-ability students, she still expected all herstudents to use MB explanations in class. Perhaps Rose felt that it was the teacher’s responsibility to help the studentsunderstand the concept in any way they could but that after it was understood, the students were expected to use MBexplanations in class. Finally, when asked if there was an explanation she would not accept from her students, she repliesthat all of the explanations are based on the same principle and therefore are perfectly acceptable.

5.1.3. Rose and her students’ enacted normsA total of nine lessons were observed over a six-month period. The contexts of these lessons varied and included such

topics as finding the least common multiple, comparison of fractions, and comparison of decimal fractions. In this sectionwe give an overall picture of the sociomathematical norms observed during these lessons.

To begin with, the social norm of giving explanations was observed throughout the lessons. Rose often turned to herstudents and asked them not only for solutions to a problem but also to explain the solutions out loud to the class. Sheoften was quoted as saying, “Our goal now is to solve the problem and explain.” Rose rarely singled out a student to answerand explain. Rather, she would turn toward the class as a whole and ask for volunteers. During the observed lessons, thesame students often participated while others were silent observers. Often, many students responded out loud at the sametime, making it difficult to observe who said what and making it difficult for anyone, teacher and students, to evaluate anyexplanations that were given. Students were not observed commenting on or evaluating other students’ explanations. Itbecame clear that a social norm had been established whereby the teacher either sorted out what had been said or ignoredthe students’ explanations, ultimately supplying the explanations herself. In this class, the participants were mostly studentswho were considered of high ability in mathematics. In summary, two other social norms observed in Roses’ classroom werethat students do not have to offer a solution if they do not feel so inclined nor are they expected to comment on the solutionsoffered.

Regarding sociomathematical norms different types of explanations, both MB and PB explanations were observed beingused in the classroom. For example, after decomposing 75 into its prime factors Rose states, “It’s an odd number because it’snot divisible by two.” This is a MB explanation.

Rose was also observed frequently using PB explanations. In the beginning of a lesson where the topic was comparisonof fractions Rose considers 4/6 and 5/6:

R: Let’s say we have a pizza that’s divided into six slices. Pizza with . . .Several students call out: Tuna. Olives. Tuna.R: Tuna? You really like pizza with tuna. The pizza is divided into sixths. On every tray [with a full pizza pie], how mayslices are there?


S1: Six.R: Six. How many slices are there on one tray [referring to the 5/6]?Together: Five.R: And on the other tray [referring to the 4/6]?Together: Four.R: Which tray do you prefer?S1: Five.R: Good. Does everyone understand?Together: Yes.

In the exchange above we see that Rose spends considerable time endearing the PB explanation to her students. Further-more, her query regarding “everyone” understanding coincides with her aim of reaching the maximum amount of studentsin the beginning of the lesson. As no one declares that they do not understand, Rose continues with a different question:

R: Now, which is bigger, one-half or three-fifths?S4: Three-fifths is bigger. Because three-fifths goes past the half and . . .R: How do I know that three-fifths is past the half?S4: Because half of five is two and a half . . . and three is bigger than two and a half.

This exchange was very interesting. The teacher had used a PB explanation in the beginning of the lesson. However,when the students are asked to explain a similar problem, one student responds with a MB explanation. Continuing withthis example, one of the students asks the teacher, “How much bigger is 3/5 than 1/2?” Rose turns to the class for a responseand one of the students responds, “One-fifth is two-tenths. And there are three-fifths so it’s six tenths. But five-tenths is ahalf. So, you have to take away one-tenth to get one-half.” Again, a student responds with a MB explanation. Next, Rose asksthe students to compare 4/10 and 3/5.

R: Who is bigger, four-tenths or three-fifths?S6: Three-fifths is closer to one whole.R: That’s one way. But, I’m not sure that I understand and I don’t see it concretely. What you said is correct. But I wantanother explanation, that will be appropriate for someone who doesn’t really understand why three-fifths is bigger thanfour-tenths.

Students, who understood and participated, were observed giving only MB explanations. While Rose gives positive feed-back to her students for participating, she infers that PB explanations may be more appropriate for the student who “doesn’treally understand.”

At the end of the lesson Rose asks to compare five-sevenths with four-sixths. One student responds, “In order for thefive-sevenths to be a whole, it’s missing two-sevenths. The four-sixths is missing two-sixths. And, two-sixths is bigger thantwo-sevenths.” The teacher then adds,

So, the one which you have to add more to, is smaller. Therefore, the one that is missing the bigger piece is smaller.Let’s say you need 100 shekels.2 If you’re missing four shekels and your friend is missing two shekels, then who hasmore money?

Once again, the student responded with a MB explanation and Rose inserts a PB explanation. First, she mentions that youhave to add a bigger “piece”. She is concretizing the fraction. Then she brings in a story revolving around money, furtherremoved from the fractions being discussed.

This pattern of the students using MB explanations and the teacher inserting PB explanations was repeated in otherlessons as well. For example, when comparing 3.79 to 3.8, Rose, with the help of the students, begins by talking about placevalue. The discussion continues with a MB explanation writing on the board the fraction equivalent of these decimals and

expanding to . And then:

R: And if it’s a matter of money, which would you prefer to have? This sum? 3.79? Or this? 3.80? 3.80. Right? It’s moremoney.S1: It’s a shekel.R: A shekel? It’s a hundredth. It’s one agurah.

After the MB explanation, Rose ends with a PB explanation.

2 A shekel is a monetary unit equivalent to 100 agurahs, similar to a dollar being equivalent to 100 cents.


Table 2Frequency of types of explanations Rose’s students would use.

Context For a friend Out loud in class

MB PB Both MB PB Both

Parity 1 2 7 5 4 1Equivalent fractions 1 4 5 6 4 –

5.1.4. Students’ perceived normsRecall that individual interviews were conducted with 10 students from Rose’s class. This section begins by reviewing the

types of explanations students claimed Rose would use in class. It then reviews the types of explanations students claimedthey would use out loud in class at the teacher’s request and those they claimed they would use as an explanation for afriend.

In general, there seemed to be no consensus regarding the types of explanations, MB or PB, students thought Rose woulduse to explain the parity of 14 or to explain the concept of equivalent fractions. Two students (one high-ability student andone average-ability student) claimed that she would use only PB explanations for both concepts and one student (average-ability) claimed that she would use only MB explanations for both concepts. Students who claimed that Rose would use bothtypes of explanations often argued that the teacher would take into consideration which student needed the explanation.For example, one student claimed that Rose would use a PB explanation to explain equivalent fractions to her (a low-abilitymathematics student) but would use a MB explanation “for someone who is smart.” Another (high-ability) student statedthat Rose would use for him “an explanation without pictures because I know it (the mathematics). But for students whoselevel [of mathematical ability] is lower, for them she would explain with pictures.” When discussing the parity explanations,a different high-ability student stated that Rose would choose an explanation that “depended on the level of the student.”On the other hand, when discussing the explanations from the fractions questionnaire, this same student claimed that theteacher would use the PB explanations because, “every time someone had a problem (understanding) she would explain itwith chocolate.” A different student also mentioned that “the teacher always uses chocolate bars. . . We asked her why shealways reminds us of chocolate. She said because everyone loves it.”

Students were asked to choose which explanation they would use for a friend and which explanation they would use inclass if the teacher asked them to explain the concept out loud to the whole class. Responses, summarized in Table 2, indicatethat when giving an explanation to a friend, both types of explanations were considered appropriate. This was largely due tothe fact that students were asked this same question repeatedly, each time naming a different friend from class. This affordedthe students to choose different explanations for different friends. In fact, only two students (one high-ability student andone average-ability student) answered consistently for both concepts. These two students both chose PB explanations forboth concepts because these were the same explanations they each thought were most convincing. Another student chosedifferent explanations for different friends “in order to vary.” A different student chose to give a PB explanation for one friendbecause “it’s easiest to explain” but chose a MB explanation for a different friend because “it’s more difficult but she canunderstand it.” Another student also chose different explanations for different friends based on the perceived mathematicalability of the friend. Choosing a MB explanation for one friend she explained “she really understands the material and it (theexplanation) goes according to level (of mathematics ability).”

When it came to choosing an explanation that they would use if they were answering a question posed by the teacher,students were more decisive. Three high-ability students said that they would give PB explanations for both the parity andfractions concepts, four students (two high-ability and two average-ability) said they would give MB explanations for bothconcepts, and three said they would give a MB explanation for one concept but a PB explanation for the other. Some studentschose an explanation based on what would be easiest for them to explain or the easiest for them to remember. One studentwould give a MB explanation because “the teacher doesn’t need pictures. She understands.” On the other hand, all of thestudents agreed that Rose would equally accept any of the explanations. As one student stated, “they (the explanations) allhave the same meaning.”

5.2. Hailey’s classroom

5.2.1. BackgroundHailey spontaneously uses MB explanations on both the parity and fractions questionnaires. After reviewing the expla-

nations presented on the questionnaires, Hailey comments that all of the explanations are more like specific examples andthat they are missing a general definition at the end of the explanation. This remark is made regarding both the MB andthe PB explanations, “If we don’t define [even numbers] then for me it’s not enough. I think at the end there should be adefinition or a rule.” Nevertheless, Hailey chooses MB1 (divisibility by two) as the explanation which most convinces herthat 14 is an even number because it is most in line with her approach.

Regarding the students, Hailey chooses both MB3 (whole number quotient) and MB1 (divisibility by two) and explains:

I find it hard to imagine that in fifth grade someone would want this explanation (PB2) or this (PB1). That is, maybebecause someone has an affinity to pictures or thinks it’s cute or something. But I don’t really see that explanations


Table 3Frequency of types of spontaneous and most convincing explanations of Hailey’s students.

Context Spontaneous Most convincing

MB PB MB PB

Parity 22 1 11 12Equivalent fractions 20 – 10 11

like these (points to PB1 and PB2) are more convincing than these (points to MB1 and MB3). Even if they would saythat this (PB1) is more convincing, I don’t believe it really is the most convincing unless they think that this (PB2) isjust cute.

Hailey believes that all of her fifth grade students would be convinced by the MB explanations. She is so positive aboutthis that she believes that any child who claims otherwise is only drawn to the external trappings of the explanation andnot the content. It seems that for Hailey, to be convinced is to be certain about some truth. However, she is not sure that herstudents would relate in the same manner to the notion of being convinced.

The results of the students’ parity and fractions questionnaires showed that almost all the students in Hailey’s classspontaneously used MB explanations for both mathematical contexts (see Table 3). However, after being presented withvarious MB and PB explanations, there seemed to be no consensus as to which type of explanation was most convincingfor either concept. Students’ conceptions of a convincing explanation were similar to that of Rose’s students describedpreviously.

5.2.2. Hailey’s endorsed normsDuring her interview, Hailey comments that for the concept of even and odd numbers, she would not use a PB expla-

nation in the fourth, fifth, or sixth grades because “it’s more appropriate for a second grade level.” This is not to say thatHailey would never use PB explanations in the upper grades, “A child in the fourth, fifth, or sixth grade needs manipu-latives for other things. Like fractions.” In other words, according to Hailey, the use of PB explanations is dependent onthe grade as well as the mathematical context. Yet, even within a context and a grade level for which PB explanationsare considered useful, Hailey still sets limitations to their use: “Most of the children who need manipulatives only needthem once. I start the topic with manipulatives. We work them for the first two lessons. Very quickly the students don’tneed them.” Later on she adds, “they [the manipulatives] are very nice as an introduction but then a child doesn’t needthem.”

When asked which of the explanations on the fractions questionnaire she would give to a whole class, Hailey starts offby considering a fourth grade class:

H: I’m debating between this (points to MB1—twice equivalent fractions) and this (points to PB2—chocolate explanationwith illustration).I: What are your considerations?H: This (MB1) stresses the intuitive meaning of a fraction and is, after all, on a higher level than this (PB2). I always strivefor one step ahead. Let’s say that I prefer to be understood by 95% [of the students] using a higher level explanation thanbeing understood by 99% [of the students] using a lower level [explanation]. Because I want to train their heads. That is, thepurpose of teaching mathematics is not only to teach a topic. It’s more than just that. Therefore, I prefer to be understoodby a few less [students], but on a higher level of explanation because tomorrow this level will become the standard leveland that way I can pull their thinking, their understanding, forward.

Hailey clearly wants to create norms whereby the students will begin to exercise what she believes are higher levelthinking skills. This can be done, according to her opinion, by using MB explanations, as opposed to PB explanations. Haileyis also asked to consider individual students. Going back to the parity questionnaire she replies:

H: There are special types [of students]. There is Debbie who always wants to draw. Not because she’s not capable ofunderstanding this (MB2) or this (MB1). She just likes this (PB1 and PB2) because she’s very artistic.I: What about Hanna (a student considered to be of average to low ability in mathematics)?H: Even Hanna. Even a student like Karen (a student with learning disabilities). This (MB2) she can certainly understand.Even though Karen really has difficulties. Even she.

When asked which of the explanations she expects her fifth grade students to use she again reverts to MB explanationsclaiming, “these are more mathematical.” She then goes on to add that she expects all her students to use these (MB)explanations, regardless of what other teachers say.

Sometimes I have disagreements with other teachers. They say a certain child is not mature enough, he’s not capableyet, maybe in a few years . . . In my opinion, you can always say a child is not mature enough or capable. But instead


you can anticipate from him all the time that he can achieve more and in the end, even a student like Karen (a studentwith learning disabilities) can understand.

From all of the above statements, we learn three endorsed norms which are important to Hailey. First, PB explanationshave a limited use. Second, Hailey expects all of her students to always give MB explanations in class. Third, the teacher isexpected to challenge her students.

5.2.3. Hailey and her students’ enacted normsA total of nine lessons were observed over a six-month period. The contexts of these lessons varied and included such

topics as finding the least common multiple, addition and subtraction of fractions, subtraction of mixed numerals, andconverting fractions to decimals. In this section we give an overall picture of the sociomathematical norms observed duringthese lessons.

By mid-January, when the first lesson was observed, it was obvious that certain social norms were established. Studentsknew they were expected to explain their solutions to different problems. Still, if a student did not spontaneously offer anexplanation, Hailey was quick to ask why, or how come. She also prompted students to give different explanations. At times,Hailey offered a solution to some problem and asked the students to explain her solution. Equally important to the socialnorm of giving an explanation, was the social norm of evaluating explanations. Although Hailey was the prime evaluator,at times she turned to the class and asked, “What is your opinion?” At times, different students spontaneously offered theirown evaluation of a peer’s explanation. Many students of all mathematical abilities were observed participating during thelessons. Another social norm which emerged from the observations was that Hailey organized whole-class discussions inwhich she construed solutions in interaction with the students, although she usually took the lead leaving the students tofill in the blanks.

Regarding sociomathematical norms related to different types of explanations, the only time PB explanations wereobserved in use was when Hailey used them to introduce decimal fractions. For this lesson, Hailey brought to class base tenblocks to concretize the notion of decimal fractions. Even when using PB explanations, Hailey chose to use a well-constructedmathematical model and not an explanation based on real-life contexts. Subsequent observed lessons on decimal fractionsdid not include reference to the base ten blocks used in the introductory lesson.

The use of MB explanations, on the other hand, was prevalent throughout the lessons observed. For example, in mid-February, Hailey began a lesson by requesting her students to solve the multiplication problem 25 × 36 “quickly and byheart.” The following classroom discussion evolved:

Several students together: 20 × 30 + 20 × 6 + 5 × 30 + 5 × 6.Hailey: This uses the distributive property.S1: Multiply 6 by 25 and then 30 by 25.Hailey: You are correct. This is using the distributive property. But I want a quicker way.

Hailey bases her evaluation not only on validation of correctness but on thriftiness and simplicity. Furthermore, shecomments on previous mathematical properties that the students have already learned, such as the distributive propertyof multiplication over addition. The conversation proceeds with other students suggesting to round the numbers and onestudent giving the same break up as the first group did. And then finally,

S2: 25 is equal to 5 times 5 and 36 is . . .S3: 6 times 6.Hailey goes to the board and writes: 5 × 5 × 6 × 6.Hailey: What law do we have in multiplication?S3 and S4: The commutative law.Hailey: What does this law say?S4: That you can change the numbers.Hailey: You can change the order of the numbers. So, what can you do? I can multiply it in whatever order is good for me.S4: 6 times 5 and 6 times 5.

In the above segment we see again that Hailey is reminding the students of previously learned mathematical properties,in this case, the commutative property of multiplication. Hailey then proceeds to ask the students for yet another way tosolve this problem based on the same property and ends with 25 × 4 × 9.

One might argue that in the lesson described above, there was no real opportunity or reason to use PB explanations.One might also argue that the above explanations were mostly procedural. On the other hand, Hailey’s encouragement ofmultiple methods to solve the problem and the emphasis placed on mathematical properties, suggest her encouragementof MB explanations. In the following segment we describe a lesson based on fractions, a concept which many teachersexplain with PB explanations. Previously, students had already learned to add and subtract simple fractions. In this lesson,the teacher introduces subtraction of mixed numerals.


Hailey writes on the board:

Hailey: I would like to begin with the following example. Let’s think together. Where should we begin?S5: The half is three-sixths. Two-thirds is four-sixths.Hailey: What is Eden doing? Raise your hand. What does he want to do?S6: Expansion.Hailey: Why does he want to do expansion?S7: Common denominator.

Hailey writes on the board:

Hailey: From which side should we begin to solve this problem? From the fractions or the whole numbers?(Students debate from where to begin.)S4: From the fractions.Hailey: We start to solve this problem from the smallest place value. What does this remind you of? Which other problemsdo we start from the smallest place value?S8: Subtraction and addition (of multi-digit numbers) written in column form.Hailey: Correct.

First we note that in the above exchange Hailey did not once mention commonly used PB explanations such as pizza piesand cakes. She then continues:

Hailey: Can we take four-sixths away from three-sixths?S9: You do three and three-sixths is two and nine-sixths and then take away . . .Hailey: What is Ron saying? You can’t do three-sixths take away four-sixths. So, where do you take from? Fromwhere?S10: From the whole numbers.Hailey: And then you’re left with?S11: Two wholes.Hailey: And you add?S12: Six sixths.Hailey: Did you notice that you understood it without me teaching? You took away six-sixths. You broke up the whole. Justlike we solve subtraction (of multi-digit numbers). Let’s look at a regular problem. Let’s say 33 take away 14. Three takeaway four?S4: Can’t do.Hailey: So, you take from the tens digit. You write here two and how much do you add to the ones digit?S4: Ten.Hailey: Why? Because in the tens digit, how many ones are there?S11: Ten.

In the above exchange, Hailey makes explicit how subtraction of mixed numerals is connected to what the studentsalready know of multi-digit subtraction. Although Hailey’s explanation may be considered largely algorithmic, Fischbein(1993) considered algorithmic knowledge, knowing procedures as well as their justifications, a legitimate component ofmathematical knowledge. Although some algorithms are specific to a limited domain of numbers, Hailey is pointing outthat there exist algorithms which may continue to be valid as students expand their world of numbers. After doing out loudanother similar example, a student raises a question:

S12: Let’s say I had the problem 32 take away 34. I would have to do an exchange. You can’t do two minus four.

Hailey writes the example in column form on the board:


S12: So can you do the exchange?

Hailey writes on the board:S12: So, 12 take away four is eight. And then two take away three.S11: Is minus.Hailey: Minus one. But this −1 is in the tens digit. So it’s minus ten plus eight. Which is?

In the above exchange we see how the students follow the example set by the teacher. When teaching subtraction ofmixed numerals, Hailey emphasized that one should look at other mathematical concepts to explain the current situation.Adrian then applied this MB explanation to subtraction of whole numbers which results in negative numbers. Although thestudents have not been formally introduced to integers, they begin to see how mathematical operations may be understoodin an expanded number system when MB explanations are applied.

5.2.4. Students’ perceived normsRecall that individual interviews were conducted with ten students from Hailey’s class. The section begins by reviewing

the types of explanations students claimed Hailey would use in class. It then reviews the types of explanations studentsclaimed they would use out loud in class at the teacher’s request and those they claimed they would use as an explanationfor a friend.

Five students of mixed abilities claimed that Hailey would use only MB explanations for both the concept of parity andequivalent fractions. As one (low-ability) student stated, Hailey would only use MB explanations because “she would notstart telling stories . . . It doesn’t matter for who.” The other five students claimed that Hailey would use both types ofexplanations. For examples, an average-ability student said that Hailey would choose MB explanations for the concept ofparity because “we’re not in kindergarten.” However, when asked if Hailey would ever use one of the PB explanations, shesaid, “Yes, because there are kids who find it difficult. Even if they are in the fifth grade.” One high-ability student also statedthat Hailey would use both types of explanations, “She would use the second explanation (a MB explanation) for kids thatunderstand math a little better and these (PB explanations) for kids that know less.” This comment is similar to a commentmade by one of Rose’s students who also related PB explanations to low-ability in mathematics.

When it came to choosing an explanation they would use to explain a concept to a friend, students’ choices were varied(see Table 4). Four students of mixed abilities said they would use only MB explanations for both concepts, including twowho said that they personally preferred PB explanations. One average-ability student claimed that the MB explanationswere “simpler.” One high-ability student said he would use only PB explanations claiming “If I used something else, I mightnot really know how to explain it and then it wouldn’t be right.” In other words, he considered his own ability to give anexplanation rather that his friend’s ability to understand the explanation. The rest would use both types of explanations. Onestudent chose PB explanations for students who were non-Hebrew speakers. Another chose MB explanations for students(high-ability students) that she perceived could understand the concept without examples, “For kids that have difficulties Iwould give this explanation (a PB explanation) with pictures.”

Regarding the types of explanations students chose to use if they were answering a question posed by the teacher,responses were mixed and depended on the context (see Table 4). Five students (high and low-ability students) claimedthey would give MB explanations, regardless of the mathematical context. One student would only give PB explanations.

The reasons for choosing various explanations were varied. Some students who chose MB explanations claimed that theywouldn’t want to draw pictures on the board for the teacher. Likewise, one student chose a MB explanation because “it’seasier to write.” However, one student stated,

She (the teacher) always want us to be on a level to know . . . and this one (a PB explanation) is a picture and moreappropriate for fourth grade. So, if I would give her this (a PB explanation) she would tell me to explain it differently.

One student chose to give PB explanations and not MB explanations because, “You have to give an explanation for thewhole class and maybe someone wouldn’t understand (the MB explanation).”

Regarding the types of explanations Hailey would accept, most students said that Hailey would accept any explanation.One student said that Hailey, “always asks for additional explanations . . . she likes lots of explanations. A different studentsaid that Hailey would accept any explanation, “but prefers them short and to the point.” Another student added that “if theexplanation is not what Hailey thinks . . . she would give a different explanation.”

Table 4Frequency of types of explanations Hailey’s students would use.

Context For a friend Out loud in class

MB PB Both MB PB Both

Parity 4 2 4 6 3 1Equivalent fractions 4 1 5 8 2 –


6. Discussion

This paper set out to investigate three different aspects of sociomathematical norms related to MB and PB explanations:teachers’ endorsed sociomathematical norms, teachers’ and students’ enacted sociomathematical norms, and students’ per-ceived sociomathematical norms. In this section, we summarize our findings and begin to examine the relationships betweenthese three aspects.

Both teachers spontaneously gave MB explanations when asked to explain the different concepts discussed in this study.Both found MB explanations most convincing for themselves. Yet the sociomathematical norms they sought to establishin their classroom differed. Rose explicitly stated that PB explanations should be used (1) when a difficult concept, such asequivalent fractions, is being taught, (2) when any concept is being presented to the whole class since the aim is to reachmost students in the quickest amount of time possible, and (3) when a low ability student asks for help. Hailey clearlyendorsed the use of MB explanations in her classroom for all of her fifth grade students, regardless of their mathematicalability. Although she believed that PB explanations have a place in the mathematics classroom, she considered their use verylimited. Hailey set out to establish the norm of using MB explanations, looking ahead to what her students might need inthe future.

Most of the endorsed norms declared by Rose and Hailey were enacted by them in their classrooms. Rose used bothMB and PB explanations in her class. She accepted MB explanations from the students but constantly added her own PBexplanations. Perhaps Rose believed that by adding PB explanations she would be able to insure that all students understoodthe concept at hand. Hailey was observed using PB explanations only once when introducing a new concept to her students.On the other hand, she was frequently observed using MB explanations. She also encouraged the use of MB explanations bypointing out the relevant mathematical properties which may be used in different explanations.

The students’ enacted sociomathematical norms did not always coincide with the teachers’ endorsed and enacted norms.In both classes, students were observed using only MB explanations. This might have been expected in Hailey’s class. How-ever, in Rose’s class, this implies that the students frequently ignored the teacher’s implicit (and explicit) encouragementof PB explanations. In other words, for the most part, there was a consistency noted between the teachers’ endorsed normsand the teachers’ enacted norms. However, the sociomathematical norms endorsed by the teachers were not necessarilyenacted by the students. One possible reason for this discrepancy may be in the role each teacher played in the developmentof sociomathematical norms. Both teachers frequently demonstrated the types of explanations they preferred. However, ateacher should also evaluate the students’ explanations based on legitimacy and usefulness (Lampert, 1990; Yackel, 2002).For the most part, Rose accepted her students’ explanations without much comment leaving sociomathematical norms tobe established almost solely by her demonstrations of explanations. Furthermore, the absence of evaluations by the teachercould have caused the students to believe that any correct explanation was equally acceptable. Hailey, on the other hand,took a more active role in establishing the sociomathematical norms in her classroom. She not only validated a student’sexplanation, but she commented on why the explanation was acceptable by either providing the warrants and backingsfor an explanation or eliciting such supports from the students. She also commented on the usefulness of an explanation.Previous studies described classroom discussions in which the sociomathematical norms were negotiated (Ball & Bass, 2000;Yackel & Cobb, 1996). This, perhaps, is indicative of classrooms where discourse is encouraged. In Rose’s classroom, therewas a distinct lack of discourse regarding validity and usefulness of explanations. Perhaps this is another reason why thestudents’ enacted norms did not necessarily coincide with Rose’s endorsed norms.

Social and sociomathematical norms also guide what it means to participate and how to participate in the classroom. Inboth classes, it was clear that students’ participation entailed explaining solutions to problems. It was also clear that studentsperceived this as a norm. It was not clear, however, that participation was conditional on the type of explanation to be givenby the student. It is important to note that in both classes, students were observed using only MB explanations. However,during their interviews, students from both classes claimed that they would use both MB and PB explanations and that bothtypes of explanations would be acceptable by both teachers.

Had we only investigated endorsed and enacted norms, and not investigated the students’ perceived norms, we might betempted to think that as long as a teacher takes an active role in establishing sociomathematical norms, students will complywith those norms. In Rose’s class, the students’ enacted norms were not consistent with the teacher’s endorsed norms. Yet,in Rose’s class, only high-ability students were observed participating. Although it is true that 72% of the students in thisclass were considered to be of high-ability, we raise the question of whether a behavior can be considered normative ina heterogeneous class if only high-ability students are observed acting according to this norm. In addition, students whowere interviewed claimed that Rose would use both types of explanations. They also claimed they would use both typesof explanations out loud in class. A few students chose explanations according to what they personally preferred, showinga relationship between individual preferences and the use of explanations in the classroom. For their classmates, studentsin Rose’s class would mostly give explanations based on ability. In other words, on a one-to-one basis for their friends, thestudents adopted Rose’s norm of choosing MB explanations for high-ability students and PB explanations for low-abilitystudents.

In Hailey’s class, where students of various abilities were observed using MB explanations, we may be tempted to thinkthat Hailey was indeed successful in developing the norms she endorsed. Yet, only one student clearly recognized thatHailey was trying to establish norms related to the types of explanations used in class. Furthermore, contrary to Hailey’sendorsed norms, two students claimed that Hailey would use different explanations depending on the student’s math-


ematical ability. In general, half of the interviewed students expected the teacher to use only MB explanations and halfsaid that they themselves would use only MB explanations. How do we interpret this information? Do we stress thefact that not all of the students perceived the norms that Hailey both endorsed and enacted in the classroom? Or, dowe say that some of the students expected Hailey to use MB explanations for both concepts and the rest claimed thatshe would use both MB and PB explanations? This interpretation stresses that in general, MB explanations were expectedto be used more often than PB explanations by both the teacher and students in Hailey’s class. Which interpretation iscorrect?

Even if we choose the latter interpretation, we should also consider the reasons Hailey’s students gave for using MBexplanations in various circumstances. Many chose these explanations because they were short, or quick, or simple. Theydid not choose MB explanations necessarily because of the mathematical content of the explanation. Nor did they claim tochoose MB explanations because they perceived those explanations as more acceptable than PB explanations. Can we saythat the use of MB explanations was normative even if the underlying reasons for using these explanations were not quitewhat the teacher had in mind?

This study has shown that intending to establish a particular sociomathematical norm may not guarantee that this normwill indeed be perceived by all students in the class. What does it mean if the norms are not perceived by the students?Does it mean that they do not exist? Are certain norms more difficult to establish than others? Certainly, the answersto these questions are beyond the scope of this paper. However, they do raise important issues that need to be furtherinvestigated.

Although the framework of research used in this study assisted in untangling the different components of sociomathe-matical norms, it also highlighted the components not investigated in this research study. The term, endorsed norms, may beused for both teachers and students (Ben-Yehuda et al., 2005). This study focused on the teachers’ endorsed norms. It did notinvestigate those norms which students claim ought to be established in their classrooms. Likewise, this study investigatedthe students’ perceived norms. It did not investigate the teachers’ perceived norms, the norms which teachers claim to be inplace in their classrooms. The complexity of investigating sociomathematical norms is compounded by the need to inves-tigate the individual participants, before, during, and after classroom observations, in order to gain a clearer picture of theclassroom culture. Future research involving sociomathematical norms should consider taking these additional perspectivesinto consideration by investigating teachers’ and students’ endorsed norms, teachers’ and students’ enacted norms, as wellas teachers’ and students’ perceived norms.

References

Ball, D., & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.),Yearbook of the national society for the study of education, constructivism in education. Chicago, IL: University of Chicago Press.

Ben-Yehuda, M., Lavy, I., Linchevski, L., & Sfard, A. (2005). Doing wrong with words: What bars students’ access to arithmetical discourses. Journal forResearch in Mathematics Education, 36(3), 176–247.

Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17, 25–64.Cobb, P., McLain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1–78.Fischbein, E. (1993). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. Scholz,

R. Straber, & B. Winkelmann (Eds.), Didactics of Mathematics as a Scientific Discipline (pp. 231–245). Dordrecht, the Netherlands: Kluwer.Hershkowitz, R., & Schwarz, B. (1999). The emergent perspective in rich learning environments: Some roles of tools and activities in the construction of

sociomathematical norms. Educational Studies in Mathematics, 39, 149–166.Israel national mathematics curriculum (2006). Retrieved December 10, 2008, from http://cms.education.gov.il.Ju, M., & Kwon, O. (2007). Ways of talking and ways of positioning: Students’ beliefs in an inquiry-oriented differential equations class. Journal of Mathematical

Behavior, 26(3), 267–280.Kazemi, E., & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. The Elementary School Journal, 102(1),

59–80.Lampert, M. (1990). When the problem is not the question and the solution is not the answer. American Educational Research Journal, 27(1), 29–63.Levenson, E., Tirosh, D., & Tsamir, P. (2006). Mathematically and practically-based explanations: Individual preferences and sociomathematical norms.

International Journal of Science and Mathematics Education, 4, 319–344.Levenson, E., Tsamir, P., & Tirosh, D. (2007a). First and second graders’ use of mathematically-based and practically-based explanations for multiplication

with zero. Focus on Learning Problems in Mathematics, 29(2), 21–40.Levenson, E., Tsamir, P., & Tirosh, D. (2007b). Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero. Journal of Mathematical

Behavior, 26, 83–95.Martin, T., McCrone, S., Bower, M., & Dindyal, J. (2005). The interplay of teacher and student actions in the teaching and learning of geometric proof.

Educational Studies in Mathematics, 60, 95–124.McClain, K., & Cobb, P. (2001). An approach for supporting teachers’ learning in social context. In F. Lin, & T. Cooney (Eds.), Making sense of mathematics

teacher education (pp. 207–232). Dordrecht, The Netherlands: Kluwer.Perry, M. (2000). Explanations of mathematical concepts in Japanese, Chinese, and U. S. first- and fifth-grade classrooms. Cognition and Instruction, 18(2),

181–207.Rivera, F. (2007). Accounting for students’ schemes in the development of a graphical process for solving polynomial inequalities in instrumented activity.

Educational Studies in Mathematics, 65(3), 281–307.Sfard, A. (2000). On reform movement and the limits of mathematical discourse. Mathematical Thinking and Learning, 2(3), 157–189.Stylianou, D., & Blanton, M. (2002). Sociocultural factors in undergraduate mathematics: The role of explanation and justification. In Proceedings of the

second annual conference on the teaching of mathematics Crete, Greece,Szydlik, J., Szydlik, S., & Benson, S. (2003). Exploring changes in pre-service elementary teachers’ mathematical beliefs. Journal of Mathematics Teacher

Education, 6(3), 253–279.Toulmin, S. (1969). The uses of argument. Cambridge: Cambridge University Press.Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21(4), 423–440.Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 22,

390–408.

http://cms.education.gov.il/


Yackel, E., Cobb, P., & Wood, T. (1993). The relationship of individual chilren’s mathematical conceptual development to small-group interactions. In T.Wood, P. Cobb, E. Yackel, & D. Dillon (Eds.), Rethinking elementary school mathematics: Insights and issues (pp. 45–54). National Council of Teachers ofMathematics.

Yackel, E., Cobb, P., & Wood, T. (1998). The interactive constitution of mathematical meaning in one second grade classroom: An illustrative example. Journalof Mathematical Behavior, 17, 469–488.

Yackel, E., Rasmussen, C., & King, K. (2000). Social and sociomathematical norms in an advanced undergraduate mathematics course. Journal of MathematicalBehavior, 19, 275–287.

Documents

Students’ perceived sociomathematical norms: The missing paradigm