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Exploring Collective MathematicalCreativity in Elementary School

ABSTRACTThis study combines theories related to collective learning and theories related

to mathematical creativity to investigate the notion of collective mathematicalcreativity in elementary school classrooms. Collective learning takes place whenmathematical ideas and actions, initially stemming from an individual, are builtupon and reworked, producing a solution which is the product of the collective.Referring to characteristics of individual mathematical creativity, such as fluency,flexibility, and originality, this paper examines the possibility that collective math-ematical creativity may be similarly characterized. The paper also explores therole of the teacher in fostering collective mathematical creativity and the possiblerelationship between individual and collective mathematical creativity.

Many studies have investigated ways of characterizing, identifying, andpromoting mathematical creativity. Haylock (1997), for example, and morerecently, Kwon, Park, and Park (2006) assessed students’ mathematical creativ-ity by employing open-ended problems and measuring divergent thinking skills.Leikin (2009) explored the use of multiple solution tasks in evaluating a student’smathematical creativity. These studies focused on an individual’s mathematicalcreativity as it manifests itself in the solving of various problems. Yet students,acting in a classroom community, do not necessarily act on their own. Ideas areinterchanged, evaluated, and built-upon, often with the guidance of the teacher.The resultant mathematical creativity of an individual may be a product of collec-tive community practice. The question which then arises is: Who is beingmathematically creative, the individual or the community? This study focuses onthe collective, not as the aggregation of a few individuals, but as a unit of study.Although some of the studies mentioned above acknowledged the effect of class-room culture on the development of mathematical creativity, and others consid-ered the creative range of a group of students, those studies did not necessarilyinvestigate mathematical creativity as a collective process or as the product ofparticipating in a collective endeavour.

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Exploring Collective Mathematical Creativity in Elementary School

This study combines theories related to collective learning and theories relatedto mathematical creativity to investigate the notion of collective mathematicalcreativity. The notion of collective creativity has been used to investigate creativ-ity in several contexts including the work place (Hargadon & Bechky, 2006) andthe global community (Family, 2003). In those cases, collective creativity wasconsidered to occur when the social interactions between individuals yieldednew interpretations that the individuals involved, thinking alone, could not havegenerated. Can the notion of collective creativity also be applied to the classroomcommunity?

THEORETICAL BACKGROUNDTwo main issues are at the heart of this study, the nature of mathematical

creativity and the collective nature of mathematical learning. This section beginsby situating the study within the extant literature of creativity and then moveson to the specifics of mathematical creativity. The next section describes litera-ture related to group creativity and more specifically, the collective nature ofmathematical learning.

Views on creativityAs opposed to studies which are concerned with the creativity of a few eminent

persons who have made a significant and lasting contribution to society (some-times known as Big-C creativity), this study is concerned with everyday creativity(little-c creativity) as it manifests itself in classrooms. Often, students experiencecreative insights as they learn a new concept. This type of creativity, calledmini-c creativity by Kaufman and Beghetto (2009), focuses on the “novel andpersonally meaningful interpretation of experiences, actions, and events” (p. 3).This view is in line with Runco’s (1996) view of creativity as “manifested in theintentions and motivation to transform the objective world into original interpre-tations, coupled with the ability to decide when this is useful and when it is not”(p. 4). Both views focus on intrapersonal insights and interpretations. Runco,however, adds the aspect of discretion. This aspect is necessary in order to differ-entiate between divergent thinking as a component of creativity and the produc-tion of random useless ideas. Divergent thinking is often measured in terms ofthe fluency, flexibility, and originality of ideas produced. Fluency may be mea-sured as the total number of unduplicated ideas generated (Jung, 2001). Flexi-bility and originality may be more dependent on the context of the problem. Forexample, in Jung’s (2001) study, three subcategories of flexibility (based onTorrance’s (1965) work) were assessed: adaption, addition, and substitution).Combined with intrapersonal evaluation processes which are critical and selec-tive (Runco & Smith, 1992), divergent thinking may lead to creative ideas whichare both original and meaningful.

Fluency, flexibility, and originality have also been cited as components ofmathematical creativity and have been assessed relative to the domain. Situatingcreativity within a domain, such as mathematics, reminds us that creativity may

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also be subject to the rules of that specific domain. Thus, evaluation processes,both intrapersonal and interpersonal, may take into consideration the rules andaesthetics of that domain. Situated within the field of mathematics education,this study adopts the view that mathematical creativity is “an orientation or dis-position toward mathematical activity that can be fostered broadly in the generalschool population” (Silver, 1997, p. 75). As such, the product of mathematicalcreativity in the classroom may be original ideas that are personally meaningfulto the students and appropriate for the mathematical activity being considered.

Several studies have investigated the promotion and evaluation of mathemati-cal creativity in the classroom. Silver (1997), for example, claimed that the use ofill-structured and open-ended problems in mathematics instruction may encour-age students to generate multiple solutions, in turn encouraging the developmentof fluency. Thus, fluency in mathematics education is quite similar to fluency inmost other fields. Leikin (2009) evaluated flexibility within the domain of math-ematics education by establishing if different solutions employ strategies basedon different representations (e.g., algebraic and graphical representations), prop-erties, or branches of mathematics. The notion of flexibility was also used wheninvestigating teacher-student interactions where teacher flexibility referred toadjusting the planned learning trajectory according to student replies (Leikin &Dinur, 2007). In other words, when a student’s response is unexpected or incon-venient for the teacher’s agenda, does the teacher adjust accordingly? Thisnotion of flexibility is also related to flexibility in creative mathematical thinking.When solving a problem, how does the student react when coming upon anunexpected obstacle or result? Does the student exhibit flexibility and perhapstry a different procedure or does he stubbornly plod along?

At times, it helps to think of flexibility in relation to its counterpart, fixation. Inproblem solving, fixation is related to mental rigidity (Haylock, 1997) or self-restrictions (Krutetskii, 1976). Flexibility is then shown by overcoming fixation orbreaking away from stereotypes. This may be a form of intrapersonal evaluation.Haylock further differentiated between content-universe fixation and algorithmicfixation. Overcoming the first type of fixation requires the thinker to (intention-ally) consider a wider set of possibilities than at first is obvious and extend therange of elements appropriate for application. This is reminiscent of Runco’s(1996) perspective on creativity being an intentional act. For example, elemen-tary school students asked to find two numbers whose sum is 18 may miss 18and 0 or may not think of fractions because they do not consider the possibility ofnumbers which are not natural. The second type of fixation relates to when anindividual adheres to an initially successful algorithm even when it is no longerappropriate. This type of fixation relates to the familiar case of a student who isrequested to calculate 20×20 and resorts to the long multiplication algorithm,though it is clearly unnecessary in this case.

According to some theories, originality is related to creating new ideas. Forexample, the systems model of creativity suggests that when an individualemploys the rules and practices of a domain to produce a novel variation within

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the domain content, then that individual is being creative (Sriraman, 2009). Withregard to mathematics classrooms, this aspect of creativity may manifest itselfwhen a student examines many solutions to a problem, methods or answers, andthen generates another that is different (Silver, 1997). In this case, a novel solu-tion infers novel to the student or to the classroom participants. Leikin (2009)measured the originality of a solution based on its level of insight and convention-ality according to the learning history of the participants. For example, a solutionbased on a concept learned in a different context would be considered originalbut maybe not as original as a solution which was unconventional and totallybased on insight.

The nature of group creativity and collective mathematical learningIn general, group creativity relates to the generation of creative ideas by groups

when the interactions and inputs of several people are considered. One line ofresearch related to group creativity stems from the group-brainstorming para-digm (Osborn, 1957) where each group member is explicitly requested to gener-ate ideas, listen to the ideas of others, and combine or build on shared ideas.Results of group-brainstorming studies have shown that it is not always the casethat more ideas are generated by the group than if the individuals sat on theirown and the sum of all ideas was collected (e.g., Paulus, Larey, & Dzindolet, 2000).In other words, the creative product is not necessarily enhanced in a group set-ting. In attempting to analyze why this is so, we may consider interpersonal evalu-ation skills which may affect creativity, individually and in the group. An individualin a group must evaluate the idea of another, select appropriate ideas, and buildon them. Paulus and Yang (2000) claimed that deficient results of group creativ-ity may be caused by the group members not being attentive to the ideas of thegroup. Individuals may find it difficult to generate their own ideas while carefullylistening to the ideas of others. It may also be that even if the individual is atten-tive to other ideas, there is not enough incubation time to reflect on those ideas inorder to integrate them with one’s own ideas.

Another perspective on group creativity considers the situation where diverseindividuals come together to solve a problem (such as in the work place). This isnot quite the same as brainstorming where individuals are specifically requestedto come up with many ideas. Instead, this situation may start with divergentthinking, but eventually, ideas must converge in order to solve the problem athand. On the one hand, the different backgrounds and knowledge base of adiverse group may contribute different perspectives for consideration. On the otherhand, diversity may be so wide as to hinder individuals as they strive to under-stand different ideas and come up with an agreed-upon solution (Kurtzberg &Amabile, 2001).

Several studies have proposed suggestions which may enhance interpersonalor group creativity. For one, a supporting social context, where the group partici-pants feel comfortable raising novel ideas, may impact on the group’s creativeendeavours (Amabile, 1998). Hargadon and Bechky (2006) proposed four types

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of social interactions that may enhance collective creativity: help seeking, helpgiving, reflective reframing, and reinforcing. These interactions help participantsto draw from and reframe past experiences in ways which may lead to new andvaluable insights. Family (2003) pointed out that a sense of responsibility bythe individual towards the collective may encourage individuals from diversebackgrounds to work together in order to solve a common problem. In thissense, collective creativity infers a coming together of ideas in order to solveone problem.

Finally, there is the role of the group leader. In his review on leadership stylesand group creativity, Jung (2001) mentioned the leader’s role in providing a safeenvironment, where group members can try out novel ideas and question theirown, as well as the leader’s, values and beliefs. In addition, intellectual stimula-tion may help group members look at a problem from a different perspective. Inhis study, Jung (2001) found that transformational leadership promotes higherlevels of group creativity, as measured by divergent thinking. This type ofleadership involves “active and emotional relationships between leaders andfollowers” (p. 187).

Within learning contexts, creativity may start as a social function mediated bycultural tools such as language (Vygotsky, 1978). Thus, in the classroom, collec-tive creativity may be considered the result of social interactions between indi-viduals. Within mathematics education, Yackel and Cobb (1996) introduced thenotion of sociomathematical norms to describe “normative aspects of mathemati-cal discussions that are specific to students’ mathematical activity.” (p. 458).They were interested in how normative aspects of mathematics discussion aredeveloped, such as what counts as mathematically different, mathematicallyefficient, and mathematically elegant. They argued that the roles of the teacher indeveloping such norms include indicating which solutions are desirable, whichsolutions need more explaining, and in general to encourage reflective activitiessuch as having the students compare solutions.

According to Martin, Towers, and Pirie (2006) collective mathematical under-standing emerges from coactions. Coactions describe particular mathematicalactions carried out by an individual but which are “dependent and contingentupon the actions of the others in the group” (p. 156). Collective understandingdoes not necessarily occur whenever two or more people collaborate or interact.Instead, “coacting is a process through which mathematical ideas… initially stem-ming from an individual learner, become taken up, built on, developed, reworkedand elaborated by others…” (p. 156). This notion builds on Sawyer’s (2000, 2004)theory of group improvisational performance which emphasizes the unpredict-able flow of ideas and actions which emerge from the performers workingtogether and responding to each other. Improvisational performance may thenbe tied into creativity in the classroom in the sense that participants are notfollowing a script. All participants, including the teacher, must be flexible.

As in group creativity studies, the leader, in this case the teacher, has animportant role in collective mathematical learning. The teacher’s role is to

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facilitate classroom discussion, providing structure to the creative process. View-ing the teacher’s role as a facilitator, Sawyer (2004) also stressed the high degreeof pedagogical content knowledge necessary for responding to unexpectedstudent queries as well as the managerial skills necessary for managing groupimprovisation.

RESEARCH AIMSAccording to Haylock (1997) there are two main approaches to recognizing

creativity. The first is to consider the cognitive process which is indicative of cre-ative thinking. Overcoming fixation is one such process. The second approach isto consider the product which indicates creative thinking has taken place. Anoriginal and mathematically appropriate solution would be one such product.Components of creativity such as fluency, flexibility, and originality, are some-times related to the process (Shriki, 2010) and sometimes related to the product(Haylock, 1997). The first aim of this study is to explore both the process in-volved in collective mathematical creativity as well as the product of collectivemathematical creativity. Specifically, it examines if and how the notions of flu-ency, flexibility, and originality used when describing individual mathematicalcreativity may also be used to describe the process and product of collectivemathematical creativity.

The collective in this study refers to classroom participants. Classroom par-ticipants include not only the students but also the teacher. Thus, the secondaim of this study is to investigate the teacher’s role in facilitating and promotingcollective mathematical creativity.

Finally, although the aim of this study is to explore the emergence of collectivemathematical creativity and describe its nature in elementary school mathemat-ics classes, it also acknowledges the significance of individual mathematical cre-ativity. Thus the third aim of this study is to explore the interrelationship betweenindividual mathematical creativity and collective mathematical creativity.

METHODTwo fifth grades and one sixth grade class participated in this study. Hailey

(not her real name) taught the sixth grade class and one of the fifth grade classes.In her sixth grade class there were 28 students, 16 girls and 12 boys. In her fifthgrade there were 32 students, 15 girls and 17 boys. She had 14 years experienceteaching fifth and six grades. Nina (not her real name) taught the second fifthgrade class. There were 28 students in the class, 12 girls and 16 boys. Shehad eight years experience teaching fifth and sixth grades. Both teachers taughtaccording to the mandatory mathematics curriculum using state approved text-books. Both taught in local public schools located in the same middle-incomesuburb of Tel Aviv (a major city in Israel). The teachers did not collaborate withthe researcher and were not explicitly implementing a program aimed at promot-ing creativity. During the school year, each class was observed approximately tentimes. During classroom observations, the focus was on students’ interactions

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with materials, other students, and teachers and the ways in which “ideas arepicked up, worked with, and developed by the group” (Martin, Towers, & Pirie,2006, p. 152). All lessons were video recorded and transcribed by the researcherwho also took field notes during the observations.

RESULTSThis section describes four classroom episodes that illustrate different aspects

of mathematical creativity. Each episode begins with a description of the mainaim of the lesson. It then continues by relaying the classroom interactions andthe mathematical creativity, individual and collective, observed. Pseudo nameswere employed to protect the identity of students.

Episode 1: Collective fluency and collective flexibility?This episode was taken from Hailey’s sixth grade class, in the middle of the

school year, where the main topic of the lesson was multiplication of decimalfractions. The class had already been introduced to this topic and had alreadypracticed the procedure for multiplying decimal fractions during previous les-sons. The teacher put the following problem on the board, __ × __ = 0.18, andasked the class, “What could the missing numbers possibly be?” Many childrenraised their hands and the teacher commented, “There are many possibilities.”She then called on one at a time:

Gil: 0.9 times 0.2.Teacher: Another way. There are many ways.Lolly: 0.6 times 0.3.Teacher: More.Tammy: 0.90 times 0.20.Teacher: Would you agree with me that 0.2 and 0.9 is the same [as 0.90

and 0.20]? I want different.Miri: I’m not sure. 9 times 0.02.Teacher: Nice. Can someone explain what she did?(The teacher and students then review the rules for multiplying decimalfractions.)

First, we note that although Gil and Lolly gave different answers both answersmay be considered similar in that they consisted of two numbers with one digitafter the decimal point. The teacher does not acknowledge their similarity. How-ever, when Tammy attempts to break the mould, the teacher does not acceptthe answer because 0.9 is equal to 0.90. This relates to sociomathematicalnorms (Yackel & Cobb, 1996) and the establishment of what it means for solu-tions to be different. In this case, the teacher is establishing that merely expand-ing a number from 0.9 to 0.90 does not qualify for difference. She then requestsa different solution, which is supplied by Miri and addresses the class asking them

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if someone can explain Miri’s solution. In other words, there is an expectation thatalthough Miri supplied the solution, others would have listened and would beable to explain Miri’s solution. This is essential if the group is going to act as acommunity and produce different solutions.

Tom: What about 0.18 times 0.1?

Tad: No.

Teacher: Why not?Mark: [The answer would be] 0.018 because there would be three

digits after the decimal point.

Teacher: Ah. Ok. Thank you. We want a number with two digits afterthe decimal point.

Gad: 0.18 times 1.

Ben: And 1 times 0.18.

Teacher: You’re using the commutative property of multiplication. But,it’s really the same as Gad’s answer.

Toby: 18 times 0.1?

Many students: 18 times 0.01.Teacher: Let’s move to another problem. (The teacher writes on the

board the following problem: __ × __ = 0.012.)

This episode illustrates how one child may have the germ of an idea but an-other child may develop it. In the first vignette, Gil, factoring 18 into 9 and 2,comes up with one solution. Tammy attempts to use the same factors as Gil, butis not successful. Miri then follows up on the idea, producing an additionalsolution. The same scenario occurs in the second vignette. Tom has the idea offactoring 18 into 1 and 18 but comes up with an incorrect solution as witnessedby Tad. Gad follows up on the idea and comes up with a correct solution. Benattempts to follow up on Gad’s idea using the commutative property of addition,but, as in the first vignette, the teacher does not accept this solution as beingdifferent from the previous one. Toby also attempts to find a solution with thesame basic factors which is then corrected by other students in the class.

In this episode we see an illustration of collective fluency which seems to havebeen promoted by the teacher. The teacher has the class working together toproduce many different solutions to the same problem. Up until this point in thelesson, the class came up with five different correct solutions. Perhaps, if moretime was available, the class may have produced more solutions.

Regarding flexibility, we note that the second solution, 0.6 × 0.3 followed moreor less the same strategy as the first solution 0.9 × 0.2. The last three solutions,9 × 0.02, 0.18 ×1, and 18 × 0.01, may also be considered similar to each other.Each example consists of one factor which is a whole number and a secondfactor which is a decimal fraction with two digits after the decimal point. Thus, it

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seems as though this problem promoted collective fluency but not necessarilycollective flexibility. Only one student, Miri, was successful in employing a differ-ent strategy, a sign of flexibility. And yet, Miri’s solution came after Tammy’s sug-gestion. Although the teacher did not accept Tammy’s solution as being differentfrom the first, it was Tammy who attempted a solution with two digits after thedecimal point. Perhaps, the flexibility manifested in Miri’s correct solution wasthe result of working on Tammy’s suggestion. Perhaps, Miri understood what wasmeant by a solution being different and had the courage to think flexibly afterTammy paved the way. In other words, it is possible that in this case collectiveflexibility refers to a collective process and not necessarily that the group, as awhole, produced solutions based on different strategies.

Episode 2: Individual or collective originality?This episode took place in Nina’s fifth grade classroom. The students had pre-

viously been introduced to decimal fractions, had learned to convert back andforth between decimal fractions and simple fractions, and had recently learned toadd and subtract decimal fractions. The main topic of the current lesson wasreviewing addition and subtraction of decimal fractions. The following problem,taken from the classroom textbook, was given as a homework assignment, and,at the request of one of the students, was reviewed in class.

Complete the following sequence:5 30____, ____, ____, ____, ____

100 100 100 100 100

The following discussion ensues:

Teacher: After 30/100, mmm hundredths, and again, mmm hundredths, andagain. They want a sequence. What is a sequence?

Uri: It continues with jumps.

Teacher: Equal jumps. The jumps must be equal. What types of jumps are there?

(A few students say out loud different numbers: 25 and 25/100.)

Teacher: That’s the size of the jump. You mean to add 25/100.

Uri: Can the jumps be in multiplication?

Teacher: Wonderful. That’s exactly what I mean. If I jump by adding 25/100then the next will be 50/100. Now, you mentioned another type ofjump. We didn’t learn that yet…it’s part of next year’s syllabus. But,there are also multiplication jumps. Who said that going from5/100 to 30/100 means that I added 25 [hundredths]. I can alsomultiply . . .

Sam: By 6.

Sarah: 180.

Teacher: 180 hundredths.

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At this point it is worthwhile to note that although multiplication of simple frac-tions and decimal fractions had not yet been introduced, Uri came up with thisidea by himself. Furthermore, although it is technically part of next year’s cur-riculum, the teacher does not dismiss this idea. Finally, two more children con-tribute to the idea by carrying out the actual multiplication

Uri: That’s what I did at first. But I thought it was a mistake.Teacher: Is that allowed?Tina: I thought that it would be a mistake.Nat: But then you get big numbers.Teacher: So, you can use a calculator.

Note that at least two children thought of multiplying but ended up dismissingthe possibility for various reasons. Nat, who claims that he would end up with“big numbers”, shows signs of content fixation. This may have been brought onby previous textbook examples which refrained from using ‘big’ numbers. Theteacher is quick to negate this excuse, keeping the door open for additional pos-sibilities. Nat then takes Uri’s idea one step further, by considering division.

Nat: Then you can also divide.Teacher: You can divide, but not here (referring to the jump from 5/100 to

30/100).Nat: You can multiply by 6 and then divide by 3.Teacher: Ok. That’s also a type of sequence. Multiply by 6, divide by 3,

and then again multiply by 6 and divide by 3.Tina: But, that’s not good. You need equal numbers.Teacher: This is a different type but it is certainly acceptable. Let’s try it.Tina: But, it won’t come out. You need equal numbers.Teacher: Let’s just say that when the textbook requests a sequence, they

generally don’t mean this type. They usually mean jumps thatare the same each time. But, this is definitely a sequence.

Tina: But, they are not all equal.Dan: You can also have more than two types of jumps. Multiply, divide,

and add.

Nat expands on the idea of multiplication and division jumps by including bothin the same sequence. This is a novel idea, which the teacher accepts. In a way, itis surprising that the teacher accepts this possibility. Recall that in the beginningof this episode, the teacher claimed that the jumps in a sequence must be equal.It could be that, in the beginning, the teacher was not so much concerned withcontinuing a sequence as she was concerned with reviewing addition and sub-traction of decimal fractions. After all, this problem had been given as a home-work assignment with the intention of reviewing the current mathematical content

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being learned. It could also be that, as she later claims, the norm of the textbookis to build sequences with equal jumps. Thus, her first instinct was to claim thatsequences must have equal jumps. Yet, she is flexible. While conceding thatthis type of sequence may not be the norm, the teacher attempts to legitimizethinking that may be unconventional, as long as it is mathematically acceptable.Acknowledging that it is mathematically acceptable to build a sequence withunequal jumps, she changes the direction of the lesson. She improvises her origi-nal plan of reviewing addition and subtraction of decimal fractions to allow thestudents time for considering different mathematical sequences. At the end ofthis vignette, Tina remains unconvinced that two types of jumps may also beconsidered a sequence. On the other hand, Dan, who has previously remainedsilent, but apparently has been listening, joins in and adds yet another novel idea.

Looking back, it is apparent that several students displayed original ideas. First,we have Uri. The children had not yet learned multiplication of fractions and yet,he considers the possibility. Then there is Nat who suggested employing bothmultiplication and division in the same sequence and raises the possibility thatthe jumps do not have to be equal. Nat raises this idea despite that the teacherhad previously said that a sequence consists of equal jumps. In other words, Natchallenges the convention. Finally, Dan suggests employing three mathematicaloperations at once in the same sequence. Perhaps, employing multiplication anddivision may be allowed because they are essentially inverse operations. But toconsider addition in the same sequence is indeed novel. So, who displayed origi-nal thought? Certainly, Uri, Nat, and Dan suggested novel ideas. Yet, looking atthe sequence of events, it also seems that each student built on the previousstudent’s idea. So, was Uri the only student to display original thinking? Now,consider the teacher. Her role, it seems, was to legitimize the novel ideas pre-sented by her students, opening the way for further solutions. Perhaps, takentogether, we can say that this is an illustration of collective originality.

Episode 3: Individual or collective flexibility?This episode is actually a continuation of the previous episode. Nina presents

another problem from the book.

Teacher: Let’s look at another problem. Build a sequence that has in it thenumbers 0.2 and 1.1.

(Four children raise their hands.)

Dan: Add 0.9.

Teacher: You’re saying to place them next to each other and then thedifference is 0.9. Then what would be the next number?

Judy: 2.

Teacher: And then?

Mark: 2.9.

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Teacher: But you can make a different sequence. Who says that the twonumbers have to be next to each other? It doesn’t say that theyhave to be one next to the other.

Tali: You can do jumps of 0.3.Teacher: Jumps of 0.3. Let’s see. What would come next? (The teacher

writes on the board 0.2, 0.5, 0.8, 1.1.)Dan: You can put the sequence in backwards order and do subtraction.Teacher: Ok. You can start with 1.1.

In this episode, the class is working on an open-ended or ill-defined task.Unlike the previous task in the previous episode where the first two numbers inthe sequence were given, in this task, two numbers are given but are not placedin any specific order. The teacher takes advantage of the situation in order topromote flexibility. In other words, she seems to be less interested in promotingfluency and more interested in trying to encourage the students to think of vari-ous ways of placing the numbers. She then raises another suggestion, moving inan entirely different direction than the ones suggested by the students.

Teacher: I have another idea. You can expand the numbers. (The teacherwrites on the board 0.20, leaves a lot of space, and then writes1.10.)

Tomer: 0.9.Shay: Nine and a half.Teacher: 0.90 so the expansion is by 10 and then I can do jumps of 0.45. Is

that allowed?Tomer: Yes.

This last part is interesting because it seems that the teacher is also displayingflexible thinking and joining the collective effort to come up with various ways ofplacing the two numbers in a sequence. If we look at the different solutions to thisproblem, we may count four solutions where each solution stems from a verydifferent way of combining the numbers into a sequence. Perhaps, we can callthis collective flexibility.

Episode 4: Extending the range of applicationThis episode was taken from Hailey’s fifth grade class where Hailey was intro-

ducing for the first time subtraction of mixed numerals. The following example iswritten on the board:

Teacher: Let’s think together. We’ll solve this problem in column form. Whatdo I have to do?

(Many students raise their hands.)Ian: One half is three-sixths. Two-thirds is four-sixths.

13 __22— 1 __3________

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Teacher: Now, wait a minute. What did Ian do? Raise your hands. What didIan want to do?

Penny: Expansion.Teacher: He wants to do expansion so both fractions will have what?Abe: So, they’ll have the same six so that you can solve it. And there

is a half which is three out of the whole — uh — six. And two-thirds is 4 . . .

Teacher: So, what is he doing?Harold: A common denominator.Teacher: So, what should I write here?Harold: Three out of six.Teacher: You lost something on the way.Miri: The half.Abe: The three wholes.Teacher: So, it’s three and three-sixths.The teacher, together with her students, brings the second fraction also to a

common denominator and rewrites the problem:

The following discussion then ensues:Teacher: From which side should we begin to solve this problem? From

the fractions or the whole numbers?(Students debate from where to begin.)Miri: From the fractions.Teacher: We start to solve this problem from the smallest place value. What

does this remind you of? Which other problems do we start fromthe smallest place value?

Lev: Subtraction and addition (of multi-digit numbers) written incolumn form.

Teacher: Correct.

The teacher then poses the problem of taking away 4/6 from 3/6 and togetherwith the students they exchange the 3 and 3/6 to 2 and 9/6 and proceed tosubtract. To summarize the procedure, the teacher once again reminds the classthat subtracting fractions is essentially the same as subtracting whole numbersclaiming, “You broke up the whole. Just like we solve subtraction (of multi-digitnumbers) . . . say 33 take away 14 . . . you take from the tens digit and add to theones digit.”

While the above vignette seems long and tedious and void of creativity, it illus-trates the role of the teacher in scaffolding the mathematics learning, in providing

33 __64— 1 __6________

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structure so that collectively, the students may construct knowledge. It also stressesthe role of the teacher as the representative of the mathematical community andthe expert participant, breaking up new content into its parts and relating newmathematical content to previously learned mathematical properties. This, aswe shall next see, possibly paves the way for mathematical creativity. After doingout loud another similar example, a student raises her hand:

Amy: I have a question but it’s not exactly related.Teacher: That’s OK. It doesn’t matter [if it’s not related].Amy: 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.The teacher writes the example in column form on the board: 32

— 34_______Amy: So can you do the exchange?The teacher writes on the board: 2 12

32/ /— 34___________

Amy: So, 12 take away four is eight. And then two take away three.Don: Is minus.Teacher: Minus one. But this –1 is in the tens digit. So it’s minus ten plus

eight. Which is?

In this episode, the teacher introduces the procedure for subtracting mixednumbers by relating it to a procedure learned in a different context, that of sub-tracting two-digit numbers in column form. Amy tentatively poses a problem,which seems to the student unrelated to the current topic. The teacher goes withthe flow and encourages her to pose the problem. Amy then inquires about ex-tending the procedure to a new domain, that of negative numbers, a domain notyet explored in the fifth grade. In other words, she overcomes what Haylock (1997)termed content-universe fixation, when she explores the idea of moving beyondthe current content (subtracting mixed numbers) into the domain of negativenumbers. The teacher, for her part, encourages this direction by placing the prob-lem on the board for the whole class.

On the one hand, this episode may seem like an example of individual math-ematical creativity. On the surface, the lesson seems routine. There is a back andforth between the teacher and various students, at the end of which, Amy exhibitsflexibility and novelty. On the other hand, we can view Amy’s idea as the continu-ation of a process begun by others in the class. First, there is the mathematicsproblem posed by the teacher, new for these students. Ian begins solving theproblem by converting one half into three-sixths and two-thirds into four-sixths.Penny names this process expansion and Harold further adds that this is done inorder to find a common denominator. While the teacher seems to be the primeinstigator of this process, it works because the students are listening to each other

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and to the mathematics. Martin, Towers, and Pirie (2006) called the high level ofattentiveness between students the etiquette of emerging understanding. Theteacher then emphasizes the mathematical reasoning behind the fraction prob-lem and how it is related to previous subtraction examples, namely subtraction ofmulti-digit numbers. It is only after this discussion that Amy comes up with heridea of taking the same process to yet another number domain, negative num-bers. Thus, while it was Amy who extended the range of application of the origi-nal problem, the collective creativity in this episode may be seen in the emergenceof a novel idea built upon the meticulous analysis of the mathematical contentconducted by the group, facilitated by the teacher.

SUMMARY AND DISCUSSIONWhen reviewing the episodes presented in this paper, it is possible to discern

both the product and process of collective mathematical creativity. This sectionbegins by reviewing the products of divergent thinking, continues by examiningthe processes, and then discusses collective creativity. As shall be seen, it isdifficult at times to separate the product from the process. Next, this sectionfocuses on the role of the teacher in collective classroom mathematical cre-ativity. Finally, it explores the relationship between individual and collectivemathematical creativity.

Collective fluency, flexibility, and originality: Products and processesFluency is one mark of divergent thinking. In the first three episodes we were

able to count four to five distinct solutions to the problem at hand. As such, thenotion of collective fluency may be similar to the notion of a collective solutionspace which is “a combination of the solutions produced by a group of individu-als” (Leikin, 2009, p. 134). Regarding collective flexibility and collective original-ity, it becomes more difficult to separate the product from the process. On theone hand, when discussing flexibility as a product, we may look at the solutionsproduced by the group which employed different strategies as opposed to merelymarking the number of solutions produced by the group. For example, in the firstepisode, flexibility regarding the product may be viewed in terms of solutions thathad a different number of digits after the decimal point. Regarding the process,we may say that flexibility was marked by a certain adaptation of previoussolutions, using the same factors but changing the places of the decimal points.Yet, can we say that the collective was flexible? Regarding originality, it seemsalmost far fetched to talk about collective originality. Originality presumes a uniqueor novel idea. If the idea is unique, is it not individual? Indeed, most studieson group creativity do not discuss group originality but rather an individual’snovel solution relative to the solutions produced by the group (Leiken, 2009).And yet, although a solution may be unique, it may be the product of a collectiveprocess of creativity. This brings us to problem of separating the product fromthe process.

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Regarding the collective process, we focus on the interplay of ideas put forthby individuals which is then woven together. Thus, even as we describe the prod-uct of collective mathematical creativity, we look not only towards the individualsolutions but towards the end product of the collective process. For example, inthe first episode, one student put forth an idea, another expanded it, anotherassessed the correctness of a solution, and the teacher clarified boundaries. Theend product consisted of five solutions brought forth by the collective process.

Perhaps collective flexibility may refer to the dynamic process by which onestudent leads the group in one direction which then reminds another student aboutthe possibility of another direction. Together, the group tries out various strate-gies and possibly produces solutions based on different mathematical propertiesor different representations. Thus collective flexibility may be used to describea process as well as a product. Regarding originality, as we noted in the fourthepisode, the student’s novel idea at the end may have been the result of thecollective process of working out the mathematics behind subtraction of mixednumerals and together tying this process to other mathematical domains.

As mentioned in the background, divergent thinking in groups has also beeninvestigated in non-classroom contexts. In some ways, those studies may berelated to this study. For example, in the first episode the teacher mentions, morethan once, that the problem has many solutions and encourages the students tocome up with more and more solutions. Recalling that group brainstorming isnot always more successful than individual brainstorming (e.g., Paulus & Yang,2000), we may first ask why the teacher chose to implement this activity with thewhole class and did not assign it to the students as individual work. It is mostlikely that the aim of the teacher was not to explicitly promote mathematical cre-ativity but rather to review a mathematical concept learned previously. Not thatboth aims cannot be met during one class session. We could argue that a combi-nation of individual work followed by group work could promote divergent think-ing and intrapersonal evaluation along with interpersonal evaluation and conceptreview. The teacher could argue that there is not enough time in class for both.

Although the nature of the activity in episode one reminds us of brainstormingstudies, the context is different. To begin with, the group of individuals in thisstudy are not as diverse as individuals in other studies. They are all the sameage and share a similar educational history – most students have been learningtogether in the same class from first grade. Thus, the diversity which may hindercommunication and understanding between individuals is minimized in this study.On the other hand, the lack of diversity may limit the group’s creativity as mea-sured by fluency. In the first episode, there are many more than five solutionswhich could be appropriate for the solution as well as for the grade level.

In discussing mathematical creativity in the classroom, we recall that this studytakes a developmental view of creativity. In line with Runco (1996) we are notonly looking for multiple solutions to a problem, or for one unique and brilliantidea, but for novel ways of interpreting the mathematics which are meaningful tothe student and can be appreciated by the student’s peer group. Evidence of this

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was seen in the second and fourth episodes. Specifically, in the second episode,one student came up with a novel interpretation of sequences having unequaljumps and the teacher, as well as another student, legitimized this idea. Inthe fourth episode, we make remark on the intentionality of the student whospecifically brings new meaning to a standard algorithm, appropriating it to anew context.

Finally, while many previous studies discussed group creativity as an aggre-gate of individuals, this study discusses collective creativity among individualswho have a shared history and an emotional connection. This study focuses onthe interactions between these connected individuals which foster new interpreta-tions (Vygotsky, 1978). In such a context, collective creativity is partly the resultof a climate that allows the free flow of ideas and a teacher who is flexible enoughto allow and perhaps foster this climate. This brings us to the roles of teacher.

The teacher is part of the collectiveWhen discussing collective creativity in the classroom, the teacher acts as both

a group member and a group leader. For example, in the third episode, we mayinclude Nina’s solution, the sequence 0.20, 0.65, 1.10, as part of the collectiveproduct to the problem of producing a sequence which includes the numbers 0.2and 1.1. We also may view her suggestion that 0.2 and 1.1 does not have to beplaced one next to the other as part of the collective process which led to onechild offering the solution of the sequence 0.2, 0.5, 0.8, 1.1. In effect, taking partas a group member, the teacher also establishes an emotional connection withthe students, motivating them to follow her example. This is an aspect of transfor-mational leadership (Jung, 2001). Yet, the teacher is not an equal to the students.The teacher is the representative of the mathematical community and the expertparticipant. Thus, as mentioned in the beginning of this article, one of the teacher’sroles is to establish sociomathematical norms which will differentiate betweensolutions which are the same and those which are different (Yackel & Cobb, 1996).This came up in the first episode. According to Leiken (2009), flexibility may alsobe expressed in solutions which employ different mathematical properties. Asthe expert participant, the teacher may provide a breakdown of the mathematicsbehind a process which may then allow students the flexibility to apply a proce-dure to other domains. This was seen in the fourth episode.

The teacher also has the role of tying new concepts to previously learnedconcepts. This is similar to reflective reframing, one of the social interactionsmentioned by Hargadon and Bechy (2006) when discussing collective creativitywithin a work environment. This element was exhibited by Hailey in the fourthepisode when she tied in subtraction of mixed numbers to subtraction of multi-digit whole numbers. The collective remembering helps participants to reframepast experiences in ways that may lead to new insights. In the fourth episode, thisresulted in Amy’s insight of expanding the procedure to signed numbers.

The teacher also has the role of group leader. As mentioned in the background,studies have shown that leadership style affects group creativity (e.g., Jung, 2001)and different styles may counteract some of the loss of ideas caused by group

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interactions. Fostering a safe environment, where students can challenge theteacher’s beliefs without fear of repercussion, is one way to counteract this loss.This was illustrated in the first episode where Miri offered her idea even as shesaid, “I’m not sure.” Another example may be seen in the second episode, wherea student directly challenged the teacher’s belief that a sequence must have equaljumps. In the fourth episode the student felt safe enough to move away from thetopic being discussed and introduce a concept not yet learned.

Besides fear of evaluation, difficulty in attending to other people’s ideas andhaving enough time to think over different ideas were also cited as possible prob-lems with group creativity (Paulus & Yang, 2000). In the first episode, the teacherspecifically counteracts this problem by pausing to review the mathematical rulesinvolved in the current content. This pause not only allows time for students tothink of what was previously suggested but directs the students to pay attentionto the mathematical content and not just call out answers which are not math-ematically appropriate. In the fourth episode, the relatively long time and detailedanalysis of the mathematical manipulations involved in subtracting mixednumerals perhaps served as an incubation period for Amy in which she was ableto come up with a novel way of using the algorithm proposed in class.

The teacher also chooses which problems to pose to the class and whichstudents’ problems will be discussed together in the classroom. Hailey, in the firstepisode, wrote a problem on the board which did not appear in the classroomtextbook. The problem was open-ended in that it had many possible solutionsand allowed the children to use different strategies to reach these solutions.According to several studies (e.g. Haylock, 1997; Kown, Park, & Park, 2006; Leikin,2009), the use of multiple solution tasks is a hallmark for promoting and assess-ing mathematical creativity. Nina, in the second episode was reviewing an as-signed homework problem given from the textbook. Based on the beginning ofthis episode and Nina’s claiming that the jumps in a sequence must be equal, it isdoubtful that she chose to review this problem for the sake of promoting math-ematical creativity. However, in the third episode, she chose to bring up a secondproblem from the textbook, although it had not been assigned previously to theclass. Perhaps, her choice of introducing this second problem was a result of thedirection taken with the first problem. Perhaps, this is a sign of Nina’s flexibility.Hailey also exhibited flexibility when she was willing to listen to Amy’s question,even though Amy thought that the question may be irrelevant. In other words,both teachers were willing to deviate from their planned activities and tend tounexpected questions from her students (Leiken & Dinur, 2007; Sawyer, 2004).Perhaps teacher flexibility encourages collective mathematical creativity. Perhapsthe teacher’s individual creativity facilitates collective mathematical creativity. Thisbrings us to the issue of the relationship between individual and collectivemathematical creativity.

The relationship between individual and collective mathematical creativityAlthough the focus of this study was on collective mathematical creativity, one

cannot help but notice the many instances of individual mathematical creativity.

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Raising the idea that the jumps of a sequence do not have to be equal, illustrateshow one student was able to break away from a stereotype. By focusing oncollective mathematical creativity this study does not claim that individual math-ematical creativity is not important. On the contrary. As noted by Martin, Towers,& Pirie (2006) a musical performance, such as a jazz improvisation, is highlydependent on the creativity of each individual player. Yet, together, the musicianscreate a performance that is more than the sum of its parts. So too, individuals,working on a mathematics problem, may each contribute insights, ideas, anddirections building eventually to a collective idea. At the end of the second epi-sode, one student came up with the original idea of creating a sequence thatemployed three mathematical operations. This idea was not put forth in a vacuum.Before him, there was a student who introduced the idea that a sequence mayemploy two operations and before him a student who introduced the idea that thejumps in a sequence do not have to be equal. In other words, the students built oneach other’s ideas coming up with many different solutions as well as some verynovel solutions. Thus, the intrapersonal creativity of one produces a creative prod-uct which is then appropriated by others, proliferating interpersonal creativity.So, can we say that the final original idea was only that of the one student? Or,might we say that the final original idea was the product of the collective?

In considering the relationship between individual mathematical creativity andcollective mathematical creativity, we might also consider affective issues. Whileevaluation apprehension may stifle creativity in a group, other affective issuesmay support creativity. For example, on one’s own, a student may lack the per-sistence sometimes necessary for creative mathematical thinking or the courageto try something new (Movshovitz-Hadar, 2008). Working as a collective mayactually encourage students to keep at it and try new ideas. In other words, bypromoting collective mathematical creativity we may also be promoting individualmathematical creativity.

CONCLUSIONThis paper combined two fields of study, collective learning and mathematical

creativity, in an initial investigation into the notion of collective mathematical cre-ativity in elementary school classrooms. One limitation of the study was that itwas based on classroom observations. A more comprehensive study, involvingstudent and teacher interviews, would most likely lead to greater insights into theprocesses of collective mathematical creativity. It also investigated a relativelysmall group of elementary school classes. Many questions remain. Will the sametypes of tasks used to promote individual mathematical creativity promotecollective mathematics creativity as well? What are the ramifications of the ageof the students? How might collective creativity manifest itself in small groupcollaborations as opposed to whole class interactions? Additional research isnecessary in order to answer these as well as other questions, including how tobetter prepare mathematics teachers for promoting mathematical creativity intheir classrooms.

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Esther Levenson, Tel Aviv University, Bar Ilan 50, Raanana, Israel 43701, e-mail: [email protected]

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Exploring Collective Mathematical Creativity in Elementary School