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Exploring Ava’s developing sense for tasks that may occasion mathematical creativity Esther Levenson Ó Springer Science+Business Media Dordrecht 2013 Abstract This study explores the relationship between participating in a graduate course aimed at enhancing teachers’ theoretical and practical knowledge of mathematical crea- tivity and one teacher’s changing perspectives regarding mathematical creativity and tasks that may occasion mathematical creativity. Results indicated that perceptions of creativity may include ideas about how creativity is characterized as well as among which students it may be promoted. These perceptions were closely related to the task features, cognitive demands, and affective issues, the teacher associated with tasks that may occasion math- ematical creativity. The teacher’s reflections on her participation in the course indicated that both theoretical and practical elements of the course impacted on her changing per- spectives. Also discussed are the advantages and limitations of providing professional development by means of university-based graduate courses. Keywords Mathematical creativity Á Tasks Á Professional development Introduction While an agreed-upon definition of mathematical creativity seems to be elusive, mathe- matics educators agree that it is important to foster mathematical creativity among stu- dents. What are they trying to foster? Mathematical creativity is often related to the generation of novel ideas, such as seeing an old problem in a new way, coming up with new questions, or finding new and useful solutions to an existing problem (Sriraman 2009). Mathematics educators are interested not only in fostering creative thinking processes, such as divergent thinking, but in promoting a view of mathematics as a domain where new and novel ideas are appreciated (Mann 2006). As opposed to studies that focus on math- ematical creativity and giftedness (Livne and Milgram 2006; Sriraman 2003, 2005), this study shares the view of Silver (1997) and others (Kwon et al. 2006; Levenson and Gal E. Levenson (&) Tel Aviv University, P.O. Box 39040, Tel Aviv 69978, Israel e-mail: [email protected]; [email protected] 123 J Math Teacher Educ DOI 10.1007/s10857-013-9262-3

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Page 1: Exploring Ava’s developing sense for tasks that may occasion mathematical creativity

Exploring Ava’s developing sense for tasks that mayoccasion mathematical creativity

Esther Levenson

� Springer Science+Business Media Dordrecht 2013

Abstract This study explores the relationship between participating in a graduate course

aimed at enhancing teachers’ theoretical and practical knowledge of mathematical crea-

tivity and one teacher’s changing perspectives regarding mathematical creativity and tasks

that may occasion mathematical creativity. Results indicated that perceptions of creativity

may include ideas about how creativity is characterized as well as among which students it

may be promoted. These perceptions were closely related to the task features, cognitive

demands, and affective issues, the teacher associated with tasks that may occasion math-

ematical creativity. The teacher’s reflections on her participation in the course indicated

that both theoretical and practical elements of the course impacted on her changing per-

spectives. Also discussed are the advantages and limitations of providing professional

development by means of university-based graduate courses.

Keywords Mathematical creativity � Tasks � Professional development

Introduction

While an agreed-upon definition of mathematical creativity seems to be elusive, mathe-

matics educators agree that it is important to foster mathematical creativity among stu-

dents. What are they trying to foster? Mathematical creativity is often related to the

generation of novel ideas, such as seeing an old problem in a new way, coming up with

new questions, or finding new and useful solutions to an existing problem (Sriraman 2009).

Mathematics educators are interested not only in fostering creative thinking processes,

such as divergent thinking, but in promoting a view of mathematics as a domain where new

and novel ideas are appreciated (Mann 2006). As opposed to studies that focus on math-

ematical creativity and giftedness (Livne and Milgram 2006; Sriraman 2003, 2005), this

study shares the view of Silver (1997) and others (Kwon et al. 2006; Levenson and Gal

E. Levenson (&)Tel Aviv University, P.O. Box 39040, Tel Aviv 69978, Israele-mail: [email protected]; [email protected]

123

J Math Teacher EducDOI 10.1007/s10857-013-9262-3

Page 2: Exploring Ava’s developing sense for tasks that may occasion mathematical creativity

2013) that an orientation or disposition toward mathematical creativity may be fostered in

the general school population among students of different mathematical abilities. The

question then becomes, ‘‘How can we foster mathematical creativity in the classroom?’’

The literature is replete with studies that indicate the importance of tasks in mathematics

education (e.g., Silver et al. 2009). Tasks present opportunities for encountering mathe-

matical concepts, ideas, and strategies and afford students the opportunity to use and

develop mathematical thinking. They also influence how students experience mathematics

(Pepin 2009). For example, open-ended tasks may encourage students to associate math-

ematics with inventiveness and to view mathematics as a domain where active and flexible

thinking are appreciated (Boaler 1998). It is not surprising, therefore, that several studies

have suggested promoting mathematical creativity by engaging students with appropriate

tasks. However, promoting mathematical creativity is not always an explicit aim of

mathematics curricula (Leikin et al. 2013), and teachers are not always aware of what types

of tasks can occasion mathematical creativity (Levenson 2013). Thus, there is a need to

raise teachers’ awareness of mathematical creativity and how it may be nurtured in the

classroom (Bolden et al. 2010; Leikin et al. 2013). There is also a need to research and

study the different ways of achieving this aim (Even et al. 2009).

This paper describes the results of such an initiative. It takes an in-depth look at one

teacher, Ava, who participated in a graduate course entitled Creativity in Mathematics

Education. As opposed to professional development that takes place in professional devel-

opment centers or in workshops, professional development in the form of a graduate course

may be less practice oriented and more theory based (Stein et al. 1999). In Israel, where this

course took place, teachers participating in courses given at professional development centers

are often expected and even required to immediately implement in the classroom what they

learned at the center. Teachers participating in graduate courses are not expected to do so. As

the course instructor, this presented me with a challenge. I was interested in not only pro-

moting participants’ theoretical knowledge of mathematical creativity, but in promoting their

practical knowledge as well. I also wanted to investigate the effectiveness of the course.

Taking into consideration the central role tasks play in promoting students’ mathematical

creativity, taking into consideration that choosing which mathematical tasks to implement

with students is central to the work of teachers (Ball et al. 2008), and finally, taking into

consideration that the practice of choosing tasks could be developed during a graduate course

while still being within the bounds of the participants’ expectations from such a course, I

chose to focus on promoting participants’ sense of tasks that could occasion mathematical

creativity and to use this developing sense as a means to investigate the course.

At three different intervals during the course, participants were asked to choose tasks

that, in their opinion, could occasion mathematical creativity and explain their choices. The

first aim of this paper is to explore the relationship between Ava’s participation in the

course and her changing perspectives regarding mathematical creativity and tasks that may

occasion mathematical creativity. The second aim of this study is to explore a method for

evaluating professional development, where the venue is a graduate-level course given at a

university, and discuss the advantages and limitations of this type of professional devel-

opment and the methodology used to evaluate it.

Theoretical background

There are two central issues of this study. The first is analyzing Ava’s developing sense of

tasks that may occasion mathematical creativity. The second issue is professional

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development aimed at promoting participants’ knowledge of mathematical creativity. This

section begins with a review of the literature related to tasks and promoting creativity. It

then continues by discussing past studies of teachers’ perspectives on creativity and related

professional development initiatives.

Promoting mathematical creativity in the classroom

Runco (1996) viewed creativity as ‘‘manifested in the intentions and motivation to

transform the objective world into original interpretations, coupled with the ability to

decide when this is useful and when it is not’’ (p. 4). When discussing mathematical

creativity in the classroom, we are not referring to absolute creativity, which refers to great

historical works recognized on a global level, but to relative creativity, which considers

creativity relative to a specific reference group, such as one’s peers or classmates (Leikin

and Pitta-Pantazi 2013). Just as creativity, in general, has been characterized by divergent

thinking and measured in terms of the fluency, flexibility, and originality of ideas produced

(e.g., Jung 2001; Torrance 1965), these three dimensions may be used to evaluate creativity

in the mathematics classroom (Leikin 2009; Levenson 2011; Silver 1997). Fluency may be

measured as the total number of unduplicated, mathematically correct, and meaningful

ideas generated, whereas flexibility and originality may be more dependent on the context

of the problem (Jung 2001). Flexibility may be evaluated by establishing whether different

solutions employ strategies based on different representations (e.g., algebraic and graphical

representations), properties, or branches of mathematics. Flexibility may also be under-

stood in terms of overcoming fixation or breaking away from stereotypes (Levenson 2011).

Haylock (1997) differentiated between content-universe fixation, not being able to consider

a broader set of possibilities than at first is obvious, and algorithmic fixation, when an

individual adheres to an initially successful algorithm even when it is no longer appro-

priate. Originality in the mathematics classroom manifests itself when a student examines

many solutions to a problem, methods or answers, and then generates another that is

different (Silver 1997). In this case, a novel solution 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 conventionality according to the learning history of the participants.

For example, a solution based on a concept learned in a different context would be

considered original but maybe not as original as a solution that was unconventional and

totally based on insight.

Different types of tasks may promote mathematical creativity in different ways. Several

researchers focused on problem-solving and the types of problems to be solved. Leikin

(2009), for example, claimed that multiple-solution tasks offer students the opportunity to

solve problems in many different ways, in turn encouraging fluency, flexibility, and

novelty. Similarly, Kwon et al. (2006) suggested that an open-ended approach in teaching

mathematics may cultivate divergent thinking among students of different mathematical

abilities. Encouraging students to come up with insightful solutions may be facilitated by

engaging them with challenging problems that are sufficiently demanding, as well as

sufficiently accessible (Mann 2006). Sheffield (2009) claimed that mathematical creativity

may be encouraged when students search for patterns and generalizations.

Some researchers suggested problem posing as a means to promoting mathematical

creativity. For example, Sheffield (2009) claimed that mathematical creativity may be

encouraged when students encounter problems that may be extended with further ques-

tioning, and Silver (1997) suggested providing students with problem-posing tasks (Silver

1997). Similarly, Mann (2006) claimed that having students do what mathematicians do,

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having them find problems as well as solve problems, and having them work on ill-formed

problems employing a variety of methods and skills may promote students’ mathematical

creativity.

The teacher has several roles in facilitating, as well as actively promoting, mathematical

creativity. For example, the teacher may provide a breakdown of the mathematical content

behind an algorithm which may facilitate students’ flexibility in applying the algorithm to

other domains. The teacher may also tie new concepts to previously learned concepts,

reframing past experiences in ways that may lead to new insights (Levenson 2011). The

teacher is responsible for fostering an environment where students feel safe, where they

can challenge the teacher’s beliefs without fear of repercussion and put forth new ideas that

may seem unconventional (Runco 1996). Finally, as mentioned previously, the teacher

chooses which types of tasks to implement and which problems to pose. On the other hand,

being willing to deviate from planned activities, modify existing tasks, and tend to

unexpected questions may also encourage students to create new ideas of their own

(Sawyer 2004). Thus, it is not only the type of activity but how the activity is implemented,

which may promote mathematical creativity.

Teachers’ perspectives of creativity and professional development

Several studies investigated teachers’ perceptions of creativity. Some of the characteristics

teachers associated with creative students were cognitive in nature, such as high intelli-

gence and being an original thinker (Aljughaiman and Mowerer-Reynolds 2005; Leikin

et al. 2013). Other characteristics were affective in nature, such as being curious and

enthusiastic about learning (Aljughaiman and Mowerer-Reynolds 2005). Some teachers

viewed creativity as an innate trait belonging to only a few people (Park et al. 2006), while

others were found to believe that creativity was related to the subject matter and that art,

music, and language, and not necessarily mathematics, are contexts that occasion creativity

(Bolden et al. 2010). Shriki (2010) found that prospective teachers viewed mathematics as

a closed domain, with little room for creativity.

Only recently, efforts have begun to focus on how to prepare teachers to nurture creativity

in their classrooms (Even et al. 2009). Working with practicing science teachers, Park et al.

(2006) described an intensive 2-week program, which included lectures on creativity and

creativity-centered science education, hands-on activities, and classroom observations. Pre-

and post-questionnaires and interviews were used to analyze teachers’ changing perceptions

of creativity. Working with prospective mathematics teachers, Shriki (2010) described how

within a methods course she was able to raise prospective teachers’ awareness of mathe-

matical creativity by encouraging them to invent new geometrical concepts and examine their

properties. The segment of the course dedicated to creativity lasted 6 weeks. Participants

remarked that the experience was new for them, as well as enjoyable and exciting. Data

collected through written reflections showed that there was a shift from viewing creativity

solely as a product to viewing creativity as process. In another study related to prospective

teachers, participants’ conceptions of mathematical creativity were investigated before and

after they participated in a general education course, which included an element related to

mathematics teaching. The course did not include specific instruction on support for crea-

tivity (Bolden et al. 2010). Results indicated that after the course, prospective teachers’

conceptions of mathematical creativity were still limited, indicating a need to specifically

address the creative aspect of teaching and learning mathematics. Most other related studies

reported on efforts of letting teachers experience what it means to learn in a way that

emphasizes creativity, but did not focus on promoting teachers’ knowledge of how creativity

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may be nurtured in the classroom (Even et al. 2009). This study describes a course that aimed

to increase teachers’ awareness of the different aspects of mathematical creativity and how to

promote mathematical creativity in the classroom.

Setting and method of the study

As with other studies that attempt to follow changing perspectives (e.g., Levenson and Gal

2013; Van Zoest and Bohl 2002; Wood et al. 1991), this study uses a case methodology

that includes collecting data at different times and in different ways. Case studies offer the

opportunity to study a situation in-depth. This is especially important when attempting to

investigate changes in perspectives, changes that might be subtle, less drastic, and yet

significant. An in-depth analysis of one participant also allows the researcher to investigate

more complex situations. In this study, the complexity involves attempting to disentangle

the relationship between a graduate course, perspectives on mathematical creativity, and

perspectives related to the specific practice of choosing tasks. While a case study might not

be generalizable to a whole population or to other situations, results can inform us of what

is possible in similar situations and what might need to be changed in future studies.

Ava was a secondary school teacher with 27 years of experience. She, along with other

graduate students, participated in an elective course entitled Creativity in Mathematics

Education. Ava was chosen as the focus of this study for several reasons. First, Ava was an

experienced secondary school mathematics teacher. As such, she had experience teaching a

wide range of mathematical topics such as algebra, calculus, Euclidean geometry, ana-

lytical geometry, trigonometry, statistics, and probability and would be able to choose tasks

without being inhibited by a narrow range of mathematical topics. As an experienced

teacher, Ava also had experience engaging students with mathematical tasks and most

likely had experience choosing tasks to implement in her classrooms. In addition, Ava

claimed at the outset of the course to have no prior knowledge of creativity or having given

any prior thought to promoting mathematical creativity in her classroom. Finally, Ava was

a secondary school teacher who prepared students for their matriculation examinations and

thus could offer some insight into whether a teacher, whose experience lies mostly in

preparing students for high-stakes examinations, could still find tasks that may promote

mathematical creativity in the classroom.

The graduate course in which Ava participated took place over a semester and consisted

of 14 ninety-minute lessons. The aim of the course was to familiarize participants with

different perspectives of mathematical creativity and discuss both theoretical and practical

issues related to promoting mathematical creativity for all students. Among the course

requirements were three assignments, given at different intervals, related to choosing tasks

that have the potential to occasion mathematical creativity. The basic assignment each time

was the following: (1) Choose a task or activity from a mathematics textbook or workbook

that in your opinion promotes mathematical creativity; (2) photocopy the task and write

down its source; and (3) write one paragraph to explain why, in your opinion, this task has

the potential to promote mathematical creativity. The first assignment was given during the

first lesson and was handed in the following week, the second assignment was given

6 weeks into the course, and the third assignment was given 11 weeks into the course. The

students had 2 weeks to hand in the second and third assignments. The second and third

assignments had the additional requirement of comparing the task currently chosen to the

previous tasks chosen by that participant, in terms of the tasks’ potential to promote

mathematical creativity.

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During the first 2 weeks of the course, students were introduced to theoretical per-

spectives related to the development of creativity in general (e.g., Runco 1996), creativity

among mathematicians (e.g., Sriraman 2009), and creativity as it relates to mathematics

education (Silver 1997). In line with Shriki (2010), participants worked in pairs during the

second lesson to invent a new geometrical concept and examine its properties. The aim of

this activity was to have students experience, even for a short time, the work of a math-

ematician. Weeks three through eight were dedicated to discussing ways of assessing

mathematical creativity through fluency, flexibility, and originality (e.g., Haylock 1997;

Leikin 2009; Silver 1997), as well as what it means to overcome algorithmic and content-

universe fixation (Haylock 1997). Lectures were dotted with samples of tasks taken from

these studies. For example, similar to an activity presented in Haylock (1997), participants

of the course were asked to draw as many possible non-congruent polygons with an area

measuring 2 square meters, on an activity sheet containing replicas of geoboards. The aims

of this task were for participants to experience what it means to work on a multiple-

solution task, to show them how a task may occasion fluency, flexibility, and originality,

and to demonstrate how these characteristics of creativity may be assessed. In addition,

students participated in small group work, in which they engaged in tasks chosen by the

lecturer, also taken from various studies on mathematical creativity. The experience of

group work was used to demonstrate how an idea may initially stem from an individual

learner, but may be taken up, built on, developed, reworked, and elaborated by others, thus

increasing the creative outcome of the group endeavor. This was discussed explicitly later

on when the notion of collective creativity was introduced. Activities introduced by the

lecturer included multiple-solution tasks geared for preschool children (Tsamir et al. 2010)

and problem-posing activities (Silver and Cai 2005). The results of these activities were

then discussed and reflected upon among the course participants and followed up with

related reading assignments. From week 9 through week 13, participants again engaged in

small group work, this time centered on multiple-solution tasks for secondary school

students (Levav-Waynberg and Leikin 2012) and open-ended tasks (Kwon et al. 2006).

Other topics discussed during this time period and that were accompanied by related

reading assignments included collective mathematical creativity in the classroom (Le-

venson 2011) and the tension between individual and collective creativity. One lesson was

dedicated to views of giftedness and creativity (Livne and Milgram 2006; Sriraman 2005).

The roles of the teacher in promoting mathematical creativity (e.g., Levenson 2011;

Sawyer 2004) were discussed throughout the course in relation to the other issues being

discussed. It was the explicit focus of the last lesson of the course, where lesson transcripts

were analyzed.

Two strands of data resulted from each of the three assignments. The first strand

included the actual mathematical tasks chosen by Ava, and the second included Ava’s

reasons for choosing these tasks and her comparisons of the tasks she chose. Each strand

was analyzed on its own. In general, the framework presented in Levenson (2013) to

analyze tasks and the reasons for choosing tasks was used here as well (see Fig. 1). The

actual tasks were analyzed in terms of their task features and cognitive demands. This part

of the framework was based on earlier studies that characterized mathematical tasks (Doyle

1988; Stein et al. 1996). Task features include the number of solution strategies; number

and kind of representations; and communication requirements (e.g., demand for students to

communicate and justify their procedures) of the task (Stein et al. 1996), the source of the

task (e.g., classroom textbook, enrichment book, internet site), and the length of the task

(e.g., one main problem or several mini-tasks) (Levenson 2013). Task features related to

creativity include the number of answers to the given problem, the number of possible

E. Levenson

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solution methods that may be used to solve the problem (e.g., Kwon et al. 2006), and the

number and types of representations displayed in the task (Leikin 2009). Cognitive

demands include the use of procedures, the employment of strategies (Stein et al. 1996),

making connections between different mathematical topics or between mathematical and

non-mathematical domains, and employing a new (for the learner) way of thinking (Le-

venson 2013). Regarding creativity, algorithmic reasoning is usually associated with

imitative non-creative reasoning (Lithner 2008). On the other hand, tasks that require

students to apply a procedure in a new way may be associated with the novelty aspect of

creativity (Silver et al. 2009). Making connections between different topics is associated

with flexible thinking (Leikin 2009).

Ava’s reasons for choosing the tasks were analyzed in terms of which task features,

cognitive demands, and affective issues she mentioned in relation to the task she chose and/or

in relation to mathematical creativity. DeBellis and Goldin (2006) named four subdomains of

affect: attitudes, beliefs, emotions, and values/morals/ethics. In their study, they described

attitudes as positive or negative predispositions whereas beliefs involve attribution of some

external truth to a set of propositions. Emotions were described as ‘‘rapidly-changing states of

feeling experienced consciously or occurring preconsciously or unconsciously…[they] are

local and contextually-embedded,’’ while values refer to ‘‘personal truths or commitments

cherished by individuals’’ (p. 135). Some of the affective issues mentioned in the framework

Task features

*Number of final answers to a

problem;

*Number of solution methods

which may be used to solve a

problem;

*Number and types of

representations displayed in the

task – verbal, graphic, numeric;

*Communication requirements;

*Source of task - Classroom

textbook, enrichment book,

internet site, teacher resource

book;

*Surface characteristics – use of

manipulatives, illustrations,

every-day context.

Cognitive demands

*Types of strategies

employed – trial and error,

working backwards from the

end to the beginning, data

organization;

*Requires algorithmic (or

non-algorithmic) thinking;

*Encourages the learner to

make connections between

different mathematical topics

or between mathematical and

non-mathematical domains;

*Requires generalization

*Requires a new (for the

learner) way of thinking;

*Challenge.

Affective issues

*Emotions – fun,

challenge,

competitiveness,

curiosity, failure,

helplessness;

*Motivation;

*Values –

providing equal

opportunities for all

students,

cooperative

learning, promoting

individuality.

Analysis of tasks Analysis of participants’ reasons for choosing tasks

Fig. 1 Framework for data analysis

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of Levenson (2013) were emotions (e.g., fun, curiosity, and surprise), motivation, and values.

Curiosity and motivation are two affective features often associated with creative thinking

(Runco 1996). Three mathematics education researchers independently analyzed all of the

data. No disagreements occurred between the researchers.

The above data analysis was used to answer the following specific questions: In what

ways were the tasks different from each other? Was there a change in the task features and

cognitive processes that Ava associated with tasks that may occasion mathematical crea-

tivity? Was there a change in affective issues Ava associated with these tasks?

A semi-structured interview was conducted with Ava approximately 2 months after the

last lesson of the course and after the course grades had been handed in for each student. The

two-month distance from the course allowed Ava to look back and reflect on the course from a

different perspective. It also allowed the interviewer to see what impressions were left on

Ava. The interview took place in the university in a private room and lasted for about 40 min.

Four topics were addressed during the interview in a semi-structured manner based largely on

the following questions: (1) Can you recall what your conceptions of mathematical creativity

were before you began this course? (2) Can you describe any change in your conceptions of

mathematical creativity and tasks that may occasion mathematical creativity that occurred

during the course? (3) Can you elaborate on the reasons you had for choosing each of the three

tasks? (4) Taking into consideration the different elements of the course—lectures, reading

research literature, tasks, and activities—which of those elements were especially signifi-

cant? The interview transcript was analyzed by first seeking Ava’s statements related to her

perceptions of creativity, including which students may be considered creative and the role of

the teacher in promoting creativity. Statements that mentioned explicit aspects of the course

were also noted, as well as statements related to the specific homework assignments.

The interview data were triangulated with the data collected during the course. Ava’s

statements related to her perceptions of creativity at different times of the course were

compared to the task features and cognitive demands of the tasks Ava had submitted at

corresponding times in the course. Recall that certain task features and cognitive demands

are associated with different aspects and perceptions of creativity. Ava’s declarations

during the interview were also compared to what she had written regarding her choice of

tasks on each assignment to see whether there was a match or a discrepancy. Finally, Ava’s

reported timeline of her perceptions and her statements regarding the course components

were compared with the actual course content. This triangulation of the data strengthened

interpretations of the findings.

Findings1

Ava’s first task: week 2

The task

The first task presented by Ava appeared on the matriculation examination for 12th-grade

students studying mathematics at an advanced level. The matriculation examinations are

given to all students in the nation studying mathematics. The advanced level is the highest

of three possible levels at which students study mathematics.

1 A subset of the findings was presented by Levenson (2012).

E. Levenson

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Triangle ABC is an equilateral triangle. Two of the vertices are A(0, a) and B(0, -a),

a [ 0. Show that the sum of the distances from the sides of the triangle to some point

within the triangle, is dependent solely on a.

As this is a matriculation examination question, it is not surprising that there is no

explicit request to solve the problem in different ways. The point is to solve the problem

and go on to the next problem. Furthermore, the matriculation examination covers topics

from several areas of mathematics, and this task came under the section heading of ana-

lytical geometry. Having a task listed under some topic heading may guide students to

search for a solution associated with the topic heading.

Reasons for choosing the task

At the outset of the course, it seemed that for Ava, a task whose solution calls for taking an

atypical route leads to creative thinking. Ava wrote:

The question was given under the topic heading of analytical geometry. Trying to

solve the problem in ‘‘the expected way’’ – using analytical geometry formulas, will

lead to an algebraic expression that is too complicated (for students) to solve.

(Ava goes on to solve the problem using analytical geometry formulas, illustrating

the complicated mathematical expressions that would result from such work. She

then continues.)

As I said, this procedure led to an expression that is too complicated to simplify and

in essence, the students that will go in this direction – seemingly the ‘king’s way’2 –

will not succeed in solving this question.

A different method is necessary for the solution. You have to think ‘outside of the

box’ and not according to the way you would expect using formulas from analytical

geometry. In short: you need to think creatively!

Ava begins by pointing out what this task is not. It is not a task that can easily be solved

in the usual way. It is not a task that employs typical formulas. It is not a question that is

easily solved. In fact, the typical route will lead students astray. She elaborates:

Creativity is expressed in the search for different solutions, from different mathe-

matical areas… when the immediate solution doesn’t work, and you attempt to look

at the problem differently…looking at the same given data from a different angle.

(Ava shows how to solve the problem.)

The different (mathematical) topic which is useful in solving this question is

Euclidean geometry and the different perspective necessary is to divide the triangle

into three triangles such that the distances take on a different role, that of the heights

of the triangle, and from that the solution path is paved.

Ava focuses solely on the cognitive demands of the task, and not on the task features or

surface characteristics. She also does not relate to any affective constructs. According to Ava,

the need to connect one mathematical topic, such as analytical geometry, to a different

mathematical topic, such as Euclidean geometry, promotes creativity. Seeing the same

2 The ‘‘king’s way’’ is an expression in Hebrew, which loosely translated can mean either ‘‘the right track’’or ‘‘the path most often taken.’’

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problem or figure from different perspectives, such as seeing that the distance between a point

and a line may also play the role of a height of a triangle, is another aspect of creativity.

Ava’s second task: week 8

The task

Ava’s second task, taken from the 12th-grade classroom textbook for students studying at

the highest level of mathematics, was:

Calculate the area of a triangle whose vertices are (3,5) (-2,-2) (-1,3).

As with the previous task, this task consists of one main problem and does not require

the solver to communicate the solution method or to provide more than one way of solving

the problem. It has one correct solution, but there exist several possible methods for

reaching this solution.

Reasons for choosing the task

Ava believes that the second task has the potential to be solved using different represen-

tations. She also mentions the cognitive demand of making connections between different

mathematical domains. As with the first task, she makes no mention of affective issues.

Regarding this task, Ava wrote:

This task promotes creativity because it can be solved in different ways using rep-

resentations from different mathematical domains, for example: analytical geometry

(linear equations, the length of a line segment, the distance between a point and a

line), trigonometry (the law of cosines and area formulas), plane geometry, integrals,

and more. That is, you can evaluate students’ answers based on the three parameters

of creativity: fluency, flexibility, and originality (on the condition that the teacher

requests the students to solve the problem in several possible ways, a request which

does not appear in the book task).

Ava’s explicit reference to three parameters of mathematical creativity can be directly

linked to the course outline and notions explicitly discussed during that time period. Her

choice of a multiple-solution task, or more precisely her choice of a task that can be solved in

multiple ways, may also be linked to the course. When discussing notions such as fluency,

flexibility, and originality, the idea of solving tasks using different methods and solutions

arose, and thus, multiple-solution tasks were present in the background. Teachers also

engaged in small group work solving multiple-solution tasks aimed at preschool children. At

this point in the course, the teacher’s role in promoting mathematical creativity had not yet

been discussed, and yet Ava realizes that much of the potential in the task to promote

creativity will only be realized if the teacher requests the students to solve the task in different

ways. In other words, Ava has begun to think of how regular textbook tasks can be revised

slightly by the teacher in order to fulfill their potential for stimulating mathematical creativity.

Comparing the tasks

Ava compares the first and second tasks and notes that the difference between the two tasks

is that the first cannot (in Ava’s opinion) be solved within the presented content domain

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while the second task can be solved within the presented content domain but can also be

solved using other mathematical domains:

In the first task I presented a question for which the solution method lies in a different

content domain than the one in which the question is situated… It was a question which

provided the student with an opportunity for coping with fixation related to content as

well as algorithmic fixation, because it could not be solved within the content in which it

was given. The question presented in the current task allows for solving the problem

within several content domains, including the one in which it was given and therefore if

you want to use it to overcome fixation it is only appropriate to use it for overcoming

algorithmic fixation. On the other hand, it allows for, as I said above, flexibility and

leaving behind stereotypes and even allows for originality.

Ava revisits the first task she chose but from a less naı̈ve standpoint. Initially, she wrote

that she chose the first task because students would have to ‘‘think ‘outside of the box’ and

not according to the way you would expect.’’ When completing the second assignment,

she adds to her initial observation that by having to search for a solution outside of the

given domain, students learn to overcome both content and algorithmic fixation. During

the course, overcoming fixation was presented as an aspect of flexible thinking. While it is

unclear which ‘‘stereotypes’’ Ava is referring to, we can infer that it goes along with her

conception of overcoming fixations and thus promoting mathematical creativity.

Ava’s third task: week 13

The task

Ava’s third task was taken from a seventh-grade textbook and involved an open-ended

inquiry-based task whereby students are requested to explore the patterns found in a

partially filled multiplication table involving signed numbers. It consisted of several mini-

tasks, leading the student to form a generalized conclusion regarding the rules for multi-

plying signed numbers. Each of the exercise questions had only one final correct answer,

although finding patterns in the table could have been carried out in several ways (Fig. 2).

Reasons for choosing the task

When explaining why she chose this task, Ava does not relate to the possibility of finding a

solution in different ways. However, she does relate to the cognitive demand of looking for

patterns. In addition, for the first time, she relates to an affective issue, the element of

surprise. Ava wrote that she chose this task because:

This task was given as an introduction to the topic of multiplying signed numbers.

The child has previously learned addition and subtraction of signed numbers and is

now requested to find by himself the rules for multiplication. This task encourages

creativity from the aspect of: to create something new. The student interacts with a

task which imitates the work of a mathematician, which is creative work. He rec-

ognizes a pattern which is the motivation for defining the product of two negative

numbers as positive, and in essence creates the definition by himself. This task

encourages novelty and originality – the student, on his own, reaches something new

and even surprising, for him. (The fact that the product of two negative is a positive

is indeed surprising.)

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Ava begins by describing the context of the task. She informs the reader of the students’

previous knowledge, stressing that this is an important factor that may affect the potential

of a task to occasion mathematical creativity. After she explains her reasons for choosing

this task, she writes how she would implement the task:

Because, in the end, there is the creation of something known only to the teacher, it

would be worthwhile for the teacher to assign this as small group work, for the

benefit of those students who may not succeed in finding the rules or who may come

up with ‘‘inappropriate’’ rules. There is an opportunity here for collective creativity

with its associated characteristics: searching for help (I didn’t find the rule, I don’t

understand the conclusion…), giving help, and gaining new comprehension.

Reflecting on Ava’s choice of task, her reasons for choosing this task, and the way in

which she would implement this task, we see a combination of ideas that may be traced

back to the course, taken from a variety of lessons. First, she claims that engaging in the

Multiplying signed numbers

Below is a multiplication table:

× 3 2 1 0 -1 -2 -3

3 9 6 3 0 -3 -6 -9

2 6 4 0

1 3 2 0 0 -2

0 0 0 0 0 0 0

-1 -2 -1 0 2

-2 -4 0 6

-3 -6 0 6

a) What is the rule in the first row?

b) What is the rule in the second column?

c) Find the rule in each column/row and fill in the rest of the empty cells.

d) (i) What is the sign of the solution when multiplying a positive number with a

negative number

(ii) Where in the table are these numbers located?

(iii) Write a multiplication example using a positive number and a negative

number.

e) (i) What is the sign of the solution when multiplying a negative number with a

negative number

(ii) Where in the table are these numbers located?

(iii) Write a multiplication example using two negative numbers.

Fig. 2 Ava’s third task

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task she chose, ‘‘imitates the work of a mathematician, which is creative work.’’ Creativity

as it relates to the work of mathematicians was discussed at the very beginning of the

course, during the second lesson. It is thus interesting that she relates to this point toward

the end of the course. On the other hand, her mention of collective creativity may be

directly related to the course content at the time when the assignment was given. During

the course, the notion of collective creativity was discussed in terms of how creativity may

be analyzed in group situations and the teacher’s roles in promoting mathematical crea-

tivity in group and whole class situations. Ava, however, raises the issue of collective

creativity in relation to values, an affective construct. In her opinion, group work may offer

students opportunities to both give and receive help.

Comparing the tasks

Ava compares the first and second tasks to the third task:

In the first task, I presented a question which required insight on the part of the solver

because the solution method did not lie in the content domain in which it was given.

It was a question which provided an opportunity for coping with content universe

fixation as well as algorithmic fixation.

The question I presented in the second task allowed for divergent production—many

solutions from different mathematical domains and different representations. That

task allowed for evaluating the students’ answers based on the three dimensions of

creativity: fluency, flexibility, and originality.

For this task (the third task) I searched for a different aspect of creativity which was

not evident in the previous tasks: creating a new mathematical concept which is not

familiar to the student. As opposed to the previous tasks, here there is no opportunity

for overcoming fixation (unless the student is not used to this type of work where he

is requested to create mathematics ‘‘something from nothing.’’ It is dependent on the

norms the teacher has inducted in her classroom.) In addition, it is not a task which

invites fluency and flexibility, but it does promote the creation of something new,

which is one of the aspects of originality.

On the one hand, it seems that Ava merely repeats what she had previously written

about the first and second tasks. On the other hand, her review of the first two tasks serves

as the jumping board for analyzing the third task. Ava acknowledges that the third task

may not promote fluency and flexibility, but nevertheless, it may still promote mathe-

matical creativity. We get the feeling that Ava’s search for a task that would promote a

different aspect of creativity caused her to look back at the first lessons of the course.

Finally, Ava’s comparison of the third task, to the first and second tasks, reveals that Ava is

able to analyze both the affordances and the constraints of different tasks, with respect to

promoting creativity. In other words, she is not only aware of which aspects of creativity

her task can promote, but she is also aware of the limitations of the task, of which aspects

of creativity her task may not promote.

Two months after the course

Two months after the last lesson of the course, Ava agreed to be interviewed. She was told

that the interview was part of a study related to teachers’ perspectives of mathematical

creativity and the tasks that might occasion creativity and that it would also help me, the

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course lecturer, in planning for the following year when the course would be given again to

a new group of students.

Changes in perspective regarding mathematical creativity

Ava confirmed that prior to the course, she had not read any literature related to mathe-

matical creativity nor had she taken any professional development course which touched

upon mathematical creativity. When asked to recall what her conceptions of creativity

were prior to the course she said, ‘‘I supposed creativity was about having sudden, great

ideas, and thinking out of the box… connected more to mathematically talented students.

This (conception of creativity) changed during the course.’’ Ava’s recall of her initial

conceptions of mathematical creativity is in line with the reasons she wrote for choosing

the first task. She chose a task from the highest level of the mathematics matriculation

examination and one that would necessitate, in her opinion, thinking in an unconventional

manner. Ava was then asked to describe more fully her changing conception of mathe-

matical creativity. She responded:

The first thing that I understood is that it (creativity) is not only for very bright

students and not only for certain ages. That is, creativity can truly be for all levels at

every age. The second (thing I understood) was that I realized that it (creativity) can

also be connected to me, with the way I teach, and that it isn’t only for some special

enrichment course, that it can be part of routine teaching, even when teaching for the

matriculation exams. I even made use of it in my teaching… Usually I’m so pres-

sured… On a test I would accept different ways for solving some problem but in

class, I didn’t leave time for this. And now, I started to make time in class… instead

of solving lots of problems, we may solve one problem but in many different ways. I

was really convinced that it’s worthwhile. What else? (Ava thinks a minute.) I

understood that it doesn’t have to be sudden great ideas, and that the whole story

with flexibility, that you can solve a problem in different ways, even if each way is

conventional, is also creativity. (Ava thinks some more.) That’s it.

This changing perspective is reflected in her second and third tasks. Regarding the

issues of age and ability, Ava’s first task was taken from the 12th-grade matriculation

examination for high-level mathematics students. Her third task is taken from a 7th-grade

textbook. In Israel, seventh grade is the youngest grade of secondary school. In addition,

while in the twelfth grade, students are grouped according to ability, in the seventh grade,

students learn mathematics in mixed-ability classes. Regarding her perception that pro-

moting creativity may also be related to her actions as a teacher, when describing the first

task, there is no indication that she considers the teacher. On the other hand, when

describing the second task, she hints at the teacher’s role in promoting mathematical

creativity when adding that the teacher must request the student to solve the task in

different ways. When describing the third task, she expands upon this issue even more

when she describes how she would implement the task with group work.

Although Ava claims to adopt some elements she learned in the course in her teaching,

there are other elements, such as having the students in her class engage in group work,

which she clearly states she would not implement.

I’m afraid of the time factor… I teach 12th grade so I’m (she laughs a little) not

capable of even thinking about not running ahead with the content material. I may

solve one or two fewer problems for the benefit of solving one problem in different

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ways, but that is very different than the time it would take to organize the students for

group work.

Yet, Ava claims that working in groups is beneficial to promoting creativity. She reflects

on her own experience during the course when engaging in such activities:

There were times when working with the group, when we started working on the

task, I was fairly rigid and fixated, but when someone brought up an idea, all of a

sudden, I opened up and the ideas flowed. I am in favor of group work.

This seeming contradiction, between Ava’s positive experience with group work and

how it may benefit the promotion of creativity, and her reluctance to implement group

work in her classroom, demonstrates the complex relationship between professional

development and practice.

Ava’s reflections on the course

In order to understand which elements of the course most affected Ava’s changing per-

spectives regarding creativity and tasks that may promote creativity, Ava was also

requested to reflect on the different elements of the course—lectures, reading research

literature, and tasks and activities—and note which of those elements were most significant

for her. She responded:

It was everything together. Each element contributed its part. Also the activities that

we did in groups. But by themselves (Ava is referring to the activities), without the

theory which accompanied them, I’m not sure the desired affect would have been

reached, or at least, not to the same degree. In other words, the theoretical parts were

also important. At least for me, I do not accept things based on that they feel right.

They need to be checked out and based on research. So, it was everything together.

On the other hand, theory, without practical experience is also… I don’t know… it

(the theory) would have been interesting but it may not have left a lasting impression.

It wouldn’t have set in. And that I can sit here and talk with you about it like it was

yesterday, when really quite a lot of time has passed, means something.

When asked to consider the impact of the specific assignments of searching for tasks

that had the potential to promote mathematical creativity, Ava responded, ‘‘My answer is

the same. Those assignments were part of the practical side of the course, which

strengthened the theoretical side.’’ In other words, Ava did not feel that completing,

essentially the same assignment, three times was a waste of time, nor did she feel that it

especially contributed to her conceptions of mathematical creativity. In general, for Ava, it

seems that all the elements contributed and in a sense, the whole was greater than the sum

of its parts.

Ava’s reflection on the tasks she chose

Last, Ava was given a copy of the three tasks she had chosen and asked whether she could

elaborate on why she chose each of the tasks as ones that could occasion mathematical

creativity. When looking over the first task she presented, she said, ‘‘Right. The need for a

sudden bolt of insight. That was my first stage.’’ When going over the second and third

tasks, she commented,

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For the second task, I searched for a problem that could be solved in different ways,

to go with issue of flexibility. After I used up the issue of flexibility, and maybe even

originality, I wanted to relate to the idea of novelty because I already related to

flexibility and originality and multiple solutions. And there was talk in the theoretical

part of the course that it’s not only… a solution that no one else comes up with and

the one who had the sudden insight. It also has to do with innovation. You see, it

really is the combination of…when I went to search for tasks, I looked at my notes

from the course and what was in the theoretical background and I searched

accordingly. So, it was always the combination.

When discussing her third choice, Ava uses three similar terms: originality, novelty, and

innovation. While the terms are similar, they are not necessarily synonymous. Ava spe-

cifically differentiates between originality and novelty, implying that originality may be

promoted and evaluated when students engage in multiple-solution tasks and novelty and

innovation may be promoted by a task that involves discovering a new idea. Levenson

(2013) noted that several researchers actually use the terms ‘‘originality’’ and ‘‘novelty’’

interchangeably. However, she proposed that each term stresses different elements. Novel

may refer to ‘‘new,’’ while original may refer to ‘‘one of a kind’’ or ‘‘different from the

norm.’’ While it seems likely that a ‘‘one of a kind’’ idea will also be ‘‘new’’ and vice versa,

it is sometimes the case that an idea, especially one raised in the classroom, may be new to

a student, but if other students have the same idea, it may not be original. It seems that Ava

makes a similar differentiation in that the third task involved the students discovering a

new idea, but of course, all the students will come up with the same idea. On the other

hand, when engaging in tasks that may be solved in multiple ways, originality may be

evaluated when a student uses a method, or comes up with a solution, different than the

others. Finally, in the above discussion, Ava refers again to the combination of theory and

practice which she experienced throughout the course and stresses that when searching for

the second and third tasks, she referred back to the theory in order to search for additional

aspects of creativity.

Summary and discussion

The first aim of this paper was to explore the relationship between Ava’s participation in

the course and her changing perspectives regarding mathematical creativity and tasks that

may occasion mathematical creativity. This section begins by summarizing the findings

and following Ava’s changing perspectives. It continues by exploring the relationship

between Ava’s changing perspectives and her participation in the course. The second aim

of this study was to explore a method for evaluating professional development, where the

venue was a graduate-level course given at a university, and discusses the advantages and

limitations of this type of professional development and the methodology used to evaluate

it. Thus, the third part of this section reflects on the affordances and limitations of the

course, and the final part reflects on the methodology used to investigate the effectiveness

of the course.

Ava’s changing perspectives

What can be said about Ava’s changing perspectives on mathematical creativity and her

developing sense for tasks which may occasion mathematical creativity? In what ways

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were the three tasks similar to each other and in what ways were they different? A

summary of the findings that guide this section is presented in Table 1. Beginning with the

task sources, we first notice that Ava consistently brought mathematical tasks from

classroom textbooks or examinations. In other words, she sought out tasks that were based

on mathematical content learned in class and did not seek out tasks from extracurricular

activity books or Internet sites. On the one hand, this was the original request. On the other

hand, in Levenson’s (2013) study, a little more than a third of the participants, when given

the same request, sought out tasks from sources other than the classroom textbook. That

Ava continued each time to seek out classroom-based texts implies that, possibly from the

start, Ava believed that mathematical creativity may be fostered using readily available

classroom tasks.

All three tasks basically had one final correct answer, but potentially could be solved in

different ways. Initially, Ava associated mathematical creativity with talented mathematics

students and not for students of all ages. Although the first task had the potential to be

solved in two ways, she expected students to correctly solve it using only one method. Her

reason for choosing this task was based on the cognitive demand of solving a task in an

unexpected way. At the time, Ava characterized mathematical creativity as thinking

unconventionally and having moments of insight.

Midway through the course, we learn that Ava adopted the view that creativity may be

characterized by fluency, flexibility, and originality, by overcoming fixations and leaving

behind stereotypes, and by finding relationships between mathematical domains. Her

chosen task reflects these conceptions. It may be solved easily, but it can also be solved in

different ways, breaking the stereotype that there is only one correct way to solve a

problem. As mentioned in the ‘‘Theoretical background,’’ solving tasks in multiple ways

was advocated by several mathematics education researchers interested in promoting

mathematical creativity (e.g., Silver 1997). It was also explicitly discussed and illustrated

during the course in which Ava participated.

Toward the end of the course, Ava recognizes that creativity may be promoted among

students learning in mixed-ability classes. It is taken from a 7th-grade textbook geared for

students learning in mixed-abilities classes. While the third task may have had the potential

to be solved in different ways, Ava did not give this as the reason for choosing this task.

Instead, she chose the third task because of the cognitive demand of searching for patterns,

which would lead students to create a new, for them, mathematical rule. Ava recognizes

that creativity may be characterized by coming up with new ideas and new definitions and

that it is related to the work of mathematicians. Her chosen task reflects these perceptions.

Although this task is similar to the first task in that it is reminiscent of the work of a

mathematician, for the third task, Ava explicitly wrote that she chose it because it allows

students to work as mathematicians. In other words, she considers not only what the

student will have to do in order to solve the task, but the mathematical and creative

disposition that may be promoted when engaging in the task.

Disposition is related to the issue of affect. When choosing the first two tasks, Ava did

not relate to any affective issues. However, when choosing the third task, Ava mentions the

element of surprise, which may be considered an emotion. She also discusses the value of

group work in helping students who might find the task difficult. Values are another

affective issue. In other words, it seems that affective issues only arose toward the end of

the course, when choosing the third task. At that point, we see explicit reference to

emotions and values and implicit reference to disposition. Affective issues did not arise

again during the interview.

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E. Levenson

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In general, as can be seen from Table 1, there was a close relationship between Ava’s

perceptions of creativity and the tasks she chose. This is an important result. While pre-

vious studies found a relationship between teachers’ beliefs regarding the nature of

mathematics, learning, and teaching, and instructional practices (Peterson et al. 1989;

Stipek et al. 2001), few studies investigated the relationship between teachers’ perspectives

of mathematical creativity and instructional practices. In this study, it was found that Ava’s

perceptions of mathematical creativity involved ideas about how creativity may be char-

acterized as well as among which students it may be promoted and that both of these

factors affected her choice of tasks. These results are strengthened by previous studies that

found that prospective teachers’ perceptions of creativity may be narrow (Bolden et al.

2010) and that a rift may exist between teachers’ apparent support for creativity enrichment

and their less supportive practice because they are unaware of the defining characteristics

of creativity (Aljughaiman and Mowerer-Reynolds 2005). Yet, enhancing Ava’s knowl-

edge of the characteristics of mathematical creativity may not have been sufficient if Ava

had retained her belief that mathematical creativity may only be developed among the most

advanced mathematics students.

The relationship noted above between Ava’s changing perspectives of mathematical

creativity and her perceptions of tasks which may promote mathematical creativity brings

to light an additional, yet more subtle change in Ava’s perceptions. Ava began with a task

that requires a moment of creativity. Her second task involved solving a problem in

different ways and promoting flexible thinking. Her third task encouraged students to come

up with new ideas on their own, developing their creative disposition. In other words, her

second and third tasks are about developing long-term creative thinking processes. Evi-

dence of this change in Ava may also be seen in the affective issues raised with the third

task. She is taking into consideration both cognitive and affective aspects, when choosing a

task.

Relating participation in the course to Ava’s changing perspectives

Of the three tasks Ava presented during the semester, the second task seems to be most

directly related to what was being discussed during the course at the time. It came as no

surprise that Ava chose a task that had the potential to be solved in different ways, and it

came as no surprise that she mentioned fluency, flexibility, and originality at that time.

What was more surprising was her third choice. Her reference to the work of mathema-

ticians brings us all the way back to the second lesson of the course. If she would have

chosen a task that more closely resembled the types of tasks being discussed during that

time period of the course, we may have said that the connection between the course and

Ava’s choice of tasks was merely trivial and possibly imitative. However, from the

interview, we learn that Ava consciously took the time to search her notes and reread the

theory in order to come up with an aspect of mathematical creativity she had not referred to

previously. When she came up with this aspect, she then searched for a task which would

explicitly promote this aspect. Interestingly, the third task was taken from a seventh-grade

textbook, even though Ava was an upper secondary school teacher, specializing in pre-

paring students for their matriculation examinations. It could be that although an appro-

priate task may have been found in the 11th- or 12th-grade textbook, Ava was considering

other factors, such as the students’ willingness to engage in such activities in the 12th

grade, or the time it would take to implement such an activity in the classroom. Recall that

during her interview, Ava mentioned feeling under pressure to use her time well in order to

meet curriculum guidelines. In fact, Tirosh and Graeber (2003) found that some of the

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obstacles to realizing the potential of professional development include the teachers’

perspectives of their students and organizational constraints. With the third task, we find

that Ava, like the researchers discussed in the beginning of this paper (e.g., Silver 1997),

has found that different tasks may promote different aspects of creativity and has learned

how to choose a task based on the aim of promoting a certain aspect of creativity. This is a

non-trivial outcome of the course. Still, it should be noted that Ava did not present open-

ended tasks nor did she present problem-posing tasks. We are left wondering if this was

because she felt these types of tasks would have promoted the same aspects of creativity

the other types promoted or perhaps she did not find any examples of such tasks in the

books she perused.

That Ava searched her notes and reread the theory reminds us of the importance of

making theories explicit during professional development. Indeed, Ava stated during the

interview that the theoretical side of the course was just as important to her as the tasks. This

is in line with Tsamir (2007), who claimed that theories may also be used as tools in

professional development. In her study, it was shown that introducing prospective teachers

to the intuitive rules theory impacted on their mathematical subject matter as well as their

ability to analyze tasks and interventions with reference to the intuitive rules theory. Sim-

ilarly, in this study, introducing Ava to theories related to mathematical creativity impacted

on her awareness of the major characteristics of mathematics creativity and enabled her to

choose tasks that would promote different aspects of mathematical creativity.

Ava also mentioned the practical side of the course, her engagement with tasks during

the course, and her search for tasks when fulfilling the assignments. Much has been said

lately about the role of tasks during professional development. In fact, in 2007, a special

issue of this journal was dedicated to the roles of tasks in mathematics teacher education.

In their introduction to this issue, Watson and Mason pointed out that tasks for teachers are

often designed ‘‘so that teachers can experience for themselves at their own level some-

thing of what their learners might experience and hence become more sensitive to their

learners’’ (p. 208). Similarly, the tasks Ava engaged in during the course enabled her to

experience some of the characteristics of creativity, such as fluency and flexibility, as well

as overcoming fixation. In her summary of that issue, Zaslavsky (2007) pointed out the

importance of having teachers reflect on tasks. Watson and Sullivan (2008) agreed and

suggested that prospective and practicing teachers can be informed about classroom tasks

through reflective engagement with those tasks. Similarly, Ava not only reflected on the

tasks implemented during the course, but reflected on the tasks she chose in light of what

she had learned in the course. For Ava, it was the combination of theory and practice which

impacted on her perceptions.

When looking back at the graduate course and its impact on Ava’s choices of tasks, one

also notices room for improvement. During the second assignment, Ava mentions how she

would need to change the task slightly in order for it to meet its potential in promoting

creativity. While this was noted in a positive light, it cannot be credited directly to the

course. Altering existing textbook tasks was not discussed at all during the course. In fact,

the only tasks teachers seriously reflected on were those that they chose as having the

potential to promote mathematical creativity. However, teachers are often given curricu-

lum materials and have little choice but to use them. In such cases, it is important for

teachers to be able to analyze given tasks in terms of how they might promote different

aspects of creativity. Not surprisingly, studies have shown the importance of analyzing

curriculum materials, including textbook tasks, during professional development (e.g.,

Nicol and Crespo 2006; Remillard 2005). And so, while the second and third assignments

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did have an element of this type of reflection, in the future, such a course might consider

additional opportunities for examining curriculum materials.

Also, missing from the course were explicit discussions and lessons related to affective

issues associated with mathematical creativity. Levenson (2013) found that teachers relate

to affective issues when asked to choose tasks that have the potential to promote mathe-

matical creativity. Ava raised the issue of affect when choosing her third task. Thus, it

seems that the issue arises among teachers even when not explicitly addressed. In addition,

studies have shown that affective issues such as motivation, pleasure, curiosity, and self-

confidence may directly and indirectly impact on creative processes (Kaufman and Be-

ghetto 2009; Lin and Cho 2011; Runco 1996). A future course might, therefore, explicitly

address affective issues related to promoting mathematical creativity.

Reflections on the graduate course

The above section noted two deficiencies in the course: the lack of explicitly discussing

how curriculum materials might be changed slightly or implemented differently in order to

promote mathematical creativity and the lack of explicitly discussing affective issues

associated with mathematical creativity. Both of these deficiencies relate to the content of

the course. While content is, of course, important, it is also important to take into con-

sideration structural features, such as the form of the activity, the duration of the activity,

and the emphasis on the collective participation of groups of teachers from the same

school, department, or grade level (Garet et al. 2001).

The form of the program was a graduate-level course given at a university. This venue

has some advantages. To begin with, participants usually come to such courses out of

choice. It could be that voluntarily participating in a course means that the participants will

be more open to new ideas. In fact, in one study it was reported that teachers who did not

wish to participate in a mandated professional development program were required to

transfer to schools in other districts (Campbell and White 1997). This does not mean that

mandatory programs cannot be successful. Programs, including mandatory programs,

which are designed for groups of teachers from the same school, department, or grade

level, offer teachers the opportunity to share ideas and develop a common understanding of

instructional goals which can then help sustain change in practice over time (Garet et al.

2001). In an elective graduate course, such as that presented in this study, there is usually a

lack of such collective participation. This is a disadvantage. Participants come from dif-

ferent cultural backgrounds, teach different school levels, and have different reasons for

participating in the course. Teachers from such a course may find themselves quite alone in

their attempts to implement what they learned. Without support from other teachers and, as

Ava pointed out, with high-stakes examinations setting educational goals, the effects of

such a course on practice may be seriously limited.

Another advantage of a course given at the graduate level is that teachers expect such

courses to be theory-laden. As such, teachers may be less resistant and more open to

pedagogy, which is informed by theory. In this study, for example, Ava mentioned several

times how theory informed her choice of tasks. On the other hand, such courses are much

less practice-based. Studies have shown that professional development that is not inte-

grated into the daily life of the school may be less effective (Garet et al. 2001; Tirosh and

Graeber 2003). Ava, like the other participants in the course, was not required to imple-

ment in her classroom what was learned. Was this a serious fault of the course? Perhaps.

Did it affect the likelihood that Ava and the other teachers would actually implement what

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was learned in the course in their classroom? Maybe. But does that mean that such a course

has no place in professional development?

The fact is such courses exist. And while they may not be considered the most effective

way of delivering professional development, we cannot ignore their reality. The question

then becomes, ‘‘Is there a way of integrating more classroom practice into such courses,

despite teachers’ expectations?’’ One suggestion might be to have participants voluntarily

implement new ideas into their classroom practice and then report back, again voluntarily,

on these efforts. Taking into consideration that the participants elected to participate in the

course, the idea that they would voluntarily implement what they learned in their class-

room is not so far-fetched. In the course described here, teachers could be requested to

choose tasks that specifically come from textbooks used in their classroom (and not just

any textbook) which they believe may occasion creativity in the classroom, and then

implement them in their own classroom. In addition, teachers might be invited to refer to

theories learned in the course and voluntarily attempt to assess their students’ individual

creativity or their classroom collective creativity. Finally, while most teachers did not

come from the same background or school, a virtual meeting place could be set up for

course participants who wish to share experiences in the classroom, in a way developing a

sense of community for those participants who wish this kind of support. In other words,

with a little ‘‘creative’’ thought, one might seek non-conventional ways to bring graduate

courses closer to classroom practice.

Studying professional development: reflections on the methodology

It is always difficult to balance teaching obligations with research obligations, especially

when the course instructor is also the researcher. Studies of teachers’ changing perspective

often employ pre- and post-questionnaires and/or interviews (e.g., Park et al. 2006).

Desimone (2009) pointed out several common biases when using these methods to assess

professional development, including the issue of relying on teachers’ self-reports. In

addition, I did not want to utilize these options for other reasons. First, I did not want

merely to investigate teachers’ perspectives before and after the course, but to follow their

changing perspectives. Second, as the course instructor, I did not want the research to

intrude on the course. Thus, I attempted to find a method for investigating teachers’

changing perspectives that would also serve as a pedagogical tool.

Basically, the method used was to request participants to carry out the same assignment

(in this case, to find a task that has the potential to occasion mathematical creativity) a few

times over the period of the course, to have participants reflect back each time on what they

had done previously, and to use those assignments to track the changes in participants’

knowledge and/or perspectives. In addition, one follow-up interview was conducted two

months after the course. Was the method successful? Did analyzing Ava’s choice of tasks

allow us to investigate her changing perspectives regarding mathematical creativity and

tasks that may occasion mathematical creativity? For the most part, as seen in the previous

sections, we learned a lot about Ava’s changing perspectives from her homework

assignments. Depending on the aim of the course, a similar method might be employed in

other professional development programs, where participants might be asked, for example,

to design lesson plans at different intervals of the course. These lesson plans may then be

used to evaluate teachers’ perspectives on, for example, group work or to evaluate par-

ticipants’ pedagogical-content knowledge regarding, for example, task design. The follow-

up interview at the end of the course lent validation to the findings. On the other hand,

depending on the aim of the investigation, more direct questions might be used to

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understand participants’ decisions and choices. For example, in this study, Ava was not

asked directly why she chose the third task from a seventh-grade textbook or why she did

not choose any open-ended tasks. We are left to speculate the reasons behind these

decisions.

And therein lays one of the problems with the method of this study. Even with the

interview at the end, much is left to interpretation. For example, from the interview, we

learned that Ava checked her course notes, and according to what was written, chose a

task. It could be that Ava did not change her perspectives regarding mathematical crea-

tivity, but rather utilized her newfound knowledge to fulfill a course requirement. While

she was able to choose tasks according to what aspect of creativity she wanted to promote,

and this is an important skill for a mathematics teacher, can we say that this action reflects

her changing perspectives? Can we say that she would employ this skill in her own

classroom?

The final project of the course required participants to choose a task from one of the

tasks introduced during the course (i.e., a task chosen by the lecturer), implement it with

either a small group of students or a whole class, and report on the results. This was not

part of the research methodology as the participants had to choose a task from those

presented and analyzed in the course, and not one freely chosen from a source of their own.

Participants could choose the age of the students and the setting according to their con-

venience. Some participants chose to try out a task on their own children at home. Some

participants chose to try out one of the tasks in their classroom. Ava chose to implement

the task with a whole classroom of students—but not her own students. She went to a

seventh-grade classroom, where the teacher agreed that Ava could try out her chosen task.

In other words, even when directed to implement some activity with students, Ava did not

implement the activity with her own students. So, was the course effective? Did the

methodology used to investigate Ava’s changing perspectives reveal this incongruence?

These questions are left open for the reader to answer. Changing one’s perspectives and

changing one’s practice takes time, and we need methods for assessing both short-term and

long-term outcomes of professional development (Tirosh and Graeber 2003). This paper

explored the short-term effect of the graduate course on one participant and a method for

exploring this effect. While this study raises some questions, both pedagogical and

methodological, as noted in the ‘‘Theoretical background’’ section, few studies have sys-

tematically investigated efforts at promoting teachers’ awareness of practices that may

nurture mathematical creativity. This study is a beginning. It can serve as a springboard for

future studies.

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