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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
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
E. Levenson
123
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,
Occasion mathematical creativity
123
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
E. Levenson
123
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.
Occasion mathematical creativity
123
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
123
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
Occasion mathematical creativity
123
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
123
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.’’
Occasion mathematical creativity
123
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
E. Levenson
123
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.)
Occasion mathematical creativity
123
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
E. Levenson
123
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
Occasion mathematical creativity
123
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
E. Levenson
123
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,
Occasion mathematical creativity
123
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
E. Levenson
123
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.
Occasion mathematical creativity
123
Ta
ble
1A
va’
sch
ang
ing
per
spec
tiv
eso
fta
sks
that
may
occ
asio
nm
ath
emat
ical
crea
tiv
ity
Tas
kfe
ature
sC
ognit
ive
dem
ands
Aff
ecti
ve
issu
esA
va’
sas
soci
ated
per
spec
tives
of
crea
tivit
y
Tas
k1
12
th-g
rad
eex
amin
atio
n;
1m
ain
pro
ble
m;
1–
2fi
nal
answ
ers
dep
enden
to
nso
luti
on
met
ho
d
Req
uir
esan
un
conv
enti
onal
solu
tio
np
ath
–A
ppro
pri
ate
for
tale
nte
dst
uden
ts;
char
acte
rize
db
yh
avin
gsu
dd
eng
reat
idea
s
Tas
k2
12
th-g
rad
ete
xtb
ook
;1
mai
np
rob
lem
;1
fin
alan
swer
wit
hse
ver
also
luti
on
met
ho
ds
Connec
tsdif
fere
nt
mat
hem
atic
ald
om
ain
s–
Char
acte
rize
dby
fluen
cy,
flex
ibil
ity,
and
ori
gin
alit
y;
ov
erco
min
gfi
xat
ion
and
ster
eoty
pes
Tas
k3
7th
-gra
de
tex
tbo
ok
;se
ver
alm
ini-
task
s;o
ne
fin
alan
swer
Req
uir
esg
ener
aliz
atio
no
fa
pat
tern
Su
rpri
se;
val
ue
of
gro
up
wo
rk;
dis
po
siti
on
Ap
pro
pri
ate
for
stu
den
tso
fd
iffe
ren
tm
ath
emat
ical
lev
els;
char
acte
rize
db
yth
ew
ork
of
am
ath
emat
icia
n,
no
vel
ty,
and
inn
ov
atio
n;
the
teac
her
has
anac
tive
role
inp
rom
oti
ng
crea
tiv
ity
E. Levenson
123
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
Occasion mathematical creativity
123
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
E. Levenson
123
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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123
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
E. Levenson
123
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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