Creative Education
2013. Vol.4, No.7, 430-439
Published Online July 2013 in SciRes (
Copyright © 2013 SciRe s .
A Model for Assessing the Development of Students’ Creativity
in the Context of Problem Posing
Atara Shriki
Oranim Academic College of E d u c a t i o n , Tivon, Isra e l
Received April 19th, 2013; revised May 20th, 2013; accepted May 27th, 2013
Copyright © 2013 Atara Shriki. This is an open access article distributed under the Creative Commons Attribu-
tion License, which permits unrestricted use, d i st ribution, and reproduction in any med ium, provid ed th e o riginal
work is properly cited.
In a changing technological society, creativity is recognized as the vehicle of economic and social growth.
Although the education system has a central role in developing all students’ creativity, it is not often nur-
tured in schools. Several conditions are offered to justify this situation, among them: external pressures to
cover the curriculum and succeed in standardized tests that generally require rote implementation of rules
and algorithmic thinking; teachers’ tendency to teach similarly to the way they themselves were taught as
school students; relating creativity to giftedness, and therefore avoiding nurturing all students’ creativity;
teachers’ difficulties in assessing their students’ creativity and its development due to a lack of an avail-
able simple tool; and more. This paper is aimed at responding to the latter condition, suggesting a coher-
ent and accessible tool or model for assessing students’ creativity and its development in the context of
problem posing. The proposed model considers 4 measurable aspects of creativity-fluency, flexibility,
originality and organization, and a total score of creativity that is based on relative weights of each aspect.
Viewing creativity as relative, the scores for these 4 aspects reflect learner’s achievements in relation to
his or her reference group. The proposed model has two flexible components—the first relates to teach-
ers’ interpretation of originality, and the second relates to the weights they may wish to ascribe each as-
pect of creativity. In addition, it is suggested to provide learners with a graphical display of their scores
and progress in order to enable them to refine their products in successive iterations. The examples in this
paper are taken from mathematics; however the proposed model can be adapted to any other discipline.
Keywords: Evaluation of Creativity; Problem Posing; The “What-If-Not?” Strategy; Self-Assessment of
Creativity is generally perceived to be a key driver of social
and economic changes. One would, therefore, anticipate that
the education system would strive to nurture students’ creativity
(Beghetto, 2006). However, although teachers believe that ma-
thematical creativity can and should be nurtured in school in
most schools it is normally not encouraged (Sriraman, 2005).
As a result most students are provided with few opportunities to
experience creative learning and thinking (Silver, 1997).
This reality is a result of a combination of several circum-
stances, among them: (i) Teachers are subjected to various ex-
ternal pressures, such as covering the written curriculum, and
help their students to succeed on both teacher generated and
standardized tests (Beghetto, 2006). The education system em-
phasizes the importance of achieving high test scores, and too
often teachers are judged based solely on their students’ success
on these tests. For the most part, standardized tests generally
require rote implementation of rules and algorithmic thinking.
Furthermore, standardized tests are not designed to formally
assessing mathematical creativity (Chamberlin & Moon, 2005).
As a result, teachers tend to focus on memorization and algo-
rithms, rather than on nurturing students’ creativity; (ii) Teach-
ers are inclined to teach similarly to the way they themselves
were taught at school (Hall, Fisher, Musanti, & Halquist, 2006).
As school students, most of them experienced mathematics
education that did not place an emphasis on developing creativ-
ity. With insufficient role models in their formative years of
schooling to draw upon as positive examples, it is reasonable to
assume that, as teachers, they may not have naturally acquired
sufficient tools and methods aimed at nurturing students’ crea-
tivity. Further exacerbating the problem is a shortage of appro-
priate learning materials, which makes it difficult for teachers
to sustain learning environments that nurture creativity (Silver,
1997); (iii) Specifically to mathematics, there is a problem that
stem from teachers’ beliefs that creative thinking is not ex-
pressed across the curriculum, but is restricted to liberal arts
and humanities. In their minds, examples of creative outcomes
refer mainly to drawing, painting, writing and acting (Andiliou
& Murphy, 2010), viewing mathematics as offering fewer op-
portunities for creativity; (iv) Many teachers relate creativity to
giftedness, and therefore avoid nurturing all students’ creativity
(Aljughaiman & Reynolds, 2005; Sriraman, 2005); (v) Teach-
ers face difficulties in assessing their students’ creativity and its
development due a lack of an available simple tool and there-
fore unable to implement a systematic approach to develop stu-
dents’ creativity (Shriki, 2010).
This paper is based on two main premises: (i) Nurturing stu-
dents’ creativity is not only possible at all ages and ability lev-
els, but that it should be an integral part of the regular discipli-
nary curriculum; (ii) Providing teachers with a practical tool for
nurturing students’ creativity and assessing their progress might
inspire them to integrate appropriate activities in their regular
teaching. Given this stance, I plan to offer a model for assessing
the development of students’ creativity in the context of inquiry
assignments that have a component of problem posing. The
problem posing approach encourages students to ask questions,
explore a range of answers, and develop a critical perspective.
It is also considered to be strongly associated with fostering
creativity. The model being suggested uses four measurable
aspects of creativity: fluency, flexibility, originality, and gener-
alization. In addition, I intend to show that teachers’ and stu-
dents’ self-assessment of their own progress in each of these
aspects contributes to their development of creativity in the
context of problem posing. All the examples in this paper are
taken from mathematics, as well as some specific references to
mathematics education; however, the proposed model can be
adapted to every other discipline.
A Brief Literature Background
Interest in fostering creativity in modern education first sur-
faced in the 1950’s (Craft, 2001). The topic has garnered sig-
nificant worldwide attention since the late 90’s, leading some to
suggest that creativity has gained acceptance as a catalyst for
social and economic change (Lin, 2011). In as much as creativ-
ity is considered essential for future success in life (NACCCE,
1999), it is incumbent upon teachers to create stimulating
learning environments that are likely to nurture students’ crea-
tivity. Indeed, research has acknowledged the fundamental role
of education in nurturing students’ creativity as part of its wider
responsibility to instill content knowledge (Lin, 2011). As a
result, educators have shown interest in exploring and enhanc-
ing creativity (Henry, 2009); assume that creativity can be de-
veloped through explicit instruction (Fryer, 1996); and that all
students possess an innate sense of creative, given that they
have a natural curiosity for trying out new things (Feldman &
Benjamin, 2006). Turning specifically to mathematics, it is be-
lieved that genuine mathematical activities are closely related to
creativity (Silver, 1997), and developing students’ mathematics
creativity ought to be one of the primary goals of mathematics
education (NCTM, 2000).
In this section I present a brief literature survey that relates to
some common definitions of creativity and its relation to posing
mathematical problems. I will then introduce the “What If
Not?” (WIN) strategy as a means for posing new mathematical
problems based on a given problem. Finally, I will relate to
students’ gains from engaging in self-assessment of their own
Creativity “seems to be one of those words that although
commonly used is not easy to define. We may use the term re-
gularly, but can struggl e if asked to put into words sp ecifically to
what we are referring” (Henry, 2009: p. 200). Educational and
psychological researchers have explored the nature of creativity
for more than a century (Plucke r, Beghe t t o, & Dow, 2004). This
has resulted in a wealth of research aimed at understanding,
explaining, and assessing the development of creativity over the
years (Andiliou & Murphy, 2010) and examining its source and
expression in human experience (Silver, 1997). Furthermore,
there are over one hundred contemporary definitions of creativ-
ity (Mann, 2006). Most of these definitions are vague or inade-
quate due to the multifaceted nature of creativity (Sriraman,
2005) and none are universally accepted (Treffinger, Young,
Selby, & Shepardson, 2002). Adding to this complexity, Henry’s
(2009) literature review indicates that creativity has been re-
searched through the lens of, at least, four different perspectives:
the creative process, the creative person, the creative environ-
ment, and the creative product.
Drawing on Torrance’s (1974) definition of creativity, in the
context of this paper I refer to four of its aspects: fluency, flexi-
bility, novelty (or or i gi nality), and organization. “Fluency refers
to the number of ideas generated in response to a prompt; fle-
xibility to apparent shifts in approaches taken when generating
responses to a prompt; and novelty to t he o rigina lity o f th e id eas
generated in response to a prompt” (Silver, 1997: p. 97). Orga-
nization refers to the number of generalizations (Brandau &
Dossey, 1979). Of the four aspects listed in Torr ance’s definition
of creativity, novelty or originality is widely acknowledged as
the most a ppropria te asp ect becau se cre ativit y is genera lly view-
ed as a process related to the generation of original ideas, ap-
proaches, or actions (Leikin, 2009; Shriki, 2010).
Problem Posi n g a nd Nurtu r i n g S tu dents’ Creativity
In order to suppo rt the development of students’ mathematical
creativity, mathematics educators should view creativity as “an
orientation or disposition toward mathematica l activit y t h a t can
be fostered broadly in the general school population.” For that
matter, teachers should implement “inquiry-oriented mathema-
tics instruction which includes problem-solving and problem-
posing tasks” (Silver, 1997: p. 75). Refraining from developing
students’ mathematical creativity might lead them to perceive
mathematics as a set of skills and rules to memorize and cause
them to lose their natural curiosity and interest in mathematics
(Mann, 2006).
While contemporary views of creativity differ with respect to
the nature of the trait and to the ability of individuals to produce
creative outcomes, there is a growing consensus regarding the
centrality of problem posing and problem solving processes
within the creative act (Silver, 1997). According to Silver, pro-
blem posing has long been viewed “as a characteristic of crea-
tive activity or exceptional talen t in many fields of human endea-
vor” (p. 76). Supporting his argument, Silver maintains that al-
though mathematicians may solve problems that have been pos-
ed by others, they generally formulate their own problems based
on personal experiences and interests. Indeed, Albert Einstein
believed that “The formulation of a problem is often more es-
sential than its solution, which may be merely a matter of ma-
thematical or experimental skills. To raise new questions, new
possibilities, to regard old questions from a new angle, requires
creative imagination and marks real advance in science” (Ein-
stien & Infeld, 1938, in Ellerton & Clarkson, 1996: p. 518).
However, this perception stands in stark contrast to school-
based math ematics where, in most c ases, problems are pr esented
by teachers and in textbooks. Therefore, in order to develop
students’ mathematical creativity, teacher should engage their
students in activities that include problem posing and provide
them with developmentally appropriate, high interest opportu-
nities to pose their own problems and suggest solutions (Mann,
Copyright © 2013 SciRe s . 431
2006). As students pose their own problems, they become in-
novative, cr e ativ e, acti ve learners. They improve their reasoning,
develop diverse and flexible thinking, and enrich and strengthen
their knowledge and problem solving skills. Some have sug-
gested that they might even change their perception of mathe-
matics (Brown & Walter, 1990), as well as increase their ability
with respect to the central aspects of creativity: fluency, flexi-
bility, originality and organization (Silver, 1997).
Problem posing involves generating new problems and ques-
tions aimed at exploring a given situation, as well as the refor-
mulation of a problem over the course of its solution (Silver,
1994). This can be done by varying and reversing the “knowns”
or givens, or by varying the constraints of the problem (Marti-
nez-Cruz & Contreras, 2002). When faced with new and novel
mathematical situations, students are required to validate their
thinking and selection of mathematical co ncepts use d to generate
their answers, thus leading them to develop and deepen their
mathematical knowledge. When students formulate new prob-
lems, they develop a sense of ow nership over the subject matter,
which results in an increase of curiosity in and enthusiasm for
learning mathematics (Cunningham, 2004).
Employing the “What-If-Not?” Strategy for Posing
In order to enable studen ts to gen erate math emati cal proble ms,
Silver (1997) suggests employing the instructional approach de-
veloped by Brown and Walter (1969, 1990), often termed as the
“What-If-Not?” strategy. This method of instruction requires
students to generate new problems based on previously solved
problem, through a process of varying the conditions or goals of
the original problem (Silver, 1997). By implementing this ap-
proach, teachers are likely to support the development of stu-
dents’ creativity in mathematics.
The “What-If-Not?” (WIN) strategy suggested by Brown and
Walter (1969, 1990) is based on the idea that modifying the
components of a given problem can yield new and stimulating
problems that ultimately may result in some interesting inves-
tigations and may lead to the uncovering of mathematic regu-
larities. This approach to problem posing leads the students
through three levels of inqu iry, starting with a re- examination of
a given problem, in order to discover new related problems. In
the first phase, students are asked to produce a list of the prob-
lem’s attributes or conditions. In the second phase, students
focus on each attr ibute in the l ist, address the WIN que stion, and
suggest alternatives to the attributes. The third phase consists of
posing new problems and questions based on the alternatives
that arose during the second phase. Going through these phases
students may develop new perspectives: “Only after we have
looked at so meth ing, not as it is but as it is turned inside out or
upside down, do we see its essence or significance” (Brown &
Walter, 1990: p.15). Implementing the WIN strategy enables
teachers to expand their teaching repertoire, thereby generating
learning environments that encourage discussion of various
ideas and demonstrate to students that there is often more than
one “right way” to solve a given problem. This approach also
enables students to consider the meaning of a problem, rather
than simply focusing on finding its solution.
According to Haylock (1986), mathematical assignments
should include problem posing, problem solving, and redefini-
tion. The WIN strategy approach includes all three of these
components. The first level, in which students list the attributes
of a given problem, provides the opportunity to rethink mathe-
matical objects and concepts. The second level, addressing the
WIN question and suggesting alternatives, necessitates redefi-
nition of mathematical situations. For that matter students have
to consider the “ logical contexts and conditions that underlie the
determination of how to select certain givens and group them
together in order to create a coherent mathematical situation
(Lavy & Shriki, 2010: p. 19). The third level, posing new prob-
lems and questions and subsequently solving at least one of them,
are closely connected to creativity (Silver, 1997).
Evaluati o n o f Crea t i vi ty
The activities of problem posing and the creative aspects of
such activitie s—fluency, flexib ility, novelty, and organization—
are well established within the practice of assessing creativity.
Both processes and products of activities that involve problem
posing can be evalua ted in order to determine t he extent to which
creativit y is pr es ent. For that mat ter, it is poss ible to exam ine the
novelty of the problem formulation or the problem solution, the
extent to which modifications were evident, and the number of
formulations or reformulations produced or the number of dif-
ferent solu tion pat hs invest igated . Classi fying c reativi ty in te rms
of fluency, flexibility, novelty and organization also provides
teachers with an eas y sch ema for evalu ating th e tra it (Brand au &
Dossey, 1979; Silver, 1997).
Self-Assessment of Creativi ty
The Standards of the NCTM (2000) recommend engaging
students in self-assessment in order to nurture their confidence
and independenc e in lear ning mat hematic s. Skill s a cqui red t houg h
self-assessment are actually life skills that are applicable to a
wide range of situations (Smith, 1997). Engaging students in
self-assessment of attaining goals they set to themselves “al-
lows students ownership over and responsibility for their learn-
ing as well as providing real choices about what they learn. It
provides students with opportunities to spend time reflecting on
their learning… through being engaged in self-assessment, stu-
dents become deeply self-motivated and independent learners.”
(p. 7). Moreover, students’ involvement in assessing their own
progress in learning adds reflection and metacognition to their
learning. Students enjoy participating in self-assessment and
observing their progress using graphical displays, and are able
to use it for articulating the value of their own study (Brookhart,
Andolina, Meganzuza, & Furman, 2004). Relating specifically
to self-assessment of one’s own creativity, Chamberlin & Moon
(2005) believe that such process in itself requires creativity and
will, in return, enable students to refine their products in suc-
cessive iterations. Therefore, it is recommended to enable stu-
dents to assess their own creativity and its development. How-
ever, in order to help students assess their performance and pro-
gress, they should be provided with clear and easily understood
guidelines (Enz & Serafini, 1995), and they need to be taught
how to self-assess themselves and get a proper support from
their teachers (Brookhart et al., 2004). Research indicates (e.g.
Enz & Serafini, 1995) that after gaining a suitable training, stu-
dents are able to keep the self-assessment with little or no as-
sistance from their teacher, and students who are trained in self-
assessment outperform their peers who do not receive such pre-
Copyright © 2013 SciRe s .
A Model for Evaluating Creativity and Its
Development in the Context of Problem Posing
Considering th e mult ifaceted nature of creati vity, it is obvious
that selecting a definition that embraces all mathematical areas
or types of mathematical assignments would be impossible. In
this current framework, the proposed model is intended to eva-
luate creativity and its development in the context of inquiry
tasks that have a component of problem posing through relating
to fluency, flexibility, originality, and organization. Drawing on
Balka (1974), Brandau & Dossey (1979), and Torrance (1974),
fluency is measured by the number of different problems posed,
flexibility is measured by the number of different categories of
the posed problems, originality is measured by the relative in-
frequency of the problems, and organization is measured by the
number of problems stated as generalizations. As pointed out by
Brandau & Dossey, all of these variables are highly significant
and positively correlated.
The view of personal creativity, as a quality that can be de-
veloped in school students, requires a distinction between rela-
tive and absolute creativity (Leikin, 2009; Sriraman, 2005).
While absolute creativity is associated with remarkable histori-
cal works of pro minent mathem aticians, r elative c reativity refers
to discoveries made by a specific person within a specific ref-
erence group. This distinction is expressed in Leikin’s (2009)
model aimed at evaluating mathematical creativity in the con-
text of multiple solutions to a given problem. Leikin (2009)
suggested the notion of “solution spaces”. She referred to “Ex-
pert solution spaces” as those “that include the most complete
set of solutions to a problem known at a particul ar time”, where
in school mathematics “expert solution spaces include conven-
tional solution spaces, which are those generally recommended
by the curriculum, displayed in textbooks, and usually taught by
the teachers. By contrast, unconventional solution spaces in-
clude solutions based on strategies usually not prescribed by
the school curriculum, or that the curriculum recommends with
respect to a different type of problem”. “Individual solution
spaces” are perceived as “collections of solutions produced by
an individual to a particular problem”; and “Collective solu-
tion spaces” refer to the collection of all “individual solution
spaces within a particular community” (Leikin, 2009: p. 133).
Drawing on this distinction and adapting it to the context of
problem posing, in the proposed model below each student
receives a total as well as relative score of fluency, flexibility,
originality and organization. The individual total score is equi-
valent to the meaning of “individual space”, and the individual
relative score refers to the frequency of a posed problem in-
cluded in the individual space in relation to its frequency in the
collective space, as will be explained below.
Guidelines for Scoring the Four Aspe cts of Creativity
The scoring process refers to the third level of the WIN
strategy, wher e learners have to relate to the list of at tributes they
produced (first level) and their possible alternatives (second le-
vel) to pose new problems. In order to determine s tudents’ score
for each aspect of creativity, and an overall creativity score, the
following steps are applied:
a. Fluency scor ing
Student’s total score of fluency is determined by totaling the
number of different new problems he or she posed, based on a
given problem. The relative score of fluen cy is det ermined in t he
following manner: the student in the reference group who re-
ceived the highest total score of fluency is given a score of 100
for relative fluency. All other students’ r elative scores of fluency
are determined according to the highest score. For example, if
the highest total score is 20 (namely, each student posed 20
different problems at the most), then those who posed 20 prob-
lems receive a relative score of 100. Those who posed, for in-
stance, 12 problems receive a relative score of 60.
b. Flexibility scoring
Student’s total score for flexibility is determined by the total
number of different categories that are constituted by the posed
problems. The relative score of flexibility is determined simi-
larly to the relative score calculation used for fluency.
c. Originality scoring
Since originality is, by its very nature, relative there is a need
to predetermine the condition for originality. If we take, for
example, all the problems posed by a third of the students to be
the “upper” limit for originality, then all students who posed pro-
blems that 33% or less of th e students posed will receive a score
for originality. Obviously, this “upper limit” can vary from one
class to another. Other stu dents wi l l receive a score of 0. The re-
lative s core of or iginalit y is d eter mined in the fol lowing manner:
Assume that the group consists of 30 students. In this case only
problems that were posed b y 10 or fewer students ar e considered
for scoring originality. The students who posed the largest num-
ber of such problems receive a score of 100 for originality. Other
students who posed problems th at are consider ed for scoring ori-
ginality r eceive a relat ive score sim ilarly to the calcul ations used
for scoring fluency and flexibility.
d. Organization (or generalization) scoring
Students’ tot al score for or ganizatio n is determin ed accordi n g
to the number of posed problems that are formulated as a gen-
eralization , and then the r elative s core is calcula ted simila r to the
above cases.
e. An overall total score of creativity
Finally, the overall score of creativity is determined, assigning
a weight for each of the four relative scores. Determining rela-
tive weighting is subjected to the teachers’ discretion, according
to the importa nce th ey ascribe to each as pect. Cle arly, the b eliefs
teachers hold about creativity is likely to influence how they
define, operationalize, and evaluate students’ creativity (Andi-
liou & Murphy, 2010). Therefore, it is important to provide
teachers with the freedom to determine each component’s
weight, adaptin g it to their p refer ences, em phasiz es and te aching
Demonstr ating the S c or i ng Process
In this section I demonstrate the scoring process, illustrating
how to set the total and relative scoring for fluency, flexibility,
and organizati on; the scoring for origin ality; and the final overall
scoring for creativity. This demonstration is based on results
taken from a study carried out by Lavy & Shriki (2010). In this
study, 25 prospective teachers were engaged in problem posing
in geometry through implementing the WIN strategy. The given
problem was taken from Hönsberger (1985: p. 81): Triangle
ABC is inscribed in circle O. D is a variable point on the cir-
cumference of O. Perpendiculars are drawn from D to AB and
AC. E and F are intersection points of the perpendiculars with
the side of the tri angle, res pectivel y. Determi ne the po sition of D
whereby EF is of maximal length. Based on this problem, the
prospective teachers posed 46 different problems that were
classified into 12 different categories. Each problem was as-
Copyright © 2013 SciRe s . 433
Copyright © 2013 SciRe s .
PT1’s total score of fluency was 16. The relative score (R.S) of
fluency was determined in the following manner: The PTs with
the highest number of problems posed received a relative score
(R.S) of 100 for fluency. Table 1 indicates that PT10 was the
one who posed more problems than the other PTs (30 problems,
T.S = 30), and thus received a relative s core of 10 0. Other scor es
were determined relatively to PT10. For example, PT1, who
posed 16 problems, received a R.S of 53 [round of (16/30) *
100]. Other R.S of fluency was calculated in a similar manner.
signed a number indicating its category. For example, category
no. 1 was “A polygon is inscribed in a circle”. Namely, the only
change related to the type of polygon inscribed in the circle.
Four different alternative problems related to this category.
These new problems were numbered accordingly: 1.1 A quad-
rangle is inscribed in the circle; 1.2 A square is inscribed in the
circle; 1.3 A pentagon is inscribed in the circle; and 1.4 N-sided
polygon is inscribed in the circle. Problems 1.1, 1.2, 1.3, and
1.4 were posed by 25 prospective teachers (100%), 20 prospec-
tive teachers (20%), 14 prospective teachers (56%), and 4 (16%)
prospective teachers, respectively.
b. Scoring total and relative flexibility
For each PT, the number of categories his or her posed prob-
lems referred used for determining the total score of flexibility.
For example, PT1 posed problems that corresponded 7 of the 12
categories, thus her total score of flexibility was 7.
Organizing the Data
The PTs with the highest number of categories to which their
problems corresponded received a relative score of 100 for
flexibility. From Table 1 we can see that PT16 posed problems
that fitted in all 12 categories, and thus received a relative score
of 100. Other R.S was determined relatively to PT16. For ex-
ample, PT1’s problems matched 7 categories, thus her relative
score of flexibility was 58 [round of (7/12) * 100]. Other R.S. of
flexibility w as calculated in a similar manner .
In order to “map” the prospective teachers’ posed problems,
Table 1 was g enerated. In the first line app ears each prospective
teacher’s (PT) number (1 - 25). The left column in the grey part
of the table is the number of each categor y (c.), 1 - 12, and in the
second left column appears the number assigned for each prob-
lem that was posed in the particular category. For each PT a
mark of “ +” appears in case he or she posed the specif ic problem.
c. Scoring total and relative originality
Scoring Fluency, Flexibility, Originality,
Organization, and Creativ ity We shall n ow exami ne the two most right columns of Table 1,
titled “Orig”. For each problem that was posed, the number of
PTs who posed it was counted and recorded in the second right
column (titled “Tot”). For example, problem 1.1 was posed by
all 25 PTs. The problem with the highest number of PTs who
suggested it was indicated by “100%” (se e the most right column
a. Scoring total and relative fluency
For each PT, the number of problems he or she posed was
counted, and this determined his or her total score of fluenc y (see
Table 1-Flu, T.S). For example, PT1 posed 16 problems, thus
Table 1.
Data relating to the problem posed by each prospective teacher, according to categories; total and relative scores of fluency, flexibility, originality,
and organization, an d a score of overall creativity.
C. 1 2 3 … 10 11 12 13 14 15 16 17 … 23 24 25 Tot%
1.1 + + + … + + + + + + + + … + + + 25100
1.2 + + + … + + + + + + + + … 2080
1.3 … + + + + + + … + + + 14 56
1.4 … + + + + … 4 16
2.1 + + + … + + + + + + + + … + + + 25100
2 2.2 + + + … + + + + + + + + … + + + 25100
12.1 … + + … 6 24
12 12.2 … … + 1 4
T.S 16 17 17 … 30 29 29 29 29 29 28 25 … 19 16 17
Flue R.S 53 57 57 … 100 97 97 97 97 97 93 83 … 63 53 57
T.S 7 7 7 … 9 9 9 9 9 11 12 10 … 8 8 9
Flex R.S 58 58 58 … 75 75 75 75 75 92 100 83 … 67 67 75
T.S 0 0 0 … 7 6 5 5 5 5 5 2 … 3 2 3
Orig R.S 0 0 0 … 100 86 71 71 71 71 71 29 … 43 29 43
T.S 0 0 0 … 1 1 2 2 1 1 0 0 … 1 0 0
Org R.S 0 0 0 … 50 50 100100 50 50 0 0 … 50 0 0
Creativity 28 29 29 … 81 77 86 86 73 77 66 49 … 56 37 44
(PT = prosp ective teacher, c = category, Flue = fluency, Flex = flexibility , Orig = originality, Org = organization, T.S = total score, R.S = relative score).
of Table 1) . Oth er problems wer e calculated relative to problem
1.1 (or other problems that were posed by all the PTs). For
example, problem 12.1 was posed by 6 PTs, and thus was indi-
cated by 24% [(6/25) * 100].
As mentioned, for scoring originality I arbitrarily chose to
include only problems posed b y a third or less of the prospectiv e
teachers. N amely, problem s recorded in the most right column as
“33%” or less. In order to determine each PT’s total score for
originality, the number of problems posed tha t were indi cated b y
“33%” or less were counted. This constituted the total score for
PT’s originalit y (se e Table 1-Orig, T.S). For example, PT1 did
not pose any such problems, thus her T.S for originality was 0.
The prospective teachers with the highest number of such
problems received a relative score of 100 for originality. From
Table 1 it can be seen that PT10 posed 7 problems that were
suggested by less than 33% of the PTs. Her total score for
originality was 7 and her relative score for originality was 100.
Other scores for relative originality were determined based on
PT10’s score. For example, PT12 suggested 5 such problems,
thus her score of originality was 7 1 [round of (5/7) * 100]. Other
scores of r elative orig inality were calculated in a similar m anner.
It should be noted that in Leikin’s (2009) model originality
was evaluated by comparing individual solution spaces with the
collective solution space of the reference group based on the
rarity of their solution, distinguishing between the types of so-
lutions: those that were suggested by less of 15% of the students,
those that were suggested b y more than 15% but less that 40 % of
the students, and those that were suggested b y more than 40% of
the students. Each type of solution received a different score for
originality. Therefore, in the case of the present proposed model
teachers might wish to distinguish between the originality of the
posed problems based on similar observation instead of adhering
to one upper limit (e.g. 33%), as suggested.
d. Scoring total and relative organization
The score for organization was assigned in accordance with
the number of problems that were posed as generalizations.
There were only 5 such problems (1.4, 5.6, 5.10, 8.2, and 9.2).
Table 1 indicates that two PTs (PT12 and PT13) posed two
generalized problems. Therefore, their total score for organiza-
tion was 2 (Org, T.S) and their relative score for organization
was 100. Those who posed 1 generalized problem received a
total score of 1, and a relative score of 50. The remainder
received a total and relative score of 0.
e. Scoring overall creativity
Finally, in order to assign a score for each prospective tea-
cher’s overall creativity, it is necessary to determine the relative
weight of fluency, flexibility, originality and organization. As
mentioned, the rela tive we ight shou ld reflect the im portance tea-
chers ascribe to each component, as well as their priorities, pre-
ferences, emphasizes given in class, and teaching goals. In this
example, merely for a matter of simplicity, I arbitrarily assigned
each component the same weight (25%), namely a round of the
sum of all R.Ss divided by 4, as can be seen from the last row of
Table 1.
Inferring Information from the Scores
As stated, tea chers face d ifficulti es in evalu ating the c reativity
of their students and its development. As a natural consequence,
questions arise when employing the model for assessing stu-
dents’ creativity and its development as well as interpreting the
numerical results. The following are several suggestions. It
should be noted that these suggestions are based on results ob-
tained from two small-scale studies that were carried out through-
out the design of this model (see below).
Graphical Display of Data
Graphical display of data can include various types of infor-
mation. For exam pl e, t he r elat iv e s cor es o f ever y s tudent in each
of the four aspe cts of creat ivity, as well as the score for cre ativity;
distribution of total and relative scores; and more. Bellow there
are two examples, based on a complete version of Table 1 (Fig-
ures 1 and 2).
Based on such graphical displays teachers can get an idea
about the strengths and weakness es of each student as well as the
entire class, and make some pedagogica l decisions regardi ng the
emphases they should put in order to nurture students’ creativity.
Similar displays can also describe cumulative results of a proc-
ess that takes place over a prolonged time, in order to receive a
feedback about the impact of their teaching.
Evaluating t he Develop me nt of Creat ivity
Assuming that teachers engage their students in problem
Figure 1.
Relative scores of every prospective teacher (1 - 25).
Copyright © 2013 SciRe s . 435
Figure 2.
Distribution of relative scores.
posing assignments for an extended period of time, how can
teacher employ the proposed model for assessing the develop-
ment of their students’ creativity in order to make educational
decisions? What indicators might serve this purpose?
The model allows producing the following four numerical
values: to tal scor e s, relative scores , averages, and s ta nd ar d devi-
ations. The strengths and weaknesses of these numerical values
for making educational decisions are as follows:
a. Total scores
The total scores for fluency, flexibility, originality and orga-
nization cannot be used for assessing the development of crea-
tivity b ecause th e num ber of p rob lems that can b e posed throug h
employing the WIN strategy, and the resulted categories, might
depend on the richness of attributes that are embedded in the
original problem. However, total scores might be used in cases
where teachers wish to make comparisons between classes that
work on the same problems. Such a comparison is valuable
mainly when teachers wish to examine the impact of different
approaches intended to engage their students in problem posing
assignments, or compare between students learning at different
levels of mathematics or different age groups with respect to the
impact of engaging them in posing problems.
b. Average relative scores
In the case of relative scores, a significant increase in the av-
erage of any of the relative scores indicates that some students
developed their creativity much more than others. In such cases,
special attention might be given to students whose relative
scores did not change much with time. Alternatively, it may
indicate that creativity is indeed innate, and students that are
more creative than oth er from the outset show a gr eater tendency
to keep develop their creativity. Minor or no changes might im-
ply that the entire class, as a reference group, either did not
progress or all students exhibited similar progress (or regres-
c. Standard deviations
Assuming that practicing the WIN strategy over and over
again will not cause a decrease in mathematical creativity, a
decrease of st andard deviat ion in any of th e measures implies the
development of creativity of the class as a whole. Obviously, this
is the most desirable situa tion. Therefore, an increase or decrease
in standard deviations should serve as a primary indicator for
examining the development, lack of development, or perhaps a
decrease of creativity attributed to the class as a group of refer-
ence. If indeed, creativity of all students can be developed
through appropriate education, as suggested by the above men-
tioned research literature, then we would expect a gradual re-
duction in standard deviations over time. However, a gradual
increase in standard deviation might support theories that main-
tain that creativi ty is inna te, and that onl y except ional peopl e can
demonstrate c reative b ehavior. It s hould be noted that in order to
be able to tra ck c hanges, there is a need to predetermine the “up-
per limit” used for scoring originality, as well as the weight of
each component of creativity, and adhere it. Otherwise, tracing
changes over time would be impossible.
Some Insights Gained from Using the Model in
As mentioned, the proposed model was designed through
carrying our two small-scale studies. In the first study 6 upper-
elementary mathematics teachers were engaged in a series of 5
problem posing activities employing the WIN strategy. All the
initial problems were taken from common school textbooks in
order to demonstrate the idea that problem posing can easily
become an integral par t of school curr iculum. B y t he end of e ach
activity th e teachers presented their posed problems and tog ether
categorized the problems in order to gain an unders tanding about
the meaning of c ategor ization . In additi on, the te achers anal yzed
the appropriateness of each posed problems (see below). Then I
generated a table similar to Table 1, and each teacher received,
personally, only the information that was relevant to his or her
total and re lative scor es in a form at of graph ical displa y. Starting
from the second activity the graphical display included cumula-
tive scores (see example in Figure 3). This graphical display
was intended to help the teachers trace their progress/withdra-
wal compared to his or her reference group. During the entire
process the teachers documented their work in a reflective jour-
nal, and referred to aspects that concerned their perspective of
themselves as learners of mathematics, their mathematical crea-
tivity, the pedagogical insights they had gained, and the role
played by their ability to self-assess their progress/withdrawal.
In the second study eac h teacher r epeated the process wit h one
of his or her classes.
The results of these studies allowed formulating the proposed
format of the model, and examine th e effectiveness of t he u se of
the model to track the development of each student’s creativity,
in all four asp ects, as well as the entire clas s as a referenc e group.
Due to space limitations full results of these studies are not
included in this paper; however I would like to shed light on
some of the insights gained from them:
Types of Problem Posing Situations
In their study, Stoyanova and Ellerton (1996) suggest that
every problem posing situation can be classified as free,
semi-structured or structured. Problem posing situation will be
referred to as free “when students are asked to generate a
problem from a given, contrived or naturalistic situation. Some
directions may be given to prompt certain specific actions” (p.
519); as semi-structured when students are given an open
situation and are invit ed to expl ore the st ructure and to complete
it by ap plying kn owledge, sk ills, concepts and relationships from
their previous mathematical experiences”; and as structured
when problem-posing activities are based on a specific prob-
lem” (p. 520). The above mentioned studies indicated that the
teachers found that it was easier to implement the WIN strategy
within a structured problem posing situation, especially due to
Copyright © 2013 SciRe s .
Figure 3.
An example of a personal graphical display of cumulative relat ive scores
(t = task).
lack of previous experience with posi ng mathematical problems.
In their opinion, such situation generates “a firmer anchor to
hang on”. There is still, however, a need for a prolonged wide-
scale study aimed at examining the impact of the type of situa-
tion on the development of students’ creativity.
Appropriate n es s of the Pose d Probl ems
There is one reservation concerning the “freedom” of posing
new problems. Given that students are scored for fluency, the
teachers noticed that they might be tempted to pose as many
problems as possible, without considering their appropriateness.
Inappropriate problems can have, for example, insufficient or
irrelevant information. Namely, posing mathematical problem
cannot stand as its own goal, and students must be explicitly
instructed how to consider criteria such as appropriateness or
correctness with regard to the requirements and constraints of
the task (Andiliou & Murphy, 2010), and not implement a wa-
tered down version of the approach in an effort to introduce
“novel ideas.”
Discussing the Essence of Original Problems
In order to develop students’ appreciation to originality, the
teachers found it important to present original problems that
were posed by their fellow classmates, and spend time discuss-
ing and negotiating issues that concern the essence of these
problems. In their opinion, such discussion might shed light on
distinctions between “regular problems” and “original prob-
lems”, and, as suggested by Silver (1997), instruct students how
to evaluate the novelty of a posed problem.
Avoid Overemphasizing the Need to Solve the Posed
As mentioned, the WIN strategy can yield new and stimulat-
ing problems that ultimately may result in some interesting in-
vestigations. However, as ev iden t from t he f irst st udy m entioned
above, overemphasizing th e need to solve the problems that were
posed might suppressed the willingness to pose, what might be
regarded as a “revolutionary problem”. In such cases, learners
might hold back, concerned about their inability to solve the
problems they themselves have posed (Lavy & Shriki, 2010).
Therefore, in order not to ‘block’ the flow of problems posed by
students, teachers should instruct them not to worry about their
inability to solve some of the problem; otherwise they might li-
mit themselves to posing trivial problems. Instead, solving “dif-
ficult” problems might be done through a collaborative class
Students’ Self-Assessment of Their Progre ss
As mentioned, learners’ self-assessment of one’s own crea-
tivity enables them to refine their products in successive itera-
tions (Chamberlin & Moon, 2005). Indeed, as was evident from
the teachers’ and the students’ reflective portfolios, the graphi-
cal displays of their gradual progress, as describe in Figure 3,
was a powerful means that allowed them to reflect on their
strengths and weaknesses as well as their progress over time,
both relatively to their previous accomplishments and their re-
ference group. It turned out that t he use of the model and “trans-
lating” personal results into graphical display had a great im-
pact on the teachers’ and students’ motivation to improve their
fluency, flexibility and originality, as well as their drive to
search for possible generalized problems.
The Flexibility of the Model
As mentioned, teachers are free to determine the upper limit
for creativity as well as the relative weight of each component.
In that sense, the model is rather flexible and actually enables
teachers to adapt it to their teaching goals. As was evident from
the first study mentioned above, the teachers attributed great
importance to this flexibility. In their opinion organization, for
example, should receive a low weight in cases where low achi-
evers are engaged in problem posing, since they have difficul-
ties in understanding the meaning of generalization. Originality,
for example, should have a lower upper limit in classes of high
achievers, in order to encourage them to pose unconventional
problems. Therefore, as stated, it is important to leave the final
decision in the hand of the teachers, allowing them to adjust the
model to the target populations and their educational philoso-
Discussion and Conclusions
In a changing technological society, innovations are recog-
nized as the vehicle of economic and social growth and as es-
sential to the welfare of all (Andiliou & Murphy, 2010). Pro-
moting these innovations necessitates creativity (Shalley & Gil-
son, 2004). When taken to its logical conclusion, and viewing
creativity and content knowledge as inseparable (Rowlands,
2011), this implies that the objective of education should not be
limited to enhancing knowledge and skills, but also to nurturing
creativity (Craft, 2009). However, creativity is not often nur-
tured in school (Sriraman, 2005). As has been mentioned in the
introduction section, several cond itions ar e offer ed to justify this
situation. This complex set of perceived obstacles suggests that
attention should be given, first and foremost, to modifying tea-
chers’ beliefs regarding the nature of creativity, as beliefs tea-
chers hold regarding creativity are likely to influence the role
they assume in relation to creative thinking as a learning objec-
tive, the instructional approach they implement for fostering
students’ knowledge and creative behavior in the subject matter
being taught, and the evaluation procedures they apply in order
to assess creative products (Andiliou & Murphy, 2010). A first
step towards achieve this goal is to assist teachers to view crea-
tivity as inherent in learning (Beghetto & Kaufman, 2009), and
inspire teachers to believe that all students can become creative—
as creativity is not an exclusive trait of the gifted (Rowlands,
2011). However, in order for teachers’ beliefs to translate into
instructional practice, they have to establish the development of
Copyright © 2013 SciRe s . 437
creative thinking as a discrete learning goal. In addition, atten-
tion should be given to strengthen teach ers’ ability and readiness
to nurture students’ creativity, and provide them with appropri-
ate tools and pedagogic approaches aimed at supporting their
capability to develop and assess students’ creativity. All of this
must be done while taking into account constraints such as co-
verage of the mandated curr iculu m and stand ardized testing out-
Although contemporary views o f creativity diff er with respect
to the nature they ascribe to creativity and to the ability of indi-
viduals to produce creative outcomes, there is a growing con-
sensus regarding the centrality of problem posing and problem
solving processes within the creative act (Silver, 1997).
In order to enable studen ts to generat e their own mathem atical
problems, Silver (1997) suggests employing the three-phase
WIN instructional approach developed by Brown and Walter
(1969, 1990). The third phase, posing new problems and ques-
tions and subsequently solving at least one of them, are closely
connected to creativity (Silver, 1997). By implementing this
approach, teachers are likely to support the development of stu-
dents’ creativity in mathematics. In addition, by discussing the
posed problems students learn to evaluate the novelty of a pro-
blem. According to Haylock (1986), mathematical assignments
should include problem posing, problem solving, and redefini-
tion. The WI N strategy approach inc ludes all three of thes e com-
ponents. Therefore, it appears that implementing this approach
systematicall y in cl asses might satisf y both develo ping studen ts’
mathematical creativity and effectively respond to the need to
adhere to curricular demands within specific time limitations.
Namely, the WIN approach guarantees that nurturing creativity
does not come at the expense of teaching the subject matter, but
rather completes it. For that matter, instead of askin g students to
solve 10 different problems that do not have anything in com-
mon, teachers might ask students to solve 10 different problems
that can be derived from the same mathematical situation. In re-
turn, arriving at conclusions and identifying generalizations will
strengthen students’ mathematic al and metamath ematical know-
ledge (Lavy & Shriki, 2008; Shriki, 2010).
In this paper I suggest a model for assessing students’ crea-
tivity and i ts developm ent in the context of problem posing, with
the aim to respond teachers’ need for having a coherent and ac-
cessible tool to serve this purpose (Shriki, 2010). Reid and Pe-
tocz (2004) state that “it is a fairly difficult exercise to discern
what is meant by the term creativity’, or to deci de wh at ma y be
interpreted as a creative object, or to describe the cognitive
traits that characterize a creative person” (p. 46). Therefore,
the proposed model is based on measurable aspects of creativity,
namely—fluency, flexibility, novelty and organization (Tor-
rance, 1974; Silver, 1997). Viewing personal creativity as a qua-
lity that can be developed in school students, there is a need to
distinguish between relative and absolute creativity (Leikin,
2009). Thus, in the proposed model, e ach student receives a total
as well as relative score of fluency, flexibility, originality and
organization. As teachers’ beliefs about creativity influence how
they define, operationalize, and evaluate students’ creativity
(Andiliou & Murph y, 2010), it is import ant to let teachers d ecide
what they perceive as the meaning of “originality” and how to
weight of each of the measured aspects for calculating final
scores of cre ativity. In th at sense, the model is rath er flexible and
actually enables teachers to adapt it to their teaching goals. In
addition, it is suggested to provide students’ with an ongoing
feedback regarding their progress, preferably through a graphi-
cal display of their accomplishments.
Taking all together, it is hoped that with careful planning and
implementation, a pedagogic approach that combines problem
posing with the proposed model and students’ self- assessment of
their progress will enhance the teacher’s ability to nurture stu-
dents’ creativity and assess its development while taking into
account constraints such as coverage of the mandated curriculum
and standardized testing outcomes.
There is still, however, a need for a wide scale experiment in
order to be able to determine which of the four mentioned as-
pects of creativity is more likely to be developed through prob-
lem posing—that is, subject to observable outside influences,
why and how. A wide scale experiment will als o en able to dee p-
en the insights regarding the meaning of numerical indicators
like averages an d standard deviations that can be derive from the
model, and the ab il ity to interpre t them.
Concluding Remarks
This paper neither suggests any conclusive definition of
creativity, nor an ultimate approach dictating how to nurture
it. Teachers need to be provided with a wide variety of in-
formation, tools, and resources in order to enable them to
consolidate their world-view regarding these issues, and al-
low them to decide which approach best suits their teaching
goals, educational values and beliefs.
Given that the final score of creativity does not provide
specific information about the scores of each component,
one might prefer to display the 4 relative scores and the fin al
score of creativity in a format of a vector such as (R.S flu-
ency, R.S fl exibil ity, R.S orig inal ity, R.S o rganiz ation, over -
all crea tivity).
The problem posing approach, through implementing the
WIN strategy, can be adapted to all school disciples, and
therefore the proposed model for assessing the development
of creativity in the context of problem posing can be em-
ployed by teachers from all content areas, and not exclu-
sively by mathematics teachers.
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