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The intersection of problem posing and creativity
The 11th International MCG Conference
Hamburg, Germany, 2019 59
THE INTERSECTION OF PROBLEM POSING AND CREATIVITY:
A REVIEW
Julia Joklitschke1, Lukas Baumanns2, Benjamin Rott2
1University of Duisburg-Essen, Germany, 2University of Cologne, Germany
Abstract. In this article, we take an in-depth look at research on the intersection of problem posing
and creativity in order to present its current state of research in a systematic review. A full search in
top journals from mathematics education and the Web of Science revealed only 15 articles from
different genres, of which 11 were included in the analysis. Those articles were sorted into two
clusters, depending on whether the articles focus on the identification or the fostering of creativity.
Key words: Problem Posing, Creativity, Review
INTRODUCTION
In the 1990’s, Edward Silver published two seminal articles in which he addressed both
mathematical problem posing and mathematical creativity. The first article (Silver, 1994)
deals with problem posing, emphasizing it as a characteristic of creative activities and
mathematical ability. In the second article, Silver (1997) takes the opposite perspective,
mainly addressing creativity and highlighting its connections to problem posing (as well
as problem solving). Both contributions are widely cited in research literature and
constitute the theoretical foundation for many studies dealing in one way or another with
problem posing and creativity (cf. Bonotto, 2013; Voica & Singer, 2013; Van Harpen &
Presmeg, 2013; Sriraman & Dickman, 2017; Singer, Sheffield, & Leikin, 2017). In a recent
handbook chapter, Cai, Hwang, Jiang, and Silber (2015) discuss the progression of
problem posing research along ten answered as well as 14 unanswered questions.
Amongst others, they ask whether it is feasible to use problem posing as a measure of
creativity, pointing at one possible connection between problem posing and creativity.
There is, however, still much work to do in this field. Working in both the field of problem
posing (Baumanns & Rott, in print) as well as in the field of mathematical creativity
(Joklitschke, Rott, & Schindler, 2018), we were
intrigued to examine the intersection of both fields (Fig.
1) as indicated by Silver (1994, 1997) or Cai et al.
(2015). Ayllón, Gomez, and Ballesta-Claver (2016)
conducted a review of this intersection. However, there
are some uncertainties (details are explained below) in
the content and it is not clear to what extent the review
fully reflects the existing research literature. Therefore,
this article presents an attempt at a systematic review
of studies dealing with both problem posing and
creativity published in highly ranked journals.
BACKGROUND
In the following, we provide a current theoretical understanding of mathematical
problem posing, mathematical creativity, and their intersection.
Fig. 1: Intersection of research
on problem posing and
creativity as focus of this paper
Joklitschke et al.
The 11th International MCG Conference
60 Hamburg, Germany, 2019
Problem Posing
Problem posing has been emphasized as an important mathematical activity by many
mathematicians (e.g., Hadamard, 1945; Cantor, 1966/1932) as well as mathematics
educators (e.g., Brown & Walter, 1983; Silver, 1994; English, 1997). As an important
companion of problem solving, problem posing can lead to flexible thinking, improve
problem-solving skills, and sharpen learners’ understanding of mathematical contents
(English, 1997). There are two definitions of problem posing, at least one of which is used
or referred to in the majority of research papers on the topic. The first definition was
proposed by Silver (1994, p. 19), who describes problem posing as the activities of
generating new problems and reformulating given problems. Both activities can occur
before, during, or after a problem-solving process. The second definition comes from
Stoyanova and Ellerton (1996, p. 518), who refer to problem posing as the “process by
which, on the basis of mathematical experience, students construct personal
interpretations of concrete situations and formulate them as meaningful mathematical
problems”. The authors also maintain a categorization for different types of problem-
posing situations and differentiate between free, semi-structured, and structured
problem-posing situations, depending on their degree of given information.
Creativity
Solving and posing complex problems often requires creative ideas; particularly in
technology and science, this development is very important. Educational research also
has an increased interest in research in this field (Singer et al., 2017; Joklitschke,
Schindler, & Rott, 2018). Research on creativity goes back to at least the psychologist
Guilford (1967) exploring the nature of intelligence. In his work, he differentiated
convergent and divergent thinking abilities, the latter encompassing fluency, flexibility,
originality, and elaboration. These dimensions are apparent in the well-known Torrance
Tests of Creative Thinking (TTCT; Torrance, 1974), which is an attempt to make creativity
measurable quantitatively. In the field of mathematics education, several researchers
draw on this composition to assess mathematical creativity (e.g., Leikin & Lev, 2013;
Pitta-Pantazi, 2017). Other researchers (e.g., Liljedahl, 2013) look at creativity using a
model consisting of the phases preparation, incubation, illumination, and verification and
thereby follow Hadamard (1945). In early research, mathematical creativity was
attributed exclusively to experts (e.g., Hadamard, 1945) and was therefore an absolute
characteristic. However, a number of researchers assume that creativity may also be
attributed to students, their processes, or products and view creativity as a more relative
construct (e.g., Leikin & Lev, 2013).
Intersection of problem posing and creativity
As we explained in the introduction, a considerable part of studies investigating the
intersection of problem posing and creativity refers to the articles of Silver (1994 and
1997, resp.), which is why we highlight Silver’s key statements in the following.
In 1994, Silver points out that various tests to identify creativity include problem-posing
situations; thus, it is reasonable to assume a connection between problem posing and
creativity. However, he states that the nature of this connection remains uncertain and
needs further investigation. In 1997, Silver considers Torrance’s (1974) categories of
fluency, flexibility, and originality as key components of creativity and provides
The intersection of problem posing and creativity
The 11th International MCG Conference
Hamburg, Germany, 2019 61
instructional suggestions how to foster creative activities in classrooms through problem
posing. Furthermore, Silver (1997) emphasizes that “the connection to creativity lies [...]
in the interplay between problem posing and problem solving” (ibid., p. 76). In the
following, we focus on problem posing and its relation to creativity and describe the
findings in this field on the basis of the following research questions: (1) What kind of
(and how many) journal articles exist dealing with the intersection of mathematical
problem posing and mathematical creativity? (2) To what extent is this intersection
conceptualized?
METHODS
For this review, we used the preliminary work of two literature reviews on problem
posing (Baumanns & Rott, in print) and on mathematical creativity (Joklitschke, Rott, &
Schindler, 2018). We focused on (a) databases from the seven A*- and A-ranked journals
(Törner & Arzarello, 2012) and on (b) the Web of Science within selected categories on
mathematics and its education. In the databases from (a), we used the search term
problem posing for all available years up to 2017. This procedure led to 332 articles. We
then read all abstracts and extracted all articles that have problem posing either in their
titles, abstracts, or keywords; this led to 48 articles. Furthermore, we consulted the
database (b) Web of Science for the years 1945 to 2017. Excluding the already considered
articles from the A*- and A-ranked journals, this led to another 81 articles. Within these
129 articles on problem posing from (a) and (b), we looked for the search term creativ*
within the titles, abstracts, and keywords to identify articles that potentially deal with the
intersection of mathematical problem posing and mathematical creativity. In total, only
15 articles (eleven from the A*- and A-ranked journals and four from the Web of Science)
remained.
In order to examine those 15 articles in a systematical and criteria-led way, each article
was carefully read and assigned to one of the following genres: theoretical contributions,
review articles, perspective or opinion, empirical research with mainly qualitative methods,
and empirical research with mainly quantitative methods. Thereafter, the articles were
examined with regard to their content. Due to our research question, we concentrated
mainly on the conceptualizations of problem posing and creativity and the
implementation of the empirical research if there is any. Thereby, the following questions
were decisive: What is the main message of the article? Which theories and
conceptualizations are cited? How is the relation between problem posing and creativity
represented? Based on these questions, clusters were formed inductively to classify the
articles into coherent groups that represent different approaches at the intersection of
problem posing and creativity.
RESULTS
Introduction of the reviewed articles (Research question 1)
In the following, the 15 articles are presented and sorted by their genres.
Theoretical contributions: Two articles from our data set – the already mentioned articles
by Silver (1994 and 1997, resp.) – were considered as theoretical contributions to the
intersection of problem posing and creativity. Since the central focuses of the articles
have already been covered above, we refer to the background for additional information.
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Both articles are widely recognized as milestones in the (back then) young research fields
on problem posing, mathematical creativity, and its intersection, respectively.
Review articles: One article written by Ayllón et al. (2016) is a review article summarizing
central results regarding the relationship between creativity, problem posing, and
problem solving. Apart from some inconsistencies and inaccuracies (e.g., wrongly
assigned contents), the article clearly fits the topic of our review. However, as the number
of articles considered to reflect the state of research of the intersection of problem posing
and creativity is limited (three of four cited studies are discussed here as well; the fourth
article does not meet our search criteria), the article by Ayllón et al. is not further
considered.
Perspective or commentary: Two articles were categorized as commentary articles as they
neither report on empirical studies, nor provide a theoretical discussion on problem
posing or creativity. (1) Haylock (1997) presents examples of tasks designed to identify
creativity in 11-12-year-old students. Highlighting overcoming fixation as a key
component of creativity and referring to Guilford’s and Torrance’s ideas of divergent
production, Haylock discusses specially designed problem-solving, problem-posing, and
redefinition tasks that have been tested in previous studies. (2) Sriraman and Dickman
(2017) discuss the use of mathematical pathologies (i.e. unpleasant counterexamples in
the sense of Lakatos) to foster creativity in the classroom. In addition to historical
examples of pathologies, the authors present examples from current classrooms in which
students explore counterexamples or “incorrect” methods leading to correct results (e.g.,
a misinterpretation of rules to deal with fractions). Sriraman and Dickman then propose
using the Lakatosian heuristic (conjecture – proof – refutation), to pose interesting
problems that productively deal with pathologies and counterexamples.
Empirical research with mainly qualitative methods: In two articles, qualitative empirical
studies are presented. (1) Leung (1997) correlates posed problems of 96 grade five
students from Taiwan working on 18 different initial situations and describes those
creatively posed problems in terms of change of content and context. (2) Voica and Singer
(2013) investigate cognitive flexibility (i.e. variety, novelty, and change in framing) in
problem posing. They analyze the products of 42 students with above average
mathematical abilities working on structured problem posing situations.
Empirical research with mainly quantitative methods: In four articles, quantitative
empirical studies are reported. (1) Bonotto (2013) investigates the potential of so-called
artifacts (i.e. real-life objects like restaurant menus, advertisements, or TV guides) to
stimulate critical and creative thinking. Additionally, she analyzes problem-posing and
problem-solving products from 71 primary school students (stimulated by artifacts),
using Guilford’s categories of fluency, flexibility, and originality. (2) Singer, Voica, and
Pelczer (2017) assess the cognitive flexibility (as an indicator for creativity) of 13
prospective teachers by analyzing the products from geometric, semi-structured
problem-posing situations. (3) Van Harpen and Presmeg (2013) investigate the
relationship between mathematical problem-posing abilities and mathematical content
knowledge among high school students from three different countries. Similar to Bonotto
(2013), they analyze the students’ problem-posing products using the dimensions of
fluency, flexibility, and originality to assess the students’ creativity. (4) Van Harpen and
Sriraman (2013) also use those dimensions to analyze problem-posing products of 218
high school students from the USA and China.
The intersection of problem posing and creativity
The 11th International MCG Conference
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Excluded articles: Four articles (in alphabetical order) had to be excluded for reasons that
are outlined below. (1) Ernest (2015) discusses social outcomes of learning mathematics
in school by presenting standard aims as well as unintended (and often negative)
outcomes (e.g., values, attitudes, and beliefs) and visionary aims (in which he emphasizes
mathematical creativity through problem posing and solving) of school mathematics.
Ernest uses neither the theoretical literature of problem posing research, nor that of
research on creativity; he mentions those terms in his discussion and emphasizes their
importance. (2) Patton (2002) uses biographical interviews to trace the recognition of
creativity in the lives of famous entrepreneurs and scientists. To interpret his data, Patton
uses the systems theory view of creativity, which suggests that creativity is not about being
unique, but about being the first or being a “problem pioneer”. He does cite literature
from research on creativity (esp. Csikszentmihalyi), however, he does not use any
literature from research on problem posing and, therefore, does not work in field of the
intersection of both. (3) In the article by Poulos (2017), the author investigates the way
an expert problem poser (a coach of the Greek team for the IMO) poses problems on the
Olympiad level by describing two interviews. The author only uses literature from
problem-posing research addressing experts’ behavior. He does not cite any articles from
creativity research and does not investigate the expert’s creativity. (4) Singer, Sheffield,
and Leikin (2017) wrote the introductory article to a ZDM special issue on creativity and
giftedness in mathematics education. Thus, they do not present research results or new
theoretical ideas in this article, but rather give a historical overview of research on those
topics.
Cluster formation (Research question 2)
In order to summarize ideas and empirical implementations that can be found in research
on a meta level, we will focus on the articles of the categories empirical research and
perspective or opinion. Articles of the categories theoretical contributions and review
articles will be held aside. The remaining eight articles can now be merged inductively
into clusters. Some articles are assigned to multiple clusters.
Cluster I: Problem-posing situations to foster creativity: The first cluster contains articles
that provide examples of problem-posing situations that are especially appropriate to
foster creativity. For Cai et al. (2015, p. 17), the question which kind of problem-posing
situations are appropriate to promote students’ creativity is still unanswered. (1)
Haylock (1997) presents tasks for his key components of creativity. For the component
of overcoming fixation, he designed series of problems in which stereotypical approaches
should be discarded. Additionally, he argues that in particular specially designed
problem-solving, problem-posing, and redefinition tasks require the component of
divergent production. (2) Bonotto (2013) investigates the potential of artifacts as semi-
structured problem-posing situations to identify (see also II.a) and foster critical and
creative thinking in the classroom. (3) Sriraman and Dickman (2017) use mathematical
pathologies, the Lakatosian heuristic, and problem-posing activities to address students’
creativity.
Cluster II: Identifying and investigating creativity through problem posing
II.a: Guilford’s and Torrance’s framework: This cluster contains all articles that – similar
to Leikin and Lev’s (2013) analyses of multiple solution tasks – use the categories fluency,
flexibility, and originality based on works by Guilford (1967) and Torrance (1974) to
investigate the participants’ problem-posing products. (1) To measure fluency, Bonotto
(2013) takes the total as well as the average number of problems created by the pupils
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working with her artifacts into account. To measure flexibility, the posed problems are
categorized with regard to the number of details presented as well as the data introduced
by the students. To measure originality, the rareness of the posed problems is considered:
if a problem was posed by less than 10 % of the other pupils, it was considered original.
(2) Van Harpen and Sriraman (2013) operationalize fluency and originality in the same
way. Flexibility is measured by the total number of categories (e.g. analytical geometry,
lengths, area, angles) the posed problems of a student can be assigned to. (3) Van Harpen
and Presmeg (2013) use the same operationalizations of all categories as Van Harpen and
Sriraman (2013). (4) The article by Leung (1997) could not be classified into this cluster
quite as clearly; although Torrance’s dimensions are mentioned and the test instrument
is also based on the TTCT, these components do not play a role in the empirical part.
II.b: Other approaches to measure creativity through problem posing: The third cluster
considers the approach by (1) Voica and Singer (2013), and (2) Singer, Voica, and Pelczer
(2017). This cluster represents another line of research on mathematical creativity that
is based on organizational-theory and discusses creativity in terms of cognitive flexibility
(composed as cognitive variety, cognitive novelty, and change in framing) as an indicator
for creativity. As a theoretical concept, they refer to the construct of cognitive flexibility
to grasp the relationship between problem posing and creativity. As a methodological
concept, they especially use the criteria of coherence and consistency of the posed
problems. Additionally, Singer, Voica, and Pelczer (2017) consider the two dimensions of
Geometric Nature (GN), and Conceptual Dispersion (CD). The GN assesses whether the
posed problem is about finding sizes or specific computation (metric), or about geometric
reasoning without computation (qualitative). The CD assesses if the posed problems are
organized within clearly defined structures or systematically exploiting a configuration
(structured), or the posed problems are e.g. disconnected from each other (entropic).
They found that cognitive flexibility is inversely correlated to metric GN and structured
CD.
CONCLUSION & OUTLOOK
In 1994, Silver stated that it is reasonable to assume a connection between mathematical
problem posing and mathematical creativity; the concrete connection, however, was
unknown. 25 years later, the gain in knowledge is still limited. In our literature review,
we found only eleven articles in the whole databases of all A*- and A-ranked journals on
mathematics education and on the Web of Science that address the intersection of
problem posing and creativity. Since our review goes back to the founding dates of
journals (e.g., ESM started in 1969, JRME in 1970, and FLM in 1980.), we realized that no
articles addressing both problem posing and creativity have been published before 1994.
Analyzing the content of the articles under review, we were able to build coherent
clusters focusing on (I) problem-posing situations to foster creativity and (II.a) using
problem posing to identify and investigate creativity via Guilford’s and Torrance’s
framework or (II.b) other approaches to measure creativity through problem posing. The
inductively built clusters (I) and (II.a) can also be deductively sustained from Silver’s
theoretical considerations. In 1997, he focuses on fostering creativity through problem
posing (and problem solving) which is the key aspect of cluster (I). Additionally, both of
Silver’s articles (1994, 1997) provide considerations to assess, identify, and investigate
mathematical creativity through problem posing by applying Guilford’s (1967) and
Torrance’s (1974) framework as in cluster (II.a). Interestingly, the assumption that
The intersection of problem posing and creativity
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Hamburg, Germany, 2019 65
problem posing can be used to measure creativity has not thoroughly been investigated,
for example by correlation studies.
Apparent limitations of this review lie in the mere description of the articles central
methodological and content-related elements. A more in-depth discussion about the
chosen theoretical foundations, the research approaches as well as the results cannot be
carried out at this point. Furthermore, the aim of this article was to look at the
intersection of creativity and problem posing, which is why we did not consider other
constructs such as problem solving or giftedness.
In order to expand the dataset of the existing articles, it would be of interest for a future
review to also consider the databases of B-Journals (Törner & Arzarello, 2012), seminal
collections on problem posing such as Mathematical Problem Posing (Singer, Ellerton &
Cai, 2015), as well as the papers of the International Group for the Psychology of
Mathematics Education (PME) and the International Group for Mathematical Creativity
and Giftedness (MCG). Furthermore, additional keywords such as innovat*, invent*, and
divergent think* regarding creativity (cf. Joklitschke, Rott, & Schindler, 2018) or task
design, and problem formulation regarding problem posing would extend the range of
articles considered in the field of mathematical creativity. This consideration may lead to
a wider data base, further clusters, and a better chance of comparing the different
research approaches.
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