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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.

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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

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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.

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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

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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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