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Chapter 9

Creativity in Question and Answer

Digital Spaces for Mathematics Education:

A Case Study of the Water Triangle

for Proportional Reasoning

Benjamin Dickman

Abstract As digital spaces evolve, mathematics educators must develop an

awareness of the ways in which these environments can facilitate discussion and

foster creativity. Question and Answer (Q&A) sites such as Mathematics Educators

Stack Exchange (MESE) provide a platform through which those interested in the

teaching and learning of mathematics can harness new technologies to address

novel queries, and engage collaboratively with others who share their interests. This

chapter aims to trace one example of a question-answer combination on MESE as

situated in the broader context of technology and creativity in mathematics edu-

cation, and to utilize the example as a lens through which we can critically examine

the current state of digital environments and reﬂect on their potential use by

mathematics educators.

Keywords Collaborative emergence Mathematical creativity Online spaces

Participatory model of creativity Q&A sites

9.1 Introduction

The ideas outlined in this chapter coincide with an evolution of the digital spaces

that can foster mathematical creativity. Geographical barriers no longer pose the

same sort of hindrance to collaboration among education researchers and practi-

tioners. Today, mathematics educators come together through social media such as

the MathTwitterBlogosphere (MTBoS), deftly navigate vast repositories of math-

ematical information such as the arXiv, and communicate directly through various

web forums.

B. Dickman (&)

New York, USA

e-mail: benjamindickman@gmail.com

©Springer International Publishing AG, part of Springer Nature 2018

V. Freiman and J. L. Tassell (eds.), Creativity and Technology in Mathematics

Education, Mathematics Education in the Digital Era 10,

https://doi.org/10.1007/978-3-319-72381-5_9

233

Question and Answer (Q&A) sites for mathematics are not new to the decade, or

even the millennium. The online Geometry Forum, which is now NCTM’sMath

Forum,

1

traces the history of its Q&A component Ask Dr. Math back to 1994.

2

Two

years later, in 1996, the similarly titled Math Doctor began at Nicholls State

University, but changed its name shortly thereafter to Math Nerds, which had “about

one hundred volunteers and answer[ed] about 1500 questions per month”as mea-

sured about a decade after its creation (De Angelis et al., 2008, p. 28). Writing out of

the Technical Education Research Centers (TERC), Rubin’s(1999)Technology

Meets Math Education: Envisioning a Practical Future Forum on the Future of

Technology in Education “insists that, rather than looking at math education from the

perspective of the computer, we must look at computers from the perspective of

mathematics education”(p. 1). In doing so, Rubin identiﬁes ﬁve powerful uses of

technology in mathematics education, among which Resource-Rich Mathematical

Communities includes the Math Forum as “the best known”resource site at the time

(Rubin, 1999, p. 8). This subsection on resource sites concludes by remarking that the

forum “has served as an important portal for mathematics educators and as a kind of

social center for the mathematics education community”(p. 9).

The social centers for mathematics education communities have continued to

exist on the Internet, but have evidently changed over the past two decades. One

feature is the inclusion of sites that allow anyone, not just those who are conﬁrmed

as experts, to answer questions about mathematics and, in some cases, mathematics

education. For example, the reddit community dedicated to socializing around

mathematics

3

has subscribers in the hundreds of thousands, and allows anyone to

sign up and post or comment about mathematical links and questions.

In this chapter, we look at a particular Q&A digital space, Mathematics

Educators Stack Exchange (MESE), which ﬁts within the broader Stack Exchange

network. Unlike many of its predecessors, MESE is organized around mathematics

education rather than mathematics proper. To gain insight into how MESE ﬁts into

the landscape of technology and creativity in mathematics education, we proceed as

follows: First, we articulate the three key ideas that will be covered throughout the

chapter, after which we look to the literature as concerns the breadth of deﬁnitions

that have arisen over the years in investigations of ‘technology’and ‘creativity’.

Next, we provide brief remarks around the connections between our speciﬁc subject

of study and the broader topics of this text: mathematics and mathematics educa-

tion, technology, and creativity. The third section summarizes our speciﬁc subject

—a question posted to MESE about a tool used in proportional reasoning—and

then investigates the ways in which responding to a reference request is an act of

creative collaboration. The fourth and ﬁnal section provides avenues for further

research by proposing three open questions related to the initial key ideas, before

closing with our conclusion.

1

(http://mathforum.org/).

2

(http://mathforum.org/dr.math/abt.drmath.html).

3

(https://www.reddit.com/r/math).

234 B. Dickman

9.1.1 Key Ideas: Q&A Sites, Deﬁnitions, and Creative

Collaboration

We non-exhaustively list here three key ideas for the chapter, which relate,

respectively, to Q&A sites for mathematics education, the importance of deﬁning

terms and scope when discussing technology and creativity, and the situating of

everyday ideas as examples of creative collaboration within a participatory model

of creativity.

Key Idea 1 Question and Answer (Q&A) sites speciﬁc to mathematics education

are a recent phenomenon, and have emerged along with a collaborative paradigm in

which users frequently serve as both askers and answerers of questions. Earlier

precursors include Q&A sites speciﬁc to mathematics, which have existed for over

two decades, yet for which the askers and answerers have sometimes constituted

disjoint, or nearly disjoint, groups.

Key Idea 2 The many deﬁnitions for ‘technology’and ‘creativity’require a certain

amount of speciﬁcity in any discussion for which they play prominent roles. We

advocate for an interpretation of ‘technology’that admits both digital technologies

(such as online Q&A forums) and domain-speciﬁc tools (such as the water triangle

for proportional reasoning). Moreover, we advocate for an interpretation of ‘cre-

ativity’that coincides with Hanson’s(2015a,b) description of a participatory

model: Rather than focusing on single ideas or identifying individual creators, we

look at how creative collaboration (e.g., through a Q&A forum) is distributed

among actors and objects.

Key Idea 3 Our particular example of a tool (the water rectangle) paired with a

necessarily incomplete account of its history does not constitute a watershed,

domain-shifting moment in mathematics education; rather, the collaborative cre-

ativity exempliﬁed by the reference request described in this chapter contributes to

an ongoing conversation about proportional reasoning, in particular, and mathe-

matics education, in general. It is a conversation that began before the modern

language of proportional reasoning existed, has continued with the predecessors for

this tool and the tool itself, was furthered by the satisﬁed reference request, and

which will continue far beyond the everyday ideas put forth in this chapter.

9.1.2 Deﬁnitions for ‘Technology’and ‘Creativity’Over

Time

Deﬁnitions for ‘technology’and ‘creativity’abound. As contemporary conceptions

of technology undergo rapid change, we begin by looking back to Hansen and

Froelich’s(1994) early attempt at articulating the variety of deﬁnitions for ‘tech-

nology’in their aptly-titled Deﬁning Technology and Technological Education,in

9 Creativity in Question and Answer Digital …235

which they remark that “philosophers, anthropologists, sociologists, historians, and

teacher educators continue to study the subject [of technology], yet a widely

accepted deﬁnition remains obscure”(p. 179). The authors continue in exploring

deﬁnitions of ‘technology’by looking to dictionaries and considering its etymol-

ogy; by looking to individual scholars from a variety of domains; by considering,

among other conceptions, ‘technology’with regard to products and processes; and

by examining technology as relates to feminism and the evolution of women’s roles

in society (Franklin, 1992). Analogously, there are elsewhere discussions about the

emergence of ‘creativity’in English dictionaries (e.g., Mason, 2003); debates about

whether creativity is domain-speciﬁc (e.g., Baer, 1998; Plucker, 1998); conceptions

that include creativity with respect to products and processes (Rhodes, 1961); and

discussions around creativity as relates to the evolution of women’s roles in society

(Bateson, 2001,2004). Beyond these parallels, in Trefﬁnger et al. (2002) the

authors remark that “Trefﬁnger (1996) reviewed and presented more than 100

different deﬁnitions [of creativity] from the literature”(p. 5), and Sawyer (2011)

goes so far as to contend that “deﬁning creativity may be one of the most difﬁcult

tasks facing the social sciences”(p. 11). Deﬁning either term is certainly no easy

task.

There is a school of thought within creativity research, originating with work by

Amabile and Hennessey (e.g., Amabile, 1983,1996; Hennessey, 1994; Hennessey

& Amabile, 1999), in which one operationalizes subjective agreement on that which

constitutes ‘creativity’with respect to particular products, rather than providing a

formal catch-all deﬁnition for the term. There are also schools of thought, more

process-oriented, that essentially identify creativity with problem solving; for

example, Weisberg (2006) writes that “it seems reasonable to adopt as a working

assumption the premise that creative thinking is an example of problem solving”

(p. 581). In a similar vein, others associate creativity with problem posing; for

example, Getzels (1975) quotes Einstein as stating that, “The formulation of a

problem is often more essential than its solution, which may be merely a matter of

mathematical or experimental skill. To raise new questions, new possibilities, to

regard old questions from a new angle, requires creative imagination and marks real

advances in science”(p. 12). Such an approach (see also: Getzels &

Csikszentmihalyi, 1976; Runco, 1994) continues the line of thought associated with

Guilford’s(1950) trait of “sensitivity to problems”as relates to creativity (p. 454).

This work of Guilford appeared in his APA presidential address, and also touched

upon the ability to reorganize, or redeﬁne, in the sense of Gestalt psychology;

relatedly, one ﬁnds Wertheimer (1959) remarking that “often in great discoveries

the most important thing is that a certain question is found. Envisaging, putting the

productive question is often more important, often a greater achievement than

solution of a set question”(p. 141).

236 B. Dickman

9.1.3 Perspectives Adopted for Investigating the Creative

Use of Technology

In this chapter, we will adopt a combination of perspectives on creativity: We will

use the Question and Answer (Q&A) format of web-based platforms to frame the

creative use of technology. The notion that both questions and answers are

important is established within the discipline of mathematics education. We may

associate ‘answers’with the process of ﬁnding solutions, and look to the vast

literature on mathematical problem solving (e.g., Polya, 1945; Schoenfeld, 1985,

2010) and, similarly, associate ‘questions’with the process of problem formulation,

and look to the vast literature on problem posing (e.g., Brown & Walter, 2005;

Duncker & Lees, 1945; NCTM, 1989,1991; Kilpatrick, 1987; Silver, 1994).

Furthermore, we assume of the reader a familiarity with the domain of mathematics

education, and, therefore, the capability to apply the subjective commendation of

‘creative’to products (cf. Baer & McKool, 2009). The sheer breadth among con-

ceptions of technology and creativity will make our own study intractable without

ﬁrst limiting our scope; we use here a case study of one, which is an approach

foreign neither to creativity research (e.g., Gruber & Davis, 1988; Gruber &

Wallace, 1999) nor mathematics education (e.g., Brizuela, 1997; Erlwanger, 1973).

Speciﬁcally, we will trace a single example of a question-answer combination

posted on the Mathematics Educators Stack Exchange (MESE) website, and unpack

from a seemingly straightforward reference request the ways in which technology

and creativity collide in a present-day digital space designed for those interested in

the teaching and learning of mathematics.

9.2 Brief Connections to Mathematics, Technology

and Creativity

In this section, we situate our subject of investigation by connecting it to the three

broad topics of mathematics and mathematics education, technology, and creativity.

9.2.1 Brief Connections to Mathematics and Mathematics

Education

The Stack Exchange network includes over 150 Q&A communities; among these

are MathOverﬂow (MO), which is designed for those engaged in research level

mathematics, as well as Mathematics Educator Stack Exchange (MESE), which is

designed for those interested in the teaching and learning of mathematics. Earlier

work by Tausczik et al. (2014) explored the collaborative problem solving that

takes place on MO, and the ﬁve “collaborative acts”of providing information,

9 Creativity in Question and Answer Digital …237

clarifying the question, critiquing an answer, revising an answer, and extending an

answer identiﬁed through a process of open coding (p. 359). Though not explicitly

connected there, the authors’research ﬁts well with the notion of collaborative

emergence in the creativity literature (Sawyer, 2011; Sawyer & DeZutter, 2009). At

present, there appear not to have been any investigations of MESE, which was

proposed as a site in 2014, and currently holds over 8000 combined questions and

answers in the domain of mathematics education.

MESE is speciﬁc to mathematics education, and the example traced here is no

exception: “The ‘water triangle’proportional reasoning task”

4

is a reference request

about the origin of a tool previously depicted on Wikipedia’sproportional rea-

soning page.

5

Proportional reasoning is a fundamental component of early math-

ematics education, and relates to work with such topics as ratios, fractions, rational

numbers, and rates (Tourniaire & Pulos, 1985; Lobato et al., 2010). The asker

suggests the tool may have been created by mathematics educator Robert Karplus in

the 1970s, but is otherwise unaware of its history. This tool inspired the con-

struction of a water rectangle in the asker’s dissertation on mathematics education,

as well as subsequent investigations presented at the ICMI-East Asia Regional

Conference on Mathematics Education (Noche, 2013; Noche & Vistro-Yu, 2015).

9.2.2 Brief Connections to Technology and Online Forums

The movement to incorporate technology into learning trajectories can be seen by the

growing presence of online classes, MOOCs, sites such as Coursera and MIT

OpenCourseWare, and web-based platforms such as Moodle and Blackboard to

supplement classroom-based courses. There are also digital spaces associated with

post-secondary programs in mathematics education, such as The Math Forum (Drexel

University, mathforum.org) and The Mathematics Teaching Community (University

of Georgia, mathematicsteachingcommunity.math.uga.edu). Mathematics Educators

Stack Exchange (MESE, matheducators.stackexchange.com) is not associated with

an academic institution, and instead ﬁts within the Stack Exchange (SE) network; the

network includes an additional site speciﬁcally for mathematics questions at the

research level (MathOverﬂow.net) and another for general mathematical queries

(math.stackexchange.com). Although SE contains over 150 different Q&A com-

munities, MESE is, at present, the only one concerned speciﬁcally with education.

In addition to the technology involved in interacting through a digital environ-

ment, both the question and answer connect to technology, as well. The question

explored is about a particular form of technology: although it is not digital tech-

nology, the water triangle is itself a tool for investigating proportional reasoning

(Kurtz, 1976). With regard to digital technology, the answer emerged from a

4

http://matheducators.stackexchange.com/q/29.

5

https://en.wikipedia.org/wiki/Proportional_reasoning.

238 B. Dickman

conﬂuence of sources: MESE, Wikipedia, ProQuest, e-mail and more. To unpack

the power of a modern technological tool such as a Q&A digital space, we will

explore both the ‘Q’and ‘A’part of the given example; more precisely, we must

remain cognizant of the types of technology that exist outside of the

computer-based forms commonly associated with contemporary conceptions of

‘tech’—a deep understanding of connections to technology emerges most promi-

nently when the digital requirement is dropped, and a broader toolbox conception is

adopted.

9.2.3 Loose Ends: A Couple of Additional Connections

to Creativity

In addition to the already mentioned problem solving, problem posing, and col-

laborative emergence, we consider two more important connections to the literature

on creativity. First, Stokes (2005,2010) discusses the development of creativity

through constraints. The digital space under discussion is designed speciﬁcally for

questions about mathematics education; this precluding constraint ensures that

questions that are deemed off-topic by other site users are either reﬁned, migrated to

another site, or closed entirely. Moreover, there exists an additional promoting

constraint with regard to novelty; namely, that new questions be distinct from

earlier ones: If the question already exists on the network, then site users may

choose either to close the new version or encourage its modiﬁcation so as to prevent

repetition. Second, Rhodes’(1961) classical framework around situating creativity

concerns the person, process, product, and environmental press. These are only a

few of the many conceptions of creativity, and, although this chapter contains a

portion narrativized as a personal recollection, our ultimate goal is to consider

creativity from a variety of perspectives; as is the case with connections to tech-

nology, a deeper understanding of creativity emerges when we adopt a broader

toolbox conception.

9.3 Collaborative Creativity Through a Reference

Request

In this ﬁrst sub-section, we detail the history of a single example of a routine

reference request, which will provide us with a lens through which, in the subse-

quent sub-section, we may examine the current state of question and answer digital

environments as we reﬂect on their potential use by mathematics educators.

9 Creativity in Question and Answer Digital …239

9.3.1 Reference Request: ‘Water Triangle’for Proportional

Reasoning

“The ‘water triangle’proportional reasoning task”was initially posted on the

Mathematics Teaching Community in 2012

6

where it remained unresolved. The

question was modiﬁed by its creator and re-posted to MESE in 2014. The question

essentially asked about the source of the ‘water triangle’depicted on Wikipedia’s

proportional reasoning page; the illustration under discussion can be seen in

Fig. 9.1.

The author of this chapter ultimately located the original source of the water

triangle, and provided an accepted answer to the query; in recounting how this

answer came about, we change voice here to the ﬁrst-person for the sake of clarity:

I began by investigating the Wikipage for proportional reasoning, and also

looked through its history to see if there was relevant information to be found in

earlier versions of the page. Earlier incarnations of the Wikipage included an

additional photograph of a physical water triangle being used by students (Fig. 9.2)

and the image had the same credited uploader as the illustrated version already

shown in Fig. 9.1.

The original question on MESE included a mention of mathematics educator

Robert Karplus; however, both of the images were credited to Barry L. Kurtz,

whose e-mail address was included, as well. I wrote to Professor Kurtz to ask

whether he was aware of the water triangle’s origins; his response message was as

follows:

I completed my Ph.D. under Bob Karplus at UC Berkeley. I was his last Ph.D. student. My

dissertation dealt with teaching for proportional reasoning. I invented the idea of a “water

triangle”to teach inverse proportions. There were all made by the workshop at the

Lawrence Hall of Science; they were not a commercial item. I doubt any exist today; I

certainly don’t have any. Thanks for your interest. You did a good job tracking me down!

I followed up on this lead by using ProQuest

7

to ﬁnd Kurtz’s doctoral disser-

tation, where the water triangle can be found on page 34; an image of the disser-

tation (Kurtz, 1976) is displayed in Fig. 9.3.

In a follow-up message, Kurtz pointed to an article based on his dissertation

(Kurtz & Karplus, 1979) and noted that it was later reprinted in Fuller’s(2002)

A Love of Discovery: Science Education—The Second Career of Robert Karplus.

6

https://mathematicsteachingcommunity.math.uga.edu/index.php/685/the-water-triangle-proportional-

reasoning-task.

7

http://www.proquest.com/.

240 B. Dickman

Fig. 9.1 The water triangle for proportional reasoning (released by its author into public domain

for any purpose, and without any terms or conditions; cf. https://commons.wikimedia.org/wiki/

File:Water-triangle.JPG)

Fig. 9.2 Students using a physical version of the water triangle (released by its author into public

domain for any purpose, and without any terms or conditions; cf. https://commons.wikimedia.org/

wiki/File:Constant-product.png)

9 Creativity in Question and Answer Digital …241

9.3.2 Collaborative Creativity: What Do We Mean

by ‘Creative’?

The previous sub-section brieﬂy detailed how a question on Math Educators Stack

Exchange, which asked about the origin of a water triangle for proportional rea-

soning, was investigated and resolved. This section aims to answer the natural

follow-up question with respect to inclusion in a textbook on creativity and tech-

nology in mathematics education; succinctly: Okay, so what?

We begin by observing the manifold ways in which technology allowed for such

a question-answer combination. Speciﬁcally, we have an educator in The

Philippines posing a question about mathematics education, for which a mathe-

matics educator in the East Coast of the United States was able to respond by

tracking down graduate research completed in the West Coast of the United States

some four decades earlier. Without a conﬂuence of technological means—

Wikipedia, which allows users to participate by uploading original content; e-mail,

which allows individuals to communicate without the geographical barriers that

would have hindered such an interaction in the recent past; and ProQuest, which

includes doctoral dissertations uploaded to a searchable database—such a query

would have been essentially intractable. Moreover, there needed to exist a digital

space that could facilitate such interactions between individuals with the relevant

curiosity and expertise: for the asker, this meant the curiosity and expertise to

formulate the question after utilizing a domain-speciﬁc tool based on the water

triangle in their own work on mathematics education; for the answerer, this meant

the curiosity and expertise to understand and investigate the question, and to use a

combination of digital technologies in order to resolve it.

Creating spaces in which members of a ﬁeld can come together to interact

meaningfully is a nontrivial endeavor. As mentioned earlier, the reference request at

the heart of this chapter was already posted to another digital space for mathematics

Fig. 9.3 An excerpt from the doctoral dissertation of Barry L. Kurtz

242 B. Dickman

education two years prior to its re-post on MESE. Internet-based Q&A digital

spaces provide both a platform and a technological tool that can be used to foster

creativity within the domain of mathematics education; such an observation points

to the requirement not only for tools and technologies, but also to users who can

intentionally and capably operate them. For the question itself, we needed an

environment that not only could house such a query, but could also do so in an

accessible manner: for example, using again the language of Stokes (2005,2010),

we needed the search space to be both broad enough to allow for a variety of

questions (e.g., by instituting promoting constraints around novelty such as insti-

tuting a reputation-based system to award credit to those who formulate

well-received questions and answers) and narrow enough to appeal to members of a

particular ﬁeld (e.g., by instituting precluding constraints such as closing duplicated

questions, or, more generally, having a site speciﬁc not to mathematics or to

education but rather to mathematics education).

Similar observations hold outside of the digital-interpretation of ‘technology’:

The original water triangle, depicted as an illustrated diagram in Fig. 9.1, is itself an

example of a tool and technology; however, its educational relevance is perhaps

more clearly depicted in Fig. 9.2, where we observe individuals interacting with a

physical model. Even still, the images themselves require additional interpretation

and exploration; to this end, we require members of a ﬁeld to push concepts further

by creating web-based content such as a Wikipage or a response on MESE, by

developing the language of proportional reasoning and explaining its

domain-relevance in the earlier dissertative work and publications, by continuing

with new ideas around proportional reasoning in more recent dissertative work and

presentations, and, self-referentially, by summarizing and connecting these various

contributions in our present account of matters as they stand today. None of this is

accomplished by individuals working in a vacuum, just as none of the ideas has

emerged ex nihilo; rather, we have the collaborative emergence (Sawyer, 2011;

Sawyer & DeZutter, 2009) that is enabled by the collision of technologies: physical

and digital, old and new.

At this point in our discussion, it is hopefully clear that Q&A digital spaces

exemplify an environment that has the potential to facilitate discussion among those

with domain-relevant expertise and interest. But our argument here is that these

spaces can foster creativity, and it is to this claim that we must now attend. What do

we really mean by ‘creativity’? In Hanson’s(2015b)Worldmaking: Psychology and

the Ideology of Creativity there are a variety of conceptions around creativity put

forth and described. Speciﬁcally, Hanson (2015a) writes that:

…the concept of creativity provides a site to explore important issues within a framework

of often unquestioned assumptions. Beyond claims of the speciﬁc theories, the amalgam of

theories have contributed to an underlying ideology. This ideology is important because it

concerns one of the most salient characteristics of our times: change. It is also a fascinating

ideology because, in keeping with the values it represents, the ideology changes over time.

The belief espoused in our own account is not that satisfying a reference request

constitutes a sort eminent or ‘big-C’Creative (e.g., Kaufman & Sternberg, 2009)

9 Creativity in Question and Answer Digital …243

achievement that shifts an entire domain (Csikszentmihalyi, 1999). Instead, we

wish not to leave the underlying assumptions around creativity unquestioned, and

adopt a view of creativity as a participatory model. Quoting Hanson (2015a) again:

Instead of focusing almost entirely on how to get people to think of new ideas, the par-

ticipatory models situate ideation within individual development, group dynamics and

historical settings. The support roles and the ﬁeld (gatekeeper) roles that people take up as

they integrate novelty become more central. Choosing, supporting, interpreting and reﬁning

ideas are as important as ‘having’an idea. Indeed, on close examination, distinctions

between ﬁeld roles and the ‘creator’role begin to disappear…Creator as curator [of ideas]

emphasizes the tasks of selecting, emphasizing, and powerfully presenting ideas that –as

always –derive from historical domains, broader culture (‘commonsense’), the artifacts of

culture and other people’s ideas.

The creativity in our discussion is distributed among the many actors and objects

involved, ranging from the research of Karplus and students on proportional rea-

soning, to the question-answer combination on MESE, to the understanding

reached (extended, challenged, etc.) by the reader of the work at hand. In most any

direction we look, there is more to be unpacked and asked about (e.g., How did

Wikipedia, and the wiki-paradigm more generally, support these interactions? How

did the availability of resources at the UC-Berkeley Lawrence Hall of Science

contribute to Kurtz’s work?). Technologies such as Q&A digital spaces allow

educators to expand and develop their own network of enterprises (Gruber &

Wallace, 1999) as they participate in projects that allow for new ways of organizing

the self—what needs to be done, what can I do, and what must I do (cf. Gruber &

Barrett, 1974)—in relation to one’s own ongoing work and the creative endeavors

of others.

Our attention to the role of Q&A digital spaces in fostering creativity is not

restricted to Rhodes’(1961) framework around the person,process,product,or

press; nor is it with respect to some sort of self-actualization (e.g., Maslow, 1943)

in becoming “Creative Educators,”or by conﬂating creativity writ large with the

separate construct of divergent thinking within out-of-the-box models (Runco,

2010), Gestaltist views on creative insight (Wertheimer, 1959), or eminence-based

theories of creativity (Csikszentmihalyi, 1999). Rather, we focus on continuous

participation in the evolving ideology of creativity (Hanson, 2015b). As mathe-

matics education develops as a domain, the corresponding ﬁeld of mathematics

educators is confronted by challenges that require the use of tools and technologies

both new and old. Even, or especially, as central concepts in mathematics educa-

tion, such as proportional reasoning, have changed over the past several decades,

we must be prepared to exchange and modify ideas and understandings within

spaces that allow us to engage in the sort of dynamic interactions that best support

our work as individuals and as groups. Neither a routine reference request nor the

novel pathway carved out in resolving it constitute paradigm-shifts in creativity

within mathematics education. Instead, these questions and answers, the digital

spaces that provide platforms for their formulation, and the many people whose

244 B. Dickman

work is inextricable from the technologies and tools that are built and used to

resolve them are all parts of a conversation around the teaching and learning of

mathematics, and all contribute to the creativity necessary to face the continuous

change found in the ever-evolving world of education.

9.4 Looking Ahead and Wrapping Up

We conclude our chapter by posing three follow-up questions that correspond to

possible ways in which our initial three key ideas can be further explored.

9.4.1 Looking Ahead: Three Open Questions

We conclude with three open questions for future investigation, consideration, and

research. Each of the three questions is intended to extend, or challenge, compo-

nents of the respective key ideas from the beginning of the chapter. None of these

questions is intended to be answered succinctly, or even directly; instead, they are

posed as questions to guide, or inﬂuence, those who wish to think further around

the subject matter contained in, or related to, this chapter.

Open Question 1 As the distinction between asker and answerer is blurred within

Q&A sites, and forums become more inclusive of participants at various positions

along a novice-to-expert continuum, how can we better ensure that creative col-

laboration will ultimately be productive both within the conﬁned digital space, and

within broader conversations throughout the domain of mathematics education?

Open Question 2 For those who adhere to, or advocate for, interpretations of

‘technology’and ‘creativity’different from ours, what conclusions can be drawn

about the impact of evolving Q&A sites for mathematics education? For example,

for those whose deﬁnitions of technology are restricted to the digital, and for whom

the participatory model of creativity is rejected in favor of an out-of-the-box

divergent thinking model, are digital spaces such as Math Educators Stack

Exchange well-positioned to support and foster creativity?

Open Question 3 Given a conception of creativity in which it is viewed as a

feature of an ongoing conversation, and the creative collaboration is distributed

across many individual actors to the extent that supporter roles and creator roles

fade away, what impact does the adoption of such a perspective have on students,

teachers, and other stakeholders in mathematics education?

9 Creativity in Question and Answer Digital …245

9.4.2 Conclusion

We have advocated in this chapter for a view of creativity that is participatory:

rather than looking to identify eminent individuals or singular insights, we chose to

examine everyday creativity as exempliﬁed by a reference request. Our view of

technology was similarly inclusive, as it admitted not only the digital sort, but also

tools in a more general sense, up to and including a physical water triangle for

proportional reasoning. As we did not concern ourselves with ﬁnding the one

person, or one moment, or one idea, to which the commendation of ‘creativity’can

be applied, we focused instead on the distribution across multiple actors, and the

ways in which they fulﬁlled their roles as facilitated by a particular Q&A site. In

doing so, we actively push back against the myth (cf. Weisberg, 1986) that cre-

ativity is strictly the work of geniuses. And in promulgating a participatory model

of creativity in which a hierarchy of creator and supporter roles ceases to exist, we

hope to shift the focus away from precisely who or what can be considered creative,

and instead to think more inclusively about the ways in which a diverse array of

individuals can productively work together, whether this collaboration occurs in the

Q&A digital spaces of today, or in yet-to-be-created spaces of the future.

References

Amabile, T. M. (1983). A consensual technique for creativity assessment. In The social psychology

of creativity (pp. 37–63). New York, NY: Springer.

Amabile, T. M. (1996). Creativity in context: Update to the social psychology of creativity.

Boulder, CO: Westview.

Baer, J. (1998). The case for domain speciﬁcity of creativity. Creativity Research Journal, 11(2),

173–177.

Baer, J., & McKool, S. (2009). Assessing creativity using the consensual assessment. In Handbook of

assessment technologies, methods, and applications in highereducation. Hershey, PA: IGI Global.

Bateson, M. C. (2001). Composing a life. New York, NY: Grove Press.

Bateson, M. C. (2004). Composing a life story. In Willing to learn: Passages of personal discovery

(pp. 66–76). Hanover, NH: Steerford Press.

Brizuela, B. (1997). Inventions and conventions: A story about capital numbers. For the Learning

of Mathematics, 17(1), 2–6.

Brown, S. I., & Walter, M. I. (2005). The art of problem posing (3rd ed.). London, UK:

Psychology Press. https://books.google.com/books?hl=en&lr=&id=xn95AgAAQBAJ&oi=

fnd&pg=PP1.

Csikszentmihalyi, M. (1999). Implications of a systems perspective for the study of creativity.

In R. J. Sternberg (Ed.), Handbook of creativity (pp. 313–338). Cambridge, UK: Cambridge

University Press.

De Angelis, V., Kang, K., Mahavier, W. T., & Stenger, A. (2008). Nerd is the word. Math

Horizons, 16(1), 28–29.

Duncker, K., & Lees, L. S. (1945). On problem-solving. Psychological Monographs, 58(5).

Erlwanger, S. H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of

Children’s Mathematical Behavior, 1(2), 7–26.

Franklin, U. (1992). The real world of technology. Concord, ON: House of Anansi Press Ltd.

246 B. Dickman

Getzels, J. W. (1975). Problem-ﬁnding and the inventiveness of solutions. The Journal of Creative

Behavior, 9(1), 12–18.

Getzels, J. W., & Csikszentmihalyi, M. (1976). The creative vision: A longitudinal study of

problem ﬁnding in art. New York, NY: Wiley.

Gruber, H. E., & Barrett, P. H. (1974). Darwin on man: A psychological study of scientiﬁc

creativity. New York, NY: EP Dutton.

Gruber, H. E., & Davis, S. N. (1988). 10 inching our way up Mount Olympus: The

evolving-systems approach to creative thinking. The Nature of Creativity: Contemporary

Psychological Perspectives, 243.

Gruber, H. E., & Wallace, D. B. (1999). The case study method and evolving systems approach for

understanding unique creative people at work. Handbook of Creativity, 93, 115.

Guilford, J. P. (1950). Creativity. American Psychologist, 5(9), 444–454.

Hansen, R., & Froelich, M. (1994). Deﬁning technology and technological education: A crisis, or

cause for celebration? International Journal of Technology and Design Education, 4(2),

179–207.

Hanson, M. H. (2015a). Worldmaking: Psychology and the ideology of creativity. New York, NY:

Palgrave Macmillan.

Hanson, M. H. (2015b). The ideology of creativity and challenges of participation. Europe’s

Journal of Psychology, 11(3), 369–378.

Hennessey, B. A. (1994). The consensual assessment technique: An examination of the

relationship between ratings of product and process creativity. Creativity Research Journal,

7(2), 193–208.

Hennessey, B. A., & Amabile, T. M. (1999). Consensual assessment. Encyclopedia of Creativity,

1, 347–359.

Karplus, R., & Fuller, R. G. (2002). A love of discovery: Science education—The second career of

Robert Karplus. New York, NY: Springer Science+Business Media.

Kaufman, J. C., & Beghetto, R. A. (2009). Beyond big and little: The four c model of creativity.

Review of General Psychology, 13(1).

Kilpatrick, J. (1987). Formulating the problem: Where do good problems come from? In A.

H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale,

NJ: Lawrence Erlbaum Associates.

Kurtz, B. L. (1976). A study of teaching for proportional reasoning (Unpublished doctoral

dissertation). Berkeley, CA: University of California.

Kurtz, B., & Karplus, R. (1979). Intellectual development beyond elementary school VII:

Teaching for proportional reasoning. School Science and Mathematics, 79(5), 387–398.

Lobato, J., Ellis, A., & Zbiek, R. M. (2010). Developing essential understanding of ratios,

proportions, and proportional reasoning for teaching mathematics: Grades 6–8. Reston, VA:

National Council of Teachers of Mathematics.

Maslow, A. H. (1943). A theory of human motivation. Psychological Review, 50(4), 370.

Mason, J. H. (2003). The value of creativity: The origins and emergence of a modern belief.

Farnham, UK: Ashgate Publishing.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for

school mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching

mathematics. Reston, VA: National Council of Teachers of Mathematics.

Noche, J. R. (2013). Conceptual and procedural knowledge in proportional reasoning of

undergraduate students (Unpublished doctoral dissertation). University of the Philippines

Open University.

Noche, J. R., & Vistro-Yu, C. P. (2015). Teaching proportional reasoning concepts and procedures

using repetition with variation. In Paper Presented at the 7th ICMI—East Asia Regional

Conference on Mathematics Education, Cebu City, Philippines.

Plucker, J. A. (1998). Beware of simple conclusions: The case for content generality of creativity.

Creativity Research Journal, 11(2), 179–182.

Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.

9 Creativity in Question and Answer Digital …247

Rhodes, M. (1961). An analysis of creativity. Phi Delta Kappan,42, 305–310.

Rubin, A. (1999). Technology meets math education: Envisioning a practical future forum on the

future of technology in education. Retrieved June 11, 2017 from http://ﬁles.eric.ed.gov/fulltext/

ED453055.pdf.

Runco, M. A. (Ed.). (1994). Problem ﬁnding, problem solving, and creativity. Westport, CT:

Greenwood Publishing Group.

Runco, M. A. (2010). Divergent thinking, creativity, and ideation. In J. C. Kaufman & R.

J. Sternberg (Eds.), The cambridge handbook of creativity (pp. 413–446). Cambridge, UK:

Cambridge University Press.

Sawyer, R. K. (2011). Structure and improvisation in creative teaching. New York, NY:

Cambridge University Press.

Sawyer, R. K., & DeZutter, S. (2009). Distributed creativity: How collective creations emerge

from collaboration. Psychology of Aesthetics, Creativity, and the Arts, 3(2), 81.

Schoenfeld, A. H. (1985). Mathematical problem solving. New York, NY: Academic Press.

Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its

educational applications. New York, NY: Routledge.

Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1),

19–28.

Stokes, P. D. (2005). Creativity from constraints: The psychology of breakthrough. New York,

NY: Springer.

Stokes, P. D. (2010). Using constraints to develop creativity in the classroom. In R. A. Beghetto &

J. C. Kaugman (Eds.), Nurturing creativity in the classroom (pp. 88–112). Cambridge, UK:

Cambridge University Press.

Tausczik, Y. R., Kittur, A., & Kraut, R. E. (2014, February). Collaborative problem solving: A

study of mathoverﬂow. In Proceedings of the 17th ACM Conference on Computer Supported

Cooperative Work & Social Computing (pp. 355–367). New York, NY: ACM.

Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational

Studies in Mathematics, 16(2), 181–204.

Trefﬁnger, D. J. (1996). Creativity, creative thinking, and critical thinking: In search of

Deﬁnitions. Sarasota, FL: Center for Creative Learning.

Trefﬁnger, D. J., Young, G. C., Selby, E. C., & Shepardson, C. (2002). Assessing creativity: A

guide for education. Sarasota, FL: The National Research Center on the gifted and talented.

Weisberg, R. (1986). Creativity: Genius and other myths. New York, NY: WH Freeman/Times

Books/Henry Holt & Co.

Weisberg, R. W. (2006). Creativity: Understanding innovation in problem solving, science,

invention, and the arts. Hoboken, NJ: Wiley.

Wertheimer, M. (1959). Productive thinking: Enlarged edition. New York, NY: Harper.

248 B. Dickman