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Creativity in Question and Answer Digital Spaces for Mathematics Education: A Case Study of the Water Triangle for Proportional Reasoning

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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 education, 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.
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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 reect 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
©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,
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 NCTMsMath
traces the history of its Q&A component Ask Dr. Math back to 1994.
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 monthas mea-
sured about a decade after its creation (De Angelis et al., 2008, p. 28). Writing out of
the Technical Education Research Centers (TERC), Rubins(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 identies ve powerful uses of
technology in mathematics education, among which Resource-Rich Mathematical
Communities includes the Math Forum as the best knownresource 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 conrmed
as experts, to answer questions about mathematics and, in some cases, mathematics
education. For example, the reddit community dedicated to socializing around
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 ts 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 ts 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 denitions
that have arisen over the years in investigations of technologyand creativity.
Next, we provide brief remarks around the connections between our specic subject
of study and the broader topics of this text: mathematics and mathematics educa-
tion, technology, and creativity. The third section summarizes our specic subject
a question posted to MESE about a tool used in proportional reasoningand
then investigates the ways in which responding to a reference request is an act of
creative collaboration. The fourth and nal section provides avenues for further
research by proposing three open questions related to the initial key ideas, before
closing with our conclusion.
234 B. Dickman
9.1.1 Key Ideas: Q&A Sites, Denitions, and Creative
We non-exhaustively list here three key ideas for the chapter, which relate,
respectively, to Q&A sites for mathematics education, the importance of dening
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 specic 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 specic 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 denitions for technologyand creativityrequire a certain
amount of specicity in any discussion for which they play prominent roles. We
advocate for an interpretation of technologythat admits both digital technologies
(such as online Q&A forums) and domain-specic tools (such as the water triangle
for proportional reasoning). Moreover, we advocate for an interpretation of cre-
ativitythat coincides with Hansons(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 exemplied 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 satised reference request, and
which will continue far beyond the everyday ideas put forth in this chapter.
9.1.2 Denitions for Technologyand CreativityOver
Denitions for technologyand creativityabound. As contemporary conceptions
of technology undergo rapid change, we begin by looking back to Hansen and
Froelichs(1994) early attempt at articulating the variety of denitions for tech-
nologyin their aptly-titled Dening 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 denition remains obscure(p. 179). The authors continue in exploring
denitions of technologyby looking to dictionaries and considering its etymol-
ogy; by looking to individual scholars from a variety of domains; by considering,
among other conceptions, technologywith regard to products and processes; and
by examining technology as relates to feminism and the evolution of womens roles
in society (Franklin, 1992). Analogously, there are elsewhere discussions about the
emergence of creativityin English dictionaries (e.g., Mason, 2003); debates about
whether creativity is domain-specic (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 womens roles in society
(Bateson, 2001,2004). Beyond these parallels, in Trefnger et al. (2002) the
authors remark that Trefnger (1996) reviewed and presented more than 100
different denitions [of creativity] from the literature(p. 5), and Sawyer (2011)
goes so far as to contend that dening creativity may be one of the most difcult
tasks facing the social sciences(p. 11). Dening either term is certainly no easy
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 creativitywith respect to particular products, rather than providing a
formal catch-all denition 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
Guilfords(1950) trait of sensitivity to problemsas relates to creativity (p. 454).
This work of Guilford appeared in his APA presidential address, and also touched
upon the ability to reorganize, or redene, in the sense of Gestalt psychology;
relatedly, one nds 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 answerswith the process of nding solutions, and look to the vast
literature on mathematical problem solving (e.g., Polya, 1945; Schoenfeld, 1985,
2010) and, similarly, associate questionswith 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
creativeto products (cf. Baer & McKool, 2009). The sheer breadth among con-
ceptions of technology and creativity will make our own study intractable without
rst 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).
Specically, 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
The Stack Exchange network includes over 150 Q&A communities; among these
are MathOverow (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 ve collaborative actsof providing information,
9 Creativity in Question and Answer Digital 237
clarifying the question, critiquing an answer, revising an answer, and extending an
answer identied through a process of open coding (p. 359). Though not explicitly
connected there, the authorsresearch ts 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 specic to mathematics education, and the example traced here is no
exception: The water triangleproportional reasoning task
is a reference request
about the origin of a tool previously depicted on Wikipediasproportional rea-
soning page.
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 askers 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, and The Mathematics Teaching Community (University
of Georgia, Mathematics Educators
Stack Exchange (MESE, is not associated with
an academic institution, and instead ts within the Stack Exchange (SE) network; the
network includes an additional site specically for mathematics questions at the
research level ( and another for general mathematical queries
( Although SE contains over 150 different Q&A com-
munities, MESE is, at present, the only one concerned specically 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
238 B. Dickman
conuence 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 Qand Apart 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
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 specically for
questions about mathematics education; this precluding constraint ensures that
questions that are deemed off-topic by other site users are either rened, 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 modication 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
In this rst 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 reect on their potential use by mathematics educators.
9 Creativity in Question and Answer Digital 239
9.3.1 Reference Request: Water Trianglefor Proportional
The water triangleproportional reasoning taskwas initially posted on the
Mathematics Teaching Community in 2012
where it remained unresolved. The
question was modied by its creator and re-posted to MESE in 2014. The question
essentially asked about the source of the water triangledepicted on Wikipedias
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 rst-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 triangles origins; his response message was as
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
triangleto 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 dont have any. Thanks for your interest. You did a good job tracking me down!
I followed up on this lead by using ProQuest
to nd Kurtzs 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 Fullers(2002)
A Love of Discovery: Science EducationThe Second Career of Robert Karplus.
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.
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.
9 Creativity in Question and Answer Digital 241
9.3.2 Collaborative Creativity: What Do We Mean
by Creative?
The previous sub-section briey 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. Specically, 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 conuence 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 databasesuch 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-specic 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 eld 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 eld (e.g., by instituting precluding constraints such as closing duplicated
questions, or, more generally, having a site specic 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 eld 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 Hansons(2015b)Worldmaking: Psychology and
the Ideology of Creativity there are a variety of conceptions around creativity put
forth and described. Specically, 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 specic 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-CCreative (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 eld (gatekeeper) roles that people take up as
they integrate novelty become more central. Choosing, supporting, interpreting and rening
ideas are as important as havingan idea. Indeed, on close examination, distinctions
between eld roles and the creatorrole begin to disappearCreator 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 peoples 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 Kurtzs 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 selfwhat needs to be done, what can I do, and what must I do (cf. Gruber &
Barrett, 1974)in relation to ones 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 conating 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 eld 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 inuence, 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 conned 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
technologyand creativitydifferent from ours, what conclusions can be drawn
about the impact of evolving Q&A sites for mathematics education? For example,
for those whose denitions 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 exemplied 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 nding the one
person, or one moment, or one idea, to which the commendation of creativitycan
be applied, we focused instead on the distribution across multiple actors, and the
ways in which they fullled 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.
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... Técnicas que puedan generarse desde la exploración y reconocimiento de relaciones y sofisticarse mediante variables didácticas, como en este caso, determinar un ' ' suficientemente grande. Asimismo, sería de gran interés generar un diseño didáctico basado en la invención de problemas, ya que este tipo de tarea está altamente relacionada con la creatividad matemática (Dickman, 2018;Sala et al., 2016). Lo que representa un reto en términos de la componente matemática que se elija. ...
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Se presenta un modelo teórico para el estudio del talento matemático, fundamentado en la Teoría Antropológica de lo Didáctico y la noción de creatividad. En dicho modelo se proponen dos componentes de la actividad matemática creativa: la Componente Matemática, que sustenta las técnicas matemáticas; y la Componente Creativa, definida por cuatro funciones: producir técnicas nuevas, optimizar técnicas, considerar tareas desde diversos ángulos y adaptar una técnica. Con base en los modelos Teórico y Epistemológico de Referencia sobre sucesiones infinitas, se genera un diseño didáctico conformado por seis situaciones problemáticas y se implementa en una institución creada para potenciar el talento matemático. El análisis de dos tareas realizadas por una pareja de niños constituye un estudio de caso, que permite ilustrar que enfrentar tareas retadoras de un mismo tipo, bajo condiciones institucionales propicias, posibilita el desarrollo del talento matemático.
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Teachers try to help their students learn. But why do they make the particular teaching choices they do? What resources do they draw upon? What accounts for the success or failure of their efforts? In How We Think, esteemed scholar and mathematician, Alan H. Schoenfeld, proposes a groundbreaking theory and model for how we think and act in the classroom and beyond. Based on thirty years of research on problem solving and teaching, Schoenfeld provides compelling evidence for a concrete approach that describes how teachers, and individuals more generally, navigate their way through in-the-moment decision-making in well-practiced domains. Applying his theoretical model to detailed representations and analyses of teachers at work as well as of professionals outside education, Schoenfeld argues that understanding and recognizing the goal-oriented patterns of our day to day decisions can help identify what makes effective or ineffective behavior in the classroom and beyond.
Updated and expanded, this second edition satisfies the same philosophical objective as the first -- to show the importance of problem posing. Although interest in mathematical problem solving increased during the past decade, problem posing remained relatively ignored. The Art of Problem Posing draws attention to this equally important act and is the innovator in the field. Special features include: •an exploration ofthe logical relationship between problem posing and problem solving •a special chapter devoted to teaching problem posing as a separate course •sketches, drawings, diagrams, and cartoons that illustrate the schemes proposed a special section on writing in mathematics. © 1990 by Stephen I. Brown and Marion I. Walter. All rights reserved.
Robert Karplus, a professor of physics at the University of California, Berkeley, USA, became a leader in the movement to reform elementary school science in the 1960s. This book selects the enduring aspects of his work and presents them for the scientists and science educators of today. In an era when `science education for ALL students' has become the clarion call, the insights and works of Robert Karplus are as relevant now as they were in the 1960s, '70s, and '80s. This book tries to capture the essence of his life and work and presents selections of his published articles in a helpful context.
The subject of creativity has been neglected by psychologists. The immediate problem has two aspects. (1) How can we discover creative promise in our children and our youth, (2) How can we promote the development of creative personalities. Creative talent cannot be accounted for adequately in terms of I.Q. A new way of thinking about creativity and creative productivity is seen in the factorial conceptions of personality. By application of factor analysis a fruitful exploratory approach can be made. Carefully constructed hypotheses concerning primary abilities will lead to the use of novel types of tests. New factors will be discovered that will provide us with means to select individuals with creative personalities. The properties of primary abilities should be studied to improve educational methods and further their utilization. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
With an increasing emphasis on creativity and innovation in the twenty-first century, teachers need to be creative professionals just as students must learn to be creative. And yet, schools are institutions with many important structures and guidelines that teachers must follow. Effective creative teaching strikes a delicate balance between structure and improvisation. The authors draw on studies of jazz, theater improvisation and dance improvisation to demonstrate that the most creative performers work within similar structures and guidelines. By looking to these creative genres, the book provides practical advice for teachers who wish to become more creative professionals.