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A Framework to Design Creativity-Fostering Mathematical Tasks

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Title: A Framework to Design Creativity-Fostering Mathematical Tasks
Language: English
Authors: Houssein El Turkey (ORCID 0000-0003-3678-7442), Gulden Karakok, Emily Cilli-Turner, V. Rani Satyam, Miloš Savic, Gail Tang
Source: International Journal of Science and Mathematics Education. 2024 22(8):1761-1782.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 22
Publication Date: 2024
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 1836369
1836371
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
Descriptors: Mathematics Education, Mathematics Instruction, Undergraduate Study, Calculus, Creativity, Discovery Processes, Creative Thinking, Teaching Methods
DOI: 10.1007/s10763-024-10449-3
ISSN: 1571-0068
1573-1774
Abstract: Fostering students' mathematical creativity is important for their understanding and success in mathematics courses as well as their persistence in STEM, but it necessitates intentional instructional actions, such as designing and implementing tasks that have the potential to foster creativity. As teaching innovation requires support for instructors who implement them, we developed a creativity-fostering task design framework that can be used by instructors of undergraduate mathematics courses. In this paper, we share this framework and its research-based development process. The framework includes research-based task features and aligns with the Creativity-in-Progress Reflection (CPR) on problem-solving, a formative assessment instrument. We share two creativity-fostering task samples for Calculus 1 courses as we notice that this course could be enhanced with such tasks. We also discuss ways in which Calculus 1 instructors utilized task features and framework as they designed their own tasks. We observed that the multiple answers and open-ended features of creativity-fostering tasks were frequently incorporated in instructors' tasks; meanwhile, the framework provided opportunities for instructors to be intentional about creating tasks that incorporate mathematical actions such as making connections and taking risks.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1447783
Database: ERIC
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  Value: <anid>AN0180804013;[3d0g]01dec.24;2024Nov13.05:14;v2.2.500</anid> <title id="AN0180804013-1">A Framework to Design Creativity-Fostering Mathematical Tasks </title> <p>Fostering students' mathematical creativity is important for their understanding and success in mathematics courses as well as their persistence in STEM, but it necessitates intentional instructional actions, such as designing and implementing tasks that have the potential to foster creativity. As teaching innovation requires support for instructors who implement them, we developed a creativity-fostering task design framework that can be used by instructors of undergraduate mathematics courses. In this paper, we share this framework and its research-based development process. The framework includes research-based task features and aligns with the Creativity-in-Progress Reflection (CPR) on problem-solving, a formative assessment instrument. We share two creativity-fostering task samples for Calculus 1 courses as we notice that this course could be enhanced with such tasks. We also discuss ways in which Calculus 1 instructors utilized task features and framework as they designed their own tasks. We observed that the multiple answers and open-ended features of creativity-fostering tasks were frequently incorporated in instructors' tasks; meanwhile, the framework provided opportunities for instructors to be intentional about creating tasks that incorporate mathematical actions such as making connections and taking risks.</p> <p>Keywords: Calculus; Creativity tasks; Mathematical creativity; Task design framework</p> <p>Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</p> <hd id="AN0180804013-2">Introduction</hd> <p>Mathematics education research highlights the importance of aligning task design and implementation processes with specific learning outcomes (e.g. Abell et al., [<reflink idref="bib1" id="ref1">1</reflink>]; Biggs & Tang, [<reflink idref="bib10" id="ref2">10</reflink>]). These outcomes include developing knowledge on specific mathematical content and fostering thinking processes such as reasoning, problem-solving, connection making, and communication (e.g. Common Core State Standards Initiative [CCSSI], [<reflink idref="bib15" id="ref3">15</reflink>]; Committee on the Undergraduate Programs in Mathematics [CUPM], [<reflink idref="bib14" id="ref4">14</reflink>]). The design and implementation of purposefully developed tasks not only impacts students' understanding of mathematical topics, but also shapes students' experiences and perspectives of mathematics in general (Henningsen & Stein, [<reflink idref="bib24" id="ref5">24</reflink>]; Pepin, [<reflink idref="bib48" id="ref6">48</reflink>]).</p> <p>Numerous research studies, policy and curriculum-standard documents call for the inclusion of mathematical creativity as a learning outcome in mathematics courses and programs (e.g. Askew, [<reflink idref="bib4" id="ref7">4</reflink>]; Borwein et al., [<reflink idref="bib11" id="ref8">11</reflink>]; CUPM, [<reflink idref="bib14" id="ref9">14</reflink>]; Levenson, [<reflink idref="bib33" id="ref10">33</reflink>]; National Science Board [NSB], [<reflink idref="bib42" id="ref11">42</reflink>]; Partnership for 21st Century Skills, [<reflink idref="bib45" id="ref12">45</reflink>]; Silver, [<reflink idref="bib58" id="ref13">58</reflink>]). For example, the Mathematical Association of America's latest guidelines for majors in mathematical sciences state that "a successful major offers a program of courses to gradually and intentionally lead students from basic to advanced levels of critical and analytical thinking, while encouraging creativity and excitement about mathematics" (Schumacher & Siegel, [<reflink idref="bib55" id="ref14">55</reflink>], p. 9).</p> <p>Despite these almost decade-old calls, explicit curricular work on including mathematical creativity in mathematics courses has been rare. Additionally, while there are various existing frameworks for the design process of tasks to address other student learning outcomes (e.g. Gravemeijer & Doorman, [<reflink idref="bib22" id="ref15">22</reflink>]; Johnson et al., [<reflink idref="bib25" id="ref16">25</reflink>]; Lithner, [<reflink idref="bib37" id="ref17">37</reflink>]; Smith & Stein, [<reflink idref="bib60" id="ref18">60</reflink>]), there is no framework that guides instructors to design tasks that focus students' attention on their creative mathematical thinking in undergraduate mathematics courses. Thus, it is essential to not only include mathematical creativity as an explicit student learning outcome in mathematics courses, but also to provide a framework for instructors wanting to design mathematics tasks that provide opportunities for students to foster creativity.</p> <p>Opportunities to be creative are often lacking in tasks given in undergraduate mathematics courses, especially in Calculus courses. For instance, a textbook analysis (Lithner, [<reflink idref="bib35" id="ref19">35</reflink>]) concluded that 70% of Calculus exercises at the end of textbook sections required only mimicking previously done examples of the same section and hence employing <emph>imitative reasoning</emph> (Lithner, [<reflink idref="bib36" id="ref20">36</reflink>]) or <emph>algorithmic reasoning</emph> (AR) (Lithner, [<reflink idref="bib37" id="ref21">37</reflink>]). On the other hand, nonroutine and unfamiliar tasks prompted high school students to apply <emph>creative mathematically founded reasoning</emph> (CMR) (Lithner, [<reflink idref="bib36" id="ref22">36</reflink>]). Given that most Calculus textbook exercises only require imitative reasoning, Lithner's ([<reflink idref="bib35" id="ref23">35</reflink>], [<reflink idref="bib36" id="ref24">36</reflink>]) results indicate a need for tasks that have the potential to foster mathematical creativity. However, Lithner ([<reflink idref="bib37" id="ref25">37</reflink>]) highlighted that designing such tasks can be challenging. This challenge motivates our exploration of creativity-fostering tasks and the creation of a framework to design such tasks. While the development of this framework was completed in Calculus 1 courses, we intentionally kept the framework broad enough so that it can be applicable to a variety of undergraduate courses to promote mathematical problem-solving and creativity. Indeed, the task design framework incorporates a reflection tool for problem-solving that does not specify content, but only problem-solving processes.</p> <p>We discuss underpinning of theoretical perspectives of mathematical creativity that guided our creation of the Creativity-in-Progress Reflection[<reflink idref="bib1" id="ref26">1</reflink>] (CPR) on problem-solving instrument (Karakok et al., [<reflink idref="bib26" id="ref27">26</reflink>]) and the development of the task design framework in the next section. Then, we provide an overview of the relevant literature on task design research and some examples of existing frameworks. Our research on developing a task design framework to foster students' creativity in problem-solving is aimed at complementing the existing literature on problem-solving and mathematics task design and frameworks by combining research-based aspects in a systematic way to provide guidance to undergraduate mathematics instructors. With our creativity-fostering tasks and task design framework, we aim to support undergraduate instructors' teaching knowledge which has been identified as an area that needs further research (Andrews et al., [<reflink idref="bib2" id="ref28">2</reflink>]). Our research aims at addressing two research questions: What are the features of tasks fostering students' mathematical creativity? In what ways do instructors utilize such features and/or our framework to design tasks to foster creativity in Calculus?</p> <p>Resulting from a larger research project spanning three cohorts of instructor participants to investigate mathematical creativity in Calculus 1 courses, we illustrate features of tasks to foster mathematical creativity with two tasks we developed. We share results of our research to depict the development of the framework (Fig. 3) from an initial form by incorporating analysis of data from instructor participants from Cohorts 1 and 2. In particular, participants' desire to incorporate the CPR in their course through their designed tasks guided our revision of the initial framework. We also share results from Cohort 3 participants who used the revised framework. In our "Discussion and Concluding Remarks" section, we identify task enactment practices (i.e. task implementation) as part of our future extension to the framework.</p> <hd id="AN0180804013-3">Theoretical Background and Relevant Literature</hd> <p></p> <hd id="AN0180804013-4">Theoretical Perspectives of Mathematical Creativity</hd> <p>There are over 100 different definitions of mathematical creativity (Mann, [<reflink idref="bib39" id="ref29">39</reflink>]) and various perspectives are utilized in creativity research efforts (Bicer, [<reflink idref="bib9" id="ref30">9</reflink>]; Leikin & Pitta-Pantazi, [<reflink idref="bib32" id="ref31">32</reflink>]). In our research studies, we operationalize mathematical creativity as a process of offering new solutions or insights that are unexpected for the student with respect to their mathematics background or the problems they could have seen before (Liljedahl & Sriraman, [<reflink idref="bib34" id="ref32">34</reflink>]; Savic et al., [<reflink idref="bib52" id="ref33">52</reflink>]). This conception focuses on mathematical creativity that is relative to students; hence, it has the essence of a relativistic perspective of mathematical creativity versus creativity relative to the field of mathematics (Beghetto & Kaufman, [<reflink idref="bib8" id="ref34">8</reflink>]; Leikin, [<reflink idref="bib31" id="ref35">31</reflink>]). Furthermore, we align this perspective with underpinnings of two other theoretical perspectives of creativity: <emph>problem-solving and expertise-based</emph> and <emph>developmental</emph> (Kozbelt et al., [<reflink idref="bib28" id="ref36">28</reflink>]). The <emph>problem-solving and expertise-based</emph> perspective emphasizes the role of an individual's problem-solving process and highlights key concepts such as problems and heuristics. In this context, problem-solving is a process in which the problem solver tries to attain some outcomes without having immediate access to typically known methods for that problem (Schoenfeld, [<reflink idref="bib54" id="ref37">54</reflink>]). Additionally, the primary assertion of the <emph>developmental</emph> theory is that creativity develops over time and that the environment plays an important role in this development. The main focus of investigation of studies through the lens of development perspective is the person's process and actions as opposed to the product created.</p> <p>The <emph>process</emph> orientation (Pelczer & Rodriguez, [<reflink idref="bib47" id="ref38">47</reflink>]) of mathematical creativity emphasizes and explores actions that students take leading up to solutions. This contrasts with examining the final products (Runco & Jaeger, [<reflink idref="bib50" id="ref39">50</reflink>]) of those processes. The focus on process not only allows researchers to observe students' mathematical creativity beyond the existing stage model of creativity (Schindler & Lilienthal, [<reflink idref="bib53" id="ref40">53</reflink>]) but it also allows instructors to inquire into student thinking (Laursen & Rasmussen, [<reflink idref="bib30" id="ref41">30</reflink>]). Finally, our view of mathematical creativity identifies aspects of creativity specific to the domain of mathematics rather than a domain-general creativity (Baer, [<reflink idref="bib5" id="ref42">5</reflink>]).</p> <p>In summary, our research is set in undergraduate mathematics courses (in particular, Calculus as discussed in this paper), where the goal is to support students in development of their mathematical creativity through problem-solving processes with carefully designed tasks. In other words, our research is embedded in domain-specific creativity with relativistic perspective on the process of problem-solving that develops over time in an intentionally designed environment.</p> <p>We incorporated these perspectives in our Creativity-in-Progress Reflection (CPR) on problem-solving formative assessment tool to provide an opportunity for students to reflect on their problem-solving process to develop their mathematical creativity. Our process-oriented view of mathematical creativity encompasses taking mathematical actions of <emph>making connections</emph> and <emph>taking risks</emph> while working on a problem. These two actions are part of the CPR on problem solving, that we created for undergraduate level mathematics courses (Karakok et al., [<reflink idref="bib26" id="ref43">26</reflink>]). The CPR is intended to help both instructors and students to explicitly discuss mathematical actions aimed at fostering mathematical creativity while guiding students on actions they can improve upon. The <emph>making connections</emph> category of the CPR (Fig. 1) identifies processes involved in connecting a given problem with definitions, formulas, theorems, representations, and examples from the current or prior courses and connecting the attempted problem solutions or approaches to each other. The <emph>taking risks</emph> category of the CPR (Fig. 2) focuses on processes of actively attempting a solution, demonstrating flexibility in using multiple solution paths, using a tool or a trick, posing questions about reasoning within solutions, and evaluating solution attempts or solutions.</p> <p>Graph: Fig. 1 Making connections category of the Creativity-in-Progress Reflection (CPR) on problem-solving</p> <p>Graph: Fig. 2 Taking risks category of the Creativity-in-Progress Reflection (CPR) on problem-solving</p> <p>The CPR is designed to allow the user to formatively assess their processes during problem-solving through three general levels: beginning, developing, and advancing, each of which serves as a marker along the continuum of a student's progress in that subcategory. This continuum among levels of the CPR communicates the possible states of growth, aligning with the theoretical constructs of the developmental perspective.</p> <p>In our earlier version of the CPR for proof-based courses, we observed that it provided students with concrete mathematical actions to focus on when solving problems and proving (Omar et al., [<reflink idref="bib44" id="ref44">44</reflink>]). It also provided language for classroom discourse and made mathematical creativity an explicit part of the course goals, all of which aligns with environmental aspects of the developmental perspective (El Turkey et al., [<reflink idref="bib20" id="ref45">20</reflink>]; Karakok et al., [<reflink idref="bib27" id="ref46">27</reflink>]). Related to this environmental aspect, in our earlier studies, we noticed the need to develop tasks that provide authentic opportunities for these mathematical actions to occur frequently and coherently throughout the course. This observation, coupled with the need to provide instructors support to design tasks (Maaß et al., [<reflink idref="bib38" id="ref47">38</reflink>]) that are aimed to address the twenty-first century skills such as creativity (National Research Council [NRC], [<reflink idref="bib43" id="ref48">43</reflink>]) motivated the development of such tasks and a design framework.</p> <hd id="AN0180804013-5">Literature on Task Design</hd> <p>We adopt Henningsen and Stein's ([<reflink idref="bib24" id="ref49">24</reflink>]) definition of a mathematical task as "a classroom activity, the purpose of which is to focus students' attention on a particular mathematical concept, idea, or skill" (p. 528). Our perspective for task design emphasizes the creation of tasks that aim to focus "students' attention" on both a mathematical concept and mathematical creativity processes that include actions such as making connections and taking risks. We will refer to these tasks as <emph>creativity-fostering tasks</emph>.</p> <p>In a summary of research articles on task design, Thanheiser ([<reflink idref="bib65" id="ref50">65</reflink>]) characterized the goals of tasks into two categories: <emph>domain specificity</emph> and <emph>domain transcendence</emph>. <emph>Domain specificity</emph> refers to tasks within a particular domain content area goal (e.g. tasks for understanding the mathematical concept of function). <emph>Domain-transcendent</emph> tasks, on the other hand, have dual purposes. These tasks not only include purposes of developing specific content but also aim to develop overarching skills (e.g. critical thinking). Our creativity-fostering tasks fit in the domain-transcendent category as they aim to foster mathematical creativity in conjunction with specific mathematical content.</p> <p>In addition to categorizing tasks by their goals, researchers identified two aspects of the task design process: <emph>design as intention</emph> and <emph>design as implementation</emph> (Watson & Ohtani, [<reflink idref="bib67" id="ref51">67</reflink>]). <emph>Design as intention</emph> focuses on the initial formulation of tasks that includes identification of theoretical frames guiding this process. According to Ruthven et al. ([<reflink idref="bib51" id="ref52">51</reflink>]), design as intention focuses on "original design and the clarity and coherence of the intentions it [task] expresses" (p. 329). On the other hand, <emph>design as implementation</emph> attends to design of tasks and their integration in classrooms which is then followed by a refinement of tasks and further implementations.</p> <p>The framework we introduce in this paper was developed based on these theoretical foundations as it prompts practitioners to identify their task's learning objectives, goals, and design intentions that aim to foster students' actions of making connections, taking risks, and evaluating their mathematical work. In the following, we first explore features of tasks aimed at fostering students' creativity and report on existing frameworks from literature to highlight the practicality of our framework.</p> <hd id="AN0180804013-6">Features of Creativity-Fostering Tasks</hd> <p>In our own implementations of the CPR and through a literature review, we observed that certain tasks provided more meaningful classroom discourse on mathematical creativity, which initiated our interest in designing such tasks. For a preliminary answer to our question "What are the features of tasks fostering students' mathematical creativity?" we compiled a list of research-based features of tasks (El Turkey et al., [<reflink idref="bib19" id="ref53">19</reflink>]). In particular, some of these research studies explored problem-solving in secondary education in which the process of solving problems that are new to students provided the means to highlight such features. This list of task features includes tasks involving multiple solutions or answers, multiple representations, different approaches (Leikin, [<reflink idref="bib31" id="ref54">31</reflink>]); posing problems and questions (Beghetto, [<reflink idref="bib7" id="ref55">7</reflink>]; Haylock, [<reflink idref="bib23" id="ref56">23</reflink>]; Lithner, [<reflink idref="bib36" id="ref57">36</reflink>]; Silver, [<reflink idref="bib58" id="ref58">58</reflink>]); promoting evaluation/justification, generalization making, conjecturing (Breen & O'Shea, [<reflink idref="bib12" id="ref59">12</reflink>]); non-routine or CMR (Lithner, [<reflink idref="bib37" id="ref60">37</reflink>]); and open-ended or ill-defined tasks (Becker & Shimada, [<reflink idref="bib6" id="ref61">6</reflink>]; Kwon et al., [<reflink idref="bib29" id="ref62">29</reflink>]; Yeo, [<reflink idref="bib68" id="ref63">68</reflink>]) that lead to incubation and uncertainty (Sriraman, [<reflink idref="bib61" id="ref64">61</reflink>]).</p> <p>Adding to these features, Levenson ([<reflink idref="bib33" id="ref65">33</reflink>]) also discussed tasks that include communication requirements, either using verbal or written methods, which align with Sheffield's ([<reflink idref="bib56" id="ref66">56</reflink>], [<reflink idref="bib57" id="ref67">57</reflink>]) communication stage. Levenson also stated the importance of surface characteristics of tasks that include use of manipulatives, models, illustrations, and every-day context, which seem to be different from previously discussed features. In particular, Levenson highlights, through her task analysis framework, that an appropriate task would not only allow students to develop their mathematics content knowledge and creativity, but also shape students' perspectives of mathematics as a creative field. These research-based features from existing research studies guided our work on task design with the purpose of enhancing students' mathematical creativity.</p> <hd id="AN0180804013-7">Task Design Frameworks</hd> <p>Few examples exist in the literature that supported our development process for creativity-fostering tasks and a design framework. One such framework provides design principles to distinguish between an algorithmic reasoning (AR) task and a creative mathematically founded reasoning (CMR) task (Lithner, [<reflink idref="bib37" id="ref68">37</reflink>]). An AR task is one for which a complete solution method is available (given or recalled) to a particular student from the start. A CMR task is a task in which "(<reflink idref="bib1" id="ref69">1</reflink>) no complete solution method is available from the start to a particular student, and (<reflink idref="bib2" id="ref70">2</reflink>) it is reasonable for students to justify the construction and implementation of a solution" (p. 941). Lithner ([<reflink idref="bib37" id="ref71">37</reflink>]) emphasizes the need for CMR tasks and highlights the challenge of creating them:it is easy to design a task that requires CMR to be solved; one must ensure only that the student does not know the solution method in advance. However, it is more difficult to design such a task in a way that is simultaneously not too difficult for the student to solve. Designing a task that requires CMR, that is not too difficult and whose solution also leads the student to construct a particular target knowledge, is even more challenging. (p. 941)</p> <p>Levenson ([<reflink idref="bib33" id="ref72">33</reflink>]) similarly emphasized the importance of "choosing appropriate tasks" (p. 273) for the promotion of creativity. Levenson designed a task analysis framework with two overlapping components: (<reflink idref="bib1" id="ref73">1</reflink>) analysis of tasks which included a list of task features and (<reflink idref="bib2" id="ref74">2</reflink>) analysis of instructors' reasons for choosing such tasks, which included a list of cognitive demands and affective outcomes. It is worth noting that Levenson's study involved graduate students training to become K-12 teachers and the sample tasks provided were intended for K-12 students. The framework listed the task length, source of task (e.g. textbook, internet), communication requirements, number and types of representation displayed, surface characteristics, number of final answers, and number of solution methods as part of task features. Within cognitive demands, Levenson included types of strategies, requirement for algorithmic or non-algorithmic thinking, encouragement to make connections, requirement to generalize, and requirement of a new way of thinking. Motivation, values, and emotions were part of affective reasons instructors to choose a task.</p> <p>Furthermore, Breen and O'Shea ([<reflink idref="bib12" id="ref75">12</reflink>]) summarized multiple frameworks (Pointon & Sangwin, [<reflink idref="bib49" id="ref76">49</reflink>]; Swan, [<reflink idref="bib63" id="ref77">63</reflink>]; Watson & Mason, [<reflink idref="bib66" id="ref78">66</reflink>]) on designing rich tasks for undergraduate Calculus courses. These frameworks were aimed at helping teachers design tasks to foster mathematical habits of mind (Cuoco et al., [<reflink idref="bib16" id="ref79">16</reflink>]) such as the ability to "conjecture, experiment, visualise, describe, invent and generalise" (Breen & O'Shea, [<reflink idref="bib12" id="ref80">12</reflink>], p. 84). To achieve the goal of fostering these mathematical habits of mind, they suggested within the body of the task use of words such as "exemplifying, specialising, completing, deleting, correcting, comparing, sorting, organising, changing, varying, reversing, altering, generalising, conjecturing, explaining, justifying, verifying, convincing, refuting" (Mason & Johnston-Wilder, [<reflink idref="bib40" id="ref81">40</reflink>], p 109).</p> <p>Overall, these examples guided our development of a design framework for creativity-fostering tasks. However, our framework's alignment with the CPR on problem-solving provides guidance for practitioners to explicitly discuss creativity with students as well as to formatively assess their tasks.</p> <hd id="AN0180804013-8">Methodology: Development of Task Design Framework</hd> <p>Within a larger research project investigating mathematical creativity in Calculus 1, we developed creativity-fostering tasks and the framework to guide instructor participants' teaching of Calculus 1 to foster students' mathematical creativity. Our research projects are designed from a social phenomenological perspective (Abakpa et al., [<reflink idref="bib3" id="ref82">3</reflink>]; Denscombe, [<reflink idref="bib18" id="ref83">18</reflink>]) on creativity. As a methodology, this perspective aligns with the relativistic view and the development perspective to allow us to examine the development of mathematical creativity of students, their perspectives, and experiences as well as instructors' development of teaching actions that foster mathematical creativity. In particular, our chosen methodology of phenomenological perspective allows us to capture instructors' experience (and perspective) with tasks, design of tasks, and implementing such tasks through qualitative inquiry. To capture participants' development processes, experiences, and perspectives, we utilized qualitative data sources such as interviews, reflection tools (e.g. CPR, journal entries) and artifacts (e.g. task designs).</p> <p>The larger research project consisted of three cohorts: Cohort 1—Spring 2019 (2 instructors), Cohort 2—Spring 2020 (6 instructors), and Cohort 3—Fall 2021 (5 instructors). The instructors attended a weekly online professional development with members of the research team to discuss the different components of the project (e.g. task design and implementation, CPR implementation, pedagogical choices, students' affect). The instructors were given the two creativity-fostering task samples (Limit and Circle Tasks) with their features. Additionally, instructors were asked to design at least four additional tasks and implement them along with the Circle and Limit Tasks, while using the CPR in their implementations. They were also asked to reflect in journals and end-of-semester semi-structured video-recorded interviews on their task design and implementation processes.</p> <p>We developed our task design framework in two stages. In the first stage, we provided the task features and an initial design framework (see next section) to Cohorts 1 and 2. In the following section, we provide our data analysis from these two cohorts that led to the revised framework. In the second stage, the revised framework was then given to Cohort 3. An analysis of data from Cohort 3 is then provided. We note that we shared the revised framework in a preliminary research report (El Turkey et al., [<reflink idref="bib21" id="ref84">21</reflink>]), and in this paper, we elaborate on its development process with an in-depth data analysis from the three cohorts.</p> <hd id="AN0180804013-9">Stage 1: Task Samples and Initial Framework</hd> <p>Our research team developed an initial list of task features (El Turkey et al., [<reflink idref="bib19" id="ref85">19</reflink>]) from existing theoretical and empirical studies, without any specific order, structure, or connections to the CPR. This list was the same as described in the "Features of Creativity-Fostering Tasks" section above.</p> <p>The research team utilized this list to create two sample creativity-fostering tasks for use in a Calculus 1 course, the <emph>Circle</emph> and <emph>Limit</emph> Tasks, that exemplify the list of features. The <emph>Circle Task</emph> poses the questions "Is there anything in real-life that is a perfect circle? How do you know if you have a perfect circle?" This task involves the mathematical concepts of infinitesimals, (possibly) limits, approximations, integrals, and arc length. The open-ended nature of this task provides opportunities for students to take risks in exploring novel ideas and to make conceptual connections to fundamental aspects of Calculus, while exploring their tolerance of ambiguity or uncertainty.</p> <p>For the second task, we modified a typical "find the limit" question to "Consider the limit: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mtext>lim</mtext><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mfrac><mrow><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></math> </ephtml> . Evaluate the limit in as many ways as possible" (see Appendix). By modifying the question from a routine exercise on limits to asking for multiple ways that are within the realm of students' knowledge base, we addressed the challenging aspect of designing a CMR task (Lithner, [<reflink idref="bib37" id="ref86">37</reflink>]) without providing a pre-described solution method. We designed this <emph>Limit Task</emph> to address the content objectives of limits and derivatives, intending it to be given multiple times during a course as students learn new content to showcase that some problems can be done using different approaches. For example, this task can be introduced when teaching limits and then reused on an assessment after a lesson on L'Hopital's rule. Additionally, by asking students to evaluate the limit in more than one way, students are pushed to think of a solution not similar to what they have seen in class (i.e. beyond AR). The task carries an open-ended feature because it does not give explicit instructions to students to approach the problem algebraically, graphically, using a table, and so forth. After students learn about the derivative, this task can be revisited to foster making connections between various concepts; for example, students could compute this limit as the slope of the tangent line (derivative) of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi></msqrt></math> </ephtml> at <emph>x</emph> = 1.</p> <p>To guide instructors' creation of tasks, we developed our initial creativity-fostering task design framework with six identifying components: (i) learning outcomes of the task, (ii) the statement of the problem, (iii) task variations, (iv) mathematics concepts, (v) creativity concepts, and (vi) instructional hints for implementation. We provide a detailed written description of the Limit Task using these six components of this initial framework in the Appendix (Fig. 4). This description was provided to Cohort 1 and Cohort 2 instructors as well as an empty template containing the six components.</p> <p>To explore ways in which instructors utilized the list of features, the sample tasks, and the initial framework to design their own tasks, we conducted a basic qualitative research study (Merriam & Tisdell, [<reflink idref="bib41" id="ref87">41</reflink>]) to explore the research question: In what ways do instructors utilize features of creativity-fostering tasks and/or our framework to design tasks to foster creativity in Calculus? We examined data from Cohorts 1 and 2 from three sources: instructors' tasks, their reflections on the design and implementation of these tasks, and their interviews.</p> <hd id="AN0180804013-10">Data Analysis and Results from Cohort 1 and Cohort 2</hd> <p>We first analyzed the tasks designed by Cohort 1 (Jo Parker, Juniper)[<reflink idref="bib2" id="ref88">2</reflink>] and Cohort 2 instructors (Professor Kirke, On the 1s and 2s, Farah Nurhalizah, Ann, Irma, and Arrow)<sups>2</sups> with a deductive approach (Patton, [<reflink idref="bib46" id="ref89">46</reflink>]) using task features as codes. We furthered this analysis by focusing on instructors' reflections and interviews to gain insight on their task design process and conducted deductive analysis on instructors' usage of task features and categories of the CPR. We summarize these results under two categories: task features from the research-based list of features and task design intentions that capture the instructors' purposes and intentions from the task.</p> <hd id="AN0180804013-11">Task Features</hd> <p>We observed that most of the participants incorporated multiple answers/solutions and multiple representations, which allowed their tasks to have an open-ended nature. In addition, tasks designed by these instructors included justification and evaluation features. On the other hand, there were fewer tasks that prompted generalization or conjecturing and posing problems or questions. A summary of task features used by instructor participants is provided in Table 1, where a checkmark denotes if the instructor designed a task with the corresponding feature.</p> <p>Table 1 Features of tasks designed by Cohort 1 and 2 instructors</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" colspan="2" /><th align="left" colspan="8"><p>Instructors</p></th></tr><tr><th align="left" colspan="2" /><th align="left"><p>Jo Parker</p></th><th align="left"><p>Juniper</p></th><th align="left"><p>Prof. Kirke</p></th><th align="left"><p>On the 1 s and 2 s</p></th><th align="left"><p>Farah Nurhalizah</p></th><th align="left"><p>Irma</p></th><th align="left"><p>Ann</p></th><th align="left"><p>Arrow</p></th></tr></thead><tbody><tr><td align="left" /><td align="left"><p>Number of tasks designed</p></td><td align="left"><p>5</p></td><td align="left"><p>5</p></td><td align="left"><p>3</p></td><td align="left"><p>2</p></td><td align="left"><p>3</p></td><td align="left"><p>4</p></td><td align="left"><p>5</p></td><td align="left"><p>3</p></td></tr><tr><td align="left" rowspan="7"><p>Task features</p></td><td align="left"><p>Multiple answers</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Multiple approaches</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left" /></tr><tr><td align="left"><p>Multiple representations</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left" /></tr><tr><td align="left"><p>Pose problems or questions</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left" /><td align="left"><p>✓</p></td><td align="left" /><td align="left" /></tr><tr><td align="left"><p>Open-ended or uncertainty</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Justification or evaluation</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Generalize or conjecture</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr></tbody></table> </ephtml> </p> <p>We noticed that tasks from Cohort 2 included ones that implored students to consider assumptions and constraints or to make assumptions and examine constraints, similar to mathematical modeling problems, which was not on the original list of features. For example, Arrow created a problem, based on a real-world need:I was trying to buy like a nightlight for my son, but I needed the base to fit on the shelf without hanging over the shelf because he's two and he'll grab it if it's over the edge. And, so the first day of class I asked this question... you have the shelf, it's a triangle and you have a lamp that has a base, it's rectangular, [with] these dimensions, will it fit on the shelf without overlapping, without coming off?</p> <p>Even though there are common textbook problems of maximizing area of a rectangle inscribed in a triangle, Arrow's setup and reframing of the problem provided opportunities for students to show creativity through making assumptions and considering constraints.</p> <hd id="AN0180804013-12">Task Design Intentions</hd> <p>To gain insight on instructors' task design intentions, we identified places in their journal reflections where they discussed the usage of the features of tasks and the categories of the CPR. When considering Cohort 1 instructors' task design process, we observed that the alignment of a designed task with the CPR was not on the forefront of their task intentions. For example, Jo Parker referenced the usage of the CPR on two occasions a week apart. In the first appearance, she wrote: "I am assigning an exam reflection assignment today. Two components of the creativity rubric are included." A week later, Jo Parker wrote: "I should have been more thoughtful when including the creativity rubric [CPR] with the exam reflection. <emph>By that I mean I should have chosen what questions they analyze</emph> [emphasis added]. Some students chose very procedural questions so [CPR] didn't really apply." These reflections indicate Jo Parker's intention to align the task she designed with the CPR and her noticing the need to be more purposeful with this alignment.</p> <p>Juniper first mentioned the CPR in her reflections on a task she designed for infinite limits. In her reflection, within the list of "Things to improve for next time," she included the item "Incorporate rubric [CPR] (maybe I'll introduce it in the next class meeting?)." This note indicated that Juniper had designed tasks without specific attention to the use of the CPR, even though as researchers, we envisioned the task design process would be informed by the CPR categories, in which specific actions were intended for students to take while working on the designed tasks. For example, Juniper referenced the CPR when she reflected on the implementation of the Circle Task. She recognized the alignment of the task features and the CPR. She later attempted this in her design of a related-rate problem for a writing assignment.</p> <p>Similar observations on the alignment of task design process with the CPR were made from the interview data. Even though Jo Parker did not explicitly refer to using the CPR during task design, she mentioned it when prompted to talk about the list of features. She recalled "The main ones [features] I remember are multiple representations; I guess it kind of overlaps with the rubric [CPR]; [also] multiple approaches to a problem." When asked if there was any particular feature that was most important to her in task design, she emphasized rigor and the use of formal definitions, but the use of the CPR and the list of the features were not highlighted. Juniper highlighted that "creativity assignments, the way I designed them, were more getting at the why as opposed to the how." In her design intentions, she was thinking of having a hypothetical class conversation "the way that I design the tasks was kind of the way I like to have the actual class discussion in some way." Similarly, our analysis of the interview data from Juniper indicated that when she designed her tasks, she did not regularly think of the CPR and its potential usage.</p> <p>The analysis of reflections data from Cohort 2 instructors revealed similar ideas. Farah Nurhalizah had reflected on their design process highlighting three components of our initial framework: content, features, and implementations where they wrote "I spent a lot of time in the beginning emphasizing the CPR and discussing explicitly how we wanted to bring creativity into the classroom, especially to scaffold the first task." From Irma's and Arrow's reflections, they wrote that they used verbiage from the CPR in classroom discourse when discussing a task, but they did not mention if the CPR was considered when they designed their tasks. In Cohort 2 interview data, we noticed similar intentions to Cohort 1 in the design processes with a focus on the features rather than the alignment with the CPR. Only one instructor, Farah Nurhalizah, mentioned that the CPR "was useful to think about how to make problems."</p> <p>Overall, the analysis of data from Cohorts 1 and 2 indicated that participants mainly focused on the list of features to design their tasks without close attention to assessment methods, particularly formative assessments using the CPR. Specifically, the instructors, when designing tasks, did not consider how the tasks could allow students to incorporate mathematical actions from the CPR, as this would make these tasks a natural fit for using the CPR for formative assessment purposes. This discrepancy guided our revision of the framework that better aligned the task features, including the newly added feature of <emph>make assumptions and consider constraints</emph>, with the CPR.</p> <hd id="AN0180804013-13">Stage 2: Revised Framework</hd> <p>The results from the analysis of data from Cohorts 1 and 2 guided our decision to structure the features of tasks in alignment with the CPR to better assist instructors and students in (a) formative assessments of the mathematical actions students take when working on these tasks and (b) provide instructors with the tools to explicitly discuss mathematical creativity in the classroom. This alignment with the CPR is the foundation of the revised creativity-fostering task design framework (Fig. 3).</p> <p>Graph: Fig. 3 Creativity-fostering task design framework</p> <p>Figure 3 contains the framework in which the left column provides a list of features based on the categories of the CPR on problem-solving and the right column indicates the connections to the categories of the CPR and literature. The first row of the framework allows the instructor to identify the learning objectives and fundamental mathematical concepts of the proposed task, whereas the second row focuses the instructor's attention on the goals of the task that are different from these content-specific objectives and are explicitly related to creativity. These task goals could include goals related to specific mathematical practices (CCSSI, [<reflink idref="bib15" id="ref90">15</reflink>]) and habits of mind (Cuoco et al., [<reflink idref="bib16" id="ref91">16</reflink>]) such as reasoning, evaluation, modeling, and generalization. Inclusion of these two rows in our framework was intentional. We aimed at providing instructors guidance on designing domain-transcendent tasks (Thanheiser, [<reflink idref="bib65" id="ref92">65</reflink>]) that attend to task goals in terms of both mathematical content and mathematical creativity. Together with the next three rows of the framework, instructors are given further guidance to create such domain-transcendent tasks. These three rows align with the CPR to allow the instructor to identify subcategories that they intend to target with the designed task to foster their students' mathematical creativity.</p> <p>When designing tasks with this framework, instructors make fostering creativity through the tasks deliberate and intentional. Furthermore, because the framework aligns with the CPR, the designed tasks have built-in implementation opportunities to explicitly discuss mathematical creativity in class while allowing students to reflect on their problem-solving processes from the subcategories of the CPR. Hence, the tasks designed with this framework focus on developing students' processes (development perspective and process orientation) on questions that have multiple aspects that turn these questions into problems for students (relativistic and problem-solving and expertise-based perspectives).</p> <p>Our research team first examined this framework with our two creativity-fostering sample tasks, Circle and Limit Tasks. The purpose of us examining our tasks with the framework was to assess our own tasks' alignment with the CPR and to make sure that the framework captured previously discussed task features which we used in the design of these tasks and shared with participants. To illustrate this work, we share the Limit Task analysis with the revised framework in the Appendix (Fig. 5).</p> <p>We provided the Limit Task, its analysis using the framework (Appendix Fig. 5), and a fillable task design template to Cohort 3 participants at the beginning of their professional development. This template had space for the statement of the task/problem, a fillable framework (with an empty second column), and prompts for instructors to reflect on their design intentions and implementation of the task such as "How did you go about designing the task? What parts of the framework did you want to capture? How did you incorporate the CPR in your implementation?".</p> <p>We examined the tasks designed by Cohort 3 instructors (Dr. Watson, Carmen Williams, BMil, Claude Louverture, and Bartholomew Jackson)<sups>2</sups> with a deductive approach (Patton, [<reflink idref="bib46" id="ref93">46</reflink>]) using task features as codes. Note that we added <emph>make assumptions and consider constraints</emph> as a new feature in this analysis. We furthered this analysis by focusing on instructors' reflections and interviews to gain insight on their design process. Our goal was to address the same research question, with a revised framework and with a different population. We again summarize these results under two categories: task features and task design intentions.</p> <hd id="AN0180804013-14">Data Analysis and Results from Cohort 3</hd> <p></p> <hd id="AN0180804013-15">Task Features</hd> <p>We observed that all the participants incorporated multiple answers in their tasks, which was similar to the tasks designed by Cohort 1 and 2 instructors. Most of the instructors also designed tasks with multiple approaches and representation features, which was again similar to what we noticed in Cohort 1 and 2 data. Even though more than half of Cohort 3 instructors attended to the feature of justification and evaluation in their tasks, this feature was observed in all Cohort 1 and 2 instructors' tasks. It was also less common for Cohort 3 instructors' tasks to have the generalization and conjecturing feature. We summarize the results of our analysis of incorporated task features in Table 2.</p> <p>Table 2 Features of tasks designed by Cohort 3 instructors</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" colspan="2" /><th align="left" colspan="5"><p>Instructors</p></th></tr><tr><th align="left" colspan="2" /><th align="left"><p>Dr. Watson</p></th><th align="left"><p>BMil</p></th><th align="left"><p>Carmen Williams</p></th><th align="left"><p>Claude Louverture</p></th><th align="left"><p>Bartholomew Jackson</p></th></tr></thead><tbody><tr><td align="left" /><td align="left"><p>Number of tasks designed</p></td><td align="left"><p>4</p></td><td align="left"><p>2</p></td><td align="left"><p>4</p></td><td align="left"><p>4</p></td><td align="left"><p>4</p></td></tr><tr><td align="left" rowspan="7"><p>Task features</p></td><td align="left"><p>Multiple answers</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Multiple approaches</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Multiple representations</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Pose problems or questions</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left" /><td align="left"><p>✓</p></td><td align="left" /></tr><tr><td align="left"><p>Open-ended or uncertainty</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Justification or evaluation</p></td><td align="left" /><td align="left" /><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td><td align="left"><p>✓</p></td></tr><tr><td align="left"><p>Generalize or conjecture</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left" /><td align="left"><p>✓</p></td><td align="left" /></tr><tr><td align="left" /><td align="left"><p>Make assumptions</p></td><td align="left"><p>✓</p></td><td align="left" /><td align="left" /><td align="left"><p>✓</p></td><td align="left" /></tr></tbody></table> </ephtml> </p> <hd id="AN0180804013-16">Task Design Intentions</hd> <p>Cohort 3 instructors used our fillable reflection template to varied degrees on the tasks they designed that were different from the Limit and Circle Tasks. Using the alignment with the CPR as a code, we deductively coded instructors' reflections. In one of his reflections, Dr. Watson highlighted how he designed the task in alignment with the CPR: "For this task, I really wanted to capture the 'posing problems' aspect of the CPR." In her reflection on one of the tasks, our participant BMil noted the alignment between her task design and the CPR "...I want to capture connection-making and risk-taking opportunities, as students often have different ways of explaining concepts to their peers that provide clarity and deeper understanding." She also stated in her reflection that she assigned the CPR as a reflection to students and "their self-assessment using the CPR suggested they felt more creative on this task than previous tasks because of the open-ended wording. It gave some of my students the opportunity to connect calculus to their other courses." In her reflections, the instructor Carmen Williams recognized that she modified a problem from previous semesters so that "they [students] would create multiple examples of functions/limits satisfying the given properties" which relates to one of the subcategories of the making connection category of the CPR and the framework.</p> <p>We analyzed the Cohort 3 interview data deductively using the framework to capture the ways in which they utilized the framework in their design process. During these interviews, when instructors described their tasks and designs, their descriptions of this process referenced various aspects of framework (i.e. language of the framework). All instructors mentioned that the identification of a certain task goal or a specific learning objective was a guiding principle in their design process. For example, Carmen Williams stated "I need them to really understand what it means to have an integral define the area under a curve" and this desire led her to create a task with this objective, thus having a content goal in mind guided her design process. She also identified having a specific task goal, such as motivating a concept, as a guiding principle, "let me ask them that and see if they can come up with something." Claude Louverture identified students' accessibility to the task as a guiding principle. Similarly, Dr. Watson emphasized that learning objectives and reinforcement of concepts were guiding principles in his task design. Other task goals or intentions identified in the analysis of interview data were adding personalization to the task where students need to come up with their own input/criteria, having minimal emphasis on algebraic manipulation, building a toolkit over the course of the semester, having tasks that are fun to grade, and generating classroom discussions.</p> <p>During interviews, participants also referenced features of tasks relating to making connections, taking risks, and evaluating (i.e. the other parts of the framework) that they intended when they designed their tasks. For example, BMil highlighted her emphasis on making connections in her last two tasks: "the last two tasks I assigned I was really wanting them to also make those connections of ideas throughout the semester" and she did not want her students to only focus on the most recent concept. BMil additionally stated her intentions in one of her tasks to foster evaluative opportunities: "they could have talked about the continuity of their function and how they knew it was continuous and evaluated some limits, or explained like how they knew, you know, their function satisfied, some of the conditions." Claude Louverture also referenced these evaluative opportunities when talking about their roller coaster task: "they [students] were supposed to write up a couple pages about why their design met the requirements and why it provided a fun ride."</p> <p>Dr. Watson talked about his modification of a task to allow students to take risks through problem posing: "now you [students] come up with the criteria and hand it to your friend and see if they can come up with the function." Similarly, when describing one of her tasks, Carmen Williams emphasized taking risks through trying different approaches "there was actually a lot of different ways that you could solve this." Bartholomew Jackson also incorporated the framework categories in their design process of "worksheets that allowed them [students] to either run into a pitfall or a roadblock... encouraged them think about it a different way... or show them a simpler way afterwards, so they can compare and contrast which one they like."</p> <p>Overall, the analysis of data from Cohort 3 indicated that participants not only focused on the list of features to design their tasks, but they also used the revised framework to guide their design process. For example, having a specific task goal or a learning objective was an important guiding principle. Additionally, there was more explicit attention given to the alignment between the designed tasks and capturing categories from the CPR or framework.</p> <hd id="AN0180804013-17">Discussion and Concluding Remarks</hd> <p>Given that creativity is highly emphasized in research, policy, and curriculum standards, it is timely for mathematics education research to explore the ways in which creativity can become an explicit part of undergraduate mathematics courses. In this paper, we shared our approach to achieve this goal in Calculus 1 courses through tasks designed to incorporate features of mathematical creativity even though designing such tasks is considered challenging (e.g. Lithner, [<reflink idref="bib37" id="ref94">37</reflink>]).</p> <p>To guide instructors of these courses in their creation of such tasks, we described our work to design creativity-fostering tasks from research-based features and provided two frameworks (initial and revised) to create tasks that intentionally foster students' mathematical creativity. The design frameworks fit under the design-as-intention category of design research. In stage 1, we provided a list of features and an initial framework that designers should consider (e.g. Fig. 4 in Appendix) during their task design process. This initial framework allowed instructors to attend to many features of creativity-fostering tasks (Tables 1 and 2). In particular, the frequent use of the multiple-answer feature suggests this feature could be an entry-point for designing and implementing creativity-fostering tasks for other instructors seeking to foster creativity (El Turkey et al., [<reflink idref="bib19" id="ref95">19</reflink>]). This result mirrors Levenson's ([<reflink idref="bib33" id="ref96">33</reflink>]) findings that when asked to find a task which fosters creativity, many participants advocated for the use of multi-solution tasks.</p> <p>Our revision of this framework (Fig. 3 and Fig. 5 in Appendix) resulted in a closer alignment between the task design process and our conceptualization of mathematical creativity within the relativistic, process-oriented, domain-specific theoretical perspectives of mathematical creativity research. This alignment was embodied through using the mathematical actions listed in our formative assessment instrument (CPR) in the framework. While our proposed task design framework shares some common features from existing literature such as Levenson's framework at the K-12 level, its alignment to the CPR and focus on problem-solving at the undergraduate level puts this framework at a unique position. As the overall goal is to bring mathematical creativity to the forefront of undergraduate mathematics courses, we encourage instructors to adopt either the initial form or the revised version of the framework when thinking about task or problem design, because both versions encourage instructors to be intentional to focus on mathematical creativity. A byproduct of this intentionality is providing opportunities for students to view mathematics as a creative field (Cilli-Turner et al., [<reflink idref="bib13" id="ref97">13</reflink>]).</p> <p>In this work, we only focused on the principles (intentions) of designing creativity-fostering tasks that have evolved into two frameworks. Implementations of tasks are not discussed in this paper. Nevertheless, we understand that the implementation of a task plays an important role in achieving its goals. Stein et al. ([<reflink idref="bib62" id="ref98">62</reflink>]), for example, noticed that the most cognitively demanding tasks (e.g. involving conjecturing, justifying, generalizing) became less-demanding activities for students during implementation for various reasons. One reason was that the problematic aspects of the task "somehow became routinized, either through students' pressing the teacher to reduce task ambiguity and complexity by specifying explicit procedures or steps to perform or by teachers' taking over the challenging aspects of the task" (p. 479). Therefore, it is important to connect intentions and implementations of tasks following Thanheiser's ([<reflink idref="bib65" id="ref99">65</reflink>]) cyclical nature of task design. In our future work, we will explore this connection and cyclical process. For example, following the second framework, we plan to explore how instructors design, implement, and revise creativity-fostering tasks.</p> <p>Regardless of which version of the framework instructors choose to use, students from Cohorts 1, 2, and 3 reported that the implementation of such tasks had positive impact on their sense of creativity and positive affective outcomes such as enjoyment, comfort, and confidence (Tang et al., [<reflink idref="bib64" id="ref100">64</reflink>]). As these outcomes do take place when the designed tasks were implemented by instructors, we plan to further explore in the future the connection between these affective outcomes and features of tasks and their implementation.</p> <p>As we agree with Sierpinska ([<reflink idref="bib59" id="ref101">59</reflink>]) that the "design, analysis and empirical testing of mathematical tasks" is "one of the most important responsibilities of mathematics education" (p. 10), our work supports the development of teaching knowledge of undergraduate mathematics instructors in response to findings from Andrews et al. ([<reflink idref="bib2" id="ref102">2</reflink>]). In particular, our framework's alignment with CPR and inclusion of learning goals that are both content specific and process oriented provide opportunities for instructors to incorporate their knowledge of topics, the structure of mathematics, and practices in mathematics (Delgado-Rebolledo & Zakaryan, [<reflink idref="bib17" id="ref103">17</reflink>]) with an eye towards mathematical creativity. Our creativity-fostering tasks and framework also provide a potential solution to the challenge Lithner ([<reflink idref="bib37" id="ref104">37</reflink>]) put in front of the mathematics education community.</p> <hd id="AN0180804013-18">Acknowledgements</hd> <p>Creativity-in-Progress Reflection on problem-solving was first shared in a conference proceeding article, Karakok et al. ([<reflink idref="bib26" id="ref105">26</reflink>]), for which the authors hold the copyright. In particular, the proceedings state that the manuscripts are "copyrighted by the authors. Permission to reproduce an article or portions from an article must be obtained from the author."</p> <p>In addition, the design framework shared in this manuscript was further developed from the report, El Turkey et al. ([<reflink idref="bib21" id="ref106">21</reflink>]), which we presented at the MCG 12 Conference. Our report was part of the conference proceedings. We, the authors, have the copyright for this work as the proceedings state, "All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, electronic, mechanical, recording, photocopying, or otherwise, without the permission of the copyright holder." Furthermore, we received a confirmation from the editor that the authors are the copyright holders for the online open-access proceedings.</p> <hd id="AN0180804013-19">Funding</hd> <p>This work was supported by the National Science Foundation under Grant # 1836369/1836371.</p> <hd id="AN0180804013-20">Data Availability</hd> <p>Sanitized data is available upon request.</p> <hd id="AN0180804013-21">Appendix</hd> <p>Figures 4 and 5</p> <p>Graph: Fig. 4 Limit Task with the initial framework</p> <p>Graph: Fig. 5 Limit Task with the revised framework</p> <ref id="AN0180804013-22"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Abell ML, Braddy L, Ensley D, Ludwig L, Soto HProject Leadership Team. Instructional practices guide. 2017; Mathematical Association of America</bibtext> </blist> <blist> <bibl id="bib2" idref="ref28" type="bt">2</bibl> <bibtext> Andrews TC, Speer NM, Shultz GV. 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  Data: <searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Study%22">Undergraduate Study</searchLink><br /><searchLink fieldCode="DE" term="%22Calculus%22">Calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Creativity%22">Creativity</searchLink><br /><searchLink fieldCode="DE" term="%22Discovery+Processes%22">Discovery Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Creative+Thinking%22">Creative Thinking</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1007/s10763-024-10449-3
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1571-0068<br />1573-1774
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Fostering students' mathematical creativity is important for their understanding and success in mathematics courses as well as their persistence in STEM, but it necessitates intentional instructional actions, such as designing and implementing tasks that have the potential to foster creativity. As teaching innovation requires support for instructors who implement them, we developed a creativity-fostering task design framework that can be used by instructors of undergraduate mathematics courses. In this paper, we share this framework and its research-based development process. The framework includes research-based task features and aligns with the Creativity-in-Progress Reflection (CPR) on problem-solving, a formative assessment instrument. We share two creativity-fostering task samples for Calculus 1 courses as we notice that this course could be enhanced with such tasks. We also discuss ways in which Calculus 1 instructors utilized task features and framework as they designed their own tasks. We observed that the multiple answers and open-ended features of creativity-fostering tasks were frequently incorporated in instructors' tasks; meanwhile, the framework provided opportunities for instructors to be intentional about creating tasks that incorporate mathematical actions such as making connections and taking risks.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2024
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1447783
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1447783
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s10763-024-10449-3
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 22
        StartPage: 1761
    Subjects:
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Undergraduate Study
        Type: general
      – SubjectFull: Calculus
        Type: general
      – SubjectFull: Creativity
        Type: general
      – SubjectFull: Discovery Processes
        Type: general
      – SubjectFull: Creative Thinking
        Type: general
      – SubjectFull: Teaching Methods
        Type: general
    Titles:
      – TitleFull: A Framework to Design Creativity-Fostering Mathematical Tasks
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Houssein El Turkey
      – PersonEntity:
          Name:
            NameFull: Gulden Karakok
      – PersonEntity:
          Name:
            NameFull: Emily Cilli-Turner
      – PersonEntity:
          Name:
            NameFull: V. Rani Satyam
      – PersonEntity:
          Name:
            NameFull: Miloš Savic
      – PersonEntity:
          Name:
            NameFull: Gail Tang
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          Dates:
            – D: 01
              M: 12
              Type: published
              Y: 2024
          Identifiers:
            – Type: issn-print
              Value: 1571-0068
            – Type: issn-electronic
              Value: 1573-1774
          Numbering:
            – Type: volume
              Value: 22
            – Type: issue
              Value: 8
          Titles:
            – TitleFull: International Journal of Science and Mathematics Education
              Type: main
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