Mathematicians' views on undergraduate students' creativity

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Mathematicians’ views on undergraduate students’

creativity

Gulden Karakok, Milos Savic, Gail Tang, Houssein El Turkey

To cite this version:

Gulden Karakok, Milos Savic, Gail Tang, Houssein El Turkey. Mathematicians’ views on undergraduate students’ creativity. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.1003-1009. �hal-01287301�

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undergraduate students’ creativity

Gulden Karakok¹, Milos Savic², Gail Tang³ and Houssein El Turkey⁴

1 University of Northern Colorado, Greeley, USA, gulden.karakok@unco.edu 2 University of Oklahoma, Norman, USA

3 University of La Verne, La Verne, USA 4 University of New Haven, New Haven, USA

There are studies investigating mathematical creativity in the primary and secondary levels, but there is still a need to explore creativity in the tertiary level. Our effort of expanding research to this level started with investigating mathematicians’ views on creativity and its role in teaching and student learning of mathemat- ics. One 60-minute interview was conducted with six mathematicians who teach courses at the tertiary lev- el and are active in research. Two themes, Originality and Aesthetics, were observed capturing participants’

views of creativity in their work, aligning with existing process and product views. In addition, all participants believed creativity could be encouraged in undergradu- ate courses and provided suggestions on how to cultivate and value creativity in courses focusing on proving and problem solving.

Keywords: Creativity rubric, proof process, undergraduate mathematics education.

INTRODUCTION

Creativity is one of the important aspects of professional mathematicians’ work. It has been documented that many mathematicians describe creativity in their work as an enlightenment that is somewhat unexpected (Hadamard, 1945; Poincare, 1958). Furthermore, creativity helps in the development of mathematics as a whole (Sriraman, 2009). However, creativity is an intricate research construct to explore, made appar- ent by the myriad of definitions (over 100 as reported by Mann, 2005). In fact, Borwein, Liljedahl, and Zhai (2014) demonstrated that many brilliant mathematicians had differing views about mathematical creativity.

Some conceptualizations of creativity focus on em- phasizing whether the end-product is original and useful (Runco & Jaeger, 2012), while others describe it as a process that involves different modes of thinking, some of an unusual nature (Balka, 1974). Liljedahl and Sriraman (2006) suggested a definition in which creativity was viewed as a personal trait of a mathematician; “the ability to produce original work that significantly extends the body of knowledge (which could include significant syntheses and extensions of known ideas)” (p. 18). In addition, Sriraman (2005) argued that creativity in K-12 classrooms is different than the kind employed by mathematicians and that students’ creativity needs to be evaluated according to their prior experiences. This particular point highlights the difference between absolute and rel- ative creativity; the former one refers to historical inventions or discoveries at a global level and the latter one is defined as, “the discoveries by a specific person within a specific reference group, to human imagination that creates something new” (Vygotsky, 1982, 1984; as cited by Leikin, 2009, p. 131). Using a relativistic perspective, Sriraman and Liljedahl (2006) define mathematical creativity at the school level as a process of offering new solutions or insights that are unexpected for the student, with respect to his/her mathematics background or the problems s/he has seen before. This particular definition acknowledges that students “have moments of creativity that may, or may not, result in the creation of a product that may, or may not, be either useful or novel” (Liljedahl, 2013, p. 256).

Despite the acknowledgment of differences in creativity between professional mathematicians and K-12 students and studies in K-12 level, the mathematical creativity research in undergraduate mathematics

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Mathematicians’ views on undergraduate students’ creativity (Gulden Karakok, Milos Savic, Gail Tang and Houssein El Turkey)

1004 education has been sparse. The purpose of our re-

search project is to explore creativity in undergraduate level teaching and learning by first focusing on mathematicians’ views on creativity. Researchers have investigated mathematicians’ views on creativity (e.g., Hadamard, 1945; Sriraman, 2005), but we contribute to the existing literature by focusing on mathematicians’ views on the role of creativity in teaching and students’ learning, especially in the case of proving and problem solving. More precisely, our research addresses the following research questions:

(1) How do mathematicians define creativity? (2) How do mathematicians view creativity in undergraduate mathematics courses, especially the ones that focus on proving and/or problem solving? (3) Can we (or should we) value and/or assess undergraduate students’ creativity in proving (and/or problem solving)?

Prior to sharing findings of our analysis, we brief- ly summarize some studies that explored creativity from mathematicians’ views and in K-12 levels.

LITERATURE REVIEW

Exploring mathematical creativity is an ongoing quest of researchers, with the earliest known attempt by two psychologists Claparède and Flournoy in 1902 (as cited in Borwein et al., 2014, and Sriraman, 2009).

Following this survey-focused attempt, which elic- ited voluntary responses from mathematicians of that time, Hadamard (1945) resumed the exploration of mathematical creativity with surveys of his own sent to prominent mathematicians across the world.

Using a psychological framework created by Wallas (1926), Hadamard theorized four stages in the process of creativity: preparation (thoroughly understanding the problem), incubation (when the mind goes about solving a problem subconsciously and automatically), illumination (internally generating an idea after the incubation process), and verification (determining whether that idea is correct). Sriraman (2005) found that Hadamard’s four stages are still applicable to modern day mathematicians by interviewing five research mathematicians. Furthermore, his study provided more detail of the stages by considering the roles of personal and social attributes such as imagery, intuition, and interaction with others. Guilford (1950), however, found Hadamard’s stages “superficial from the psychological point of view” (p. 451). His concerns were that these stages were not informing us on the mental processes that occur and the stages were not

testable. He suggested some testable factors such as fluency, flexibility, production of novel ideas, synthesizing and analysing ability, and evaluation ability.

His list was refined to fluency, flexibility, originality, and elaboration, which were expanded and used in forthcoming creativity research by others (e.g., Balka, 1974; Leikin, 2009; Torrance, 1966; Silver, 1997).

Fluency in general refers to the quantity of outputs.

Silver (1997) defined it in the problem-solving setting as the “number of ideas generated in response to a prompt” (p. 76). Flexibility is “shifts of approaches tak- en when generating responses to a prompt” (Silver, 1997, p. 76). This could mean that a student approached a certain task, was not successful with finding a solution or did not feel the approach was going to be fruitful, and changes to a new approach. Originality (or novelty) is described as a unique production or an unusual thinking (Torrance, 1966). Elaboration refers to the ability of producing detailed plan and generaliz- ing ideas (Torrance, ibid). These factors of creativity have been used in K-12 levels to determine students’

creativity. For example, Leikin (2009) focused on fluency, flexibility and originality to create a creativity rubric (using a point system) that evaluated how creative a student was when s/he produced solutions of certain tasks. Similarly, Yuan and Sriraman (2011) integrated the same three aspects to measure participants’ creativity in mathematical problem posing.

Recently, Chamberlin and Mann (2014) proposed a fifth aspect of creativity, iconoclasm, which “entails the penchant of mathematically creative individuals to dissent from commonly accepted principles and solutions” (p. 35). They suggest the possibility of ob- serving iconoclastic behaviour in individuals who are considered to have a high degree of creativity.

Even though exploring students’ creativity in K-12 level is a common practice, such efforts are sparsely expanded to undergraduate mathematics level. There have been on-going efforts of implementing new ped- agogical strategies (such as inquiry-based learning or problem-based learning) to improve undergraduate students’ skills that are related to creativity (such as investigating ideas, providing multiple solutions, analysing others’ strategies). However, we know little about how to explicitly value or assess undergraduate students’ creativity in courses involving proving, or in more traditional teaching settings. To expand our understanding of how creativity can be cultivated while learning mathematics, we conducted a quali-

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tative research study investigating mathematicians’

views of creativity and their perspectives on its role in teaching and learning of mathematics.

METHODOLOGY Participants

Participants of this study were six mathematicians who are active researchers in their mathematical are- as and teach undergraduate and graduate level mathematics courses. Two of the participants are from a mid-size Ph.D. granting, but predominantly teaching mid-western university, and four participants are from a large Ph.D. granting and research dominant mid-western university. Two are tenured associate professors and four are tenured full professors, and one of whom is female. Participants had 8 to 30 years of teaching and research experience. Research fields vary from algebraic geometry, nonstandard analysis, geometry and topology, representation theory, signal analysis, and number theory.

Data collection

We conducted one 1-hour interview with each participant in his/her office. Interviews were audio- and vid- eo-taped and transcribed. The semi-structured interview had three parts. Participants were asked to talk about their views of creativity in their mathematics work in part one. In the second part, participants were asked to comment on a given a set of creativity definitions from different theoretical perspectives. The third part of the interview focused on participants’

views on teaching and learning. To elicit their initial thoughts, participants were first asked to talk about their perspectives of teaching creativity and its po- tential role in students’ learning. After, we gave three proofs constructed by three students (Birky, Campbell, Raman, Sandefur, & Somers, 2011), one at a time, and asked them to comment on creativeness of these proofs (See Appendix). All participants were then given a Creative Thinking Value Rubric (Rhodes, 2010) to evaluate the same proofs. This rubric was chosen by the authors because it claims to assess undergraduate students’ creative thinking across disciplines.

Analysis

We employed grounded-theory methodology (Strauss

& Corbin, 1998). Researchers independently read the transcripts several times to select passages “that ex- press a distinct idea related to [our] research ideas”

(Auerbach & Sliverstein, 2003, p. 46). After identify-

ing these relevant texts, we searched for repeating ideas that could be combined into themes. The emergent themes describing mathematicians’ views of creativity were Originality and Aesthetics. These two themes match with the process view and the end-product view in the creativity literature, respectively. Quotes that highlight the characteristic of each theme are shared with brief discussions in the next section. We also present our analyses of the mathematicians’ views on teaching and student learning in relation to creativity.

More precisely, we describe our participants’ views on how to cultivate students’ creativity in proof and problem solving courses, and how to identify, evaluate and value student creativity in such courses.

RESULTS

Views on creativity

Though participants were asked for their definitions during the first part of the interview, they stated different aspects of creativity as they responded to questions in other parts. For example, some participants would refer back to their definitions of creativity as they explored given students’ proofs or as they discussed given definitions. We also noticed that some participants would contradict their definitions when discussing the students’ proofs, so we asked, “How does this particular idea you mentioned align with your previously mentioned view of creativity?” For these reasons, we analysed each participant’s entire interview to uncover his/her views of creativity, thereby generating a more holistic view of his/her perspective.

Overall, we observed two main themes in our participants’ views of creativity: Originality and Aesthetics.

Since the first theme shared some similar aspect with the originality (novelty) described in the previous literature, we used the same word. All of our participants mentioned creating a “new way”, “new approach/strategy” or “new trick” when they described creativity in mathematics, generally or in their own work.

Dr. B So for me the creative aspect is you intro- duce a new way to look at the problem.

Dr. C [While talking about his creative moment] So, it wasn’t that creative in the sense that there was already stuff out there that I didn’t have to think about it

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1006 myself, but in the process of applying it

I think I created something new.

Dr. E I think the definition of creativity is ap- proaching a problem from a different perspective or with different tools.

Within the theme of Originality, we noticed some sub-themes that the participants referred to as they described the process of creativity: (i) Making Connections, (ii) Attempts, and (iii) Insight.

All of our participants highlighted the importance of making some sort of connection. For example, Dr.

A mentioned seeing connections between the task at hand and other theorems, whereas Dr. D mentioned making connections between various topics of mathematics to approach a task from a new way. Similarly, Dr. F stated,

Dr. F For example, notic[ing] that some equa- tions result in geometry, with some geometry connects to some algebra, doing something unusual.

Participants also emphasized that in the process of creating something original, having several attempts, even incorrect ones played an important role.

Dr. A If it did happen [an incorrect attempt], then it is creative in some sense because exploring a wrong answer helps.

Dr. D Well, let’s see the most recent example I can think of is when my creativity took us in a wrong or took us in a negative direction.

Further in the interview when Dr. D was asked about the role of creativity it was observed that s/he again mentioned the importance of making attempts by say- ing, “You have to be willing to try something.”

We observed that some of our participants mentioned the role of intuition or insight in the process of creativity. This particular sub-theme, Insight, was not shared by all of our participants. The quotes below demonstrate ideas from these participants.

Dr. A But having the idea [of a proof in his research] was the spark…It’s that initial moment that is the creative part, not the actual carrying out the thing.

Dr. D I think […] there are people that have more and less or different types of creativity. There are mathematicians who are very intuitive, […] who can see a broad result.

We also noticed that some of our participants stated the importance in making conjectures. For example, Dr. A So, I was writing a paper with a colleague just last week. So I was working on that and there was an implication that the colleague had proved, A⇒^B.

And it occurred to me…is the converse true? Does B also imply A?

Even though this idea was not repeated by other participants, this quote from Dr. A provides a valuable insight for encouraging students’ to make conjectures in courses.

The first theme of Originality and its sub-themes thus far speak to the process of proving rather than the end-product proof. The latter view is encompassed in the second theme: Aesthetics. Almost all of our participants mentioned some aspect of creativity that relates to the look of the final proof. In the following quotes, we underlined some code words that helped us create this particular theme.

Dr. B [when talking about a “creative proof”]

But there is a notion…there is a notion of economy, of something surprising that you would not expect in that proof, and something lovely.

Dr. C There’s kind of an element of aesthetics involved, or beauty, and so when I think of creativity in mathematics I think of people that are able to pull that out of themselves and come up with nice problems to solve that are attractive, some- how. Or follow a line of thought that is attractive.

Dr. E [when evaluating a student proof] Wow!

Yeah, that does seem more creative, that is cute. That is really cute!

Dr. F [when discussing creativity in teaching]

When I’m teaching classes for example, sometimes I find a cute way of doing the proof, or I find an elegant way of doing computation.

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Role of creativity in teaching and learning To understand our participants’ perspectives on the role of creativity in teaching and learning undergraduate mathematics, we focused on the responses that were given during the third part of the interview. In this part, participants were first asked to talk about their ideas on how to cultivate students’ creativity when teaching and then were given three student-constructed proofs to assess the creativity. The analysis of summaries from each participant yielded similar ideas.

All of our participants believed that creativity could be encouraged in undergraduate courses. They provided similar teaching ideas, which could foster creativity. For example, providing problems and let- ting students “play” with them and discuss different solution techniques would be one way to cultivate creativity. Another example was to show students different proofs of the same theorem and discussing the ones that are more creative and why they would be considered more creative.

Participants also discussed possible ways of evaluating creativity in proofs. They were given three student-constructed proofs (See Appendix) to the theorem “If n is an integer such that n ≥ 3, then n³ ≥ (n+1)².” All of our participants determined that the first student’s proof was not creative due to the fact that the induction proof technique was an expected method to implement in this particular question. They thought the second and third proofs were more creative, and some participants discussed the importance of using prior knowledge and making connections between the tasks and the student’s existing knowledge as they talked about these two proofs.

Dr. E [referring to the second proof] It is possible that they had a course where this kind of trick was really looked at.

Furthermore, participants acknowledged the mistake in the third proof, which started a conversation about the role of correctness in creativity. All participants thought incorrect attempts could play an important role during the process of creativity. However, some of our participants thought that such incorrect ideas should be fixed in the final product.

Dr. F I will risk it and say that [a proof] doesn’t have to be correct to be creative. But at

least it should be fixable. It can happen that you have an original idea and you mess up details, which is not surprising because if it is an original idea then it means that you haven’t practiced that, you would make mistakes.

Some participants stated that they do give “higher”

points to students’ proofs or solutions to problems if they thought that the approach was original or unexpected. Other participants were hesitant to give extra points to “creative” proofs or solutions but said they would provide written encouraging comments to students’ work.

When participants viewed and tried to implement the Creative Thinking Value Rubric (Rhodes, 2010) to evaluate three proofs, they designated applicable and inapplicable categories to mathematics. For example, all participants agreed that Taking Risks, Innovative Thinking, and Connecting, Synthesizing, Transforming categories would be applicable. They believed that the Solving Problems category by itself was what they expect their students to do in mathematics so it would not be applicable. Some participants thought that there were too many levels provided in this rubric (Capstone, Milestones, and Benchmarks).

DISCUSSION AND CONCLUSION

The purpose of this particular study was to explore participating mathematicians’ views of creativity and its role in teaching and student learning at tertiary level. The first question relating to participating mathematicians’ definitions of creativity is described by two observed themes: Originality and Aesthetics. All six mathematicians’ views of creativity highlighted the notion of Originality. That is, we noticed that our participants discussed the process of creating ideas and mentioned the importance of making connections, trying or attempting different solutions, and having an insight or “spark,” which Wallas (1926) called the

“illumination” stage of creativity. In the process of creating ideas, connections between different mathematical knowledge require the individual to understand and absorb many previous definitions and theorems, which Wallas (1926) called the “preparation” stage of creativity.

With our second research question, we investigated the actions and thoughts of mathematicians with

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1008 regard to the teaching and learning of mathematical

creativity. We found an emphasis on the process of proving when participants were asked to evaluate students’ proofs based on creativity. They thought in order to evaluate a final product or final proof, they either judged them against one another, or, similar to Sriraman and Liljedahl (2006), they needed to know the student’s prior knowledge and thought process.

Some mathematicians thought that the proving process might reveal mathematical creativity aspects of a student or a mathematician not revealed in the final proof. For example, the student that created proof 1 may have tried proofs that resemble proofs 2 or 3 and realized that they may not be as fruitful as the technique for proof 1. Also, having courage to take a risk and create an attempt or multiple attempts was determined to be an aspect of mathematical creativity. Therefore, valuing multiple attempts might encourage students to be creative in their proving process (Leikin, in press). But, as Dr. E stated, there is some caution in how to value those attempts; a student might make multiple attempts that have “zero chance of working.”

Finally, we attempted to answer the third research question by utilizing the interviews and previous creativity rubrics (Leikin, 2009; Rhodes, 2010) to create the Creativity-in-Progress Rubric (CPR) on proving (see Savic et al., 2015, and Tang et al., 2015, for more details.) This rubric is a formative assessment to explicitly value undergraduate students’ creativity during the proving process in proof-based courses at the tertiary level. Given the heavy emphasis the mathematicians placed on the process of construction, the CPR focuses on assessing the process of proving (Originality) rather than the final product of the proof itself (Aesthetics). As a formative assessment tool, the rubric has three major categories Making Connections, Taking Risks and Creating Ideas. These categories align with our participants’ views of creativity and its role in undergraduate mathematics. In addition, the use of the rubric as a formative tool has merged from participants’ suggestions to encourage creativity in the classroom. The greatest use of the CPR on proving, we believe, is that it can start the discussion of creativity and the proving process in the classroom. Mann (2005) states that avoiding the acknowledgment of creativity could “drive the creatively talented underground or, worse yet, cause them to give up the study of mathematics altogether” (p. 239). Since there is an increased need for students to have research-like experienc-

es (e.g., Research Experiences for Undergraduates [REUs] (Garcia & Wyels, 2014)), valuing mathematical creativity may bridge the gap between undergraduate mathematics and research mathematics. In particular, using formative assessment tools, such as the CPR on proving would help in such efforts.

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APPENDIX - THREE STUDENTS’ PROOFS FROM (BIRKY ET AL., 2011)

Proof 1: If n=3, then n³=27 and (n+1)²=16, so n³ >(n+1)²=16.

Now assume that k³ >(k + 1)², for some integer k≥3. On the left-hand side, we add (3k²+3k+1) and get k³ +3k² +3k+1=(k+1)³. On the right-hand side, we add the same

thing to get (k+1)² +(3k²2+3k+1)=(4k²+5k+2). We see that (4k²+5k+2)>(k²+4k+4) because 4k² >k² and, since k > 2, 5k+2>4k+2+2=4k+4. Thus we see that (4k²+5k+2)>(k+2)² and so we have (k+1)³ >(4k²+5k+2)>(k+2)². Therefore, by the principle of mathematical induction, n³ >(n+1)² for all integers n≥3. QED

Proof 2: Assume n≥3. Then n>2, so 1>2/n, and n² >1, so 1>1/n² also. This means that n≥3=1+1+1>1+2/n+1/n². So

n>1+ 2/n+ 1/n² . If we multiply each side of this last in- equality by n², we get n³ >n² +2n+1. Thus n³ >(n+1)². QED.

Proof 3: Assume n≥3. Then, since n-2 is a positive integer, (n+1)² <(n+1)²(n-2). Thus, (n+1)²<n³-3n-2=n³ -(3n+2)<n³. Therefore, n³>(n+1)². QED.