# Reflections on Teaching for Mathematical Creativity

By Gail Tang (University of La Verne), Emily Cilli-Turner (University of La Verne), Milos Savic (University of Oklahoma), Houssein El Turkey (University of New Haven), Mohamed Omar (Harvey Mudd College), Gulden Karakok (University of Northern Colorado, Greeley), and Paul Regier (University of Oklahoma)

What surprises you mathematically in your classes? When do you witness students’ creative moments? How often does this happen?

When instructors develop an environment where students are willing to put themselves “out there” and take a risk, interesting moments often happen. Those risks can only build one’s creativity, which is the most sought-after skill in industry according to a 2010 IBM Global Study.

How do we get students to be creative? And how does that balance with the content we are required to cover? Below, past and present members of the Creativity Research Group present reasons on why and how we each teach for creativity. We all have different but synergistic teaching practices we engage in to foster creativity in our students. Gulden focuses on having students making connections, while Milos has students take risks through questioning and sharing wrong answers. Emily focuses on tasks that have multiple solutions/approaches; Gail emphasizes the freedom she gives students in exploring these tasks. Mohamed provides time for students to incubate their ideas. Houssein and Paul reflect on their teaching practices and how teaching for creativity has been integrated into theses practices. There is also the thread of opportunities for student self-reflection woven throughout these stories.

One common aspect is that we try our best to saturate our courses with chances for students to be creative from beginning to end. These stories are our attempts at being creative about fostering creativity. Enjoy!

Gail Tang

I recently watched an Ugly Delicious episode where world-renowned chefs talked about their definitions of pizza and what it meant to them to make pizza. These chefs fell into two types: those who stuck to traditional ways to make pizza and those who departed from these ways. Those who ended up leaving the traditional ways behind did so because they felt stifled to operate under the strict rules of making pizza; they felt their identities were being compromised. With the courage of their convictions, they left to forge their own pizza paths. These chefs rejoiced in their freedom; finally they made pizza in a way that paid tribute to their identities. Chef Christian Puglisi said “If you only look at how it used to be done, or how it’s supposed to be done, you don’t allow yourself to move it forward.” This episode really resonated with me; replace “pizza” with “math” and “make pizza” with “teach math,” – you get the same story of suppressing innovation in the name of tradition. The idea of teaching others in the same way I was taught suffocated me. I was not interested in producing generations of students who could mimic my every mathematical move.

I started with baby steps in Calculus. I found exercises with more than one solution path to the same answer and assigned these without any direction. Students wrote different solutions on the board. Students were not used to seeing each others’ creations let alone creating their own solution paths. The energy in the room was thrilling. Unfortunately, I did not collect this data at the time, but fortunately Houssein El Turkey (see his narrative below) has an example of three students’ work on computing the limit below.

Letting students try problems on their own with little direction has the opportunity to have a profound impact on their mathematical identities. For example, one student started my Calculus 1 as a biology major and ended Calculus 1 as a math major! She wrote in her Calculus 2 weekly reflection:

I think having math ‘done to me’ rather than getting to explore it and have fun with it in high school is the reason I didn’t enjoy math in high school. I love how in your classes we get to try problems our own way and don’t have to use your method. I also think it’s super cool that you encourage using different methods. I would never have considered myself a creative person until I started working with numbers. I’m not anywhere near as creative as I should be, but I feel like math is helping me become more creative. The other night I did a problem with a method that I knew was going to be wrong, but I just wanted to see what happened. It actually helped me understand why that method doesn’t work.

Emily Cilli-Turner

A turning point in my thinking about students’ potential for creativity and how to foster it in the classroom happened while I was teaching a Linear Algebra course using the inquiry-oriented linear algebra materials. One task asked for the solution to a system of three equations in two variables; if there was no solution, find the “best” approximate solution. This task purposefully did not define the word “best” so the students would be forced to think about what qualifications the best approximate solution would have. Every group graphed the linear equations (which bounded a triangular region) and most presented “best” solutions as averages of $x$ and $y$ values of intersection points. However, one student was able to find the exact least squares solution by using optimization of functions of two variables techniques from calculus. Once this student presented his solution to the class, the other students were intrigued and could see the drawbacks of their own solutions. This was very unexpected for me. It drove home the point that students can and will be creative when we give them the tools and the freedom to hone their creativity.

In my mind, teaching for creativity has two main components: task design and collaboration. A large part of teaching for creativity is providing students with tasks that involve multiple approaches/solutions. The above episode would have turned out very differently if I had given the definition of the least squares solution and then several problems finding the least squares solutions of a system. The student would have never had a chance to find his original solution because all of the mystery would have been taken away with a provided definition. Yet, if the students had not been working on the task in groups, bouncing ideas off of each other, and using the group whiteboard to do scratch work, I think this vignette would not have happened. Collective creativity can be greater than that of the individual, and the students’ discussions helped that individual student refine his ideas and come up with the idea of using optimization to solve the problem.

Milos Savic

I believe teaching for creativity addresses a lot of issues in mathematics education. When a student is trying to be creative, there are many side effects, including more saturation with the content, greater mathematical confidence, and the ability to manipulate mathematics in different or new ways. Also, I believe mathematical creativity allows a student to be more of themselves instead of more like me. Finally, solving problems in STEM fields requires creativity, and I strive to create authentic experiences so that students can engage in being creative. These beliefs either are expressed explicitly (“Class, I want you to play with this idea”) or implicitly through tasks, quizzes, tests, and other requirements.

To generate curiosity, I have students ask two questions for every homework assignment; one is intended to be about the concepts, and the other is about their mathematical processes. I also give many routine problems with little twists. For example, I pose problems that require a student to go backwards instead of forwards (e.g. what non-linear function, when integrated from 1 to 2, is 17?). I also ask them to provide their own definitions or theorems using what they know. For example, using what they knew about groups and semigroups, a student created an anti-identity (the additive inverse of the multiplicative identity) and anti-inverses. My actions support these tasks; I celebrate many wrong answers in the class in order to show the process of mathematics and the creative moments within it.

Houssein El Turkey

It is interesting how my teaching has evolved to focus on more than just covering the basic learning outcomes in the classes I teach. I have become aware that teaching mathematics can and should include discussions with students on the novelty and flexibility in problem solving (or proving). Now I seek different approaches from my students to show them that there are often multiple ways to solve a problem. I also point out to my students when we build on something we discussed a while back to show that making connections is crucial. Another action I have taken is to explicitly show how certain processes generalize to a bigger picture.

Seeing an AHA moment from a student is one of the best highlights from a class. For example, when I asked my students to factor $x-1$ many of them were baffled but with hints and guidance, some of them came up with: $x-1 = (\sqrt{x}-1)(\sqrt{x}+1)$. The majority were taken by the simplicity and/or originality of the solution and the look on their faces was priceless. These AHA moments have been occurring more than before and they are constant reminders that they don’t have to be ground-breaking in order to have significant impacts on students. To a first-year college student, these simple tricks generating AHA moments can be crucial to show the originality aspect of doing mathematics even though instructors might find these tricks standard.

I also noticed that teaching to foster creativity has lifted my expectations of my students. I now see more potential in them and I work harder to get the best from them as I challenge them with tasks that require incubation and effort. An example of such a task that I used this semester was finding the limit $\lim_{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$ in three (or more) different ways. Students struggled in their first attempt in class to see it as a slope of a tangent line but I left it for them to think about it at home and we picked it up the next class. I was happily surprised that some students had made that connection. Whether it was due to incubation or someone suggested it to them remains unknown to me.

Assigning challenging tasks was accompanied by emotional support through my continuous encouragement and emphasizing that it is OK to fail or not complete a task and it is OK to struggle because that shapes the learning process.

Mohamed Omar

Many students come to mathematics with an inherent curiosity, but traditional teaching practices tend to suppress this. It is essential that we tap into the curiosity that students have as a way to make mathematics come alive for them. Infusing creativity in the classroom fosters this in a way unlike any other.

To facilitate creativity, I ask open-ended or open questions on assignments. For example, a central theme in one course was counting the number of regions that a set of hyperplanes partition an $n$-dimensional space into, and determining what data about the set of hyperplanes are sufficient to answer that question. The related open-ended question asked how these results change if we used circles or other geometric objects instead of hyperplanes. I gave students plenty of time to play with these open-ended problems, and supported them along the way. The key to unlocking creativity was to create a structure where students were rewarded for their efforts and diversity of approaches, rather than their final output. To facilitate this, I required a three-page reflection using the Creativity-in-Progress Rubric (To read more about the rubric, please see our short article in MAA Focus Feb/Mar 2016). Students had to submit all their scratch work and all partial results, and subsequently use the rubric to reflect on their problem solving process. For instance, if a student used definitions and theorems from the course in conjunction definitions and theorems from outside resources, they could provide direct evidence for this in their work and comment on how the conjunction occurred. This allowed students reflect on their thinking processes, facilitating the pathway to creativity.

Gulden Karakok

Through active learning teaching practices, I plan learning situations that provide opportunities for discovery and making connections. Making connections has been an important component of my teaching, as I believe this process facilitates learning of the new topics and also allows students to transfer learning to other situations. Making connections comes in many forms in my courses — connections between definitions, theorems, various solution approaches, examples, and representations. I often ask my students if they have seen a similar topic, definition or example before. My goal is to discourage compartmentalization of ideas and topics. Unfortunately, our education system seems to train students to see ideas in disparate categories. To address this concern and foster creativity, I have been pushing for the process of making connections. I think asking students to find similarities and differences between ideas from their perspectives and background not only helps students to “own” these ideas but also develop sense of “usefulness” of them. With this ownership, students will be more equipped to be creative.

One example of how making connections promotes creativity comes from my preservice elementary math content course. During a class discussion on definitions of even and odd numbers, one student raised her hand and asked how we can determine “quickly” if a number in base 5 is even or odd (e.g., Is 123 base 5 even or odd?). This particular student was making connections to different bases discussed during the first week. Students worked on this problem and came up with several generalizations. We then discussed connections between those generalizations.

Paul Regier

Far too many students are afraid of math. I believe the antidote to their fears concerning math is experiencing mathematics by their own creativity. I suspect that in removing creative exploration from teaching mathematics, we run the risk of damaging our students. Just as the processing of modern food removes most of its nutritional value, removing creativity from math robs students of the most significant benefits they can gain from studying mathematics. Although a few students may appreciate the sugar rush of an already processed solution presented to them, it does not nourish them. What they gain does not last. It does not stick with them.

I teach for creativity by thinking creatively myself! When I lesson plan, after I have some kind of basic connection to the material and to past experience, I try to incubate for at least a day before I think about how I’m going to structure the class. Then I ask myself, “How little time can I spend presenting an idea, so that students are motivated and ready to start thinking about it themselves? What do I subconsciously withhold from students’ experiences (the joy of discovering and creating mathematics for themselves) that I can give conscious attention to?” In my experience, it is much easier to facilitate this kind of awareness by having students work together in groups with limited (but carefully planned) guidance and encouragement from me. It’s may be easy to give students quick answers, but this often takes away the student’s creative drive. Thus, I am learning to acknowledge students’ own thinking to be able to better provide opportunities for their own creative discovery.

Conclusion

As this blog ends, we hope that our stories serve as the beginning to your adventures in teaching for creativity! The Creativity Research Group has recently been awarded an NSF IUSE Grant (#1836369/1836371), “Reshaping Mathematical Identity by Valuing Creativity in Calculus”. To learn more about how to participate, or to communicate any of your ideas about fostering creativity, email us at creativityresearchgroup@gmail.com.

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