# Applying Design Thinking to Mathematics Research

By Jennifer S. Pearl and Padmanabhan Seshaiyer

In recent years, design thinking (DT) has become a widely-used methodology to generate innovative solutions to so-called “wicked problems” – problems plagued by incomplete information or complex interdependencies, often with a human factor. DT has been used in product design and urban planning, as well as more theoretical or policy-based settings. Upon learning about DT, we were immediately struck by its parallels with the process of mathematical discovery. Although design thinkers come from various backgrounds, it is hard to find many that are trained as research mathematicians. We hope to introduce DT to the SIAM community and offer an example of its successful use in helping students develop interdisciplinary mathematical approaches to eliminate animal poaching.

DT is a human-centered approach to problem solving that employs an iterative process of discovery, ideation, and experimentation to address and innovatively solve real-world challenges that focus on human needs. Companies like IDEO and Stanford University’s d.school have extensively used and popularized the technique. Jeanne Liedtka, Tim Ogilvie, and Rachel Brozenske present a particularly nice formulation of DT [2], denoting the DT steps with four questions:

- What
*is*?
- What
*if*?
- What
*wows*?
- What
*works*?

We see parallels between the DT steps “What *is*?,” “What *if*?,” and “What *works*?” and the steps that many research mathematicians employ in their jobs. “What *wows*?” is not as critical to mathematicians and falls under “What *works*?;” if it works for us, it wows. While some proofs are more elegant or insightful and some algorithms are faster, cheaper, and more optimal than others, we only wish to show what is true.

The first step, “What *is*?,” focuses on a thorough comprehension of particular examples from the problem under study. The emphasis is not on sifting through large volumes of data, but rather understanding key examples very, very well. DT literature provides various techniques—often low-tech and involving direct observation—to help do this. In a famous example, a group of students at Stanford aimed to solve mortality problems of premature and low-birthweight babies in developing countries. These babies seldom had access to incubators, so the thinking at the time revolved around the development of lower-cost incubators. Instead of jumping on this bandwagon, the students visited Nepal to observe births in Kathmandu and the surrounding villages. By taking the time to carefully understand many birth situations, they found that most of the premature and low-birthweight babies were born in rural areas and would never make it to a hospital. Thus, cheaper incubators would not solve the problem. They realized they had to “reframe” the problem, a key step in the “What *is*?” stage of DT.

This reframing mirrors the way a mathematician might attack a problem. Mathematicians first generate many examples to get a sense of the phenomenon under study, and then attempt to identify and categorize those that are significant, determining which aspects are crucial to interpreting the underlying structure. Mathematician Arnold Ross famously said, “Think deeply of simple things.” Understanding a few key examples often yields real insight, but determining which examples are key is not easy. Furthermore, mathematicians are familiar with the crucial moment when they realize that the problem they are solving is actually not the right one to tackle. This notion is parallel to the aforementioned “reframing.”

In the “What *if*?” step, design thinkers utilize tools to generate creative possibilities in a carefully-articulated way. The idea is to “push beyond simplistic expressions of new possibilities” [2] by following brainstorming rubrics that build on the observations in the “What *is*?” stage and do not allow for premature judgment. Mathematicians also formulate theories or hypotheses by looking at examples and attempting to generalize the specific behavior they observe. They draw pictures to explain the ideas to others (or to themselves) and further their thinking processes. Many hypotheses are typically generated, nearly all of which are wrong. But the process helps hone in on the underlying structure they seek to uncover.

In the “What* works*?” stage of DT, practitioners test conceived possibilities by developing rough prototypes and using tools and games to understand assumptions and limitations. Mathematicians often behave similarly, employing strategies to determine which hypotheses might be true or which algorithms will work. Looking for counterexamples is one strategy, while another is proof by contradiction. These two widely-used mathematical approaches are reflected in two tools described in [2]: “Worst Idea” and “Contra-Logic.” After discarding several alternatives or prototypes, the real problem’s essence begins to show. It is often at this stage that we return to the original problem and reframe it again. Determining the “correct” problem statement is generally the crux of the whole issue, and the answer comes easily after this is done. The Stanford students investigating infant mortality with DT techniques reframed their problem to focus on keeping babies warm without reliable electricity. This led them to design Embrace, a cheap, portable, and reusable baby sleeping bag that has been used by over 200,000 babies to date.

Since DT steps closely mirror our thought processes as mathematicians, it seemed natural to formally introduce some DT techniques in a multidisciplinary student research project. In 2015, a group of researchers including faculty, graduate and undergraduate students, and high school teachers and students from the U.S. and Tanzania investigated the poaching of elephant tusks and rhino horns. This collaborative project, led by co-author Padmanabhan Seshaiyer and supported in part by the National Science Foundation, helped participants engage in the discovery phase for “What *is*?,” “What *if*?,” and “What *works*?” The “What *is*?” stage consisted of directly talking to all stakeholders, including students, teachers, faculty, park rangers, and the broader community, about the problem of poaching. Using written and online surveys and pre- and post-assessments, students developed a needs assessment for poaching and helped us (as problem solvers) *empathize*—by observation and review—with the audience for whom we wish to design solutions.

We then identified several problems during the “What *if*?” step, including the purpose, process, logistics, and current approaches for poaching. From these, we *defined* a specific problem that involved developing new engineering, mathematical, and scientific approaches to stop poaching. In the next phase of “What *if*?,” the team brainstormed using a variety of *ideation* approaches to develop an early-warning alarm consisting of an intelligent sensor-based tracking system combined with a mobile and satellite network that game rangers can employ for tracking and monitoring. The “What *works*?” process then allowed us to *prototype and test* unmanned air vehicles, such as *quadcopter drones*, that the tracking system could utilize to understand the process of illegal poaching.

The DT framework also provides an exciting opportunity for an integrated science, technology, engineering, and mathematics (STEM) experience for faculty and students at all levels. Specifically, the poaching project allowed the team to address the following questions: How does one build a drone? What mathematics and physics are involved? What types of engineering concepts and control processes must be employed? What kinds of mathematical calculations should be performed? What type of next generation technology should be incorporated? How do we promote awareness of this project’s integrated STEM education to the next generation of students?

Understanding the quadcopter dynamics and simulation requires the development of mathematical relations to describe inertial and rotational equations of motion for the rigid body dynamics. For example, one can employ Newton’s laws of motion to describe the relationship between the displacements of the quadcopter, given in terms of the various forces \(\textrm{F}\) and the mass \(m\). These equations must be solved with rotational equations of motion that express the rotation about the center of the quadcopter. The rotational components of angular acceleration \(\omega\) are expressed as a function of components of the inertia matrix \(\textrm{I}\) and torque \(\tau\). One may also consider a suitable Proportional Derivative (PD) control, with the component proportional to the error between our desired and observed trajectories and its derivative (see Figure 1).

**Figure 1.** Nonlinear dynamics and control of a quadcopter. Image courtesy of [1].

Additionally, mathematicians can tackle the search problem associated with the drone’s decision-making on whether the target is present in the search region, and if so, where exactly it is located. They can study the associated search problem using a Bayesian framework with the objective of improving the decision as the search pattern continues through the evolution of a belief function. One could consider a binary detection random variable in the analysis, representing whether a specific target in a given cell has been detected. Combining this with a given sequence of observations allows for the computation of individual cell belief probabilities to iteratively identify target location. Figure 2 depicts a summary of a target detection algorithm, along with a closed formula obtained as part of our research for the belief function of a specific case.

**Figure 2.** Probabilistic approaches to target detection algorithm for the quadcopter. Image courtesy of [1].

Comparison of DT processes and mathematical research has limits. Not all of the DT steps carry over to mathematics. Although mathematics can model human behavior in critical ways, the problems mathematicians solve are not always human-centric. And although mathematicians tend to value succinct communication and can often be found scribbling equations on the backs of napkins, we don’t write formal “napkin pitches” like design thinkers do. However, we do think that the ideas of “What *is*?,” “What *if*?,” and “What *works*?,” along with the mantras of empathize, define, ideate, prototype, and test, carry over quite nicely and can be useful both in the research and teaching of mathematics.

*Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.*

**References**

[1] Baez, A., Mclane, K., & Pradyuta, P. (2016, April). *Council on Undergraduate Research: 20th Annual Posters on the Hill*. Washington, D.C.

[2] Liedtka, J., Ogilvie, T., & Brozenske, R. (2014). *The Designing for Growth Field Book (a step-by-step project guide)*. New York, NY: Columbia Business School.

Jennifer S. Pearl is a program director in the Division of Mathematical Sciences at the National Science Foundation (NSF). She learned about design thinking through coursework at the University of Virginia and the Brookings Institution, and tries to use it to develop creative solutions for the NSF. Padmanabhan Seshaiyer is a professor of mathematical sciences at George Mason University and currently serves as a program director in the Division of Mathematical Sciences at the NSF as well as chair of the SIAM Diversity Advisory Committee. He has developed a course titled DICE: Design Thinking, Innovation, Creativity and Entrepreneurship in STEM, which has helped foster student and teacher interest in solving global challenges using STEM tools.