SIAM News Blog

Teaching Students to Explain Their Thought Processes Through Communication Practices

By Jillian Kunze

An important aspect of education is the development of critical thinking skills that help students convince other people that their ideas are sound and their advice is worth taking. To make these arguments, individuals must be able to clearly walk through their process of starting from a problem prompt and arriving at an answer. However, students are often accustomed to simply producing the correct answer in math classes and moving on.

“We’re trying to break them out of that habit,” lieutenant colonel Leann Ferguson of the United States Air Force Academy said. “We really want to see that thought process.” During a minisymposium presentation at the 2024 SIAM Conference on Applied Mathematics Education—which took place at the beginning of this week in Spokane, Wash., concurrently with the ongoing 2024 SIAM Annual Meeting—Ferguson presented communication practices that she gives her Calculus III students to help them build the skill of constructing clear and forthright explanations of mathematical processes.

Figure 1. The difference between the clear, logical, and well-explained solutions that teachers will demonstrate and the murky solutions that students may produce. Figure courtesy of Leann Ferguson.

When demonstrating practice problems on the whiteboard during class, math teachers should provide commentary on how they write solutions that are coherent, logical, well-executed, well-communicated, and correct (see Figure 1). Communication practices encourage students to pursue this “ideal” solution, which addresses the question, builds an argument that the answer is correct, and reflects an understanding of the relevant mathematical concepts. The focus is on description and communication (in addition to accuracy), such that others who read the work are able to understand the steps to the solution.

“I want to work on that focus, so I’m using a holistic grading practice,” Ferguson said. She provides a rubric for the communication practices to help students learn to assess themselves and understand how they will be graded in order to improve their work. Students perform peer evaluations based on this rubric; while the student who is writing the solution knows what they want to say, their peer only know what is communicated on their paper, so their solution must stand on its own. “We want to see, where is this making sense, and where is this not making sense?” Ferguson said.

Figure 2. Bloom’s taxonomy, highlighting the four levels that are relevant to the communication practices: understand, apply, analyze, and evaluate. Figure courtesy of Leann Ferguson.
To help students understand and apply the process and then analyze and evaluate solutions, Ferguson presented a series of communication practices based on four of the levels within Bloom’s taxonomy (see Figure 2): an educational framework for learning goals. As a first foray, Ferguson uses a problem that students have already completed for homework in the online portal WebAssign, so they already know the answer. In class, the students attempt to write out an understandable solution to the problem, then trade papers with their peers and compare with an ideal that Ferguson provides. The students grade each other and write out feedback based on the rubric, and the class then discusses how to improve their explanations. This first attempt usually takes up about half of a class period; Ferguson later grades the solutions herself and provides ample feedback before giving an exam on the material a few weeks later using the same rubric.

The next iteration of the communication practice reuses a problem from a previous year’s exam that the students have not seen before, so that now they must apply what they learned while worrying about the correctness of their solutions as well. After that, students analyze an answer that a participant in a previous class wrote for a problem in which they simply had to set up an equation in response to the prompt scenario. And finally, students evaluate an entire given solution based on the rubric and create their own answer key. Ferguson also encourages students to write out their solutions in this way for homework problems, even when it is not required, as it provides a record of their methods that will be useful for studying.

“Anecdotally, this has been very effective for the Calculus III classes,” Ferguson said. Students acknowledge the value of communication and feel more prepared for their future classes and for giving military briefings — which will be an important duty in their future careers. Feedback indicates that students experience some major lightbulb moments during these exercises, uncovering and rectifying their own misperceptions and procedural missteps and improving their overall performance. Over the course of the semester, students are able to create more coherent and logical solutions that demonstrate their understanding of the mathematical concepts (see Figure 3). “They are so much better at giving these arguments, not matter the context,” Ferguson said.

Figure 3. The progression of a Calculus III student’s work throughout the class. Figure courtesy of Leann Ferguson.

This past year, Ferguson mentored two students on a capstone project in education to investigate quantitative and qualitative aspects of the pedagogical technique. The student researchers made recommendations based on Bloom’s taxonomy and looked at exam scores over time in order to explore which types of practice problems worked best. They found that the students typically performed better when they did not already have the answer to the problem ahead of time, and had to find the solution while also writing out their thought processes. So, it tended to be most efficacious to use exam questions from previous years in these exercises. The analysis is still ongoing to investigate the best methods for teaching the students of the Air Force Academy.

This approach was featured in Calculus III classes for this ninth time this spring; it also appeared in Calculus II classes for the second time and Calculus I classes for the first time. Ferguson is working on ways to help students build these skills in the earlier calculus courses without overwhelming the instructors, who often do not have much instructional experience. Since many of the enrollees in Calculus I and II do not want to major in technical domains, the communication focus of these activities is especially relevant and will help students elucidate their thoughts no matter what path they pursue.

The views expressed in this presentation and article are those of the speaker and do not necessarily reflect the official policy or position of the United States Air Force Academy, the Air Force, the Department of Defense, or the U.S. Government.

  Jillian Kunze is the associate editor of SIAM News


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