Tag Archives: Richard Brown

Beyond the Proof: Bringing Writing into Real Analysis I

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[Guest post by Richard Brown, Teaching Professor, Department of Mathematics, Johns Hopkins University]

Mathematics isn’t really just about solving problems—it’s about understanding and communicating how and why those solutions make sense. The ability to clearly communicate mathematical ideas can deepen students’ conceptual understanding, and also prepare them to write in different formats, for different purposes, and for different audiences, both within and beyond the mathematical community. Despite its importance, mathematical writing is rarely explicitly taught at the undergraduate level. Instead, students are often expected to develop these skills “along the way,” simply through the process of studying and doing mathematics.

With this in mind, I planned a redesign of AS.110.405 Real Analysis I, a core requirement for mathematics majors, as a writing-intensive course. As one of the first person's hand writing in a notebookcourses in which students transition from calculation-based mathematics to more conceptual, proof-based work, Real Analysis I seemed like a natural fit for incorporating writing instruction. In the spring of 2023, I was awarded a Teaching Innovation Grant from the Center for Teaching Excellence and Innovation (CTEI) which helped fund the course redesign.

In alignment with the goals of Johns Hopkins University’s Second Commission on Undergraduate Education (CUE2) and the Krieger School’s emphasis on writing within STEM disciplines, the redesign aimed to expand opportunities for students to develop as mathematical writers without sacrificing essential course content. I partnered with the University Writing Program (UWP) to ensure the course met UWP criteria for qualifying as writing intensive. I also incorporated the principles of Universal Design for Learning (UDL) to reduce barriers to learning and create inclusive opportunities for students to engage with course material. Informed by the Scholarship of Teaching and Learning (SoTL), I kept the focus on student learning, using evidence-based teaching practices to guide course activities, assignments, and assessments.

The existing version of Real Analysis I was taught in standard lecture style format with instructor-led lectures, weekly problem sets, midterms, and a final exam. The revised version was designed to be more engaging and included several active learning strategies:

  • Small group activities to practice definition writing, direct vs. reverse engineered proof-writing, example and counterexample development brainstorming, pop-math and other audience-specific expression.​
  • Opportunities for short presentations of writing samples with peer review.​
  • Writing workshops led by teaching assistants (who were trained by UWP staff) to focus on writing for different audiences. ​
  • Lecture elements, worksheets, assigned readings, and handouts highlighting stylistic elements involved in mathematical presentation, proof-writing, and concept development.​

The midterms and exam were replaced with a mix of small-stakes, no-stakes, and large-stakes writing assignments that support both effective mathematical communication and conceptual understanding:

  • Problem Sets: weekly proof-writing exercises to demonstrate understanding of mathematical concepts as well as refine writing style.
  • Diary: short low and no stakes written assignments to reflect on growth in understanding.
  • Final Project: 3-5 page paper, written in peer-reviewed journal style on a mathematical topic related to the course (i.e. Taylor’s Theorem, Picard Iterations, etc.).  Students worked in pairs to facilitate collaborative writing in a manner common in mathematical research.
  • Presentation: presentation to the class of final project allowing the authors to practice the transition from formal writing to formal presentation of mathematical material.
  • Portfolio: a set of polished artifacts geared toward a variety of audiences (e.g., peers, professor, textbook style, etc.).

Challenges
Although the course went well overall, there were several challenges in implementation:

  • Writing assignments can be particularly time-consuming to grade.
    • The grant funded two additional course assistants and the UWP provided substantial help developing rubrics, but training TAs to evaluate student writing and grading the work itself still required significant time. The variety of assessments further increased the grading workload.
  • There were so many moving parts to the course.
    • Students sometimes felt like they had something due every day, which was challenging to manage alongside the demands of other courses. The workload may have diminished some of the course’s intended benefits, and certain components, such as the Diary, received less attention as the semester progressed.group of student working around a table
  • The variety of assignments and activities sometimes left students unclear about expectations.
  • Implementing this approach more broadly would require substantial training for faculty, TAs, and students who are new to the process.

Results
Midsemester and end-of-course student surveys were excellent, reflecting students’ strong appreciation for the course design and learning experience:

  • 90% agree: The quality of the content of the overall design and organization of the course was good.
  • 90% agree:  Compared to a standard lecture course in mathematics, I found this course to be a much more valuable learning experience.

Student quotes:
“I like this course because you are not just teaching us math.  You are teaching us how to be mathematicians.”

“I love this class, the content, the structure, and everything involved.”

“I appreciated that this course went beyond the usual lectures + exams because it felt like it prioritized an understanding of how to think about math.”

Outcomes
The department has established an ad hoc committee to examine writing within the discipline and identified courses across the curriculum in which to incorporate writing instruction and exercises. Elements of the course were adapted for use in several classes, including critical writing exercises in second-semester Analysis and Algebra, Topology, and Introduction to Proofs. Other examples include a mock journal in Topology, portfolios in Differential Equations and Vector Calculus, and written group project work in Calculus I and II for Biology and Social Sciences majors.

TA instruction has been expanded to include additional training on teaching, assessing, and grading mathematical writing, with continued collaboration with the UWP on rubric development. More formal training has been incorporated into the TA training program, and writing exercises have been implemented more broadly in section meetings. These efforts have helped foster a more active culture of TA-led instruction.

Conclusion
Ultimately, the redesign demonstrated that writing can and should be an integral part of learning mathematics, helping students develop not only their understanding of mathematical concepts but also their ability to communicate as mathematicians. While the approach required additional time, coordination, and support for both instructors and TAs, the positive student feedback and the department’s subsequent adoption of writing activities across the curriculum suggest that the benefits extended beyond a single course. By integrating writing into existing mathematics courses and strengthening TA preparation, the department continues to build a culture in which mathematical writing is taught intentionally rather than left for students to develop on their own.

Richard Brown
Teaching Professor, Mathematics
Johns Hopkins University

RIchard Brown is the Director of Undergraduate Studies and a teaching professor in the Department of Mathematics. He has been a member of the faculty at Johns Hopkins since 2005. His research interests include Hamiltonian dynamical systems, surface mapping class groups, character varieties of surfaces, and symplectic geometry.

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Rethinking Oral Examinations for Undergraduate Students

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Oral examinations, also called viva voce, have long been associated with graduate studies, but many years ago, when I was an undergraduate, oral exams were not unheard of. All undergraduates at my university were required to write a thesis, and many of us took comprehensive written and oral examinations in our fields. I had several courses in my major field, art history, which held oral examinations as the final assessment of our work. At the time, this practice was not uncommon in British and European universities for undergraduates. Since then it has become a rarity both here and abroad, replaced by other forms of assessment for undergraduate students.

Stack of triangular, caution-type road signs with red border and the word TEST in the white center.Recently I learned that Richard Brown, Director of Undergraduate Studies and an associate teaching professor in the JHU Department of Mathematics, had experimented with oral examinations of his undergraduate students in Honors Multivariable Calculus.

Some background: Honors Multivariable Calculus is designed to be a course for students who are very interested in mathematics, but are still learning basics. Students must have the permission of the instructor to enroll. They are likely to be highly motivated learners. In this instance, Brown had only 6 students in the class—five freshmen and one sophomore. For the freshmen, this fall course was their first introduction to a college math course. They came in with varying levels of skill and knowledge, knowing that the course would be challenging. The course format was two 75 minute lectures a week and one hour-long recitation (problem solving) session with a graduate student teaching assistant. This is the part of the course where students work in an interactive environment, applying theory to practice, answering questions, and getting an alternate point of view from the graduate student assistant instructor.

Assessments in the course included two in-class midterms (written and timed), weekly graded homework assignment (usually problems), and the final exam. As Brown thought about the final exam, he realized that he had already seen his students approach to timed and untimed “mathematical writing” in the midterms and homeworks. So, why not try a different environment for the final and do an oral examination? He discussed the concept with the students in class and allowed the students to decide as a class which option they preferred. The students agreed to the oral exam.

Brown made sequential appointments with the students, giving them 20 minutes each for the exam. He asked them different questions to minimize the potential for sharing information, but the questions were of the same category. For example, one student might be asked to discuss the physical or geometric interpretation of Gauss’s Theorem, and another would be given the same question about Stokes’s Theorem. If a student got stuck in answering, Brown would reword the question or provide a small hint. In contrast, on a written exam, if a student gets stuck, they are stuck. You may never identify exactly what they know and don’t know. Another advantage, Brown discovered, was that by seeing how a student answered a question, he could adjust follow up questions to get a deeper understanding of the student’s depth of learning. He could probe to assess understanding or push to see how far the student could go. He found the oral exam gave him a much more comprehensive view of their knowledge than a written one.

In terms of grading, Brown noted that by the end of the semester he knew the students quite well and had a feel for their levels of comprehension, so in many ways the exam was a confirmation. He did not have a written rubric for the exam, as he did for the midterms, but he did take notes to share with the students if they wanted to debrief on their performance. He saw this as a more subjective assessment, balanced by the relatively objective assessment of the homeworks and midterms.

Following up with students after the exam, Brown found that four of the six students really liked the format and found it easier than anticipated. Only two of the students had planned to become majors at the start of the course, but ultimately four declared a mathematics major. Brown noted that he would like to use the oral examination again in the future, but felt that it would not be possible with more than 10 students in a class.

After talking with Brown, I searched to find recent literature on undergraduate oral exams. Two papers are worth reading if the concept is of interest:

Oral vs. Written Evaluation of Students, Ulf Asklund and Lars Bendix, Department of Computer Science, Lund Institute of Technology, Pedagogisk Inspirationskonferens, Lund University Publications, 2003. A conference paper detailing advantages and disadvantage of the two formats. The authors, based on their experience, found that oral examinations are better suited than written for evaluating higher levels of understanding based on Bloom’s Taxonomy.

Oral versus written assessments: A test of student performance and attitudes, Mark Huxham, Fiona Campbell, and Jenny Westwood, Assessment & Evaluation in Higher Education 37(1):125-136, January 2012. This study of two cohorts of students examined “…[s]tudent performance in and attitudes towards oral and written assessments using quantitative and qualitative methods.” Many positive aspects of oral examinations were found. See also a SlideShare Summary of this paper. Possible benefits of oral assessment included: “1) Development of oral communication skills 2) More ‘authentic’ assessment 3) More inclusive 4) Gauging understanding & Encouraging critical thinking 5) Less potential for plagiarism 6) Better at conveying nuances of meaning 7) Easier to spot rote-learning.”

A site to explore is the University of Pittsburgh’s Speaking in the Disciplines initiative. “…committed to the centrality of oral expression in the educational process.” Detailed information for those considering oral examinations is provided, including benefits (“direct, dialogic feedback,” “encourages in-depth preparation,” “demands different skills,” “valuable practice for future professional activity,” and “reduced grading stress”) and potential drawbacks (“time,” “student resistance and inexperience,” “subjective grading,” and “inadequate coverage of material”).

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Macie Hall, Senior Instructional Designer
Center for Educational Resources

Images source: Pixabay.com