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Transitioning a Small Seminar Class into a Lecture with Recitation—Milena Stanislavova (2010)

chalkboard covered in math equationsOverview

To address increased enrollment, a professor supplemented lectures with recitations to strengthen student engagement and performance. Readers of this portfolio will learn ways to

  • Design courses to meet demands of increased class size
  • Align concepts presented in lectures, discussed in recitations, and applied in homework problems
  • Evaluate students’ critical thinking skills

MATH 220 (Applied Differential Equations) was originally structured as a single small seminar for approximately 25 students. Over time, class size grew, and I found that the increased size was straining this structure, with students having less access to one-on-one assistance. Therefore, I decided to change the course structure with the hope that combining both a large and small class setting would improve student understanding and performance.


In Fall 2010, I altered the course format from small seminar to lecture combined with small recitation, which allowed for further changes within the course. Students gained greater access to one-on-one assistance and a greater chance to practice larger concepts, while I could focus on application of skills.

Student Work

With greater emphasis on application, for the three course exams I focused on student ability in relation to word problems as an indicator of whether or not the implemented changes were positively affecting student understanding and performance. While student performance on the first exam generally ranged from those who were able to solve the given word problem (strong performers) to those who were unable to solve the problem (weak performers), by the final exam all students were able to at least attempt the word problem, although a range of abilities still existed.


Generally, students responded positively to the new course structure. By adding a recitation section, students were able to engage in small group problem-solving and discuss lecture topics in greater detail than if they had only a single lecture. In the future, I will revisit both the technology used (finding a better balance in solving examples) and organizational details, such as questions of attendance and group project assignments. Overall, however, I believe the students were very successful in this format and that I was able to better emphasize the course’s important points compared to previous versions of MATH 220.

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professor stanislavova explains a problem on the boardBackground

Applied Differential Equations (MATH 220) (pdf) is an undergraduate-level course that, in fulfilling a University general education requirement, typically draws a large number of non-majors. Since the course is focused on application of material, it particularly draws students from such areas as engineering, architecture, and other sciences.

There are two general course goals:

  • Gain knowledge of various types of differential equations and how to solve them.
  • Understand how to apply this knowledge to “real world” situations.

Previously, my colleagues and I had taught MATH 220 as a single small lecture seminar with approximately 25 students, which worked quite well. However, over time the class size grew to 45 or 50 students, which made it more difficult to provide one-on-one time that would help ensure student learning. This growth was due, partly, to an increasing emphasis on concept application, which brought more students from other departments to the course. At that point I had the idea of changing the course structure: instead of a single seminar, there would be one large-group lecture (meeting twice a week) plus a smaller, teaching assistant-led recitation group meeting once a week.

In Fall 2010, I offered the course using this structure. This would be an experiment not only for my specific class but also for the department as a whole. Would student performance improve under this model? If so, what further changes might an analysis of student work indicate?

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professor stanislavova explains a problem on the boardImplementation

MATH 220’s new configuration meant introducing several structural changes. By adding recitation sections, I also gained a graduate teaching assistant. This gave students greater access to assistance. Additionally, by holding class in a larger lecture hall, I was better able to implement various technological aids. For example, I could show the class a graph while lecturing. Finally, the recitation group provided students with an opportunity to go through more examples than could be done in lecture. I also used the recitation sections as an opportunity to assess student learning and performance, periodically giving quizzes at the section’s start. These quizzes allowed students to encounter the same sort of problems they would find on the exams.

I tried to shift the focus, to some extent, to applications of math skills. The textbook was helpful in this, following each new type of differential equation with a mathematical modeling section. These sections highlight the questions: what can you do with these equations, how do you write your model, how do you set up your unknown functions, and at the end how do you interpret your results? This “real world” application usually is hard for students. By having lectures and recitation sections, I was able to spend a significant amount of time discussing these applications, problems, and points, which could then be further exemplified in the recitations, and reinforced a third time in the homework problems.

The other methods used to assess student performance were exams (two during the term and a final). In these exams, while part of each problem rested on developing solutions, another larger part forced the student to apply the material in question to show a true understanding. My GTA and I evaluated the exams using a traditional grading scheme. Prior to each test, students were given access to a practice exam (pdf) and, following each test, to the grading scheme used to evaluate their work.

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students working in classStudent Work

Traditionally, math exams are structured so that students have straightforward problems to solve. However, in an attempt to place greater emphasis on application, MATH 220 exams included at least one word problem per test. In these problems, students not only had to perform multiple steps to gain the solution, but they also had to apply critical thinking skills to the problem itself.

In the Fall 2010 iteration of the course, I particularly focused on student ability in relation to the word problems as an indicator of whether or not the implemented changes were positively affecting student understanding and performance.

For the first exam (pdf), the one word problem (problem number five) was the most challenging for students. Those students who performed well on the test were able to solve problem five as well as the other problems, perhaps with minor mistakes, and possibly solve the bonus question. Students who gave a weaker performance, generally, were able to solve the first three problems with some minor errors; however, they were unable to solve the word problem and, in some cases, problem four as well.

Exam 1

Example 1 (pdf)

Example 3 (pdf)


Example 2 (pdf)

Example 4 (pdf)

The second exam (pdf) followed the same format as the first. The students had to solve five problems, one of which was a word problem, with the opportunity of earning bonus points.

The final exam consisted of ten problems, with the one word problem (number five) and problem number eight (a fourth order equation) providing the hardest challenge. In the end these two problems generally determined the student’s grade. Most students were able to give at least an idea of how to work problem five, even those who, on the overall exam, were low scorers.

Final Exam:

Example 1 (pdf)

Example 5 (pdf)


Example 2 (pdf)

Example 6 (pdf)


Example 3 (pdf)

Example 7 (pdf)


Example 4 (pdf)

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professor stansislavovaReflections

Although restructuring a course from small sections to a larger lecture might seem an unexpected step, I feel that using a combination of lecture and recitation format worked quite well for this course. Student responses and an improvement in overall grade distribution seem to support this feeling.

Most students openly commented that they liked having a teaching assistant lead the recitation portion of the course. The recitation gave them an opportunity to participate and engage in both solving problems in small groups and discussing in-class examples. The recitation groups also allowed for a slower pace when we worked on theoretical aspects of the course and gave students a different point of view. By having a teaching assistant, students had daily help available if needed between the professor’s and the teaching assistant’s office hours.

These days, one can get a lot of help solving even very complicated differential equations from easily available, free online technology, and we did not hide this from our students. On the contrary, the media rooms that we used were ideal for sharing our experience with using computers and software, and all students were quite proficient in using these tools by the end of the course. I find this one of the greatest unexpected outcomes of the new structure of the course.

That said, I might have to work some more to try to find the right technological balance: how much technology is enough without relying too heavily on computers to solve the examples? The fact still remains that many students struggle with solving the required integrals, and we don't want to free them completely from having to learn this skill. But the benefit of being able to write the model, solve it, and then analyze it to produce real world results is priceless.

I also would need to improve some organizational details, such as making sure that students attend both lectures and recitations, as they might have the tendency to miss class when the room is big and they feel their absence will go unnoticed. The large lecture format precludes me from assigning group projects and, therefore, I would not use this structure for the honors version of the course where a project is an important feature.

The main mission and challenge was to see improvement in how students handled the word problems. I worked hard to identify examples from different fields that the students should have been able to handle and to classify the problems in several basic types. We made sure that everyone was capable of handling these basic steps; then, we worked on modifications and step-by-step were able to assign more difficult examples on the homework sets and final exam. I saw definite improvement in that area compared to previous semesters; everyone was able to at least start the problem correctly, and there were almost no students who completely skipped the word problems. In addition to this, there were more A’s and B’s in the course, and all of these high performing students were able to completely solve the word problems in the midterms and final exam. This is a skill that will be invaluable in their future engineering courses and in their careers. Overall, I find that the students were very successful in this format and that I was better able to emphasize what is important in the course when compared to previous versions of it.

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