The final project of the Certification in College Teaching Program is the mentored teaching experience. It involves asking a teaching and learning question, developing instruments to collect data, collect the data in the context of a course at MSU, and document the findings.
Due to my teaching experience at Michigan State University (a list of my teaching experience can be found here), I decided to implement my teaching project in the context of the Calculus sequence. These courses (Calculus 1 - Calculus 4) are a requirement for many science majors. Students often struggle with the pace of the class and require additional help, either at the Math Learning Center, or in private tutoring sessions. Therefore I decided to offer a workshop for students, that would give them the opportunity for additional practice.
Transitioning from Calculus 1 to Calculus 2, students also have to handle the change from a small class setting in Calculus 1, to a big lecture in Calculus 2. Most sections of Calculus one meet three times a week for 50 minutes in classes of about 30 students, and most sections of Calculus 2 are taught in large lectures of about 200 students, meeting three times a week for 50 minutes and additional weekly recitation for 50 minutes as well. This change away from direct interaction with the instructor can be challenging for students. Hence I chose to design a workshop for students in Calculus 2.
The decision on timing and topic for the workshop proved to be difficult. First I offered a workshop on u-substitution in the beginning of the semester, which had very poor attendance. I believe that this was mostly due to the timing. In the beginning of the semester most students are still on top of their studying and therefore don't want to attend an additional workshop. In conversation with instructors for Calculus 2 (MTH 133) another topic for a workshop became obvious: trigonometric substitution. This is the last topic being taught before the first midterm in Calculus 2. I lead a workshop on trig-substitution on February 16th, 2017 and had over 20 students attending.
Knowing that students often struggle with the complexity of trig substitution (first they have to recognize that they need to apply a trig-substitution, then they have to pick the right function as a substitute, work through the integration, and lastly they often need to draw reference triangles for resubstitution), I designed a workshop to specifically train students in these steps with one worksheet for each step. In particular I wanted to evaluate if active learning techniques have a positive impact on the students understanding of this mathematical ability.
My research project was IRB exempt.
After this workshop students to be able to apply trigonometric substitution to solve integrals. This goal splits into multiple objectives:
What is the effect of active learning techniques to achieve the teaching and learning goal?
The section of Calculus 2 that I chose for my workshop was taught in a big lecture setting with about 200 students. The class met three times a week for 50 minutes. Additionally the students met once a week for 50 minutes with a teaching assistant in a smaller setting of about 30 participants. During these recitations they took quizzes. Additionally the students had to finish weekly online homework on WebWork. They had two midterms and a final exam.
I announced the workshop myself during the lecture and I made sure that students were aware that this was just an additional opportunity to get practice on trig-substitution, but it would not contribute to their grade. Students learned trig substitution in MTH133 as their last topic before their first midterm, so that many students were excited to get that additional training.
The workshop was divided in the following parts, following the teaching and learning goals:
After the midterm (about a week after the workshop) I contacted participants to schedule interviews. I conducted three interviews, that I recorded. The interviews were half-structured, i.e. I had questions prepared, but inquired if the interviewee said something interesting. The questions and notes from the interviews can be found below.
To evaluate the effectiveness of the workshop, I let students work through one complete trig-substitution problem and collected the sheets. This way I could identify if and which common misconceptions persisted through the workshop. I could also determine, if they used methods we developed together, or if they used other techniques to solve the problem.
I further had an evaluation sheet, that students filled out. It had the following questions:
I also conducted interviews with students after their midterm to further explore their way of working in groups and their way of thinking of trig-substitution problems. The interviews were half structured. My questions could be divided in three parts:
With a general rating of 8.05/10 from the participating students and an increase of 43% of students perceived understanding of the material, the workshop was a success. Students seemed to enjoy the additional opportunity to review and study the material. Most students also enjoyed the group work. In addition 15/22 participating students solved the trig-substitution problem correctly, two only made minor mistakes. The interviews I conducted were very interesting, since they gave me more insight into students' thinking. It became obvious that different ways of teaching work better or worse for various students. Even though all my interviewees contended that they liked the group work in the work shop, none of them worked in groups for their homeworks for reasons ranging from bad handwriting, over missed opportunities to make friends with classmates to just preferring to work alone.
It became obvbious during the workshop that the students enjoyed the group work and that it was helpful to them to discuss their thoughts with their peers. As mentioned in the introduction above I believe that the transition from a small class, to a big lecture setting is challenging for many students. With this change they don't only loose the direct contact with the instructor, but also the opportunity to interact with their peers. It was impressive to me that one of the interviewees mentioned how hard it is to find classmates to work with.
One of the greatest difficulties for students in Calculus 2 is how to pick an appropriate method of integration. Therefore I designed the first handout. I believe that this problem stems partly from the fact that students' don't have to decide which method of integration to use before they start reviewing for the exams. In their homework and on quizzes students know which section(s) they are working on, and therefore know which method of integration they are supposed to use. Many workshop participants commented, that they particularly enjoyed this part and that they learned something they didn't know before. For the future it might be a good idea to offer a workshop later in the semester, once all integration methods are covered, and concentrate on picking the "right" method of integration.
One of the things I tried to foster with the second worksheet was that students should realize why the choice of a certain trig function made sense. In the interviews it became obvious that this had failed. All interviewees memorized a table to determine which substitution to pick. Moreover they did not remember where the table was coming from. My fear with this technique of memorization is that after the course students will have a hard time remembering it. I am not sure, how to address this issue in a workshop. Maybe a discussion with the instructor would be a good way to go about it.
A common critique from students was that the practice problems weren't hard enough. By this they mean that the constants were too easy. For me, as an expert on trig substitution, the constants don't determine the difficulty of the problem. The students, as novices in the field, can get thrown off by these little changes. To address this issue, it might be a good idea to alter one or two problems on the second worksheet.
Many students wished for more full problems in the workshop. As an instructor it is always a struggle to decide what to spend time on. After the workshop I realized that even though one student made a mistake during the resubstitution, the students' comments indicate, that they felt fairly confident about trigonometric resubstitution and drawing reference triangles before the workshop. Therefore I suggest to replace the last worksheet by another full problem for students to work on.
One of the first things I realized organizing the workshops was that timing was extremely important. The first workshop that I held in the very beginning of the semester only had one student attending, whereas the one a couple of days before the midterm had very good attendance. Students obviously need the external motivation to attend this workshop in their "free" time. A difficulty for the organization of the workshop was that I didn't know the number of participants prior to the workshop itself. Therefore there weren't enough chairs in my second workshop, and the room was too crowded. This could be solved by using a sign-up sheet prior to the workshop. However I would be scared that this might initmidate studenents who decide to participate spontaneously.
If I were to conduct a more rigorous reasearch project to determine the effect of similar workshops, I would use two groups of students, offering the workshop to one group, and not to the other. Then I would compare exam or quiz results for both groups to see which group had the higher average. Additionally it would be interesting to interview more students on their work habits and their attitude towards group work. I would like to determine if this attitude changes throughout the different Calculus courses.