Math to the People

In my teaching, there are two main models I use to frame my instructional approach. In the “lecture-based” model, I tend to spend most of my class meeting time talking. There are interruptions to my lectures of course: questions my students ask me, questions I ask my students, and 5-10 minute breaks from the lecture in which students are solving practice problems. But, in my lecture-based classes, about 75% of class time is spent with me talking. This is the approach of teaching I am the most familiar with, as it is the one by which I was taught.

I have also, over the past 8 or 9 years, been developing a different teaching approach using the “flipped classroom” model. Under this model, most of the lectures are outsourced to videos which students watch before they come to class, and much of class time is spent by the students working on problems in groups, and most of my time talking in front of the class is me directly responding to student questions by solving problems on the board. The idea behind the flipped classroom model is that class meeting time is the only time during the week when students are able to directly interact with their instructors, and that in-class time is better spent on activities in which student-teacher interaction is more effective, while lectures which require little interaction from the students are better completed outside of class.

In the Fall of 2022, I conducted a small classroom experiment. I had been assigned to teach two sections of Math 100, one after the other, and I decided it would be worthwhile to teach one section using each instructional model to see which one was more effective. In my pre-2020 teaching at Alexander College I almost exclusively taught using the lecture-based model, however, when lectures moved online in 2020 due to the pandemic, I adopted the flipped classroom model to better reach those students who had internet connectivity issues that prevented them from taking full advantage of synchronous online lectures. I felt that the transition back to campus was a good opportunity to compare these two models to see which one would be a more effective approach for my particular set of students.

The Two Different Models

Both sections of Math 100 in the Fall of 2022 were evaluated in the same way. Both had the same number of open-book quizzes, and the same number of midterm tests, and both wrote the exact same final exam. Both completed the same assignments, and both sections were given links to the same free online textbook. Both sections were given links to the same set of video lectures I had made, although only the flipped-class section was explicitly asked to watch the videos as homework. Both sections were graded on in-class participation in solving practice problems. However, the lecture-based section had less class time allotted to solving practice problems than the flipped-class section.

In the lecture-based sections, about 75% of class time was occupied by my lectures, with the other 25% devoted to students solving practice problems during class time. These practice problems were presented immediately after the material that they were related too, so students had an opportunity to see me solve an example immediately before they were asked to solve one themselves.

In the flipped-class sections, I reduced the time spent lecturing to about 30% of class time. These lectures were summaries of the video lectures, rather than full lectures in themselves. Another 40% of class time was devoted to students solving practice problems. About 30% of class time was students working on problems in groups, while 10% was devoted to games such as Kahoot. I presented these practice problems all at once rather than after each portion of the lecture, so that I could free myself up to circulate around the class and help students with those problems. The final 30% of class time was spent by me going over solutions to problems students had been asked to complete. I did this both for the practice problems I had asked students to complete in class and also for the problems from the weekly assignments.

Observations

One of the things I noticed through direct classroom observation was that the flipped-class students had much more difficulty solving the problems in class than the lecture-based students did. I believe one of the reasons for this was that the problems presented in the flipped-class section were genuinely harder problems than the practice problems given to the lecture-based section. The intention was to give students a familiarity with exam-difficulty questions to prepare them for the midterms and final exam. The hope was that the experience of solving harder problems in class would better prepare the flipped-class students for the final exam.

I also believe that another reason for the greater ease with which the lecture-based section completed their practice problems had to do with the way in which class time was organized. Since the practice problems came immediately after I had solved similar examples, a lot of students were able to solve the practice problems quickly and easily by simply copying the example I had just done. Students trying to succeed through ‘rote learning’ where they solve problems by copying an example is something I have observed as a common pitfall that students at Alexander College succumb to. Many students do not grasp the difference between true mathematical understanding, and the ability to get the right answer by copying a previous example. I hoped that, the flipped class students could learn better how to truly understand what the problems were asking for by having a separation between my examples and the practice problems.

I also noticed a difficulty staying on task in the flipped-class section. Many of the students would take a 10-15 minute break when given a 5 minute break, and when given time to work on problems in groups, would spend that time talking about non-mathematical topics. Many students would complete only 1 or 2 of the 5 problems they were assigned, and many would seem to be completely unfamiliar with the material when they came to class time, suggesting that they had not in fact watched the assigned videos.

I will also remark here that, in the flipped-class section, I did not receive as many request from students for help as I would have liked. I spent a great deal of the time I had allotted to helping students with their practice problems waiting for students to ask me for help. Students would ask their friends for help, but that help would usually simply amount to copying their friends’ answer rather than the friend actually explaining to them how to solve the problem. I believe that many of my students, perhaps due to the academic culture of their high school studies, or due to their difficulties with the English language, were not confident asking me for help when they needed it.

Statistical Results

At the end of the semester, I tabulated the students’ grades from the two sections to get quantitative results I could use to compare the two approaches. I computed the students’ average grade (out of 100) for their homework, quizzes, and participation, and recorded their grades on each of the two midterms and the final exam. I then computed the mean, median, and standard deviation of each grade category for each section.

Grade Category	Lecture Mean	Lecture Median	Lecture St. Dev.	Flipped Mean	Flipped Median	Flipped St. Dev.
Midterm 1	54%	49%	29%	51%	52%	23%
Midterm 2	46%	41%	33%	40%	37%	29%
Final Exam	47%	50%	33%	37%	37%	24%
Homework	77%	83%	28%	83%	92%	28%
Participation	63%	63%	30%	66%	71%	25%
Quizzes	58%	63%	33%	56%	61%	26%
Final Score	53%	54%	31%	49%	50%	24%

One thing I will note at this point is that part of the reason for the higher standard deviation in the lecture-based section was due to the fact that the lecture-based section had 7 students who did not write the final exam versus 5 students in the flipped-class section. There are always a number of students who stop coming to class If we remove these students, the higher mean and median scores in the lecture-based section becomes even more apparent.

Analysis

The flipped-class section had higher scores on homework and participation than the lecture-based section. Both of these are likely due to the greater amount of class time devoted to students solving problems on their own and in groups. The greater amount of time devoted to practice problems indicated to students that these problems were important, so more students attempted them. Also, students were instructed to start working on their homework assignment if they finished the practice problems early, which likely contributed to the higher homework grades.

At the same time, the scores on the quizzes and exams were notably higher in the lecture-based section than in the flipped-class section. It seems clear, that despite my intention to use in-class practice problems to prepare students for tests and exams, students in the flipped-class section were not as prepared for their tests and exams as students in the lecture-based section. There are a number of possible reasons for this which I will outline below.

One possible explanation is that the students in the lecture-based class simply took better notes than those in the flipped-class section, and because they had better notes, they had an easier time students for tests and exams. Most students are not used to taking notes from videos, while taking notes from an oral lecture is a skill most students are familiar with. Moreover, many students in the flipped-class section appeared not to be watching the videos I had assigned, and therefore were not getting the benefit of any lectures at all.

Another possible explanation is that the students in the lecture-based class took the class more seriously than those in the flipped-class section. Due to the amount of time spent by students working at their own pace on practice problems, and the fact that many students were able to start their homework assignments during class time, I believe that many students in the flipped-class section may have felt that they didn’t need to do much studying outside of class. Rather than spending time outside of class watching the video lectures I had assigned and then completing their homework assignments, I believe that many students skipped the videos and solved most of their assignments in class, leading them to spend less time on the course than other students. This meant that, in the end, they were simply less prepared for the tests and exams.

A third explanation I will propose is that the difference in test and exam scores between the two section is not in fact due to the different teaching model used, but is due to a pre-existing difference between the two sections. I have noticed that, when I teach two sections of the same course with one section in the morning and the other in the afternoon, the afternoon section tends to perform better in terms of grades. I believe this is due to the priority registration system used at Alexander College where stronger students get first pick of which section to register in, and that afternoon sections are generally more popular. The flipped-class section was a morning section while the lecture-based section was an afternoon section. I would need to compare two morning sections or two afternoon sections to eliminate this explanation.

Conclusion

While it is possible that the difference in grades between my two sections was due to a pre-existing difference between the two sets of students, evidence still suggests that my lecture-based approach is the more effective one under the current circumstances. This is not to say that I will completely abandon the flipped-class approach, but that I will need to rethink the approach to ensure that students take my flipped-class sections as seriously as they do my lecture-based sections. This may mean requiring students to write online quizzes to prove that they watched the videos, or creating a more structured format for in-class activities. While I hope to test out a more effective flipped-class format in the future, for now I am used the lecture-based format for my in-person classes.