By far, my most influential and consequential work has been the creation and development of the UC-wide online calculus sequence for Math, Science, and Engineering: Math 19A, Math 19B, Math 23A, and Math 23B between 2012 and 2018. This work was done in close collaboration with my colleague and friend in the Mathematics Department, Distinguished Professor Anthony (Tony) Tromba. This page acts as the home page for the online calculus project at UC.
a. Motivation for Moving to an Online Model
Our motivation to create online courses was first and foremost to improve student learning outcomes and student success. Instructing ever larger numbers of students in substandard classrooms at unsuitable times of day became unacceptable, as well as professionally irresponsible. We needed to act. The traditional route of hiring more faculty in order to create smaller class sizes, as has been recommended by many departmental review committees, has not been supported on this campus (as opposed to some other UCs) for over 50 years. To be precise, in 1970, the Mathematics Department had 12 FTE and UC Santa Cruz about 3,000 students. Today, we have 19 FTE and almost 20,000 students at UC Santa Cruz.
We decided to apply for a course development grant offered by the newly formed UC online at the UC Office of the President (Fall 2011). With the success of Calculus I came the grant money for Calculus II; then, as part of ILTI (Innovative Learning Technology Initiative) by Governor Jerry Brown, we were awarded the grant money for Calculus III and Calculus IV, with a total of approximately $500,000 awarded for development of the four courses. All four courses are now offered as part of the UC-wide cross-enrollment system.
b. Structure of a Seamless Online Learning Experience
Creating the learning resources for these courses was a massive undertaking. We created close to 400 tablet-based lecture videos for our four courses. A few selected videos will be made available here. The scripts for these videos were written by Prof. Tromba, implemented and recorded by Prof. Bäuerle in a custom-designed recording environment either on campus or at Prof. Bäuerle’s home, and edited by a professional videographer and/or instructional designer. Often there were several iterations of review and revisions.
Selected Tablet Lecture Videos are included on a google drive for this web site.
• Introductory Video to Vector Calculus
• A Second Look at the Area of a Triangle
• The Hyperboloids and Paraboloids
• Applications to Physics: Force, Work and Escaping the Earth
• Cavalieri’s Principle
• The Cycloids
• The Field Concept. A Philosophical Discussion. Part II: Maxwell and Einstein
Equally daunting was our creation of a state-of-the-art readable E-Book for each of the four courses, which included progress check questions, interactive art and applets, historical context, exercises for quizzes, and homework assignments—all seamlessly integrated and synchronized with the video lectures. Interspersed throughout the book, the progress check questions are designed to provide students with formative feedback on their comprehension of the material they have just read. The applets are designed to aid student visualization.
For a short demonstration of some of the features of our E-Book see here.
To our knowledge, there is no other four-part calculus course in the world that integrates all these components.
c. Success of the Online Calculus Learning Experience
Ultimately, the quality of our courses must be judged by their acceptance by students and by student outcomes. In our view, these four courses have been successful beyond our expectations, with their capability to increase student access, to effectively address the needs of students with disabilities, and to level the educational playing field. There was no question that online learning will be part of the future of a first-rate university education in our minds before the pandemic, and the last years with strikes, fires and the pandemic have clearly shown the importance of the ability to offer courses in multiple modalities.
Some evidence of successful outcomes and student migration to online education:
1. IRAPS Study on Efficacy of Online Calculus
In 2017, Dr. Julian Fernald, director of UCSC’s Institutional Research and Policy Studies (IRAPS), conducted a study investigating the outcomes of student performance in courses taken after calculus and compared the performance of students who took Calculus online vs. face-to-face. In an e-mail, Dr. Fernald communicated the following:
“Regression analysis indicated that there is no relationship between whether upstream classes are taken online or face-to-face after taking into account the effects of preparation (SAT Math and previous course performance. In fact, only students’ grades in the upstream courses consistently predict grades in the downstream courses.”
In short, students who took online calculus perform as well in subsequent courses as students who took courses face-to-face.
2. Equity Analysis by Course Modality (Our own and by IRAPS)
In early 2020, we initiated an equity analysis of the student learning outcomes of our courses by looking at student performance in Math 19A, 19B, 23A, and 23B. We focused our attention on classes offered between Fall 2016 and Fall 2019 and considered pass rates, GPA, and B or better rates as well as retention of women. Here is an excerpt of our findings, and more data will be added later. The table below shows student performance (as measured by GPA, Pass Rate, B or better rate) by race in Math 19B, represented with cumulative enrollment data from Fall 2016 through Fall 2019, then disaggregated by mode of instruction. A course offered in face-to-face (F2F) mode is the traditional brick-and-mortar, in-person version; a course offered in the online mode is co-taught in the instructional model developed by us.
|
|
Count |
GPA |
Pass |
B or |
||||||||
|
All |
4663 |
2,475 |
2188 |
2.58 |
2.58 |
2.57 |
79% |
76.61% |
81.90% |
45.7% |
47.96% |
43.24% |
|
African-American/Black |
149 |
75 |
74 |
2.02 |
1.85 |
2.19 |
62.42% |
54.67% |
70.27% |
28.86% |
29.33% |
28.38% |
|
American |
29 |
21 |
8 |
2.21 |
2.19 |
2.25 |
65.52% |
66.67% |
62.50% |
37.93% |
38.10% |
37.50% |
|
Asian |
1,730 |
903 |
827 |
2.66 |
2.69 |
2.63 |
81.45% |
79.07% |
84.04% |
48.61% |
51.83% |
45.10% |
|
Latinx |
840 |
452 |
388 |
2.36 |
2.31 |
2.41 |
73.57% |
69.47% |
78.35% |
36.55% |
38.72% |
34.02% |
|
International |
600 |
234 |
366 |
2.82 |
2.8 |
2.83 |
85.50% |
82.91% |
87.16% |
56.83% |
57.69% |
56.28% |
|
Native |
37 |
30 |
7 |
2.34 |
2.46 |
1.8 |
75.68% |
80.00% |
57.14% |
35.14% |
43.33% |
0.00% |
|
Unknown/Not |
84 |
43 |
41 |
2.76 |
2.73 |
2.79 |
84.52% |
86.05% |
82.93% |
50.00% |
48.84% |
51.22% |
|
White/Caucasian |
1,194 |
717 |
477 |
2.56 |
2.62 |
2.47 |
78.48% |
77.82% |
79.45% |
44.81% |
48.12% |
39.83% |
As the data indicate, significant equity gaps exist in all measured categories, particularly for African-American students (African-American students earn an average GPA of 2.02; all students earn an average GPA of 2.58). One also can see that, on average, the equity gap (as measured by GPA and pass rate) is significantly smaller for students who took the class in the online mode. Latinx students perform about as well as the white students in the online mode (78.35% vs 79.45% pass rate) whereas in the F2F mode, the pass rate varies significantly (69.47% vs 77.82% pass rate). Interestingly, the overall GPA for all students in both instructional modes, with similar cumulative enrollments, is almost identical (2.57 and 2.58, respectively).
We decided to approach IRAPS to conduct a similar equity analysis before publicizing our findings. IRAPS is able to conduct regression analysis and can take into consideration factors such as preparation (as measured by SAT or ACT Math scores), which we did not have access to nor do we have the expertise in data analysis that IRAPS has. The IRAPS study also focused on pre-pandemic data and chose to consider URG students as a single group rather than disaggregated by individual groups. There are advantages to that approach since the number of students can be small in some groups, but a disadvantage might be that particular differences between students in certain groups or combinations of groups could be missed. Here is an excerpt of an e-mail exchange between us and IRAPS staff:
Our question: “1) Your analysis for comparing performance of URG students vs non-URG students by course modality considered URG students as a single group. In our initial data analysis we observed that equity gaps for African American students and Hispanic students have narrowed significantly (see my April 18 e-mail below for some details on this). How does this relate to your study, and what conclusions can we draw from our observations relative to your analysis?”
IRAPS response: “Regarding question #1, we collapsed the disaggregated race/ethnicity categories into URG and non-URG categories as the cell sizes start to get quite small otherwise, particularly in the courses with smaller class sizes. We have also used the URG status-based analysis in other IRAPS reporting where more disaggregated categories would become too small for analyses. The course performance by modality patterns in the disaggregated race/ethnicity data from the initial analysis share similarities with the findings from the follow-up analysis. For example, in the followup analysis, we found that for Math 19B and Math 23A, URG students who took the course online had higher course GPA than URG students who took the course face-to-face.”
3. Students Are Choosing Online
Case Study 1, Math 19B Winter ‘19: In many people’s minds, the traditional face-to-face lecture model is the gold standard of instruction. Many incoming UC students have never taken an online course, so they don’t necessarily know what they are comparing when choosing between online and face-to-face versions of a course. But when they do know the difference, it appears that they choose our online courses by a far larger margin. In Fall 2018, Math 19A was offered face-to-face as well as online with roughly equal enrollments. When the sequel, Math 19B was again offered in both versions in Winter 2019, students overwhelmingly enrolled in online (by a margin exceeding 2:1). Due to enrollment balancing by the department, the final overall enrollment is less lopsided, but enrollment still clearly favors online.
|
Date |
Math 19B Online |
Math 19B |
Maximum Enrollment in Each Class |
|
Winter 2019 |
417 |
259 |
420 |
Case Study 2, Math 23B/Calculus 4 enrollment: The table below lists the quarterly enrollments for the class during the 5 academic years preceding the pandemic. The online offerings are highlighted.
|
Academic Year |
Fall Quarter |
Winter Quarter |
Spring Quarter |
|
19-20 |
105 |
167 |
142 |
|
18-19 |
146 |
131 |
103 |
|
17-18 |
103 |
119 |
100 |
|
16-17 |
104 |
107 |
100 |
|
15-16 |
113 |
92 |
102 |
The three online offerings clearly have the largest enrollments.
Case Study 3, Math 19B/Math 23A enrollments in Spring 2022:
In Spring 2022, the department offered Math 19B and Math 23A in the online format (open to cross-enrolled or XC students from other UC’s) and also in the traditional face-to-face format. The course enrollments are listed below.
|
Class |
Math 19B |
Math 19B |
Math 23A |
Math 23A |
|
UCSC Enrollment |
203 |
25 |
347 |
8 |
|
XC Enrollment |
63 |
|
15 |
|
|
Total Enrollment |
266 |
25 |
362 |
8 |
These numbers are probably influenced by the pandemic, as it is likely that many of our students were still not in Santa Cruz during the quarter. However, it was interesting to see that there were more students from other UCs taking calculus online than UC Santa Cruz students taking the classes in person.
4. The Explosion of Summer Session Enrollments
Summer session enrollments in Math 19A, 19B, 23A and 23B: These enrollments have exploded since the courses went online. Before online, the enrollment in these courses was just enough to justify one offering of each class per summer. In fact, Math 19A was cancelled several summers due to low enrollments in the decade preceding online. This situation has fundamentally changed with the advent of online calculus and instantly the online classes had substantially higher enrollments than the face-to-face predecessors. Since summer 2020, Math 19A, 19B, 23A and 23B are now offered in both sessions, and in summer 2022 we had a combined enrollment of 524 students.
|
Average Cumulative Summer |
Math 19A |
Math 19B |
Math 23A |
Math 23B |
Math 19AB |
|
|
|
|
|
|
|
|
Face-To-Face |
16 |
31.4 |
25.4 |
26.5 |
99.3 |
|
|
|
|
|
|
|
|
All |
97.6 |
117.8 |
154.9 |
83 |
453.3 |
|
|
|
|
|
|
|
|
Online |
117.3 |
112 |
153.3 |
83 |
465.6 |
|
|
|
|
|
|
|
5. Student Feedback
We receive much positive feedback from students in office
hours, student evaluations, and unsolicited e-mails. Below is a small selection of unsolicited testimonials.
a) “I greatly enjoy the online class setup…I think I have learned far more from my online classes because we have more access to the Professor and TA’s. It seems that in math and science based face-to-face lecture formatted classes, I have to wait in this massive line to ask a question. Then, I have to deal with the scoffing of other students as I ask my question because they think I am asking a "dumb question". It’s really disheartening…I wish I could take most of my math and science classes online. It makes the educational process for me less stressful and I appreciate the access we have to learning resources online.”
– Jaime M., UCSD student, Math 19B, Summer 2014
b) “Hello Professor Bauerle,
I have just finished my final and I would like to take the time to thank you for your efforts in this class. The class was well-organized and the lectures were presented in a way that made it easy to understand. I think that the organization played a big role improving my studying habits and it is reflected in me getting an above average score on the midterm. Anyways, thank you for your help and I hope to take Math 23A with you during the fall!”
-Chidera O., Math 19B, Summer 2017
c) “Hey Prof. Bauerle,
I’d like to thank you for teaching the best college math course
I’ve ever taken, even though it was an online course. It is rare to find a
teacher that cares as much about math or about his students.
It’s been a pleasure,”
– Layla G., Math 19A, Summer 2019
d) “Dear Professor Bauerle,
I just wanted to say thank you for your lecture videos in the Math 19B course. Your notes are very clear and the examples are well explained. I can see how much care you put into these videos and really appreciate them! Have a good week.
Sincerely,”
– Erica S., Math 19B, Winter 2017
e) “Professor,
I am writing to inquire on my final grade for the course. I also wanted to thank you, Professor Tromba, the T.A’s, the web developers, and everyone involved in the construction of this online series from 19A to 23A.
This series more than exceeded my expectations, and I feel as though I have a new grasp on mathematics as I apply to graduate schools. This course was challenging and well thought out from a theoretical and practical perspective, and the series maintained an excellent continuation in regards to previous material. Above all, the teaching staff was incredibly responsive and timely.
As an alumni of UCSC, I was extremely grateful to have access to my alma matter whilst forward deployed, and this entire series has reinforced the pride I feel as a Banana Slug and the quality of education the UC system offers.
Thank you for all the hard work,”
– Don W., Class of 2015, UC online student in Math 19A, 19B and 23A.
f) “Greetings,
I just wanted to say thank you for making this quarter of calculus so amazing;
and that I really appreciate the effort that went into the lectures, practice
questions, piazza, and the homework. I had taken 19A last quarter I enjoyed it
is as much as 19B! It was tough at times, and I remembered being nervous before
every test; but now that I finished I appreciate Calculus in a whole new light.
I know students don’t usually thank instructors like this, but I really
appreciate how this course changed my mindset for the better. A couple of months
ago I kept telling myself that I wasn’t smart enough to understand calculus,
but now that I’m here I cant help but say thank you for making it possible.
Sincerely, “
– Jorge G., Math 19B, Fall 2020
g) “Thank you all so much for the Calculus courses. I really appreciate how well these courses were put together. The lecture videos, the responsiveness on Piazza , the ebook homework and the practice tests/quizzes all made these courses super accommodating. It was a great experience for me. I got a lot out of the entire sequence.
Thank you all so much, Phil”
– Phillip J., Math 19A, Math 19B, Math 23A, Math 23B Summer 2020 -Winter 2021