Math 105L/106L Post-COVID Changes

Executive Summary

Over close to two decades of teaching, thinking about teaching, and interacting with students, I have come to the realization that for my particular set of learners, largely students from minoritized and underprivileged backgrounds, the most important component of instruction is the creation of environments in which it is safe to learn mathematics.

During the past few semesters, I have overhauled my classroom environments and assessment methods in order to facilitate better learning. Remote learning as a response to COVID-19 gave me time and space to reimagine the classroom in ways that were previously near-impossible. Following the return to in-person learning, I have kept many of the same restructurings, after having seen their positive impact on students.

I have permanently and completely eliminated all tests and timed assessments in favor of an ongoing, recursive, and continuous system of work. This relies on a large team of around a dozen instructors, undergraduate assistants, and graders supporting a highly personalized feedback system for student work.

The following document outlines this in much greater detail, and includes substantial feedback from students at its end.

Teaching Philosophy and Practice

Over close to two decades of teaching, thinking about teaching, and interacting with students, I have come to the realization that for my particular set of learners, the most important component of instruction is the creation of environments in which it is safe to learn mathematics. As instructors, it is imperative that we recognize the reasons our students fear the math classroom and work to overcome the past trauma they have experienced regarding its study. In particular, it is critical we understand that women, non-binary students, students of color, students from low income backgrounds, and public school students are all minoritized in an environment such as Duke (and even more so in the Math Department), and that this requires specific strategies to facilitate their learning. In my course design and my classroom engagement, I take it as a given that systemic discrimination has impacted my students’ learning in the past, be it through implicit bias or lack of access to resources. Therefore, my teaching methods aim to remove the pressures imposed by prior systems that so often hinder progress in mathematics, pressures that so many students see as intrinsic to mathematics classrooms. I aim to show students that there is a different way of learning math, one that allows deep dives and non-rote learning, one that maintains rigor, while taking away the drive to memorize, cram, and reproduce force-fed material and problem solving methods.

Remote learning as a response to COVID-19 gave me time and space to reimagine the classroom in ways that were previously near-impossible. Over the past few semesters, I have overhauled my classroom environments and assessment methods in order to facilitate better learning. Following the return to in-person learning, I have kept many of the same restructurings, after having seen their positive impact on students.

I strongly believe that at this level and for these groups of students, high-stakes timed assessment and a focus on grades hinders rather than improves learning. I have therefore permanently and completely eliminated all tests and timed assessments in favor of an ongoing, recursive, and continuous system of work. My classes meet every weekday, and students work on and submit multiple assignments a week, making their learning a non-stop experience. The constant engagement with critical ways of thinking about mathematics, combined with the ability to seek and receive quick help and feedback from multiple sources allows and requires students to learn in ways they have rarely to never experienced in their prior schooling. Rather than cram for tests, students are required to engage and re-engage with material, strengthening and shoring up their knowledge and skills without pause, and with more-or-less equal intensity throughout the semester. They are allowed to misunderstand, receive feedback, correct, and get on track over and over without fear of any one component destroying their ability to pass the class. The removal of the pressure to perform is transformative: students consistently report that they are able to engage with math in ways that facilitate critical thinking. Instead of fearing having to learn the reasons behind formulas, students see these deeper dives as useful in their learning and growth. Many calculus students are fearful that if they don’t completely internalize the ideas behind a given proof or formula, they will fall behind their peers on the next test or quiz. By removing those entirely, students are able to take their time and space to learn as they wish, making it safer for them to learn. Thus more students end up doing that deep learning that is so very essential to success in math.

I believe that college mathematics classes should serve as a gateway to the sciences, not as a barrier to entry to them. Many of my students arrive at Duke with giant ambitions, having done exceptionally well in their prior academic work, especially in high school. When they get here, they confront an overwhelmingly wealthy, white, and privileged environment. For many, the shock of being directly confronted with the disparities in access to resources between their own high school experiences and those of median Duke students is overwhelming. As an early-college educator, I believe it is my duty to meet these students where they are: rather than throwing students in the deep end and telling them to swim, I aim to scaffold a learning structure that enables them to discover and rediscover their abilities. For example, many of my students never learned how to efficiently manage and structure their study time, having had little focus on this from high school educators. To meet them where they are, I created a mandatory, intensive, highly regular cycle of submission and resubmission of assignments, thus creating a strict structure for their study. Constant and ongoing view and review has become a hallmark of my classes. Despite the very high workload, this regularity is highly lauded by students.

Calculus has traditionally been taught as a bag of tools, with little attention paid to what it is used for beyond the highly artificial toy examples that textbooks and classes are full of. It is clearly important to teach these tools. Yet, feedback from students is so often a frustrated ‘what is this good for?’. Since the 1980’s, Laboratory calculus at Duke has attempted to enrich the toolbox with extensive application to data and real-world situations. The relatively small class sizes of Math 105L and 106L allow more of a focus on lab work than most other classes. Every year or two, I create new labs to replace aging ones. Students are required to work in groups using generic and custom-designed online tools and do their own research. Recent examples include labs on COVID-19 hospitalization data in NC, on growth of children, on global warming, and on sustainable fishing policies. These examples are complex, requiring students to confront and understand subtle ideas through mathematical argument and analysis. In maintaining the successes of Duke Lab Calculus while updating it to the needs and wants of 21st century students with rather different backgrounds than those of four decades ago, I aim to show students that the toolbag we teach them to use is far from arbitrary, and that its use gives real insight into the world in which they live. While lab work, especially in groups, is highly challenging, student feedback overall is very positive, expressing a deep appreciation for the application of math to their world.

The intensive structure of my classes requires a lot of support for student learning outside the classroom. The Mathematics Department has for many years run a calculus help room, open for over 40 hours a week, in which students can get help from fellow undergraduates. In addition, I run 3-4 hours of office hours a week. These take place on Zoom, allowing students easy access to support. Following each and every assignment, students receive comments from one of the large team of graders and teaching assistants. Each student is assigned the same grader throughout the semester, in part so that they have a person to talk to about their work who is not the main instructor.

I have also worked extensively with the Academic Resource Center to refer students who have need for extra support to small learning groups through the SAGE program. These groups, pioneered in past years by the Chemistry Department, have been extended and expanded to support Mathematics students as well. Feedback from students lauds all these highly flexible and extensive support structures, with many stating that the availability of constant and consistent access to assistance enables better learning.

Duke is a highly resourced institution. Especially in recent years, there has been a focus at the highest levels of the administration to direct a substantial amount of its resources toward students from minoritized and underprivileged backgrounds. In my work in Math 105L and 106L, I aim to create structures that support learning, remove barriers, provide access, and enable just these students to succeed in the highly ambitious goals they bring to Duke. The opportunities for reform created by a leadership at the departmental and university level that encourages teaching faculty autonomy, as well as by COVID-19 have given room for a rethinking and revamping of my classes. As a result, the DFW rate in Math 105L and 106L has dropped from 15% to less than 5%, and the continuation rate from 105L to 106L has risen substantially. Through careful and critical cycles of learning and feedback, I believe that math at this critical level has begun to become a gateway, not a barrier.

Class/Lab and Assessment Structure

  • In-building contact time - Class: 50 minutes, MWF; Lab: 50 minutes, TTh. Class lectures are videoed and posted online.

  • 30-35 worksheets, all started in class during and following lecture. Worksheets are handed in the Monday following class. Worksheets are commented upon by myself, TAs or graders, and returned by Friday. Students revise and resubmit by the following Monday (From Spring 22: Tuesday, by student request). Revised worksheets are graded and returned for pass/fail ‘grades’. Students must pass all but four worksheets to pass the class.

  • 8-9 lab assessments, done in groups. These range from essays and projects to short question forms. Like worksheets, these are commented upon, revised, and then graded. Students must pass all but one.

  • 3 take-home midterms (known as ‘Big Assignments’). Students have 8-10 days to complete each summative assessment. These are then graded numerically and commented upon, then returned for revision. Students receive up to half credit back for correct revisions. Students must pass at least two of the three, and make substantial effort to submit and resubmit all three.

  • All materials (with the exception of the Big Assignments) are posted on the class webpage before the semester begins.

Support Staff

Two undergraduate teaching assistants are present in every class session, and one in every lab session. In class and lab, these assistants work alongside me with students to complete worksheets and lab work. In addition, I hire 3-4 extra graders. Myself, each TA and each grader are assigned to a set of 4-6 students. They comment on and grade work for that same group throughout the semester. By default, each TA and grader also spend time in the calculus help room. TAs and graders are hired from a pool of students who have taken math at Duke, with preference given for those who took 105L and 106L, as well as those who have worked these classes in the past. Thus, I have created a pipeline of rewarding strong students with the opportunity to support those who come next.

Opportunities for Future Assessment

The class strucutre described above is a first for Duke Math, and is very unusual. Its ability to exist relies heavily on the resources a place like Duke can provide, as well as the flexibility and autonomy given to teaching faculty in the Math Department. Assessing the success of this model is challenging, as the class is S/U by design, and most students do not take further math classes at Duke post-106L. It would be of interest to consider the design of assessment of this particular rather unique methodology: minoritized students at highly resources institutions form an important community of interest. If results are postive, spreading the ideas underlying the design of these classes elsewhere at Duke and beyond would be highly desirable.

Such assessment is beyond my field of expertise. A math department colleague has been collecting pre- and post-semester data on student attitudes and perceptions towards mathematics, as well as math test anxiety. This is one avenue for such assessment. Another could be partnering with Pre-Health Advising to talk about the impact of 105L/106L on persistence in the pre-health track. Yet another would be the collection of testimonials from students who discovered a love of math after taking these classes in order to celebrate their growth and achievement.

I would very much welcome the opportunity for study and feedback.

Feedback from Students

(from Mid-Semester Surveys and Course Evaluations)

This class taught me to think in a much deeper way than before. I have never learned math in this much depth, but I think it gave me much better understanding of all concepts. We learned the actually theory behind the math which allowed me to understand why something was true rather than just being told it is true. I would say I’ve developed evidence based thinking because of this. I have learned to question why something in true rather than just accepting that it is. I believe this concept fosters greater understanding of the whole subject. I was also taught to handle multiple assignments at once which tested and improved my time management.

I think the learning environment of this class was very positive and encouraging. The enthusiasm from the instructor created an attitude of a desire to learn. We were congratulated on our success and taught in our confusing. There was a lot of encouragement to work through confusing ideas and grow through that learning process. We were not just given easy answers but rather encouraged to work for understanding. I would keep this enthusiasm and positive environment for future years. I makes hard concepts much easier to understand and work through. I also appreciated all of the help available from instructor office hours, TA office hours, and the help room. I felt very supported and that there were many resources for my success. I would definitely keep this in future years.

I enjoyed that the class utilized the take-home test structure with the Big Assignments. This was helpful for me to work through the problems at my own pace and actually learn the material, rather than anxiously rushing through a test.

I felt like everyone was included in the lectures and labs. No student was left behind if they were struggling.

Pass/fail structure of class helped alleviate stress and served as motivation for me to learn more about the concepts and enjoy my experience in the course.

The structure and components of this course facilitated my learning by always providing a long-term plan of what was going to happen in class, having effective synergy between the provided homework, worksheets, and in-class instructions, and the availability of office hours and math help room.

This class was welcoming and inclusive because the professor always encouraged people to participate in class and ask questions about the content presented. Additionally, even when students got questions wrong, no judgment or negative comments were permitted.

The instructor and format of this course facilitated my learning by always providing well-described and well-planned assignments and due dates, allowing me to anticipate periods of rigorous work and plan accordingly. I would keep this structured and well-planned approach to learning in the future.

It was stimulating because this class didn’t allow me to become complacent. I always had to work to make sure I understood.