The questions are:
- Can you expand on the context at your institution for your course and what led to your teaching this summer?
- What kind of course materials did you use?
- What was your technology setup?
- How would you describe the synchronous components of your course?
- For those of you who had asynchronous components, how would you describe your engagement with your students and their engagement with each other?
- What was your assessment structure?
- Did you have any concerns about academic honesty?
- What challenges did you encounter in your summer course and what do you wish you did differently? What changes do you plan to make for your next online course?
- Do you have any personal anecdotes or final thoughts you’d like to share?
The table below gives an overview of each colleague’s summer teaching experiences.
|
Professor |
Institution |
Course |
Online Format |
|
Kyle Evans |
Trinity College |
Mathematics and Politics (general education) |
Synch |
|
Roser Giné |
UMass Lowell |
Calculus II (4 sections) |
Synch |
|
Jean Guillaume |
Sacred Heart University |
Mathematics for Liberal Arts |
Synch |
|
Rachel Schwell |
Central Connecticut State University |
Calculus II |
Semi-synch |
|
Ileana Vasu |
Holyoke Community College |
Calculus I |
Asynch |
Kyle Evans (KE): I taught Mathematics and Politics for the past four semesters and this is a general education course that I “rebranded” rather than fully redesigned to create a meaningful theme of politics through the four content units of Voting, Apportionment, Redistricting, and Game Theory. Summer courses are traditionally not common at Trinity, but the pandemic created a desire to offer a more robust selection of (online) courses to provide students with opportunities to stay on track, get ahead, or simply fill their summers. As a result, I chose to offer a course I was very comfortable with to ease my prep workload.
Roser Giné (RG): The format of Calculus II at UMass Lowell has typically been an in-person, lecture-based course. Often there are multiple sections of the course during the academic year, with one course coordinator who provides homework assignments and a common final exam. During the summer sessions, instructors have a bit more freedom around the course design; however, given the fast pace of the course, it is difficult to build in much inquiry based work, even when meeting face to face. This summer, I was originally scheduled to teach one section, but the course filled up and additional sections were added, resulting in my assignment of 2 sections during each of two intensive, six-week summer terms.
Jean Guillaume (JG): As the name suggests, the course has been instituted as a basic math course requirement for Liberal Arts students. It is intentionally designed to be computation-free and aimed at activating mathematical thinking, logical reasoning, and problem-solving skills. To achieve these goals, it resorts to self-contained topics such as set theory, graph theory, symbolic logic, voting theory, and so on. This summer, the course was set to meet face-to-face twice a week for a 3-hour period each time. As COVID continued to impose its will, it became quite clear that the face-to-face aspect was not feasible and the School ultimately determined that it was best for the course to be delivered fully online, leaving it up to the instructor to decide whether to make it synchronous or not. Since my decision to keep my Spring courses synchronous after transitioning online had received high praise, I opted to maintain the course with this summer joint. The only tweak I made was not to mandate attendance after the first meeting.
Rachel Schwell (RS): In Calculus II, I typically do a mix of an introductory active lecture followed by group worksheets. Ideally the amount of time spent on the latter exceeds the former. (As a disclaimer, I don’t necessarily believe this is the best possible course model, but I have always found a traditional Calculus II curriculum, in particular the techniques of integration, to be the most challenging to convert to a pure active form.)
As a summer course, it would traditionally have met for eight hours a week for eight weeks. The decision was made by the registrar toward the end of the spring semester that all summer courses would just be listed as “online”, with no synchronous times included. Therefore, if I wanted to meet synchronously, it was up to me to find times that would work for everyone once we began. When I saw the first doodle poll my heart sank because, with busy summer schedules, there wasn’t even one 30-minute time slot where all students were available. In fact, there wasn’t a single time where more than 60% of the class could meet! So, this forced me to make the choice that attendance wasn’t required, which then forced me to have enough content and resources available independent of class meetings. Unfortunately, with only about a week to plan, I didn’t have the time I would have liked to develop a rich asynchronous component.
Ileana Vasu (IV): Traditionally, summer Calculus at HCC had been offered in a 7 week format, on campus. This is the first time a Calculus course from the Calculus I, II, III sequence has been offered online due to COVID. My school chose the course to be entirely online, with no synchronous component feeling that this would maybe help enroll more students., I was asked to move the course to summer session II, which was only five weeks and just ended on August 10. Since the format of the course was asynchronous online, I decided not to try any synchronous meetings for the teaching side since I felt that this would be most equitable. I did offer a variety of office hours by zoom, made and selected videos on sticking points.
What kind of course materials did you use?
KE: When I taught this course in the traditional semesters I utilized guided note packets with built-in practice problems to engage in some form of active learning and for the summer class I tried to maintain that structure but with my slides/notes replacing the guided note packets. This means that there is no additional cost to the students for course materials and opportunities to practice on their own are built within my types of assessment.
RG: For our synchronous meetings, I knew that I would need a variety of learning modes to reach and engage students during a four-hour period. Notes were done synchronously: I explained ideas and wrote them on Sketchbook, asking orienting and probing questions of my students. At the beginning of each class, I also used warm-up problems. These were often “visual proofs” related to the day’s content. Students would examine a diagram, and in their chat boxes they would post ideas and questions, which were then released at the same time (a “chat storm”, serving a similar purpose as a gallery walk).
During each class session, students worked in break-out groups at least twice per session, where they completed one or two problems related to our course content. These problems were always posted within our week’s folder in advance (pre-typed worksheets).
We also made use of various online applets. I would either share the applets with students and ask questions while exploring together, or students worked on them individually, responding to prompts I posed in advance (see Calculus Applets using Geogebra; Sequences and Series Applet in Geogebra). When teaching volume, I used demos from the Wolfram Demonstration Project, to assist students in visualizing three dimensional objects. I also used various applets (primarily Geogebra) for exploring power series and Taylor and Maclaurin polynomials and series.
A few times during the semester, students were able to engage in some discovery-based work. I created discovery-based worksheets that students completed either during class or between our class sessions (topics uncovered through an exploratory approach included: the Fundamental Theorem of Calculus; understanding distance traveled versus displacement; and various tests for convergence of series).
During each week, students also watched a few videos I created in advance. These consisted of derivations or additional examples that illustrated specific topics of the course.
Homework was completed through Pearson’s MyLab Math. This online homework platform also included an e-book adopted for this course, University Calculus, Fourth Custom Edition for UMass Lowell, by Hass, Thomas, and Weir.
JG: In addition to the e-text available on Pearson MyLab Math (online platform used for homework), lecture notes prepared on my iPad using Notes Plus were made available in advance (see sample below). In these notes, one can also find problems and extra credit questions. Empty space is intentionally left after each problem to write down the solution worked out together during lectures. Complementary to these notes are short videos (around 5-min long) highlighting outcomes and main points. When deemed necessary, parts of virtual lectures were also recorded and posted.The following is an example of a short video: https://youtu.be/6dcD77xxFjA
RS: The textbook was Hass with MyMathLab, which I used for online homework and quizzes. I created the videos myself as well as the practice sheets used during class. The practice sheets were mainly just that: practice problems. But, I always made a point of including a variety of problems, only a few of which whose “type” we had actually gone over together. I considered this the most inquiry-based portion of the course.
IV: I wanted the students to have free access to the course, so I sought out free resources. There was a required OER text available in Moodle as pdf Calculus Volume 1 (Strang and Herman).
The course homeworks were designed using a MyOpenMath course platform. I learned that MyOpenMath can be embedded in Moodle (or Blackboard or Canvas) so it is seamless, and so students do not have to go to a different place to do the homeworks, which worked out nnicely. Exams were designed in Moodle and had a written component students had to submit in a submission box within half an hour of completing the exam. Projects and course reflections were designed so students could access them through Moodle and collaborate within this space on their work.
What was your technology setup?
KE: I used my laptop for my camera, audio, and Zoom controls and then joined the meeting with my iPad (a purchase I made in March immediately following the announcement of remote learning) and shared its screen containing prepared slides in Notability so that I could write on them.
RG: I used my laptop with a webcam and computer audio. UMass Lowell uses Blackboard as a learning management system, and I met synchronously with students through this platform, within Blackboard Collaborate Ultra. I found the accompanying white board application to be very limited, so instead I used Sketchbook for Macs (an application often used in illustration). I purchased a Wacom tablet (Intuos Pro) to connect to my laptop and taught my class on my Sketchbook application shared through Blackboard Collaborate Ultra. I also set up all of the materials for the course within Blackboard so that students could access documents needed for each class from the same platform through which we met. A few times, I tried using my ipad with a pencil and Explain Everything, and while I really liked the application, my handwriting would show up slightly blurred on Blackboard when mirroring my ipad (I used Mirroring 360 to project my ipad onto my laptop). I stuck with Sketchbook when working synchronously with my students.
As mentioned above, I also recorded about three or four videos per week to supplement our synchronous sessions. I filmed the videos using Screen Cast-O-Matic along with the Sketchbook application and Wacom Tablet. These were uploaded to YouTube so that the videos would meet accessibility criteria for my university (YouTube automatically captions video). When I had a chance to play around a bit with Explain Everything, I downloaded this application to my ipad and used it to create the videos. I then uploaded these additional videos to my YouTube channel.
JG: Blackboard Ultra was set up to host lecture notes and videos, online discussions, quizzes, exams, and reminders. I used weekly modules on Ultra to guide students through the week’s activities and to support executive functions. Synchronous live lectures were dispensed via WebEx. These meetings were hosted on my laptop and I then joined the meeting with my IPhone 11 for better cam and audio quality. For chats during lectures, I used my laptop keyboard.
RS: I recorded videos on OneNote on my iPad using a compatible stylus and the built-in recording feature (which was clunky - there are definitely good free apps that would have been better such as Screen Cast-o-Matic as mentioned by Roser above). I posted links to videos and class recordings on Piazza, and had students post all questions on there as well. The synchronous class meetings took place in Webex Training which has breakout rooms. In order to access my iPad during class meetings I just joined as another participant. It can be a bit clumsy transferring the presenter and host rights back and forth between the two versions of yourself, but I got used to it after a few sessions.
IV: I designed and integrated assignments and course content in Moodle, MyOpenMath, and Zoom. I used a variety of videos, some created by me, and some provided via the OER course I was using. During Zoom sessions, I connected the Ipad to my computer so I could project my ipad. I recorded Zoom sessions was able to employ Explain Everything for writing on my Ipad. After each session, I downloaded the pdfs from class work, along with videos of sessions, and posted them on our course website. I also used the whiteboard available in Zoom. At various times the course necessitated student use of Desmos and a TI84.
How would you describe the synchronous components of your course?
KE: My class met twice a week for 5 weeks and while we had a scheduled 3-hour block for classes, my average class meeting was around 2 hours long. With only 10 course meetings, the class was naturally fast-paced and the sessions themselves were not as interactive as I wanted them to be. As I moved through my slides, very few students had their cameras on, there was little response to questions I posed, and they did not ask me many questions. For most classes, I built in practice problems into the slides and utilized breakout rooms for students to work together on them and compare thoughts and answers. These worked relatively well as more voices were heard and all students turned their cameras on without any prompting. They were also much more willing to interact with me as I bounced between the rooms.
RG: My classes met twice a week during a six-week period (Monday-Wednesday, or Tuesday-Thursday), with two additional sessions per course scheduled on Fridays. Each class session was 3 hours 50 minutes in length. The first ten minutes involved some kind of warm-up problem related to big ideas of the course. After a short discussion on the warm up, we typically spent half an hour to forty five minutes reviewing homework problems. I then introduced new material through a lecture format, where I asked questions of students as we developed new concepts. Within the first two hours of class, students spent at least half an hour in break-out groups, completing a few problems with their peers. During break-out group assignments, I circulated among the groups to check in on any difficulties encountered. The second half of each session was similarly structured, with question-based lecture and additional break-out groups activities.
JG: Over a 5-week period, the class was scheduled to meet twice a week, more specifically on Monday and Wednesday, from 6 to 9 pm. Despite announcing in the course syllabus that students get rewarded for class participation, always the same crew of 5 students showed up to these virtual meetings. Typically, I started each meeting the same way: I took questions on assigned homework, listened to concerns and did some housekeeping; then I proceeded to give a recap on our last meeting’s materials and lay out the agenda of the day and expected outcomes. During lectures, I refrained from using the posted notes though the order in my delivery of the contents remained the same for the most part. Most importantly, I showed consistency in my highlights and punchline(s). As a group, we discussed the problems left unsolved in the notes and wrote down their solutions. Since the meeting took place around dinner time, we often established a relaxed atmosphere. For example, we sometimes had virtual dinner together, which I often used as an extension of the lecture. I recall using the set of foods we had as a group to explain the concept of the number of subsets.
RS: In the class meeting that followed each set of videos, we would do some follow-up discussion or examples interactively as a class, during which I would call on attendees by just going down the participant list in order. Since there were not many attending, each student was called on several times. We would then break up into groups of 2-4 students to work on problem sheets. In practice, this often meant just one breakout group, since with the scheduling issues I might only have three or four students in attendance. It was still worth the extra step of putting them in the breakout group, however, because they had the whiteboard to collectively write on. I also think it psychologically “passed the torch” to them as the presenters.
There was one class meeting where we were working on a particularly stubborn series that seemed to resist all obvious test attempts, and so we transitioned to group work by my sending them off to see if they could solve it themselves. This is probably the most engaged and eager they ever were in diving into group work, so I wish I had done it more than just once!
IV: This was an online course, but several students came together regularly to discuss course concepts, and to work together on problems, reflections and group projects.
RS: My “asynchronous” aspects were really just a result of the fact that my head was still very much in a synchronous mindset from the spring, yet I could not require attendance at the synchronous sessions. Therefore, the asynchronous side was not developed beyond the videos, and making sure that every class meeting (the whole-class portions) was recorded and posted. For each topic, there were ~30-40 minutes’ worth of videos broken up into several parts. I found myself very torn in the videos between wanting to keep them short, but also ensuring there was enough content for students who would literally not attend class at all.
IV: Due to the nature of an online class we had no
specific class “meeting times”, a consistent, easy to understand and
inviting structure was important. The course was housed in Moodle
and the course home featured an welcoming message about the course to the
students followed by a video where the professor walked the students through
both the requirements of the course, and navigating the course in Moodle.
Although there were not lectures, there were typical assignments built into the
course structure. The course relied on MyOpenMath for the assignments and
an OpenStax textbook. Assignments consisted of two parts: pages to read
and problems to solve. Students were encouraged to discuss homework
assignments with classmates in the Homework Discussion Forum in Moodle, but
they had to write and submit their own solution. My introductory video
can give you some insight as to how the course structure: Ileana's introductory video
I scheduled around six hours of zoom help sessions a week, at various times of the day, and recorded some of the help sessions.
I organized the course in six blocks:
- A Course Information: containing syllabus, tentative schedule, and course policies.
- Assignment Dropbox: containing submission boxes and all non-homework assessments, and the homework discussion forum
- Additional Resources: zoom recording of extra explanations, derivative tables, solutions to selected problems
- Module I: Derivatives
- Module II: Applications of the Derivative
- Module III: Integration
- Applications of Integration (optional for those who wanted to read something extra)
- Functions and Graphs (for those who wanted a review).
The modules contained reading, videos, and homework assignments.
What was your assessment structure?
KE: Last year, I converted this course from a traditional grading system to a mastery grading system. (I also wrote a series of blog posts on my website about this.) This assessment structure worked well with the sudden shutdown in the Spring because I was able to use new weekly module assignments I had created as the primary mechanism for demonstrating mastery as opposed to traditional in-class quizzes and exams. One issue I had with my previous version of my mastery system is that I felt the grain size of my objectives was too small as I had a total of 80 across the four units. In preparing for my summer class, the first thing I did was consolidate my objectives and ended up with a total of 25 instead of 80 without making any changes to the content.
In my new list of objectives, 20 corresponded to mathematical content and the remaining 5 were writing (“political context”) assignments in which students respond to and reflect on readings that explicitly connect the course content to political systems. (I have always included versions of these assignments in the course, regardless of grading system.) For the content objectives, I created a series of “mastery quizzes” and parallel “practice quizzes” in Moodle (our LMS). The practice quizzes were auto-graded so that students could immediately check their understanding, could be attempted an unlimited number of times, and to promote productive struggle, I had the system withhold the correct answer(s) of any (sub)questions that students answered incorrectly. The practice quizzes were optional, but were attempted by just about every student before each corresponding mastery quiz.
The mastery quizzes were set up on 1-hour timers from the time students opened the quiz and all questions were “essay” type, meaning that students had to type or upload their reasoning and any steps of calculations and that I had to manually grade them. Students were allowed to use any notes and resources during the quiz because I wouldn’t have a reliable way to observe them and I increasingly believe that the application and synthesis of course content is a more authentic form of assessment. There were generally two questions on a quiz (often multiple parts per question) and quizzes were scored on a 5-point scale, with a 4 or higher resulting in a “mastery check.” A score under 4 meant that the students could elect to retake the quiz by using one of their five digital “tokens” for the course. (For more information on “tokens”, please see the blog posts on my website.) To keep the final grading system simple, a final letter grade was determined based on the number of objectives a student had mastered.
RG: Assessment for the course included the following elements:
- Attendance: 5%
- Participation: 10% (this included in-class participation, break-out group work, as well as weekly discussion board posts, where students shared problems or I posed big-idea problems or questions. For example, one discussion board prompt asked students to examine an alternative approach to partial fraction decomposition in integration, and then to solve and post a problem from the homework using this approach).
- Homework: 15%
- Quizzes: 5% (We had 3 quizzes that students completed during class through MyLabMath; these typically lasted about 15 minutes)
- Exams: 40% (Two exams, each worth 20%; not proctored, with open notes. Students signed an honor statement on their exams stating that they would not use online assistance and would not give nor receive assistance from any other person)
- Final Exam: 25% (proctored,
using Respondus Lockdown Browser; students were provided with a note sheet
as reference)
JG: The course grade distribution is displayed below. Materials covered in the notes came with a set of online assignments on MyLab Math, carefully crafted to deepen understanding. As mentioned earlier, weekly quizzes consisting of True/False, Fill-in-the-blank, and open-ended questions were administered under a fixed time through Blackboard Ultra. Quizzes must be completed in one sitting and features which prevent backtracking and which only allow one question to appear at a time were used. In addition, students underwent two untimed midterm exams that ran largely unproctored besides the 4 students randomly chosen in advance to take it via WebEx. At the end, the course is capped with a timed cumulative final exam. Note that all exams were released and submitted via Blackboard.
RS: My assessment ended up being extremely traditional, even more so than it would usually be because I could not require a participation portion by mere class attendance, participation, and group work as I normally would. (Of course, there are many asynchronous ways to require and measure participation, but I was not able to develop those in time for the start of my course.) I therefore just had the following simple structure:
The exams, including the final, took place in real time via a Webex meeting with video feed required. I posted them on Piazza, students either printed them or simply wrote out the problems on scrap paper, and used a scanner app to scan and post them back to Piazza immediately after finishing. Before I added a given student’s name to the exam post, they were required to show me their entire workspace, including their hands and papers.
IV: My assessment structure was largely motivated by course goals. Most online classes are solitary and students work by themselves and never get to engage with one another. I wanted this course to be different. To address these goals, I included multiple activities and assessments that were substantially weighed in the course average:
- 25%: Homework (MyOpenMath)
- 20%: Active participation: Homework Discussion Posts (10%), Module Reflections (10%)
- 10%: Group or Individual Project
- 10%: Quizzes.
- 10% Midterm
- 15%: Final Exam
- 10% Essays
Aside from exams and homework, each week students received a participation grade based on their active engagement with the course - discussion forums, group work, etc. Each student had to write two essays related to calculus, diversity in mathematics, or their experience in the course, and a mathematical autobiography at the beginning of course. Each of the three units (Derivative, Applications of the Derivative, and Integration), were followed by Module Reflections where students in groups reflected on the big ideas of that module. Groups were selected at random or intentionally. My first assignment was a math autobiography students had to post where they shared their experience with mathematics, and presented their vision as to how this course may help their career. I made sure mine was posted too.
Students were expected to make 2 discussion posts in Moodle per week, throughout the semester, as part of their grade. Discussion posts could be either asking or answering a question. Since everyone had different numbers for the homework problems, students were asked to provide the original problem with your numbers in the discussion post. For the title of the forum students put the assignment name and question number. For example: Homework Derivatives of Polynomials #2. This way, all could easily find others who were working on the same problem. When posting a problem they were struggling with, students needed to show some of the steps they have tried (the more the better). If people were stuck I responded to any question posted in the Discussion Forums that has gone unanswered by other students for 24 hours. Students responded substantially to most posts but on occasion for students who were behind others with their posts, or for those who were way ahead others, there was less activity.
My course tried to address the goals (and meta-goals) below, and I attempted to balance the learning objectives and assessments to address the following goals:
- Knowledge Base in Mathematics - knowledge and understanding of the course content that includes, but not limited with functions and their properties, limits, continuity, differentiability, shortcuts to differentiation, implicit differentiation, linear approximation, using the derivatives in problems (velocity and slope, optimization and modeling, related rates), the integral ( definite integrals and the fundamental theorem of calculus). This was assessed by all the seven grading components of the course.
- Problem Solving and Critical Thinking: engage students in problem solving by investigating, forming conjectures, verifying, applying, evaluating, and communicating mathematical concepts and ideas.This goal was assessed through homework, exams, discussion posts, and module reflections.
- Mathematical Modeling and Technology: use technology to investigate, visualize, and justify the course content in a wise and proficient way. This goal was assessed in the midterm and student projects.
- Communication: communicate the material clearly in multiple ways including written, orally and visually. This goal was assessed through the homework discussion posts, module reflections, student projects and essays.
- Collaboration and Active Learning: apply the content in the course in interdisciplinary relations, to investigate, solve and interpret applications from physics, engineering, chemistry, biology and other disciplines; collaborate with their classmates in an effective and self sufficient-manner when engaging in mathematical practices. This goal was addressed through the homework discussion session, collaborative module reflections, and student projects.
Did you have any concerns about academic honesty, particularly in the entirely remote format?
KE: Given my mastery grading structure, my biggest concern was attempting to maintain the integrity of the online mastery quizzes. There were deadlines for these, and students could attempt them at any point with a 1-hour timer so that it would be completed in one sitting and hopefully mitigate opening the quiz to see the questions and then talking to other students. They were allowed and encouraged to use resources which I think helped to curtail collaboration but really it just seemed that I got lucky and not many of the students had existing friendships within the class. This is potentially one plus of running a class that wasn’t very interactive, as students didn’t get many chances to develop relationships with each other which could have led to working together on the mastery quizzes. Trust me I still would have preferred interactive classes but that point was worth making.
In the Fall HyFlex format I am leaning towards using the synchronous class times for mastery quizzes because no campus students would be much more likely to work together otherwise. I will still allow the use of resources and I believe these approaches will be the most equitable for the in-person and remote students.
RG: I had concerns around academic honesty when students completed our two unproctored exams. While students signed an honor statement, I found similarly unusual approaches to problem solving on some papers that were consistent with problem solutions found online. One one occasion, I did confront two students who both admitted to having copied from posted solutions online.
I also found that students performed very poorly on the proctored final exam in comparison to how they performed on the two open-notes, unproctored exams. It is difficult to tease this apart since notes were allowed, but I found it very discouraging. While I know that exams only give one slice of a student’s true understanding, I still questioned whether and how much students had learned in the course. I believe that more varied types of assessment might have better reflected student learning. Class activities, discussion posts, and break-out group work provided some information on student understanding, but I still felt that I did not have a clear enough picture that could explain the large gap in grades between unproctored exams and the proctored final.
JG: I certainly had concerns about academic honesty and this is the very reason why I adopted somewhat stringent testing conditions such as one-sitting completion format, only-one-question display at a time, and no backtracking for quizzes. For exams, I chose open-ended and mastery-oriented questions that require students to analyze, infer, and explain their reasoning. Unfortunately, all of these extra steps did nothing to deter a student from seeking to have their tutor do the work for them. In my case, the evidence was compelling: I saw some amazing displays of advanced mathematical notations in some of my students’ works. By the way, I have a bizarre anecdote about tutors coming your way.
RS: Yes, I most definitely did. I suspected academic dishonesty on homework and quizzes but didn’t feel particularly motivated to do anything about it since I expected that to come out in the wash on exams (which it did). More significantly, I had two blatant cases of cheating on the first exam, even with the live proctoring set-up with video. (So the lesson there is that live proctoring doesn’t necessarily prevent cheating but it makes it easier to catch!) These two cases were easy to spot for several reasons: (1) their eyes were not on their papers for significant amounts of time (one was constantly looking at the screen of his computer and the other was looking off at some object out of the frame); (2) the work they presented was clearly produced by either Wolfram Alpha or a live expert, given the jumps in reasoning and choice of notation; (3) I had gotten to know them via their participation during class, in particular during group work in the breakout sessions, and their perfect exam solutions were far from what they had been able to produce in those settings. For each of the two students I held a recorded one-on-one meeting in which I asked them to justify certain gaps in their answers, which they were not able to satisfactorily do. Neither one ultimately fought my decision. A positive is that one of the students was so scared to be kicked out of school that he stopped cheating on quizzes (or so it seemed from the sudden drop in his grades).
IV: Yes, I worried about the authenticity of the work done by students, so I incorporated various ways to assess students’ integrity:
- Quizzes and mastery quizzes required selected hand-written work to be uploaded in a submission dropbox in Moodle within a half hour of the assessment.
- Module reflections on the big ideas in the course required students to explain key ideas and their conceptual relevance, not just present answers.
- Homework posts required students to post solutions, explanations and solutions.
On two occasions, I confronted students with correct answers for lacking details in their handwritten work and when they could not explain the reasoning, I had to negotiate a lower score. I plan to use ProctorU for the fall semester, but I feel that using a variety of assessments has been helpful to provide me with a more comprehensive understanding of what the students learned.
KE: As an educator that takes pride in active learning, opportunities to discover and learn from mistakes, and inclusive classroom practices, I’m almost ashamed at the “lecture-style” that emerged as the norm for my class sessions. Upon reflection, this lack of interaction comes from no time being spent in the first class on setting norms. This is something I always think carefully about in physical classrooms and it wasn’t necessary in the Spring because I made this class asynchronous and my Calculus II class norms naturally transitioned into my synchronous remote classes without needing to be addressed.
One small adjustment I made along the way in my summer course that did help was to stop at purposeful locations in the slides and not continue until they asked me questions via the chat. In fact I even encouraged them to chat with me privately to make them feel more comfortable and remove the anxiety some students have about contributing in class. (A nice advantage to an online format!) I would make sure to acknowledge and answer all questions and I would never attach their names to the questions asked privately. I’m not sure how I’ll manage the online dynamic in my Fall HyFlex classes, but when I teach Statistics online in the 5-week Winter Session I know I will spend more time on class norms, interactive practices, active learning, and building class community.
Making personal connections with students is a key feature of my standard teaching practice and online formats definitely make that more challenging. I did my best to make connections through breakout rooms of 3-4 students and occasional individual office hour meetings to at the very least get the message across that I care about them as people and as learners despite the barriers. I also struggled with most cameras being off because it is difficult to gauge student learning in real time and going forward I plan to make the use of cameras a class norm (while allowing for exceptions in cases of privacy, connection speed, etc).
RG: Making connections with students. The two Calculus II courses that I taught during the first summer session each had 27 students. At UMass, we can only suggest that students keep their cameras on during our course meetings (respecting students’ privacy). Not seeing students’ faces while teaching was challenging for me; much of the success I feel that I have had in teaching has relied on forming meaningful connections with my students so that I can support their academic growth. Depending solely on hearing students’ voices felt very disconnected for me, and many times I could feel my own frustration in teaching my lessons because I was unable to see (or sometimes hear) my students. Some students resorted to the chat box in our platform, rather than responding verbally to the questions I asked. Overall, this posed a great challenge as I could not determine whether students were engaging with the course material.
Active student participation (lack of). I also found that students had difficulty working together in the break-out groups. When I visited the groups, students were primarily working individually without sharing ideas or even their final solutions on problems.
The same 5 students spoke in each class. There were a few students who were willing to have discussions in class, ask and answer questions, and participate within their small groups. Unfortunately, the same five to ten students spoke in class, while others stayed silent.
Assessment and proctoring. Because I was generally following the same structure as the face-to-face Calculus courses at UMass Lowell, assessment consisted of exams, quizzes, homework, attendance and participation. As mentioned above, I was able to incorporate some discovery-based activities, but these were few and far between. I also piloted a proctoring browser for our math department, but only on the final exam. Students performed much better on the first two non-proctored exams than on the final, as expected. There were some cheating instances on the first couple of exams, as some students used online sources to solve problems.
There was a one-week break between the two sessions. During this week, I had to really reflect on the course and the difficulties I faced teaching online. While I tried to be forgiving of myself given my lack of experience teaching fully online, I still felt like a failure, and knew that I would need to make some changes for the upcoming courses, both, to ensure student learning and success, and also to improve my own practice so that I might feel some level of success as a teacher.
Changes implemented during the second summer session:
Something I know is very important to me and my teaching style is to form connections with students. I feel strongly that when I learn more about my own students, I am better able to support their academic growth. I also believe that when students trust their instructor and understand the care ethic that the instructor brings to the classroom, they invest more deeply in their own learning. To get to know my students during the second summer session, I allowed ample time during the first class meeting for introductions. Pairs of students interviewed each other and presented their partner to the class, and then we switched partners to do the introductions once more, with the caveat that students had to share new information with their new partners. While this was time consuming, we were already laughing on the first day.
I also introduced the warm-up problems during the second summer session. These typically involved the examination of a visual or a mathematical statement, with time for posting a chat and releasing all chats at the same time. We then reviewed and synthesized one another’s ideas. I found this to be a great way to connect with each other at the beginning of class, through a learning structure that also seemed fun to the students.
Discussion board
After a disappointing first summer session with respect to student participation, I decided to try the discussion board in our learning system. I felt that writing within the discussion board would provide more silent students the opportunity to share their ideas in a safe environment. I created one prompt each week, and asked students to respond to the prompt and to two other students’ posts. This work contributed to the participation grade for the class. The types of prompts I used were varied so that students could be creative in some instances, or learn ideas that we would be unable to cover in class. For instance, one post was a cartoon of various amusement park rides and students were asked to caption the rides using the language of calculus. Another post involved students sharing their thoughts on a reading about the Calculus “wars” and to consider how mathematical discoveries are made over time. A third type of prompt was very open ended, asking students to post their work on a problem and ask questions of their peers around the specific problem shared.
Check-ins
Prior to teaching college, I taught high school mathematics for twelve years in reform-oriented schools (member schools of the Coalition of Essential Schools). Knowing students well was one of the guiding principles of this network of schools. During the first summer session, I found it difficult to get to know my students. This was in part due to the inability to “see” their faces in the virtual environment and my limited repertoire of skills for connecting with students in this new setting. I realized that I needed to be very purposeful if I was going to establish caring relationships with students that might encourage their own investment in learning. I decided to just check in with students at different times during the class, by simply asking them questions about their lives. For example, I checked in with my students by asking them to share something they were looking forward to in the next day, or week, or month; or I asked them about something interesting that happened to them that day; or I wondered about what might be causing anxiety for them, and just asked. I felt, especially in the circumstances in which we find ourselves, with a surging pandemic and a very uncertain future, that students might find comfort in connecting with a caring adult and with peers who share the ambiguity of this time and who are also experiencing a drastic shift in their educational journey. While I have no hard data that this was beneficial for students, all of the changes for session 2 did result in increased participation and in more student voices being heard during each class.
Changes in break-out group problem structure
I learned that it was more efficient to use fewer problems while students worked in small groups, and to provide scaffolded prompts for each question or problem students worked on. During the second session, I also became comfortable providing students with more time in their break-out groups (between 15-30 minutes). I often told students in advance to be ready to share their work and to ensure that everyone in their group understood concepts in focus. While students worked in groups, I visited each group at least once. I also found that an ideal group size was between 3 and 4 students: 2 students were too few since sometimes there were connectivity issues, and more than 4 made it difficult for each student to take on some role within the small group. Finally, I let students know that the only time I would call on them would be after break-out group sessions, having had the opportunity to share ideas and practice with their peers.
JG: Evidently, I could have done a lot more to make the course more interactive and to get the entire class engaged. I believe something simple as online group discussions would have made a difference. Instead, I let the circumstances of that time dictate my approach. The reality was that these students just came off an “unusual” semester (to put it mildly). Based on some preliminary information I had about the group, I concluded that students would benefit more from a “hands off” approach. To compensate for this lack of interaction, I issued targeted homework (more than I usually do) and provided rooms and options for executive functions. One-to-one interaction with the instructor was encouraged and timely, mastery-oriented, and individualized feedback was given. All was intended to encourage self-regulation and engagement. Unfortunately, the end result was not the desired one: more than half of the class chose not to even attend a single lecture.
One issue that I was not prepared for is having to deal directly with a student’s tutor, who happens to also be a college professor. To be clear, this student’s authorized accommodations allowed for this interaction to take place. Interestingly, the student in question did not pay attention to any notes and videos. Unfortunately, he was not the only one; most submitted works display notations (often advanced) that are nowhere mentioned in the course. Inconsistencies between homework and “sporadic” proctored assignments are flagrant. Most students missed on the extra credit questions sprinkled all throughout the notes until the last days of classes when they reached out for extra credits. A student’s parents even complain to the School that their kid was unfairly targeted for being asked to take the exam live on WebEx. Unbelievable!
Moving forward, I will add “scavenger hunt” activities that encourage students to read notes and watch videos. Online group discussions will be a must.
RS: With my course set-up, I felt some regret in knowing I was leaving behind some of the important IBL principles. I felt this the most in the videos, in which I was painfully aware of how much I was simply telling. It shouldn’t have come as a surprise to me that a significant portion of the students didn’t retain much from them, knowing what I know about active vs. passive learning. (And yet, somehow it still did…..) I do believe that videos could be done much more interactively, with prompts throughout that are even left unanswered, to be discussed as a group in class. That unfortunately felt far beyond my level of energy and time commitment to plan at the time.
I came away with two main lessons learned:
First, as I previously mentioned, I really learned that videos need some sort of overlying structure such as an accompanying worksheet, or checkpoints within the video itself. I see now that just as most students seem to need to be taught how to read math, they need to be taught how to watch it too. Many of them seemed to be watching the videos like any other youtube video, just watch it once all the way through and you’re done.
I also really learned that there will be remnants of spring semester still with them: since we were all in such a crazy state at that time, we know that standards were somewhat dropped (including widespread pass/fail options). I believe this also means many were able to cheat their way through to a passing grade, unfortunately. It seemed that a non-trivial portion of the class believed this class would be the same, with looser standards and easy opportunities to cheat. With this comes the fact that students are likely to be missing important concepts from a prerequisite class taken in the spring since they didn’t learn them that well if at all. As mundane as it sounds, I believe it will be important to emphasize in the fall that they are now taking a “real” course, with real expectations and a normal time commitment (which should be spelled out and quantified). We know that isn’t sufficient, but I am convinced it’s necessary.
I am in fact teaching Calc II again in the fall, online but fully synchronous, so I will immediately be able to try to address the issues I had this summer. One positive outcome is that I ended the course extremely motivated to make significant changes in order to do so. Below is a list of ideas I am currently considering. I’m presenting them to you in brainstorming, train-of-thought form because that is still how they currently exist in my mind! :)
- No videos (not because they can’t be done well, but because I don’t have to, and I know the time and effort required to do them the way I would like would be more of a commitment than I want to make)
- Student presentations, possibly via GoReact which my school has just adopted. Within these presentations, I am considering requiring at least one deliberate mistake that the audience must find (with bonus points to the presenter if no one finds it, to encourage more subtle mistakes?)
- Required discussion board posts (using Ileana’s ideas - see her portion of this blog post!)
- Possibly splitting up the topics into groups as discussed in one of the summer meetings. For example, one group dives deeply into integration by parts, another into trig substitution, etc., and then they all report back to each other (or groups are remixed so each one has at least one “expert” in each topic)
- More alternative-type questions: given a solution with gaps (maybe directly from Wolfram Alpha), fill in the gaps; or, given a detailed solution line by line but scrambled, put in the correct order (concerned about when….class time already feels so full - what will it replace?)
- Bring back a worksheet system I used a long time ago (a set of “challenge” problems each week to go with the more straightforward online homework). Might want to include some ways to manage cheating: make them include their scrap work, or do a few randomly assigned face-to-face interviews each week to discuss their work. (Or use these as the presentation problems, to avoid scanning and posting papers every week.)
IV:
Participation
One of the main goals of this course is helping students become better communicators of mathematics. Had the course been full semester, I would have had students present solutions to problems during the course of the semester, using a presentation rubric, and at least some of the presentations could carry a video component in which students explain their solutions. Limited to a five week course, I relied on the Homework Discussion Posts, exams, unit reflections, and course projects,to assess the way students communicated mathematics. About two thirds of the students responded well to this process. Some students however were a little more shy, and some got behind, so they felt embarrassed to post questions on problems that others had posted on way earlier. So for these students I had to scramble to find alternate assignments to earn participation points, and I felt I had unintentionally excluded them from part of the conversation. I intend to expand on the ways I assess participation in the future.
Housekeeping
Part of authenticating student work submitted on exams, students had to submit written work on the exams they completed. Submissions were to be in Moodle in an assignment dropbox. Despite clear explanation that written work had to be submitted get credit for the scores faded by MyOpenMath, some students had to be pestered to turn in this work.
Cheating
I am sure some of my students cheated at least some of the time. Due to other assessments, I had a pretty good idea of who may have been cheating, but it was difficult to prove.
Moving forward:
Many of the practices adopted in this first foray in Calculus I online instruction were successful.
- I would like to add one other way for students to actively be able to participate in the course content beside the Homework Forum, and maybe include an opportunity to extra credit, to promote more engagement. For example, posting a problem or a concept once a day, and giving credit to the first two or three students who submit a correct answer or solution may be one way to achieve this. While the homework session was great for students who were on track and motivated them to stay competitive, those who got more than a little behind could not have participated well in this discussion. Also, in an online environment, I could also grade participation in Zoom office hours, either my own or those at the tutoring center.
- To further authenticate student work, I plan to include Oral Exams or Student Interviews in my future online course. This is something I have done in face to face courses, but now realize I can adapt to online environment.
- I am still unsure how to proceed with group work for those groups who did not collaborate well. I have included rubric for student assessment of group work, and I usually select group task mindfully, the reality is that some groups are still not working well, especially in the online environment. There are still other questions around student voices and ensuring that all students are empowered to make contributions to course group assignments.
Do you have any personal anecdotes or final thoughts you’d like to share?
KE: So much of teaching, as in life, is learning through experiences and teaching my first entirely online class was no exception to that. I think I did my best to make proper adjustments for assessment and interacting with individual students but I know the class sessions could really use some improvement and I can now take these experiences and make purposeful changes to my next online class.
Remote teaching has been less rewarding for me on a personal level because of the lack of personal connections, classroom community, and seeing less of the “aha” moments. On the other side, many students that I’ve talked to also prefer the “live” experience which is encouraging for the future of education. Even when we ultimately get to the other side of the pandemic I think remote teaching and learning in some form is here to stay and that’s not necessarily a bad thing! For example, I actually MUCH prefer recording notes on my iPad as opposed to a chalkboard or whiteboard. For me it’s all about balance and experience so if the role of online courses eventually shifts back to winter/summer sessions to balance out the traditional semesters then I could see myself gaining more satisfaction from the online experience as I continue to learn and implement new digital practices.
RG: I remain unsatisfied with how my classes went during the summer. However, experiencing difficulty in teaching has enabled me to more carefully consider how I wanted my classes to look in the virtual environment. I want all of my students to engage with the material, to think about mathematics, to learn from me and from each other, to feel heard, and to feel supported in their own learning and in their growth. Some of the tools I implemented during the second summer session allowed me to work towards these goals. The discomfort I feel teaching online also inspires ideas, motivates me to reach out to colleagues for advice, and prompts me to research and implement approaches that might improve my virtual teaching practice. I know that the more open I am to learning, the better I will be able to meet the needs of my own students.
JG: Let’s start with this anecdote: one of the students who was in constant communication with me before and during the first four weeks of class decided to drop the course at the last minute; more specifically after the student found out that he would be taking the final live on Webex. I knew that he had access to a tutor the entire time, for he told me on day one. What makes the situation bizarre was that he had a course average well above 75% going into the final. When I reached out to discuss his decision, he was adamant that he made the right decision not to take the final and kindly asked me to respect his choice.
Here is my take on remote learning: it certainly motivates all of us who care about teaching excellence and for our students to explore the many resources available out there. I must say that before Covid I was stuck in my old world made of LaTeX, python, microsoft office, and textbooks. During the last four months, I have learned much more about technology than I did my entire life prior to that. To me, that is very exciting. Also, what Covid makes me realize is that I held many misconceptions about online learning; one of them has to do with the lack of interaction between teacher and students.Though I agree that there is no substitute for face-to-face learning, the online experience can be as gratifying if we invest the required amount of time, which is a lot more than usual.
In a nutshell, as soon as I find an effective way to combat the tutor dilemma, I would be more than fine.
RS: I echo some of the sentiments expressed by my co-bloggers, in that I have regrets about the course and feel some shame in having produced a course so lacking in inquiry and certainly lacking in targeting of meta-goals. With only about a week to prepare this course, I ended up making quick and easy decisions and couldn’t find it in me to do more. However, I also feel I gained a lot from the experience in that it really motivated me to make significant changes for the fall, and so I really dove into the summer lunch meetings with an eagerness and thirst for ideas. I don’t know that I necessarily would have been as motivated had I not had the reality check of the summer. I think that once I teach this course on ground again, it will be forever changed for the better.
I also feel less nervous about starting a remote course from scratch. Like Roser, I also opened the first class meeting with introductions: name, school, major, hometown or where they were currently, and a fun fact about themselves. While it was indeed time-consuming, I agree with Roser that it was worth it.
I’m confident now that I can build a rapport, and I realize that students will be hungry for it. It is possible to create an online camaraderie with students you’ve never met! (And who’ve never met each other.) The break-out sessions seemed to be an important factor in this, in particular, continuity of group-mates. After a few class sessions they started trading phone numbers and creating group chats, without any prompting from me. In fact, two of them decided to actually meet in person for ice cream the day after the final exam and sent me this picture! And invited me to join them the next time!
IV: One of the suggestions I have is that instructors should use multiple ways to assess student learning and encourage them to collaborate with each other to create meaning. While my course was not as inquiry oriented as I had hoped for, which I certainly regret, I realize that even in a purely synchronous context, it is possible to create a pull participatory and empowered environment. While I was not always able to entertain inquiry with all students, there were many moments where students proved that they were becoming mathematicians. Here is an example of a student query in a reflection where she is asking and her own questions and trying to answer them:
I was looking through the questions in the book for section 4.3 Maxima and Minima, and came across this question:
"Is it possible to have no absolute minimum or maximum for a function? If so, construct such a function. If not, explain why this is not possible."
It caught my eye at first because I thought it was funny how it's basically a trick question to make students forget about linear equations (the easiest answer), but I started thinking more and thought a more interesting question would be: do all even functions (with a domain of -infinity to +infinity) have an absolute max/min?
1) all odd functions have at least one zero. Even functions would always give an odd function derivative, and it needs to have a zero for there to be an absolute max or min. This one doesn't seem too hard to prove for polynomials. But I am not sure how to prove in general. I'm more so asking about the second criteria--
2) The range would need to be [y, inf) or (-inf, y], not (-inf, inf) since there needs to be a definite top or bottom. I know even functions are symmetric and both ends go either up or down in the same direction, but I don't know how I'd like ... prove that. And is there some way for the middle of a function to be at the opposite infinity? I can't really process how a function like that could be continuous, even if that's a real function that you can do things with. If the two ends of a function don't meet until infinity, would that have an asymptote (at x = 0, in the case of the x^2 - inf example)? I have lots of questions! Haha.
I suggest that if you plan to have students discuss mathematics outside class, the homework discussion forum is a great place to start. Because participation in the Homework Discussion Forum counted a lot in my grade, and because students saw me enter participation grades weekly, students participated in this forum. As a consequence, they learned that they can be vulnerable in an open forum in front of other classmates and still learn. This helped tremendously with both student understanding and engagement. In a reflection at the end of the semester, students described it as a “saving grace”, and one of their “most valuable resources.” In a class of 26 students, over the five weeks, there were over 70 different discussion threads (posts about different questions) and over 150 overall student responses dialoguing on these topics. I was struck by the quality of work that was being posted and the willingness of students to explain and do mathematics. I was also struck but the casual banter and the sense of community that some students developed.
Here is a typical Homework Discussion Post
Title: Section 3.10, problem 11
Student Question: I am not sure I understand what to do with logarithmic differentiation on this problem:
f(x) = some rational function with lots of factors to powers in numerator and denominator given as
screenshot.
Student Response:
The idea is to actually make the problem easier by separating everything using properties of logs. Like, for that first one, you could probably just do the product, quotient, and chain rules, but that would be a really long problem. Basically, multiplication can be separated by addition, division by subtraction, and exponents get moved to the front as a coefficient. And then at the end of the problem, you need to multiply your solution by the original equation.
Here's a picture of my work on these problems, hope it helps. (I forgot to write on the paper, the second one is using the product rule, with the x times the ln().) I honestly still feel pretty shaky on this myself since I'm not very familiar with logs overall. But once you figure out the log expansion part, it's not too much different from the other stuff we've been doing.
Although the student paper does contain an improper use of the equal sign, it does illuminate her understanding of the deeper ideas and her ability to explain them to someone else.
Development of Deeper Ideas
For those who worked well together, the three collaborative module reflections (derivative, applications of the derivative and integration) also contributed to student engagement and community, and helped them talk about the central ideas of the course often reminding each other of what they learned in the chapter, and even clarify some misconceptions during the conception. The two essays along with the project on Calculus and their area of study turned out to help students make connections between their life or experience and mathematics. Projects varied from calculus and economics, harmonic oscillators, drugs and dosages and Noyes Whitney Equation, Calculus of variations and geodesics, and more. I was impressed many students took these assignments seriously and did not just slap some stuff together last minute.
Collaboration, sense-making, and community can happen even in this completely remote asynchronous environment. My next steps are to more carefully orchestrate more inquiry oriented projects for my fall classes, and be more strategic about how I am asking students to collaborate, present and critique each other’s work.
Authors and Contact Information
Kyle Evans Roser Giné Jean Guillaume
kyle.evans@trincoll.edu Roser_Gine@uml.edu guillaumej@sacredheart.edu
Rachel Schwell Ileana Vasu
schwellrac@ccsu.edu ivasu@hcc.edu
