In this section, I describe my experience as an instructor at the University of British Columbia. After listing the courses I taught, I describe some instructional designs that I have worked with and conclude with a reflection on identifying student difficulties.
List of courses taught
My teaching experience as an instructor has allowed me to get a firm grasp of what it takes to impact the learning of students in introductory service courses. In contrast with my experience with extremely strong and motivated mathematics students as a teaching assistant at the EPFL in Switzerland; the students I taught in these courses had little interest in the discipline itself and very often (especially in MATH 110) a negative emotional to mathematics.
MATH 110 – Differential Calculus – University of British Columbia
This a 6 credit service course designed for students who do not pass the standard prerequisites for the other differential calculus courses offered. This two-term course is designed to allow students to strengthen their understanding of pre-calculus topics and expose them to the content of a standard calculus course.
I taught one of the three sections of this course in the 2010-2011 academic year (September to April); enrolment was 80 students.
MATH 200 – Calculus III – University of British Columbia
This is a 3 credit service course in multivariable calculus. I taught one of the six sections offered in Fall 2009 (September to December); enrolment was 68 students.
MATH 104 – Differential calculus with applications to commerce and social sciences – University of British Columbia
This is a 3 credit service course in differential calculus aimed for students in commerce, business and social sciences. I taught one of the eight sections offered in Fall 2008 (September to December); enrolment was 70 students.
List of course designs prepared for students
Having started with a traditional lecture style teaching design, I gradually took on to experiment with other designs in the “active learning” family. Throughout this experimentation, I have built experience with group work, the use of personal response systems (iClickers) and online interactions (wikis and WeBWorK).
MATH 110
For this course, I took on to experiment with several new designs. Given the feedback I obtained from my MATH 200 course, I implemented group work at the core of my design and chose to create outside-of-class interactions with a wiki. On top of these choices, the course was designed with two elements that were new to me: weekly workshops for students and the use of WeBWork. Having only three instructors for this course, we met on a weekly basis and heavily collaborated on the preparation of materials for our students.
Workshops
This course offered a 90 minutes workshop that students attended on a weekly basis. Instructors did not attend these workshops but we collaborated on the creation of all the activities (usually rotating the responsibility and discussing the prepared materials at our weekly meetings).
WeBWorK
The online homework system WeBWork (http://webwork.maa.org) was used for weekly assessment of student work. Using the large library of available problems, I created several additional homework assignments to complement the weekly assignments (created by the instructor in charge).
Interactive group work
During class, students were sitting in pre-established groups of 4 to 5. Class time was split between group work on worksheets that I designed, group or class discussions and targeted lectures. I used a personal response system (iClickers) to gather immediate feedback from students on various tasks or questions (on average 5 questions per hour).
Wiki
Outside of the classroom, a wiki was setup and used for two main purposes: an interactive course webpage and collaborative space for student work. Each week, students had a group-based homework assignment (aside from their written individual homework and individual WeBWorK assignments) that was to be completed on the wiki. The tasks varied from solving given problems to creating summary pages related to important pre-calculus topics to review; to presenting uses and applications of calculus in areas of interests to them.
MATH 200
Having practiced with the traditional lecture style in my first course, I was interested in experimenting new strategies in this course. The main decision was to use a personal response system (iClickers) which led to three design choices: enforced pre-reading, the use of worksheets in the classroom and group work.
iClickers
Students were all required to obtain a iClicker (already widely available on campus). Their use of this system was made part of their grade (mostly for their participation and not their performance).
Pre-reading
Students were requested to read parts of the textbook in advance of class. Guidelines and accompanying questions were offered. Additionally, each class started with a 10 minutes “warm-up” which consisted on using clicker questions to check on their understanding of the material they had read. This allowed to free class time by letting students begin their interaction with new definitions or concepts on their own and instead spend more time in class working on more complicated aspects.
Worksheets and group work
To engage students with the material and prepare them for target amounts of lectures, I designed worksheets that they were to engage in (at some times individually or with their neighbours).
MATH 104
For this first teaching experience, I practiced the traditional lecture style. I prepared notes on the weekly topics (as prescribed by the curriculum given by the instructor in charge), created pertinent examples and delivered this on the blackboard with some questions to the class.
Identifying student difficulties
Clickers
The use of personal response systems (iClicker) has had a profound impact on my classroom practice and my understanding of student difficulties. I know plan those questions throughout the class time in order to explore my student’s understanding and very often create questions on the spot out of my interaction with the students.
Here is an example that I have found transformative: While teaching MATH 110, I asked my students to identify the occurrence of a horizontal asymptote. We already had a discussion about the idea that a function can actually cross a horizontal asymptote, so I displayed some examples. One with a vertical asymptote instead a horizontal one; a classic horizontal asymptote (something like 1/x); a rational function crossing the horizontal asymptote (something like (x-1)/(x+2)^2); and a functions that has a horizontal asymptote and keeps intersecting that asymptote (something like sin(x)/x).
Students successfully identified the horizontal asymptotes except for the last function (sin(x)/x). Walking in this discussion, I was convinced that out of our previous chat about being allowed to cross asymptotes they would not get stopped by that example and so what was supposed to be a 2 minutes “checking-in” turning into a 10 minutes discussion. What I discovered is that students’ mental model of asymptotes had gone from “the functions can never touch the asymptote and goes really really close to it” had now shifted to “the function can cross the asymptote a few times, but eventually reaches a state where it doesn’t anymore and gets really really close to it”.
What was missing was a deeper understanding of what a limit is and through that work they saw the idea of “continually getting closer to” without having to “monotonously” getting closer to. This was a “aha” moment for many students and so much worth the time spent on this. It provided me with a more detailed understanding of the type of scaffolding that it takes to transform students’ understanding of horizontal asymptotes and confirmed my practice of remaining curious about my students’ understanding when using clicker questions.
Assessment examination
Homeworks, midterms and final exams provide a great opportunity to understand student difficulties. Using some light data analysis and looking for bottlenecks in those assessment often leads me to make different instructional choices (as well as reconsider the effectiveness of assessment questions themselves).
For example, while studying the mistakes my students made in a midterm question on related rates, I discovered that around a third of them were making mistakes in the implicit differentiation process by lack of clarity on the role of variables: in a cone-shaped tank type problem, the radius r is a function of time and so (d/dt)r = dr/dt; while in a cylinder-shaped tank, the radius is actually a constant and so (d/dt)r = 0. Since then, I have changed my approach with related rates problem and spend more time with students not only identifying variables but their different roles (as a function, as a constant). I don’t have hard evidence of improvement, but strong anecdotal evidence: through my collaboration with Warren Code on our paper “Teaching Methods Comparison in a Large Calculus Class” and through one-on-one tutoring with students.