Eleven Teaching Ideas

• Learning Contract: A learning contract is a document that is written by the instructor and the student. The student writes down his or her goal for the course (e.g. my goal is to get a final mark of A). Then, both individuals detail t heir objectives for the course and make a list of the things they need to do. It’s much more formal than simply writing down "I will work hard". A learning contract increases accountability and provides both individuals with constant feedback and motivation. Students write down a detailed action plan, where each goal is specific, measurable, and realistically attainable. And the instructor does the same thing.

• Warmup Problem: as soon as a student walks into my class, he or she is greeted with a warmup problem on the board. Usually this problem is directly relevant to the day’s topic, but other times it is just a fun puzzle. The purpose of t he warmup is to have the students walk in, sit down, and immediately begin doing mathematics. Also, I start discussing the warmup problem exactly at the start of the class, so students are motivated to come to class several minutes early, rather than sev eral minutes late. This helps get the class off to a strong, focused start.

• Think-Pair-Share: in large classes, I often put up a problem on the board and ask my students to think about possible approaches to solving it. I ask them to think for a minute on their own, then pair up with someone sitting nex t to them, and then share their ideas with one another. By sharing their ideas, students become accountable to their partner, and hence are more likely to stay on task. In the past, when I asked students to think about something for a minute, som e used that as an opportunity to nap for sixty seconds. Now I get them to pair and share, to encourage group collaboration and ensure accountability.

• Cooperative Learning + Jigsaw: cooperative learning is an extension of the think-pair-share activity. The students are divided into groups of three or four, and they spend a large part of the class working on a list of problems. I’m m ainly there as a facilitator, to answer any major questions the students have and making sure everyone is contributing. However, the students are primarily learning from one other. They are doing the mathematics, rather than listening to me do the mathe matics for them. We then take up the solutions together, as a class. The jigsaw is similar to cooperative learning, except that each group is working on a different subset of the problems, which they become the "experts" in. I walk aro und, helping each group solve the problems that they have been assigned. Then the students break off to different groups, so that every group has an "expert" for each problem. The students completely teach each other the mathematics, with vir tually no interference from me. My role is that of a listener and supporter.

• Students Presenting Solutions: since mathematics is not a spectator sport, I encourage my students to come up to the front and present solutions on the board. Often this is unsuccessful at the beginning of a course, but after several cooperat ive learning and jigsaw activities, students become more comfortable with one another and lose their trepidation about coming to the front. Students who present solutions become empowered by their efforts.

• Students Inventing Problems: whether I am teaching high school students or undergraduates, I always make my students invent an original problem. My favourite problems are typed up and distributed to the students, with the promise that several of those original problems will appear on the final examination. Thus, the students develop their creativity by making up an original problem, as well as becoming empowered by having their own problems appear on their final exam. The students take owne rship of the material.

• Using Technology: I feel that technology can be used effectively to introduce students to important mathematical concepts and help students reinforce them. For example, I always do demonstrations in Geometer’s Sketchpad to illustrate key theo rems in Euclidean Geometry to my students. The dynamic nature of Sketchpad allows the students to see the existence of the Euler Line for infinitely many triangles, whereas on a blackboard I can only demonstrate it for one (poorly-drawn) triangle. Using technology provides my visual learners with a greater appreciation and understanding of the material.

• Formative Evaluations on Assignments: since I feel that process is more important than product, I do not like having assignments used strictly for evaluating students. When I give an assignment, I also allow my students the option of showing me the assignment two days before the due date, strictly as a formative evaluation. Thus, I go through the solutions with the student, commenting on what is good, and what needs to be improved.

I critique the student’s solution-writing skills and suggest ways to improve, and also check to see if their solutions are mathematically accurate. My students appreciate having an opportunity to learn from their mistakes without being penalized for t hem. The focus is on learning and understanding, and as an added bonus, the students get a far better mark from participating in this formative evaluation. Karl Dilcher does something very similar with his analysis class: after the students receive thei r marked assignments, they must re-submit any problem for which they did not get a perfect mark. Karl then remarks their second submission and that becomes their mark for that assignment. (So the first mark is thrown away).

• Using the Web: since every student has internet access, I put each of my courses on-line. Every single handout is typeset in LaTeX and put on the course webpage as a .pdf file. Thus, if a student misses a class for whatever reason, h e or she can just print off the handouts from that class. Solutions to all assignments and exams are typed up and put on the webpage as well. I also put small hints to assignment problems online, which the students find extremely convenient. I actually encourage my students not to take notes in class, because it is tremendously difficult to simultaneously write, listen, and try to understand the mathematics on the board. This is especially true for non-auditory learners. Instead, I want my stud ents to be thinking about potential solutions and contributing ideas than copying down notes from the blackboard, because students should be active participants rather than passive observers. In return, I ensure that the notes from every class (i.e., my s olutions combined with the contributions of the students) are put on my course web page later that day, and the students can download those pages at their own leisure and use them to study for tests.

• Soloman-Felder Index of Learning Styles (ILS) Test: the Soloman-Felder test is a simple 44-question test that measures the learning preferences of each student (e.g. visual vs. verbal, global vs. sequential). In small classes, I get my studen ts to complete this on-line test at the beginning of each course, so that I can understand the unique learning preferences of each student, and modify my teaching to accommodate it.

• The Two-Minute Drill: in the last two minutes of each class, my students answer three questions: one (simple) mathematical question based on the key ideas in that lesson, as well as two questions that help reveal how well the class went. Such feedback questions include "what was the most important thing you learned today?", or "what is the foremost question or concern you have about today’s material?", or "how were you keeping up with the content? Was I going too fas t or too slow, or was the pace just right?". The two minute drill achieves three objectives: first, it gives students a chance to reflect on the class and process what they have just learned. Secondly, students learn that their comments and opinion s are valued and important. And finally, I get regular feedback from my students which helps me improve my own teaching skills to give my students a more meaningful learning experience.