Katelyn Grayshan, a friend and former collaborator in our group at Notre Dame, introduced me to wordle. This program generates a ”word cloud” from a block of text, where each word is sized according to the frequency in which it is used. Below is a word cloud generated from my course evaluation feedback.
While at Notre Dame, I have taught Math 10260, and am currently teaching Math 10250. These two courses constitute a two semester sequence of calculus for arts and letters, architecture and business majors. These courses places an emphasis on conceptual learning rather than mastering technically challenging problems. They also emphasis applications to real world problems students will face in modern society. Students learn series as the solution to how to calculate interest paid over the lifetime of a home mortgage, rather than series as a mathematical construct with a possible application in a homework assignment.
I have also worked as a teaching assistant for 5 semesters, in classes ranging from a first semester calculus for engineering majors to linear algebra. These involve two to three 50 minute tutorials each semester in which I would answer questions and break students into groups to complete worksheets. Some semesters I also gave weekly quizzes with questions taken from previous exams.
There is no subject which is easy to learn. Mathematics is especially unforgiving, as it is the most difficult subject to work in with only partial knowledge. Thus instructors of mathematics more than other subjects must make learning engaging. Our purpose is to motivate and stimulate our students to learn.
I have spent some time watching more senor professors lecture, and have attended numerous seminars on teaching effectively. Through these experiences, the most important lesson I have learned is that there isn’t a right way to teach effectively. Effective teachers find a way which works best for themselves through trial and error and a concerted effort to reflect upon what they have accomplished, their failures and successes. Some professors have suggested the best way to keep a class focused, is to tell lots of jokes. Others insist that to teach effectively one must break a 50 minute lecture into two pieces, with a quiz or group activity in the middle. What I have noticed from all effective teachers is that they find a way to keep a class stimulated and engaged. I prefer asking questions, and breaking the monotony of lecture by creating a conversation with my class.
Most lectures I begin with a conversation in which we review the previous lecture. Rather than tell them what we did, I ask them to remind us what we have done; in particular, we discuss or write down any formulas we derived, or theorems we have proved. I think this helps focus the class and prepares students to learn mathematics. I then break up the act of lecture by asking my class questions. On hurried days, this may mean asking them to check my algebra, and waiting for a response. If I have more time, I will ask them more significant conceptual questions, or more complicated technical steps which they may have seen only once before. This method of lecturing also elicits interruptions to ask questions by my students, which I love.
For me, the easiest way to measure whether my students are learning, is if they are asking questions. When a student asks a question, I immediately know where the struggle to learn is, I can pinpoint whether or not the difficulty is technical or conceptual, and I can address the gap in knowledge appropriately. This is also why I think office hours are so important.
I also believe technology, if applied well, can aid tremendously in learning. As part of teaching the business calculus sequence, I have maintained course webpages and constructed a online quiz system using Sakai. The online quiz system consists of end of chapter quizzes due approximately 3 lectures after the last lecture on the material. These quizzes consist of approximately 10 questions each, with questions ranging from multiple choice to calculation of integrals or interpretation of graphs and diagrams. Using random number generators, we ensured that no two students would get the same question with the same numbers.
The purpose of the quizzes was to mirror the types of questions they had on their homework, in a format more closely related to an exam. They provided instantaneous feedback, and allowed for multiple submissions so students could learn from and correct their mistakes. I think the most valuable asset they provided as a learning tool, however, was that they provided an extra exposure to the material after lecture and homework before they took their exam.