Talking is not teaching.Lecturing to students is not the same as teaching them. A lecture is one tool at a teacher's disposal, but only that. When I teach, I use a range of tools to help students grasp a topic, but they must become actively involved in the learning process.
Several years ago, for example, I developed a two-semester Calculus sequence for students in science and engineering majors that included using a computer algebra system (CAS) in both lecture and laboratory sessions. Each semester I served almost a third of the students enrolled in that university's Calculus course (i.e., about 150) with the assistance of two graduate student TAs. In other terms, these TAs worked with other faculty on "traditional" forms of the course. The TAs and I were interviewed (separately) by a group of regular faculty members looking into the impact of technology-based courses.
When asked about any particular challenges I found in the course I had created, I said I had been surprised to find that, with most of the assignments submitted electronically, it took me twice as long -- nearly three weeks -- to learn everyone's name. I had not realized how much I use handwriting clues on homework papers to help me associate names with specific students. The panel was surprised that I learned the names of 150 students in a lecture. My strategy and success speed might vary according to whether the class had 25, 100, or 400 students , but I consider the task important in communicating with the students all term. This led us into a comparison of what took place in our various lectures.
I do present information there, but I also pause to ask questions and wait for student answers. My questions might come in parts, with each one building on the previous response, and they might require that students have time to work a while and talk things over with their neighbors before making a response to the entire class. I use such moments to discuss collaboration in learning: I expect each of them to be responsible for their own individual learning, the mere trading of answers is considered cheating, but open discussion and comparison of ideas can ease and improve the learning process.
I plan times when students are encouraged to ask me questions, and I make it clear that they are welcome to ask questions as needed. I try to answer their questions in a clear and respectful manner: my answers are intended to address the question and to serve as models for the behaviour I expect of them.
Teaching is a skill to be learned and practiced.My courses based on using a CAS, in some ways, had more in common with other undergraduate laboratory courses than they did with the typical math course, so I started to contact faculty in other departments who had experience in labs. When the math department then had me teach the use of a CAS in a new "laboratory" course that had the traditional calculus course as a co-requisite, my lab-savvy colleagues were very helpful. We compared notes regularly. Sometimes I felt lucky: for example, my labs did not involve toxic materials or hazardous equipment like those in science or engineering. Other times it was clear what I needed to learn: for example, I tried to avoid the view that an answer was either right or wrong and to build on the motivational approach in computer science that a program was either working or simply not working yet. Most times, however, we had many common concerns, such as how to effectively coordinate labs with lectures, how to help students learn to write meaningful lab reports, or how to get students to practice skills and maintain knowledge between lab sessions that met only once each week.
In the review panels for my technology-rich courses, my TAs were asked, "How does the time you spend TA-ing for Carol's class compare to what you spend on a traditional class?" I am told that a typical reply included the comment that, "We spend more time on Carol's classes, but that has nothing to do with the computers. She makes sure we do the teaching well, and she'd do that no matter what course style was being used. She spends as much time as needed to make sure we are able to meet our responsibilities, to help us learn ways to handle various situations, to check on our progress. She spends time helping TAs from other courses when they ask her! Even though it takes more time to be part of her projects, it's worth it because she helps you learn things that will be useful in any course you teach after that."
Set your sights both high and realistically.Some students will learn no matter what most of their teachers do; other students really will not learn certain topics in spite of everything we try to do. The vast majority of students, however, fall in between those two extremes, and will learn as much as we realistically expect of them.
I find it particularly rewarding, myself, to work with students in "entry level" college courses. At that time, they are making important decisions about whether or not to actually pursue careers that require mathematical logic, scientific reasoning, engineering analysis, etc. If we teach these courses with an emphasis on rote memorization, then we promote and retain students with sheer memorization abilities, and weed out the creative thinkers from the group that chooses to proceed. Is it any wonder, then, that later on we sometimes ask, "Why can't these students really think?"
I require my students, whenever possible, to look at problems both qualitatively and quantitatively, to consider both visual and algebraic approaches, to present their answers both formally, with equations, and informally, with words. They must not merely show their work and a final result, but also explain how much they believe in their result, whether they have been able to confirm it via another approach, or whether there appear to be discrepancies that they have been unable to resolve. I tell them that I consider those who make some mistakes and indicate that particular answers may be wrong, even if they are not sure exactly where they went astray, to be in much better shape than their classmates who may make a "less serious" mistake but have no clue there is anything wrong.
Requiring discussion seems to give me additional insight into the concepts that students have and have not really understood or mastered; to ease the process of starting a "Socratic dialogue" where students can be guided, first, to discover and resolve problems and, later, to build up new knowledge much more independently; and to offer students a way to learn how to develop and present sound, if informal, arguments before they move on to the formal notation and terminology that is so often a challenge.
If machines can compute the answers, what do students need to learn? I do not believe that any particular technology will resolve all our educational problems, but our students are going to find ways to use technology to serve their own purposes, whether or not we support its use. I do not fully understand why many members of the math community, in particular, remain doubtful of its impact. I do believe that we should help students to learn appropriate uses for the various tools at their disposal.
Teachers must be learners too.My advisor in graduate school believed that all faculty members should be required to take at least one course in a new area at least every other year, across a range of disciplines. He believed it was important to remember what it was like to be a student: learning something new and trying to meet course requirements while balancing them against life's other demands.
I have followed his advice in principle, though not always in its specific form. In addition to teaching my own classes, I have also had the privilege over the years of working with faculty, teaching assistants and student teachers, and with students in many disciplines and at many different levels. Through consulting, observing, advising, sharing ideas, and striving with them to improve the educational process from within, I continually reinforce my appreciation for the art and craft of teaching, my commitment to improving my own skills as an educator, and my desire to continue in this specific profession.