MY EVOLVING TEACHING
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ALAN FELZER - CAL POLY POMONA
Like most new instructors, I began my teaching career using those techniques that I found most useful as a student. What helped me the most in my Electrical Engineering classes was detailed problem set solutions and detailed notes. So for about my first twenty years of teaching
I didn't grade homework for correct answers because some students would then simply copy. I wanted the focus of the homework to be on learning.
- I put together and handed out detailed solutions to problem sets.
- And I wrote on the board detailed lecture notes for the students to copy. My goal was to include not just equations and graphs but also descriptions of what was being done, why it was being done and how it was being done. The goal was to tell the "story" of what was going on including where the equations came from and how they followed from one another. I was trying to avoid the problem I had as a student when instructors would make "brilliant" and "insightful" comments that I didn't understand well enough during the lecture to know what to write down.
My assessment of how things were going consisted of
My overall sense was that the students were doing ok. I felt that all the time I was spending revising my notes, writing problem sets and writing problem set solutions was time well spent.
- What I thought of my lectures as I gave them
- The questions the students asked
- How the students did on the homework
- How the students did on the exams
But there were clear indications that there was room for improvement. First of all lectures are invariably plagued by the fact that many - if not most students - are tuned out at any given time. This of course is obvious to anyone who has ever walked by the open door of a lecture and seen the looks on the students' faces. Even the Chaplin at Harvard - a very dynamic speaker - says he only has the attention at any given time of about one third of those "listening" to his sermons. In addition there was the problem that most students were doing the weekly problem sets at the last minute. And not looking at my solutions until just before the exams. Now, despite all this, there was learning going on - but only in spurts. And as a result many students weren't really understanding as much as I would have liked.
Be all this as it may I wasn't really looking to change my teaching. But then I heard a presentation on active learning by Clarence Stephens (now retired) from the math department at the State University of New York at Potsdam in which he described how he would assign proofs for the students to study at home and then discuss in class in groups - proofs he wanted the students to completely master. Stephens said that at the beginning of the course the weaker students not only contributed less to group discussions but also brought fewer questions. But as time went on and they got a better understanding of the material they asked more and more questions and got more and more into group discussions.
I was very impressed with Stephen's approach and decided to try it. My first attempt consisted of having the students read material at home and then bring questions on what they didn't understand to class for group discussion. It didn't work. For whatever reason the students did not bring in questions that resulted in stimulating eye opening group discussions. Instead they got frustrated and upset. They felt I wasn't teaching them.
This was a very frustrating time. I had a real dilemma. I was convinced that active learning was better than lecturing. But my teaching clearly wasn't working. For me - at least - active learning was easier said than done. And the students were staying away in droves. What came out of all this - after much gnashing of teeth - was the realization that for active learning to work for me I had to be the one asking the questions. At least the questions that were guiding the development of the basic course material.
I was very lucky at this time to come across the High School geometry book Discovering Geometry by Michael Serra. The key idea I got from this book was a way to develop the concepts and results of the class through investigations. The idea is to first ask the students what they think will happen in a given situation, then have them do the analysis and then have them compare their original conjecture with their analytical and/or graphical results and reconcile differences.
And so I started writing problem sets in the form of investigations. Their goal was to function as guided tours leading the students through the development of the results of the class and how these results could be used in applications like calculating the delay times of digital gates and the frequency responses of filters. Class time was then spent as follows -
Things definitely improved but there were still problems -
- Class would start with a problem on the board for the students to do as a group which I would then go over.
- Students would then get in groups to go over their investigations.
- I would assist groups that asked for help. But I made a point of not trying to listen in on groups. I felt it would be intimidating - I know I don't like being watched when I'm in a group.
REVISING THE INVESTIGATIONS
- Many of the questions in the first investigations were not very good - too open ended, too vague, not really relevant.
- Asking students to make conjectures on their homework didn't work as well as in class because the students would invariably do the analysis first and then go back and "make their conjectures".
- Students would relax too much after the group problem they did at the beginning of class.
The big challenge was to improve the quality of the investigations. Several presentations I went to during the summer of 1993 gave me some valuable insights. One was by Eric Mazur, a physics professor at Harvard. First of all he made the point that by hook or by crook most students will learn how to do the basic calculations needed for exams. It's not necessary to spend a lot of time on sample problems. What really requires development is concepts so the students can solve problems like finding how a circuit's response will change as various parameters are varied. I also learned from the Harvard Calculus consortium to make a point of not only asking analytical questions but also graphical questions and questions that ask the student to explain and describe concepts and results in words. I also learned from experience that it really helps when a class has a theme that can be related to a practical application.
Given all these observations I set about the never ending process of revising my investigations. At this point in time I have written investigations for the following six classes:
Here is an example illustrating how I write investigation problems. The basic idea is to lead the students through specific examples and then generalize on the results.
- ECE 109/129L - Introduction To Linear Resistor Circuits
- ECE 207/252L - Circuit Analysis I
- ECE 209/253L - Circuit Analysis II
- ECE 307 - Circuit Analysis III (really systems analysis)
- ECE 308 - Introduction To Digital Signal Processing
- ECE 315 - Introduction To Probability and Random Processes
I introduce Kirchhoff's Current Law with a problem illustrating the fact that equivalent positive charges enter nodes at the same rate they leave. I do this by giving them nodes with currents that have been measured correctly as follows
and also nodes in which the currents have been measured incorrectly like these
I then ask the students to draw pictures showing the flow of equivalent positive charge as follows
and describe what characterizes the correct measurements.
Now in addition to the quality of the investigations themselves the format of the class is also of great importance. At the present time my class organization has evolved to the following -
THE INEVITABLE TRADEOFFS
- Each class starts with about a two minute wakeup question for the students to do on their own. I go over the answer after the students have handed them in. I check all the papers but do not give grades on them The goal of the question is to make sure everybody has gotten the main idea of the investigation.
- I then give about a ten minute transition lecture with the goal of summarizing the main results of the day's investigation the students did for homework and how they're related to what we've been doing in previous investigations.
- The students then get in groups of three and four to go over the investigations they did for homework. The students use pencils to do their work on the investigations at home and blue pen for the work they do in class. I give the students a checklist for their investigations. I want them to make clear what each problem is and how they're getting their answers. I don't want just a bunch of equations.
- I interrupt the group work once or twice during the period with conceptual and computational nuts and bolts questions for the students to first work on alone in pencil and then discuss with their groups and correct in blue pen. I mark all the questions but do not give them grades. I got the idea of asking several shorter questions during the period - particularly conceptual questions - from Eric Mazur. This works much better than one longer question at the beginning - it really helps keep the students focused on getting through the investigation during the class.
- And finally I bring all the groups together for the last ten minutes to answer their questions from the investigation. One advantage of having the students work in groups is that when a student asks a question it implies that nobody in his group knows the answer - he's not the only one.
- Grades are given for the following: one investigation per week evaluated mostly on the quality of the writeup; quizzes given at the end of the second, fourth and eighth weeks; a midterm given at the end of the sixth week; a final; and a notebook of all a student's work with a grade based mainly on how conscientiously the student corrected mistakes.
Now there are always tradeoffs, of course, to whatever methods are used to teach classes. Some of the advantages of using investigations as outlined above are as follows:
But nothing comes for free. Investigations are a fair amount of work. They invariably take a long time to write, they invariably take the students a fair amount of time to do and then correct and they invariably take a fair amount of time to grade.
- Most students come to class with a reasonably good start on the day's material. They may not have all the answers but they typically have a pretty good idea of where they need help.
- Students are directly participating in the development of the class results.
- Students have ample opportunity to ask questions of their group and the instructor.
- The instructor gets detailed information on how well the students are understanding the material by how well they're doing the investigations. And so the instructor gets detailed information on what needs improving.
- Class time invariably goes quickly. And with a good mix of students the classes can be particularly lively and enjoyable.
- Another thing I've been doing in my circuit analysis classes ECE 109, ECE 207 and ECE 209 is follow each of the three class meetings with an hour of lab. I feel this has really helped reinforce the investigations.
Now the big question, of course, is how well do these investigations really work. Are they worth all the work. One thing I started doing this past year - with several other instructors - is to get an instructor not teaching the class to write one to three problems for our finals based on the course outline. I particularly like this scheme because of its simplicity and the fact that it's completely voluntary. Instructors can either use or not use the problems, they can reword them and they decide how much to count them on the final. Instructors do not get the problems until after the last lecture of the quarter. So far I've done this in ECE 109 and ECE 209. In both cases I would have liked my classes to have done better - but they certainly did good enough. And both experiences stimulated good ideas for revising my investigations.
Alan Felzer, January 2001