I could have benefited greatly from Steven Krantz's tips in 1962 when I taught my first class. In fact, I can see that over the years my lecturing style and techniques evolved to be remarkably similar to those Steven Krantz (SK) suggests. I was a very popular lecturer and recently won an MAA sectional award for distinguished teaching based in no small part on the lecture courses I gave at Illinois between 1968 and 1988. But for the last ten years, I have completely abandoned the long lecture method.
My last lecture effort was calculus in 1988. I thought I did a bang-up job, but the students did not respond with work anywhere near the level I was used to and have become used to after I gave up on introductory lectures - despite the fact that I had been giving the lectures largely in harmony with SK's recommendations.
Simply put, today's students do not get much out of long lectures, no matter how well they are constructed. The material comes too fast and does not sink in well. The students of the past responded by becoming quiet scribes. Today's students demand more action and accountability. That's why many students cut class and even when they come they often ask hostile questions such as "What's this stuff good for?" They do not read their texts. Some students even disrupt lectures. And as SK notes, many professors ask the questions:
And then they shrug it off saying to themselves: "If only I had taught at Harvard things would be different. I would have bright and eager students." or "Students these days are impossible."
It is the lecture method of teaching that is impossible - the method of teaching via long lectures is crumbling under its own weight. This is true not just in mathematics. Across the University of Illinois, there is a major controversy about whether professional notetakers may take notes and sell them to students who would rather not attend lectures. One of the first to note that the lecture system needed to be replaced was Ralph Boas in 1980: "As a means of instruction, lectures ought to have become obsolete when the printing press was invented. We had a second chance when the Xerox machine was invented, but we muffed it."
Many math instructors are trying to teach today's students using only yesterdays tools and approaches. And neither the instructors nor the students pleased with the results.
Introductory lectures are not (and probably never have been) a particularly effective vehicle for introducing students to new material. A few strategically timed and strategically placed short follow up lectures (sound bites) can be very effective. But the problem with introductory lectures is that they are full of words that have not yet taken on meaning and full of answers to questions not yet asked by the students. A further problem is that many lecturers fall into the trap of believing that their job is to think for the students. This effectively shunts the students to the sidelines - making them into mere scribes who verify in the homework and tests the math truths promulgated by the lecturer. As Bill Thurston put it:
"We go through the motions of saying for the record what the students 'ought' to learn while students grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material 'covered' in the course, and then grading the homework and tests on a scale that requires little understanding. We assume the problem is with students rather than communication: that the students either don't have what it takes, or else just don't care.
Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs."
In summary, I do not disagree with SK's approach to lectures, as he gives some great advice, which I used to follow as well. However, I do question the necessity, importance, and educational quality of lectures as a method for students to learn mathematics.
Another piece of wisdom from Ralph Boas: "Suppose you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractable claws, a distinctive sonic output, etc.: I'll bet not. You probably show the kid a lot of different cats saying 'kitty' each time until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience."
Today, my calculus, differential equations and linear algebra students get the experience they need through Mathematica-based courseware written by Bill Davis, Horacio Porta and myself. The basic ideas are laid out in interactive Mathematica Notebooks in which new issues arise visually through interactive computer graphics. With this courseware, limitless examples are possible almost instantly. If the student doesn't get the point right away, then the student can rerun with a new example of the student's own choosing. They can use the courseware to touch and see the math "kitty" as many times as they want to. They see for themselves what the issues are before the words go on and generalizations are made. One of our favorite techniques is to give a revealing plot and ask the students to write up a description of what they are seeing and to explain why they see it. In these courses, conceptual questions are the rule and students answer them. Contrast this with the typical student problems assigned in traditionally taught mathematics courses.
Here is the story behind the evolution of our courseware and the way it is used: In 1988-90, when Horacio Porta, Bill Davis and I were developing the original version of the computer-based course Calculus&Mathematica, Porta and I offered regular introductory lectures at Illinois. We noticed poor attendance and asked the students why. The students uniformly replied: "We don't need them. We can get what we need from the computer courseware when we need it. What we do want is a follow up discussion from time to time."
We followed their advice and have never seen the need to go back. Our students taught us how to teach. Over the years, almost all teaching of Calculus&Mathematica (and sister courses DiffEq&Mathematica, Geometry&Mathematica, and Matrices) has evolved to this model (sometimes known as Studio learning, a term coined by Joe Ecker for his Maple-based calculus course).
All the student problems are freshly written with the idea of engaging the student's interest. Assignments are made on Thursday. Students work on each assignment for one week. One day before the assignments is due, a classroom session is held to discuss what the week's work was all about. Students come armed with questions and if they don't fill up the whole hour, then the instructor gives several pointed mini-lectures addressing points that the students should have picked up during the last week. All other class meetings are in the computer lab with the instructor answering student questions as they arise - at the ultimate teachable moment.
This lab interaction between teacher and student (which is sometimes done via email) is very important. No longer are the students the professor's audience; students are the professor's apprentices.
The students' weekly assignments count for at least half their semester grades. Because there are no other lectures, the whole course consists of student work. In this model, it is what the students do that is important rather than what the professor says and how the professsor says it. Still the influence of the instructor is pervasive and the course ends up satisfying Gary Jensen's and Meyer Jerison's criteria: setting pace, teaching students to read, and fully engaging the student in the learning process.
This learning model cannot be accomplished with traditional printed texts and traditional lectures and lends itself rather well (but not perfectly) to Internet distance education. NetMath centers offering, via the internet calculus, differential equations, and linear algebra courses for university credit supported by live mentors have formed at several colleges and universities. Here is a reaction from a high school teacher who sponsors NetMath Calculus&Mathematica in Alaska: "Jessica has really enjoyed the course, and her father, a veteran of traditional calculus courses, is very impressed with the understanding of the mathematics that this method imparts. He has done all the problems and loved it. There have been some loud arguments--most of which she has won."
The trouble with the lecture system is compounded by the fact that our undergraduate courses, for the most part, have been frozen in the past and have become unable to adjust to modern demands. Undergraduate mathematics courses today are nearly indistinguishable from the undergraduate courses I took in 1960. Peter Lax put it this way in 1988: "The syllabus has remained stationary, and modern points of view, especially those having to do with the roles of applications and computing are poorly represented."
When I look over mathematics undergraduate courses during this century, I see a smooth evolution of new ideas and better mathematics through the period 1900-1960. Topics of limited interest such as haversines, common logarithms, Hoerner's method, latus recta, involutes, evolutes, and Descartes's rule of signs all had their time in the sun but were deemphasized in favor of more important topics. After which, content became frozen.
There is a whole list of 20th Century topics that have been, by and large, rejected in today's mathematics classroom. A short list: the error function, singular value decomposition for matrices, Unit step functions and their "derivatives," the Dirac delta functions in differential equations, using the computer to plot numeric solutions of differential equations, Fast Fourier Transforms, wavelets.
There is plenty of what Peter Lax calls "inert material" in most of our current mathematics courses. It's time to get rid of it and open the door to some fresh, important material.
My bet is that the underlying cause of this is our current fanaticism about having one-size-fits-all uniform texts chosen by central committees who often lack the expertise to make significant changes. They just go on tinkering with what was done the year before. It seems the central committees do not trust their own individual faculty members, so they shackle them with obsolete material.
Publishers respond in kind. And the publishers stay away from texts for modern courses because new, modern, original texts are unlikely to sell well. This is the reason that most well-selling traditional calculus texts are clones of George Thomas's calculus course of the 1950's.
The trend is for engineering, biology and science departments to begin teaching the mathematics their students need. Mechanical engineering departments are teaching lots of advanced calculus and differential equations. Electrical engineering departments are teaching lots of probability and complex variables. According to a source inside Stanford Computer Science, they have decided:
No wonder Sol Garfunkel and Gail S. Young wrote: "Our profession is in desperate trouble - immediate and present danger. The absolute numbers and the trends are clear. If something is not done soon, we will see mathematics department faculties decimated and an already dismal job market completely collapse. Simply put, we are losing our students."
Are mathematics professors and departments in extreme denial? I wish SK had dealt with these serious issues.
Here are some sources for those who want to reexamine their ideas about teaching of mathematics:
If you read this book, you will not forget the experience!
And: "Unfortunately competitive examinations often encourage [an educational] deception. The teachers must train their students to answer little fragmentary questions quite well, and they give them model answers that are often veritable masterpieces and that leave no room for criticism. To achieve this, the teachers isolate each question from the whole of mathematics and create for this question alone a perfect language without bothering about its relationships to other questions. Mathematics is no longer a monument but a heap."
Copyright © 2006 Calculus & Mathematica at UIUC
|
|