How technoogy influenced me to stop lecturing and start teaching

J. J Uhl

University of Illinois at Urbana-Champaign

1. Why I gave up long lectures

I can see that over the years my lecturing style and techniques evolved

to be remarkably similar to those of todays's good lecturers -

enthusiasm, humor, content. 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. Many insructors

ask the questions

->"Why won't my students talk to me?"

-> Why is class attendance so poor?

->Why won't students do their homework?

->Why do they preform so poorly on exams?"

And then they shrug it off saying to themselves:"If only I had taught

at Harvard tor a fancy private high school, 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

followup 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."

2. How effective use of technologu made it possible for me to stop

lecturing and start teaching

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 rectile claws, a

distinctive sonic output, etc.: I'll bet not. You probably show the 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 me. The basic ideas are laid

out in interactive Mathematica Notebooks in which new isssues 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 usethe 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 and Matrices,

Geometry&Mathematica) 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 a several

pointed mini-lectures addressing points the students should have picked

up during the last week.of the next 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 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 eqautions and linear algebra courses for

university credit supported by live mentors have formed at several

universities and colleges. 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."

3. Content issues

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. Some of the math of

1930 remains important but not all of it. Many mathematics courses

today are nearly indistinguishable from the courses I took from 1954 to

1962 (college and high school). 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 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, Descartes's rule of signs all had their time in the

sun but were demphasized in favor of more important topics. And then

then 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 numerically 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 materiall and technlogy makes it all accessible!

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 the initiatives of

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.

Are mathematics instructors in extreme denial?

With technology at the beginning of the learning process, we can get

students intrigued by seeing the math come to life before the strange

langauge and the proofs go on. For experience, i can say that many

students actually do want to understand why things work out the way

they do. But they don't want to be told this until they ask for it.

Answers to questions not yet asked stifle student learning. ut when

students see and feel mathematics coming to life on their computer

screens, they do get excited. Good use of technology is a great

motivator. Get the students' iterest and then get out of their way-

constantly supporting (but never puling) them as they advance

4. Some specific remarks:

Many instructors have the idea that before a student can touch a

computer or a calculator, the student must have mastered the process by

hand.

I say: "Why not before? " Is this a moral issue? Certainly this is not

an educational issue. Students (and research mathematicians) learn lots

from examples. Here is my version: Have the students to play with

graphics, first varying a and coming up with a conjecture of what the

influence of a is. Then ask them to explain why their response is

correct. Then ask them to explain how changing the value of a affects

the first and second derivatives in a certain way and how this is

reflected in corresponding plots. In this way the students engage

completely in Saunders Mac Lane's sequence for the undertsanding of

mathematics::

" intuition-trial-error-speculation-conjecture-proof." I don't care

how many buttons students press; if it helps them to think critically

and analytically, I'm all for it.

Many instructors think that their students cannot know something unless

it was covered specifically in class.

I say "Bull." Many instructors have fallen into the trap of trying to

think for their students - to get them ready for the test. This

approach is in no small way responsible for the failure of many math

classes. Students pick up programming without classes,; they can learn

a lot of math in the same way.

Many instructora try to deliver "math skills."

Rote skills should not be confused with mathematics. Mathematics is

something much bigger. In spite of what political groups in California

say, mathematical science is our attempt to undertand why things work

out the way they do. Let the computers -not the students become Saxon's

automatons. As MIT mathematician Gian-Carlo Rota

said,"<fontfamily><param>Helvetica</param>An axiomatic presenation of a

mathematical fact differs from the fact that is being presented as

medicine differs from food. It is true that this particular medicine is

necessary to keep mathematicians from self-delusions of the mind.

Nonetheless, understanding mathematics means being able to forget the

medicine and enjoy the food."

Let's use good doses of technology at the beginning of the learning

process to give our students a math appetite. Once hungry, they will

enjoy the food.