In choosing the topic, I looked back at almost 40 years of teaching --I stretched that a bit by beginning with my first class as a teaching assistant-- and tried to think of the one thing I have done that has seemed to be the most effective. I think it's fair to label it collaborative learning (you can judge for youself later in my talk), although at the time I started doing it it certainly wouldn't have had such an impressive title.
I would like to make some general remarks first, to set the context. Everyone seems to agree that we are in the middle of a crisis in the teaching of mathematics, at all levels. A great deal has been written about the crisis, there have been numerous conferences and reports. At the collegiate level there are "reformed" courses, and "reformed" textbooks, and a lot of ideas have been put forward. The National Science Foundation has even awarded a lot of grants in the attempt to investigate and address the concern. To add some perspective, I'd like to read from a recent article [3] that quotes one report on the situation.
The author of the article, who is responding to this report, is William Mueller, who works for a software company called Mathsoft, which is the developer of Mathcad, which is one of the software packages used by our College of Engineering. The author goes on to say:
The report's conclusions are hardly a revelation, however, to the many mathematicians and educators whose impassioned critiques of popular mathematical textbooks and instructional methods, especially those used in calculus, have appeared in publications such as this one.
The article is from the American Mathematical Monthly, Feb 2001. The Monthly is the publication of the Mathematical Association of America, which is directed at college teaching more than at the research community.
At least, it seemed that way in 1890. That is the date on the report from the Department of Education (when it was called the Bureau of Education). The president of Cornell University was C. K. Adams, who delivered his remark to the annual meeting of the New England Association of Colleges and Preparatory Schools, in 1888.
Mueler proposes a simple question:
I couldn't resist the chance to have a bit of fun by quoting the article.
Seriously, is there really a crisis?
--Yes, I think so.
Has it lasted a whole century?
--Yes, probably so.
Will it soon be solved?
--I rather doubt it.
The problems begin at the elementary level,
with the way we train our teachers,
and the way our textbooks are written,
but include the environment at home,
and various troubling aspects of our society as a whole.
To expect our schools to turn things around by themselves
is asking a great deal.
I don't believe that the situation is nearly as desperate
at the university level,
and it seems appropriate that I should concentrate
my remarks on what I see in our NIU courses.
I think that the crisis in the teaching of mathematics might actually be called a crisis in the learning of mathematics. Learning mathematics has never been particularly easy, but it seems that the recent generations of students have been less inclined to do the necessary hard work. Of course, you can say, with good justification, that each generation believes that the succeeding ones aren't as dedicated to learning. But in my lifetime I've seen the advent first of television, then of computers and computer games, and now, of course, the Internet, with Napster, and many other diversions. Students just aren't as willing to spend their weekends doing Calculus. Some are going home to work, rather than forgetting everything at Amnesia.
Other things are happening, too, that I don't pretend to understand. Why does it seem to be socially acceptable to say something like "Oh, I was never any good in math." Maybe that is on par with throwing up your hands and saying "I've never been able to program my VCR". Sometimes I wonder if both are not just complaints about the increasing complexity of our lives. When you investigate, both of those statements usually turn out to be exaggerations. And here I can't avoid putting a lot of blame on our teaching. When students come to my office for help I often ask about their experiences, and almost always they say something like "I liked math and I was good at it until..." and then they tell me their story. It might be about a 5th grade teacher they didn't get along with, or it might be about changing schools and being placed in an honors class they weren't really read for. That is the stuff nightmares are made of-- being in a class where you believe everyone else but you knows what is going on, and the teacher is going to call on you next.
Is mathematics different from other disciplines? At least to some extent. Because mathematics has had such a long history of development, and because it builds on such a long chain of results, if anything goes wrong, at any point, it is really hard to recover. Learning math is sort of like setting out on a sailboat on Lake Michigan. I've never done it, but I've heard it can be very treacherous. Eveything seems to be going smoothly when of a sudden you realize that the weather crept up on you, and you're in the middle of an unexpected storm. And if you are swept overboard all you can do is watch that sail continue on over the horizon. The boat rarely comes about, and it may take some kind of a miracle to survive. I see that happen over and over again in individual courses, as well as over the 12 or so year of elementary, middle school, and high school.
How ever you look at it, there's a lot of hard work involved. The university could start by being honest with our students. It does take a lot of hard work. When I tell my classes that traditionally we expect two hours of work outside of class for every hour in class, and that studying is a full time job, I get some very interesting reactions. Particularly in light of these student opinions, we have to carefully guard against a "consumer" mentality. We have to remember that as a university we are less like McDonalds and more like a Driver's License Bureau. Our business is to issue diplomas that certify that our students have certain skills, and a certain level of education. At least here in Illinois we understand very well what happens when those licenses are just put up for sale.
In response to this crisis in teaching mathematics, many "solutions" have been proposed. It is hard to understand how there can be so many fads, and how there can be so many about faces in this long march. Maybe part of it is that to get a grant you almost certainly have to propose something radically different. I know I would never get NSF support to fund my solution: we just have to do everything better. I think we really know what to do, but it is not easy, and it is quite expensive. Maybe a good analogy is that we all know that we need to eat well and get lots of exercise. It just seems hard to do, and so we tend to try the latest dietary fad. It works for a while, until we lose interest. And of course, it wouldn't sell books to just say "eat right and exercise".
Where can we start? One thing we need to do is to look at the learning process itself. We need to look inward, not necessarily outward, at technology and "reform" methods of teaching. Student motivation is a crucial issue. I often hear that we need to do interesting applications to motivate student interest. One problem is that interesting applications are usually rather hard. But more basicaly, I think that the most important factor in motivation can be stated very succinctly: it is success in the task at hand. I see that in myself. If I try something that catches my interest (yes, we need to catch student interest) and I seem to have some success, I stay caught up in it. If not, I lose interest very quickly, and need some very strong outside influences to keep at it. I think that collaborative learning can have this important aspect. The method I'm going to describe directly contributes to success. Admittdly, it does so by starting small, like starting a snowball rolling downhill.
To play the role of cynic about my own presentation thus far, I guess what I've just done is to try to whittle away at your expectations. Now maybe I can claim to be successful if I can make just a small change in the classroom. I don need to come to the real point of the presentation. I'd like to talk very briefly about collaborative work, in general.
I've read several of Richard Felder's papers on teaching, and on cooperative learning, in particular. I think his work is excellent. I'd like to quote from his paper [2] "The future of engineering education II. Teaching methods that work"
One issue is whether the collaborative work will be done "in class" or "out of class". For me, "out of class" works best in upper level and graduate courses. In my department we already know how to do this--give some hard problem sets, (which might be called projects in other fields) and give the students a week or more to work on them. Our graduate students have always been good at working together (they work out of several large offices) and so in general this has worked without requiring very much organization. On several occasions, I've tried to organize work in class, but it hasn't been very effective. I think that perhaps the steps in the work are too big. Individuals need to do some background work before they can collaborate effectively. At that level, a mathematics course in generating genuinely new material all the time. So the collaborative learning I'm interested in focuses on classes at the first and second year level, algebra through calculus. The work can be done effectively in class because it can be given in relatively small increments.
I need to talk about the appropriate technology. In order to actually collaborate, there must be group work areas, where the work is easily visible to each member of the group. Actually, I want it to be visible to the entire class, as well. It must be possible for each member of the group to easily change the work--so it has to be highly interactive. Finally, mathematics has special problems with symbols, and typesetting equations, so there must be an effective way to write mathematics.
The poster for today's talk included a picture of students working at a computer. It points up the importance of technology. As a very pleasant surprise, we do have appropriate technology already available on campus--the traditional blackboard (and I don't mean Blackboard.com, for which I do have high hopes). If you think about it, I can have 5 or 6 groups of 3 students working at the board, and their work is clearly visible to everyone. It is interactive--I see students erasing each others work, and making corrections. I do the same. There isn't any problem with typsetting symbols, or handwriting recognition. And one of the biggest benefits is that on the scale of a blackboard, you get to walk around the room, rather than sitting at a desk.
You may be having the same reactions my students do, the first time I say that I'd like everyone to go to the board. The first reaction is "Come off it, that's high school." At a committee meeting on Tuesday, I was giving a preview to Jule Scarborough, from Engineering, and her first reaction was that she didn't like to be forced to be the center of attention for the class. I didn't either, so I've been using an approach that is very low key. I have everyone go to the board, so that there aren't any spectators, and I have everyone work on the same problem.
So here is the outline of what has worked for me.
The main goals of my approach are
(1) to actively involve the students (there should be no spectators), and
(2) to make sure that they are successful in class.
It is cooperative, supportive, and provides a positive atmosphere.
In my experience, each of the following points is crucial.
1. Everybody goes to the board at the same time.
2. Students work in groups of two or three.
3. All of the groups work on the same problem at the same time.
4. My last requirement is that I must check each problem before it can be erased.
Perhaps what I'm describing is guided collaboration. But I'm really thinking of myself as a collaborator. I try to make suggestions as a colleague. True, I don't hesitate to say something is incorrect. I also say to the class that my goal is to make sure the each student in the class understands every problem. So in this sense, the entire class is involved in the collaboration.
How did I arrive at this technique? It does differ in several important ways from techniques that I know some of my colleagues in the math dept have been using. One time honored technique is to single out one student to present a solution to the class. That is an effective technique, under the right circumstances, but it certainly isn't cooperative learning, and doesn't meet my other goals. It is also true that I hated to have to do that in high school. A lot of other people I talk to have the same feeling. Tuesday I was giving a short preview to a somebody in another department, and that was exactly the reaction I got.
Another technique, that doesn't put as much pressure on individuals, is to have teams go to the board and put up solutions to various homework problems. That is cooperative, but it doesn't accomplish my goal of immediate learning--students copy down the solution and (supposedly) go home and try to understand it. That's one of my rules--I don't let students copy down a solution. That is just an avoidance mechanism, wanting to postpone understanding it, and that is not terribly likely to happen. So if there is a student solutions manual, it helps to do problems on the board that are in the solutions manual, so that I can just tell students to go look up the solution later, if they can't remember how to solve the problem.
How did I get started, and when have I used this effectively? Before coming to NIU, I taught for two years at the church college at which I did my undergraduate work. The first semester I had a calculus class with 75 students, which was scheduled to have discussion sections (15 students each) on Thursday. The problem was that there were no teaching assistants, and so the help sessions were part of my assignment--five hours in a row. Even fresh out of grad school I couldn't handle it. My voice would be gone by the end of the day, so I decided that as a matter of survival I had to find something else to do. I didn't like to be put on the spot myself, so I didn't want to send students to the board one by one, and that's why I thought it was easiest to just have everyone go to the board. There wasn't enough room for 15 students to work individually, so I had them work in groups of 2 or 3. I actually found that I could introduce new material by giving them a problem for which they didn't have the requisite technique, letting them work for a while, and then suggesting the new technique they needed. Once again, that works when it is only a minor variation that gets introduced.
During my tenure at NIU, I've taught in the CHANCE program on a number of occasions. Most recently I taught in the early 90's, when our class size was 18 to 20 students. Working at the board proved to be very effective. I think it helped the success rates, and the retention, in Math 108 and Math 109. The first few time, students were reluctant to go to the board. But when they saw that it was a supportive atmosphere they soon started asking if they could go to the board. I tried to plan my syllabus so that I could have students go to the board for most of one class period per week. If I thought I could manage it, I would try to spend the last 20 minutes at the board during one or two more classes each week.
This is the place to mention cost, because my technique only worked in the CHANCE classes due to extra help. We have a Supplemental Instruction program in place, so I had an SI leader to help with the groups. Much of the time I also had a prospective high school teacher who could also help. That meant that we had a ratio of 1 instructor (leader) to 2 working groups. At that level, that was probably the right ratio. On average, that meant each group was helped 50% of the time. At about the same time, I taught Honors Calculus, and although my class was too large in Calc I, by the time we did Calc II I had 9 students. They were capable of working more independently, so the ratio of 1 instructor to 3 working groups was effective. I don't think it can be done with a higher ratio. To scale this up to larger sections would require a lot of help, and some pretty big blackboards. But a combination of a faculty member and some upper level undergraduates could be used very effectively in classes of size 20 to 30.
My final comment about the CHANCE classes is that I had to make sure the door was closed. There was a lot of activity and some very intense involvement. I was worried that some of my colleagues would walk by and see the chaos. But out of that activity came some real learning. I didn't think I needed to do any critical evaluation. It was so obvious that the students were involved and were changing their attitudes toward mathematics and toward their ability right before my eyes.
I think this method can replicate, to a high degree, the one-on-one tutorials I give during office hours. It is a particularly effective way to get students back on board if they have had an unfortunate experience somewhere along the line. It produces success, albeit small, one step at a time, but it does lead to higher motivation and involvement.
I'm afraid I've spent a long time saying some obvious things. But then, at the beginning, I only offered to tell my own story. I don't know how to scale this up to large classes, and I'm not sure how to translate it to other disciplines. I think I see in the technique elements of what I would expect to see in the lab in Physics or Chemistry or Biology or Engineering. Although the work occurs in groups, the guidance seems to me to be like what I think happens in a studio art class. I hope that I can stimulate some discussion that goes across disciplines.
In closing, I'd like to Bob Wheeler and the Provost' office, and Murali Krishnamurthi and the Faculty Development and Instructional Design Center. The support that they provide for teaching is essential to the university.
[2] Felder, R.M., D.R. Woods, J.E. Stice, and A. Rugarcia, The future of engineering education II. Teaching methods that work", Chem. Engr. Education 34 (2000) 26-39.
[3] Mueller, William, Reform Now, Before It's Too Late, Amer. Math. Monthly 108 (2001) 126-142.