The following document was endorsed by the Undergraduate Committee of the Department of Mathematics at The Ohio State University on May 14, 1996.

Changing Teaching In The Math Department

Teaching in departments of Mathematics throughout the country is facing hard times. Faculty are unhappy with the students in classes at all levels, and students are openly critical of their courses and teachers.

There are probably as many reasons given for the current situation as there are people involved in the discussions. They range from the low level of preparedness of students, through the ineffectiveness of individual teachers, to criticism of the techniques and methods, and even questions about the dedication of everybody involved: teachers and students. Without doubt, there is some truth in each of the explanations presented. It is time to try to change the situation, both the perception and the reality. As we discuss each of the causes, we should cast our discussions in the context of possible remedy, not blame.

In this report, we suggest a beginning plan for action, to be implemented immediately. The steps we believe are necessary are:

  1. Return class sizes to manageable numbers. Use the smaller classes to implement different teaching methods.
  2. Continue and augment special teaching initiatives currently going on in the department. This should include assessment of the value of such efforts.
  3. Re-examine and articulate clearly the purpose and goals of each of the undergraduate math offerings. Eliminate certain courses if appropriate, introduce new ones if needed, and clarify the foci, goals, and desired outcomes of all courses.
  4. Provide continuing exposure and training to all teachers, faculty, TA's, lecturers, casuals and teaching aides, related to teaching styles and methods. This should include fundamental ideas about how people learn.
  5. Augment the teaching staff. This includes new hiring of faculty, enhanced TA recruitment, and the employment of new resources for teaching, including a cadre of trained undergraduate teaching aides.

The teaching staff in the Department of Mathematics is dedicated to teaching and learning. Most members of the faculty try to do an effective job, and put high value on teaching. Virtually everyone is frustrated by perceived weaknesses in their audience and by the results of their efforts. A large number of faculty is skeptical that changing what we do can have a reasonable favorable outcome. Many others believe that we can change the outcome by doing some things differently. Most people are afraid that Òthe institution' will decree changes in our teaching that will have deleterious effects on our overall professional activity. We hope, in this plan, to begin to address these issues, and to initiate discussions which move us toward change which resonates with our total professional mission.

I. Understanding the Problem

One opinion that is universally held by college and university teachers of mathematics is that the students who enter our classes directly from high school are under-prepared, and that the situation is getting worse. Should we question this conclusion and launch into a grand study to verify or negate it? Probably not. The question we must ask is whether or not anything we do can overcome the weakness. A common response is that there is nothing we can do. One point of view is that we should only concentrate on that small number of students who come to us with appropriate skills and attitudes toward mathematics. The book, ÒHow to Teach Mathematics', by Steven Krantz, and published by the AMS, is dedicated to this point of view. This is, in fact, quite a popular and historic view in our department. We have for many years provided special programs and special treatment for mathematically talented young people. This activity must continue, of course. However, we might also ask what we can do to change the effect of poor preparation on the majority of, or at least large numbers of, incoming and continuing students. Some optimists believe we can change things substantially.

Students in courses beyond the very beginning courses demonstrate the same weaknesses as beginning students. People complain that students entering junior level classes can't remember simple facts from earlier courses, and that they aren't able to reason. (These are separate issues. That's part of what we should come to understand.) Whose responsibility is that weakness? We are the people who passed the students through the elementary courses and into the subsequent ones. At some point we must begin to understand that and accept responsibility.

Central questions remain: What does learning mathematics mean? How can we know that a student has learned the mathematics we want them to know? The prevalent view about the second question seems to be that knowledge and facility with calculational facts is what we should seek and test. The evidence belies this belief. High school students are tested extensively on computational skills. They pass. When they come to us, we test them again on those same skills, and they are placed into our courses according to their performance on these tests. We test students in our beginning and more advanced courses, and they pass. However, they continue to lack both the skills and basic understanding we hope for when we meet them in our classrooms. Is our basic premise wrong? Apparently it is.

This conclusion isn't radical. It has been espoused by teachers of mathematics through the ages. In our lifetime, one of the great teachers of mathematics, George Polya, told us about it in his book, ÒHow To Solve It'. He tells us that people learn mathematics by doing mathematics, and by wrestling with the concepts as well as the procedures. In other words, he tells us that students should learn mathematics much in the same way we professional mathematicians do as we try to do new mathematics. After all, for most of our students, most mathematics is new.

What does this have to do with the problems at hand? It should influence how we teach, what we expect, and how we determine success or failure.

II. Changing The Ways We Teach: The Need

We mathematicians tend to believe that most of our responsibility in teaching comes down to explaining the truth in well prepared lectures, and then expecting the students to assimilate that truth. If that were the way it worked, we would be singularly successful in this endeavor, because we do that well. People who become mathematicians are very good at learning under that model. We tend to forget the process we went through to gain the ability we expect of our students, and we fail to remember that most of our students have neither our ability nor our motivation. We also misunderstand that some of the abilities we expect students to have aren't there because our whole culture has placed such low value on individual responsibility and critical thinking.

Many of our lower division courses are taught according to rigid syllabi (which consist of lists of topics as defined by textbook authors) in large lectures with attendant recitation sections. The lecturer covers (and uncovers) topics, and the recitation instructor tends to do little more than solve homework problems for the students. The situation isn't much different in small classes. We expect the students to do the homework problems and to be able to reproduce solutions to homework problems on quizzes and exams. What is missing is the real involvement of students in the process. (See Polya) We expect too little of our students, and we get what we expect. We don't ask students to understand a problem in the way Polya describes. We present the students with templates, expect them only to manipulate the templates, and then register great surprise when they can do no more than that.

Most of the changes people suggest for improving learning of mathematics center around placing more responsibility on the student. The changes show up as changes in the types of problems students work on and in the type of solution and explanation students are expected to produce.

These changes, to be effective, cannot occur within the current structure for teaching in our department. There are models for alternative teaching methods which can work. Some of them are being tried in our own department. Some are very radical, and others are rather mild from the point of view of the teacher. Most require further attention and additional resources.

Teaching will remain ineffective and impersonal with current class sizes and current teaching resources. The first changes have to address class size, both in individually taught sections and in lecture format courses. This has immediate personnel implications. We must have more faculty. We must be very careful in how we increase the teaching staff. Some believe that the increase should come in the form of working mathematicians who are able to, and who will, teach well. Others will propose forming a sub department, or even a separate department devoted to teaching lower division courses. Begin serious discussions on these questions immediately.

III. Changing the Way We Teach: Some Models

The basic model many people believe is effective is this: Students spend some portion of their learning time working together on serious problems in an environment which provides challenges, support, and direction. This can happen in many different ways.

The most radical approach has the students spending all of their class time in working groups, grappling with problems, seeking help from faculty, TA's and trained assistants. Lectures, except for occasional highlighting of material, disappear completely. Students spend their time working together in groups, and the units which deal with the teachers are, for the most part, groups rather than individual students. The environment is staffed by learning aides and tutors for a large amount of time each day, for example, from early morning until late in the evening. Students are encouraged to use the room and the assistants during all hours. This model exists in some computer based courses here at Ohio State, and in dramatically different courses at other institutions.

At the other end of this spectrum, the same sort of room is provided, and the work students do in this room becomes the center of the course rather than the place students go to get people to do their homework for them. Challenging problems are provided, assistance is available, and students are expected to use the room for their primary contact with the subject. In a lecture based course, the role of the lecture itself changes from the central driving force for the course to the major support resource for the activity in the learning community. The most visible example of this model was developed by Uri Treisman at U Cal, Berkeley, and has been exported to many other institutions. The faculty member's role in this is to supervise the activity in the learning community, set the direction for the work there, and be certain that the lectures are pertinent to the learning going on in the working sessions. An experiment based on this model was proposed for the Math 148 course here.

A new phrase crept into the previous paragraph: learning community. This is a euphemism for the set of all participants working toward students' learning in a specific course or collection of courses. In typical teaching situations, that community consists of the students taking the course, and the teacher dispensing the truth and checking on the progress of students. In the examples cited above, the learning community consists of the faculty, TA's, and trained assistants serving as peer mentors for the students in the course. Additionally, though, students in the course are strongly encouraged to help each other. Students with particular talents are expected to share their insights. If such a learning community has a physical home, perhaps a room like CH 240, students from different courses will come to work, and can receive help from all of the people above, as well as from students from other courses who use the same room.

In yet another model, a faculty member teaching an individual section of a course can decide that employing some of the ideas of cooperative group learning, and Socratic methods, even as described by Polya, would contribute favorably to the students' learning. This can be done in the regular classroom, and/or in additional working labs as described above. In the latter case, the faculty member poses interesting problems for students to attack together, and in working sessions, students interact with each other, with TA's and trained undergraduate assistants in their quest for learning.

The common feature of all of these is that students are expected to work together with their peers, other students, and to get help and direction from TA's, faculty, and trained mentors.

We are clever enough to design and explore other models as well. It is clear that different faculty members bring different talents and expectations to their classes. Everyone must be given the opportunity to employ whatever teaching methods are most appropriate for them. We only ask that people are given the opportunity to do better.

IV. Necessary resources

Ken Wilson and Bennett Daviss, in their book Redesigning Education, support the fundamental ideas of learning communities outlined in the previous section. In their learning communities, though, they advocate the use of frequently overlooked and underutilized learning resources. Among them are the peer groups of students, the expert peers, the trained student assistants, and many others. The 'people' resources are available. What is needed is financial support from the institution, and a willingness from the faculty to try.

First of all, we must have enough faculty ready to move toward change. Below are some concrete proposals for courses run in individual sections and for courses to run in a lecture/learning community model.

Second, we introduce a new kind of staff member, the expert undergraduate teaching aide. In the learning community, this teaching aide acts as mentor and tutor, and can contribute to the evaluation of the work of students. Their presence in the learning community is extremely valuable, and very inexpensive when compared to the cost of the time of TA's and faculty. In addition, they have proven to be highly effective in courses here, in other mathematics departments, and even here at Ohio State in other departments, notably English. An ideal situation would have one such assistant for each section of a math course. (Their actual work assignment might be connected with hours spent in a particular learning community rather than with a specific section, though.)

Learning communities need homes. That means rooms. For evening hours, this shouldn't present insurmountable obstacles, since classrooms can be used for these groups. In fact, students from different courses can use the same sites, and that should be built into any such plan. The difference from our existing tutor rooms is that we expect heavier use, and more carefully directed activity in those rooms. Rooms like our current tutor rooms can work quite well. An ideal situation would have several more such rooms open and staffed at all open hours. Students will come to these rooms for directed study sessions which replace typical recitation meetings, and for directed help. The primary difference between current tutor rooms and these rooms will be in what goes on there. Since the level of problems will have changed from memorizing formulas and templates to understanding and analyzing the mathematics at hand, there will be very little problem solving done for the students by the staff in the room.

One problem with all of the change proposed is that it requires a deep commitment from several groups of people. Well designed teaching methods will require accommodation for the valuable time of faculty and graduate students. Also, someone has to organize and coordinate these activities. There are a few people in the department who do that sort of thing now in special courses. We are proposing to extend the same sort of development and coordination to a much broader slate of courses and activities. We should provide the opportunity for such effort. This certainly means providing time and appropriate rewards (see below) for such people, and may, in fact, mean hiring some new faculty members whose primary interest has turned to education.

V. Training

The addition of a new kind of resource, the undergraduate aide, into the teaching staff demands a new element of training. These people need to be given clear direction related to effective interaction with students, how to interact with students who ask questions you can't answer, how to help students find their own ways to solutions of problems, and even how to make working together effective. In fact, we should (and, in fact, do) offer some such training to our incoming graduate students. In some instances, regular faculty members would also like to learn more about how to use some of these methods.

We propose the design and implementation of training programs for the aides, for TA's, and for interested faculty. The training should occur each summer, and/or just before classes begin in the autumn. Further, there should be continuing training seminars throughout the year in which participants review what is and isn't working, and discuss particular problems which arise in teaching in different ways. For example, who hasn't encountered a student who really doesn't understand what simple trig identities are about (even if you ignore the fact that they don't remember the formulas)? Such a difficulty encountered in an upper division course might well influence changes in lower level courses.

The department has several experts in this sort of activity. Among them is our recently retired colleague, Tom Ralley. We should bring him out of retirement and charge him with organizing these training activities.

VI. Faculty Support and Rewards

We move to some of the most difficult issues which come with a serious commitment to change in teaching. How do faculty members get rewarded for such activity? This needs much deeper, thoughtful discussion. As a starting point, though, here are some reflections gathered from people across the country who have made serious commitment to change in teaching mathematics.

Of utmost importance, even more important that an overt award system, is the assurance that faculty who move into this arena and do effective work should not be penalized for doing this work. A common experience is that salaries suffer. We ignore the far more difficult questions of the role of this activity in promotion and tenure decisions.

The second item of deepest concern is departmental and institutional support for the activity. For example, one of our colleagues is currently reconsidering doing an experiment employing learning communities in a 100 level course because of technical difficulties in arranging room use, computer use, and training of attendant help. That should never happen. We expect the administration of the department, college and university to provide this technical support. Ken Wilson would undoubtedly list the department, college and university administration as components of the under-utilized resources. Another colleague who has been running a successful program for six years finds himself, for the seventh year in a row, in the process of writing proposals to the university, the federal government, and private agencies for continuing support of the courses. Faculty members should be writing and assessing materials, reviewing and assessing the teaching, and rewriting materials for the remote learning program, not spending most of their time arguing for funding and other resources.

VII. Evaluation and Assessment: Setting Goals and Expectations

We must clarify what we really want our students to learn. The way most of us do this is to write down lists of formulas and techniques we consider essential for success in subsequent courses. There is good reason to believe that current goals fall far short of the mark, and that students can satisfy such goals without any understanding of truly basic mathematical concepts underlying those facts. As a frightening example, one of our colleagues encountered the following in a calculus course. Students were asked to measure the area of a square one of whose edges was the line from a point on the x-axis to a point on a curve y = f(x) directly above that point. The students, of course, could say how to find the area of a square, they simply didn't know how long that line would be. It's not uncommon.

We have all given tests on Friday on which students receive passing, even good, grades, only to find that the entire class is unable to use the facts from that test the following week. What many of us don't know is that this is a natural phenomenon related to the way the brain functions. In other words, setting routine memorizable tasks as the goals for specific courses will of necessity produce such results. This alone should cause us to reassess the ways we set goals for student performance.

Evaluating student performance and understanding is also hard. We have been raised in a culture which places virtually all of the burden on individual performance on examinations. It is interesting that we don't measure each other that way, isn't it? In fact, whose knowledge is measured that way outside school? More and more, people are trying to measure students' performance based on a more global view of their work, including homework, projects which demand clear and concise discussion, individual and cooperative work, and examinations. Why, for example, do we equate working together with cheating?

We will make mistakes. Some things we try will fail, and others will not be effective enough to justify the costs in time, money and commitment necessary to keep them going. It is our job to evaluate what is done. That is very difficult. However, it must be done. Fortunately, help is on the way. The MAA has devoted a great deal of effort to assessment. Who will assess specific projects, and who will determine which survive and which die?

VII. Current Efforts

There are several projects in our department which focus on changing the way we teach. We suggest that the department should determine which of these will get continuing support, for what term, and to what extent.

Summary of the Plan

We will re-examine the goals of courses and use our conclusions to revise expectations for students. That means that we will write down the expectations for students completing the course: the skills, the concepts and the understanding required for passing the course. This will require participation of many members of the Faculty, and will probably require many of us to take a fresh look at our old attitudes about goals and outcomes.

We will give opportunities for the teaching staff to teach with modified schemes which we design to respond to our new views of student goals. For the majority of faculty, graduate students and undergraduates, we realize that time is an extremely valuable commodity, and that teaching remains only one part of what we are expected to do. Modified teaching methods must keep the mission of the department clearly in view.

The teaching staff will be augmented by the addition of paid, trained undergraduate teaching aides. This requires new money, as well as focussed training.

Designing, implementing and assessing change requires a very large commitment on the part of some people. The department, college and university will provide resources and support necessary for such development.

Change and evolution demands training and continued professional development. We recommend that some faculty member be charged, as part of a regular assignment, with organizing and modifying that activity.

In courses for which the learning community model is deemed appropriate, at least as a major component, space must be provided as the home for the course and/or related courses.


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