Table of Contents
1. Teaching Philosophy and Goals
I base my teaching on the belief that the only way to learn mathematics is to do mathematics. While the process of reading examples and proofs in textbooks and from lecture notes is valuable, the real learning comes through one's own efforts at solving mathematical problems, either computational, theoretical, or both. This is achieved mostly through class assignments, but also through in-class discussions and exercises. I view my role as a facilitator for this process. I must design the framework in which learning can take place, and then stimulate and nurture the students' development, giving help in terms of knowledge, techniques, and encouragement.
My goals in teaching are not
just to promote learning of the subject matter. I also try to
help the students learn to think logically, learn problem-solving
methods and techniques, and improve writing skills (writing clearly
and concisely, explaining step-by-step processes, providing valid
reasons for logical arguments). In addition, I try to help students
see the course material in a holistic context by requiring them
to synthesize the various concepts of the course by applying them
together.
Courses taught: I have taught a wide variety of courses, from the precalculus through graduate levels. At the graduate level, I have taught Linear Algebra, the Real Analysis sequence, and the Functional Analysis sequence. Upper division courses include all of the Numerical Analysis courses, Advanced Calculus, Topology, Complex Analysis, Partial Differential Equations, Linear Algebra, and Abstract Algebra. In the lower division, I have taught the engineering Calculus sequence, including computer-enhanced sections, Differential Equations, precalculus Algebra and Trigonometry, and the large lecture Discrete Mathematics course.
I have taught Calculus in several different ways: traditional, computer-enhanced (via class projects using Mathematica), and with graphing calculators. Programming and use of numerical software is a large component of the Numerical Analysis courses.
I have served on numerous Masters and Ph.D. committees. I have given reading courses in Functional Analysis in Summer 1995, Fall 1996, and Spring 1997.
Grading: I view the purpose of grading as mostly motivational, not judgmental. By requiring students to demonstrate knowledge of course material, this motivates them to do the necessary work required to learn mathematics. I do not grade on a strict curve. I usually curve each exam and also the total assignment scores and quiz scores. The curve is based on a mixture of class performance, percentage correct, and comparison with past classes' performances on the same material. Thus, it is possible for everyone in the class to get a good grade, or the opposite. For example, I have had classes in which everyone received a C or better, and others with only one A. It is rare for a student who attends class regularly and does all of the class work to receive a grade below a C. On the other hand, to receive an A, one must score consistently high marks on exams and assignments.
I generally have high expectations of my students. When grading midterm exams, I try to strike a balance: be fairly strict while at the same time avoid discouraging the students. First of all, this gives a signal to students who are not doing the work, or for whom the course material is too difficult, so that they can either drop the course or make the decision to devote more time and effort to the course. Second, it gives students a good idea of where they stand in the course, so that there are no surprises at the end. And finally, it allows me to take other factors into account when I make up final grades, resulting in increases for some students at the end of the course. I spend quite a lot of time in the process of figuring and deliberating final grades. As a result, I very rarely have any student who thinks that his or her final grade is unfair or unexpected. Other information related to grading is included in the Exams and Homework sections below.
Feedback: I have tried to increase communication and feedback both to and from my students. I encourage the use of office hours, and I make myself available at other hours as well. I have instituted a World Wide Web page for each class, which allows students access to all sorts of class information (see the Course Syllabi and Information section below), and which also includes a feedback form for students to submit suggestions and comments. I collect the email addresses of all students at the beginning of the class so that I can quickly notify them of changes or hints on assignments, temporary changes in office hours, etc. I also publish this class list on the class Web page so that the students can contact each other for notes and group study. Also, in graduate and upper division courses (except Math 473, which is quite large), I have a small conference with each student about midway through the semester. I tell them how I view their work so far and try to give them suggestions for areas in which to improve, and I find out from them about any problems they are having.
Availability:
I always hold five office hours per week, and I make myself available
at other times as well. I tell my students they are welcome to
come by my office at other times, and unless I have an immediate
deadline or meeting or class, I will be able to help them. My
detailed schedule is posted on my Web homepage.
Class sessions - lecture and discussion: I try to begin each class with a brief summary of the previous class session, and a reminder of where we are in the topic we are currently working on. At this point I usually ask if there are any questions from the reading, homework, or previous class. In lower division classes I usually assign homework and reading every day, so I spend a substantial amount of class time discussing homework problems. After this discussion, I usually give a lecture on new material. I try to begin the lecture with a brief outline and a list of objectives, and I try to always include examples during the lecture. I always encourage questions and pause in the lecture to answer them. Depending on the time and topic, I may then have an in-class exercise, probably involving cooperative learning (see the next topic), with a follow-up discussion to end the class.
There is usually more lecture in upper division classes, and more discussion and in-class exercises in lower division classes. However, in upper division and graduate classes I often incorporate discussion into the lecture. For example, in theory classes, rather than simply writing out a proof on the board, I try to first motivate the proof (see if the result seems to be intuitively true, look at some examples), and then have the class participate in the writing of the proof (provide the next step, fill in details, etc.). I also have found that group problem-solving (i.e., proof-writing) sessions are very effective in these types of courses.
As much as possible, I try to present course material in analytical, numerical, and graphical contexts. This approach of course depends on the particular topic, but it is particularly valuable in calculus, differential equations, and numerical analysis courses. I am especially conscious of using pictures and graphs to help illustrate different concepts, as most students can then at least intuitively understand the concepts even if they have trouble understanding the analysis.
Cooperative learning: For many years I have assigned group projects in some of my classes (see the next topic). And in the past several years, I have incorporated more cooperative learning techniques into the class sessions, especially in lower division classes. These techniques usually involve working in pairs or groups of three on a short problem, with specific instructions on how to share ideas and come up with a common solution. While the groups are working, I can move around the classroom to help various groups, and at the end we compare and discuss the various groups' solutions. Sometimes I have a more complex group exercise (usually a sequence of connected problems) that takes up most of the class session. In order to learn more about cooperative learning techniques, I have participated in a workshop and a seminar series on cooperative learning, and have also read extensively about these techniques.
Homework - level of difficulty, projects, group work: As stated at the beginning of this document, I believe that the only way to learn mathematics is to do mathematics. For this reason, I usually assign a lot of homework. I try to assign a mixture of routine and challenging problems so that I can stimulate the more advanced students but still enable the poorer students to at least learn the basics of the course material. In small classes, usually upper division, I try to grade as much of the homework as possible. However, in large classes, usually lower division, it is not possible to grade all of the routine assignments. In the latter classes, students can check their work on routine assignments by using solutions manuals, the help room, and posted solutions, and by asking questions in class.
In larger classes, I also often assign more involved projects, and I always grade these. The projects involve a writing component, and also often involve the use of calculators or computers (see the next topic), but not always. Usually they are completed in groups of three or four. In this case, each group has a "scheduler" who is directed to arrange meetings and make sure the group has a plan for completing the assignment. Another group member, the "historian", turns in a paragraph which details the meetings, who attended, and any problems encountered. A third member, the "assembler", checks to make sure that all problems are completed, that the solutions are assembled in the correct order, and that all group members have photocopies of the solutions. These jobs rotate for each assignment. After several assignments, I usually mix up the groups, and at this time I ask the members of the old groups to rate each other. This will help prevent one group member from just coasting on the work of the others. Overall, I have had good success with group projects. I am able to assign more challenging and interesting problems, and the students help teach each other through working together. Also, the students often find the group experience more enjoyable than individual work, and I have consequently received mostly positive comments about group projects.
Recently, in small theory classes, I have started a policy of allowing students to rewrite incorrect proofs. This encourages the students to take my suggestions, look at the problem again, and work through a new (hopefully correct this time) argument. I think this idea promotes much more learning since it require more active work by the student, rather than the more passive approach of simply looking at the comments and reading the written solutions.
Exams: When I create an exam, I try to follow several guidelines. First, I try to test over a reasonable range of class material, and I try to stress the important concepts. I don't include unimportant items or problems which require some "trick" that the students may have only seen once. I also include problems of varying difficulty. However, I usually do not include trivial problems. Before each exam, I spend some time in class discussing what topics will be covered and which are most important. I usually give more detail for undergraduate courses, especially the lower level courses. I also try to be careful not to make the exams too long, but I sometimes fail in this regard. I find this is especially hard in higher level theory courses, where it is very hard to judge how long it will take students to do the creative thinking necessary to come up with a correct proof. Consequently, in these courses I do not expect students to solve all of the problems on the exam.
I always keep old exams, and I notate them if they are too long or if a certain problem was not worded well or was not a good question for this particular class (i.e., if most of the class missed it). When I teach the same course again, I do not usually ask the same questions, but I often ask similar ones. So in this way, I keep some uniformity when I teach the class again, and the exams should get better, in theory at least. I know that I have a reputation for giving difficult exams, but I have observed that over time my exams have been getting easier. One reason is that the students' overall abilities and preparedness, especially in the lower level classes, has decreased over the years, and it makes no sense to me to give an exam on which most of the students do very poorly. For one thing, this would be extremely discouraging for the students. Also, after all, part of the purpose for an exam, especially a midterm exam, is to help the students learn. It shows the students what they do and do not know, so that they can go back and work more on the areas in which they are having trouble. For the same reason, it also makes no sense to give an exam which is too easy and which gives no indication of problem areas.
However, I should note that I do not grade exams on a strict curve. If the entire class does well, then everyone gets a good grade, and conversely if everyone does poorly. My grading of exams usually follows a percentage scale which is modified by class performance and also by comparison with previous sections of the same course.
Uses of technology: I have used computers for demonstration purposes in many courses. I have also taught several "computer-enhanced" calculus sections, and one calculus section which used graphing calculators. In the computer-enhanced sections, I assign several group projects which require the use of Mathematica. In the calculator sessions, students use the calculator daily to help with in-class exercises and homework. Some of the homework assignments are designed to be solved with the help of a calculator.
Programming and use of numerical software is a large component of the Numerical Analysis courses. Most assignments are designed to be completed using Matlab, but Fortran or C is required on a few problems. In addition, a working knowledge of Unix on Project Vincent is needed.
All courses make use of the
World Wide Web for informational purposes and some distribution
of class materials (see the Course Syllabi and Information section
below). In real analysis classes, I have also used interactive
Web-based instruction.
4. Course Syllabi and Information
At the first class meeting, I hand out a syllabus which gives the basic information for the course. It lists ways to contact me: office number, phone number, and email address, and also the address of my Web homepage and the homepage for the class. It also informs the students about the prerequisites, text, topics to be covered, the number of exams and quizzes, information on homework assignments (approximate number, types of problems, grading policy, possibilities for revision of incorrect solutions, etc.), and grading policy. If the class requires use of a calculator and/or computer, the syllabus includes a section describing how they will be used in the class and what will be expected from the students in the areas of programming and calculator/computer expertise. Finally, the syllabus also includes a section on use of the class Web page.
During the first week I announce my office hours (also listed on my Web homepage, along with my more detailed schedule). I do not give out a list of all homework assignments for the semester as some instructors do. I prefer the flexibility of changing the problems and due dates during the semester as I get to know the strengths and weaknesses of the students. This allows me to tailor the assignments to the needs of the class.
Use of the World Wide Web: Each of my classes has a World Wide Web page which allows students access to all sorts of class information. The syllabus and classlist is available here, as well as lists of homework assignments and reading assignments (updated as they are assigned), and a list of materials placed on reserve at the library. As each exam approaches, I include information about the exam that they will need for studying. I have a "Latest News" section that informs the students of upcoming deadlines, new postings outside my office (usually homework solutions), etc. There is also an anonymous feedback form for students' suggestions, compliments, and criticisms. Additional information is available depending on the class. For example, for numerical analysis classes I have included a lot of helpful information on computer use (see the link for math473.html below). For real analysis, I have added links to interactive Web-based instruction, which acts as an alternative supplementary source for the course material. For the large lecture Discrete Mathematics course, I have a Help page which has hints for improving performance in the class and sources for obtaining help, another page which gives instructions for doing some of the class problems using calculators and computers, and also a page which lists all of the scores and grades for each student, along with the detailed results of each multiple choice exam. The following three links are examples of class Web pages I have used in the past:
Summary of student evaluations:
On a scale of 1
(high) to 5 (low), my student evaluations have fallen in the following
ranges:
Evaluation | # of courses | # lower division | # upper division | #graduate |
1.00-2.00 | 21 | 6 | 8 | 7 |
2.01-3.00 | 24 | 15 | 9 | 0 |
3.01-4.00 | 1 | 0 | 1 | 0 |
4.01-5.00 | 0 | 0 | 0 | 0 |
average | 2.08 | 2.36 | 2.02 | 1.37 |
As the chart shows, I have received excellent evaluations in graduate and upper division courses, and a bit lower evaluations in lower division courses. See Appendix F for a detailed listing of my evaluations.
Students generally comment positively on my class presentation, organization, fairness, and availability. Some of the negative comments are that I go too fast in class and try to cover too much material (of course, for some courses I have no choice in this matter), exams are too hard, and there is too much homework. I used to get complaints that the class was boring and that I showed little enthusiasm, but I have rarely received these types of comments any more. I have included various samples of comments below. See Appendix F for a more extensive collection of student comments.
Sample comments from graduate students:
"The instructor taught
very well and clearly. He always gave us lots of homework. But
now that the term is going to end, I think I learned the material
very well."
"Excellent lecturer!
Very clear, concise, well prepared. Very thorough coverage of
the topic - I learned a lot!"
"Even though I found
the course material very difficult and sometimes tedious, Dr.
Wagner's enthusiasm made the course very enjoyable. I felt a camaraderie
in the class which made it fun and interesting. Dr. Wagner was
very helpful and always available outside of class, and I think
most students took advantage of this. He was always well-prepared
and I'm extremely amazed at his knowledge and level of understanding
of this material."
"Dr. Wagner did an outstanding
job in my opinion; in fact I cannot think of a professor who has
done a better overall job here at ISU. The information was clearly
presented, Dr. Wagner was available to discuss problems at least
4-5 hours every day, and I know that he did everything he could
to encourage my success. It is hard to think of anything that
could be improved…In any case, I appreciated Dr. Wagner's
efforts very much. He is a very careful, concerned professor,
who has given a lot to his students."
"He really helps students
to understand this difficult subject."
"Dr. Wagner is one of the best instructors in this department. His presentation of the material was outstanding!! It was clear, organized and complete, yet I didn't feel as if we were "spoon-fed" the material."
Sample comments from undergraduate students:
"I learned an extraordinary
amount and found the class rewarding and worthwhile…On the
whole, I really liked the course, and I applaud the rewrite opportunities
- they are very instructive…I think you teach very well -
your presentation is clear, direct, and not hard to follow…I
felt there was great camaraderie among the students."
"Wagner did a fine job
teaching this course."
"The grading is tough,
but that's good, because I need to know my mistakes, even the
little ones…The rewrites are good. They really help us learn
from our mistakes."
"I really enjoyed being
in his class."
"This was a difficult
class for me, but I feel I have learned quite a bit."
"The class was extremely
well done, except sometimes the lectures went too fast."
"I think the instructor
did a good job presenting the material in a clear way…The
WWW page was also very nice - good current technology."
"I thought the web pages
were a great idea, and I used them as a resource."
"Web pages summarized
all the material we needed; this was nice."
"I liked doing homework
in groups, it made learning easier. "
"I liked the group assignments
because I believed I learned better from explaining to team members
and asking questions about the idea…The web pages were great!"
"The group assignments
were helpful. It allowed me to understand problems better."
6. Teaching Improvement and Future Plans
I have been working very hard in the last several years to try to improve my classes. The primary focus in this process is providing an environment which promotes better learning. This means looking at all aspects of the course, not just trying to improve my lecturing skills. I have found that it is especially important to remain flexible, and modify the teaching methods to fit the students in the class. Some students thrive on lectures with lots of theory, others need lots of examples and visual materials. Some students learn well in groups, and others prefer more individual attention. It is important to talk to the students and grade some assignments and/or quizzes early in the course to get an idea of the various students' strengths and weaknesses. Then I can try to adjust my teaching methods accordingly, and in the process hopefully benefit all of the students.
To help learn more about teaching, I attended an Effective Teaching Workshop in 1993, and in 1995 I participated in a Cooperative Learning Seminar. I also have read several books and newsletters covering all aspects of the teaching process.
Here are some of the changes I have recently incorporated into my classes:
In particular, I have been trying to consciously slow down and not try to cover so much material in class. I want to do a better job covering the most important topics rather than a poor job covering all the topics. I can leave more material for the students to learn on their own, with my guidance and direction.
In the future, I will continue to try to find better methods in class for determining if the students really understand the current topic. I will also experiment with more variety of methods of disseminating class materials, such as with interactive Mathematica or Matlab notebooks, or interactive web-based materials using HTML or PDF files. Finally, I will continue to try to find ways to improve the role of exams in the learning process. I would like to make them more of a learning tool rather than just a method of judgment.
I also have several projects in mind for specific courses:
7. Other Teaching-Related Activities
Invited Talks:
Conferences and Seminars:
Curriculum Development and Support:
Grants:
Appendix A: Sample course syllabi
Appendix B: Sample assignments
Appendix C: Sample exams
Appendix D: Sample in-class exercises and demonstrations
Appendix E: Recent class experiences
Appendix F: Student
evaluations and comments