II. Statement of Teaching Philosophy and Goals

Teaching at any point in time should have reference beyond merely the course one is teaching. Examples of this include follow up courses, courses in ones major, ones career, or the general activities of rational and quantitative reasoning. A course should not lose all sight of these goals. As something like ABET evolves, its impact will trickle down to courses like freshman mathematics, and an appropriate relationship will need to be worked out. If learning is a lifetime activity, then part of what one comes out of a course with must a better ability to learn, to think and understand in new ways. The average freshman will be in the work force until approximately 2040 and most things that his or her job demands will not be anything we have envisioned at this point in time; they will have to learn on their own what they need. Getting people to think in new ways is rarely easy; we all like to fall back on what we already know. A class should be a constant occurrence of communication, both from me to the student, as well as the other way and between students. This happens best through discussion, question and general feedback. I try to do whatever I can to promote an environment which makes this possible. I always learn a great deal from any question or comment; absence of such makes for a poor class. I would rather discuss math with students than lecture on it; answering questions is a very enjoyable part of class for me. I would also rather try to lead a student to the answer in a Socratic manner than simply provide it; I am trying to get people to understand and think, not dispense information. Large format classes are a challenge and problem as they discourage this - a student in a room of 120 is not as likely to become involved as one in a class of 25. Students need motivation. Many students are sometimes underachievers because they do not believe a course has any intrinsic value. A certain degree of salesmanship is sometimes needed. I often work from the perspective of the most cynical student in the class who asks "so what's this stuff any good for?". Very often students would like to know how the material in a course relates to their major or to the world they are entering. We take it on faith that they believe we have answers, but we rarely provide them despite asking the student to put in 15-20 hours per week into a course.

Students also need help with what is often called "study skills", whether they will admit it or not. WPI, thanks to people such as Lance Schacterle, has put in a considerable amount of effort in this area. I was part of an "Academic Success Skills" program in 1996 which had mixed success. However, I keep making efforts in this area.

If nothing else, I firmly have come to believe that to get anything out of a course, the student simply must be there. As WPI has a fair number of large section courses (that is, over 100 students in the same lecture), this is a problem. Last year, I took attendance daily in two courses each of which had over 125 students. I simply compared the average attendance with the grade for each letter group. The data is easily summarized as follows:

The students that we have are reasonably gifted mathematically, if their SAT scores are taken seriously. If they perform poorly in a course, it is far more often due to poor study skills (i.e. attendance, time management) than inability to comprehend the material or master the skills required. In short, I believe if I can get them there, I can educate them. With this as a goal, I have undertaken a number of small incentives. First of all, I point out the data just displayed to them (in fact it is now in a Web page), and in a positive manner challenge them to pick out the grade they would like; this data shows how to achieve it. Last year, I gave a 5 point bonus for the course to anyone who had perfect attendance. I was surprised at the number of people who dragged themselves to an 8:00 am class in quest of those five points! In general ,attendance was up about 15% due to this, and NRs were down substantially. (During term B of this academic year, Peter Christopher of the Mathematics Department was also teaching linear algebra at 8 am and tried the attendance/5 pt incentive system and reported that attendance was up substantially as a result.) Years ago, I probably had the traditional philosophy of assuming the responsibility of the student to come to class; that was his/her problem - mine was to teach. I would probably say at this point that the ends justify the means; whatever gets them there and gets them a better education is fine. I have no qualms about sending an email to someone who should be in class and reading them the riot act. I found not long ago that the same immature 18 year old who could not come to class somehow became a very mature and responsible 22 year old, working long hours and weekends to make a job successful. In short, the maturity would come in time - I had to do whatever it took to get the job done in the meantime. I would say at this point that philosophical speeches are well received but 5 weeks later have little effect. Concrete incentives like bonuses for attendance, extra credit from attending conference and other similar acts produce more results. Finding a means is a part of the job.

C. Efforts to Improve Teaching

1. Conferences attended.

• 1994 - Teaching Portfolio Workshop - WPI

• 1997 - Lilly Conference on Education - UNH

• 1995 - various CED workshops at WPI

• various MAA meetings (1993 - )

2. Literature

3. Continuing Education

I try to regularly audit courses outside of the mathematics department with the idea of broadening my background as well as coming up with gems of mathematics which might be worked into one of my courses. While at WPI, I have audited courses in Industrial Robotics, Expert Systems, Computer Image Processing, and Discrete Mathematics. I also took a graduate computer science course, Theory of Computation, for credit. In each case, I have been able to take something back to mathematics courses I teach. As an example, the instructor of the Industrial Robotics course went over the design of an assembly line he had personally developed for a Massachusetts electronics company. The line fed IC chips to a robot arm which inserted them into a circuit board. It fed them through use of a vibrating bowl. The bowl had to be carefully designed. Using fairly traditional vector and trigonometric methods, he showed that the efficiency of a well designed bowl might be some 40% greater than that of a hastily designed bowl, with the difference coming from carefully applied trigonometry. Since the assembly line is sequential, the slowest part of it impacts the entire line's productivity. Back in the mathematics course, the story provides ammunition for the skeptic who looks at trigonometry and says "so what's any of this good for?" (see remarks elsewhere).

Writing this document is good in that it reminds me that I have not audited a course in over a year, and doing this is an integral part of my personal strategy for maintaining a background appropriate for a mathematics instructor (that being awareness and knowledge of subject matter outside of mathematics).

I just finished auditing PH 1121, Electricity and Magnetism, during term B.

As a part of my teaching philosophy, I believe that one should seek, beyond the confines of mathematics departments, literature, and courses, a broader sense of how mathematics is used and by whom. This, I believe, cannot help but enhance one's teaching.

4. Educational Research

About three years ago, I was asked to provide some tutoring in statistics for a person in the Management Department who needed it for his doctorate. It turned out he was analyzing data from WPI's participation in a Davis Foundation Grant, the basis of which was use of Peer Learning Assistants in undergraduate courses. We both found we had many common interests educationally, and it has been valuable for me to see the assessment side of education, which I believe is mostly in its infancy, at the college level. Since that time, we have jointly written one paper and are beginning a second. This research, I hope, will provide a necessary balance to my career.

5. Make Effective Use of Technology to Support Courses

I have made the conversion in the last two years from having paper based courses to those where much material is on the local Web. Paper saving considerations aside, I see value in the hyper-link approach to documents, as well as the fact that material can be updated without having to distribute new versions.

I have become proficient with Geometer’s Sketchpad, a very noteworthy piece of software which is used by high school geometry teachers. I pass this on to the students in the MME program when they are in MME 518, the Geometry course. I have also had one student do her thesis based upon implementation of Sketchpad into her geometry course. Assessment was also done and she found a significant (in the statistical sense) improvement in performance of her students due to the presence of material utilizing Sketchpad. This was one fairly rate instance when the impact of technology was measurable.

In general, I am cautious of implementing technology for technologie’s sake; not only does it take a lot of work and expense, but all too infrequently are there measurable improvements. More often one simply ends up doing a task "differently" as opposed to better. One of my MME students is just finishing his thesis/project on applications of technology and while doing the literature search, he found it very hard to come up with hard, statistically verifiable case studies which really showed a positive impact of technology (graphing calculators in his case at the secondary level).

During term B, we linked calculus 4 and Physics 1121. One of the requirements of the physics instructors was to have the students turn in notebooks with handwritten sketches and diagrams. It occured to me that in the process of doing this, alot of valuable thinking goes on, even for the attempts which get crumpled up and tossed out. Had these been made with software, I seriously doubt the same thinking would have gone on. In that environment, the task would have changed to deciding what menu options, what button to push and where to click. While one may argue that these are skills, too, they are not skills that have anything to do with understanding physics or math. In summary, computers may complement thinking by doing some tasks quickly and more accurately, but they are never any substitute for thinking. This point is often lost on students.

I believe I have come to utilize Maple in a reasonable manner in MA 2071 (Linear Algebra). I have taken the approach that it is merely a tool to make one faster and more accurate; nothing more. If the student wishes to use it, they can. This has been fairly successful and it has been a productive tool for people with less than the usual amount of complaining about it (Maple). (note to reader: Maple is a computer algebra system used at WPI and other schools. It is used in the calculus sequence and is the frequent target of complaints. The name comes from the University of Waterloo)

D. Redevelopment of Existing Courses

Starting in 1994 with MA 2071, Linear Algebra, a course with an average enrollment of 130 non math majors, I have developed an extensive collection of "projects" to augment the course. By a "project" I mean an endeavor worked on for at least a week, with a central theme or application, by a "team" of 2-3 students, involving not only mathematics but interpretation and writing. This is consistent with the WPI approach to education. My goals in doing this were: show the students meaningful applications of the material, have them work in teams and do writing and interpretation, relate to the "real world", motivate them, and (often) use technology. My personal goals were to dip into a variety of areas and interests, and to be creative; to put my own personal stamp on the course at that moment in time. I now have enough projects that a student can follow a sequence of them through which relate to their particular major. This means that, while the course is in a "large" format and thus somewhat generic, it can still be tailored to an individual students needs and interests.

Examples of projects offered in this course include:

• managing an oil refinery
• regression analysis
• market share in the auto industry
• marketing computer operating systems
• Richardson's Arms Model
• fractals and chaos
• development of a productivity index for business
• energy flow in an ecosystem
• graph theory and linear algebra applied to communication networks
• Fourier Series
• the New England Fishing Industry
• structural analysis of bridges
• linear control
• Lyme disease

(see actual projects). These projects also require continual maintenance and improvement; topics get stale, new opportunities come along, new approaches become apparent. As they are all on the Web, utilizing appropriate software and technology is also a need.

Based upon student course evaluations and general levels of effort put in, these have been well received.

During 1997, I became involved in an NSF sponsored project initiated by Judy Miller and Art Heinricher called "bridges". The basic idea of bridge projects is to link two courses together which a student is concurrently taking, for the betterment of one or both courses (the whole is greater than…). During the fall of 1998, I took over the "general" section of MA 1023 which has a population of 175. In line with the bridge notion, I sought to introduce a link to another course for as many students as possible. A survey on the first day of class showed three possibilities: CS 1005 (introductory programming), CH 1010 and 1020 (Chemistry I and II), and PH 1110 (Physics - introductory mechanics; non calculus based). My goal was to see if, through a series of 5 weekly projects, I could cause them to have a more productive experience in the "other" course. Thus the projects had the computer science students writing C programs to study math, the physics students studying vector methods as they applied to mechanics, and the chemistry students studying equilibria from both a chemical and mathematical view.

Not wanting to leave out the rest of the students, I also wrote a sequence of projects of a generic nature for them, whose theme was statistics, from a mathematical viewpoint (integral calculus) and how it applies to problems in quality control in business.

Thus I ended up with 4 "tracks" of 5 projects each, one per week. I also covered the lab periods myself ( 7 periods per week). This proved valuable despite the time required as it gave me a chance to interact with students who I otherwise would not be able to, and to discuss their experiences with regard to the "other" course.

Project work is only a fraction of what this course consists of, however (they constitute 30% of the grade). The rest is mostly infinite series and approximations, the topics of lecture, exam and homework. My redevelopment effort here was to use the weekly conferences cooperatively by writing up materials for the students to work on in teams. Conferences in math are not always productive (last spring, mine were a disaster), so this was an accomplishment. I gave a 3 point bonus each week to the team which got the most problems right during its conference; this may have helped attendance and enthusiasm. The cooperative approach was both more productive as it got students involved and also took the load off the "PLAs" running the conference, who are not experienced in front of a classroom.

I am just beginning the process of data collection to see if the bridging produced any quantitatively verifiable improvements. This involves getting grades from the instructors of the "other" courses and seeing how the "bridge" students compare to the general population. All of the above is only so much good intention if no improvement can quantitatively be shown.

The original place I implemented the bridge philosophy was also in Calculus III during the first term of 1997. The section I had here was "pure" in the sense that all 64 students in it were also taking PH 1111, mechanics, traditionally a very hard course with a high failure rate. Merely setting up the section was a challenge as the students were initially enrolled in a large section of perhaps 200.

By rearranging the syllabus, focusing on applications to physics, making consistent notation a rule, and timing material correctly, we made some improvement. The first two exams in PH 1111 showed averages well above that of the previous year. Toward the end of the course, we had to cover mathematical topics required of the course which did not complement physics. At that point, the average on the third physics exam dropped below the previous year; there are a variety of possible explanations. Overall, the data from the first two exams provided valuable, quantitative support for the bridge notion, support that should help to keep it alive in years to come.

Follow this link for assessment data on all bridged courses (http://www.wpi.edu/~goulet/bridge_data.htm)

MME 518 is entitled Geometrical Concepts and is a required course in the colleges Masters in Mathematics Education (MME) program. I have taught it twice now and have developed what both I and the students feel is an extensive, enjoyable and informative course. Without reproducing the entire syllabus, we now cover 3 major areas:

• classical geometry (Euclidean and conic sections)
• neo classical geometry (differential and Non Euclidean)

• modern geometry (vectors, fractals)
• historical view of the evolution of geometry

Technology is also an important component of the course - students learn how to use the popular and powerful Geometer's Sketchpad package for plane geometry. I also provide some fractal software that I have written.

As all participants are also teachers, they develop materials to use in their own courses as a requirement.

At the college level, we often find students (freshmen) background in geometry is lacking, so the opportunity to influence a collection of high school teachers is a valuable one.

I do not teach the course again until the summer of 2000 but I have already gathered some valuable material to improve the course once again.

I have just finished developing a bridge for term B between calculus IV and PH 1121, Electricity and Magnetism. The general approach will be similar to the one discussed above (see item D3). The following link describes this course

E. Information from Students

I have stacks of envelopes with hundreds of student evaluations in them and, frankly, no organization to this information. The evaluations have two components: a numerical response to some 17 questions, and 5 written comments. The numerical responses come with a tally sheet. The data I have is seemingly quite "good" however so are everyone else's (virtually no one is "below average") and so an outsider might very well question the validity of the data. The written comments, at least in my opinion, are far more informative and interesting. Summarizing this in any useful way is a problem. If one "picks and chooses" then the objectivity is subject to question.

A statistical summary of all of my course evaluations may be found by following link . This data encompasses 47 courses and 1420 students. Of note to me is the data which says that over 95% of my students found me to be either "above average " or "well above average" as an instructor, with 56% finding me to be "well above average".

A comment on a calculus IV project in fractals perhaps illustrates what I am hoping to accomplish:

The following comment came in an unsolicited letter from a recent graduate of the MME program:

" I also want to say thanks for all of the time, effort and kindness you gave throughout the two years I spent at WPI. It was evident how much you cared about all of us as students as well as teachers. It was also appreciated how much information in terms of mathematics and education, in general, you impacted upon us."

And this from a letter attached to a recent survey:

"John, I think I speak for all students in the MME program during the last two years when I say it is abundantly clear that you truly care about the program and each student. I wish you continued success in your "new position" as head of the program".

These perhaps illustrate that all students need consideration as individuals, and that in some cases I have succeeded in that regard.

I also have evaluation letters from the two other colleges I taught at (see Appendix); these are of much the same nature as the comments I have gotten at WPI.

G. Teaching as a Dynamic Process

Each time I teach a course, it should represent a potential improvement over the previous time. There are many areas where this improvement might occur – delivery, quality and availability of materials, addressing the needs of the students in the course, motivation, preparation for exams, availability to students, supporting examples and cases, and so on. By critically examining what is done at any one time, it should be possible to plan improvements, and then implement them. There is no magic to this process; it is a combination of clarity of vision and goal, and attention to detail and concern for the education of the participants. One thing this portfolio may accomplish is to document how I feel I go about this process.

H. Service to Teaching

• peer review – I served on a committee last spring to review the teaching in Tensor Analysis of Rob Lipton. In addition to visiting his class, I interviewed some of his students and wrote up the results of committee deliberations.
• coordinator - MME program. Essentially acting as academic and thesis advisor for these students. Helping them to generate and carry out their thesis projects.
• current project: evaluate mathematics preparation of incoming freshmen to WPI. Each August, the math department administers a math test to all freshmen, the goal of which is to place them into the correct calculus course. This year, with the help of a graduate student, I hope to summarize their efforts on a question by question basis and look for correlation along geographic, SAT, ethnic or other lines. That done, I plan to track their progress through their freshmen year in mathematics courses again looking for correlation with performance on the exam. This may help, a year from now, in constructing a better exam and doing a better job of placing students. Judy Miller has supported this work.
• I am currently developing a series of workshops for high school teachers to attend at WPI on Saturdays in the Winter and Spring. These would offer topics such as dynamical geometry, fractals, project development and linear algebra, and would be as close to "free" as overhead allows.
• will be in charge of WPI's annual mathematics meet for high schools in 1999.
• possibly assist younger faculty in development of teaching portfolios

I. Publications Relating to Teaching

"Making Students Independent Learners - Fractal Projects" Vol 4, No. 1,

April 1997 Creative Math Teaching.

This paper describes project work in MA 1024 (Calculus IV) in the fall of 1995 where students worked in teams to find applications of fractals relative to their majors.

"Calculus, Bell Shaped Curves and Global Competition" Vol V, No. 1,

This paper describes project work in MA 1022 (Calculus II) which uses improper integrals to study statistical quality control and introduces the notion of global competition in manufacturing.

Linear Algebra Projects (for Prentice-Hall):