A Course Portfolio in Mathematics

Orin Chein, Mathematics, Temple University

 

 

 

In some sense, my involvement with portfolios dates back to the spring of 1990, long before I had even heard the term. At that time, my service as department chair was coming to an end, and for a number of reasons (none of which had anything to do with superior knowledge about the art of teaching) the dean had selected me as the new director of the College of Arts and Sciences Teaching Improvement Center. I had always taken pride in the belief that I was a good teacher, but, as is the case with most college teachers, I had no for­mal training in teaching. I knew nothing about teaching theory or developmental frameworks, and so I attended a variety of conferences and workshops on the subject as prepara­tion for my new position. At these, I learned many things; in particular, I became familiar with the notion of a teaching portfolio. When I was nomi­nated, several years later, for Temple University's Great Teacher Award, I put this new knowledge to work by developing my own teaching portfo­lio (and was selected for the award).

 

I mention all this because I think it is relevant to why I became involved with a group of colleagues assembled by AAHE in a project to explore a new‑version ‑ a subset really ‑ of the teaching portfolio, "the course portfolio." Having already had a successful experience with the teaching portfolio, I was eager to join others in exploring this related form of documentation.

 

Choosing the Course

 

I knew from the start the course I would choose as a focus for my port‑folio. Basic Concepts of Mathematic (Math 141) was a course that I had developed and introduced into the curriculum many years before, dur­ing my first few years at Temple, any one that had quickly become a requirement for all math majors. However, for two antipodal reasons, not all of my departmental col­leagues were as convinced as I was of the importance of this requirement. Some felt the course was superfluous because students eventually learn the same material when they take other courses. Others thought that the course was important but that it didn't go far enough in achieving its goals (to teach students how to thin and write mathematically; to present topics that reappear often in later courses; to provide students with an early indicator of whether or not mathematics is really the field for them). In any case, Basic Concepts of Mathematics was a course that had become identified with me more than any other in the curriculum, and it seemed the natural one to choose. As I write in my introductory remarks to the portfolio,

 

By preparing this portfolio, l can showcase exactly how 1 think this course should be taught, what it should cover, what should be emphasized, how different aspects of the course hang together. This pre­pares a foundation on which further discussion and negotia­tion with my colleagues can be based.

 

 

Thinking About Purposes

 

As noted above, I undertook the process of developing my course portfolio as part of a group of col­leagues from around the country. At one of our early meetings, we dis­cussed two essential questions: What should go into a course portfolio? and Why would a person want to develop a course portfolio? We decid­ed that the answer to the former question might well depend on the answer to the latter, and we then identified three primary reasons that one might want to develop a portfo­lio: for use in personnel decisions, as a pedagogical tool, and as an intro­spective tool. Each of us agreed to develop our portfolio along one of these lines, but I decided to keep all three purposes in mind.

 

Although I already was a tenured full professor and I had received an award for the quality of my teaching, I thought that preparing a course portfolio might prove useful for future personnel decisions. Our col­lective bargaining agreement with the university allocates a pool of money annually for merit awards, and merit for teaching is one catego­ry for which those awards are made. I thought that developing this portfolio might be useful in supporting a case I might make for such an award. I also was considering applying for a differ­ent teaching award (from the College of Arts and Sciences, rather than from the university as a whole), and I thought the course portfolio might be helpful for that, as well.

 

The pedagogical usefulness of this portfolio is evident in the reasons for which I decided to focus on Basic Concepts of Mathematics, as noted above. I saw the portfolio as a vehicle through which I could make a case for why this is an important course ‑possibly the most important course in our curriculum. It also gave me an opportunity to present my own thinking about how the course should be taught. I was eager to do this because I believe that at least part of the criticism from those who feel that the course does not do enough to prepare students for some of our more advanced courses can be attributed to the fact that not every­one who teaches the course (at least 10 different people have taught it in recent years) emphasizes what I con­sider to be the appropriate topics or covers everything that I think the course should cover.

 

(Now that I write this, I realize that although my portfolio, in its current version, usefully presents what I cov­er when I teach the course, it needs to say more about why. I mention this to make the point that a course portfolio is a living document, one that can be revised or embellished as often as one feels inclined to do so.)

 

The third reason for developing a portfolio ‑ its usefulness as a tool for introspection ‑ has turned out to be the most important one for me. I have had to examine my goals in teaching this course and the ways in which I hope to accomplish them. In the process of doing so, I developed a number of new teaching tools and techniques. Unfortunately, most of the work on my portfolio took place after the fall of 1995. As I did not have an opportunity to teach this course again until the fall of 1997, I was not able to put some of my new ideas to the immediate test in this course. However, I was able to experiment with some of them in other courses. I am pleased by the results and plan to continue using the ideas in appropri­ate courses in the future. I will dis­cuss some of these further below.

 

 

Starting With Reflection

 

During the fall of 1995, I taught Basic Concepts and immediately began the process of developing my course portfolio. My first task was to write a reflective statement in which I dis­cuss my goals when I teach the course and what I try to accomplish, how I conduct the class and why, my view of the role of a teacher, my phi­losophy of homework and testing, and a number of innovations and experiments I planned to try, espe­cially to attempt to get the students more actively involved in the learning process. Writing the reflective state­ment also provided me with an opportunity to expand and elaborate on the syllabus I distribute to the class so that students can know exactly what will be expected of them and so they can acquire in advance a fuller appreciation of what the course is about.

 

Sampling Student Work

 

Having evaluated some of our in ­progress early drafts, my colleagues and I in the AAHE Course Portfolio Working Group all agreed that a port­folio should include samples of actu­al student work. But the question was how much. Since my course is an intensive writing (IW) offering (that is, it is one of a number of courses at the university that satisfies a writing­across‑the‑curnculum requirement), students are required to do a consid­erable amount of writing. This includes not only several examina­tions and a group term paper but also regular weekly homework assignments and occasional revi­sions, as well.

 

As a result, I collect a large amount of written material from my students. I included much of this in the student work section of the first draft of my portfolio. I included the actual student papers with my com­ments about what was wrong, what was well done, suggestions for rewriting, etc. (I returned this work to students as it was corrected throughout the semester, but I col­lected it again in a student portfolio at the semester's end.)

 

When the Working Group met in the fall of 1996 to exchange draft portfolios, it was generally agreed that mine was too bulky and that only a sampling of student work should be included. But how should that sampling be selected? Should it be the best work of the best student; or should it be selected from the whole spectrum of student performance? And how large a sample shoal be included? While we never really reached agreement on the latter issue, we did agree that one approach to this problem would be to identify a random sample of stu­dents at the beginning of the course and to include their work in the port folio.

 

The Question of Quality

 

During the discussions of the Working Group, it became clear that one of the issues we would have to address (at least for portfolios intended for use in personnel deci­sions) is the question of quality: Ho,. that is, do our portfolios present evidence of and allow judgments about teaching effectiveness? Student evaluations, which can be included in our portfolios, give evidence of what our students think of our teaching ability and approach, but they don't present evidence of what the stu­dents have learned. Samples of stu­dent work, including homework, term papers, and/or examinations (the kind of things I included in my portfolio) provide more direct evidence about student learning, but drawing conclusions about the quali­ty of teaching is nevertheless prob­lematic, especially for readers from outside the field. For teachers who regularly teach mufti‑section courses, effectiveness might be assessed through the use of uniform final examinations, with comparisons of student performance made over a period of several years, rather than based on a single semes­ter. For those teaching introductory courses, effectiveness might be meas­ured by tracing student performance in subsequent courses. But neither of these approaches was practical in my situation. Instead, I settled on several techniques that I have now incorpo­rated into all of my other courses; as well. These assessment techniques are described in detail in my portfo­lio, but I will simply summarize them here.

 

1. At the beginning of the semester, I administer a pre‑test on which I list all the terms and concepts I hope to discuss during the course of the semester. Using a five‑point scale ranging from "never heard of it" to "understand it well enough to teach it to someone else," stu­dents are to rate their confidence in their knowledge of each topic. The same test is then adminis­tered at the end of the semester (as a post‑test), and the results are compared. (Students were assured that their performance on these "tests" had no bearing whatsoever on their grades.) A discussion of this comparison is included in the portfolio.

 

2. After each class, each student is required to prepare an index card in accordance with a format described in the course syllabus. These index cards are collected weekly to provide me with a pic­ture of what students think they are learning or what they think they do not really understand. They also provide students with an occasional opportunity to display their grasp of the "big picture" in mathematics. I return these cards to the students, with my comments and suggestions, and then I Collect them again at the end of the semester. I includ­ed a sample of these cards in the portfolio because they not only give a picture of what students learned, they also help illuminate how I teach.

 

3. At the end of the semester, again in accordance with specific instructions contained in the course syllabus, students are required to write a brief (two­ page) paper summarizing what they feel the course was about and what they learned. These summaries are included in the portfolio.

 

4. Students are also required to write a description of their group expe­rience and an evaluation of the other members of their group, fol­lowing a framework of specific questions that I put forth. These, too, now appear in the portfolio.

 

These assessment strategies do not, I'm aware, completely and unam­biguously answer all questions about quality, but they are surely an improvement over the current pre­vailing dependence on student satis­faction ratings alone. I feel that my portfolio gives a much more com­plete and reliable picture of the effects of the course on students' learning than I was able to achieve without it.

 

The Impact of the Portfolio on Teaching Practice

 

In addition to motivating me to adopt some of the assessment tech­niques discussed above, working on the course portfolio has led to several other innovations in my teaching. As a result of having to examine exactly what I would like students to get out of the course, I have collected a list of terms, theorems, and techniques that I would like them to know and mas­ter. I have turned this list into a study, guide, which I now distribute to my class. (I have subsequently done the same for other courses, as well.)

 

As I mentioned above, because this is an IW course I have always required a substantial amount of writing on the part of my students. As a result of my work on the portfolio, I have rethought my role in helping students develop their writing skills. I have restructured some assignments and added others. Students now have to rewrite and resubmit several of their weekly writing assignments. At the end of the semester, they have to submit a portfolio containing their best and most improved work, as well as their weekly index cards and sever­al other items. In addition, each week two students in the class serve as class secretaries. They write up the notes for the week and submit them to me for comment and correction. They then have to rewrite them for ultimate distribution to the class.

 

Finally, students‑have to partici­pate in a group project that requires them to write a paper on material that time constraints have prevented me from covering in class. Samples of this collaborative student work are contained in my portfolio. Group work is another area in which my teaching has been devel­oping. While I started experimenting with group projects even before the AAHE project, I did so with only United success. Many of the better students resent "wasting their time" when they feel they would be more efficient and just as successful working on their own. Groups with primarily weaker students often flounder around, having no idea where to start or how to continue even if I help them get started. Nevertheless, I full believe that active learning results u better understanding and longer retention.

 

I continued to think about these issues as I worked on my portfolio, and I decided that what is needed a~ well‑designed "discovery" exercises, which actively involve the students in discovering new results for them­selves and which students can attack as individuals or in a group, as they see fit. Designing such exercises is not easy, and none was available when my portfolio was last assem­bled. This semester I have develop a few so far, and I will include them in my next edition of the portfolio.

 

The Time Commitment

 

One of the questions I am asked me frequently is how long it took me to assemble my course portfolio. Truthfully, I do not know the answer I would say that I certainly spent in excess of 50 hours, and maybe ever as many as 100. However, this includes not only the time I spent writing introductory remarks and n reflective statement but also the tin I spent expanding the syllabus, developing the study guides, creating an analyzing pre‑ and post‑tests, developing discovery exercises, and doing the many rewrites that resulted fro: my evolving sense of what would make the portfolio (as well as the course) most effective. With one course portfolio now under my be] think I will be able to complete my second one (which I plan to do soon) in much less time ‑ maybe as few as 20 hours.

 

Responses From Readers

 

A question of concern common to many of us developing course portfo­lios is, How can we make portfolios most helpful to readers, especially to those who might use them to help make personnel decisions? Aside from creating a good table of con­tents and using separators and color ­coding and the like, the real question is, What should the portfolio contain to make it readable and useful to someone who might have to read many such portfolios? We agreed that the portfolio needed "narrative glue" to make it readable; that is, it had to tell the story of the course. But we also wanted to know what sense readers would make of this story. To answer this question, I selected three people from my institution who agreed to read my portfolio: my department chair, the chair of the College Personnel Committee, and the provost. Along with my portfolio (sans student work), I sent each of them the following set of questions:

 

i. Does the portfolio provide you with insight into the way I teach this course? What else would you need to see in order to get such insight?

 

2. If you were on a search committee or a promotions and tenure com­mittee or an awards committee, would this portfolio help you judge the quality of my perform­ance as a teacher? What if your field of interest or expertise had no relationship to the course in question? If many or all candi­dates under review by such a committee were to submit portfolios such as this, would that pre­sent a problem? If so, do you have any suggestions?

 

3. If you were assigned to teach this course for the first time, would reading this portfolio be helpful to you?

 

4. Would a portfolio such as this be useful to make an argument about the value of a course such as this and/or the way in which it should be taught?

 

5. What, if anything, could be done to make the portfolio more acces­sible? Is there material that should be omitted or additional material that should be included? Should the portfolio be organized differ­ently? Would a cover sheet of guidelines for the reader be helpful?

 

6. With regard to samples of student work, would it have been helpful to you if I had included some with the material I sent you? Should they be included with an actual portfolio?

 

7. Do you have any additional com­ments or suggestions that would be helpful?

 

As of this writing, I have received only one response ‑ from the provost, who happens to be a mathe­matician (and who has since left the institution). In his written feedback to me, he said that the portfolio reveals much about the course and how I teach it, but that he was not sure that he had learned enough to form an opinion with regard to the controversy concerning the place of the course in the curriculum. He sug­gested that a clearer statement of the goals of the course and how these goals relate to the rest of the mathe­matics curriculum would be helpful.

 

The provost reports that the ele­ments of the portfolio that he found most useful were "the extremely detailed syllabus"; the supplementary exercises and exams, which "showed the range of mathematics you includ­ed in the course, as well as what a successful student should be able to do at the end of the course"; and the pre‑ and post‑tests together with the results. He added that with some examples of student work (which I had not included in what I provided, but do appear in the actual portfolio), the portfolio should "go a long way towards helping make a good evalua­tion of the quality of teaching and the amount of student learning in this course."

 

Additionally, the provost suggest­ed that one item about which he would have liked more information was the question of whether I had experimented with different coopera­tive learning techniques, and if so, how well each of these techniques worked. His final remark was that he was "relieved by the length of [the] portfolio." He said he had feared that "the volume of paper per portfolio would overwhelm my capacity to deduce useful information from them." Apparently, this proved not t be the case (although I can't help wondering whether he would still feel the same way if 20 people had simultaneously submitted portfolios for 1: perusal).

 

Conclusion

 

As you may surmise from the open­ing section of this case study, the history of my involvement with the course portfolio is closely tied to m3 own personal circumstances and interests. I do not claim that others would have my experience. Never­theless, judging from the case studies of my colleagues whose work is rep­resented here, I believe that the per­sonal growth and development I experienced is likely to be shared by any readers of this volume who are willing to devote the time and effort needed to develop a course portfolio of their own.