Reflective portfolios help students assess their own growth. Project portfolios identify their interests and tackle more ambitious assignments.
Background and Purpose
We discuss our use of portfolio assessment in undergraduate mathematics at Occidental College, a small, residential liberal arts college with strong sciences and a diverse student body. Portfolios are concrete and somewhat personal expressions of growth and development. The following elements, abstracted from artist portfolios, are common to all portfolios and, taken together, distinguish them from other assessment tools:
These defining features also indicate assessment goals for which portfolios are appropriate.
Our use of portfolios is an outgrowth of more than six years of curricular and pedagogical reform in our department. We focus here on two types of portfolios reflective portfolios and project portfolios which we use in some second-year and upper-division courses. We also describe how we integrate portfolios with other types of assessment.
Reflective Portfolios. In a reflective portfolio, students choose from a wide range of completed work using carefully specified criteria. They are asked to explicitly consider their progress over the length of the course, so early work which is flawed may nonetheless be included to illustrate how far the student has progressed. A reflective portfolio helps students assess their own growth.
The collection of portfolios can also help a teacher reflect on the strengths and weaknesses of the course. It can point out strong links made by the students, and indicate struggles and successes students had with different topics in the course. It can further open the window to student attitudes and feelings.
Project Portfolios. Project portfolios are one component of a "professional evaluation" model of assessment [1, 2]. The other components are "licensing exams" for basic skills, small but open-ended "exploration" projects, and targeted "reflective writing" assignments. These different components, modeled on activities engaged in by professionals who use and create mathematics, have been chosen to help students develop a more mature approach to their study of mathematics.
Students choose projects from lists provided for each unit of the course. Completed projects are included in their project portfolio. This method of assessment helps students identify their interests, produce work of high quality, and tackle more ambitious assignments which may take several weeks to complete. Collecting work from the entire course encourages students to look beyond the next test or quiz.
Method I. Reflective Portfolios
To increase students' self-awareness of how their understanding has developed requires that information about this understanding be collected by the students as the course progresses. Keeping a journal for the course is a useful supplement to the usual homework, classwork, quizzes, and tests. For example, in a real analysis course, students are asked to make a journal entry two or three times each week as they, among other possibilities: reflect on their readings or problem sets, discuss difficulties or successes in the course, clarify or connect concepts within this course or with those in other courses, or reflect on their feelings toward this course or mathematics in general. The teacher collects and responds to the journals on a regular basis.
Near the end of the semester, students are asked to prepare a reflective portfolio. The assignment says:
The purpose of this portfolio assignment is to allow you to highlight your own selections of your work and give an analysis of them in your own words. I will focus on two specific things in evaluating your portfolio: (1) an understanding of some key concepts in real analysis and (2) a self-awareness of your journey (where you started from, where you went, and where you are now).
Select three pieces of work from this semester to include in your portfolio. These pieces can include journal writing, homework assignments, tests, class worksheets, class notes, computer experiments, or any other pieces of work you have produced in this class. Your analysis should explain your reasons for picking these pieces. As examples, you might consider a selection which shows the development of your understanding of one key concept, or a selection which shows your growing appreciation of and proficiency with formal proofs, or a selection which shows your connection of two or more key concepts.
The most important part of the portfolio is your reflection on why you chose the pieces you did, how they show your understanding of some key concept(s), and how they show a self-awareness of your journey through this class. You should definitely write more than one paragraph but no more than five pages. The portfolio is 5% of your final grade.
Method II. Project Portfolios
The project portfolio corresponds to papers and other completed projects a professional would include in his or her curriculum vitae. Two sorts of themes are used for projects in a multivariable calculus course with a linear algebra prerequisite. For example, "Vector Spaces of Polynomials" concerns fundamental ideas of the course, while "Optimization in Physics" is a special topic for students with particular interests. All students are required to complete certain projects, while in other cases they choose from several topics. Here is an example of a portfolio project assignment.
Vector Spaces of Polynomials. What properties define a vector space over R? Let P3 denote the space of polynomials over R of degree less than or equal to three. Show that P3 is a vector space over R with natural definitions for "vector addition" and "scalar multiplication." Find a basis for P3 and show how to find the coordinate representation of a polynomial in P3 relative to this basis. Show that "differentiation" is a linear transformation from P3 to P2, where P2 is the space of polynomials over R of degree less than or equal to 2. Find a matrix representation for this transformation relative to bases of your choice for the input space P3 and the output space P2. To check your work, first multiply the coordinate representation of a general polynomial p(x) in P3 by your matrix representation. Then compare this result with the coordinate representation of its derivative, p´(x). Discuss what you learned from this project. In particular, has this project changed how you think about vector spaces, and if so, in what way?
Project reports must include: a cover page with a title, author's name, and an abstract; clear statements of problems solved, along with their solutions; a discussion of what was learned and its relevance to the course; acknowledgement of any assistance; and a list of references. Reports are usually between five and ten pages long. They are evaluated on both mathematical content and quality of presentation. First and final drafts are used, especially early in the course when students are learning what these projects entail.
For the typical project, students will have one week to produce a first draft and another week to complete the final draft. Students may be working on two different projects in different stages at the same time. Allowing for test breaks and longer projects, a completed course portfolio will comprise seven to ten projects. Students draw on this portfolio as part of a self-assessment exercise at the end of the course.
The collection of reflective portfolios can serve as feedback of student mathematical understanding and growth through the course. It can sometimes also give us a glimpse of the joy of learning, and thus the joy of teaching. One student wrote:
Without hesitation I knew the selections I was going to analyze. I think one of the reasons why this worksheet and these two journal entries came to mind so quickly is because they reflect a major revelation I had in the course. Not often in my math classes have I felt so accomplished .... These selections represent something I figured out ON MY OWN! That's why they so prominently came to mind.
The first drafts of projects for the project portfolio are a rich source of information on how students are thinking about important topics being covered in the course. The final drafts reveal more clearly their depth of understanding and degree of mastery of mathematical language. We find that these projects help students achieve a better conceptual understanding of important aspects of the course and become more mature in presenting mathematical arguments. The discussion sections of their reports offer students some chance for reflection. For example, one student wrote the following in his report on vector spaces of polynomials:
I am fascinated to see how much we can deduce from an abstract space without being able to visualize it. While the concept of a polynomial vector space still amazes me it is very interesting to see how we can make this abstraction very tangible.
Use of Findings
Reflective portfolios document the development of mathematical knowledge and feelings about mathematics for each student in a particular course and class. This information has been used to realign class materials, spending greater time on or developing different materials for particularly difficult concepts. In addition, the evaluation of student journals throughout the semester allows for clarifying concepts which were not clearly understood. As an example, one student discussed an in-class worksheet on uniform continuity of a function f(x) and did not understand the role of x in the definition. Seeing this, class time was set aside to clarify this concept before more formal evaluation.
First drafts of project portfolios may highlight difficulties shared by many students and thus influence teaching while a course is in progress. Reflecting on completed portfolios can also lead to changing how we teach a course in the future. The projects themselves may be improved or replaced by new ones. Special topics which were previously presented to the entire class may be treated in optional projects, leaving more class time for fundamental ideas. These projects can also lead to more radical reorganization of a course. For example, one year a project on Fubini's Theorem revealed that students had trouble understanding elementary regions for iterated integrals. The next year we first developed line integrals, then introduced elementary regions through Green's Theorem before treating iterated integrals. Teaching these topics in this unusual order worked quite well.
The reflective portfolio, journal writing, and the project portfolio are all writing-intensive forms of assessment. Clarity, time and patience are required of both teachers and students in working with assignments like these which are unusual in mathematics courses.
The reflective portfolio is little more than a short paper at the end of the semester. The time is put in during the semester as the teacher reads and responds to the journals. Cycling through all the journals every few weeks is probably the most efficient and least burdensome way of handling them.
Some students enjoy reflective writing while others may feel awkward about it. In the real analysis class, for example, the female students used the journal more consistently and produced deeper reflections than did the males. Teachers must also allow for reflections which may not be consistent with the outcomes they desire.
Several techniques can make correcting first drafts of project portfolios more efficient and effective. Comments which apply to many reports for a given project can be compiled on a feedback sheet to which more specific comments may be added for individual students. Peer review will often improve the quality of written work. A conference with a student before his or her report is submitted may make a second draft unnecessary. Students often need to complete several projects before they fully appreciate the degree of thoroughness and clarity required in their reports. Sharing examples of good student work from previous years can help communicate these expectations.
 Knoerr, A.P. "A professional evaluation model for assessment." Joint Mathematics Meetings. San Diego, California: January 8-11, 1997.
 Knoerr, A.P. "Authentic assessment in mathematics:
a professional evaluation model." Ninth Annual
Lilly Conference on College Teaching West,
Lake Arrowhead, California: March 7-9, 1997.