Student Assessment in Calculus: A Report of the NSF Working Group on Assessment in Calculus

This report is a product of the National Science Foundation (NSF) Working Group on Assessment in Calculus, formed at the summer 1992 meeting of the NSF Calculus Program. The report is definitely a must if you teach calculus or related courses, or if you are interested in thinking about student assessment in order to gain a deeper understanding of student learning or to improve instruction. It outlines the state of the art of assessment in calculus. It provides valuable information on methods for assessing student understanding, illustrating these methods through examples of student work. It also maps out key areas in teaching, learning, and assessment that require further research. It is certainly a worthwhile reference for anyone who is interested in researching mathematical thinking, or who is concerned about student learning and ways to assess it.

Preceding the body of the report is an introduction providing a brief history on curriculum reform in mathematics education in the United States over the past two decades, and a user's guide to the report. The body of the report begins by describing the framework for assessment of student understanding in mathematics and its application to calculus, with a focus on NSF-supported calculus projects. This is followed by a general description of what is known about assessment, the assessment issues faced by the calculus projects, a research and development (R&D) agenda for calculus assessment, and finally, assessment examples. The report concludes with a reference section which provides a good bibliography for mathematics teaching, learning and assessment, and an appendix section which comprises project abstracts for 1988-94 awards in the NSF Calculus Program.

The framework for assessment which is described is handy for developing an assessment package for any mathematics program or course, or for examining the package that one has in place, in terms of the philosophical or pedagogical approach and the assumptions represented, content focus and balance, expected thinking processes and skill competencies, student exposure to a variety of problem situations, diversity, access and differential performance, circumstances that affect performance or production of work, and perceived value. Its application to calculus provides instructors with a good assessment package on which to model their calculus courses.

Four categories of assessment are outlined as follows:

• category 1 - pencil and paper (multiple choice items, short-answer items, open-ended items, and student-constructed tests);
• category 2 - performance assessment (performance tasks, investigations, projects, observations and interviews);
• category 3 - portfolio assessment;
• category 4 - self assessment.

The pros and cons of each category of assessment are highlighted, as well as the technical issues raised by each of these categories of assessment. This is valuable information for K-16 mathematics educators, particularly those making decisions about which category or combination of assessment categories should be used for a particular mathematics program or course.

Assessment issues faced by the calculus projects include layered activity structures and corresponding layered assessment structures, assessment as a signal of what is valued, assessment of group work as enhancement of what is valued, issues of reliability, validity and fairness, cost and other constraints. The discussion of these issues provides information which is useful for designing and developing mathematics courses, particularly calculus and related courses.

Six categories are outlined in the R&D agenda for calculus assessment as follows:

• category 1 - combined R&D into (a) understanding student understanding and (b) generating relevant assessment items in concert with that understanding;
• category 2 - studies of the effects of the uses of various kinds of assessments on the students and faculty involved;
• category 3 - the creation of assessments as levers for change;
• category 4 - studying the effects of various assessment formats on student performance (e.g., open-book, closed-book, group work, use of technology, time pressure, etc.);
• category 5 - the adaptation of known assessment types to calculus assessment;
• category 6 - creating assessments that are consistent with expected student uses of mathematics, and the conditions under which they might be expected to learn it and use it.

In the discussion on category 1, two complementary perspectives necessary for understanding what a mathematical domain is about are highlighted. The first deals with mathematical content (topics, themes, mathematical aspects and cognitive aspects) and the second with cross-cutting mathematical issues (role of contexts, access to mathematics, communication, writing, reading, group work, community, background, intellectual context, mathematics as a profession, and symbol manipulation). The discussion in this section as a whole, and the illustrations given (e.g., student understanding of the concept of function), provide a deeper understanding of the assessment process and of the tools for assessing student learning and improving instruction, particularly in calculus.

A variety of examples of assessment activities, from narrow to rich (a computation, a project, two laboratories, an extended instructional/laboratory sequence, two cumulative assessments, a sample end-of-term examination, and a sample two-day take-home), and samples of student work on these activities are presented and discussed in the context of the given framework for assessment. These examples provide instructors of calculus with a repertoire of practical tools to use in their courses, and give them an insight into student learning. They also serve as guideposts for instructors to use in developing assessment activities for calculus and related courses.

This is a very detailed and thorough report on student assessment in calculus. Despite the book's focus on calculus, the assessment issues discussed extend to all fields of mathematics and all levels of education. For any mathematics educator, it is definitely worth reading at least once. As indicated in the user's guide, "an enhanced understanding of student learning and ways to assess it will serve us all well." Read this report for such an understanding!

Nkechi Madonna Agwu is an assistant professor at the Borough of Manhattan Community College, City University of New York (BMCC-CUNY). Her professional interests are curriculum development and assessment, teacher preparation, history of mathematics, teaching/learning of statistics using student generated data, life experiences and real world problems, and student retention. Her email address is nmabm@cunyvm.cuny.edu.