Focus on Calculus
A Newsletter for the Calculus Consortium Based at Harvard University
Winter 1999, Issue No. 16

Modular Precalculus, Clustering, and Reform Mathematics
Nancy C. Marcus

Seventh Annual Conference on the Teaching of Mathematics
The Rest of the Story: Putting the US Twelfth Grade TIMSS Mathematics Achievement Results in Perspective
William H. Schmidt and Leland S. Cogan

On-Line Assessment with Wiley Web Tests
John Lindsay Orr and William J. Lewis

Upcoming Chautaugua Courses
From the Publisher
About this Newsletter

On-Line Assessment with Wiley Web Tests

John Lindsay Orr and William J. Lewis, University of Nebraska - Lincoln

We started using computers to give automatically-graded basic skills tests in calculus in order to make sure that students in our reform calculus course had strong enough a grasp of the core algebraic skills to satisfy our various constituencies (see our article, "On-line Gateway Exams in Calculus"; Focus on Calculus, 14). Although our focus then was on "Gateway Exams," it was clear to us from the start that the software we were developing (and which is now marketed for calculus classes as "Wiley Web Tests for Calculus") had the potential to assist a wide variety of styles of assessment. In this article we want to describe the experiences we, and some of our colleagues at other schools, have had using the Wiley Web Tests system in a variety of ways over the past three years.


Because the stakes are high for our Gateway Exams, students must take the test for credit under a proctor's supervision. In situations where the consequences are lighter, such as homework assignments, it makes sense to allow the students to take on-line assignments from anywhere they can connect to the Internet. The Wiley Web Tests system supports this kind of test in the "Unproctored" mode.

For example, Bill Ziemer at Indiana University wrote his own bank of test questions (using the LaTeXtest-authoring tools described below) which he uses to give his students on-line homework assignments. For him, the advantage of an on-line system is that it enables him to assign homework to large lecture classes and know that all the homework will be graded. He reports that previously, because of the size of his classes, graduate TAs could only grade a fraction of the students' total assignment.

At the University of Nebraska, we have used the test bank from the Wiley Web Tests for Calculus to give review assignments at the beginning of courses. For example, some instructors have assigned a selection of differentiation questions as review at the start of Calculus II, or assigned integration and differentiation questions for homework in multivariate calculus.

At Arizona State University, Matthias Kawski, who ran Gateway Exams from the system in his Calculus I class, also experimented with an on-line test in which, unlike the Gateway Exams, partial credit was available. He found that even though there was no absolute requirement to reach an 80% score in the second test, his students wanted the extra credit enough that they took the exams repeatedly anyway, until they all scored 80% or better.

Going further, regular on-line homework assignments offer an exciting possibility of changing the paradigm of how we use classroom time. Students can be given regular reading assignments, together with on-line mastery homework on the basics of the material in their reading, before any classroom discussion takes place. The students' first contact with the material is in their private study, and they measure their progress by retaking the homework assignment until they can demonstrate mastery of the basic skills of the topic. Only at this point, after all-or at least most-of the class have reached a common level of competency, is there a classroom discussion of the material. Going into class, the instructor knows from the scores in the on-line grade book that the class has reached a level of familiarity with the basics of the topic, be they names and dates, or definitions and techniques of computation. The classroom time need not be taken up with introducing this material, or with talking about it as if students were seeing it for the first time. Instead it enables more productive, high-level classroom discussion. Colleagues in our Psychology Department who are developing course material for the Web Tests system using this paradigm report exciting improvements in student results.

New Preview Feature

As a key feature, the Wiley Web Tests system contains questions that ask the student to enter a formula. The student's answer is interpreted by the machine as a mathematical expression when it's graded, so that two equivalent answers such as "(x+1)^2" and "x^2+2x+1" are graded the same. Students enter the formula for their answer in a fairly intuitive syntax, which is compatible with, for example, the TI-86 calculator, and so is something with which most students have experience.

Nevertheless, in the first few semesters that we gave Gateway Exams using this type of "open response" question, we saw quite a number of students who clearly knew how to answer the question correctly, but were graded wrong because of small slips. A typical example would be omitting parentheses-for example, typing "1/x^2+1" in place of "1/(x^2+1)". Although the computer can recognize and warn about syntax that is flat wrong ( such as unbalanced parentheses), this type of error is frustrating for the student but difficult to provide help with.

Our solution was to provide a capability for the student to preview an answer in standard mathematical notation. In the latest version of the system software (Version 0.4), open response questions come with a "Preview" button beside the answer entry box. When the student has typed an answer, she presses the button and a screen pops up showing her properly typeset answer. If the student really does know the answer, she can spot what she's done wrong as soon as she sees something like in place of .

Version 0.4 of the software, which is included with the Wiley Web Tests for Precalculus and College Algebra, is available as an upgrade to users of the Wiley Web Tests for Calculus from the URL below.

Interactive Study Sessions

In addition to testing, the Web Tests system can be used in "study" mode to provide students a set of interactive exercises, or Study Sessions. In a Study Session, the system asks the student a series of questions, which the student works one at a time. The machine provides immediate feedback on each question as it's answered, and offers hints or worked solutions. If the student gets a question wrong, she can return to the question and try again, or go on to a new question. This dialog continues for as long as the student desires. On-Line Resources

The Web site at contains links to resources for the Wiley Web Tests system. Licensed users of the Calculus system can download an upgrade to their system software from this site with a password. There are also links to a FAQ page, and software tools to help administer the system.

Users who want to write their own test questions can download a set of LaTeX macros that automate the process of preparing test questions. These macros- which we used to write the test banks for the Calculus and Precalculus systems- make it easy to include mathematical notation, and to organize and proofread large numbers of questions.

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