Gateway Testing
in Mathematics Courses
at the University of Michigan


The purpose of this document is to describe briefly the gateway testing program in the Mathematics Department of the University of Michigan. More detailed information can be found in the article

The Lotus 123/TeX gateway testing software mentioned in that article is described in more detail in UM Math Gateway Test Software, another WWW document. That document contains a button you can push to have the software delivered directly into your disk directory.

Gateway Tests

For the purposes of this document, a gateway test is a test that has these characteristics.

Reasons for Using Gateway Tests

The primary purpose of a gateway test is to assure that some particular collection of skills has actually been learned. Traditional forms of testing do not always do so. For example, a precalculus student who finds laws of logarithms confusing and difficult may manage to avoid learning them, yet still get a satisfactory grade in the course by learning other material solidly enough to survive the tests. Such a student may then get into trouble in later courses having a precalculus prerequisite in which knowledge of laws of logarithms is assumed.

The difference between traditional testing and gateway testing is that if a student performs poorly on a gateway test over a particular collection of skills, the student cannot compensate by learning other collections of skills well enough to overcome the effect of the poor performance on the gateway test. Instead, the student must go back and relearn the particular collection of skills well enough to pass the gateway test. The test does not go away until the student has learned the material well enough to pass it. (Therefore such a test represents a barrier to successful course completion if the material is not learned, for which reason David Smith's Project Calc group originally called such tests barrier tests; they later substituted the less ominous word gateway for barrier. Gateway tests have been around for a long time, under a number of different names.)

Gateway tests can be used in a course to achieve either, or both, of the following two objectives.

Some Observations and Suggestions

We have been using gateway tests in our precalculus and calculus courses at the University of Michigan for the last several years. Here, in no particular order, are some of the things we have learned about them.

The University of Michigan Precalculus Gateway Test

We give a gateway test near the beginning of our reformed precalculus course, Mathematics 105, that tests skills that should have been learned in high school courses. The skills tested include:

The test has 25 multiple-choice questions for which the student is allowed one hour. Passing is 20. Students typically require three or four tries to pass it.

A sample copy of the test is available in TeX dvi format by selecting the following icon. Depending on how your Web browser is set up, either the dvi file will be delivered to you or a dvi file viewer or printer will be started when this icon is selected.

[select here] Select this icon to receive the two-page sample copy of the test.

Each question on this test was selected randomly from a group of ten similar questions, and the question number on the test corresponds to the number of the selected question in the test bank. See the WWW document UM Math Gateway Test Software for more on the software used to create the test.

The University of Michigan Calculus Derivative Gateway Test

We give a derivative gateway test near the end of our first calculus course, Mathematics 115, to assure that the standard techniques of differentiation have been mastered. These skills include the generalized power rule, product rule, and chain rule, as well as rules for the differentiation of trigonometric, exponential, and logarithmic functions. Each test has formulas for seven functions, of which six must be differentiated correctly to pass. The student is allowed 15 minutes for the test. Students typically require two or three tries to pass it.

A sample copy of the test is available in TeX dvi format by selecting the following icon. Depending on how your Web browser is set up, either the dvi file will be delivered to you or a dvi file viewer or printer will be started when this icon is selected.

[select here] Select this icon to receive the one-page sample copy of the test.

Each question on this test was selected randomly from a group of twenty similar questions, and the question number on the test corresponds to the number of the selected question in the test bank. See the WWW document UM Math Gateway Test Software for more on the software used to create the test.


Last modified Fri 12 May 1995 18:26 EDT

Bob Megginson
Department of Mathematics
University of Michigan
meggin@math.lsa.umich.edu