# 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.

• The test covers some collection of routine mechanical skills, rather than concepts.
• The test is comprehensive and rigorous.
• Pass/fail grading is used, but a high score is needed to pass.
• The test may be retried repeatedly without penalty, as long as it is eventually passed. If it is not eventually passed, some substantial course penalty is incurred.
• A new version of the test is available for each attempt.

## 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.

• To assure that students have the prerequisite mathematical skills needed for the course. For example, gateway tests are often given near the beginning of second semester calculus courses to assure that the students really have mastered the mechanics of differentiation and elementary integration covered in their first semester calculus courses. Such gateway tests over prerequisite knowledge can also give early warning to students whose preparation for the course is too inadequate for them to survive it.
• To assure that students have mastered routine skills taught in the present course. The following are some of the advantages of using gateway tests for this purpose, then focusing more on concepts, multiple-step problems, and applications in the traditional tests for the course.
• The skills really will get learned.
• By de-emphasizing rote mechanical skills on the traditional tests, the students are less likely to view the learning of such skills as the main point of the course. Students tend to treat gateway tests as something they just have to do, while viewing the material covered on the traditional tests as the actual source of the course grade and therefore the heart of the course. (We all know students who believe that the main content of a first calculus course is rules for differentiation and integration. Unfortunately, final examinations for such courses often provide supporting evidence.)
• The instructor does not have to expend so much class time on routine material. Students are more likely to accept the idea that it is up to them to master the routine skills outside of class, rather than to have the skills drilled into them during class, if the students are given multiple chances to demonstrate mastery of those skills with no detrimental effect on their course grade so long as that mastery is ultimately displayed.

## 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.

• To accomplish its purpose, a gateway test really does need to be rigorous. If most students pass it on their first attempt, then the test may not have done what it was intended to do. (But maybe it has. If 70% of a class has really mastered a skill, and it is important enough for the other 30% to master it to give a gateway test over that skill, then it is entirely reasonable that 70% of the class will pass the test on the first try.) Keep in mind that, unlike traditional tests, students will sometimes not prepare for their first attempt at a gateway test, but rather just use that first attempt to get a feel for the level of difficulty of the test. It is not uncommon to have most students not pass on the first attempt, but have a dramatic surge in the number of passes on the second try once the students realize the performance level expected of them.
• However, it is also necessary to be reasonable about the level of difficulty of a gateway test. If most students are taking five or six tries to pass, then the test is probably much too difficult.
• Since the purpose of a gateway test is to assure mastery of some set of skills, rather than just borderline competency, it is important to require a high score for passing. The author once believed that 100% is appropriate, on the theory that anything less allows the student to avoid learning some particular skill being tested and yet still get through the test. However, many students tend to view a test as being grossly unfair if they have to retake it just because they have made one trivial error. You may not enjoy the consequences of requiring a student to retake a gateway test on logarithms when the student has worked nine questions correctly and did the following on the tenth:
ln(e^2) + ln(e^3) = 2 + 3 = 6.
If you are tempted to require 100% for passing, consider making the test a bit harder and requiring 90% for passing.
• It is often appropriate to offer the gateway test the first time during class, then have the students make appointments during office hours for further attempts. If hundreds of students are involved, an alternative is to have scheduled gateway tests in large lecture halls, perhaps in the evening.
• As has already been mentioned, a different version of the test must be available for each attempt. Each version of the test should test the same skills. It is not a good idea to have the test get progressively more difficult with each attempt (and you may find yourself accused of this if material is introduced on a version of a test that was not on previous versions, even if the resulting test is really no more difficult).
• There must be some time limit specified for passing the test. Two or three weeks is often appropriate. If this is not done, then you should prepare for a flood of students in your office taking the test the last few days of the semester, when they really should be studying for the final examination.
• Students should not be allowed to retake a test immediately after failing it. It is better to have them study the material for an evening and then retake the test the following day.
• It is important to have help available for students who just cannot seem to get through the test. This help can come from the instructor or from an institutional tutoring service, but it is important that the availability of such help be spelled out to the students. A student who keeps trying the test without success can feel like a drowning person with no life preservers anywhere in sight. It is important to point out the life preservers.
• Occasionally, a student will attempt to get through a gateway test by adopting the strategy of not studying for it at all, but rather trying it time after time in the hope of getting lucky. If a student appears to be doing this, then it is time to intervene, perhaps by asking the student to work out correctly all the problems missed on previous attempts before being granted another one.
• Many good students are accustomed to doing well on tests, and can become discouraged if they do not pass a gateway test on the first attempt. When a student says sorrowfully that he or she has just "failed" the gateway test, it is very important to counter immediately by saying "No! You haven't failed this test at all! You just haven't yet passed it." This may sound contrived, but oftentimes it really does cheer them up.

## 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:

• Simplification of compound fractions.
• Laws of exponents and radicals.
• Solution of equations involving rational expressions.
• Solution of simple inequalities, including those involving absolute values.
• The distance formula.
• Solutions of simple 2x2 systems of linear equations.
• Composition of functions.
• Linear equations.
• Logarithms (but only very basic material).
• Domains and ranges of very simple functions.
• Multiplication, division, and factoring of polynomials.
• Rationalizing denominators.
• Solution of very simple word problems.

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 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 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.