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 Test A - Prealgebra and Elementary Algebra (9th Grade) Test B - Geometry (10th grade) and Advanced Algebra (11th Grade) Test C - Precalculus including Trigonometry (12th grade)

Each student takes two tests: either A and B, or B and C depending on the student'shigh school preparation.

Each test item is constructed to correspond to an objective chosen from a detailedobjective list. Test objectives and items are chosen because they reflect skills that areprerequisite to a given entry level college mathematics course. The content of each testcovers a large segment of the objectives list for that test. The content and objectivescovered changes slightly from year to year. Larger changes can take place to reflectcurriculum changes.

Before any item is used on the Placement Test, it is piloted extensively and modifiedif necessary to insure that it tests the objective and that it is statistically useful,i.e., to insure that it discriminates well.

## Use of the Tests

When the UW System Mathematics Placement Tests were developed, they were written to beused strictly as a tool to aid in the most appropriate placement of students. They werenot designed to compare students, to evaluate high schools or to dictate curriculum. Thetest data made available to the development committees does not allow for inappropriateuse of the tests. The way an institution chooses to use the tests to place students, thesubtests used and the cutoff scores on these subtests are decisions made by eachinstitution. The Center for Placement Testing can and does help UW System institutionswith these decisions.

Each campus continues to analyze and modify its curriculum and hence the way that ituses the placement tests to place students. Cutoff scores may be changed over time, as maythe subtests chosen by a campus to reflect the prerequisites for its curriculum. Follow-upstudies are made to determine the effectiveness of placement procedures. Contact ismaintained with high schools so that modifications in the curriculum in both the highschools and the UW-System can be discussed.

## Future Directions

As the mathematics curriculum continues to evolve, the UW System Mathematics PlacementTests will evolve with it. Since the members of the UW System Mathematics Placement TestCommittee are faculty who regularly teach the entry level courses, they have a directimpact on the evolution of these courses, and the creation of new courses. In this way theUW System Mathematics Placement Tests can change immediately with the curriculum wherenational tests will have a lag-time of up to several years. An indication of this is theuse of calculators on the UW System Mathematics Placement Tests, which were allowed beforecalculators were permitted on national tests.

## Preparing Students for Placement Tests

The best way to prepare students for the placement tests is to offer a solidmathematics curriculum and to encourage students to take four years of college preparatorymathematics. We do not advise any special test preparation, as we have found that studentswho are prepared specifically for this test, either by practice sessions or the use ofsupplementary materials, score artificially high. Often such a student is placed into ahigher level course than his or her background dictates, resulting in the student eitherfailing or being forced to drop the course. Due to enrollment difficulties on manycampuses, students are often unable to transfer into a more appropriate course after thesemester has begun.

Significant factors in the placement level of a student are the high school coursestaken, as well as whether or not mathematics was taken in the senior year. Data from otherstates indicates that four years of college preparatory mathematics in high school notonly raises the entry level mathematics course, but predicts success in other areas aswell, including the ability to graduate from college in four years.

## High School Preparation for College Mathematics

### Calculus

The number of high schools offering some version of calculus has increased markedlysince the UW System Mathematics Test Committee's first statement of objectives andphilosophy, and experience with these courses has shown the validity of the Committee'soriginal position. This position was that a high school calculus program may work eitherto the advantage or to the disadvantage of students depending on the nature of thestudents and the program. Today, it seems necessary to mention the negative possibilitiesfirst.

A high school calculus program not designed to generate college calculus credit islikely to mathematically disadvantage students that go on to college. This is true forall such students whose college program entails use of mathematics skills, andparticularly true of students whose college program involves calculus. High schoolprograms of this type tend to be associated with curtailed or superficial preparation atthe precalculus level and their students tend to have algebra deficiencies which hamperthem not only in mathematics courses but in other courses in which mathematics is used.

The positive side is that a well conceived high school calculus course whichgenerates college calculus credit for its successful students will provide a mathematicaladvantage to students who go on to college. The Mathematical Association of Americahas studied high school calculus programs, and listed features which characterizesuccessful ones. These features include the following:

1. they are open only to interested students who have completed the standard four year college preparatory sequence. A choice of mathematics options is available to students who have completed this sequence at the start of their senior year.
2. they are full year courses taught at the college level in terms of text, syllabus, depth and rigor.
3. their instructors have had good mathematical preparation (e.g., college mathematics major) and are provided with additional preparation time.
4. they are taught with the expectation that their successful graduates will not repeat the course in college, but will get college credit for it.

A variety of special arrangements exist whereby successful graduates of a high schoolcalculus course may obtain credit at one or another college. A generally accepted methodis for the students to take the Advanced Placement Examinations of the College Board.Success rates of students on this exam can be a good tool for evaluation of the success ofa high school calculus course.

### Geometry

The range of objectives in this document represents a small portion of the objectivesof the traditional high school geometry course. The algebra objectives represent asubstantial portion of the objectives of traditional high school algebra courses. Theimbalance of test objectives can be explained in part by the nature of the entry levelmathematics courses available at most colleges. The first college mathematics coursegenerally will be either calculus or some level of algebra. A choice is usually based onthree factors: (1) high school background; (2) placement test results; (3) curricularobjectives. One reason for the emphasis on algebra in this document is that virtually allcollege placement decisions involve placement into a course that is more algebraic thangeometric in character.

There are reasons for maintaining a geometry course as an essential component in acollege preparatory program. Since there are no entry level courses in geometry at thecollege level, it is essential that students master geometry objectives while in highschool. High school geometry contributes to a level of mathematical maturity which isimportant for success in college.

### Logic

Students should have the ability to use logic within a mathematical context, ratherthan the ability to do symbolic logic. The elements of logic that are particularlyimportant include:

1. Use of the connectives "and" and "or" plus the "negation" of resultant statements, and recognition of the attendant relationship with the set operations "intersection" "union" and "complementation".
2. Interpretation of conditional statements of the form "if P then Q," including the recognition of converse and contrapositive.
3. Recognition that a general statement cannot be established by checking specific instances (unless the domain is finite), but that a general statement can be disproved by finding a single counter example. This should not discourage students from trying specific instances of a general statement to conjecture about its truth value.

Moreover, logical thinking or logical reasoning as a method should permeate the entirecurriculum. In this sense, logic cannot be restricted to a single topic or emphasized onlyin proof-based courses. Logical reasoning should be explicitly taught and practiced in thecontext of all topics. From this, students should learn that forgotten formulas can berecovered by reasoning from basic principles, and that unfamiliar or complex problems canbe solved in a similar way.

Although only two of the objectives explicitly refer to logic, the importance oflogical thinking as a curriculum goal is not diminished. This goal as well as otherbroad-based goals are to be pursued despite the fact that they are not readily measured onplacement tests.

### Problem Solving

Problem solving involves the definition and analysis of a problem together with theselecting and combining of mathematical ideas leading to a solution. Ideally, a completeset of problem solving skills would appear in the list of objectives. The fact that only afew problem solving objectives appear in the list does not diminish the importance ofproblem solving in the high school curriculum. The limitations of the multiple choiceformat preclude the testing of higher level problem solving skills.

### Mathematics Across the Curriculum

Mathematics is a basic skill of equal importance with reading, writing, and speaking.If basic skills are to be considered important and mastered by students they must beencouraged and reinforced throughout the curriculum.

Support for mathematics in other subject areas should include:

 a positive attitude toward mathematics attention to correct reasoning and the principles of logic use of quantitative skills application of mathematics curriculum.