Mathematics Placement Examinations

The UW System Mathematics Placement Test consists of three tests labeled A, B and C. Each test analyzes a student's abilities in a specific set or level of objectives. The content of these tests is:

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's high school preparation.

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

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

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

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

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

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

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

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

__A high school calculus program not designed to generate college calculus credit is
likely to mathematically disadvantage students that go on to college__. This is true for
all such students whose college program entails use of mathematics skills, and
particularly true of students whose college program involves calculus. High school
programs of this type tend to be associated with curtailed or superficial preparation at
the precalculus level and their students tend to have algebra deficiencies which hamper
them 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 which
generates college calculus credit for its successful students will provide a mathematical
advantage to students who go on to college__. The Mathematical Association of America
has studied high school calculus programs, and listed features which characterize
successful ones. These features include the following:

- 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.
- they are full year courses taught at the college level in terms of text, syllabus, depth and rigor.
- their instructors have had good mathematical preparation (e.g., college mathematics major) and are provided with additional preparation time.
- 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 school calculus course may obtain credit at one or another college. A generally accepted method is 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 of a high school calculus course.

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

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

Students should have the ability to use logic within a mathematical context, rather than the ability to do symbolic logic. The elements of logic that are particularly important include:

- 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".
- Interpretation of conditional statements of the form "if P then Q," including the recognition of converse and contrapositive.
- 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 entire curriculum. In this sense, logic cannot be restricted to a single topic or emphasized only in proof-based courses. Logical reasoning should be explicitly taught and practiced in the context of all topics. From this, students should learn that forgotten formulas can be recovered by reasoning from basic principles, and that unfamiliar or complex problems can be solved in a similar way.

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

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

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 be encouraged 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. |

The impact of the computer on daily life is apparent, and consequently many high schools have instituted courses dealing with computer skills. While the learning of computer skills is important, computer courses should not be construed as replacements for mathematics courses.

There are occasions in college math courses when calculators are useful or even necessary (for example, to find values of trig functions), so students should be able to use calculators at a level consistent with the level at which they are studying mathematics (four-function calculators initially, scientific calculators in pre-calculus). A more compelling reason for being able to use calculators is that they will be needed in other courses involving applications of mathematics. The ability to use a calculator is very definitely a part of college preparation.

On the other hand, students need to be able to rapidly supply from their heads -
whether by calculation or from memory - basic arithmetic in order to be able to follow
mathematical explanations. They also should know the conventional priority of arithmetic
operations and be able to deal with grouping symbols in their heads - for example, to know
that 9(5/3)^{2}=25 without recourse to pushing buttons. Moreover, students should
be able to do enough mental estimation to check whether the results obtained via
calculator are approximately correct.

The use of calculators is allowed on the UWMPT. The tests are designed to accommodate the use of calculators in such a way as to minimize the effects on placement due to the use or nonuse of calculators. Exact numbers such as will continue to appear in both questions and answers where appropriate.

Use of calculators is optional. Each student is advised to use or not use a calculator
on the UWMPT in a manner consistent with his or her prior classroom experience.
Calculators will __not__ be supplied at the test sites. The test is not designed to
accommodate the use of __graphing__ calculators by some students and not by others
without giving an advantage to some students, and therefore their use is not allowed.

Although university curricula are in a state of flux, with many basic issues and philosophies being examined, the normal entry level courses in mathematics remain the traditional algebra and calculus courses. Therefore, the placement tests must reflect those skills which are necessary for success in these courses. This is not intended to imply that courses stressing topics other than algebra and geometry are not valuable to the high school mathematics curriculum but rather that those topics do not assist in placing students in the traditional university entry level courses.

Probability and statistics are topics of value in the mathematical training of young people today that are not reflected on the placement test. It is the Committee's feeling that these topics are important to the elementary and secondary curriculum. They are gaining significance on university campuses, both within mathematics departments and within those departments not normally thought of as being quantitative in nature. The social sciences are seeking mathematical models to apply, and these models tend to be probabilistic or statistical. As a result, the curriculum in these areas is becoming heavily permeated with probability and statistics.

Mathematics departments are finding many of their graduates going into jobs utilizing computer science or statistics. Consequently, their curricula are beginning to reflect these trends.

- Prealgebra

Students should be able to:- Perform basic arithmetic operations with whole numbers, rational numbers and decimals, and reduce to lowest terms fractions involving single-digit whole numbers for numerator and denominator.
- Convert between fractions, decimals, and percents.
- Arrange fractions, decimals, and integers in order of size.
- Set up and solve simple verbal problems involving percentages, mixtures, unit conversions, and distance/time/rate.
- Find areas and perimeters of geometric figures composed of rectangles, triangles, and circles.
- Find the surface area and volume of a rectangular prism.
- Solve problems involving the measures of angles of triangles.
- Exhibit knowledge of basic geometric vocabulary.

- Elementary Algebra

Students should be able to:- Perform the basic arithmetic operations with integers using symbols of grouping.
- Solve linear equations containing simple fractions and literal numbers.
- Manipulate formulas containing fractions and several variables.
- Simplify algebraic expressions involving multiplication and division of exponential expressions.
- Simplify exponential expressions with negative bases.
- Find products of binomials.
- Collect like terms, and perform simple monomial factoring.
- Use order of operations to simplify polynomials and rational expressions.
- Evaluate polynomials at specific values of the variable.
- Set up and solve verbal problems involving linear equations.

- Geometry

Students should be able to:- Recognize and use congruence and similarity correspondences.
- Recognize and use conditions that imply congruence and those that imply similarity for triangles.
- Recognize and use symmetries of isosceles triangles and equilateral triangles.
- Set up and solve proportions derived from similar triangles.
- Recognize and use conditions that imply that two lines are parallel or two lines are perpendicular.
- Find the area of a sector of a circle given the measure of a central angle.
- Find measures of angles by using complementary and supplementary relations.
- Use Pythagorean relationships.
- Recognize and use the fact that the contrapositive of a conditional statement has the same truth value as the conditional statement.
- Given a generalization, identify examples and counterexamples.
- Use incidence and parallelism properties in three dimensions.
- Classify sets of objects by set inclusion, e.g., the set of squares is a subset of the set of rectangles.

- Advanced Algebra

Students should be able to:- Simplify (reduce) rational expressions.
- Add, subtract, multiply, and divide rational expressions.
- Factor completely: a quadratic trinomial, difference of two squares or the sum or difference of two cubes.
- Recognize the relationships between coefficients of perfect-square trinomials.
- Evaluate numerical expressions containing integral and fractional exponents.
- Simplify radical expressions.
- Rationalize numerators and denominators of rational expressions containing radicals.
- Rewrite radical expressions using fractional exponents.
- Add, subtract, multiply, and divide radical expressions.
- Simplify algebraic expressions with positive, negative, and zero exponents.
- Convert between decimal and scientific notation.
- Find the slope and y-intercept and be able to graph a straight line given its equation.
- Graph a parabola given its equation.
- Evaluate numerical expressions and solve equations involving absolute values.
- Recognize the graph of y = |x|.
- Graph the solution of inequalities such as Ax + B <0 on the number line.
- Graph the solution of inequalities such as Ax + By + C <0.
- Solve quadratic equations (includes use of quadratic formula).
- Solve fractional equations, discarding extraneous roots.
- Solve a system of two linear equations in two unknowns recognizing cases of no solutions or infinitely many solutions.
- Graph a system of two linear equations in two unknowns.
- Read, analyze, and solve verbal problems.
- Add, subtract, multiply and divide polynomials.

- Precalculus

Students should be able to:- Use the definitions of the trigonometric functions of angles.
- Evaluate trigonometric functions for special angles, e.g., 30
^{o}, - Use the standard formulas and identities: Pythagorean formulas, quotient and reciprocal formulas, complementary angle formulas, sums and differences formulas, and double- and half-angle formulas.
- Solve trigonometric equations (linear and quadratic).
- Sketch the graphs of trigonometric functions.
- Evaluate expressions involving inverse trigonometric functions.
- Solve triangles (acute, right and obtuse).
- Determine the domain and range of a function from a defining statement or graph.
- Use function notation, perform operations on functions and evaluate functions.
- Work with exponential and logarithmic functions and their graphs.
- Solve exponential and logarithmic functions and their graphs.
- Solve inequalities of the type |ax + b| <c.
- Solve quadratic inequalities one variable, and graph the solutions.
- Represent complex numbers geometrically and perform complex number arithmetic.
- Use remainder and factor theorems to find the zeros of polynomials.
- Solve systems containing linear and/or quadratic equations in two variables.
- Find the distance between two points.
- Determine from the equations of two lines whether they are parallel or perpendicular or neither.
- Convert between the equation of a circle in standard form and data on the center and radius.
- Convert between the equation of a parabola in standard from and data on its vertex and axis.
- Distinguish equations of conics.