9
	Extended Response Tasks in International Contexts
	John Dossey
	Illinois State University

Mathematical tasks with a strong connection to today's workplace call for students to use procedures drawing on knowledge of recursion, numerical approximations, exploratory data analysis, and statistical hypothesis testing. Such problems are realistic, authentic, and representative of the real world. They can involve a great deal of good mathematics. However, they can also be quite different from what one sees in the examinations given to high school students of other countries (Britton & Raizen, 1996; Dossey, 1996).

An International View

     School-leaving examinations (given to all students at the end of high school) in England and Wales, France, Germany, Japan, Sweden, and even the United States, tend to focus much more on the narrow "taught curriculum" of traditionally conceived concepts and procedures related to the road to calculus--and to the university. This canon of work is generally limited to various forms of algebra, geometry, elementary functions (including trigonometry), and introductory calculus. An analysis of representative examinations from these countries for 1991 and 1992 (Dossey, 1996) indicates that the heaviest concentration of test items falls in the areas of calculus (applications of the definite integral, applications of the derivative, and the use of the derivative in finding maxima and minima), functions (interpreting the graphs of functions, trigonometric equations, and trigonometric identities), and probability distributions--notably, the normal distribution (also known as the bell curve) and its properties.

     Little evidence was seen in the study of these school-leaving (or college-entrance) examinations of items that reflect rich ties to contemporary real-world problems. For example, when recursion appeared, it did so in a relationship between trigonometric functions or in the context of a radioactive substance which decayed exponentially, both rather theoretical settings. There were no tasks involving recursion as rich in contemporary connections as you will see in Lottery Winnings (p. 111), Drug Dosage (p. 80), or Buying on Credit (p. 87). In a like manner, no international evidence was seen of tasks concerning back-of-the-envelope calculations, estimation, rounding, or general number, operation, and symbol sense--all of which occur in various tasks in this volume.

     The school-leaving and college-entrance examinations at the international level showed a strong predisposition to focus applications in one area: classical applications of mathematics to motion and mechanics problems from physics. Consider the following item taken from the 1991 University of Tokyo examination for students applying to enter the university in science. It presents a non-routine but classically oriented problem dealing with kinematics (Wu, 1993):

     Let a, b, and c be positive real numbers. In the xyz-space, consider the truncated plane R consisting of points (x, y, z) satisfying the conditions

     |x| ¾ a, |y| ¾ b, and z = c.

     Let P be a source of light moving once around the ellipse

     x² + y² = 1

     a² b²

     in the z = c + 1 plane. Sketch and calculate the area of the shadow projected by R on the xy-plane.

     Only the examinations from Germany reflected applications to business. These applications tended to deal with quality control and applications of probability to production problems. For example, the 1992 abitur (an examination given college-bound students) in the state of Bavaria contained the following item (Dossey, 1996):

It is given that 8% of the golf balls made by a particular manufacturer are considered unusable by golf players. From past experience, it has been shown that 5% of the golf balls delivered by the manufacturer are returned because of defects. For every returned ball, the manufacturer takes a loss of 0.80 DM (Deutsche Marks), and for every ball not returned the manufacturer will make a net profit of 1.20 DM. What is the probability that the manufacturer will make a net profit of at least 210 DM on a 200-ball delivery?

     One should be careful about criticizing the focus of most school-leaving examinations on applications of mathematics to classical problems in physics. These problems remain valuable for many students who need the ability to translate and formulate problems drawing on the common languages of mathematics and science. These skills are inherently important for students continuing with the study of physics, engineering, and advanced mathematics.

     Through a much broader and potentially richer set of mathematical activities for motivating a more diverse set of students, we can show all students the power of mathematics to effect change through decision-making in settings close to their lives. The ability to balance such traditional and contemporary forces in both curriculum and assessment will remain a challenge for years to come.

A U.S. View for All Students

     The 1992 National Assessment of Educational Progress (NAEP) mathematics assessment evaluated the performance of a random sample of U.S. twelfth graders on a mixed set of tasks (Dossey, Mullis, & Jones, 1993). One task was somewhat similar to, but much simpler than, the motion problem given in the University of Tokyo examination:

     The darkened segments in the figure on the left below [Figure 9-1] show the path of an object that starts at point A and moves to point C at a constant rate of 1 unit per second. The object's distance from point A (or from point C) is the shortest distance between the object and the point. In the graph paper on the right, complete the following steps:

a. Sketch the graph of the distance of the object from point A over the 7-second period.

b. Then sketch the graph of the distance of the object from the point C over the same period.

c. On your graph, label point P at the point where the distance of the object from point A is equal to the distance of the object from point C.

d. Between which two consecutive seconds is the object equidistant from points A and C?

Figure 9-1: Diagrams from a NAEP problem

     A second NAEP problem, set in perhaps a more real-world setting, asked students to consider the following situation involving a proposed income tax:

     One plan for a state income tax requires those persons with income of $10,000 or less to pay no tax and those persons with income greater than $10,000 to pay a tax of 6 percent only on the part of their income that exceeds $10,000.

     A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusion.

     Students' performances on these two tasks were evaluated using a 6-point partial-credit rubric according to the categories of no response, totally incorrect work, minimal work, partial work, satisfactory work, and extended correct work. Only 1% of U.S. twelfth graders achieved one of the top two scoring categories (satisfactory or extended credit) for the particle-motion task. The performance was not much better for the tax item, where only 3% rated a satisfactory or extended score. These data are based on the performances of a random sample of U.S. students still attending school at the twelfth-grade level. The performance percentages would undoubtedly be lower if all youth of that age cohort were sampled.

Summary

     Contextualized extended-response items like those shown from NAEP are very demanding for U.S. students. However, they in no way reach the level of complexity expected of students of similar ages in the terminal year of secondary school mathematics in countries that are our economic peers. The tasks in this volume push the envelope further for U.S. students by requiring that our students be able to draw on information from outside the traditional mathematics/science connection. However, additional focus must be given to communicating the expectation that U.S. students be able to deal with non-routine problems in contextualized settings. The ability of mathematics educators, curriculum specialists, and assessment directors to coordinate this forward movement will require a great deal of effort.

     Comparisons with school-leaving examinations in other countries are always difficult to interpret (Gandal & Dossey, 1997). The NAEP examinations are designed to collect information on a random sample of American 12th graders, regardless of the secondary school mathematics they have taken. The University of Tokyo examination is an exceedingly challenging entrance examination given to select the best of the excellent students applying for entrance to that university. Regardless of that difference in purpose--status of the system versus selection of individuals--the data and analysis from comparisons of examinations (Britton & Raizen, 1996; Gandal & Dossey, 1997) suggest that contextualized, non-routine tasks are not commonly included even on the mathematics examinations in other countries, although a great deal has been written concerning the need for such tasks. The U.S. experience with NAEP tasks requiring such work shows that students will need a great deal of support in formulating, solving, and communicating their results.

     These cautions notwithstanding, the broad applications of mathematics in daily life, the need to motivate and retain students in mathematics, and the importance of reporting the ability levels of students in mathematics relative to the demands of the world all require that we begin to move both instruction and assessment to include tasks such as those illustrated in this volume. To do less is to abandon significant opportunities to relate the real world to the classroom while strengthening student problem-solving and modeling skills. The trick will be to balance this instruction and assessment with the concepts and skills that define the traditional core of mathematics. This is the real-life problem confronting the classroom teacher and the curriculum specialist.

References

Britton, E. D. & Raizen, S. A. (Eds.). (1996).

Examining the examinations. Boston, MA: Kluwer Academic Publishers.

Dossey, J. A. (1996).

Mathematics examinations. In E. D. Britton & S. A. Raizen (Eds.), Examining the examinations, (pp. 165-195). Boston, MA: Kluwer Academic Publishers.

Dossey, J. A., Mullis, I. V. S., & Jones, C. (1993).

Can students do mathematical problem solving. Washington, DC: National Center for Education Statistics.

Gandal, M. & Dossey, J. (1997).

What students abroad are expected to know about mathematics: Exams from France, Germany and Japan. Washington, DC: American Federation of Teachers.

Wu, L. E. (1993).

Japanese university entrance examination problems in mathematics. Washington, DC: Mathematical Association of America.

John Dossey is the Distinguished University Professor of Mathematics at Illinois State University. He is Chair of the Conference Board of the Mathematical Sciences and has served as President of the National Council of Teachers of Mathematics, and as Chair of the U.S. National Commission on Mathematics Instruction at the National Research Council.