10
	Fitting Tasks to Curriculum
	Zalman Usiskin
	University of Chicago

When I first taught high school, I used to tell my students--even the average ones--that the real test of learning was not whether they could answer questions like those they had seen in their textbooks but whether they could apply their knowledge to new situations they had not encountered. This aphorism is only partially true and was patently unfair. In applying the principle of the aphorism, when I would make up a test, I would purposely choose items that students had not encountered, items for which they would not have studied. Those items were not a test of what had been learned from the class but what had not been learned from the class. They tested some natural or acquired competence beyond the course.

     Those who wish students to apply, synthesize, analyze, and evaluate (to use the language of higher mental processes found in Bloom's Taxonomy of Educational Objectives [1956]) have always found it difficult to invent representative items. Those for whom a problem is "a situation which we want to resolve but for which we do not have an algorithm" (to use the common researcher definition) have a similar dilemma, for once a problem is solved, the astute solver has an algorithm to use for the next problem of that type. Inventing good problems has always been an art.

     The quandary presented by the desire to have students apply their knowledge and not just parrot it has been felt by all those whose goals involve more than routine skills. In the 1970's, when in a reaction to one of the weaknesses of the "new math" we began to design curricula in which a main goal was to have students apply what they knew in real-world situations, the same dilemma appeared in only slightly different clothing. We felt strongly that students were not able to apply algebra because they were not taught the applications. But if we taught the applications, then were we not changing "application" from a higher level process to a lower one?

     We decided that the goal of learning to apply was more important than how that learning had been attained; that is, we decided to teach the applications. For example, consider the following problem, introduced in Algebra Through Applications with Probability and Statistics (Usiskin, 1979).

In Chicago there are two monthly rates for local telephone service. Choice 1 has a base rate of $11.25 for 200 calls plus .0523 for each call over 200. Choice 2 is $24.50 for an unlimited number of calls. How do you decide which plan is better?

     Students were asked to write a sentence that would help them decide. The goal was to think of the sentence 11.25 + .0523(x ­ 200) > 24.5 (When is choice 1 better?) or 11.25 + .0523(x ­ 200) < 24.5 (When is choice 2 better?). This is not an easy task for students who have never studied problems like these. But we wanted to make solving such problems routine because they abound in the real world. The lesson contained similar items involving teacher salaries (compare $9,000 plus $500 for each year's experience with $9,750 plus $350 for each year) and rental cars (compare $15.95 a day plus 14¢ a mile with $12.95 a day plus 15¢ a mile). Fitting the title of the lesson, "Decision-Making Using Sentences," students were not asked to solve the sentences they wrote. The problems were employed to motivate the next lesson, in which students were shown an algorithm for solving ax + b ¾ cx + d, and were given additional problems of the type.

     "Problems of the type" is an important phrase to consider. What type is involved here? A current view is that it is unwise to sort problems by their context, such as has been the tradition in algebra with coin problems, mixture problems, distance-rate-time problems, age problems, and so on. Yet, on the other hand, Polya's advice is also commonly accepted: "If you cannot solve the proposed problem try to solve first some related problem" (Polya, 1957). When is a problem to be considered as "related"? How should we group problems for study?

     The consequences of grouping related problems reach far beyond explication of types. With respect to problem solving, the power of mathematics lies in its ability to solve entire classes of problems with similar techniques. The Chicago telephone-cost problem is not an earth-shaking context for mathematics, but it exemplifies a class of constant increase problems that lead to equations and functions involving the algebraic form ax + b. Put another way, if we expect students to come up with a mathematical model for a real situation, they need to know the attributes of the situation that would cause a particular mathematical model (linear, quadratic, exponential, sine wave, etc.) to be appropriate.

     So, in developing the University of Chicago School Mathematics Project curricula that give strong attention to applications, we have often begun with the mathematical concept and sought the key mathematical models of that concept (University of Chicago School Mathematics Project, 1989-97; Usiskin, 1991). In a few instances, the content is standard in the curriculum, as with growth and decay models for exponential functions. In other cases, the mathematical conceptualizations of the topic need to be broadened; as with angle, for example, which in geometry is traditionally "the union of two rays," but which in applications may be better conceptualized as a "turn" or as a "difference in directions." Freudenthal (1983) has done many analyses of this kind.

     In a few cases, we have found that the standard approach to the problem type to be inhibiting. Consider the following problem, which originates from an actual situation:

A city charges 8% tax and a restaurant in the city gives a 5% discount for paying cash. Is it better for a diner if the discount is given first and the tax charged on the discounted price, or if the tax is charged on the discounted price, and then the discount taken?

     Students are customarily taught that taxes (discounts) are added to (subtracted from) original prices to determine total cost. Thinking this way, working from a meal with original cost M, the first option is represented by the expression

     (M ­ .05M) + .08(M ­ .05M).

     If, instead, students are taught to think of taxes and discounts as factors, i.e., to think multiplicatively, that same option is represented by 1.08(.95M). The multiplicative representation is not only simpler but makes transparent the desired generalization from doing this sort of problem: it makes no difference what the specific discounts and taxes are; if they are fixed they can be done in any order.

     Fitting tasks to curriculum involves more than assuring that the scope of the curriculum is broad enough to accommodate the tasks. There is also the question of the sequence of topics. The mathematics you will see illustrated in the Lottery Winnings (p. 111) task involves the general idea of annuities, which can be viewed as the sums of compound interest expressions, which themselves trace back to the same multiplicative idea in the restaurant example given immediately above, which in turn requires that a student have the notion that multiplication by a number larger than 1 serves to enlarge a quantity, and multiplication by a number between 0 and 1 serves to contract it.

     In the past, the mathematics curriculum has been carefully sequenced either by algorithmic considerations (to perform long division, you must be able to subtract and multiply, so these operations must precede division) or by logical considerations (one proof of the Pythagorean Theorem involves similar triangles, so these must be studied before the Pythagorean Theorem can be considered). The above analysis suggests that the development of problem-solving among the populace would be aided by the development of sequences of models and problems that range over many years of study.

     Here is an example of such a development. Begin in the primary grades with the use of subtraction for comparison and the specific example of change. When division is introduced, cover the wide range of rates such as students/class, km/hr, and people/mi² . In middle school, use negative numbers to represent measures in situations that have two opposing directions, such as gain and loss, up and down, or north and south, and picture them on the number line. Introduce ordered pairs, not only for cataloguing the locations of objects but also for recording pairs of data. Then, by asking how fast something has changed, introduce the concept of rate of change, picture this in the coordinate plane, and use both the application and the picture to lead into the idea of slope. In high school, study situations in which the rate of change is not constant. Use these to consider limits of rates of change. There is reasonable evidence that such an approach is far more effective in leading to understanding of the pure and applied mathematics involved than traditional approaches, in which the idea of slope is introduced by a definition as (y₂ ­ y₁)/(x₂ ­ x₁) with no prior buildup or connection to rate of change.

     Another example is geometric. In the elementary grades, use the familiar coordinate square grid to obtain areas of rectilinear figures and associate the product xy with the area of a rectangle with dimensions x and y. But also modify the square or rectangular grid to generate tessellations. Point out that a two-dimensional object that tessellates can be cut from a large sheet without wasting space. In the middle grades or early high school, use finer and finer grids to provide better and better estimates of the areas of regions. In high school, graph the speed of a car or other object over time, and interpret the area between the graph and the x-axis as the product of the speed and the time, i.e., as the distance traveled. This paves the way for the many situations representable with integrals.

     It is significant that the long sequences described in the preceding two paragraphs are embedded in the traditional content of arithmetic, algebra, geometry, and elementary analysis. We have yet, however, to develop long sequences for the teaching of statistics, as it has had a shorter lifetime in the high school curriculum. To incorporate tasks like those in this volume into the experience of students is a curricular problem that is currently being undertaken by some of the mathematics reform curricula.

     Even with the analysis of individual tasks and their setting in the curriculum, there remain two particularly knotty curricular problems. First, there are tasks that involve a range of mathematics too wide to be classified by a single mathematical model or even a family of related models. Incorporating these tasks into a curriculum is on the one hand easy because they can fit in so many places. On the other hand, without such a broader context in which to embed them, such tasks become unwieldy if students are not well versed in the prerequisites to them.

     Second is the issue with which this essay began. While a fundamental goal of mathematics education must remain for students to acquire the competencies to solve simple and complex problems they are likely to encounter in their lives, students must also have opportunities to approach problems the likes of which they have not seen before. A task for curriculum developers is to accommodate these two competing needs. The corresponding task for philosophers and policy makers is to consider whether it is fair for everyday classroom assessments to test students on the latter.

References

Bloom, B. (Ed.). (1956).

Taxonomy of educational objectives. Handbook I: Cognitive domain. New York: David McKay.

Freudenthal, H. (1983).

Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Polya, G. (1957).

How to solve it. (Second ed.). Princeton, NJ: Princeton University Press.

University of Chicago School Mathematics Project. (1989-97).

Transition mathematics. Algebra. Geometry. Advanced Algebra. Functions, statistics, and trigonometry. Glenview, IL: Scott Foresman.

Usiskin, Z. (1979).

Algebra through applications with probability and statistics. Reston, VA: National Council of Teachers of Mathematics.

Usiskin, Z. (1991).

Building mathematics curricula with applications and modelling. In M. Niss, W. Blum, & I. Huntley (Eds.), Teaching of mathematical modelling and applications, (pp. 30-45). London: Ellis Horwood, Ltd.

Zalman Usiskin is Professor of Education at the University of Chicago and Director of the University of Chicago School Mathematics Project. He is a member of the Board of Directors of the National Council of Teachers of Mathematics, of the Mathematics/Science Standing Committee of the National Assessment of Educational Progress, and of the United States National Commission on Mathematical Instruction. He has served as a member of the Mathematical Sciences Education Board.