Chapter 18 Overview

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 18: CONCEPTUAL UNDERPINNINGS OF CALCULUS

All students will develop their understanding of the conceptual underpinnings of calculus through experiences which enable them to describe and analyze how various quantities change, to build informal concepts of infinity and limits, and to use these concepts to model, describe, and analyze natural phenomena.

Meaning and Importance

How quantities change, and, in particular, their rate of change, is a central theme of mathematics and its application to the real world. Mathematical descriptions of change, what happens in the long run, and the fascination of infinity are concepts that are accessible and even necessary for all students. On the other hand, it must be stressed that this standard does not suggest that every student should take a course called "calculus."

K-12 Development and Emphases

Children at the elementary levels should understand that regular fixed savings in a non-interest bearing account results in linear growth (repeating patterns); these ideas and their extensions can be introduced and mastered without using the mathematical formalism of functions which is introduced later in the middle grades. These themes may be developed through the grades, beginning in elementary school, when children can record their savings and their heights on bar charts, recognize the pattern of linear growth in their savings, and compare that to the more irregular pattern of growth in their heights. Similarly, children in elementary school can appreciate exponential growth (growing patterns) through the discovery that if a pair of rabbits produce two new rabbits each month then in less than three years there will be a million rabbits!

Middle school students should be moving beyond the concrete and pictorial representations used in the elementary grades to more symbolic ones, involving functions and equations. They should use graphing calculators and computers to develop and analyze graphical representations of the changes represented in the tables, and to produce linear and quadratic regression models of the data. They should apply their knowledge of decimals to solving problems involving interest, making use of a calculator to determine for example the yield of a given investment or the length of time it would take for an investment to double. In high school, students can apply their knowledge of exponents, algebra, and functions to solve these and other more difficult interest problems algebraically and graphically.

Throughout their school years, students should be examining a variety of situations where populations and other quantities change over time, and use the mathematical tools at their disposal to describe and analyze this change. As they progress, the situations considered should become more complex; students who experiment with constant motion in their early years will be able to understand the motion of projectiles (a ball thrown into the air, for example) by the time they complete high school.

Similarly, students should be aware of the effect of change on measurement, such as the effect that changes in the linear dimensions of an object have on its area and volume. In the early years children should learn through appropriate hands-on experiments; for example, they might find that doubling the diameter of a circular can increases the volume four-fold by filling the smaller can with water and emptying it into the larger one. By the time students are familiar with variables, this intuition will provide them with the information they need to understand formulas such as those involving volume.

In many settings, the kind of change that takes place over time is repetitive and an important question that should be discussed is what happens in the long run. The principal tool for understanding and discussing such questions is the concept of infinite sequences and the types of patterns that emerge from them. Thus a second central theme is that of "infinity".

There is nothing more fascinating to all students than the mysteries of the infinitely large and, later, the infinitely small. Children are excited about large numbers and "infinity," and that excitement should be nourished and be used, as with other "teachable moments," to motivate the learning of more mathematics. Primary students enjoy naming their "largest" number or proudly declaring that there is no largest! In the early years, large numbers and their significance should be discussed, as should the idea that you can extend simple processes forever (e.g., keep adding 2, keep multiplying by 3).

Once students have familiarity with fractions and decimals, these notions can be extended. What happens when you keep dividing by 2? By 10? Can you find a fraction between 0.499 and 1? What decimal comes just before 1? Such applications can be related to compound interest and population explosion. Students should explore and experiment with infinite repeating decimals and other infinite series, where they can make tables and look for patterns. They should learn that by repeated iteration of simple processes you can get better and better answers in both arithmetic (with increased decimal accuracy) and in geometry (with more accurate estimates of the area and volume of irregular objects).

Although the concept of a limiting value (or a limit) may appear inaccessible to K-8 students, this basic notion of calculus can be explored through the process of measuring the area of a region. Students can be provided with diagrams of a large circular (or irregular) region, say a foot in diameter, and a large supply of tiles of different square sizes. By covering the space inside the region (with no protrusions!) with 4" tiles, then with 2" tiles, then with 1" tiles, then with .5" tiles, students can get an appreciation that the smaller the unit, the larger the area. They will recognize that the space cannot be filled completely with small tiles, yet, at the same time, the sum of the areas of the small tiles gets closer and closer to that of the region.

In summary, these kinds of experiences will provide a good foundation for the notions of limits, infinity, and changes in quantities over time. Such concepts find many applications in both science and mathematics, and students will feel much more comfortable with them if we begin their development in the early grades.

This introduction duplicates the section of Chapter 8 that discusses this content standard. Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in Chapter 1, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences. Many of the activities provided in this chapter are intended to convey this message; you may well be using other activities which would be appropriate for this document. Please submit your suggestions of additional integrative activities for inclusion in subsequent versions of this curriculum framework; address them to Framework, P. O. Box 10867, New Brunswick, NJ 08906.