STANDARD 15 - CONCEPTUAL BUILDING BLOCKS OF CALCULUS
The conceptual building blocks of calculus are important for everyone to understand. How quantities such as world population change, how fast they change, and what will happen if they keep changing at the same rate are questions that can be discussed by elementary school students. Another important topic for all mathematics students is the concept of infinity - what happens as numbers get larger and larger and what happens as patterns are continued indefinitely. Early explorations in these areas can broaden students' interest and understanding of an important area of applied mathematics.
Meaning and Importance
Calculus is the mathematics used to describe processes evolving in space or time. How quantities change - the velocity of a car as its position changes over time or the area of a square as its sides lengthen - and what happens in the long run are central themes of mathematics and its application to the real world. Calculus is used to describe an exact result as the limit of a sequence of approximations. Calculus is essential for understanding the physical world and indispensable in economics and industry. Engineers and physicists use calculus to calculate motion in response to forces. Businessmen and economists use calculus to find optimal solutions to pricing and production. An intuitive feel for the mathematics of infinity, limit, and change are accessible and necessary for all students.
Although some students will go on to study these concepts in a formal calculus course, this standard does not advocate the formal study of calculus in high school for all students or even for all college-intending students. Rather, it calls for providing opportunities for all students to informally investigate the central ideas of calculus. Considering these concepts will contribute to a deeper understanding of the function concept and its usefulness in representing and answering questions about real-world situations.
K-12 Development and Emphases
Students at the elementary level should understand the concept of linear growth (constant increments). For instance, a savings account grows linearly if equal deposits are made at regular intervals and the account bears no interest. This idea and its extensions can be introduced and mastered without using the mathematical formalism of functions, which is introduced later in the middle grades. Beginning in elementaryschool, children can accumulate records of processes which exhibit linear growth. Some examples are mileage accumulated by traveling between home and school, cumulative expenses for school lunches, and cumulative volume of cereal consumed if everyone in the class eats a bowl every morning. Children should learn to recognize linear growth and compare it to the more irregular pattern of increases in their own height, the height of a bean plant, cumulative rainfall, class consumption of paper, and real expenditures. Children in elementary school can be introduced to exponential growth (ever-increasing increments) through the discovery that if every pair of rabbits produces two rabbits each month (one new rabbit for every existing rabbit) then in less than two years there would be more than 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 compound 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 problems, with applications to growth in economics and biology (e.g. population explosion), 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 how changing the linear dimensions of an object - such as its height, length, or diameter - affects its area and volume. In the early years children should be involved in hands-on experiments which illustrate this; for example, they might find that doubling the diameter of a circular can (of fixed height) increases the volume four-fold by filling the smaller can with water or rice 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 cumulative, 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.
Students are fascinated with the mysteries of 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 one 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 0.999 and 1? What decimal comes just before 1? Students should explore and experiment with infinitely 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 decimalaccuracy) 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 gain an appreciation that the smaller the unit used, the larger the area obtained. 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 smaller tiles gets closer and closer to the actual area 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 they begin to develop these concepts in the early grades.
Note: 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 the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.
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