Please send comments to Martin Flashman through E-Mail: flashman@axe.humboldt.edu

Most recent general update: 11-14-2000
 
 

Introduction to the Sensible Calculus Program (SCP): This project reflects work of over ten years. (See [MAA,4].) The Sensible Calculus Program addresses the need to improve calculus learning by redesigning the college precalculus function/algebra course as well as continuing to complete work on a sensible first year calculus course. The result will be a precalculus and a calculus textbook and program with a consistent, thematic, and conceptual approach. The program will prepare students in the precalculus component for learning calculus by providing motivation, background, concept development, and opportunities for real learning. The calculus component will build on these foundations with an approach that extends the precalculus maturation into calculus. Two (among several) distinctive features of the SCP approach are the extensive use of transformation figures to visualize functions and the early and consistent treatment of modelling situations and especially continuous probability to explore the power of mathematics. These will be discussed in later sections in more detail. A partial list of features in the Calculus Component of the SCP is also included at the end of this description.

The Problems. Many projects have tried to improve the quality of calculus instruction with approaches mixing technology, pedagogical tactics, and curriculum revisions.

1. Define clearly organizing concepts and principles for the content of the course;
2. Provide a consistent distinction between motivation, interpretation, and the mathematical concepts of calculus;
3. Connect the numerical, visual, and symbolic aspects of the calculus ( called by some the "rule of three");
4. Encourage students to communicate as they solve problems;
5. Use current technology to enhance concepts, not merely to facilitate mechanical approaches to computation, graphing, or symbolic manipulation;
6. Use applications, projects, and history to weave together the many strands of mathematics.
7. Provide opportunities for students to control their own learning and understanding and to be creative while expressing their own views; and
8. Improve student backgrounds and motivation before calculus.

The SCP Approach: The SCP focuses on the themes of differential equations [FL] and estimation throughout the first year of calculus, using modelling as a central motivation for applications of the calculus [FL2]. An important and unique feature of the text is that interpretations of calculus concepts consistently refer to both a geometric/tangent view using the graph of a function and a dynamic/motion view using transformation figures to visualize functions. (See Figure 1- an example of a transformation figure.)

Figure 1- A transformation figure for q(x) = x² - 6x + 5

[If you have Java you can look at a dynamic transformation figure example made with  JavaSketchpad.]

This approach is particularly helpful for students who are still developing a basis for understanding the function concept. The figure visually distinguishes the argument of a function from its value by placing them on distinct source and target lines. It also is a great aid in making sense of several calculus concepts and results. To make effective use of these visual tools the SCP employs them regularly in a variety of situations [FL1]. This is central to the visually and conceptually balanced approach of the SCP. With an early discussion of differential equations, the text introduces its first visual approach to DE's through tangent (direction) fields (see Figures 2 and 3 ). [FL2] A combined numeric and visual approach to DE's is developed using Euler's method [FL1]. These complement the more conventional symbolic discussions of the indefinite integral and separation of variable techniques.
 

Examples of tangent fields for differential equations.

Figure 2 Tangent Field for y' = x -1.

Figure 3 Tangent Field for y' = x - y.

Models: The text uses models to motivate transcendental functions with differential equations. For example, a heuristic model for learning introduces a differential equation situation based on the premise that the rate of learning should be a positive yet decreasing function of time, for instance L'(t) = 1/t with t > 0. After considering choice of units and time scales the boundary condition L(1) = 0 is established. Questions such as what will be the short and long run behavior of the learning function are investigated using tangent fields and Euler's method, as well as the derivative form of the Fundamental Theorem of Calculus. All this leads to a treatment of the natural logarithmic function. The object of this approach is not to give formal rigorous treatments of these functions, but to show how common functions arise and/or remain significant because they solve problem situations modelled with differential equations. Likewise population (predator-prey) models motivate the exponential and trigonometric functions. Inverse trigonometric functions are treated with differential equations as well as with their traditional definition. Second order differential equations and population models motivate the hyperbolic functions for student investigations.

Taylor Theory. The text approaches Taylor theory as a distinct and important part of calculus, not a mere appendage at the end of a year's course. The SCP development is based on differential equations with estimation issues. The calculus of Taylor polynomials (not series) appears as a tool for approximating difficult definite integrals such as ò₀¹e ^{-x^2} dx with a sensible control on the error. Estimating the solution to differential equations such as y'' = -y or y''= y with y(0) = 1 and y'(0) = 1 [ which characterizes the cosine or the hyperbolic cosine functions] provides additional motivation for the convergence questions of infinite series. Infinite sequences and series analysis discuss Taylor theory examples from the beginning along with the traditional examples of geometric and harmonic series. The problems we want to solve are placed  in the text before the techniques or theory.

Probability. The SCP includes continuous probability in the development of concepts of both differential and integral calculus. The text starts with a simple experiment of throwing a dart at a unit circle and arrives at the distance from where the dart lands to the center of the circle as a random variable for further investigation. Trying to understand the distribution function for this random variable eventually leads to its density function, i.e., its derivative. The text treats probability concepts frequently as interpretations of the calculus. The evaluation form of the Fundamental Theorem of Calculus shows the integral relation of the distribution function to the density function. Other probability concepts, such as the median, mode, and mean are introduced with an appropriate calculus concept applied. This unique approach follows the philosophies of both the NCTM Standards and the California Framework that call for the exploration of probability concepts as a regularly encountered strand in the fabric of mathematics. Whether a student is planning to be an engineer, a physician, or some kind of scientist, one of the most useful parts of their mathematics training will be probability and statistics. Making probability more relevant throughout the course allows students to see the interaction of calculus with this universal application. It enriches their understanding of both subjects as well as showing the application of continuous mathematics to nondeterministic situations.

Technology. Though the SCP is not committed to any specific technology, it presumes access to technology capable of graphing and computation as well as some programming features in the event that computation or visual features are not predesigned in the technology.The text is not driven by technology, but it does presume (through its problem sets) that students and teachers will use technology to explore and enhance concepts as well as to solve problems.
 

Pedagogy:

• Consistent and balanced use of transformation (mapping) figures to visualize functions and calculus concepts.
• Learning builds from experience with examples to generalization and abstraction.
• Meaningful use of history to introduce and develop concepts.
• Technology used to enhance exploration of concepts with rule of three.
• Different assignment formats: team assignments, Journals, outside reading / research assignments, oral presentations (not yet incorporated in the text).
• Synthesizing applications that put together many tools to explore a particular application in detail. [ The integral of exp(-x ²)]

• Early tangent fields and integral curves to visualize D.E.'s and solutions.

Example: Draft of Section on Tangent Fields.
• Euler's method for numerical estimation of solutions for differential equations visualized with tangent fields and transformation figures.

Example: Draft of Section on Euler's method .
• Differential equation and area motivation for the definite integral.
• Methods of integration
• connected to applications and modelling problems
• visualized using tangent fields
• Model motivation (DE's) for transcendental functions (exp, ln, trig, and inverse trig)

Examples: Draft of Section VI.A connecting DE's to the exponential function and Section VI.B connecting DE's to the natural logarithm function. Draft of VI.C  connecting the natural logarithm and exponential functions.

• Taylor Theory applications without reliance on infinite series

Example: Draft of Section IXA introducing Taylor Theory for e^x with applications.
• Analysis of infinite sequences and series explains prior calculus experience.
• Series focus on applications to themes of estimating functions, definite integrals and solution to D.E.'s .

Example: Draft of part of Section  X.B Series: Some Key Examples.

• Probability as a consistent interpretation for the calculus.
• The distribution of a continuous random variable as the primitive concept for probability.
• The density function as the derivative of the distribution function.
• Median and mode related to calculus concepts.
• The Fundamental Theorems of Calculus interpreted with density and distribution functions.
• The mean of a random variable introduced without weighted averages.
• Expected value and variation.
• Normal Curve parameters
• Probability and infinite series.

[FL1] Flashman, Martin. "Computer Assisted Calculus And Themes for A Sensible Calculus," in Calculus and Computers: Toward a Curriculum for the 1990's, The Report of A Conference on Calculus and Computers Held at the University of California at Berkeley, August 24-27, 1989.

[UMAP] Flashman, Martin. "A Sensible Calculus.," The UMAP Journal, Vol. 11, No. 2, Summer, 1990, pp. 93-96.

[FL2] Flashman, Martin. "Using Computers to Make Integration More Visual with Tangent Fields," appearing in Proceedings of the Second Annual Conference on Technology in Collegiate Mathematics, Teaching and Learning with Technology of November 2-4, 1989, edited by Demana, Waits, and Harvey, Addison-Wesley, 1991.

[FL3] Flashman, Martin. "Concepts to Drive Technology," in Proceedings of the Fifth Annual Conference on Technology in Collegiate Mathematics, November 12-15, 1992, edited by Lewis Lum, Addison-Wesley, 1994.

[FL4] Flashman, Martin. "Historical Motivation for a Calculus Course: Barrow's Theorem," in Vita Mathematica: Historical Research and Integration with Teaching, edited by Ronald Calinger, MAA Notes, No. 40, 1996.

[MAA,1] Committee on the Undergraduate Program in Mathematics, "Recommendations for a general Mathematical sciences program," Mathematical Association of America, Washington,1981.

[MAA,2] Douglas, Ron, editor. Toward A Lean And Lively Calculus. MAA Notes, No. 6, 1986.

[MAA,3] Steen, Lynn, editor. Calculus for a New Century: A Pump, Not A Filter. MAA Notes, No. 8, 1988.

[MAA,4] Tucker, Thomas W., editor. Priming the Calculus Pump: Innovations and Resources. MAA Notes, No. 17, 1990.

[MAA,5] Solow, Anita E., editor. Preparing for a New Calculus: Conference Proceedings. MAA Notes, No. 36, 1995 (?).

[MAA,6] Tucker, Alan C. and Leitzel, James R.C., editors. Assessing Calculus Reform Efforts: A Report to The Community. MAA Reports, No. 6, 1995.

[MAA,7] Roberts, A. Wayne, editor. Calculus: The Dynamics of Change. MAA Notes, No. 36, 1996.

[S&J] Swann, Howard and Johnson, John. Prof. E. McSquared's Calculus Primer, Janson Publications Inc., 1989.
 
 

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