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Preface for Students

Let's Face Reality

You are taking a course in calculus. If you are a typical student, you expect to become skilled at using procedures to answer questions at the end of each chapter.

If you have taken calculus before, you probably know about derivatives and integrals and expect to become highly skilled at differentiating and integrating functions the textbook presents to you.

However, if you hope to enter a field requiring calculus, then you should know the reality of what you face.

Any company that might hire you already has a $350 computer program or handheld app, or even worse, free access to Wolfram Alpha, that can solve equations, simplify, differentiate, and integrate functions faster and more accurately than you ever will! In other words, if all you know is procedural calculus, companies will buy a computer program instead of hiring you. On the other hand: • This computer program will not be capable of deciding what equations to solve, what functions to differentiate or integrate. • This computer program will not be capable of interpreting the results it produces in relation to the problem you are trying to solve. • This computer program will not be capable of judging the reasonableness of its results. Each of the above requires a human who can conceptualize situations and model them mathematically. Each requires a human who understands the calculus well enough to justify and explain the solution he or she creates using this computer program. These are our aims: • That you come to understand the ideas of calculus and become proficient in using them. • That any company gladly turns you loose with its$350 computer program to solve problems important to the company.

Even if you do not plan on a career that requires calculus, you will benefit by being literate in one of the greatest intellectual achievements of modern humanity.

But you must be patient about learning the ideas of calculus. They are very deep, and they come from understanding many elementary concepts in ways that might be new to you. Again, be patient.

What is A Rigorous Approach to Conceptual Calculus?

The meaning of rigorous is uncontroversial. "Rigorous", according to the Oxford English Dictionary, means "extremely thorough and careful".

The meaning of conceptual varies among people using it. We use it to mean having to do with ideas.

Regarding a rigorous approach to conceptual calculus, we should first state features conceptual calculus does NOT have:

• Conceptual calculus is not fuzzy, no-math calculus.
• Conceptual calculus is not hand-wavy calculus.
• Conceptual calculus is not watered-down calculus.

Rather, to understand calculus conceptually means:

• To understand why ideas of calculus arose, to understand problems people faced that gave rise to ideas of calculus.
• To understand ways people conceptualize relationships among quantities whose values vary by accumulating at some rate.
• To be able to represent these relationships symbolically and to interpret symbolic statements in terms of relationships they represent.
• To tie together all the above in developing methods for solving entire classes of problems about situations involving quantities whose values vary.

So, conceptual calculus is simply calculus, understood well. A rigorous approach to conceptual calculus is the attempt to support you in understanding calculus well.

Using This Textbook

Read Michael Larner's description of a common disconnect between students and professors. If Larner's remarks do not apply to you, then skip this section and be happy they do not.

If these remarks do apply to you, even partially, then take what follows seriously and strive to attain the goals we state.

Examples and Exercises

It is common that students approach a mathematics textbook thinking its end-of-section questions are most important and examples are the key to answering them. They see the textbook's prose as being largely irrelevant.

You must approach this textbook differently. Examples are not meant to be mimicked. They are meant to be understood. Exercise sets are are not places to practice memorized procedures. Our goal is that exercises help you understand the textbook. Exercise sets are constructed to exercise your understanding of the ideas the textbook proposes.

Studying

With an ever-increasing understanding of meanings and ways of thinking we propose, you will become able to answer complex and sophisticated questions by reasoning meaningfully about them. You will answer questions successfully because you understand the ideas and relationships the questions entail. But to reach this state you must study the text, not just look for examples to mimic.

Studying this textbook attentively will be more important than you realize. It is written to help you learn to think about important ideas. It does this by giving you opportunities to think about important ideas. But the textbook cannot think for you. To learn to think these ways you must try to think these ways. Studying the textbook attentively, striving to understand its sentences and its examples, will give you those opportunities to improve your thinking skills.

Animations

The textbook contains many animations. Do not simply watch them like you would a television show. Instead, strive to understand them. Pause them midway to analyze what varies, and how those variations are related to other things that vary.

Every animation is situated in discussions of it. Attend not just to the animation itself. Think about (reflect on) the animation in relation to its related discussions.

If you cannot explain the ideas and meanings an animation illustrates, and how it illustrates them, then you do not understand the animation. Take this as a sign that you must speak with your instructor or TA.

Reflection Questions

Each chapter contains blocked text, in red, labeled Reflection xx.yy.zz, where xx is the chapter number, yy is the section number within that chapter, and zz is the reflection number within that section.

Pay attention to Reflections. They are important for your understanding of particular points just made. Reflections bring out nuances in ideas easily missed without your attention being drawn to them.

Using the Computer Program called Graphing Calculator

You might be accustomed to using your hand-held graphing calculator in mathematics courses. Graphing calculators today are powerful tools for numerical and symbolic calculations.

We will require you to use a computer program called Graphing Calculator (GC), and we do so with a specific goal.

GC allows you to type statements that appear on your screen as they would had you written them on paper. The difference between the mathematics on paper and the mathematics on your screen is that the statements on your screen are "live". GC will interpret the mathematics you have written according to standard mathematical conventions and meanings.

Statements you type in GC will represent a mathematical process or a product of a process. This is a powerful mathematical idea--that you think of mathematical statements as being representations--representations of relationships, processes, and products that processes produce.

But this power can lead to confusions.

GC will report an error if you type a statement that is mathematically invalid. Or, GC might produce something (a number or a graph) you did not anticipate when you type a statement that is conceptually faulty.

Do not fall into the trap of thinking "GC is hard to use". Focus instead on whether what you are trying to say is conceptually sound and whether you stated it validly in symbols. GC is easy to use. Writing valid and conceptually coherent mathematical statements can be hard.

If you have difficulty formulating what you are trying to say in symbols, then STOP and take that as your problem. State your thoughts in words before trying to represent them symbolically.

On the other hand, GC is a computer program, just like Microsoft Word. It has conventions built into it (e.g., press ctrl-L to get a subscript; press ctrl-9 in defining a function) that you must remember to use it effectively.

When GC does not work as you intend:

• Do a quick scan to see if you've violated a GC convention.
• Have an open attitude that an error in GC reflects an error in your thinking.

The "fix" for conceptual errors is to reflect on what you typed, what it means to GC, and whether that matches what you intended.

Thoughtful Precision

The senses of "precision" in the Oxford American Dictionary center around ideas of exactness of execution or calculation. However, there is a sense of "precision" that is even more important in mathematics. It is precision of meaning.

An important goal of this textbook is that you develop the habit of expressing yourself precisely, whether in words or symbols. To express yourself precisely, though, requires that you develop precise meanings.

How does it feel to have precise meanings? How does one develop them?

This might sound circular, but people develop precise meanings by reflecting on their meanings and by reflecting on what other people might mean. They attend to distinctions. They examine what they have said or written from the perspective of how others might interpret it.

This is what you must do to develop thoughtful precision--you must take time to interpret what you say or write from others' perspectives. This will seem effortful at first. The payoff, though, is cumulative. Your thinking will be clearer the more care you put into expressing yourself precisely and to attending to the precision of language and imagery in the textbook.

To use GC productively requires that you attend to what you mean by expressions you type. Your interpretations of animations will require that you attend to what you understand they show. Your work on chapter exercises will require that you reflect on your understanding of what they request and that you reflect on what you mean in your responses to them. Your effort in all this will, over time, lead to precision in your thinking.

Learning From Your Activity

There are, in principle, two ways to approach a problem.

• Follow your nose, thinking only about what you have just done and what you might do next.
• Constantly keep the original problem in mind, including the question asked and what the different elements of the problem mean.
• Understand the problem, its elements, and goal before trying to solve it.
• Keep in mind the problem's elements and goal as you work.
• Keep in mind the reasoning youâ€™ve done to get to this point.
• Pause occasionally to mentally scan your interpretation of the problem and the reasoning you have done.

The second approach requires more effort, especially if you are unused to doing it. But the second approach will lead to significant learning if you use it consistently. The first approach might lead you to getting an answer, but you will not have learned much by getting there.

A Common Complaint

We often hear the complaint, This is not mathematics. There is too much thinking. Reflect for a moment on the preconceptions of mathematics a statement like this entails.

On the other hand, this complaint is inaccurate only in claiming, "This is not mathematics." You indeed will think. We hope you come to appreciate the true nature of thinking mathematically.

Office Hours

Finally, do not hesitate to use your instructor's or TAs' office hours! Avoiding office hours can lead to a terminal disease.

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