This class will be different from mathematics courses you've had in the past - especially high school math courses. To survive, the student must be a much more active learner (as opposed to a passive learner). To help you cope with all that, we present

Modeling with Calculus Survival Guide

The Main Guideline:

Spend about 30% of your study time organizing the material, and 70% of your time doing actual homework problems.

Note 1) Organizing the material should consume paper and ink: think of writing an outline.
2) Do homework problems you have identified as relevant in (1).

Organizing: Readings | Cataloging tools | learning facts and terminology | Why?

Readings:

This is the place where your college math experience will differ the most from prior math experiences. You do have to read the text. That can be a challenge, because in the typical high school math class, reading the text is usually discouraged. In this class, reading the text is essential to your survival. The good news is that the number of assigned pages will be lower than in your other college courses. Balancing that is the fact that mathematical readings are far denser, i.e. they hold much more content per page.

Tips:

Always read mathematics with a notebook or notepaper and a pencil in hand. At first, outline the main points on a very rough level, say around three points per page of reading.
Try to fill in the steps that the reading skips over. If you do this from the first day of class, it won't be too hard, but if you skip this practice for a few weeks, it will be difficult indeed.

If you get stuck, skim ahead first, to see if the text doesn't fill in those steps in the next paragraph.

It is important that you attempt to rephrase the main points in your own words. Merely copying down items highlighted in the text will have only limited value.

Your first attempts will have errors. Do not be alarmed, but do come back and improve your summaries as your understanding improves.

Cataloging mathematical tools:

This is more familiar - it is more closely related to doing problems, but it is not the same as doing problems.
Make a catalog of methods and techniques used in the class. It is most important that you put things in your own words rather than merely copying someone else's. This phrasing exercise is an essential step in digesting and understanding the course content.

In each section you read, and for each class session, identify the main methods or techniques.

Attach a name to each method, inventing your own if necessary.

Be descriptive, e.g. use names like "finding a linear equation from two points" rather "that line thing."

Describe in your own words both:

What sort of problem the method solves, and
how one carries out the method.

Writing out an example is a good idea.

You are building your "toolbox"; you'll want to learn what your tools are called, what they can do, and how to use them.

Learning definitions, examples, theorems, and terminology:

This "learning area" is similar to learning the vocabulary in a language course. You have to spend some time doing the memorization.There is, though, more than just memorizing the words (or definitions): you need to know how things fit together.

For example, take the concept of a linear function. You need to know the definition, of course:

DEFINITION: A function f defined on the real numbers is linear if f(x) can be written in the form f(x) = ax + b for some constants a and b.

...but you also need to have digested this. Knowing just the verbatim definition above won't help much with a question like "Give examples of three distinct forms that a linear function can take".

In addition to the formal definition of something, one should know:

any alternate definitions that are easier to understand

for example, y = f(x) is a linear function

if Ax+By+C=0 for some constants A,B,C,

or if y-y₁=m(x-x₁) for some constants m,x₁,y₁.

examples, particularly of special cases that one might miss, such as f(x) = 5, a constant function
"anti-examples," e.g. examples of things that aren't linear functions, such as g(x) = 1/(ax+b).
uses (what are things one can use linear functions for?)
properties, e.g. linear functions have the same slope between any two points.
relevant theorems
connections to other objects/ideas/concepts

Most importantly, you need to know how all this fits together. First of all, that's what mathematics is all about - how things fit together - which mathematicians call the structure of something. Secondly, psychological research shows that it's far easier to remember something that has some cohesive organization than something that is made up of lots of unconnected pieces. So think about how your examples and anti-examples illustrate the definition or properties, for instance. Here's a metaphoric image to give you the idea:

[Most of the ideas in this section come from a talk given by David Lay of the University of Maryland at the Joint AMS/MAA Meeting in 1998. In particular, the flower-metaphor and building what he calls a concept image out of these various parts are his ideas. Professor Lay requires his students to write out all these components for each major concept in the course, though perhaps not in flower form.]

Some cognitive psychology

So... why don't your calculus professors just give you a list of all this stuff? First of all, we can't organize your thoughts. Every student needs to find their own way of understanding a topic, and while some students may share perspectives, the way a student thinks about a topic is fundamentally very personal. Psychologists say that people build a cognitive map or mental model in order to understand a subject. The point here is that this model or map is quite personal. Typically, it isn't even correct at first; we make adjustments as we build our "maps." The process of building your mental map is in fact the process of learning a topic. The only new thing is that we are encouraging you to be aware of this model you are building.

In doing the organization exercises above, you will produce an invaluable resource to use when reviewing for exams. Make sure you do actually produce that resource. Beware of relying on fellow students' outlines - even though sharing outlines can be a good idea, you will derive little benefit from reading others' outlines when compared to the benefit you will gain by writing one.

Remember, if you want to learn how to do a thing, watching someone else do it isn't enough. If it were, lots of us would be successful professional athletes. Organizing your (own!) thoughts on a topic is a way to start doing the topic, which brings us to:

Actually doing problems

This is the familiar part: practicing routine (and not-routine) calculations. Remember the general guideline:

Spend about 30% of your study time organizing the material, and 70% of your time doing actual homework problems.

Note 1) Organizing the material should consume paper and ink: think of writing an outline.
2) Do homework problems you have identified as relevant in (1).

When you do the organization alongside the actual problems, you will find mathematics far less bewildering.

Good luck,

Last modified August 27, 2012.
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