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

Note 1) Organizing the material should consume paper and ink: think of writing an outline. |

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.

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.

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 A
*x*+B*y*+C=0 for some constants A,B,C, - or if
*y*-*y*_{1}=*m*(*x*-*x*_{1}) for some constants*m*,*x*_{1},*y*_{1}.

- 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.]

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:

Note 1) Organizing the material should consume paper and ink: think of writing an outline. |

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

Good luck,

*Last modified August 27, 2012.*

Prof. Janeba's
home page | *Please mail comments or questions to:* mjanebawillamette.edu

Willamette University
Home Page