Math from Dr. Del

Reading Mathematics Tip 4

2006-01-21T12:59:00.000-06:00

Reading Mathematics

Tip 4 from Dr. Del
Learn the vocabulary.

Why are definitions important?
If a reader does not know requisite definitions mathematical statements become meaningless, textbook presentations are confusing at best, and lecture explanations have absolutely no value to the learner. In addition, if the learner does not know requisite definitions his/her own written or spoken statements are incorrect and in many cases completely nonsensical.

Every mathematical term in the following sentence, taken from an elementary Algebra textbook, has been replaced by a randomly chosen word from a foreign language.

To natus one desenvolvemos by another, natus each consectetur of the first sagte by each geschiedensboek of the second liever and nesciunt dignissimos hálito.

Clearly no one can learn mathematics from statements such as this. However, that is precisely what a reader attempts when definitions are not learned. After a few attempts it becomes clear that “the book doesn’t help at all” and the student stops reading the text and simply tries working problems.

There are of course other reasons why definitions are important in the study of mathematics. Here is what professor Stephen Maurer of Swarthmore college writes about the role of definitions in mathematics.

“Most disciplines don’t need to make definitions explicit nearly so often as maththematics does – they don’t need to be so precise nor do they deal so regulary with situations outside common experience.”[Maurer]

Definitions play a much more important role in mathematics than they do in any other area of study. A word may (in fact probably will) have a different meaning in mathematics than in normal discourse. One of the dictionary definitions for the word function is: “the action for which a person or thing is specially fitted or used or for which a thing exists”. The definition of function in mathematics is very different. That mathematical definition will be the subject of study in Chapter 2 of our College Algebra course.

Definitions in mathematics form a solid and completely adequate foundation upon which we base all our mathematical reasoning. A mathematical definition of a concept gives necessary and sufficient conditions for a thing to be an instance of that concept.

For example the definition of a prime number is: A number is a prime number if and only if it is a natural number greater than 1 with exactly two divisors. From this definition it is possible to conclude that 7.32 is not a prime number because it is not a natural number and therefore violates the necessary condition that a prime number be a natural number. From this definition we can also conclude that 6 is not a prime number because it has four divisors and therefore violates the necessary condition that a prime number has only two divisors. From this definition we can also conclude that 7 is a prime number because it is a natural number greater than 1 and it has only two divisors and therefore satisfies the necessary conditions stated in the definition.

Non mathematical concepts on the other hand are frequently defined in a hazy and flexible manner. Find any definition of prime real estate and note that it does not give necessary and sufficient conditions. Rather it will be hazy, deliberately quite flexible, and open to personal interpretation.

In an article about teacher preparation, Professor H. Wu from the Mathematics Department at Berkeley claims that precise definitions form the basis of any mathematical explanation. He also correctly states:

“logical explanations – the essence of mathematics no matter how mathematics is defined – cannot be given without precise definitions.”[Wu].

Finally, I should point out that many/most of the exercises in College Algebra are simple one or two step logical consequences of a definition. For example, a firm understanding of the term “graph” will be the key step to answer many questions. Just one example of this is the fact that an understanding of the word “graph” provides the basis for finding the points of intersection of two graphs.

[Maurer] Advice for Undergraduates on Special Aspect of Writing Mathematics
http://www.swarthmore.edu/library/cornell/WRITE_PRIMUS.pdf
[Wu] What Is So Difficult About the Preparation of Mathematics Teachers?
http://math.berkeley.edu/~wu/pspd3d.pdf

Reading Mathematics Foundation 1

2006-01-13T11:59:00.000-06:00

Reading Mathematics

Foundations from Cognitive Science
The Learning Process
There are two classifications of conditions for learning. There are external conditions of learning which are controlled by the instructional developer or teacher. There are internal conditions which derive from the stored memories of the learner. The following is a simplified glimpse of some aspects of the internal conditions of learning.

The following diagram is a widely accepted model of the processes involved in the act of learning.

Information stays in short-term memory for a very short time (measured in seconds) except during an activity called rehearsal. Information which is to be remembered for use at a later time, must be semantically encoded and stored in long-term memory.

During the learning process, information is retrieved from long-term memory into short-term memory where it combines with other items in short-term memory to bring about new kinds of learning.

Suppose the learner has previously learned the meaning of the two terms “mathematical statement” and “equal sign”. To say he has learned this information means it has been semantically encoded and stored in long-term memory and is ready for retrieval into short-term memory.

When this student is presented with the definition “An equation is a mathematical statement which contains an equal sign”, the two previously learned bits of information are retrieved from long-term memory into short-term memory where they are combined with the new defintion to be encoded and stored in long-term memory. At this point the student has learned the meaning of the word equation as it is used in mathematics. I should point out that the process is actually a bit more complex but this example illustrates what must happen during the learning process.

What can go wrong in the above process to prevent learning from happening? There are many potential problems. We have absolutely no control over some, but the learner, the instructional developer, and instructor can prevent some problems.

In the above example, it is clear that the learner must retrieve two items from long-term memory. If that retrieval does not take place, learning does not happen. Retrieval might fail because the two necessary items are not in long-term memory or they are in long-term memory but not available for retrieval.

Careful sequencing of courses and topics in mathematics education is an attempt to insure that the learner has previously learned requiste material and has it stored in long-term memory ready for retrieval.

Availability for Retrieval
“Semantic encoding is the process involved in moving information from short-term to long-term memory. This process involves making the information meaningful by tying it to previously learned information structures (schemas) or establishing new structures. Linkages of this sort would seem to be facilitated through the use of concept maps whereby the learner is enabled to see the structure of the material to be learned.” [Gagne p.68]

“As in the case of individual facts, the learning and storage of larger units of organized verbal information occurs within the context of a network of interconnected and organized propositions previously stored in the learner’s memory.” [Gagne p.84]

Without the use of concept maps some information in long-term memory is simply not available for retrieval. It can’t be remembered.

There are four common kinds of concept maps with a few other specialized maps that help in certain situations. The concept map which is most commonly associated with mathematics is called the Hierarchy Concept Map. Here is a simplified picture of a Hierarchy Concept Map which might be used for mathematics.

The entries in this concept map are the course names, textbook titles, chapter titles, section titles, concept names, etc. The student who pays attention to these various titles (and the associated heirarchy) is constructing a concept map which is essential for efficient retrieval of mathematics information from long-term memory.

That is why my previous tips for reading mathematics encourage you to pay attention to the organizational titles. They help build the very essential concept map.

Another View
Another way of visualizing how your long-term memory organizes information might be the following.
There is a “room” reserved in your LTM (long-term memory) for mathematics information. The room is divided into “sections” labeled “Algebra”, “Geometry”, “Analysis”, and so on for each of the major segments of mathematics.

Each of these sections of the room contains numerous “file cabinets”, each reserved for a subset of the room section. So in the “Algebra” section of the room, among others there will be a file cabinet for Functions. Inside this file cabinet are folders for topics such as Zeros of functions, Linear functions, Quadratic functions.

This organization permits efficient recovery of information from any one of the folders.

When the learner uses no concept map, it is comparable to having all the millions of mathematics facts strewn about on the floor of the mathematics room. Obviously in such disarray, virtually nothing is available for retrieval and therefore very little learning can take place.

References:
[Gagne] R. M. Gagne, L. J. Briggs, W. W. Wager, Principles of Instructional Design, 1992.

Reading Mathematics Tip 3

2006-01-10T22:08:00.000-06:00

Reading Mathematics

Tip 3 from Dr. Del
Read, think about, and remember each section title as you study the material in the section.  At the same time read and think about any subsection titles as you study that material.

For example, when you start studying Chapter 1, keep in mind that the first section deals with Graphs of Equations (the section title).  Next you should pay attention to the fact that the section is divided into subsections named:

     The Graph of an Equation.
     Intercepts of a Graph.
     Symmetry.
     Circles.
     Application.

As you study the material in the text you should always be aware of the Chapter, Section and subsection in which you are currently studying.

Attention to and actually doing the recommended activities in Tips 1, 2, and 3 have very real actual impact on how well you learn mathematics.

In the next post to this blog, I will try to explain why Tips 1, 2, and 3 are important to learning mathematics.

Reading Mathematics Tip 2

2006-01-10T19:26:00.000-06:00

Reading Mathematics

Tip 2 from Dr. Del
Read, think about, and remember each chapter title as you come to it.  At the same time read and think about each of the section titles for that chapter.

For example, when you get ready to start studying Chapter 1, look carefully at Page 77 where you find the chapter title; Equations and Inequalities.  That tells you that the next 94 pages of this textbook are devoted to topics which are related in some way or another, probably very directly, to the study of two mathematical creatures; Equations and Inequalities.

Well, if those two creatures are important enough to warrant 94 pages of instruction, they are probably pretty important.  It would seem natural to decide that one of the first things you want to discover is a PRECISE definition for each of these words.  That gives you at least two learning objectives when you turn to Page 78.

Now look at the eight section titles.  It looks like the concept of inequality will not be taken up until the last two sections.  It is not at all clear what Section 5 has to do with equations.  You should just decide to let the authors tell when you get to Section 5, where you certainly expect to learn the PRECISE definition of Complex Number.

The other section titles make it pretty clear that you will want to learn the PRECISE definitions of variable, graph, linear, and quadratic.

You have now several learning objectives for your study of this chapter.
     1. Learn the definition of equation.
     2. Learn the definition of inequality.
     3. Learn the definition of variable.
     4. Learn the definition of graph.
     5. Learn the definition of linear.
     6. Learn the definition of quadratic.
     7. Learn the definition of complex number.

Make sure that you complete each of these learning objectives as you study the chapter.

Trust your authors and your teacher to reveal other learning objectives as you progress through each of the sections.

Reading Mathematics Tip 1

2006-01-10T16:13:00.000-06:00

Reading Mathematics

Tip 1 from Dr. Del
Read and remember the title of your textbook as well as its authors.
In this College Algebra course at Meramec Community College we use the sixth edition of a text named College Algebra written by Ron Larson and Robert P. Hostetler.

Dr. Larson is a professor of mathematics at Penn State Erie.  He has authored several math textbooks.  He took his Ph.D. from the Univ. of Colorado in 1970.

Dr. Hostetler is an emeritus professor of mathematics at Penn State Erie.  He has authored several math textbooks.  He took his Ph.D. from Penn State University in 1970.

Why should you take note of the book title ?
I will have a bit more to say about this in a later post, but for now you should observe that this title indicates that the topic is ALGEBRA and that it is at the COLLEGE level.

So in this course we study the subject of algebra.  Not arithmetic; not trigonometry; not geometry; not calculus.  We will study ALGEBRA.   We will not study it the same way you might have studied it in the past, because this is COLLEGE Algebra not HIGH SCHOOL Algebra.

Welcome

2006-01-09T10:12:00.000-06:00

Welcome to College Algebra.
You have lots of resources available to help you to master this mathematics course.

With the help of:

Textbook

Lectures

Website

Blog

Tutors

Office Hours

and lots of hard work you should surely master this subject.