Mat 417

As promised on the first day of classes, we again survey our understanding about groups to evaluate how well we met the goals established at the beginning of the quarter.

How well do you feel you can now explain the following terms or concepts to your friends? As briefly as possible, write-out what you now understand to be the mathematical meaning of each of the following terms:

1. Function
Very
Somewhat Not Very Not at All.

2. Binary operation

Somewhat

Not at All.

Very

Not Very

3. Closure

Not Very

Somewhat

Not at All.

Very

4. Identity element

Very

Somewhat

Not Very

Not at All.

5. Inverse element

Very

Somewhat

Not Very

Not at All.

6. Associativity

Not Very

Not at All

Somewhat

Very.

7. Subgroup

Somewhat

Very

Not Very

Not at All.

8. Coset

Very

Not at All

Not Very

Somewhat

9. Normal Subgroup

Somewhat

Not at All

Very

Not Very

1O. External Direct Product

Very

Somewhat

Not Very

Not at All.

We also claimed at the beginning of the quarter that studying the concepts and methodology of abstract algebra contributes to your critical thinking and problem solving abilities. This is important for you not just as a student in this class but as a life long learner. We also pointed out that becoming proficient in seeing mathematics as a process takes time and advised you to budget your time accordingly. Taking into consideration classroom instruction, course organization, and the time you really put in on the course, how well do feel we met that goal this quarter?

Note: This class depends upon your skills in set theory and logic. As such, you not only need to know and be able to apply the concepts and theorems covered, but also, how to prove them using sound mathematical principles.

Theorems, axioms, or statements may be proven in many different ways. For example, the statement, "All squares are triangles" is false, so you need to be able to explain why it is false: Simply saying, "A square is a quadrilateral," is a true statement but says nothing as to why a square is not a triangle. Some valid responses using only the basic definitions or properties of quadrilaterals vs. triangles include: "Both geometric objects are polygons but a square has four sides, while a triangle has only three sides;" or "A square is a quadrilateral having four right angles and a triangle can have at most one right angle;." Both of these responses are direct comparing the basic definitions or properties of quadrilaterals vs. triangles.

In Mat 310, we covered several forms of proof, in/not direct, in/deductive, and all basic logic proofs. In this course, you are expected to validate all statements that are not questions or subjective. "I like triangles," is a subjective statement that may or may not be true. Your opinions, without a logically valid mathematical argument to back them up, are nothing more than declarative statements that may or may not be of interest to anybody else.

Comments on Pre-Self-Confidence Survey directed to 491 pre-credential teachers pg. 2, 8/11/97

There is a direct proportion in the quality of your own learning skills and how effective you will be as a teacher. As a teacher it will be your responsibility to help your students learn to problem solve and successfully cope with this complex and ever-changing world. The mathematics you learned in elementary school is not the same as the mathematics you will be teaching. Mathematics is a process that changes as our abilities to think and calculate change. We no longer use solid objects such as rocks or wood to count. Past cultures used such tools as rocks, an abacus, tables, a slide rule, and in the seventies, four function calculators to do the calculations necessary for us to make keep track of and make sense of our world affairs. We no longer have to teach the elaborate borrow and carry algorithms we learned to do five digit by five digit multiplication or division. All we need now is to know our basic number facts such as one digit by one digit or two digit multiplication. This is all we need in order to readily calculate small numbers and be able to tell whether the calculator answer to larger numeric problems is reasonable. The eighties brought us graphing calculators that draw graphs in an instant more completely that we could ever plotted them In any time frame using pencil and paper. The algorithmic capabilities formerly only able to be done on high powered computers are becoming more and more a part of the overall calculator scene. Solving problems in several unknowns is now no longer any more difficult with matrix capable calculators than the single variable problems of the fourth grader.

Continue discussion in whatever vein the students tend to be interested, but not for no more than seven to ten

minutes. The discussion is just a prelude to the play, not the execution of the it. "