# Mathematics

## MATH 155 WAY OF THINKING

SYLLABUS - FALL 2003

Section 02:

MWF 2:10-3:00

T 2:00-2:50

FC 204

Instructor: Dr. Milan Luki´c

Office: MC 521

Office Hours: MWF 10:00-10:50, or by appointment.

Phone: (608) 796-3659 (Office); 787-5464 (Home)

e-mail: lmilan@execpc.com

WWW: http://my.execpc.com/˜lmilan

Course Description and Objectives: From the catalog:

An investigation of topics including the history of mathematics, number systems, geometry, logic, probability, and statistics. There is an emphasis throughout on problem solving. Recommended for General Education.

For some of you this course might serve to satisfy the math competency requirement, for others this will be just one of the mathematics courses required by your major/minor program. In any case, the main goal of this course is to help you develop and strengthen the foundations of your analytical thinking.

Every day, we are faced with numerous questions, such us: Should I run through this yellow light? What should I eat today? Which courses to enroll next semester? . . . We often have to resolve those questions, make appropriate decisions, and then act according to those decisions. The thinking process required for resolving all kinds of questions, puzzles, problems, is known by the name of analytical thinking. We can use mathematics as a convenient tool for working on the analytical thinking skills. In order to achieve this goal of developing and strengthening your analytical thinking skills, via mathematics, our focus will be on the following questions:

• What does it mean to do mathematics?

• What does it mean to think mathematically?

• What does it mean to understand a piece of mathematics?

In other words, I expect you to:

• Do some mathematics;

• Use mathematical reasoning, i.e., ask questions such as:

– What does (something) mean?

1

2 MATH 155 WAY OF THINKING SYLLABUS - FALL 2003

– How did we get from A to B?

– Is this (a statement, claim, formula, . . . ) correct?

– How do I know that it is correct?

• Strive to understand every idea, concept, problem, solution that we encounter in this course.

The mathematical content which we will use to achieve these objectives will expose you to a variety of areas of mathematics, and thus give you an idea of the importance of mathematics in today’s world and a multitude of ways it is being used in practice. We will learn some elements of mathematical logic, set theory, geometry, statistics, probability, consumers mathematics, and some basic algebra.

The General Education aspects of this course. The content and the methods of this course are designed in accordance with general education objectives and the work in this course should help you in developing a number of skills included in the NCTM (National Council of Teachers of Mathematics) ‘standards” for mathematics education, and also being among the general education objectives at Viterbo. The main emphasis throughout the course will be on problem solving and developing thinking skills. This includes: (a) writing numbers and performing calculations in various numeration system (INTASC 1), (b) solving simple linear equations (INTASC

1), (c) exploring the mathematical model of simple and compounded interest rates, and learning how to use those ideas in solving the problems of loan payments (INTASC 1), (d) exploring a few major concepts of Euclidean

Geometry, focusing especially on the axiomatic-deductive nature of this mathematical system, including a variety of different proofs of the Pythagorean Theorem (INTASC 1), (e) develop an ability to use deductive reasoning, in the context of the rules of logic and syllogisms, i.e., learn how to make/recognize a valid argument (INTASC 1), (f) some basics of probability and statistics . . . (INTASC 1) Potential benefits of the course. Mastering this material requires to learn how to reason mathematically, and also how to communicate mathematics.

In learning how to do so (on exams, essays, portfolio, and in oral presentations), you will also develop a confidence in your ability to do mathematics. This way you will strengthen your ability to solve problems, analyze arguments, understand abstract concepts.

Other benefits of this course include: cultural skills (appreciation of the history of mathematics and its role in today’s world, learning how to handle simple loans, etc.), appreciate the beauty and intellectual honesty of deductive reasoning, thereby adding to life value and aesthetic skills.

I encourage you to read the text at: http://my.execpc.com/~lmilan/viterbo.html - the Viterbo critical

thinking web page Text: Robert Blitzer, Thinking Mathematically, Prentice-Hall, 2003.  Format: Class sessions will consist of lectures, work in small groups, exams, and individual presentations. I expect students to work out the recommended practice problems and ask for help whenever needed.

MATH 155 WAY OF THINKING SYLLABUS - FALL 2003 3

Resources: Please do not hesitate to contact me for any question you might have; do not let a feeling such as “I am lost . . . ” to last. Other resources include:

• Internet and the Blackboard software. There is a lot of material on my web page. There will be some quizzes given using the Blackboard.

• The Learning Center.

• The library. Note that both a video set and a CD set that covers your textbook exist.

You can use either of these to hear a lecture again, or just to see/hear another explanation of a particular topic.

Grading: The final grade is based on homework, exams, presentations, portfolio, and a (cumulative) final exam. There will be opportunities for a small amount of extra credit. The following grading scale applies to individual exams, and to the overall grade as well:

A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%.

The following exceptions to that scale are possible:

• An A on the final exam (more than 180/200 points) will raise your grade up, one letter, i.e., a B will turn into an A, a BC will become AB, . . . .

• An outstanding presentation, or an outstanding portfolio can raise your grade up a half letter, i.e., a C will turn into a BC, . . . .

• If one is failing the course by the end of the semester, but has over 40% average on exams, and earns at least 55% points on the final, he/she can get a D for the final grade.

• If one is passing the course by the time of the final exam, but earns less than 30% points (a score less than 60/200), that will result in an F for the final grade.

Assignments:

• Recommended practice: First 10, middle 5 and the last 5 problems from each Practice Exercises set in each section that we cover; at least one or two of the Application Exercises, at least one of the Writing in Mathematics Exercise, and at least two of the Critical Thinking Exercises.

These practice problems will not be graded. However, fell free to ask me for help with any difficulty you might have with those problems.

• Four essays, 20 points each:

(1) Essay I - Autobiography: Introduce yourself to me in a 1- 2 pages essay. State your name, and where (city/state) you are coming from. The reason you are taking this course, and what mathematics courses you have had before. What was your experience from those courses and what are your expectations, if any, from this course? This assignment is due Friday, August 29.

IMPORTANT - for this, and the other essay assignments:

4 MATH 155 WAY OF THINKING SYLLABUS - FALL 2003

(2) Essay II - A mathematical story. In order to make the connection of the first assignment (the Autobiography) with the main goals of the course more explicit, I would like you to recall some of your specific experiences of doing mathematics, and tell me a short story (1-2 pages) about it. In particular, I would like you to address the following questions in this story:

– Try to recall an experience of you actually doing mathematics. Give an example. Describe, make a story about it.

– How about an experience of reasoning mathematically? It would be great if you could give a simple example, and even better if you have had an opportunity to communicate your mathematical reasoning to somebody else.

– Did you ever truly understand a piece of mathematics? Give an example. Describe. Explain.

If your answer to any of the questions above is negative, i.e., you have not had such an experience, then please try to explain how is that possible.

Due: Due Tuesday, September 2, 2003.

(3) Essay III - World without mathematics: another 1-2 pages

20 points essay.

Try to imagine, and describe, a world without mathematics.

Due: Friday, September 5, 2003.

(4) Essay IV - Me, a Mathematician. In this essay, you should answer same questions as in Essay II, but in relation to the material covered in this course. More precisely, the questions are:

– Did you ever, during the work in this course, have an experience of actually doing mathematics. Give an example. Describe. Explain.

– Did you ever, during the work in this course, have an experience of reasoning mathematically? Give an example. Describe. Explain.

– Did you ever truly understand at least one piece of mathematics encountered in this course? Give an example. Describe. Explain.

This last essay is due Monday, December 1, 2003 (the last week of class).

Homework: At the end of each chapter, there is a Chapter Test. Each one of those tests will be due second class period after the corresponding chapter is covered, and each problem on the “test”is worth 1 point.

This rule is a tentative one. Sometimes, I give a different problem set instead of those Chapter reviews.

Exams: There will be three in-class exams, worth 100 points each. An exam will typically cover three chapters worth of material. The exams will be closed notes, closed book. However, a calculator and a formula sheet (but not any worked out problem) is allowed.

MATH 155 WAY OF THINKING SYLLABUS - FALL 2003 5

Before each exam, I will give you a take-home practice exam, which will be very much like the actual exam coming. I will grade (25 points) the first one of those, i.e., the “Exam 1 - Practice”, but not the others. I will also allow a makeup (up to 50%) of the lost credit for one of the exams. It will be Exam 1 this time. This makeup will be oral, and will apply to those under 90/100 points on the test, and is to be done within two weeks after the exam. Final Exam: Final exam is a 2-hour, cumulative exam, and is worth 200 points. Portfolio: It should consist of 5 problems, but no two problems should be of the same type (from the same section). Format: You state a problem, write a complete/correct solution to it, and then write a paragraph (or more) explaining why did you choose that particular problem, what did you learn from it, etc.. The portfolio will be worth 50 points.

The problems you choose for the portfolio should illustrate the progress in learning mathematics, the change of the perception (if any) of what mathematics is about, the change (if any) in your perception about your abilities to do mathematics. In other words, the portfolio should provide you with an opportunity to demonstrate what you have learned in the course and the progress you made in that process.

In-class Presentation: The presentation of a proof of the Pythagorean Theorem found on the Internet. Typically, the explanations you will find on the Internet are a bit sketchy. So, part of your job will be to make sure you really understand the proof you are going to present (including filling in the gaps, i.e., the reasons not entirely spelled out in the Internet write-up), and then to clearly explain that proof to your classmates. Sometimes, some people, may find this part quite difficult. Of course, I am here to help you understand and overcome those difficulties, and so please do not hesitate to ask me for help.

You should also be prepared for the questions from the audience (myself and/or other students), and it is expected that you listen closely to other presentations and ask any question you might have.

The presentation will be worth 35 points. In addition to that, one certain problem for one of the exams, or for the final exam, is going to be: State and prove the Pythagorean Theorem.

Usually, I would reserve the last three weeks of class for the presentations. This time, we will try a different model. Starting the last Monday in September, each Monday will be devoted to the presentations. Then, we will take the last week (or more, if necessary) to do just the presentations. The presenters will be determined by a (computer generated) random drawing.

Project: A project worth 50 points will be assigned some time in the second half of the semester. The details will be given then.

Important University Policies: Those are Viterbo’s policies on Attendance, Plagiarism, and Sexual Harassment. You can find the statements at: http://my.execpc.com/~lmilan/viterbo.html

Disability: Americans with Disability Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for

6 MATH 155 WAY OF THINKING SYLLABUS - FALL 2003

this class, please see me andWayneWojciechowski in Murphy Center Room

320 (796-3085) within ten days to discuss your accommodation needs.

1. Schedule outline

Week Section Week Section

Aug. 25 12.1 - 12.4 Sep. 1 12.4; 9.1-1.3

Sep. 8 10.1-10.3 Sep. 15 10.4-10.6

Sep. 22 Exam 1, 2.1-2.4 Sep. 29 2.5; 3.1-3.2

Oct. 6 3.3-3.5 Oct. 13 3.6 - 3.7, Break

Oct. 20 4.1-4.4 Oct. 27 Exam 2; 5.1-5.2

Nov. 3. 6.1-6.3 Nov. 10 8.3; 11.1-11.2

Nov. 17 11.3-11.4 Nov. 24 Exam 3

Dec. 1 Presentations

Important dates.

Classes begin: August 25.

Midterm break: October 17.

Thanksgiving Vacation: November 26 - November 30.

Last day of class: Friday, December 5.

No classes, due to my conferences: Friday, November 7;

Final Exam: From the Final Exam Schedule:

The final exam for all Monday 2:10 P.M. classes - on Tuesday,

December 9 from 12:50-2:50 p.m.

This syllabus is tentative and may be adjusted during the semester.

Have a good semester !