# Mathematics

## MATH 155 WAY OF THINKING

SYLLABUS - SPRING 2004

Section 01:

MWF 9:00-9:50

T 10:00-10:50

MC 402

Section 02:

MWF 1:10-2:00

T 1:00-1:50

MC 553

Instructor: Dr. Milan Luki´c

Office: MC 521

Office Hours: MF 2:00-2:50, T 12:00-12:50 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?

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In other words, I expect you to:

• Do some mathematics;

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

– What does (something) mean?

– How did we get from A to B?

– Is this (a statement, claim, formula, . . . ) correct? What does it mean to say that “something” is 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:

MATH 155 WAY OF THINKING SYLLABUS - SPRING 2004 3

http://my.execpc.com/~lmilan/viterbo.html - the Viterbo critical thinking web page

Text: Miller, Heeren, Hornsby, et al.: Mathematical Ideas, Expanded 10th edition, Addison Wesley, 2004.

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.

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. Grading: The final grade is based on homework, exams, presentations, portfolio, in class participation, 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: 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, January 16.

IMPORTANT - for this, and the other essay assignments:

(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:

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• 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, January 20.

(3) Essay III - World without mathematics: another 1-2 pages 20 points essay. Try to imagine, and describe, a world without mathematics. Due: Friday, January 23.

(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 of the last week of class. Homework: Some of the HW will be a group HW. I will place you in small groups based on a random arrangement. For other HW, I will expect you to do the work entirely on your own, not asking for help anybody but me.

What material, and how, it will be covered in this class depends very much on the input you provide in class. The HW will be tailored to what we cover in class and it will be prepared as we go along. Exams: There will be two or 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.

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.

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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. On a more detailed scale, these 50 points are divided as follows:

• The choice (the quality) of problems - 10 points;

• Correct, clear statement of the problems - 5 points;

• Correct solutions - 30 points;

• Reflection - 5 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.

The portfolio is due the last day of class. 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 February, each Monday will be devoted to the presentations. Then, we will take the last week (or more, if necessary) to do just the presentations. I prefer volunteering to do a presentation. In the absence of volunteers, the presenters will be determined by a (computer generated) random drawing.

Project: One or two projects 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 this class, please see me andWayneWojciechowski in Murphy Center Room 320 (796-3085) within ten days to discuss your accommodation needs.

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1. Schedule outline

The material for Exam 1 will include Chapters 1, 9, and 13. Chapters 3, 7, and 14 will be tested on Exam 2.

Chapters 12 and 16, with perhaps some other chapter will be tested on Exam 3. Important dates.

Classes begin: January 12.

Humanities Symposium: February 2-4.

Midterm break: February 28 - March 7.

Easter Vacation: April 8-12.

No classes, due to my conferences: Friday, April 16;

Last day of class: Friday, April 30.

Final Exam: Section 01: Thursday May 6, 9:50-11:50.

Section 02: Monday May 3, 9:50-11:50.

2. Recommended Practice

• Chapter 1:

– Section 1.1, Exercises 1-5, 15-20, 34, 42, 44, 50, 55, 60-62.

Most of these problems ask you to recognize a pattern, and based on your guess what that pattern is determine the next term in a given sequence of numbers. You can challenge yourself a bit and try also to guess the hundredth term of a sequence in question. Even better, you may want to try to find a formula that would tell you how to calculate any term of that sequence.

Try also problems 24-30 in this section. These are a bit more difficult, especially if you try the followup questions I suggested above.

– Section 1.2: Exercises 5-10, 20, 24, 28, 30, 40, . . .

– Section 1.3: Exercises 4, 7, 8, 56, 60, . . .

• Chapter 3.

– Section 3.1: Exercises 4, 5, 16-18, 26, 28, 37, 38, 48, 50, 54, 56.

– Section 3.2: Exercises 28, 44, 66-70, 75, . . .

– Section 3.3: Exercises 2, 8, 14, 19, 38, 53, 60, 68-72, 90, 96.

– Section 3.4: Exercises 22, 24, 36, 42.

– Section 3.5: Exercises 10, 18, 20.

– Section 3.6 Exercises 10, 30, 38, 48, 50.

• Chapter 7.

– Section 7.1: Exercises 40, 60, 70.

– Section 7.2: Exercises 30, 32, 38, 44, 72, 76, . . .

• Chapter 9.

– Section 9.1: Exercises 10, 42, 66, 74.

– Section 9.2: Exercises 1-30, 42, 48.

– Section 9.3: Exercises 16, 24, 28, 30, 32, 50, 54, 62, 68, 80.

– Section 9.4: Exercises 4, 12, 16, 18, 24, 34, 52, 60, 70, 72, 76, 82, 88.

– Section 9.5: 18, 38, 40, 56;

• Chapter 12.

– Section 12.1: Exercises 5-8, 18, 34, 52.

– Section 12.2: Exercises 16, 20, 32.

– Section 12.3: Exercises 14, 28, 30, 72.

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– Section 12.4: Exercises 8, 10, 26, 32.

– Section 12.5: Exercises 8, 12, 18.

– Section 12.6: Exercises 4-8.

• Chapter 13.

– Section 13.1: Exercises 1-10, 34.

– Section 13.2: Exercises 5-15, 54, 60.

– Section 13.3: Exercises 4-10, 28, 46.

– Section 13.5: Exercises 4-10, 28-32, 42, 46.

• Chapter 14.

– Section 14.1: Exercises 14, 30, 72.

– Section 14.4: Exercises 1-10, 16, 32.

• Chapter 16.

– Section 16.1: Exercises 10, 14, 40.

– Section 16.2: Exercises 1-5, 12.

– Section 16.3: Exercises 1-6.

– Section 16.4: Exercises 2-10, 40.

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

Have a good semester !