MATH 160 — Introduction to Contemporary Mathematics
Introduction to Contemporary Mathematics, will be based on the Text, Excursions in Modern Mathematics, 4-th Edition, by Peter Tannenbaum and Robert Arnold, (Prentice Hall, Upper Saddle River, NJ 07458, 1998, ISBN 0-13-5983355-5). According to the line schedule, enrollment in this particular section (REF #14510, MWF 8:30) is restricted to Elementary Education majors only. However, if there is space available, I will admit students from outside this curriculum, after having first given priority to elementary education majors.
While this course carries College Algebra (MATH 100) as a prerequisite, I don't anticipate this as being strictly necessary. Not having taught any version of this course in many years, I don't have a fixed syllabus, though one will fairly quickly evolve as we get into the semester. As things stand at the moment, I would like to cover as much as possible of Parts 1 and 2 (the mathematics of Social Choice and the mathematics of Management Science,) together with Part 4 (Statistics). Given many recent events—not the least of which being the recent presidential election—the topics that we shall take up should give us ample evidence that mathematics undergirds much of what happens in our everyday lives.
It is important to realize that the three sections of MATH 160 are run independently of each other. In particular, the syllabi and grading criteria for these three sections need not be the same. Thus, the present syllabus is valid only for students in the MWF 8:30 section of MATH 160.
In the broadest terms, this course is intended as a “math appreciation” course. However, rather than trying to get students to appreciate mathematics for its intrinsic beauty (which is probably a far too lofty goal), I hope to get the students to heighten their awareness of the ubiquity of mathematics: we shall find mathematics rearing its head in some very pedestrian, but genuinely interesting circumstances. For future elementary school teachers, this is an especially important theme, as children need to be apprised as early on as possible of both the beauty and the utility of mathematics. For this to happen, it is logically necessary for the teachers to have also fully embraced this important fact. We see that as human beings, we are confronted on a daily basis by “problems.” For example, we shall begin the semester by considering various voting schemes. This is appropriate if only because of the recently disputed presidential election. A problem is to find one that is “fair,” or to find one that is the “most fair.” What mathematics shall do for us is to give us means (more than one, in fact, and this is where life gets interesting!) by which we can say precisely what “fair” means. The beauty here is that we don't need rocket-science mathematics to accomplish this. Grade-school arithmetic will do. But we do need to develop a little maturity and sophistication to genuinely appreciate what's going on. At the same time, the future teachers among you should already be thinking ahead to significant related classroom activities for your future students. Note that in this particular instance, well-thought-out lesson plans for your students will result not only in heightened mathematical awareness for your students, but also in heightened social awareness.
For the students in this course, the objectives are two-fold. First, I set a baseline requirement that the students can carry out the necessary mathematics (simple arithmetic for the most part) to make simple deductions. Secondly, and this will separate the wheat from the chaff, I will ask the students to think enough about what they are calculating in order to interpret their calculations. This will often take the form of making a reasoned conclusion about the real-life problem, given in the language of the problem, rather than in the language of mathematics.
A typical scenario is likely to be of the following genre. Imagine that you're working in some organization, and that in the spirit of the Dilbert comic strip, your boss is a complete idiot. In particular, he knows no mathematics (and probably little of anything else). Anyway, he's apt to ask you questions that probably are not only devoid of mathematical content, they are almost devoid of any meaning at all. Yet, you work for him and you need to produce a measured response. You must
In other words, what I hope to accomplish here is for the students to see the mathematics that is otherwise hidden in the background.
As for the tangibles of this course, i.e., your grades, I will consider each student's work from a variety of sources. First, I expect to have two or three in-class, one-hour examinations, together with the scheduled final examination. Each exam will be preceded by selected review questions to help you focus your study. If I feel that the students' work is in need of further evaluation, I will consider having short in-class quizzes. However, such quizzes will be announced ahead of time, as I won't give pop quizzes.
Homework will be turned in each Monday, starting with Monday, January 22. There is a homework mailbox which is labelled MATH 160 and has my name on it. The deadline is set for 6:00 p.m. each Monday. I have a grader who will grade five problems each week, and will give either 2 points for a problem correctly worked (or nearly so), 1 point for a solution with mistakes, but worthy of partial credit, and 0 points otherwise. Each homework batch (see below) will consist of several problems—those that are bold faced represent problems from which the graded problems shall be selected. I will let the grader know each Monday which of the bold-faced problems are to be graded.
I will give each student ample opportunity to demonstrate that the objectives of the course are being met. Thus I will allow myself some subjective flexibility. One very good way to mitigate a less than satisfactory score on an exam is to ask good questions in class, or to offer up incisive remarks. I want you to show that you have put some serious thought into the subject matter; there can be a variety of means for you to demonstrate your efforts.
Please bring your texts to class. I do intend to pull quite a bit of material directly from the text and your having the book in front of you will facilitate our discussions immensely. This is especially true when referring to some of the graphics in the book, as they are difficult to reproduce on the blackboard.
There are a couple of dates to keep in mind here.
There will be additional information posted here so check here often. You have my wishes for an intellectually stimulating semester.
Walking: 1, 2, 3 (in each of the exercises (a)-(d) you are to represent the voting system in the form [q;w1,w2,w3,w4]; that is, you need to determine q, w1,w2,w3,w4 in each case), 4, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 29, 32, 33, 34, 35, 37.
Also think about this one: Suppose that we have a set A={a1, a2, a3, ..., an} of n elements. Explain why, for any number k, where k is between 0 and n, the number of subsets of A with exactly k elements in them is exactly n! / (k! (n-k)!).
Running: 63. (You'll need to refer to Exercise 60 for the definition of equivalent voting systems.) Don't worry if you don't get this problem—I just want you to give it some thought.
Fair Division. Page 99 ff:
For the really interested: As we've commented several times in class, the notion of "fairness" in the fair-division problem is predicated on the fact that if there are n people to divide up the continuous property, then each person believes that (s)he got at least 1/n of the whole. What this doesn't rule out, however, is that one player might, despite feeling that (s)he got at least 1/n of the whole, feel that someone else might have gotten considerably more. To this end, think about the possibility of this happening with the two dividers, one chooser method. Either one of the dividers might feel that the chooser walked away with more than 1/3 of the loot. When this happens, we say that there is envy involved. So a more stringent question becomes this: Is it possible to divide a cake up among n players in such a way that every player feels that (s)he got at least as much as everyone else. Thus, this division is called an envy-free division. You can get a good idea of what's involved in envy-free division by looking at Exercise 70 on page 118. The mathematics of envy-free division is a bit involved. A paper where this is discussed is Oleg Pikhurko, On envy-free cake division, Amer. Math. Monthly 107 (2000) 736–738. (See also the references contained therein.) I don't really expect you to read this as you don't have the mathematical prerequisites; rather I just want you to know that this problem has been addressed in the literature.
For example, if the true value, N, of a population is estimated as n, then the sampling error (as a decimal) is |n – N|/N. As a particular example, if the estimated value of a population of N=1,243,101 is n=1,137,003, then the sampling error is
or roughly an 8.5% sampling error.), 5 (both parts), 6–8 (for #7 the sampling error is the largest of the individual sampling errors), 9–12, 13, 14 (both parts), 15–16, 21, 23, 24, 25–26, 27–30, 35, 36 (c), 37–38.
Descriptive Statistics Page 497 ff.
Assigned Problem: Please consider Example 4 on page 481, and make the additional assumption that of the total U.S. population, one third (on average) watches prime-time television. What percentage of the teenage poputation watches prime-time TV?
Walking: 1, (a),(b), (c), 3, 5, 6, 10 (In this question, assume that the 5 gumballs all come out at once and in no particular order. Can you find N?), 11, 12, 14, 15, 16, 19, 20, 21, 22, 24, 25, 30, 32, 35, 37, 38, 39 (What would you say is the probability that the spinner falls exactly on the line separating two regions?), 41, 42, 43, 44, 46, 49 (a)–(d), 50, 51 (a),(b), (c), 52, 54, 55 (a)–(d), 56 (a)–(d), 57, 58, 60, 63, 64.
Walking: 1, 2, 3, 4, 6, 8, 9, 12, 13, 14, 17, 18, 19, 22, 25, 28, 29, 30, 31 (a)–(d), 32 (a), (b), (c), (d), 34 (a)–(d), 37 (a)–(d), 38 (a), (b), 39 (a)–(c), 40 (a), (b), (c), 41 (a)–(f), 43 (a)–(c), 44 (a), (b), (c), 47 (a)–(d), 48 (a)–(c), (d), 53, 54.
Logic Puzzles . Here are some logic puzzles to attempt in your spare time. Give them a try—if you can solve any of them, email me with your solution. All responses will be recorded and factored into your grades as positive subjective bias.
Ladies or Tigers. These puzzles have been extracted from the wonderful book, The Lady or the Tiger, by Raymond Smullyan, (Alfred A. Knopf, New York, 1983, ISBN 0-394-51466-1). I highly recommend this book for the serious recreational reader.
The idea of each puzzle is that a King has captured a prisoner, who is required to make a choice between two doors. Behind each door is a lady or a tiger; if the prisoner chooses a door behind which is a lady, then the prisoner is entitled to marry the lady (and live happily ever after). However, if the prisoner chooses the door to a room containing a tiger then (presumably) the prisoner will be eaten by the tiger—an unhappy ending, indeed! The prisoner doesn't know beforehand what's behind each door. It might, for example, be the case that there are ladies behind both doors, in which case the prisoner is fated for happiness regardless of which door he chooses. On the other hand, the sinister king might have placed tigers behind both doors, and the prisoner is surely doomed!
| Door I: In this room there is a lady, and in the other room there is a tiger | Door II: In one of these rooms there is a lady, and in one of these rooms there is a tiger |
The prisoner asks, “Is it true, what the signs say?” The king, known for his veracity, replies, “One of them is true but the other one is false.”
How should the prisoner choose?
| Door I: At least one of these rooms contains a lady. | Door II: A tiger is in the other room. |
The prisoner asks, “Is it true, what the signs say?” The king replies, “They are either both true or both false.”
How should the prisoner choose?
| Door I: Either a tiger is in this room or a lady is in the other room. | Door II: A lady is in the other room. |
The prisoner asks, “Is it true, what the signs say?” As in Puzzle 2, the king replies, “They are either both true or both false.”
How should the prisoner choose?
How old? Maybe you've seen ones like this one. “I am twice as old as my brother was when I was the age that he is now. The sum our present ages is 49. How old are we?”
Tennis Tournament. We have five tennis players: Allen, Bott, Klee, Vandermeer, and Cohen who are to play two single-elimination rounds of a tennis tournament, with the following information given:
Of each of the four pairings above, only one had the same win-loss result in each of the two rounds. (In other words, it might happen that Allen defeated Bott in each of the two rounds, but in all remaining cases, the matches were split.)
Who was the winner of Round 2?
The Odd Card. Dennis, Doug and Diane are playing cards with a deck of 35 cards. This deck consists of 17 pairs (say, (1,1), (2,2),..., (17,17)), together with the Joker. Dennis deals the cards, one by one, in the order Doug, Diane, himself, Doug, Diane, ..., and so forth, until all the cards have been dealt. After the cards have been dealt, the players proceed to discard all of the pairs that they hold, retaining only the unmatched cards (which might include the Joker). Here's what we observe at this point:
Who's holding the Joker?
Recommended Reading. I very much recommend reading the review of the book, Knowing and Teaching Elementary Mathematics, by Liping Ma, which appeared in the September 1999 issue of the American Mathematics Society Notices. This penetrating review was written by Roger Howe, himself a first-rate mathematician at Yale University. For your convenience you can get this review here. You'll need acrobat reader to read this file.
Ma's book brings out some of the fundamental differences between the teaching of elementary-school mathematics in China versus its delivery in the United States. Recommendations are given (both in the book and in Professor Howe's review) for improving the mathematics education in our elementary schools.
Recommended Reading. The February 2001 issue of the American Mathematical Society Notices has what should be a good article for your consideration: “Mathematics for Teaching,” by Al Cuoco. The link I've provided takes to directly to the Notices ; from there you can link directly to the feature article in question.
I now invite you to take The Wason Test, after the British psychologist Peter C. Wason. There are in fact quite a few internet links related to this elementary test; there is also an account in Keith Devlin's marvelous book, The Math Gene, (Basic Books, 2000, ISBN 0-465-01618-9), p. 116. (This is the book that I've referred to on occassion in class; you'd all be very well served by reading this book.) However simple this test is, of the 128 college graduate candidates initially given this test by Wason himself, only 5 got it right! How did you do?
If you didn't get the Wason Test correctly solved, don't feel bad, I'll give you another chance. This is also taken from Devlin's, The Math Gene (and is referred to in the above link, as well). All you have to know is that the drinking age here in Kansas is 21 years of age. Imagine that you are hosting a party where alcoholic beverages are being served, and you see four people sitting in a row, drinks in their hands. You ask them to place their drivers licenses on the table in front of them. Here's what you see:
| Guests | Guest 1 | Guest 2 | Guest 3 | Guest 4 |
| Drinks | Coke | beer | can't tell by looking | can't tell by looking |
| Driver's License | upside down | upside down | age 22 | age 19 |
Exactly which driver's licenses must you turn over and inspect, and exactly which drinks must you “sniff for alcohol,” in order to determine that these four guests are “legal.” Is this easier that Wason's test? Why? Aren't they really the same? Hmmm.