Applications Über Alles: Mathematics for the Liberal Arts

 

Edward F. Aboufadel

Department of Mathematics

Southern Connecticut State University

New Haven, CT

 

(This article appeared in the December 1994 issue of PRIMUS.)

 

Abstract: This paper presents a fresh approach to a general education mathematics course. The basic idea is to turn the customary mathematics class on its head by focusing on applications first, through a reading of articles from magazines and newspapers, and then turning to the technical mathematical details. A description of the topics that were covered in my course, along with references to various publications, is given. A course such as this one is important because it conveys how mathematics is serving the goals of society.

 

Key Words: applied mathematics, general education courses, discrete mathematics, mathematics in government, mathematics in industry, probability, statistics, coding theory, geometry.

 

A Mathematics Class Turned Upside-Down

 

This past spring semester, I taught a mathematics course that was deliberately messy, incomplete, and topical. In my syllabus, I wrote, "You have probably never taken a mathematics course like this one!" The themes of the course can be heard in the new AMS National Policy Statement [4] and in the NCTM Standards [28]. The course, a general education requirement at Southern Connecticut State University (SCSU), is Mathematics 103: Mathematics for the Liberal Arts.

 

The basic idea was to turn the customary mathematics class on its head. Typically, what we consider as "the mathematics" is taught first (usually abstractly), and then several applications are included. In my class, a discussion would be initiated by looking at articles from publications such as the New York Times and Science, and at pamphlets from organizations such as the United States Department of the Treasury. Then, in order to make sense of a real-life situation, students learned (or recalled) key mathematical ideas. For example, one week we studied a case of scientific fraud known as the Baltimore Case [16]. In order to understand this case, a conversation about normal distributions from statistics was in order. Other articles led to discussions about discrete mathematics and geometry. In my class, an understanding of the application was the most important, with an appreciation of the mathematics to follow.

 

Attempts at introducing students to the nature of mathematics often fail because there is too much of a focus on the "beautiful", abstract nature of mathematics. Students may not care. But students are interested in the way the world works, be it politics or banking or gambling. And through these topics, we can bring the students the dirty, applied nature of mathematics. Is that wrong? Not if we want to accomplish the goal, well-stated in the new AMS National Policy Statement, to convey "... to the public, and to policy makers the nature of the mathematical sciences, how they are serving the goals of society, and how in the future they will serve these goals in new and remarkable ways." [4] We are not going to convey anything if we don't get people's attention first, and the way to get their attention is to talk about topics they are interested in.

 

Joseph Rosenstein at Rutgers University (in a conversation with me) summed up his attitude about a similar course at Rutgers as, "the one math course you would want your congressman to take." I wanted my course to be one that you would want everyone to take, to get them to realize that mathematics is important, that it is everywhere, that "why do we have to learn this?" has an answer. Below I describe first, in general, the way the course is set up at SCSU, and then, in particular, the topics discussed in my class and how well they were received. Following the outline of the topics, I present the activities and quiz questions that I used with two of the topics. I conclude this paper with some discussion.

 

Mathematics 103 at My University

 

Students at SCSU have to complete several general education requirements, including an approved mathematics course. General education courses fulfill certain university-wide objectives; for example, a course should "influence students to develop a willingness for intellectual exploration and the risk this involves." Different students fulfill the requirement in different ways. For instance, business majors enroll in Mathematics 124: Quantitative Analysis, which features linear programming and probability. Scientific majors complete Mathematics 150: Calculus I. The liberal arts majors (e.g. History, Theatre, English) enroll in Mathematics 103: Mathematics for the Liberal Arts. Many of these students do not like this requirement, and they will let you know it.

 

At SCSU we spend a lot of time in committees discussing and writing course outlines. For each course, we collectively draw up a list of topics, including a designation of how much of the course should be devoted to each topic. When we actually teach these courses, we do a reasonable job of sticking to these outlines.

 

The course outline for Mathematics 103 is different. Instead of a list of topics, the outline describes a "General Philosophy" and a "Perspective" for the course. From the outline:

 

General Philosophy:

 

A. Mathematics 103 should attempt to teach students how to handle some abstract ideas and how to generalize specific ideas and concepts. It is desirable to demonstrate structure and pattern abstraction and to specifically require that students do some abstracting themselves.

 

B. Mathematics 103 does not seek to remedy the skill deficiencies (arithmetic and algebraic) evident in many students.

 

C. There should be some unifying element in the course. It is desirable to "answer" the question "How does the material we are discussing fit together and how does it fit into the larger scheme of things mathematical?"

 

The second statement above, concerning skill deficiencies, is both a headache and an opportunity for instructors of this course. Since the prerequisite of Mathematics 103 is a passing grade in Intermediate Algebra (basically 11th grade Algebra) or a sufficient score on our placement exam, the typical student is not terribly strong when it comes to skill activities such as multiplying binomials. So, if mathematics is a forest, then, for this class, the trails in the forest are not in good shape. Make no mistake, though, these students are not stupid, so the opportunity arises to start again at the edge of the forest and to blaze totally new trails in fresh directions.

 

The course outline goes on to describe three possible perspectives in order to fit this philosophy: a historical perspective, a pure mathematics perspective (mathematics as an art), and an applied mathematics perspective. I was interested in the applied mathematics approach, for which the course outline states:

 

An applied mathematics perspective: general remarks on how mathematics is applied in modern technology and in social and economic problems, such as linear programming, probability, statistics, set theory, logic, and Boolean algebra.

 

The course outline for Mathematics 103 gives instructors a lot of freedom. No specified topics need to covered; no particular landmarks need to be reached. Mathematics for the Liberal Arts is meant to be a rousing conclusion to the formal mathematics instruction for a plurality of our students, and as I prepared to teach it, I decided that a rousing was just what the students were going to get. After chatting with more established department members to make sure I understood the ground rules correctly (a few wanted to make sure I followed the "unifying element" idea in the General Philosophy), I pulled out my file of articles that I had been collecting for the past five years, and I created a course.

 

My Class: Applications Über Alles

 

To give a little structure to the course, I divided it up (roughly) into units on discrete mathematics, government issues, probability and statistics, coding, and geometry. As indicated below, I used many different teaching strategies during the semester, from hands-on manipulatives to group work to straightforward lecture. Early topics stressed looking for patterns (a key item in the NCTM Standards [28]). In order to not shock the students, it wasn't until almost a month into the course that we started reading articles from magazines and newspapers Here is a list of the topics, with comments and references. (In the next section of this paper, I present more specifics on how two of these topics were developed in my class, including group activities and quiz questions.)

 

Nim. Nim is a two-player game in which each player chooses a number of objects (such as pennies) from a pile (or set of piles) [18]. The goal is to reduce the pile to nothing and the winner is the one who takes the last object. I had the students play for a while and instructed them to look for a strategy to win. Most of the students were active, but the strategy wasn't obvious, so the students got a bit frustrated after a while. After a class discussion, though, most students began to see the pattern and Nim turned out to be one of the most popular topics of the semester. We came to calling different game set-ups "winners" or "losers".

 

The Mailcarrier Problem, Shortest Paths, and Shortest Connected Networks. For almost three weeks we worked through three different units in the second volume of Mathematical Investigations [33]. The activities in this book are organized to guide students to find patterns. Much of this work was done in groups during class and I was pleased to see how much the groups accomplished. Since many students in Mathematics 103 think that mathematics equals algebra, I asked the class whether or not these activities in graph theory were really mathematics. Many of them were not sure.

 

Logic Problems. One day, in groups, I had the students work on logic problems from a GRE practice test [12]. Although it was a good group activity, I really wasn't sure where to go with it. These problems did not really fit into the unifying element of the course: applications of mathematics.

 

Money. At the end of 1992, Mexico began using the new Peso with the formula that 1000 old Pesos equaled one new Peso. A trip to San Antonio in January 1993 (for the Joint Mathematics Meetings) yielded a pamphlet [35] from the Mexican government about the new currency. Some interesting (yet straightforward) mathematics came up, issues such as rounding and exchange rates. Then, after studying a pamphlet [38] from the United States Department of the Treasury, we turned to codes on United States money (e.g. "B" = New York City) and the new security features known as the security stripe and microprinting. Although there was little mathematical content in these security features, this was a good way to introduce the role of the Secret Service in investigating counterfeiting, a role which re-emerged later in the course. The students really liked this topic.

 

Congressional Redistricting. In the history of the United States, three different methods have been used to apportion representatives to the different states [10, 29]. A lot of interesting mathematics goes into these methods. The algorithms (Hamilton's, Jefferson's, and Huntington's) are a little tricky at first, but they only require addition, division, and square roots. The students practiced the methods working in groups, and during lecture I pointed out the statements in the Constitution concerning the Census and the debates in Congress and the courts about apportionment and gerrymandering.

 

The Monty Hall Problem. This problem was made famous in the mathematical community after an article in Parade magazine [37] by alleged genius Marilyn vos Savant. Our source for the problem was an article in the New York Times [36] that surveyed the problem, quoted Monty Hall himself, and discussed the ugly interaction that occurred between mathematicians and vos Savant. I had the students postulate the best strategy for the problem, and then, to get their hands dirty, the students worked in pairs doing several trials of the problem. Combining all of the trials together, we saw empirically what was the better strategy. This topic was wrapped up with a discussion of empirical probability versus theoretical probability.

 

The Lottery and Pascal's Triangle. About a month earlier I missed a class due to a talk I was giving elsewhere and my colleague Susanna Fishel filled in and gave a lecture on Pascal's Triangle. Now I brought up the lottery, in particular, the pick-6-out-of-44 Lotto that is available in almost every state. Through a number of examples, the students began to see the following circle: to understand the probability of winning the lottery, use binomial coefficients. Binomial coefficients make up Pascal's Triangle. Pascal developed his triangle (actually, rediscovered, since it was know in China a millennia earlier [7]) during a series of discussions with Chevalier de Méré about gambling. The students were quite interested in this, although they complained that they just didn't like Pascal's triangle. I found that computing the binomial coefficients without getting bogged down in the factorial notation was a good idea.

 

Is the Bet Fair? A natural sequel to the topic of the lottery, this was another unit from the Mathematical Investigations series [34]. With jai-alai and also the new casino here in Connecticut, many of the students were quite interested. Some students were overwhelmed trying to determine if a bet was fair in a particular case.

 

The Baltimore Case. This is a story [16] about an investigation into scientific fraud that involved, among others, Nobel prize winner David Baltimore, scientist Thereza Imanishi-Kari, her postdoc Margot O'Toole, Congressman John Dingell, and the Secret Service. Part of the investigation had to do with whether or not a set of digits were random (and hence, true), or non-random (fraudulent). This led to the topic of statistical distributions. Some lecturing by me on IQ (Intelligence Quotient) helped students get the gist of it.

 

ISBN Codes. Every book that is published today has a 10-digit ISBN (International Standard Book Number) code [22], but not every 10-digit code can be used. This brought up the idea of reducing the code into a one-digit checksum, an idea which came up again in later topics. The checksum is used for error-correction, which is particularly helpful when two people with different native languages are attempting to communicate about a book.

 

The Numbering Crisis in World Zone I. This topic dealt with the ideas that have gone into assigning phone numbers and area codes [19]. World Zone I is basically North America, and the system that was set up in the 1950's -- a system that was supposed to work for 300 years -- has failed because we have run out of numbers. A new system has now been implemented by the Bell Communications Research (Bellcore), and understanding this system, mathematically, requires some principles of counting. This is yet another topic where the students can make use of their prior knowledge and experience to make sense of things.

 

Electronic Time-Stamping. (This topic is also called the Digital Notary.) Stuart Haber and Scott Stornetta, mathematicians at Bellcore, have developed a way to "notarize" computer documents. This notarization involves crunching the document into a hash number (which is akin to a checksum), and combining these hash numbers together into a "root value". The "root value" is published in the Sunday New York Times in the classified advertising of the Metro Section. These mathematicians have set up their own company (Surety Technologies, Inc.; you can call them at (201) 701-0600; they are quite helpful). In class, we started with a useful article about electronic time-stamping from Science magazine [9], computed hash numbers by hand from simple documents, and started wondering if everything mathematical in the world is connected either to Bellcore or the Secret Service. One idea in the Science article was that this "notarizing" didn't require anyone to trust anyone else, as long as the computations were correct. This idea took us on a tangent into Koblitz' idea of mathematics as propaganda [23].

 

Passports. An article on passports [11], introduced the students to checksums that are on every passport. The article brought up an interesting point: the International Civil Aviation Organization decided to list basic data about the traveller (e.g., birth date) on the passport along with the checksum code, so that travellers wouldn't worry about secret information coded on the passport. (On the other hand, do you know what information is coded on the magnetic stripe on your ATM card?) About the same time that we were discussing these codes, an item in the Wall Street Journal [25] indicated that many people filing their tax forms refused to use the address stickers provided by the IRS because they were afraid of secret codes that were on the labels.

 

Mathematics and Religion. For one class period I ventured onto thin ice. This topic involved two readings. The first was an article "Varieties of Religious Infinities" [6], written by a theologian with a mathematics degree, who attempted to describe how humans were countably infinite while God was uncountably infinite. Students found the idea of countably and uncountably infinite absolutely mind-bending, and the extension of this idea to God and man was equivalent to a computer crash. The other reading was a quote that appears twice in the Bible [1] concerning how pi equals 3. Although it has been a long time since these students studied geometry, I was pleased by how many of them remembered the relationship between the diameter and the circumference of a circle.

 

Rates. Perhaps feeling that the class hadn't done enough mathematical algorithms lately, I turned to a unit in Mathematical Investigations [33] on Rates. The goal here was to challenge their intuition: if you travel at 60 mph for 100 miles and then 30 mph for the other 100 miles, what is your average speed? (Hint: Not 45 mph.) Students had a "Ugh, story problems" reaction to this topic.

 

Topographical Maps. Here, the class observed how to represent three dimensions with two dimensions [33]. A steep hill near campus, represented on a topographical map of New Haven, made the idea of contour lines clear to most students. Another useful example was a weather map from the newspaper showing different isotherms. Students reacted warmly to this topic.

 

Flatland. Not exactly an application of mathematics, this book [3] nevertheless shows how mathematics is used in literature and how it can be used to reflect what is going on in society. (I was surprised by the initial hostility of the class to the assignment of reading this book, as they felt it didn't fit into the unifying element of applications.) Although we only read a few sections of the book, it is amazing how much is going on in this book mathematically. Interior angles of polygons, limits, and the relationship between two and three dimensions are just some of the topics that arise. Of course, there is also the whole sexist subtext (women as lines, men as polygons) that is downright scandalous in this day and age, but it is interesting to point out how mathematics is being used here for decidedly non-mathematical ends. Rather than lecture, I came up with a long worksheet of questions for students to talk about in groups during class, and I was quite pleased with the work the students did. We also had some interesting discussions about four-dimensional objects.

 

Let's Go To The Videotape. The Audio-Visual department rolled in a VCR on the last day of class. The first video was an animated film that won an Academy Award in 1989. It is called Balance [26]. A few minutes before I showed the film, I reviewed with the students the basics of the teeter-totter. The film is a clever and dark fable about the folly of not cooperating. Balance was followed up with a selection of the Multiplication Rock videos [32], which I found surprisingly fresh despite the fact that they are 20 years old. The students were amused, too.

 

Roads not travelled. With Flatland, the semester ended, but there are many other topics that I feel would have fit into my class: encryption codes such as PGP and RSA [8, 24, 27], a two-dimensional bar code [14], the cost of pizza [30], arguments about decibel levels in New York City [31], "The Power of Three" [17], statistical distribution of colors in a bag of M&Ms, the use of estimates by the Census Bureau to supplement actual counting [21], and what it means that Ivory Soap is 99.44% pure [2]. An article in a recent issue of Consumer Reports [13] concerning the risk of electromagnetic fields has also caught my eye, in particular because of this conversation-provoking passage: "A second problem with the epidemiology is mathematical: Childhood cancer is rare to begin with. The Swedish study came up with 39 cases of leukemia, over a 25-year period, among 127,383 children who had lived within 300 meters of high-voltage transmission lines. When researchers are working with such a small number of cases, it can be difficult, mathematically, to separate chance variations from real effects."

 

As Usual, the Devil Is in the Details

 

Although all of us can think of wonderful topics in mathematics that we would like to teach to our students, the difficult question is to create effective ways to teach mathematics in the classroom. Here are the details for two of the topics from my course: "Money" and "The Monty Hall Problem"

 

Money. We spent two days on this topic. The first day I conducted class in "Informal Lecture" mode, meaning that there was a body of material that I wanted to convey to the students, but there was opportunity for student comments and anecdotes. The first part of class dealt with the old Mexican Peso. I passed around an old 2000 Peso note (worth about $1 when I was in Mexico) and an old 500 Peso coin and asked students to imagine what it was like to make simple purchases with such denominations. Children in Mexico must have been adept with larger numbers, since candy, for instance, would have cost hundreds of pesos. I asked students to speculate how a currency can become so devalued. One student had spent a lot of time in Mexico and added her insight into the matter.

 

Next we turned to the new Mexican peso. As of January 1, 1993, the Mexicans have a new currency, with 1000 old Pesos equivalent to 1 new Peso. (They use the $ symbol for Peso and N$ for Nuevos Peso, and they divide the new Peso in to 100 cents.) The new currency will have the same designs and colors as the old, so as to avoid confusion, but the denominations will be different. Now, prices of goods in Mexico resemble prices in the United States (e.g., N$14.67). The new five-cent coin is the smallest denomination, so the Mexican government had to legislate what kind of rounding would occur. The law is that rounding would only occur on cash purchases and only on the total amount due, and not on individual prices. I worked through a few examples with my students and asked them to imagine what it would be like if the United States followed the Mexicans example.

 

Next we considered the currencies of Canada and Great Britian. I passed out samples of these currencies, including the Canadian dollar coin and the British pound coin, and we discussed why these countries have eliminated the one dollar/pound banknote, and what it would be like if the United States did the same. This was a natural transition to United States currency. We studied some of the numbers on the each note, including the Federal Reserve Bank number, the serial number, and the plate numbers. While lecturing on these numbers, I passed around some United States currency. (I got it all back!)

 

This was also an opportunity to talk about counterfeiting and the role of the Secret Service to prevent it. New security features were introduced in 1990 to prevent counterfeiting. One in particular, microprinting around the portraits, makes it more difficult to use a computer and a scanner to create bogus bills. This led to a much too short discussion about how computers can store images.

 

To complete this already packed hour of class time, I lectured briefly about exchange rates and demonstrated a few computations. One student had the day's Wall Street Journal available, and we worked from the rates posted in that paper. I concluded class by distributing these "Working Questions":

 

 

1. Why did Mexico come out with new currency on January 1, 1993?

 

2. Suppose you are in Mexico and you buy 3 products costing $9.46, $12.93, and $4.44. How much do you pay if you were to pay in cash? How much if you use a credit card?

 

3. $2000 old Pesos in Mexico is equal to how many new Pesos?

 

4. Name a country with a $1 coin. What is one reason to have a $1 coin instead of a $1 bill?

 

5. When did the U.S. first issue Demand Notes? When was the phrase "will pay to the bearer on demand" dropped from the notes?

 

6. Describe the Security Thread feature on U.S. money. Which other country did we discuss in class that has a similar feature?

 

7. Money from the Federal Reserve Bank of New York in marked with a "B". What number are they also marked with? Similarly, the Federal Reserve Bank of Chicago is marked with a "G". What number is money from that bank marked with?

 

8. Which agency of the government is responsible to control counterfeits?

 

9. On November 30, 1993, the exchange rate listed in the Wall Street Journal for Mexico under "Currency per U.S. $" was 3.1035. If you had $100 in U.S. money, how many Pesos would that be equal to?

 

10. What is a "star note"?

 

11. Where do they put the Face Plate Number on U.S. money? Why do they put this number on the currency?

 

 

 

On the second day of this topic, I went through these questions with my class and went through a few more calculations with the new Peso and with exchange rates. In the following class, I asked these questions on a quiz

 

(1) Suppose you are in Mexico and you buy three products costing $7.77, $3.21, and $2.94. How much do you pay if you pay in cash? How much do you pay if you are writing a check?

 

(2) a. Name a country with a $1 coin; b. Name one of the women on the British L10 note; c. $6700 old Pesos is how many New Pesos?

 

(3) Describe the Security Thread feature and the Microprinting feature on U.S. money.

 

(4) You have a ten dollar bill from the Federal Reserve Bank of San Francisco, which is code letter L, and on the bill is the number "9" in four different places. This bill is counterfeit. Why?

 

The students did quite well on this quiz (average score 18 out of 20), which is probably not surprising given that the only mathematics needed for this topic was basic arithmetic. However, the topic "Money" served my purpose in that it set the stage for the next topic (congressional redistricting -- a prime example of mathematics in government) and for "The Baltimore Case", which also featured the Secret Service and issues about counterfeiting. Further, I recently came across an article in Science about currency redesign [20] which filled me with glee and gave me ideas for the fall semester. Here is a quote: "But a fractal pattern developed by [Mitchell] Feigenbaum, a pioneer of chaos theory at Rockefeller University, is one of the security features being considered by the Department of the Treasury to foil counterfeiters armed with ever more powerful computers, color photocopiers, and electronic scanners."

 

The Monty Hall Problem. Few students are familiar with this problem, which can be stated as follows: "There are three doors. Behind one door is a new car, behind the other two doors are goats. Monty Hall knows where the car is, and asks you to try to select which door has the car behind it. You pick door #1. Monty then opens either door #2 or #3 (depending on what Monty knows) and shows you one of the goats. He then asks you if you want to stay with door #1 or if you want to switch to the other remaining door. Is there any advantage to you staying or switching, or does it not matter?"

 

On the first day for this topic, I walk in with a deck of cards, pull out two jacks (representing goats) and one ace (representing a car), place the three cards on the chalkboard sill, and play this game twice with volunteers from the class. (Invariably, one student stays with door #1 and the other switches.) I then ask students to speculate on whether there is any strategy to this game. After a few minutes of guessing, I ask the students to get in groups of two, I give each duo three cards (two of one rank, one of a different rank -- I find that one deck is just enough) and a worksheet.

In each pair, one student is Monty Hall and the other is The Contestant, and they play the game fifteen times. Then the students switch roles and play fifteen more times. When they are done, I ask them to complete the part of the worksheet with the totals. Organizing the groups, re-iterating the rules, and conducting the trials used about thirty minutes of class time.

 

Once the trials were completed, we had data from over six hundred trials to work with. At the blackboard, I collected the data by asking each group for four totals: stay and win; stay and lose; switch and win; and switch and lose. The students began to see that there was something going on. After talking very informally about conditional probability, we completed the worksheet based on our data. (Like the first class on "Money", I was rather pressed for time at this point.) I concluded this class by assigning the article from The New York Times [36] and passing out the following "Working Questions"

 

1. Describe the Monty Hall problem. Who is Monty Hall and why is this problem named after him?

 

2. Describe the interaction that occurred between Marilyn vos Savant and some mathematicians.

 

3. As is asked in the article: should you stick or switch?

 

4. Explain in your own words your justification for your answer in problem #3.

 

5. What is experimental probability? What is theoretical probability?

 

6. In the article, when Monty Hall is playing at the dining room table, is this an example of experimental probability or theoretical probability?

 

7. How is this problem different if there are four doors instead of three?

 

During the second class, we went through these questions and discussed the article in the Times. I wanted to make clear the distinction between experimental probability and theoretical probability by giving a theoretical argument for switching doors which can be summed up by this chart:

 

In a later class, I asked the following questions on a quiz:

 

(1) Describe the Monty Hall Problem.

 

(2) Describe the difference between empirical probability and theoretical probability.

 

(3) The Warden at the SCSU Jail has put you in a private cell which has three doors (a red door, a blue door, and a yellow door). The Warden is a twisted man, and he has decided to give you a chance to get free. One of the doors opens out onto a city street, and if you pick that one, then you can just walk away free. Behind the second door are two prisons guards who are ready to put you in handcuffs. Behind the third door is a hungry tiger.

Each door has a buzzer, and to play the game, you must buzz the door that you want. So, you buzz the yellow door. Suddenly the red door opens and the two prison guards run out. But a voice from a loudspeaker says, "No, not today, fellas", and the guards just stand still. Then the voice says, "Buzz the yellow door if you want to stay with the yellow door. Buzz the blue door if you want the blue door." Which one do you choose? Why do you choose that door? What could happen when the door that you choose opens?

 

Some students did not like the last question much. Many students did not address the fact that if you switch to the blue door, you might still become tiger lunch.

Deliberately Messy, Deliberately Incomplete

 

A series of educational issues arose in conducting this course: How are students to be evaluated? What kind of special responsibilities did I have teaching this course? How deep is the mathematical content of this course?

 

Between two and five class periods out of a total of forty-eight, were used for each topic. I administered a twenty minute quiz approximately once a week to give the students an opportunity to absorb a topic before going on to the next one. There were also comprehensive examinations at midterm and at the end of the semester which required students to keep organized the facts, ideas, and algorithms from the different topics. About a third of the questions were algorithmic in nature (e.g., compute the checksum; apply an apportionment method), a third were short questions of fact (e.g. name three facts about the Women in Flatland), while a third of the questions were expository in nature (e.g. describe the Baltimore case; in electronic time-stamping, what is one-way about a one-way hash function?).

 

This course was a lot of work for me. Unlike the average mathematics course, this course had no "sacred" textbook to guide the course. Rather, the students purchased one of the Mathematical Investigations books [33], Flatland [3], and a collection of articles that I made available at the local copier store [1, 5, 6, 9, 10, 11, 14, 16, 19, 29, 30, 35, 36, 38]. One consequence of this was that, other than from the Investigations book, there were no readily available questions to assign as homework. As a result, I had to create what I called "working questions" for most of the topics covered during the semester. Each topic had about ten of these questions assigned to it, questions that were either algorithmic, factual, or expository. The problems on the quizzes and the examinations were usually based on these questions.

 

Beyond the "working questions," this course required a lot of preparation. The information in newspaper and magazine articles is woefully inadequate in a mathematical sense. For example, see the passage above on high-voltage transmission lines from Consumer Reports [13]. Here are two more examples. First, from the article on the Baltimore Case [16]: "When Science subjected this distribution [of digits] to a chi-squared statistical analysis, the result suggested that such a skewed distribution has only one chance in 1032 of occurring randomly." Students with only a high school mathematics background are going to need some guidance unraveling that statement. Similarly, in the article about decibels levels in New York City [31]: "While all the numbers crunching [sic] might seem like a whole lot of fuss over a couple of measly decibels, because the scale is logarithmic, the difference between 95 and 100 and 102 is actually significant." Of course, the writers don't explain what the difference is technically, and that was one purpose of my class.

 

As a result of working with articles rather than a textbook, the course was at times rather messy and incomplete. Sometimes instruction in the mathematics itself came at the end of a topic; sometimes there was no formal mathematics instruction at all. Rather than probing deeply into binomial coefficients or minimal network path algorithms, there was just enough discussion to connect the reality to the mathematics. Sometimes a mathematical idea didn't make sense to anyone, so we moved on. (A few times I realized, in retrospect, that we moved on too fast, and that some topics take more than a week to digest.) I was determined not to get bogged down in definitions, special cases, and formalisms, and in that I believe I succeeded.

 

During the unit on graph theory and networks, I asked my students, "Is this math?", and this question weighed on me all semester. The answer depends on what you believe that "math" is. No, I did not spend a lot of time in Mathematics 103 training students to do algorithms or getting deeply involved in abstraction. This course focused on mathematics' utility, rather than its beauty. It focused on specific applications, rather than general exercises (preparing for a war that never comes). During the spring semester, the students saw how mathematics impacts society today. To understand this impact, the students needed to understand some mathematics. Yes, this course was a mathematics course, just not one we were used to.

 

This course, in many ways, challenged my students notion of what mathematics is and what learning is. First, the students finally got to see that there is more to mathematics than just the "schoolmath" (David Fowler's term [15]) that they have seen in high school (and perhaps again in remedial courses in college). Second, the students found themselves learning not from the well-digested facts in a textbook or a lecture, but from raw or poorly-seasoned information available at the newsstand, and from activities related to that information. I had confidence that my students could learn that way, and they did. Finally, the students had to use more than just numbers and symbols to tell me what they were learning. They had to write sentences relating ideas, and, in general, they were up to the task.

 

In closing, here is a response from one of my students to the question, "What did you learn in this course?" The student writes: "I learned that math really does affect your life in some way every day. The presentation of the math concepts in this course has made the class more tolerable for me. Math has always been a subject I have never really liked, but now I seem to understand its applications a little more." Well, that's all that I asked for.

 

 

References

 

[1] 2 Chronicles 4:2 and 1 Kings 7:23.

 

[2] "99.44 Percent Pure What?" New York Times Magazine (May 22, 1994), p.15.

 

[3] Abbott, Edwin, Flatland: A Romance of Many Dimensions (fifth edition, revised), Harper & Row, 1983 (original edition published c. 1884).

 

[4] American Mathematical Society, National Policy Statement 94-95, 1994.

 

[5] Bern, Marshall W. and Graham, Ronald L., "The Shortest-Network Problem," Scientific American (January 1989), p. 84-89.

 

[6] Bohler, Carolyn Stahl, "Varieties of Religious Infinities," source unknown.

 

[7] Boyer, Carl B. and Merzback, Uta C., A History of Mathematics (second edition), John Wiley and Sons, 1989.

 

[8] Bulkeley, William M., "Cipher Probe: Popularity Overseas of Encryption Code Has the U.S. Worried," Wall Street Journal (April 28, 1994), p. A1.

 

[9] Cipra, Barry, "Electronic Time-Stamping: The Notary Public Goes Digital," Science 261 (July 9, 1993), p.162-163.

 

[10] "Congressional Redistricting and the 1990 Census", Congressional Digest (October 1992), p. 228-229.

 

[11] Connor, Steve, "The Invisible Border Guard," New Scientist (January 5, 1984), p.9-15.

 

[12] Educational Testing Service, GRE General Test - No. 8, Warner Books,1990.

 

[13] "Electromagnetic Fields," Consumer Reports (May 1994), p. 354-359.

 

[14] Feder, Barnaby J., "For Bar Codes, an Added Dimension," New York Times (April 24, 1991), p. D1.

 

[15] Fowler, David, "What Society Means By Mathematics," MAA FOCUS, April 1994,

p. 12.

 

[16] Hamilton, David P., "Verdict in Sight in the "Baltimore Case", Science 251 (March 8, 1991), p. 1168-1172.

 

[17] Handy, Bruce, "The Power of Three," New York Times Magazine (April 24, 1994), p. 108.

 

[18] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (fourth edition), Clarendon Press, Oxford, 1960.

 

[19] Hayes, Brian, "The Numbering Crisis in World Zone I," The Sciences (November/December 1992), p.12-15.

 

[20] Holden, Constance, "Currency Redesign", Science (July 29, 1994), p.608.

 

[21] Holmes, Steven A., "Census Officials Plan Big Changes in Gathering Data," New York Times (May 16, 1994), p. A1.

 

[22] Knill, George, "International Standard Book Numbers," Mathematics Teacher 74 (January 1981), p. 47-48.

 

[23] Koblitz, Neal, "Mathematics as Propaganda," in Mathematics Tomorrow (Lynn Arthur Steen, ed.), Springer-Verlag, New York, 1981.

 

[24] Kolata, Gina, "100 Quadrillion Calculations Later, Eureka!" New York Times (April 27, 1994), p. A13.

 

[25] "Label Phobia," Wall Street Journal (May 4, 1994), p.A1.

 

[26] Laurenstein, Christoph and Laurenstein, Wolfgang, Balance, (1989) available on The International Tournee of Animation (Volume 4), Expanded Entertainment, 1991.

 

[27] Lefton, Phyllis, "Number Theory and Public-Key Cryptography", Mathematics Teacher 84 (January 1991), p.54-62.

 

[28] National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics, 1991.

 

[29] Olivastro, Dominic, "One Nation, Indivisible", The Sciences (September/October 1992), p. 51-53.

 

[30] "Pizza Pie Prize for Properly Applying the Principal [sic] of Pi," New Haven Register (November 11, 1990), p. A16.

 

[31] Rubin, Mike and Levy, Joe, "Stands for Decibels," Village Voice (May 3, 1994), p.84.

 

[32] School House Rock -- Multiplication Rock, Western Publishing Company, Racine, WI, 1987.

 

[33] Souviney, R., et. al., Mathematical Investigations -- Book Two, Dale Seymour Publications, Palo Alto, California, 1988.

 

[34] Souviney, R., et. al., Mathematical Investigations -- Book Three, Dale Seymour Publications, Palo Alto, California, 1988.

 

[35] The New Peso, Hacienda and Banco de Mexico, Mexico,1992.

 

[36] Tierney, John, "Behind Monty Hall's Doors: Puzzle, Debate and Answer?", New York Times (July 21, 1991), p.A1.

 

[37] vos Savant, Marilyn, "Ask Marilyn", Parade (Sept. 9, 1990).

 

[38] Your Money Matters ..., Department of the Treasury of the United States, Washington, 1990.