Instructive Math Instruction

Over the weekend I had the opportunity to visit with a friend and colleague at another university. We lamented the fact that students could come out of high school with so little understanding of the math they had seen and so little enthusiasm for the subject.

In discussing this we wondered how a teacher could present material in a way that would give the students a wider perspective on the techniques they were learning. The idea being that thus engaging them on a deeper level should allow them to appreciate and retain the material more successfully.

On a seemingly unrelated note, during my train ride back home I began noodling around with some computations to better understand the economic notion of competitive advantage. In working through the problem, however, I realized that this might work as just the kind of project my friend and I had been discussing.

I present it here for your use, amusement, or enjoyment. Take your pick. I would also be pleased to receive comments, suggestions, and even additional problems or problem ideas.

A Factory Problem

Suppose Alice and Bob both work at a toy factory making plastic dogs and cats. For whatever reason, the factory owners want the number of dogs and the number of cats produced each day to be equal, thus they pay each employee a set amount for each dog and cat pair that the employee produces. (For instance, a dog by itself is not paid for, it must be accompanied by a cat.)

Alice is able to make either 40 dogs per hour or 20 cats per hour. Bob, on the other hand, is only able to make either 30 dogs per hour or 10 cats per hour. We shall assume that Alice can only be making one of the two types of toys at a time, but that she can switch between tasks without any loss of efficiency. The same rules apply to Bob.

• How should Alice divide up her 8 hour work day to maximize her paycheck for the day?
• How should Bob divide up his 8 hour work day to maximize his paycheck for the day?
• How many total cats (or equivalently, dogs) do Alice and Bob (combined) produce in this way?

After working this way for a while, Alice and Bob begin to wonder if they can make more toys by redistributing the workload between them and submitting their day's production together rather than separately. That is, they wonder whether they can make more toys by having one of them make more dogs than cats and the other make more cats than dogs, but in such a way that the total number of cats matches the total number of dogs.

• Find some way to divide up each person's work day so that the total number of toys produced is greater than before, while keeping the total number of dogs equal to the total number of cats.
• Find an optimal division of labor so that the total number of toys is as large as possible, while still producing equal numbers of cats and dogs. (How many toys are produced this way? Allow non-integer answers.)

Suppose Alice and Bob engage in the plan discovered above. Now that they are making more toys, their combined paycheck is larger than before, and the question of how they should split their earnings arises.

• Describe some "fair" methods of sharing their combined wages. Are there any methods which are optimally fair? (This question is intentionally vague. In particular, no definition of "fair" is provided. I would specifically want the students to struggle with this notion. A later problem will point to a possible resolution.)

Extended Problem

There is a third worker at the factory, Charlene, and she sees that Alice and Bob have increased their paychecks, and so she wants to join in as well. Charlene produces toy dogs at a rate of 50 per hour and cats at a rate of 20 per hour. The same rules apply to Charlene that apply to Alice and Bob.

• How many cat and dog pairs can Charlene make in a regular 8 hour work day, working by herself?
• What is the maximum number of cat and dog pairs that Alice, Bob, and Charlene can make all together?
• How should the group fairly split their total wages?
• Suppose one person in the group is absent one day. How should the remaining two individuals divide up their work duties? (We have already asked this question when it is Charlene who is absent.)
• In the above scenarios, how should the remaining two workers (fairly) divide up the wages?
• Allowing the three workers to work together or not, in any way they choose, and considering the total output, which grouping of workers produces the largest number of toys?
• Which combination or combinations of workers is favored by the various "fair" wage division schemes that you considered? (That is, assuming that 1) each worker wants to earn as much as possible each day, 2) nobody is forced to work in a group if it doesn't benefit them, and 3) no group is forced to take on an individual if it will reduce any other group member's pay, then how will the workers organize themselves under each wage division scheme?)
• Which wage division scheme(s) encourage the largest output of toys?
  

(Of course, all wage questions could be answered by deciding how many cats and dogs each worker would take from the combined pool of cats and dogs produced, allowing for fractional toys if necessary.)

For Further Exploration

• How sensitive are the calculations above to adjustment in the different working rates?
• Which wage distribution agreements give (better) incentives to participate in groups?
• Is there some easy way for a manager to decide how to distribute assignments? For instance, if she can only allow her workers to work alone or in pairs, how can she decide who should be paired with whom?

There it is. Thinking about this problem gave me some new insights into competitive advantage. I hope you have fun working through these questions, too.

Read More . . .

Outstanding Plaque Completed (finally)!

Oh, the puns could flow, but I shall resist. Below is a picture of the "Outstanding Math Major" Plaque hanging in the Department. The first entry is from 1989-90, and it was updated (as far as I can tell) regularly until 1996. Then it just hung around for 12 years...

So we finally got our act together and completed the entries from 1997 on forward, relying on our own memory and that of the University archivist (Mrs. Sybil Novinski) and the Provost's office (Mrs. Bette Manzke). But done we got it.

So without further ado, I present the plaque of honor:

As you can see, it's very shiny. And (leave it to us clever mathematicians) it's a trapezoid! Ok, that's just the distortion from my camera phone.

Finally, if you see any mistakes or know of anything missing, please let me know. We'll get it fixed up quick (i.e. in another 10 years or so --- just kidding!).

Some things never change...

There is a wonderful article by Jöran Friberg in the latest issue of Notices of the AMS entitled "A Remarkable Collection of Babylonian Mathematical Texts" [pdf file]. It explores the mathematics in several cuneiform texts from the 4th millennium to the 1st millennium B.C., much of it from the Old Babylonian scribe schools. Some highlights:

• We all know (!) that the Babylonians used base 60. However, they actually used a hybrid of base 10 and base 60: they would group by 10s until they reached 59, and then would start grouping by 60s.
• They had amazing calculatory powers: one of tablets has the computation of 15 to the 12th power.
• The school tablets are clearly from students, and we can "follow in detail the progress of [the] ... students in handwriting and computational ability from the first year student's elementary multiplication exercises written with large and clumsy number signs to the accomplished model student's advanced mathematical problem texts written in a sure hand and with almost microscopically small cuneiform signs."
• One of the tablets shown has so many mistakes that an exasperated teacher (apparently) drew a big "X" across the entire tablet! As I said, some things never change...
  

Charity Week

Charity Week is underway right now. Ya'll remember that, dontcha?

Colloquium for Fall '08

Just a quick post to point out UD's official Math Department web page can be found at www.udallas.edu/math. Also, check out our Colloquium Schedule, which I will try to keep up to date as the semester goes along. If you are in town, we would love to have you stop in to see a colloquium or two.