The Joy of Math

This course is not a content driven course where you ingest a large amount of material because someone thinks it is good for you, although you are not always sure what its application is. This course is an attempt to give you a peek into the real world of math where one can pursue some interesting mathematical question and have fun with it, that is, if you think it is possible to have fun with math, which might have some likelihood of being true since you were somehow attracted to math in the first place.

The intent of the authors of our text is to give some structure to some areas of mathematics where technology can help you explore questions that could be interesting, depending on your particular mathematical prejudices of course. The introductory main worksheet and the W worksheets for each chapter give you examples to draw on in extending them a bit in the A assignments worksheets and in the end of chapter problems. Although specific questions are asked, no one prevents you from going a bit further with any of these problems where it is easy to do so.

The example worksheet 4-5-5 was my way of setting an example of how one can push a problem further, where my curiosity pushed me to see if I could actually derive the probability that the problem is studying by a frequency experiment. Another example could be problem 9, which asks a question about the frequency of 9 versus 1 as a leading significant digit of the product of 3 random positive numbers. It mentions that 9 occurs much less. Initially one might think all digits should somehow be on the same footing, so this might seem surprising, but if you think about it, doesn't this have to do with the actual numbers that come out of multiplication tables? Suppose one restricts the problem to the product of 3 random positive one digit numbers. There are only 9^3 = 729 possibilities to check, and one can count the actual frequency of each of the digits 1 through 9 that occur as leading significant digits in their products to get an idea about the more general problem. This is actually a very good hint about the observed frequencies if you run a number of trials and average them for the actual problem. The next step  might be to try 2 digit numbers, but MAPLE is overwhelmed here. There are 99^3 about 90 some thousand possibilities to check, which seems reasonable for MAPLE to be able to handle, but an "object too large" warning derails the calculation after 40 some minutes (I gave it a shot). Still the one digit case is very useful to give some idea about the general problem.  In writing this all up one comes out with a nice little mathematical report, which is another goal of our writing enriched course, to be doing some mathematical writing on our own (as opposed to editing an existing worksheet and adding a few comments).

I don't want to pressure you since to some extent that defeats the purpose of the course, but we must try to think about how much time outside of class is usually devoted to a traditional 3 credit advanced math course. Are we giving this course its fair shair of time by that measure? Moving into the last part of the course, I need to be able to assess your work to assign some reasonable grade based on your effort and your success in this effort, as measured by your ability to build from the worksheet examples to extend them to related questions, and then explore more deeply some particular questions of interest. The end of chapter problems provide a nice starting point for writing mathematical reports where we can follow our interest a bit and show some mathematical ability by the way in which we discuss the question and insert some of our own imagination into it by extending the question a bit. We have the end of section problems for chapters 4,5,6 and then some end of book project suggestions to use in this regard. We seem to have gotten slowed down considerably in chapter 4, which is okay if we are learning something from our experiences here.

To help give our laboratory course a little  more structure, I think it is wise if we regularly meet outside the class in my office to communicate individually how we are investing our time and what results we are getting. It is too easy to slight a course which is not pushing you with immediate deadlines and traditional workload. Please find the time about once a week or so to meet with me. You can also email me about your progress to keep me informed.