The highly linked hypertext architecture of the World Wide Web matches the highly linked nature of human knowledge and provides a solid, even exciting, basis for highly linked learning. Mathematics' role as a language for expressing, communicating, and manipulating ideas in every area means that students should move seamlessly to and from mathematics and other areas. For example --
It requires just a few keystrokes for someone writing material in mathematics to include biographical information about mathematicians from the MacTutor History of Mathematics Archive.
We can use real data from the United States Census Bureau and other sources.
Mathematicians have included exercises in their courses about dropping bombs for years but those exercises are more meaningful when they come from Cockpit Physics developed by the Department of Physics at the United States Air Force Academy.
As this is being written, Mark McGwire has just hit his 64th home run and is one home run ahead of Sammy Sosa. You can find a great deal of statistics and data relevant to the home run race on the Web. Click here for one example. The table below from the Chicago Cubs Web site summarizes the situation as this is being written on Saturday morning September 19, 1998.
Even the most casual baseball fans are caught up in the drama of the 1998 home run race and asking questions like --
4th or 5th Grade
This example can be used in fourth or fifth grade OR in a teacher preparation course OR in any course in which we want to talk about the relevance of mathematics.
We can approach these problems by setting up a game. Begin by dividing the class into two equal groups -- the Mark McGwires and the Sammy Sosas. Each student should make a name tag identifying himself as a Mark McGwire or a Sammy Sosa. Have each team look up the number of home runs their player has hit so far during the season and the number of games their player has played so far during the season. Then have each student make a spinner like the spinner shown below.
To construct their spinners they need to find the likelihood that their player hits a home run in any particular game. They do this by dividing the number of home runs their player has hit so far by the number of games their player has played so far. Then they carefully color the background of their spinner so that the fraction of the circle that is red is the likelihood that their player hits a home run in a particular game.
Now divide the class into pairs with one Mark McGwire and one Sammy Sosa in each pair. Each pair will "play" one home run race. The two players in each pair stand side-by-side. If one of the players has more home runs on the day that this experiment is being done then that player takes one step forward for each home run by which "he" is ahead. For example, on the morning of September 19, 1998, the Mark McGwires would take one step. Of course, you need to make sure that everyone takes steps of the same size.
Then each player spins "his" spinner one time for each of "his" remaining games. For example, on the morning of September 19, 1998 each Mark McGwire would spin 8 times and each Sammy Sosa would spin 7 times. Each time a player "hits a home run" (their spinner stops on the red color) "he" takes one step forward. Once again you need to make sure that everyone takes steps of the same size.
Now count the number of pairs in which Sammy Sosa is ahead; the number of pairs in which Mark McGwire is ahead; and the number of pairs in which the players are even. This gives us a rough experimental idea of how likely each player is to win the home run race and how likely they are to finish in a tie.
The experimental results will be more meaningful if you have a large class or if each pair plays 10 or even 20 home run races and keeps track of all the results. The same ideas can be used to study the other questions listed above.
Note that this is a very rich situation for discussion. For example, a student might raise the possibility that in real baseball a player sometimes hits two or more home runs in a single game. The game the students are playing doesn't have this possibility. This might lead to a discussion of a better model -- that is, a better game for simulating the real home run race.
Working with graphing calculators
This is a wonderful venue in which to illustrate the power of graphing calculators for working with probabilistic models. As an example, we work with the TI-92. The TI-92 function rand produces a random number between zero and one, possibly zero but never one. The TI-92 screen below illustrates this function.
Using this function, the TI-92 can simulate the spinners used in the 4th or 5th grade game above. The screens below show two functions, ssspin and mmspin, that illustrate how this can be done. Each of the two functions uses two parameters, ssgames or mmgames, for the number of games played by the player so far and sshomers or mmhomers for the number of home runs hit by the player so far. Each of the functions returns 1 if the player hits a home run and 0 otherwise. The two functions are named ssspin for Sammy Sosa's spinner and mmspin for Mark McGwire's spinner.
The two functions below each simulate one home run race for one of the two players. We use two new constants, sstogo and mmtogo for the number of games remaining for each player.
The screen below shows the values of the parameters on the morning of September 19, 1998.
The function, races(n), below simulates n home run races and keeps track of how often Sammy Sosa wins the home run race; how often Mark McGwire wins the home run race; and how often they finish the season in a tie.
The screen below shows one results of 1,000 simulated home run races. Notice that based on this analysis, on the morning of September 19, 1998 we would have estimated Sammy Sosa's probability of winning the home run race at roughly 14%; Mark McGwire's probability of winning the home run race at roughly 70%; and the probability of a tie at roughly 16%.
Better estimates can be obtained by doing more simulations -- either by using one calculator to do more simulations or by using several different calculators. With problems like this, it is often very effective to have each student in the class run the same number of experiments and use the results individually to illustrate experimental variation as well as use the combined result to obtain a better estimate based on more simulations.
The TI-92 programs above can be downloaded to your TI-92 if you have the TI-92 Graph Link 92 program and cable. Each program is available in two forms -- as a TI .92f or .92p file and as a uuencoded TI .92f or .92p file. The first form is easier if your operating system, browser, and TI-Graph Link 92 are all set up to handle it but the second form works on more combinations of hardware and software. Click here for help downloading TI-92 programs.
More advanced approaches
The problems above can also be studied using Markov chains.
Copyright c 1998 by
Frank Wattenberg, Department of Mathematics, Montana State University,
Bozeman, MT 59717