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Outreach is an important part of the Woodrow Wilson Institute. What we have seen and done this summer has inspired all of us to make some changes in our outlook and teaching for the next year. This seed has been planted but it now becomes our task to plant this seed in other teachers' minds. For each teacher we touch and encourage to change, we create change for many students.
This is an outline of some basic ideas and elements to help you create a brief presentation for a group of your peers.
The 1993 Woodrow Wilson National Fellowship Foundation math institute had as its topic "change." As the participants considered the aspects of studying change from a mathematical point of view, they also considered changes within the math curriculum, changes of both content and teaching strategies.
As we discussed our goals for a workshop, we decided that it would be overwhelming to share with any group of teachers even a synopsis of all that we had done. So what we have listed here are some possibilities. But as you prepare, remember not to plan too much and to keep it simple and do-able.
As an opening, you might use a few personal anecdotes about where math education is, where it needs to go and why it needs to go in that direction. Here are some ideas, if you need to supplement your own.
• Once a friend's child (about 8 years old) asked if I could tell her what 89 x 74 was. I said "Sure, let me get a piece of paper." Horror flooded her face and she said "you mean you're a math teacher and you still don't know your 89's?" From her perspective, math was memorizing multiplication tables ad infinitum. Our perspective of math might not be quite as limiting as hers was, but perhaps it is more limited than it should be.
• After hearing that the remedial math education program at a school consists of doing up to 2 years of individualized learning sheets on adding, subtracting, multiplying and dividing whole numbers, fractions, decimals, percents and doing some measurement, another participant replied, "Certainly we can do better for our students than a $5.95 education" (the cost of a 4 function calculator that will do the same thing).
A recipient of the Field Prize in mathematics in being interviewed was asked to describe what a mathematician does. He responded "I fill wastepaper baskets." He was referring to the fact that he thought up problems and then kept trying them and trying them until he came up with a solution. This is probably the total vision of most teachers as well as students of what a mathematician does. During the institute this summer, the vision was expanded. This might be shared, not by going over any of the problems that were presented in great detail, but by explaining just briefly one of the relevant problems and the impact that mathematicians have had on the solution of this problem. Here is a brief summary of a couple of the problems that might be useful in helping you to prepare for your presentation. Attached also are the pertinent graphs.
You might want to summarize these quickly and show the graphs associated with them, as they are excellent examples of how mathematical models can work in the real world. The Suez canal problem is very do-able by students if you want to extend these ideas.
Some of the topics relating to AIDS that have been studied by mathematicians are the data on world wide cases and U.S. cases, past and projected, the financial impact and models used in predicting the spread of AIDS. When the statisticians first looked at the data, they thought that they had a purely exponential model.
Figure 1
After several years, when they added the more recently gathered data and built a mathematical model, they found they could make much more accurate predictions. Among the things that they have discovered based on looking at these new models were the effects of limiting sexual partners, the discovery and use of a vaccine, a drug for prevention and a drug to prolong the incubation period. While a cure for AIDS may be very hard to achieve, it seems probable that extending the incubation period will improve both the quality and quantity of life for affected individuals. All of Dr. Griffiths models were based on studies of the homosexual population; however, similar models could be created for other populations. Dr. Griffiths emphasized, "The biggest single advantage of simulation modeling is that it's possible to interrogate the model. That is, we have a situation in which we can ask `What if?' "
Dr. Griffiths was employed by the Egyptian government to determine how to increase the capacity of the Suez Canal from about 65 ships per day to 110 ships a day. After fully investigating the history of the canal, the canal structure and the present scheduling of ships, Dr. Griffiths constructed a graph of the situation and began to study the model. He discovered that by changing the schedule to a 48 hour cycle rather than a 24 hour schedule, he could significantly increase the capacity of the canal.
Figure 2
Dr. Millie Johnson has studied the flow patterns of rivers and found several principles governing these patterns.
1. No river runs straight for more than 10 times its width.
2. The radius of a river bend is 2 to 3 times the width of the river.
3. The wave length is 7 times the width of the river.
Dr. Johnson discovered that a function called a sine-generated curve closely approximates the natural path of a river. Apparently as a river flows it is constantly trying to minimize its change of direction, its total effort in turning and the erosion that occurs. When a river is allowed to flow and meander in this fashion, it remains a stable entity. When the natural flow is disrupted, major problems like the Mississippi river floods occur. By using the sine-generated curve, mathematicians can help predict more stable, future river paths and plan for them.
It might be helpful to look at recursive or iterative models for functions. We can work with the explicit function f(x)=2x + 1. We plug in the values we wish for x to find f(x). Using an iterative model for the same equation, we use the most recent answer for x to find the new value of x. So, if I let my seed, x0=1, then
x1= 2x0+1
x2=2x1+1
x3=2x2+1 and so on.
Keystrokes Display
You can do this in your TI-81 or TI-82 by 1 enter 1
*2 +1 ans*2 +1
enter 3
enter 7
enter 15
enter 31
Let your participants try one themselves. f(x) = + 1 with x0=1
They should get these values x0=1
x1=1.5
x2=1.75
x3=1.875
x4=1.9375
x5=1.96875
Notice that in the first function, the values of f(x) blew up and that in the second function the values get closer and closer to 2.
It is important that teachers walk away from this workshop with some very clear ideas. It is also important they have a hands-on experience. Here is a suggestion of what you might do.
1. Create a fractal through iteration.
*a. If parts of a figure are small replicas of the whole, then the figure is called self-similar. These self-similar figures are called fractals. Fractals are generated by an iterative process of scaling, rotation and substitution. We are going to work with a special kind of fractal called a Koch snowflake.
*b. You need a sheet of isometric graph paper to do this activity. On your isometric dot paper, draw a line 27 units long.
*c. To create the "seed" of the Koch snowflake, think of another line of 27 units, shrink it by a third, put it on isometric dot paper. (An isometric dot paper master is available in Graph Paper from Your Copier). Then take another one third line, attach it to the endpoint of the first line and rotate it 60 degrees counterclockwise. Take another one third line, directed the same as the original one third line, attach it to the endpoint of the second line and rotate it 300 degrees clockwise. Now, you'll need a fourth one third line, attach it to the endpoint of the third line. This will be our seed for the Koch snowflake.
*d. Now take the figure you created in *c and follow all the directions from *c using this figure *c instead of the line as the thing that you are copying. This is your first iteration.
e. Take the figure you had at the end of *d and copy it using the directions in *c. This is your second iteration.
*f. You can keep repeating this process as often as you like.
Figure 3
2. After doing this it would be very useful to use the software, Fract-o-graph, to display further iterations with the teachers as a group.
3. If you are doing a longer in-service, you might want to look at "Fractal Geometry" at this point for extensions.
1. A look at the program Chaos and Dynamics
2. A look at some fractal application programs
3. Extensions of iteration
A presentation of various lengths can be developed by varying the amount of emphasis given to the different sections of the outline. The introduction is a good place to stress the need to include more than the computation component of mathematics and to identify the motivating aspects of teaching through applications. Looking at examples of mathematical models and, also, looking at problems dealing with "real world" data can renew a teacher's interest in reaching beyond the teaching of computation and abstract procedures.
To understand fractals, and eventually, to understand dynamical systems, requires some expertise with iteration. The number of examples of iteration and the complexity of the examples you select will depend on the length of the presentation and the type of audience. The length of the presentation will also determine how many fractals can be demonstrated and how much hands-on work can be done. The availability of computers or graphing calculators will extend the possibilities for demonstrations with fractals.
Even though the 1993 Woodrow Wilson National Fellowship Foundation Math Institute focused considerable amounts of time on the study of dynamical systems beyond fractals, this workshop outline is directed toward developing background concepts . In fact, most of the problems in this book on Change can be used with students who have a good working knowledge of Algebra II or Pre-Calculus. This book, combined with the technology now available, provides a new way of exploring problems that prior to this may have been purely in the venue of a research mathematician. It is exciting to realize that the knowledge and resources for investigating a "frontier" of mathematics are now in the reach of many high school students.
All the resources alluded to in this project are listed in the bibliography at the end of the 1993 WWNFF Math Institute book on Change.
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