The ppmmc series will resume on Monday as we will take a mild diversion since today is Friday.

Previously, I gave a short introduction to Monte Carlo methods, specifically, the “Hit-or-Miss” method. Now, the nice thing about the hit-or-miss method is that conceptually it is very easy to digest. So, I thought that the following may be of interest to some instructors when they first introduce their students to geometry.

Students are taught the basic formulas for the classic (convex) polygons and for the circle and are given exercises to reinforce some of the geometric concepts. I thought that it may be a fun in-class exercise to do things a bit out of order.

Here is the basic plan:

1. Before introducing any formulas, explain, generally, what the notion of area is. A classical way would be to consider squares of unit size and then generate any shape that can be constructed from unit squares. Thus, that shape’s area is the number of unit squares it contains.
2. Now, we can extend this and construct any shape and ask how many unit squares can be entirely contained by the shape and how many “spill over.” This can be a simple computational exercise.
3. Next, if the shape were put on a standard Cartesian coordinate grid, the instructor can ask other questions like
   • What is the perimeter?
   • What is the area of the shape in proportion to the grid on which it lays?

There are a few benefits here.

• No formulas are introduced and so students will have to work with only the conceptual understanding of area. Thus, the hope is that figuring out area of a two-dimensional object before having any formulas will help to solidify what area actually means.
• Students will get to “feel” the problem since they will have to make actual measurement, which is actually quite realistic.
• Students will have to learn how to make estimates, since shapes aren’t going to have “nice” angles (eg, they won’t all be right triangles), again realistic.
• Eventually students will take a Calculus course and will be introduced to the notion of integration via Riemann. Conceptually, this is no different from the notion of filling a shape with squares (and the new thing that students will have to grasp is how limits really work).

So where does Monte Carlo tie in? Grab allgeos.pdf. In this file, there are fives images of triangles sitting on a square, gridded canvas. Each image is succeeded by another image with 5000 random points strewn about the canvas. Where those points landed gave an estimate for the area of the triangle as a proportion of the gridded region. Additionally, an exact calculation for the proportional area is also provided.

So, the real exercise?

• Don’t show any formulas about how to calculate area. Instead explain it conceptually only.
• Provide the images of the triangles without the hit-or-miss method and ask students to estimate the proportional area for each triangle, by whatever means possible.
• Explain the hit-or-miss method.
• Compare and discuss the advantages / disadvantages of the method the student chose and the hit-or-miss method.

Since the hit-or-miss method requires a random process (actually, we can go one step further and discuss quasi-Monte Carlo methods, but it’s a bit of out scope here, look for a future discussion on this), a good discussion to have with students is about what the requirements for randomness are and how we can “tell” if something exhibits random behavior. What would happen to the area estimate if all the random numbers were in the top left quadrant of the grid? Would we consider this to be random? What kind of spread should we expect from random numbers? What would happen to the quality of the area estimate if more numbers were used? If fewer numbers were used?

Mathematical subjects are interconnected. They are not a bunch of disparate topics and ideas glommed into something we call mathematics. The construction of a “good” sequence of random numbers requires a heavy amount of number theory and strong statistical tests to ferret out specific patterns. The entire question of finding area is something that can be addressed via calculus. The formalities of probability theory can be laid out via Lebesgue measure theory which is a few steps up from what students learn in a standard Calculus course.

This exercise should be a good demonstration that random processes and geometry can be tied together and that mathematics isn’t just about formulas.

Anyhow, I’d be interested in hearing from you if you think this can work in the classroom. I had a reasonable amount of success demonstrating this in a standard college Calculus II course. I’ve never had the chance to test it out in a classroom where students are being introduced to geometry for the first time. I know that some of the things I mentioned about using unit squares, etc. have been a standard didactic method, but I am curious to know how students would receive the idea of the hit-or-miss method.