I was at the Anja S. Greer Conference on Mathematics and Technology at Phillips Exeter Academy this summer, and I loved it. Not only did I get to think about math and math teaching all day, but then Happy Hour meant refreshment and more great problems. Like when Frank Griffin posed this problem:
How many regions are formed inside a circle when you draw chords from a given number of points on the circumference?
With one point on the circumference, there are no chords and one region inside the circle.
With two points on the circumference and there is one chord and two regions.
Add a third point and draw all possible chords. How many regions? There are four regions inside the circle.
How about 4 points? And so on. At some point there are multiple answers, so we are always looking for the maximum number of regions you can create for a given number of points. Can you predict how many regions for 10 points? 45? n points?
That’s the problem. As I began to work on it, I found myself on the same kind of path I often follow when working on a rich and challenging problem (in fact, it is the reason that we often refer to such problems as “rich”, like a rich vein of ore): not a direct path to the solution, but a branching course, with a lot more trial and error than my students probably imagine math problems are supposed to involve. I decided to document my path through the landscape of this problem, both for myself and as an example for my students to follow. So what follows will involve a lot of spoilers. Stop here if you want to work on the problem on your own.
For my students, I want to describe the process of mathematical investigation just a little further. It is like trying to cross an unfamiliar landscape, where some paths may take you from your camp to a cave on the ridge, but many more will not. You have to find the right path. So you look up at the ridge you want to reach, and look at what is just below it. Then you make a guess: what looks like a likely path to the top? Now you have a new task – try to get to the bottom of that path. In order to get to the bottom of the path you have to cross the river right in front of you. So now the problem becomes how to get across the river. You can see some rocks to step on close by, but you can’t see what is beyond that. So you walk out as far as you can…
And that is how it goes. You keep changing your focus, from far away to the step right in front of you. You make guesses about what might be useful, and you make discoveries as you journey further along that may or may not change your path. Your one question – How do I get from here to there? – sprouts numerous smaller tasks and problems, and you try to solve those. In the meantime, depending on how curious you are and how much time you spend exploring, you get to know the lay of the land, more than just the path you are searching for. This is how all the various subjects of mathematics (in fact, we often refer to these as “fields of mathematics”) have been discovered and developed, by people following the breadcrumb trail of questions that lead to further questions. It literally never ends.
[THE SPOILERS START HERE]
So, the first thing that I discovered was that the pattern that appears is not the pattern it appears to be. 1, 2, 4, 8, 16, 31… Wait a second. How does that work? In looking at the differences between the terms, and looking at those differences, I realized this was not a geometric function, but it was a polynomial of degree 4. Okay. That’s weird. But it also was my first discovery and new idea… or rather my first new question. This suggests that just because I have 7 terms of 2^n, or even 17, that doesn’t mean I have the function 2^n. And here come the questions: is that generally true? Can I make the change at any term? Can I change that term to any number? Can I do it to any power sequence? This seems to suggest that polynomials can approximate a geometric sequence to arbitrary accuracy… is that useful to anyone? I don't know, but it is some mathematics that I understand in a different way because I walked here by myself. I haven't yet walked much farther down that path, but I may come back to that someday.
How many regions are formed inside a circle when you draw chords from a given number of points on the circumference?
With one point on the circumference, there are no chords and one region inside the circle.
With two points on the circumference and there is one chord and two regions.
Add a third point and draw all possible chords. How many regions? There are four regions inside the circle.
How about 4 points? And so on. At some point there are multiple answers, so we are always looking for the maximum number of regions you can create for a given number of points. Can you predict how many regions for 10 points? 45? n points?
That’s the problem. As I began to work on it, I found myself on the same kind of path I often follow when working on a rich and challenging problem (in fact, it is the reason that we often refer to such problems as “rich”, like a rich vein of ore): not a direct path to the solution, but a branching course, with a lot more trial and error than my students probably imagine math problems are supposed to involve. I decided to document my path through the landscape of this problem, both for myself and as an example for my students to follow. So what follows will involve a lot of spoilers. Stop here if you want to work on the problem on your own.
For my students, I want to describe the process of mathematical investigation just a little further. It is like trying to cross an unfamiliar landscape, where some paths may take you from your camp to a cave on the ridge, but many more will not. You have to find the right path. So you look up at the ridge you want to reach, and look at what is just below it. Then you make a guess: what looks like a likely path to the top? Now you have a new task – try to get to the bottom of that path. In order to get to the bottom of the path you have to cross the river right in front of you. So now the problem becomes how to get across the river. You can see some rocks to step on close by, but you can’t see what is beyond that. So you walk out as far as you can…
And that is how it goes. You keep changing your focus, from far away to the step right in front of you. You make guesses about what might be useful, and you make discoveries as you journey further along that may or may not change your path. Your one question – How do I get from here to there? – sprouts numerous smaller tasks and problems, and you try to solve those. In the meantime, depending on how curious you are and how much time you spend exploring, you get to know the lay of the land, more than just the path you are searching for. This is how all the various subjects of mathematics (in fact, we often refer to these as “fields of mathematics”) have been discovered and developed, by people following the breadcrumb trail of questions that lead to further questions. It literally never ends.
[THE SPOILERS START HERE]
So, the first thing that I discovered was that the pattern that appears is not the pattern it appears to be. 1, 2, 4, 8, 16, 31… Wait a second. How does that work? In looking at the differences between the terms, and looking at those differences, I realized this was not a geometric function, but it was a polynomial of degree 4. Okay. That’s weird. But it also was my first discovery and new idea… or rather my first new question. This suggests that just because I have 7 terms of 2^n, or even 17, that doesn’t mean I have the function 2^n. And here come the questions: is that generally true? Can I make the change at any term? Can I change that term to any number? Can I do it to any power sequence? This seems to suggest that polynomials can approximate a geometric sequence to arbitrary accuracy… is that useful to anyone? I don't know, but it is some mathematics that I understand in a different way because I walked here by myself. I haven't yet walked much farther down that path, but I may come back to that someday.
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