Reading ahead: Question about diagonals of a polygon

•October 11, 2012 • 5 Comments

Dan Pearcy asked this question in a comment:

I realize that I’m going a bit further on here but I’m really keen to discuss a question set on page 30.

Lockhart poses the question, “Do the diagonals drawn from one corner of a regular polygon always make equal angles?”

Firstly, I’ve never really thought about this before so, for me, it is a nice question to think about. Secondly, after jotting down a few things I’ve realized that I have an “argument” for why this works for any n-sided polygon. It’s quite interesting because it isn’t really a proof in the normal sense. It’s more like a set of instructions that you can just apply over and over again depending on how many sides your polygon has. I guess a bit more like a flow diagram with a loop in it.

I’m intrigued to know how other people have shown/proved this?

Reading assignment 2: pages 8-14 Introduction to proof

•September 28, 2012 • 8 Comments

Thanks so much for the fantastic discussions thus far. Here is the second reading assignment and questions. Please don’t worry about feeling a need to keep up—we’re going to keep the assignments small (5-7 pages/wk) and you can jump into any discussion you want.

We’ve also tried to create a variety of questions that are appropriate for many different levels of math background, so please don’t feel like you’ve got to have some level of expertise in order to participate. I think the message of this book is that the beauty of math should be accessible to everyone, and this is a point I hope we will emphasize in our discussions.

  1. Lockhart tells us “don’t be afraid that you can’t answer your own questions—that’s the natural state of a mathematician.” I think this is the natural state of just about any expert—I think really good guitar players, or athletes, or artists know exactly what it is they can’t do, and are also working hard to be able to do that thing.

    What I wonder is why is it that the world of mathematics, novices find not knowing so distressing? Guitar players don’t seem to mind being terrible at first, nor do swimmers, writers or many other types of novices. It seems like getting all learners to better embrace Lockhart’s “natural state of a mathematician” might go a long way toward helping increased people’s appreciation of math. Do you ever embrace not knowing? What helps you to do this?

  2. Suppose I could experiment with 3000 different triangles, measuring the concurrency of their medians, and concluded that all of those triangles medians met in a point (or wrote a computer program to try 3,000,000). How is this idea different from the one Lockhart proposes?
  3. Lockhart lays out two proofs for the medians of a triangle meeting at a common point. One invokes a general symmetry of flipping the triangle across an vertical axis, as shown on page 11. The other involves considering that the triangle must be rotationally symmetric, and arbitrary flipping the triangle along an axis of symmetry.
    • How would you describe these proofs in your own words?
    • Is the second proof preferable to the first? Which do you prefer and why?
    • Is there a better argument out there than the two Lockhart presents?
  4. In describing his proof, Lockhart says “It has a lot of elements mathematicians look for: elegance, simplicity. So it’s probably true.” What does Lockhart mean here?

  5. Can you give a radically different proof of the concurrency of triangle medians than the one Lockhart provides?
  6. How far up the triangle is the intersection of the medians?

Reading assignment 1: pages 1-7 Mathematical Reality

•September 20, 2012 • 27 Comments

At the moment, I’m envisioning that this group will try to read a few pages of Measurement this book every week. This will likely mean we won’t finish the book in the fastest fashion, but that isn’t really the goal of this conversation anyway.

To help guide our conversation, I, along with the help of a few other math teachers (Michael Perhsan, Shawn Cornally, and Paul Solomon), have put together some reading questions which will hopefully serve as good starters for conversations in the comments of this post, but please do not feel constrained by these questions—we would much prefer that you ask your own questions than simply respond to ours.

If our conversations become too complex to be handled by the simple commenting system here, we can always move them to a different forum.

Introduction: Reality and Imagination

  1. What does Lockhart mean by mathematical reality? How does he see this world as different from physical reality? Do you buy it? Where do you think the author is going with this distinction?
  2. Some people say that there are perfect circles, while others say that everything in the world is imperfectly round. Which side do you agree with? Why?
  3. What do you think of when you hear the term “mathematical reality”? What experiences have you had exploring mathematical reality?

Developing Mathematical Arguments

    1. For the problem of proving that connecting the medians 1 of a triangle meet at a point—can you draw some sketches with pen and paper to see for yourself that this seems to be the case? If you’re really adventurous, try making a drawing with a computer geometry system like Geogebra.

Here’s a quick video tutorial to show how easy it is to get started with Geogebra:

    • How can this problem be tweaked? Is this a narrow version, or part of some larger truth about all shapes?
    • Is this true for other shapes? What happens when you draw the diagonals and connect the midpoints of a rectangle—do they meet at a point? Does this hold true for any 4-sided object? Feel free to explore n-sided shapes—pentagons, hexagons, etc.
  1. Lockhart mentions that the best problems are you own. In that vein, what’s a mathematical problem that you have some interest in?

1A median is a line from the vertex of a triangle to the midpoint of the opposite side.

Welcome to the Lockhart Reading group

•September 18, 2012 • 36 Comments

This is a blog dedicated to exploring the beauty of mathematics. We will be doing this in a somewhat unconventional way—this blog will house a conversation about the book, Measurement, by Paul Lockhart. The conversation is open to anyone. Initially, 30 people ranging in age from 14 to 55, from all around the world and a variety of mathematical backgrounds will try to learn together by reading this fascinating book.

Perhaps it is best to begin with some short introductions. If you’re planning on engaging this conversation, please add a short comment telling us a little bit about yourself. Who you are (at least your first name), where you’re from, and in a couple of sentences, why you’re interested in learning more about the beauty of math.

I’ll also let the book’s author, Paul Lockhart, give you a sense of where we’re going.