Pick’s Theorem

Happy 2018!  Here I am posting on this infrequently-updated blog, and who knows, maybe I’ll keep it up.

In the past few years, I’ve seen Pick’s Theorem alluded to in various places. This theorem gives a surprisingly simple way to calculate the area of a polygon drawn on a lattice (most people imagine a Geoboard) based on the number of points on the polygon’s boundary and the number of points on the polygon’s interior. I won’t post the formula here in case you’d like to discover it for yourself, but here is one of many websites illustrating the theorem.

For example, the polygons below each have 5 boundary points and 3 interior points. Despite their different shapes, Pick’s Theorem predicts that each will have an area of 4.5 units.

Picks Theorem blog
I wanted to explore Pick’s Theorem with our Math Circle, a group of about 8-14 middle schoolers (mostly 6th graders). After examining lots of other math-circle Pick’s Theorem explorations, I handed the students the following much simpler version:

Math Circle – Picks Theorem

The handout ends right about where you would want to start making conjectures, with students being asked to find polygons with a given number of boundary points, number of interior points, and area. The last question is deliberately impossible.

The ideal math circle as described by its founders, the Kaplans, is a relaxed session in which students are presented with an interesting mathematical situation, then ask questions and make conjectures and discoveries together. I find that my group is a little young for that kind of Socratic seminar style: they really want to be doing things. So my aim last time was to given them lots to do, but eventually put them in a position where they would begin to generate some questions and ideas of their own.

Last time, most students got to various places on the last page. For a few, their interest waned before the end of the session. Next time will be tougher to manage! I’d rather not make a handout for the actual discovery of Pick’s theorem because I’d like the investigation to begin with student questions/observations. This means more “talking together” time, which will tax their attention spans.

My plan is to ask students to continue working on the last page, and then whenever enough people grumble about the problems being hard or impossible pool all of our data and see if we can find a pattern that might help us determine which polygons are possible. Depending on what they come up with, I might ask everyone to figure out (B,I,A) for rectangles with dimensions of their choosing, and try to generalize. Or, I might ask them to generate a bunch of polygons all with the same number of boundary points, but to vary the number of interior points, and see what happens.

I’ll bet we can come up with the formula, and at that point some students might lose interest (and I can give them a 1to9puzzle), those who are still with me can think about why the theorem might be plausible. We can come up with a good explanation for rectangles. I’m not sure if we can get beyond that, but the students may surprise me!

We’re hiring!!!

Hello, virtual world! It’s been a long time since we’ve updated this blog, but in case you’re still reading, we are once again looking for a thoughtful, enthusiastic math teacher to join our community. If you like our curriculum, maybe you’d like to try teaching it with us!


At Park, we write about our students at the end of each quarter. This feedback is meant to be global feedback that goes beyond performance on any one assignment.  This almost always takes the form of a paragraph or two about each student, and possibly some sort of rubric.  Parents love this evidence that teachers understand their students’ personalities and learning styles.  However, teachers wonder if it really makes sense to do this four times a year, and if it’s really worth all that work.  By the fourth time you are writing about a student, it can be hard to think of new things to say.   So we’re now looking at mixing it up and maybe doing something different for some of those four times. 

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The Mathematics of Fountain Design

Park teacher Marshall Gordon has an upcoming article in Teaching Mathematics and its Applications: An International Journal of the Institute of Mathematics and its Applications.  The article details the experience he had teaching a project-based unit on the mathematics of fountain design.  In the process of designing their fountains, students were naturally motivated to explore the different parameters affecting the trajectory of a parabola.

Click the link to read the article: The Mathematics of Fountain Design

Not My Best Moment

This happens more often than I’d like: we start by having a normal class discussion and end up trying to resolve something that doesn’t engage the students’ interest, or even their understanding.  It’s the exact opposite of the way I’d like my classroom to feel.

Last week in 9th grade class we were discussing the following problem, which had been assigned for homework.

Write an equation for a rule a?b, so that the answer is odd only when both a and b are even.

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Thinking beyond Conventional Standards: A Tool to Face the Challenges of a New Century

In order to find clues about how to face the challenges that the new century present us, we can take a look at the man’s mental development, as well as the revolutionary moments in the history of humankind, in particular the history of mathematics. We can examine more closely the different mental stages we all go through (according to Piaget), in order to have a better sense of human potential. Two of the innovative moments in the history of mathematics are the creation of non-Euclidean geometry by Nikolai I. Lobachevsky (1793-1856) and the formalization of the concept of infinity and the transfinite numbers by Georg F. Cantor (1845-1918). These achievements were the result of efforts performed by minds working against traditional ways of thinking, freed from the concrete reality where so many mathematicians before them had been stuck. As mathematics teachers at the beginning of the most demanding century ever, we ought to better know our students’ potential, and grant those students who think differently all the attention and support that likely creators of changes in history command.

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Polyhedral Crafting

I started the year once again feeling unsatisfied with the spare, utilitarian look of my classroom.  So, having given up on finding math-related posters I liked, I decided to head over to Math Monday to look for some cool looking thing I could make this weekend.  The result:

Tensegrity polyhedra, made from 3/16″ dowels and standard rubber bands, based on this little article by George Hart.  The coolest thing about them is that no two sticks are actually touching each other (which makes me wish I’d used different color rubber bands).  Or maybe it’s that they can collapse like this…

…and then snap back into shape.  Or maybe that they bounce. (Yep.)

In any case, they were really fun to make–the dodecahedron turns out to be a great spatial reasoning puzzle as you get close to the end–and I think they’ll make good toys or decorations.   And there’s tons more inspiration for mathy crafts at Math Monday (as well as at georgehart.com and vihart.com).  Maybe I should crowdsource this by offering it as extra credit–that ought to get the classroom looking good in no time 🙂

Getting it right this time?

Today is the last day I can treat myself to the luxury of sitting in a coffeeshop on a weekday morning/afternoon.  I came here to think about what I wanted to do in the early days of my 9th grade class.  In practice, this has translated into my spending most of the time solving and thinking about the “Tinker” problems.  This has worked remarkably well to help me set priorities.

I wanted to teach 9th grade this year because I realized that I was not doing nearly as much as I could be to teach students how to be learners.  Assigning students nonroutine problems has its drawbacks: though we have great class discussions and kids learn to see math for the open book that it is, students also have the perfect excuse to say, “I just didn’t know how to do this homework problem”, and teachers have the perfect excuse to give large hints that don’t empower kids to feel that they could have solved the problems themselves. 9th grade seems like a good opportunity to focus on changing some of my practices.  We’re all making a fresh start.

Here are the things I most want to work on with my students this year, things that we need to establish in the earliest days.

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A Unified Approach

No matter what our decisions, opinions, or ideas may be, they are always related—consciously or unconsciously—to the way we see the world. It is clear that we break up the world into small pieces, and call these pieces disciplines or sub-disciplines. Furthermore, these small portions of the world are often so sealed off from each other that it is hard from the perspective of each of them to have a sense of the unity they are coming from.

This fragmentation implies a divided and incomplete knowledge of reality. It is understandable that people have had the need to divide the world into small pieces, so that they are able to approach some kind of knowledge of it. However, this increasing division and subdivision of the world, while deepening our knowledge of smaller pieces of the universe, increases the disconnection between the components of that universe. In fact, it is evident that one can be very knowledgeable about a particular field of study, while hardly being able to understand any other field or, more importantly, how our actions within our field of work may affect the rest of the world.  Therefore, the problems that we face as a consequence of this specialized disconnection ought to make us take a look at the multiple pieces of a totality that should be restored some way.

This divided vision is even experienced within a particular discipline. As a matter of fact, it happens in the way we believe mathematics should be taught. For many years now, there has been a controversy between those who endorse teaching mathematics through real-world problems and those who favor an emphasis on basic skills. In my experience teaching mathematics for more than twenty years, I have observed the limitations that overemphasizing either skills or problem-solving brings to a true conceptual understanding—understood as the connection between a problem at hand and a more general theory from which this problem is a particular case. On the one hand, when the emphasis is only on skills, a mechanical approach to the solution of problems makes it unnecessary for the students to reach a true understanding of the situation involved. On the other hand, emphasizing only problem–solving without enough attention to the development of skills may deprive the students of the power needed to take mathematics—and ourselves—to upper levels of development. Also here I see the need for a unified approach: Skills, problem-solving, and concepts are all of necessary importance.

The evolution of mathematics shows that it moves backwards in a retrospective reflective abstraction—going deeper into earlier mental processes, looking for the roots of mathematical concepts, as the formulation of new systems of axioms shows—and also forward, formulating more powerful theories, which are a generalization of the structures formed throughout  the retrospective reflection mentioned above. More than a static frame to be applied to common situations, algebraic procedures are the formalization of general properties of multiple particular past cases. We cannot deprive the students of the power of mathematics by withholding from them either a true understanding of the problems at hand or the retrospective look of reflective abstraction contained in algebraic procedures. Therefore, our work as mathematics educators must be that of engaging the students in activities that keep them deeply focused on what they are doing, as well as in activities that make them reflect on the current and previous procedures, abstracting general properties from them.

Our Own Marshall Gordon in the Journal of Curriculum Studies

Dr. Gordon argues that Habits of Mind should be the focus of mathematics instruction for students of all ages, and especially for students who will become teachers.  “Mathematical Habits of Mind: Promoting Students’ Thoughtful Considerations” appears in the Journal of Curriculum Studies, vol. 43, no. 4, pp. 457-469.  An full abstract is printed below.

Colleges in the US are being compelled to rethink what the First Year Experience or Seminar ought to be for students who have difficulty with mathematics, and what ought to be the mathematics education of teachers, K-12, given the minimal success most students are experiencing. It will be argued here that toward ensuring a more successful education for all students learning mathematics, and most especially for those who will become teachers, the inquiry process must be made explicit so that the productive practices of a mathematically-inclined mind are considered as content. That is to say, the classroom conversation needs to include discussion of the actions mathematically able thinkers use to gain insight into a problem; such as: considering a simpler problem, tinkering, taking things apart. This paper will make an argument why this is an essential consideration for promoting a robust society, and include instances of how mathematics may be presented in this framework.

If you would like to access the article online, free of charge, send an email to parkmathblog@parkschool.net.  Provided you’re one of the first 50 people to ask, we’ll send you the link.