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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. 

Continue reading

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

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 and  Maybe I should crowdsource this by offering it as extra credit–that ought to get the classroom looking good in no time 🙂

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  Provided you’re one of the first 50 people to ask, we’ll send you the link.


Let’s Ban the “Distance Formula”!

One of the things that can be hardest for kids learning algebra is to be able to understand the value of abstraction and of using symbols to help one analyze and think about a problem.  I myself remember learning algebra (way back in 1978)  from Dolciani/Wooton, a textbook that valued formal manipulation above all else; there was almost no motivation given for where the rules came from, just lots of practice in learning how to manipulate symbols correctly.  Indeed, for a number of years afterward I thought that formal manipulation was all there was to algebra.

The idea that there are ideas to be discovered in algebra was completely foreign to me.  I knew in 9th grade that perpendicular lines had slopes that were negative reciprocals, but if you had asked me why, I would never have known, or even thought I should have known.  Dolciani gives a proof, but I would bet a lot of money that very few students read it, as it is, to be frank, inappropriately abstract for a high schooler learning the subject for the first time. 

Our curriculum ( ) approaches the topic by having students first draw a line on graph paper with a slope of 2, and then try to figure out experimentally what slope a line perpendicular to it should have.  Working out concrete examples using lines of different slopes can lead to a much deeper understanding for a 9th grader than an algebraic proof that they have little chance of following, never mind retaining.  When students are older and have more experience with algebra, of course, a formal proof can make good pedagogical sense.  But for a freshman? 

Which is why I think we should ban(!) teaching the “Distance Formula”, at least for the large majority of 9th and 10th graders.  Why, you might ask, if I am trying to teach them the value of abstraction? Continue reading

Our department is hiring!

Our department is hiring!  Would you like to teach intellectually curious students using our problems-based curriculum and collaborate with thoughtful colleagues? Click here for details.

Park School Math in the “Mathematics Teacher”

We’ve been fortunate enough to have an article published in the February 2012 issue of the Mathematics Teacher, “Geometry in Medias Res”.  We teach Geometry in a fairly unusual way, we think, so we decided to write about it and see what people thought about our approach.

One of our main ideas is that we want students to encounter interesting problems on the first day.  So we ask them non-trivial questions right away (e.g. can every triangle be circumscribed?), and in the process of them discussing/arguing with each other, we start to develop with the kids the necessity for a standard of proof other than “it really seems like it to me!”. 

There’s a lot more, but the idea is to have a more natural and intuitive introduction to the axiomatic nature of Geometry than one usually finds.  We are big believers that proof is completely accessible to all levels of students, but that it has to be introduced gradually, as a way of resolving questions students have, not as a forced superstructure like the way it was often taught in the past.

If you have a chance to take a look at the article, we’d love your feedback and thoughts.