What is a polygon?

A comparison of two different approaches

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This is the second of my newsletters reflecting on what I learned during a recent visit to Sweden. You can read my thoughts on using vertical, non-permanent surfaces here.

During the trip, I was joined by some maths teachers from the US. They also spent Day 1 watching lessons in a Swedish school. On Day 2, they had the dubious honour of receiving a day’s CPD from me, followed by a debate on various aspects of teaching and learning on Day 3.

During that debate, we fell upon the subject of teaching polygons. Whilst this may seem a rather niche example, polygons are an example of what Kris Boulton would call a Category Atom. If you can ask the question: Is this a ___?, and the answer is Yes or No, then you have a Category Atom. Other examples of Category Atoms include:

• Is this an integer?

• Are these like terms?

• Could this value be a probability?

• Are these two fractions in a form ready to be added?

I approach the teaching of Category Atoms in the same way each time, and it was interesting to compare this approach to that of my American colleague - an approach I regularly see during my lesson visits in England.

What follows is not an attempt to argue that my approach is better. Instead, it is intended to provide some food for thought.

The approach of my American colleague

She begins by giving the students a bunch of shapes to sort - ideally physical shapes, but they could be pictures of shapes. Something like this:

At this stage, her students are not given any additional guidance—they are free to apply any criteria they like during the sorting process. While this happens, the teacher circulates the room, conversing with students to find out their sorting rules—perhaps some have gone for colour, others for 2D and 3D—and then chooses some students to share their rules with the rest of the class.

The teacher then sorts the same shapes into two groups—polygons and non-polygons—not sharing this terminology at this stage. Students are invited to share their thoughts on how they have been sorted.

With the key features identified, students are presented with a new shape and asked to reason and discuss which group it belongs to.

My approach

I start by presenting five or six examples to the class. There are always 3 positive examples and 2 or 3 non-examples, depending on the critical features I want to draw my students’ attention to. In the case of a polygon, I need 3 non-examples as I wish to show that polygons cannot:

1. be open

2. have curved sides

3. be 3D

Because of the specific requirements of the following sequence, my examples are created on PowerPoint rather than physical models.

The first example is a non-example:

I present this on the board, pause for a few seconds, and then say, "This is not a polygon." I do not say any more, and my students sit in silence.

I then make a single change to this shape to make it an example:

I either rub part of the original example out and draw the new bit, or use PowerPoint’s morph feature to merge the examples.

I leave the new shape visible for a few seconds to allow my students to reflect on my change, and then say: This is a polygon. No further explanation and no talking or questions from my students.

Next, we have another positive example, this time with several key features changed:

This is a polygon.

This sequence continues in the same vein until we have:

For my fellow geeks, I have written about the principles that govern example choice and the pedagogy of delivering such sequences here.

Because neither I nor my students ask any questions, this sequence of examples and non-examples takes around 30 seconds to present.

Now, I need some data. So, I show my students a selection of other shapes and ask them if each is a polygon. Crucially, I do not call for volunteers or Cold-Call individual students. Instead, every student indicates their answer, usually by thumbs up or thumbs down, on the count of 3.

If students get an example wrong, I apologise and explain it is my fault for not choosing an appropriate sequence of initial examples. In these cases, I think of a new example on the spot to get us back on track and then improve my initial sequence after the lesson.

What these two approaches have in common

Notice that both approaches leave the definition until the very end. Definitions often have little meaning to students, as they are full of abstract concepts of technical vocabulary: A polygon is a plane figure with at least three straight sides and angles.

The most important similarity is the use of non-examples to convey meaning. I still cannot believe I taught for around 12 years without realising just how vital non-examples are.

Questions to consider

However, there are differences between these two approaches. Here are some questions to consider:

1. Which approach is quicker?

2. In which approach do the students think more?

3. In which approach is that thinking the most visible?

4. Which approach would your students most enjoy?

5. Which would you most enjoy as a teacher?

Feel free to share your thoughts in the comments below.

What do you agree with, and what have I missed?

Let me know in the comments below!

🏃🏻‍♂️ Before you go, have you…🏃🏻‍♂️

1. … checked out our incredible, brand-new, free resources from Eedi?

2. … read my latest Tips for Teachers newsletter about Inside the Black Box?

3. … listened to my most recent podcast about the Do Now?

4. … read my write-ups of everything I have learned from watching 1000s of lessons?

5. … considered purchasing one of my new 90-minute online CPD courses?

Thanks so much for reading and have a great week!

Craig