Like magic, the fish turn into birds and then back into fish. M.C. Escher's tessellations have a way of grabbing your attention and forcing your mind to make sense of the impossible figures on the paper. The Merriam dictionary describes tessellations as, "a covering of an infinite geometric plane without gaps or overlaps by congruent plane figures of one type or a few types." A geometry book I have on hand describes tessellations as geometric forms that make use of all available foreground and background space in two dimensions by repeating one or more different shapes in predictable patterns.
To tessellate a single shape it must be able to exactly surround a point, or in other words, the sum of the angles around each point in a tessellation must be 360. This means that every quadrilateral and hexagon will tessellate. Of course we see this every day in floor tiles, windows, and walls as the regular shapes repeat themselves.
Using the six steps listed below, tessellated thinking might be a way to help students make order out of the mental chaos our young learners often experience:
Step 1: Routines and Predictable Patterns
Just like Escher's tessellations repeat common forms in interesting ways, good educators also help students to tessellate what they do in the classroom by repeating interesting thoughts and behaviors. For example, students thrive on predicable patterns doing the warm-up exercise everyday, or routines of passing out papers, or collecting work. Unfortunately, it is quite a bit more difficult to establish routines of thought or predictable habits of mind. The typical disorderly thought processes of students would be the opposite of tessellation or chaos (while serendipitous things can result from chaos, too much is at stake to wait around for this to happen). In pondering about tessellations I wondered if students could use the same process to organize their thinking.
Step 2: Create Habits of Mind
For example, students frequently ask why do they have to study algebra if they are never going to use it. A true math teacher might respond differently, but one of the reasons I believe that we study math; an abstract, logical subject, is to train our minds to think analytically and teach it to follow predictable processes. According to Daniel Willingham, a cognitive scientist who wrote, Why Students Don't Like School, our brains are constantly seeking to make sense (order) out of disorder, and they are pretty good at it as long as they are given adequate information.
Step 3: Rotate Perspectives
The first step might be to adjust perspectives and get a 360-degree view. Just as the mathematical definition of tessellations of an object require it to be rotated so that it forms a complete 360-degree image around a point, mental tessellations require students to look at a concept from multiple perspectives with both literal and figurative eyes in order to get the full picture. As an illustration of tessellation of thinking, I was discussing "luck" with my son Gideon and recounted research done by Richard Wiseman regarding the different perspectives of being lucky and unlucky.
Participants that either felt they were lucky or felt they were unlucky were asked to respond to a scenario where they are in a bank and the bank is robbed. Shots are fired and they are injured in the arm. One man though he was the most unlucky person as he thought of the astronomical odds of him being in a bank at precisely the same time it was being robbed and being injured. The other thought he was the luckiest guy in the world because even though he was held at gunpoint in a bank robbery, he was not seriously harmed and was still alive to tell about it. It would be an interesting exercise to look at other perspectives of the same scenario: bank robber, police, manager, guard, etc. Perspective matters, but it matters most when students can tessellate thinking by rotating and repeating their perspectives.
Step 4: Don't Accept Chaos
Educators can help student tessellate their thinking by instructing students on different patterns of thought: analytical thinking-deductive and inductive, critical thinking, and creative thinking. These become habits of mind through consistent explicit expectations and then not accepting anything less. For example, left to themselves students will not answer in complete sentences, but over time, given the expectation and persistent do-overs, students can learn to express their thoughts in complete sentences. Providing sentence stems for each kind of thinking helps students to begin organizing their thoughts and words so that they answer in complete sentences. Some examples: the difference between prokaryotic and eukaryotic cells are... Completing the square is more valuable than the quadratic equation because... I think that if you combine the poor crop harvests with the economic instability... . The expectation of good thinking has to be constant and consistent or students will not take it seriously. If they do not meet the standard, the consequence is that they do it over until they do.
Step 5: Fill in the Gaps
Educators can help students to not only assess what they do know, but to analyze what they do not know yet, or what they need to know in order to solve a problem. For example, I told my students a short story and gave them a task to solve as follows: Two people were found dead in a cabin, high in the Rocky Mountains. How did they die? Students were allowed to ask questions. At first they began by guessing the causes of death: overdose, lack of air, too cold (and being middle-school students, abominable snowmen, and aliens). Then I asked them to identify the key words. "High" and "Rocky Mountains" were listed. I asked them why the word cabin was included and they agreed to add that word in our analysis.
Here are the probing questions I used with pauses to let them figure things out:
What is the relationship between the word "high" and "Rocky Mountains"? How high were they in the Rocky Mountains? What would the two people be doing high in the Rocky Mountains? Now add "cabin" into the mix. How does that change that relationship? How many different types of cabins are there? Why would a cabin be so high in the Rocky Mountains? Isn't a cabin for protection from the elements? If so, how would they die? Why were they in the cabin when they died? How did they get high in the Rocky Mountains? What method of transportation could they use to get there? And the final question... How did the cabin get there?
Step 6: Repeat
Once the student has done one tessellated thought, then rotate and do another. To illustrate this, my daughter Mercedes performed Pirelli in the dark musical, Sweeny Todd. Just when I thought I had figured out the plot, it rotated based on a simple thought: How to dispose of Pirelli's dead body. Mrs. Lovett, owner of the pie store below the barbershop came up with the gory solution, and the whole storyline changed. The tessellation of revenge and bakery profits rotated perfectly around the simple question of what to do with the bodies. In this regard, any good storyline is a tessellation because it fills in the space completely with interesting patterns that perfectly fit.
Just as mathematically speaking, tessellations fill the infinite plane, getting students to think has been the mantra of educators from the beginning -- and it always will be. How do you help your students tessellate? Please share in the comments below.