Tuesday, October 20, 2015

The Calculus of Friendship: A Book Review

Recently I have had the pleasure of reading a great book by Steven Strogatz called The Calculus of Friendship. In his book Strogatz writes about the letters that he sends back and forth with his high school calculus teacher, Mr. Joffary. The book contains over 30 years of correspondence, however the interesting twist is that the letters are not personal in nature, they are about mathematics. Strogatz, after high school and college, goes on to teach at MIT and also Cambridge, and during this time Mr. Joffary sends Steven math problems that his students present him. Strogatz then in turn sends Mr. Joffary proofs to the questions he asks and poses new questions for his students. Throughout their years of correspondence they touch topics such as infinity and limits, chaos, bifurcation, Hero's Formula, and relativity. The book contains a mixture of letters, proofs, and explanations about the topics discussed, and as each page turns the reader is able to see the relationship between teacher and student blossom.

Now I won't lie, a lot of the proofs presented in this book were beyond my level on mathematics, but the reason I kept turning the page was for the friendship that was growing between Steven and Mr. Joffary (or Joff as he would later call him). Both Steven and Joff go through a lot of ups and downs in life, however these are never present in the letters, only math. I got the feeling though that both Steven and Mr. Joffary knew that they had a lifelong friend, even if they never discussed it.

I think that the reason that I loved this book so much is because Mr. Joffary reminded me of my grandfather. My grandfather, Brian, taught math at Muskegon Community College for over 30 years and he is just as kind, thoughtful, and smart as Mr. Joffary. Even the jokes that Mr. Joffary includes in his letters, and his mannerisms reminded me of my grandfather and I couldn't help but smile. This was a book that not only makes you think, but also warms your heart. It is about the bond that is formed between a teacher and a student. and how sometimes the student becomes the teacher. If you are looking for a story about friendship and integrals, I suggest you pick up a copy of The Calculus of Friendship. I will leave you with my favorite quote from the book, and a video of Steven Strogatz talking about his friendship and his letters to Mr. Joffary, you can clearly see how much this relationship meant to him.

"This was a different kind of infinity than Joff had ever encountered before, a higher order of infinity. While I explained it to him, the sun began to set. We sat together on the beach and solved the problem, surrounded by the waves in Long Island Sound."

Sunday, October 4, 2015

How to Create Better Classroom Discussions

One of the struggles I have been dealing with this semester while I do my teacher assisting is trying to create better classroom discussions. I am doing my teacher assisting at Grandville Middle School in two Math 8 (pre-algebra) and two algebra classes. The atmosphere of the classroom is very collaborative (the students sit in groups of four and spend a lot of time doing small group work) but I want to become better at facilitating rich classroom discussions. I am currently reading NCTM's Principles to Actions and I just finished a section on facilitating meaningful mathematical discourse and in that section it outlines five practices for using student-responses in whole class discussions. I also found an article by Margaret S. Smith, Elizabeth K. Hughes, Randi A. Engle, and Mary Kay Stein called "Orchestrating Discussions" where they propose the same five practices, called The Five Practices Model. The five practices are:

  1. Anticipating student responses prior to the lesson
  2. Monitoring students' work on and engagement with the tasks
  3. Selecting particular students to present their mathematical work
  4. Sequencing students' responses in a specific order for discussion
  5. Connecting different students' responses and connecting the responses to key mathematical ideas

In this post I want to take these five practices and apply them to a problem introducing the concept of the variable that my Math 8 students will be working on next week. The problem comes from Ana Stephens's article called "Developing Students' Understanding of Variable" The rest of this post is a reflection on how I would apply the five practices to Stephen's problem.

The Problem:
Ricardo has 8 pet mice. He keeps them in two cages that are connected so that the mice can go back and forth between the cages. One of the cages is blue and the other is green. Show all the ways that 8 mice can be in two cages. 

In my opinion the point of anticipating is so that you are prepared for what your students are about to do. This can range from anticipating student mistakes or misconceptions, to anticipating different strategies that students will use to solve the problem. For this activity I had two groups of anticipations, one was related directly to the problem the other were anticipations about my students' understanding of variable in general. 

Anticipations about Variables:
  • I anticipate that some students will view variables as labels rather than a symbol that represents a quantity. So in this problem some students might use b and g for variables just because they are both the first letters of the words blue and green. 
  • I anticipate that some students will not understand that variables are changing quantities and they do not have only one specific value. In the context of this problem students might not realize that b (the number of mice in the blue cage) can range from 0-8 at any given time.   
  • I anticipate that some students will believe that two different variables must have two different values. So in this problem students might think that b (the number of mice in the blue cage) and g (the number of mice in the green cage) can never have the same value.
Anticipations about Students' Strategies:
  • I anticipate that students likely begin this problem by drawing a picture or creating a table. Some students may also use a graph or an equation but I don't expect either of this to be their first step. 
  • I anticipate that some students might not consider having zero in the blue or green cage.

During this monitoring time it is my job to be walking around and trying to understand my students' thinking, I will most likely do this in the form of questioning. In the book Principles to Actions there are four major questions types. I will list those types and they provide a couple examples of questions I might ask students during this activity.

Gathering Information: students recall facts, definitions, or procedures. 
Examples: "So if there are ___ mice in the blue cage, how many are in the green cage?", "How many mice will there always be?", "Is it possible to have zero mice in a cage?"

Probing Thinking: students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task.
Examples: "When you drew this first picture, what does this situation represent?", "What made you decide to create a table for this problem?", "I noticed you wrote down b=3, what do you mean by that?"

Making the Mathematics Visible: students discuss mathematical structures and make connections among mathematical ideas and relationships.
Examples: "How does the table you made relate to how many mice are in each cage?", "Can you use this information you generated in this picture to create a graph or equation?" 

Encouraging Reflection and Justification: students reveal deeper understaning of their reasoning and actions, including making an argument for the validy of their work.
Examples: "So I see you wrote an equation b+g=8, can b ever be equal to g. Why/why not?", "How can you check your equation to make sure it is true?"

What students' work you select to show all depends on what your mathematical goals, or learning targets are for the activity. Your goal is what is going to drive your selection process and decide whether you will select a misconception to show, or maybe a unique way of solving the problem. For this activity my mathematical goal is to show how variables are changing quantities, so therefore I would select students' work that represents this idea. I would want to show a student who represented this problem with a graph, a student who did it using a picture, a student who used a table, and finally a student who wrote an equation or expression. 

Now that I have decided what student work I want to show, I have to decide what order I want to present it in. Again, the order that I choose should reflect my mathematical goal(s) of the lesson. Sometimes you might want to start with a common misconception and explain why it does not work that way, other times you might want to start with the strategy that the majority of your students used and then go on to the more unique strategies. For this activity I want to sequence the student responses like building blocks, so I would start with the picture, then show the table, then the graph, and finally the equation. This way I am working from more concrete thinking to abstract thinking. In the article "Orchestrating Discussion" they give a great suggestion of creating a table that can organize your sequencing. I have taken that idea and created a table that I would use while I am monitoring this activity. 

This is arguably the most important practice of the five practices. This is where the students consolidate what they just did and they can see the "big picture" of what it is important. This normally will take the form of a whole class discussion after presenting the sequenced answers. As I was thinking about this activity these are some of the discussion points I would want to present to the class:

"So what I am noticing from a lot of your work is that the number of mice in both cages isn't always the same. How can we represent the number of mice in each cage in a general way?"

" So if we let b equal the number of mice in the blue cage and g equal the number of mice in the green cage, how can we write  relationship between the two quantities? Is there more than one way?"

"Are b and g always, sometimes, or never equal? What makes you say that?"

"So in math we call b and g variables. Based on what we have discussed what is a definition we can come up with for variable?"

At the end of this activity I would hope that my students would be able to understand that variables are changing quantities, that we can represent with letters or symbols. Each of the five practices above reflects my goal, and every move I would make in this activity would be intentional. I think that the five practices are a great way to structure mathematical discussion in a classroom. They do require some planning in advance but it will help to direct the discussion towards the mathematical goal of the lesson. I believe that by incorporating these practices into my teacher assisting placement I will begin to see better whole-class discussions in my classes.   

Tessellations...Are They Math?

The more math I learn the less it fits into the conventional definition I once had of mathematics. This semester I am taking my capstone class, The Nature of Modern Mathematics, and this class has been pushing the boundaries of what I consider mathematics. Our discussions have ranged from talking about mathematicians such as Archimedes, Brahmagupta, and Leonardo of Pisa (also known as Fibonacci) to talking about the Pythagorean Theorem and arithmetic in Roman numerals. I have been learning such a rich history of mathematics but there is one topic in particular that has stood out to me, Islamic tessellations.

So what exactly are tessellations? Tessellations are tilings of the plane using one or more geometric shapes, called tiles. The important part though, is that those tiles cannot overlap, nor can there be any gaps between the tiles. In order to understand Islamic tessellations, it is important to understand some beliefs of the Islamic faith. Most interpretations of Islamic law discouraged the portrayal of humans or animals in art for fear that it would cause people to idolize those humans or animals. Therefore Islamic art centers around three main elements: calligraphy in Arabic script, floral and plant-like designs, and geometrical designs.

The Islamic geometric designs often included a lot of repetition and variations. Even though the designs may have only consisted of a couple shapes, the patterns those shapes created couldn't have more variety. The Islamic tessellations also included a lot of symmetry, and many designs included translations and rotations of tiles as well. Here are some famous examples of Islamic tessellations.

The Alhambra in Granada, Spain (Google Images)

The Alhambra in Granada, Spain (Google Images)

The Alhambra in Granada, Spain (Google Images)

Alcazar in Sevilla, Spain (Google Images)

Alcazar in Sevilla, Spain (Getty Images)

Alcazar in Sevilla, Spain (Getty Images)

Alcazar in Sevilla, Spain (Getty Images)

There is no denying that these tessellations are masterpieces, but are they mathematical? After some research on tessellations (check out this awesome Wikipedia page!) I have discovered that there are three types of tessellations: regular, semi-regular, and irregular.

Regular tessellations have both regular tiles and identical regular corners and vertices. There are only three shapes that can form regular tessellations, equilateral triangles, squares, and regular hexagons.

Semi-regular tessellations are tessellations that use regular tiles of more than one shape with every corner identically arranged. There are eight semi-regular tilings.

Irregular tessellations are tessellations made from shapes such as pentagons that are not regular. M.C. Escher was an artist who was famous for creating tessellations that used tiles shaped like animals, humans, or other objects.

Two Lizards (1942) M.C. Escher

Johannes Kepler was one of the first people to study regular and semi-regular tessellations in 1619. In 1819 Evgraf Stepanovich Fedorov marked the unofficial beginning of the mathematical study of tessellations when he discovered that every periodic tiling contains one of seventeen different groups of isometrics, now known as wallpaper groups. Some people claim that all seventeen of these wallpaper groups are represented in the Alhambra Palace in Granada, Spain but it is still being debated.

There is no doubt that there is a lot of sophisticated patterning behind tessellations. Not only are they visually stunning but they are also rich in mathematics. I encourage you to try creating some of your own tessellations, here are some great resources to get you started!

John Golden: Islamic Tessellation Helper
Victoria and Albert Museum: Maths and Islamic Art & Design
The Metropolitan Museum of Art: Islamic Art and Geometric Design

Also here are links to isometric and square paper.

Happy Tessellating!