Algebra Teaching Game Plan

I have had so many thoughts running through my mind this summer that it became necessary to put them into some kind of organized manner.  Being that I am a visual person I put them together into a Google Drawing with links to the resources.

The Main Idea
I'm starting to see Algebra as really a course in Number Sense and Problem Solving.  I very much wish I could change the name of it but, alas, that will not happen.  My relationship with the teaching math has changed.  I see it as having three personalities that mesh together to form the study of mathematics (or at least the teaching of it). There is the "Conceptual" or all things important to truly understanding mathematics. There is the "Procedural" or the mechanics of math.  And kind of sandwiched between the two is the "Relational" aspect of algebra.  I see that as the ideal kind of relationship students have with mathematics that sincerely supports their potential to understand it's study.

Playing a big part in how I see teaching algebra differently is using the study of patterns as a foundation. There are so many inherent benefits to teaching algebra from this perspective that it cries for it's own post (I'll get to that soon).  There is the use of questioning to develop many conceptual and relational aspects of algebra. I have a theory that students struggle in math classes because they don't know how or feel comfortable asking questions.  There is much more to how I see teaching algebra (or maybe even math in general) that it is too much for one post.  For now, here is what has been brewing in my brain.

For the image with active links go here: Game Plan Link

Patterns Make Me Feel Stupid: How I'm Turning That Into a Positive For My Students

Having just completed Jo Boaler's How to Teach Math course thought Stanford's MOOC I am left contemplating the importance of studying patterns in Algebra. I understand they are more meaningful than empty equations. But as I study them for myself while trying to develop how I will use them in the classroom I have two reactions I didn't anticipate; 1) They make me feel stupid, and 2) I don't get how they relate to the real world.

Why They Make Me Feel Stupid

I'm talking specifically about working with visual patterns like those found on Fawn Nguyen's website.  I typically have little problem wrapping my head around how the patterns grow. My difficulty comes in finding the 43rd pattern and modeling the growth with an algebraic expression to find the nth case.  As soon as I recognized my internal voice commenting, "I hate these" and "I'm not good at these" I immediately realized that is how many of my students feel. So what was making me think these things? Im a teacher for goodness sake and I should know better. The answer; because they take forever for me to figure out. Epiphany! I was falling into the same trap my students do; if I can't figure it out in a few minutes then I must not know how to do it.  As much education as I have in teaching mathematics I still fall into that crappy old trap. How easy it must be for my students to fall into the same trap!

How Do They Relate to the Real World

Now I get that there is a great deal of learning that goes on in understanding patterns and using algebraic expressions to model them but how does that really relate to the real world? As i pondered the answer I took a walk on the beach during a family vacation. I found myself looking at the footprints and, noticing that some were partially washed away by the waves, I wondered how many waves it took to wash them away? And how does that differ with the varying depths of the footprints?  And why do some footprints vary in their depth? Then I wondered what caused the shoreline to jut out in certain areas and not others? 

So then I began to question whether the study of patterns was really a study of science rather than mathematics. Wasn't it really a question of the physics behind the waves that I was really wondering about? Then it hit me. I was asking the questions because I had been studying patterns.  thats what caused me to look for the patterns in the world around me.  And it is the existence of patterns and our wonderment about their cause that is the basis of science! Epiphany number two!  Happenstance does not give birth to scientific theory. Patterns do!  When we see things that occur regularly and with form; that is when we ask why.  Why does that happen? What is the cause? Can we recreate it?  By looking at patterns and working to model them with algebra we lay the foundation for scientific discovery and wonderment. That is how it relates to the real world. 

In the Classroom

In the end I am sticking with my desire to make the study of patterns much more prevalent in my classroom. I have decided that even though I am not so great at writing algebraic models for patterns, I will learn a great deal by sharing that journey with my students. We will truly be learning together. I will share with them how studying patterns initially made me feel and how important it is to remind ourselves that time and intelligence are not linked. And I will share with them how that created a curiosity in myself to see patterns everywhere and how to use them to explain the world around me. Ultimately I have come to realize that studying patterns is the seed to cultivating curiosity.  And that is more important than than my own feelings of inadequacy.