Math from Multiple Perspectives (Part 1)

Over the summer I took a MOOC through Stanford University called "How To Learn Math". It was led by mathematics education professor Jo Boaler. Although I prefer face-to-face discussions over discussions via online discussion forums (especially when over 6,000 people are taking the course), the course was chock full of interesting and relevant research related to teaching mathematics. More importantly being engaged in this course reminded me of what it really means to teach mathematics. Math isn't about right answers and algorithmic processes. It isn't about memorization and speed.  What math is really about is creativity, connections, deep thinking and attacking problems from more than one angle.  In fact, these ideas are necessary to forge a lifetime of learning in any discipline!

In addition, making mistakes and asking questions are two of the best ways to learn.  Students know it is ok to make mistakes as they are learning a challenging piece of music, but for some reason they don't feel that way about learning math.  Not all student answers are perfect, but we can learn from all student answers.

Here is an interesting quote from Harvard College Office of Admissions relative to what students need to know about mathematics for higher education.  "You should acquire the habit of puzzling over mathematical relationships.  When you are given a formula, ask yourself why the definition was made that way.  It is the habit of questioning that will lead you to understand mathematics rather than merely to remember it, and it is this understanding that your college courses require."

So, this year I will be focusing on two main goals for improving my teaching: 1) encourage a culture of questioning as a means to learn and 2) designing lessons/activities to help students see math from multiple perspectives.

At the end of "How to Learn Math", I created the following poster. I shared it with my students on the first day of class and we even looked at a problem that reinforced this foundational idea. Here is the poster:

And this is the problem we worked on during the first day of class.

(adapted from the course "How to Learn Math"):

How many squares do you see along the border?  Don't count them one by one.

This is a very accessible problem and all of my students were able to figure out a way to solve it.  After a student gave the first solution, I asked more questions related to the task at hand.  What is another way to solve this problem?  Will your method always work for different size grids?  Which method would you prefer to use for a 20 by 20 grid and why?  How do we know a particular method will always work?  What evidence would convince you that two methods are equivalent?

Stay tuned for Part 2 of this blog for answers to the following questions: What were the five different ways my students solved this problem?  How did I incorporate Desmos into this lesson?  What new insight did I see on the third time this lesson was taught?