Math from Multiple Perspectives (Part 2)

In my previous blog post, I talked about a problem that I showed to my students on the first day of school.  This problem was to get them to think about math from multiple perspectives and it was taken directly from the Stanford University MOOC that I took over the summer called "How to Learn Math".

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

First, I told students to think about an answer and put their thumb up in front of them when they thought they had it. (A great idea I learned from the MOOC.)  Students didn't shout out answers in excitement and ruin it for others!  And those who normally would shoot their arm up in rapid fire speed didn't intimidate others in the classroom!  Next I called on students to explain how they found the answer.  The five different solutions that are shown here were obtained by by 3rd period class. 

I could tell by the expressions on student faces that several were surprised that there was more than one way to think about this problem.  Next, I asked some follow-up questions with some unanticipated results. 1)  Which method would you use if you had a 20 x 20 grid?  Most students thought the third way was easier to use.  And I always thought this method was easier, too.  After all, who would want to do 18 squared in their head?? Not me.

But then it dawned on me! (It only took the third time for me to teach this lesson to see it.)
20² - 18² wasn't that bad if you thought of it as (20 + 18)(20 - 18), a difference of squares!  My surprise was genuine and the fact that I didn't notice this right away wasn't so bad.  My third period students had a chance to see me think aloud through an idea, and they had a chance to see that I still get excited about math when I see something "new" that I hadn't seen before!

2)  How do we know these solutions will always give the same answer?  Students easily explained how to generalize each solution based on an n x n grid, where n was the number of sides on a square.  (This can be seen on the left.) When I asked how we might show the expressions were equivalent, I was thinking that my students might simplify each expression to show they were equivalent. However, in each of my three classes, they suggested setting 2 expressions equal and solving for n. (You can see the circled expressions that we set equal in this particular class.) In each case, an identity resulted and they could explain what that meant, that the expressions always yielded the same result for any value of n.

3)  How can graphing equations show these are equivalent expressions?  This one stumped my students and unfortunately, we ran a bit short on time for me to make sure they really got this idea.  However, in two of my classes I showed Desmos as a quick graphing tool.  [I became more acquainted with Desmos at NCTM in Denver in 2012. Thanks Desmos folks for the cool trivia prizes! I love my mint green Desmos t-shirt!]  We graphed y = 4x - 4 and y = x² - (x - 2)². A screenshot of the graph is seen here.

You might be wondering why only the first equation is graphed.  That is because I kept turning the second graph off and on in order for students to see that these two equations really produced the same graph. 

Could I have incorporated other ideas, such as domain or the meaning of slope, into this lesson?  Yes.  However, my goal was to have my students begin to see math from multiple perspectives.  That goal was met. 

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?