Meaningful Math

Last Thursday I was doing some classroom visits with Nipmuc Principal John Clements and we couldn't help but notice a true conversation piece that was erected in the library media center.  It looked like this:


Standing 8 feet tall, it certainly was an impressive sight.  Math teacher Rob Messick and his two classes of sophomores in Fundamentals of Geometry constructed this 3-D figure, known as Sierpinski's pyramid, over the past week and a half.  They carefully cut thousands of equilateral triangles and used plenty of scotch tape to construct the repeating geometric pattern of a tetrahedral (4-sided) pyramid.

When Mr. Clements and I went into the library the students were admiring the fruition of their efforts.  They enthusiastically answered questions about the pyramid, sharing with us the fact that the figure was perfectly symmetrical and that the base pattern kept repeating in an exponential fashion (each unit referred to as a "stage").  The students were able to tell that this figure had five stages and if they were to construct a sixth stage, it would be 32 feet in height.  They shared that if this pyramid was 100 stages, it would be thousands of light years tall. They were also able to cite the volume of open versus closed spaces in the figure based open the volumes of each successive stage they created.

It turns out that Sierpinski's pyramid contains plenty of higher level mathematical concepts, including tessellations, fractals, exponential magnitude, and symmetry to name just a few.  These concepts clearly speak to higher order thinking, i.e., where students must analyze, predict, integrate, and synthesize difficult content so it makes sense.  Here is a animated visual showing the repeating various stages Sierpinski's triangle:


To put it in perspective, here is the mathematical algorithm that goes with this repeating pattern:


The area of the Sierpinski Triangle approaches 0. This is because with every iteration 1/4 of the area is taken away. After an infinit number of iterations the remaining area is 0.

The number of triangles in the Sierpinski triangle can be calculated with the formula:

Where n is the number of triangles and k is the number of iterations.

Okay, not the absolute most difficult math here... but without a visual cue, probably not the most fascinating either.  What Mr. Messick was successfully able to facilitate was true project-based learning for his students.  He gave them a challenge (they had no idea what the final outcome of their construction would be) and allowed them to discover the mathematical concepts as they constructed the figure, guiding their hypotheses as they progressed.  This type of learning is the most impactful for students because they will truly understand because they are engaged in hands-on learning.  As the Chinese proverb states:

A Fitting Tribute

Last Friday there were many heavy hearts at Nipmuc as a group of well over 150 teachers, staff, administrators, parents, and members of the Class of 2012 gathered in the school's courtyard to commemorate the life of Jack Street on the one year anniversary of his tragic death.  The program featured opening comments by Principal John Clements and an outstanding reflective prayer from classmate Matt D'Innocenzo where they recalled Jack's gentle nature, great sense of humor, and his great passion for life.  A highlight of the program was when Class of 2012 President Dan Consigli unveiled the class gift to Nipmuc, a beautiful bench (pictured above) made from Jack's skis and dedicated in his honor.  The bench will be permanently installed in the Nipmuc courtyard in the spring.

Another highlight was the performance of the original song, "Memories and Milestones" by Jack's classmate David Byer.  The lyrics are the following:


Even though I couldn’t find it, I was told to cherish the small
And now that it’s gone I can see it all

I’ll shed a tear and I’ll say a prayer
In hopes that you find all you dreamed of up there
Cause even though we’d never know, it’s the little things that mean the most
You were always so happy, just being you
There’s not a thing we could do, to ever change you
So please rest easy while back home you are not lost
In our prayers, and in our thoughts

Even though I couldn’t find it, I was told to cherish the small
And now that it’s gone I can see it all

Memories and milestones are going to trickle down my cheek
When there’s only silence is when you’ll hear us weep
But these won’t be tears of sorrow, just tears that show you
How grateful we all are that we were blessed just to know you

Even though I couldn’t find it, I was told to cherish the small
And now that it’s gone I can see it all

Yeah I can see it all, yeah I can see it all
Yeah I can see you here, with that smile
The one we all know and love, the smile that’s now above
But know your smiling at better days, and you’re smiling in a better place

This touching song may be heard here:



A common theme that was touched upon Friday afternoon was the strength of community, and how the unspeakable pain of the past year was eased a bit by students and community members being there for Street and Murphy families... and one another.  May that comfort and sense of community continue for the Jack's family.  Most of all, may Jack's memory continue to be alive and be a sense of joy to all in our community.