The "Ah-Ha" Moment

Over the last month I have been going to different schools and doing different activities with students in grades ranging from 7^th – 12^th grade.  I have ran 3 different activities during this time and had the pleasure of experiencing the “Ah-Ha” moment with students.  The three different activities were “Tower Building”, “1=2!?”, and “Predict a Pair.”

The first activity tower building is the same activity that I did in my first summer course in my STEMS^2 master’s program.  The activity consists of building a free standing structure made only of tape and straws (in my summer course we did uncooked spaghetti noodles and marshmallows).  All the kids have a blast doing this activity.  After they build their towers I give them a minute to look at everyone else's towers including their own and ask students to think about what they thought about the activity, what worked well, what was a challenge, what did they notice about the tower or materials.  I guide the discussion a little when then say things like, "The straws were colorful," or "The straws had the bendy part,"  I ask them did those things help or hinder you.  The answer we came to was that it all depends.  A few students said, "The colorful straws made their towers prettier," most students said, "The color straws didn't help or hinder their building."  So now what about the bendy part of the straw?  Some students said the bendy parts in the straw made their tower weak at those points.  Other students said it helped because they used the bendy parts to make angles to construct the tower or they could extend the bendy part of the straw to make the tower just a little taller.
Now I say, "So it's pretty important to recognize what materials were given to you and how they can be used?" Of course this question is a leading question, but nonetheless they all say yes. This is where I begin to drop the math bomb.  I tell them, "Just like when you do math it is important to recognize what is given.  You need to look at a problem and see what are the givens and how you can use the givens to solve a problem.  This also applies to life, in life when you have a problem it is important to look at what you have, what resources are available to you, and use those givens to solve your problems."  The students weren't expecting this activity to relate to math and then have math relate directly to life.  I could see in their faces that I had their attention.
One of the best moments for this activity came when we talked about how every team used some sort of shape in their structure.  I asked, "What shape is the strongest?"  In every class a hand full of students knew it was a triangle.  So I ask them if triangles are the strongest shape why do we build houses and buildings with squares and rectangles?  Many students are unsure, one said maybe its because using rectangles give you more space.  Students, including myself, starting thinking like yeah that's a good point.  I tell them I thought about another point.  I asked what would happen if I drew a diagonal line from one corner of a rectangle to the other?  I used the white board as my rectangle and drew a diagonal line.  A kid blurts out before I even finish drawing the diagonal, "It makes TWO triangles!"  I said, "Yeah it does! Maybe it is for these 2 reasons we mentioned that we build houses with rectangles instead of a triangle even though a triangle is a stronger shape.  I am not sure, but its interesting and worth thinking about."   Leaving that class with the little spark of interest and wondering made all the work leading up to that point worth it.
1=2!? is a brain teaser where you lead the reader to believe that you have proven 1=2.  The Brain Teaser is below.

So I go through all the steps with the students to make sure they see how the whole processes takes place.  I ask them so does "1=2?"  They are usually baffled at this point and I say, "Well the answer is obviously no right?  Would it be equal if I gave you one dollar and expect you to give me two?"  Long story short. I show them that in step 8 we did something not allowed which is we divided by zero since in step one we said a^2=ab.  I point out that this is why we say a number divided by zero is undefined we get fallacies.  When we divide by zero we can get weird things like proof that 1=2.  All the students went "oooohhhhh!"
The last activity predict a pair is a magic card trick where you predict the value of two playing cards (not the suit) that students pick at random from a standard deck of 52 playing cards.  How it works is you let a student pick one card from the deck.  Then the student multiplies that cards value (picture cards are worth 10 and aces are 1) by 2, then add 5, then multiply by 5.  Next the student picks another card from the deck at random and adds the second cards value to the total the student got from the previous steps.  The student then gives me the grand total.  And I "magically" predict and tell him/her the value of both cards and which card he/she pulled first and which he/she pulled second.  When I did this with a student during a tutoring session I looked at him when I told him his cards he said I was right, but just kept looking at his cards.  I thought, "oh man he's totally not impressed and or he thinks its completely lame."  Then a few seconds later he looks up at me and goes, "OK how'd you do that?," with a big grin on his face.  I smiled back and said, "Your job is to figure out how I did it."  He goes, "oh man come on just tell me."  I respond, "Nope you go home work on it and next week you show me what you worked on and then I'll tell you how I did it."  He replied, "Deal!" then shook my hand and left.

In all of these instances I feel I had the great opportunity to affect the lens in which students view  the sense of place as it pertains to their math class.  I got to generate genuine interest and help develop an understanding of why we say or do certain things in math,  It made me happy to think that I had some hand in changing a very negative space to a positive place, even if it was just for a little while.  This in turn made me realize that maybe we also need to be aware that we cannot affect ones sense of place without inadvertently effecting our own.  If I ran all these lessons and was met with disappointment I would be worried about the next time I ran a lesson with the students.  Lately my lens for math and my sense of place in the math world has been indifferent at best.  However, these students have positively affected my sense of place.  At the end of this experiences I feel the math classroom became a positive ideological and political space.  We generated interesting conversations where people expressed idea's and debated freely with no anxiety of being wrong.  A place where everyone can feel respected for their opinion.  They used their physical sense to help make sense of problems.  Its hard to describe the feeling or to accurately describe what took place, but being aware of sense of place and the affect we have on others as well as others has on ours makes teaching and learning more meaningful and enriching.