Beetles and Math?: Lessons in Binary Oppositions

I will start with a personal reflection. When I was completing my undergraduate degree at SFU I had to work part-time. The money I was making was suppose to go to my coursework and books; however I always managed to find an old VW Beetle that caught my eye, and I had to buy it! I loved working on them in my spare time. I have owned quite a few over the years. There is something I have always enjoyed about taking things apart and then trying to put them back together again, even though I didn't really know what I was getting myself into. 

"the before - destroyed version"

"the finished - created version"

On one particular 1968 Beetle, I took the original tail pipes off and put on a ferrari style exhaust system. I thought it looked and sounded pretty cool. When I put the gas pedal down, you could hear the rumble (with a few extra pops hear and there, but I thought that added to the effect). One day when I was stopped at a stop sign, I revved the engine a little and some guys passing me on the other side said, "Nice flames buddy!". WHAT?! The next chance I had, I passed by a set of massive windows at the mall and gave it gas. I was shocked and amazed at what I saw! There were literally flames shooting out of my pipes! I realized at that moment I must have not put it on properly. I wondered what those extra bolts were for! And so, the next weekend I had my beetle back on the blocks and disassembled the whole thing, got some new gaskets and put it back together again. It worked well after that, a little less exciting, but I knew I would stay alive at least.

"my exhausting beetle project"

One of my very favourite Beetles was a 1959 convertible, a very rare find! It wasn't even for sale, but I convinced the owner to sell it to me. Everyone has their price! I drove this beetle with pride until one day I the split-case transmission seized on me. For months I wondered what I would do. I decided to pull the engine and transmission and put another newer set in. While I was at it, I figured I would completely restore it from the ground up. So I pulled every bolt and piece apart that I could find, the roof came off, wheels, fenders, seats, everything was in boxes. I never saw that car hit the road again. I sold it for a third of the cost for something even more magnificent that caught my eye...I was getting married and needed the cash! I am happy to report that it was well worth it, but I still think about that beetle from time to time.

"in the process of dismantling my baby"

So some dreams are not realized, but they lead to other unanticipated, wonderful adventures. I would have never imagined when I started that project that I was about to get married. The trajectory of my life changed and I let that idea go and pursued another, better one. Likewise, in teaching I find that I may have an idea in mind of where I want to go and I jump in fully and start dismantling old ways I use to teach or even questioning theories behind how concepts are presented in textbooks and teacher resources. It's a little scary at first, but there is always something positive that comes out of this. We don't have to be perfect and sometimes we have to go back and refit things if there is a problem. The Imaginative Education Theory uses binary oppositions as a core concept for engaging students imaginations with the curriculum. I can use the destroy / create binary to make sense of what I was doing...and this is something I believe can be used in the classroom. 


Binary oppositions help us to make sense of the world by comparing and thus mediating our understanding. Just this week I taught a math lesson using the binary: small / big. At first glance this doesn't seem so important: however a binary opposition can be very powerful as a basis for a lesson or even an entire unit. In fact binaries are one of the earliest forms of understanding numbers. In this lesson on Place Value, I wanted to expand students' understanding of larger numbers. So I had them draw the base ten blocks that they are familiar with. One, ten, hundred, thousand...what comes after that? What does a ten thousand look like? A hundred thousand? A million? I designed a lesson that I hoped would allow them to use their imaginations to think of what they knew from the smallest units and move them to a point of experiencing and thinking about numbers beyond their current knowledge. Here is a student's work for base ten models:

"base ten models to 1 000 000"

I had students do a rough copy idea for homework the day before, then in class we tried to figure out how big these blocks would actually be. I had students come up to the white board and draw in their ideas, and we constructed an actual to-scale version in the centre of the classroom using meter stick and students. At the end I took the single cube and dropped it into the centre of the million cube and said, "There are a million of these inside here." To which I heard a few gasps. I asked if we should order some of these million cubes or do they even exist? Where would we store all of them? I knew I was starting to get their attention. Then I asked them, "So what is bigger than one million?" Okay, million, billion, trillion...gazillion - is that a number? 


Next I told them the story of Googol. A mathematician named Edward Kasner once wrote a 1 followed by 100 zeros. So I had the students do this. It took most of them about 2 minutes, if they didn't lose count that is. When he wrote this giant number, he asked his nine-year old nephew, Milton Sirotta, to give ti a name. Milton thought for a while and then said, "Googol". Ever since it has been known by this name. And later of course someone came up with the Googolplex. I will let you research that one. Having students understand that numbers are symbols that have been named and created by human beings throughout history connects them in a more meaningful way to what they are doing in their number work. I had also previously used the story "One Grain of Rice" by Demi. A colleague at my school who is currently completing her PhD using IE in mathematics, suggested this story as a way to engage students with the wonder of the size of numbers.  


Using stories and imaginative activities in math is not a waste of time. I wanted the students to deepen their connection to the concept of really big numbers. It's not just about naming them, it's the emotional connection and realizing we are a part of the knowledge that has been constructed that is necessary I think.  Vygostky (2003) said, 


"It is precisely human creative activity that makes the human being a creature oriented toward the future, creating the future and thus altering his own present. This creative activity, based on the ability of our brain to combine elements, is called imagination or fantasy in psychology. Typically, people use the terms imagination or fantasy to refer to something quiet different than what they mean in science. In everyday life, fantasy or imagination refer to what is not actually true, what does not correspond to reality, and what, thus, could not have any serious practical significance. But in actuality, imagination, as the basis of all creative activity, is an important component of absolutely all aspects of cultural life, enabling artistic, scientific, and technical creation alike. In this sense, absolutely everything around us that was created by the hand of man, the entire world of human culture, as distinct from the world of nature, all this is the product of human imagination and of creation based on this imagination."


Having students come up with "numbers that might not exist" and reading stories about people and numbers is an important contributor to their conceptual understanding and it creates ownership. Following this I asked the students three questions:


1. What have you realized about the size of numbers through some of your place value work?


Here are some student responses: 
"That numbers can get huge and have so many zeros."
"Some numbers can be really, really, really big and some can be small."
"working with big numbers could be really fun...my feeling is happy."
"This math that were doing has made me realize that even when the numbers are bigger it is still easy. I think that this math is easy."


For the next question I wanted to see if using the cognitive tool of "collections and sets" would help them make more connections with the concept of really big numbers. I told them about the different collections I have had over the years, and even my 6 Beetles I have owned could be considered a collection if I added them all up (this would be considered a small collection of a large object). I also gave them a few examples and showed them images on the whiteboard of ridiculously enormous collections. It's amazing what people will collect and the amount of time they put into their collections.


2. What is the biggest collection you have ever had or heard of before?


Here are some student responses:
"a girl somewhere has 1, 734 or something like that of pokemon stuff."
"my collection is silly bands is 472 sillybands and i got 15 rare ones."
"The biggest collection my friend had was a pencil shaving. He has 3 bags of all fulled pencil shavings. I think he has like one billion."


And then I really wanted to push their affective, imaginative connections.


3. Imagine a situation where you encounter a very large number. What would you be thinking, feeling, doing?


Here are some student responses:
"If a tsumai coming towards me and there is 1000000000000000000000000 water drop I'll feel too scared that I can't even move.
"What if I'm in a desert and 100000 elephants are around me I would hide under one and crawl under there legs."
"Imagine if you where sitting at a park eating a peanut butter sandwich then 50,000 rabid squirrels came and attact you!"


I love reading all the ideas my students come up with. This lesson started with the concept of small / big and I can see their use of the "size" of things in their responses to my questions. Binaries, stories, and collections are all used purposely to create an emotional connection and deepen students understanding and make learning more fun and engaging.