Counting Circle

Welcome! Welcome to my Magical Mystifying Mathematical Metacognitive Memoir!

How is it magical and mystifying? Well, it's not. But by labeling my blog as such, I'm making a statement on how math can seem like magician's work. Just as a magician mystifies their audience by procuring rabbits out of thin air, a math teacher can mystify students by procuring solutions out of a jumble mess of numbers, symbols, and words. In both scenarios, the audience and the students wonder, "How did they do that? It was just like magic!"

But once the magician reveals their secrets and shows all the tricks behind-the-scenes, the act is no longer magical or mystifying. The same holds true for math. Once the math teacher explains the thinking, the behind-the-scenes process they took to solve the problem, then the math is no longer magical or mystifying. And that is the point of my memoir - to demystify the magic surrounding math by explaining my thinking behind my solutions to mathematical problems, and illuminate any connections and lessons-learned along the way, so I and others can learn how to explore, communicate, and teach math more effectively.  

Act 1: The Counting Circle

The first day of school, the class played a game called the "Counting Circle." The class forms a circle around the room, and the first student gets a starting number and a counting number to add to it. Then the next person has to add the counting number to the last person's answer, and so it continues around the circle with each student adding the counting number to the previous person's answer. What's great about the counting circle is that it can be utilized in various ways for different math topics, not just adding natural numbers. The first one that came to my mind was for a Pre-Calculus class, which would combine the counting circle with the unit circle. Instead of counting with numbers, students would count in pi, radians, or angles. But for our first try, our class did simple addition...with big numbers. According to my recollection, the starting number was 235 and the counting number to be added was 97.

I was toward the end of the circle, so I had plenty of time to think. As the counting went around the circle, I was trying to come up with the answer for each person. I was trying to add the ones digits first, then add the tens digits, and then the hundreds digits, which is the way I was taught in elementary school. But I was so slow; I could not keep up. Others were adding the numbers in their heads so fast, (as if by magic!), and I wondered how they could do it, since I assumed they were adding the way I was.

Then it came my turn, and I struggled. First, I added the ones,  then carried the one to the tens digits, got that answer… Wait, what was my answer for the ones digit? Did the ones over again. Then the tens digit again… Thought I had the wrong answer. Needed to start over again. And so forth. Finally, after about a minute, I got an answer. Then I realized there was a better way to do it than the way I was taught in elementary school. Add 100 and then subtract 3, instead of adding 97, since 100-3=97.
At the end, we were asked how we got our answers. Many did the add 100, subtract by 3, trick. Others added in groups, such as: add 65 to 235 to get 300. Then subtract 65 from 97 to get 32. Add 32 to 300 and get 332.

Next, we were asked this question: if we continued around the circle, what would so-and-so's answer be? To solve this, I first counted how many people stood between the last person of the circle and so-and-so. I think it was 7 people. Then I modified the "add 100, subtract 3" trick, and multiplied 7 by 100 to get 700. Then added that to the last person's answer. Then I multiplied 7 by 3 to get 21. Subtracted that from the answer I got after I added 700, and got my answer. The equation would look something like this:
[last person's answer + (number of people x 100)] - (number of people x 3) = answer.

When asked how we got our answers, the majority mentioned the way I did it. A few did it some other way, but it wasn't as streamlined or elegant as the above process. In fact, some mystified me. And that's OK. In fact, that's the point. The counting circle gives students an opportunity to see different ways of looking at and solving the same math problem, so they can learn different strategies, discover which ones make sense to them, and apply those strategies to similar problems later… or for when the next counting circle comes along.