My brand new Rubik’s Cube arrived in the mail this week and I couldn’t wait to start cracking at it! I took a picture of it before I got started because I knew that it would be ruined shortly after and I didn’t have a way to make it look that good again anytime soon. I wanted to make sure that it was as “mixed up” as I possibly could make it, so I began twisting and turning rows and columns until it looked “jumbled” enough to be confusing. My first learning attempts this week were completely experimental, giving me the opportunity to try and solve it without looking up any strategy guides or doing any research. Although I did make some good progress, I hit quite a few roadblocks as well.
After the first weekend, I was able to perform the following procedures in, what I thought was, a relatively short period of time (within two to three minutes):
Solving the corners on one face:
I recognized that I could twist the left face clockwise (see my fingers, left) to bring that particular square into the location that I wanted it to end up in. If I turned both bottom front faces counter-clockwise before I made that move, then I could twist the original two squares back to their starting locations (without affecting the positions of the other squares).
Getting multiple “rows” of the same color on one face:
There were usually at least a few squares together already, so making arbitrary twists and turns for the third color in a row became fairly easy over time. The trickiest parts for me were the corners…
Solving One Face:
Eventually I was able to move a particular square to any position I wanted on one face. It was simply a matter of moving the original squares to any face that would not be affected by the one that I needed to twist (typically the opposite side). For example, if I wanted to move a square from the front to the top on the right face (but keep the original squares on the top right in place after the move), then I would have to twist the originals to the left face first and then twist them back to their original positions.
After I solved this first face (on the first day), I couldn’t get anywhere close to solving a second face without messing up the first! One square away from two faces was the closest I could get to making progress. My process was also more of a visual “feel” than a repeatable procedure (hence why this was so hard to describe) and it was taking me several hours per trial to try to get further or fix the first face again. So I decided to begin research in Week 2 to find out how to get over this hurdle and solve that second face.
I love hearing Dan Meyer speak; he always “hits the nail on the head” in a simple, creative, and eye-opening way. A recurring educational topic around the Common Core standards for mathematics has been the overemphasis on computation and the under emphasis on mathematical reasoning. Many of us have learned math in the traditional way; we are excellent at computation (repeating procedures over and over again until we know the concept like the back of our hands), but struggle with reasoning and exploration. The five symptoms that educators are doing math reasoning wrong in their classrooms are areas where mathematics teachers are challenged on a daily basis.
The eagerness for a formula (#5) is something that I have noticed in my classroom, especially with high-achieving students. Dan says that they have an “impatience with irresolution,” becoming frustrated when they can’t get the answer quickly. This can be a destructive mentality because they expect to solve simple problems as quickly as possible rather than fully appreciating or understanding the why behind it.
Furthermore, this mentality has actually made our kids weaker communicators. They are used to being given notes and procedures, then using those procedures to compute and calculate, then discussing what they have done at the end of class. According to Dan, this process is backwards; the math needs to serve the conversation, the conversation doesn't serve the math!
Curriculum modification is an area that I am now spending a great deal of time on in my own practice as well. I focus on structuring problems in a way that leads to students exploring and creating mathematics on their own; they are given the opportunity to be patient problem solvers and utilize math reasoning to solve real-world problems.
Step 1: Eliminate all the sub-steps of a problem→ students need to create these on their own and figure out what they need to do.
Step 2: Get rid of distractions with specific information that you will need LATER to solve the problem→ Ask the students, “What really matters here in this diagram?” or “What information do we really need?”
Step 3: Ask the question in the shortest and simplest way possible.
Step 4: Innovate for the 21st century by taking a picture of the actual scenario in real life.
Above and Beyond: Record a VIDEO of someone filling the tank up. The pace of the video is agonizingly slow and students begin to wonder, “How long is this going to take?” There is the hook!
The beauty of problems like this is that all students can engage with the curriculum; practically everyone wants to guess in the beginning. Dan says that kids are, “no longer intimidated by math because we are redefining what math is” in our classrooms. From a practical standpoint, teachers no longer need to get our answers from the back of a textbook either; we just watch the end of the movie to see what happens!\
"Dan Meyer at TEDxNYED - YouTube." 2010. 15 Feb. 2015 <http://www.youtube.com/watch?v=BlvKWEvKSi8>