Continuing my professional learning July, I had the privilege of attending a week long institute devoted to learning more about instructional strategies for better teaching Common Core Algebra. The session was led by two professors at Towson University, and the insights they shared, especially pertaining to instructional tasks, was invaluable.
To start the first day we were paired with another attendee and instructed to place yarn handcuffs on our wrists so that the respected yarn strands crossed one another. We were then tasked with figuring out a way to uncross our yard without cutting or removing the cuffs. This served as a nice awkward ice breaker, as my partner and I tried everything that we could think of, including stepping inside of each other's yarn to see if that would allow us to get uncrossed. This was a nice problem solving exercise as it had a solution, but it was far from obvious and required some outside of the box thinking to solve. After a while, the instructors started providing hints and support. This was very much needed, and it helped the groups to be able to support each other. As one group figured out how to do it, we could then support other groups who were still trying to figure things out.
The yarn handcuffs exercise taught me two main lessons. 1) It is important to balance when to provide support, so as to maintain student sovereignty of their solving process, while preventing frustration and shut down, and 2) utilizing student ambassadors to help support other groups in the classroom is a valuable technique to make sure that support is more widely spread throughout the class while allowing students to have the opportunity to teach others, which solidifies their own understanding.
After the handcuff activity we watched a video about Austin's Butterfly. It was about a third grader who was tasked to draw a picture of a butterfly as a scientist would. He drew the first picture, and while it was clearly a butterfly it was not precise. From his first attempt his classmates gave him specific and focused feedback on how he could improve his drawing. The emphasis here was for the students to kindly provide constructive feedback with the goal of Austin drawing the most accurate depiction of the butterfly. He continued to adjust his drawings, completing four drafts before adding color, and through the process using the feedback to improve from one draft to the next, Austin was able to produce a much more scientific drawing of the butterfly.
The lessons learned from Austin's Butterfly that were emphasized throughout the week were to create an environment where students are comfortable giving a receiving feedback, for feedback to be constructive and focused, and begin with a focus on what is going right, keep in mind the goal of any project, and attempts should be thought more of as drafts that are meant to be constantly improved.
The next opportunity to problem solve with my peers came in the form of a sort of puzzle. The set up is there is a sheep named Eric waiting in a line to get shorn. There are 50 sheep in front of Eric, and he is very impatient. Each time that a sheep is shorn, Eric skips ahead of two sheep. If this continues, how many sheep will Eric need to wait for before he is shorn?
Working through this puzzle was a terrific experience. Not only was the puzzle interesting, especially the extensions available if he skips more sheep or if more sheep are shorn at a time, it was great to be able to work through a problem with other teachers. It was very rewarding to be able to bounce ideas off of one another, and I think that as teachers we take that for granted too often. Another great lesson from this task was the availability of having manipulatives to work with. Being able to model the scenario using chips proved invaluable. While getting students to act out the situation is engaging and helps to introduce the problem, it is not efficient for them to work through multiple cases. I also learned about floor functions, as well as alternate scenarios for the problem such as a lunch line or traffic setting for the problem. The big question remaining after working through the problem, is there a function that would represent the relationship between the number of sheep Eric would have to wait for given n sheep shorn and k sheep skipped.
The remainder of day one was spent working through developing a task within groups. While working to develop a task we focused on trying to come up with a puzzle or trick that the students were trying to figure out, and trying to make sure that there was a need for students to work through the math. For instance, if I wanted students to use a table or come up with an equation, I need to provide a need for those strategies to be useful.
On the second day we spent time talking about exponents, fractions, and ratios. The goal in it all was to work to get to the why behind certain properties that have traditionally been overlooked or taken for granted. Students need to see why things are as a means to build their conceptual understanding. In this way math will no longer seem to be a sort of magic to students, and they can be active participants in the process, rather than passive observers of a mystic power in the hands of experts. The theorem we tried to prove was that a/b < (a+c)/(b+d) < c/d . A good endeavor that any teacher of math should be able to execute.
From there the focus turned to the eight standards of mathematical practices. The two main tasks that we had worked through encompassed all eight math practices, and they gave us the opportunity to experience what it means to persevere, to reason, to construct arguments, to model, to use tools, to be precise, to look for structure, and to look for patterns. Seeing and doing those things in a focused and targeted way, helped us as teachers to realize what it is that we want our students to do, use, and think about as they become proficient in making use of the math practices. It was also emphasized that the practices to not take a back seat to content standards, as they go hand in hand in developing mathematically literate problem solvers.
We were given a presentation regarding the new PARCC assessments. This was informative as it provided us with details regarding the format and layout of the new tests. I like how the new tests will provide students with a way to show their thinking, and I like that the rigor of the test is meant to match the rigor that should be taking place in the classroom. There will definitely be an adjustment for teachers and students to get used to the different types of questions and the new technology meant to assess students' proficiency, but I hope that teachers will focus more on the content standards and standards of math practice more than they will on the format of the assessment tool.
Another problem solving task that we worked through was about two lines of people. Two lines of three people on their own block with a center block between the two lines, and the two lines facing each other. The rules of the game are that people can only move through a slide or a jump. A slide can be done one block at a time, and a person can only jump over one person at a time. Each move requires that the people are still facing the same direction, and the goal is for the two respective groups of three people to be on the opposite side of the center block in as few moves as possible.
This was an interesting puzzle, and it allowed for the group to work through a multitude of problem solving strategies. We tried to work out a simpler problem, starting with one person, then two, and so on. We modeled the problem working with chips instead of continuing to move people around. We made sure to record our findings, and we looked for patterns that we noticed that might help lead to a solution. Once we found the correct order in which to execute the jumps and slides, we worked to find a function that might represent the relationship. This was a nice exercise as it yielded a non-linear function, and with the shift in common core Algebra, non-linear functions are going to be emphasized more.
The other puzzle that we worked through dealt with the famous Tower of Hanoi problem. we tried to find the relationship between the number of moves as a function of the number of puzzle pieces. This was another nice problem because it dealt nicely with recursion, as well as exponential relationships. Both of which are also focused heavily in the common core standards.
The major lesson learned from a lot of the tasks and other demonstrations we had was to try to provide as many visual representations of concepts as possible. A great example of utilizing visual and tactile representations of concepts was the use of algebra tiles to demonstrate factoring and completing the square. The presentations make me want to invest in Algebra Tiles, and the entire institute makes me want to get chips and other items that students can make use of as they work through problems.
The final day of the institute was spent presenting our tasks that we developed in our respective groups. This was a terrific experience as there were six products that were presented that I am planning on using with my students. Most everyone tried to make use of the similar puzzles and problems that nicely incorporated multiple solution paths and techniques. There was a dice trick, systems stations, the locker problem, an outbreak of Algebritis, a field trip, and profit comparison all of which provided me with ideas that I can take back to my classroom.
The entire week served to invigorate my creative flow, and it makes me very excited to try out different approaches while working to hone my execution of the approaches that already know and use. I am excited for the next institute pertaining to the Common Core Number and Quantity standards, and I can't wait to see the new ideas and tasks that I can bring back to share with my colleagues.