This week we continued to discuss the same portion of *Principles to Actions* that we read last week which focused on the varying levels of cognitive demand in the classroom, as well as a little bit of reasoning and sense making. A few of the specific types of problems that we have discussed in class that have a higher cognitive demand for students are Shift Problems, and Three-Act Problems. Shift problems are aimed at creating a sequence of tasks from one specific task in a textbook, replacing many of the traditional homework problems in a textbook with these sequences. Three-Act problems are the brainchild of Dan Meyer, and are mathematical “stories” in which the teacher introduces the conflict of the story within the First Act, the protagonist overcomes obstacles in the Second Act, and the conflict is resolved and a sequel/extension is set up in the Third Act. In Meyer’s explanation of the Three-Act problems, he discusses that historically, teachers have seen their role in the Second Act. However, with the tide of mathematical classroom layout changing to flipped classrooms, and tools such as Khan Academy becoming more and more useful, the mathematics teacher’s role is changing too. It is seeping into the First and Third Acts as well. These two types of problems are new ways of looking at mathematical tasks. Similar to the ticket annulus problem that was performed in class, these types of problems provoke a higher cognitive demand from students– they have to have a clear understanding of the mathematics behind the problem in order to accurately develop a solution. It asks them to make connections between previous topics, use deductive reasoning, and more as they strive to find a solution to a problem.

While reading, particularly about the flipped classroom in Dan Meyer’s blog, I began to think about my own experience in both high school and college. Though I never had a flipped classroom, I have thought about both the advantages and disadvantages of a flipped classroom. The collaboration that allows students to problem solve and find solutions to any given problem is harder to find in a flipped classroom. Furthermore, students may not get as deep of an understanding of the material as if they were in an inquiry-based classroom. However, talking to one particular teacher that does have a flipped classroom, there are advantages to flipping when compared to a traditional classroom. This teacher asks students to watch the videos while they are in class. They then have an opportunity to go more at their own pace, and ask questions as needed. I believe the distinction of the advantages of a flipped classroom does have to made in that is advantageous compared to the traditional classroom, but perhaps not when compared to the classroom that we are striving to create that inspires reasoning and sense making, with tasks that have high cognitive demand. In the future, I would hope to have a blend of sorts where some tasks are embedded into technology that allows students to first work through a problem by themselves.

Although not a Three-Act Problem, I was able to find a problem from NCTM called “Thunder and Lightning” based on how two people in different locations hear the sound of thunder at the same time. In this problem, students combine their knowledge of science and mathematics to find both where the lightning struck, where the people that heard the strikes were located, and to find where three different people would be. To do so, the students use the Geometer’s Sketchpad, and properties of triangles and circles. The activity sets the stage for looking at equidistance, prompting students to discover properties of the perpendicular bisector. There is a worksheet associated with the activity, but this can easily be changed so that students are working collaboratively. A teacher could distribute different scenarios to each small group of students, and ask that students present their findings for their particular solution to the rest of the class at the end of the period. There are various modifications that a teacher could make to this activity as they see fit to change the layout from a worksheet to a more collaborative activity.