Steering Students Away from the “Wheel of Operations” and Providing Meaningful Problem Solving Experiences
“How are you going to get started?”
“Multiply. Wait, no. Divide?”
“What do you plan to multiply or divide and what does that help you find?”
“Umm.. I’m not sure.”
“Okay, well let’s go back to the problem and figure out what information you are given.”
Students are very good at playing the wheel of operations game. They are used to having problems that require them to use a few operations that lead them to the correct answer. With further prompting I often discover that they are not aware of why they are using these operations or what the resulting numbers mean. To lead them away from this guessing game, I have tried implementing two strategies that encourage students to explain what they know, what tools are helpful and why, and what their chosen process will help them find.
Strategy 1 — Take away the numbers… at least initially
In a recent geometry lesson, I encouraged students to problem solve through completing a suspension bridge design. Students were given the details and design of a suspension bridge, but as they read through the problem they realized that they were not given any numbers. Not only did this eliminate the wheel of operations guessing game, but this also required students to prioritize what information they needed.
I asked students to determine the minimum amount of reasonable information needed to solve the problem. By reasonable, I meant that someone would need to go out to the bridge site and take measurements over a body of water for a bridge that was not built yet. I also had some great discussions about reasonable information that an architect may already know and use in their bridge design. For example, an architect could determine the height of the pillars needed to support a bridge of “x” length given what they knows about physics, tension, and safety regulations. I had predetermined what I thought the minimum requirements were for being able to build the bridge, and asked groups who wanted more information than necessary to rethink about the relationships between the triangles and bridge pieces. This required them to determine that the triangles were similar, which allows them to use proportions all before having measurements to work with.
It was encouraging to hear students replacing numbers and operations with contextual language, such as, “We can use the ratio between the height of the pillar and this section of the bridge to solve for the other cables.” I also had in depth discussions with groups about the various approaches to the problem (a few of which I hadn’t thought of myself), and at the end of the lesson I gave each group a chance to share what information they got from me and how they used it to complete the bridge design.
Strategy 2 — Provide a structure of questions to help students understand and persevere through the problem solving process
The second lesson strategy I have tried is a modified version of Dan Meyer’s “Three Acts”. This process is designed to engage students in problem solving similar to what is seen in plays/movies. Students are encouraged to take ownership over what information they have or need and discuss how to use that information to work toward the solution. I have modified these acts slightly so that I could apply them to some of the great problems already provided in our curriculum. I also decided to structure the acts as questions for students to answer in hopes that when they approach open ended questions in the future they have a questioning strategy to guide them. The “acts” that I used are as follows:
Act 1: (Understanding the problem) What do you know? What are you asked to find?
Act 2: (What process and why) What tools can you use to answer the question? What other information do you need?
Act 3: (Applying process and re-examining situation) Solve problem. Was your solution correct? What questions remain/arose?
This questioning process has helped give me structure to large problems. I often try to pause the class after they have discussed Act 2 to make sure we are on the right track and discuss any misconceptions so far. It also gives me some piece of mind to know that groups are thinking in a useful direction instead of wandering to far off track or giving up.
I am still working on presenting the problems in engaging ways and ensuring that the initial information given is not bogged down by facts or numbers. I want that information to be discussed and asked for in Act 2. Again, there is value in waiting to give numbers until a plan of action has been developed. It is difficult to present problems in such an open ended manner and allow students to take control over what they want to solve, especially when you think about all the learning targets you are attempting to cover for the day. If you are nervous about jumping right into a full three-act problem such as the ones Dan Meyer has provided, this modified version might give you a bit more structure to get started.
Within these two problem solving strategies there are many opportunities for me to walk around and have deep discussions about the content. During the discussions about what tools to use or what measurements to ask for, I like to ask groups for justifications. While students were debating what bridge values to get from me, I asked them how they could relate the length of the bridge to the triangles, or how the large triangles would help them get values for the cables in the smaller triangles. My favorite questions of the day were: “How are the triangles related?” and “Why do you know that?” I also made sure students gave me very specific answers. I wanted to know which triangle similarity condition they were using and which angles were involved. I also encouraged students to use the vocabulary we learned throughout the triangle units.
Although these strategies are not perfect they have helped me gain some insight on how to best support my students in the problem solving process. I hope to continue giving students authentic problems that require them to take control. If that means taking away numbers so they are forced to think deeply about what the problem is telling them, then I will continue to do so. Giving students goals to work toward throughout the problem, such as getting information (numbers) from me and listing what tools or strategies they can use in the problem, helps students persevere through the difficulty. My goals along the way are to encourage deep discussions and finally get rid of the wheel of operations guessing game.
