I chose to enact the “Create the Problem” teaching experiment as the culminating activity for the Word Problems investigation in our 7th grade “Expressions and Equations” unit. In this unit, students were introduced to solving word problems without being asked to use a specific strategy. Throughout the investigation, students learned the guess-check-generalize method of solving, building into developing the skills to generalize and create an equation to represent a word problem using key words that indicate mathematical operations. The “Create the Problem” teaching experiment seemed to be an ideal conclusion to this unit, as it requires students to apply their knowledge of generalizing using key words to execute the process in reverse by taking an equation and creating a corresponding word problem. I also selected this activity to assess how students worked together to overcome a challenging problem, as it requires a different sort of creativity than is typical in the mathematics curriculum that these students have experienced. My goals for this activity centered around creating and solving single variable equations and understanding and utilizing precise mathematical languages, specifically as it relates to mathematical operations in contextual problems. To enhance student progress toward my learning goals, I deconstructed the activity into three phases: problem creation, peer editing, and problem solving. In the first phase, students worked with a partner to write a word problem that could represent a given equation. I scaffolded this process by giving students a checklist (see Appendix A) with criteria such as “Solve your equation. Does the answer fit the context of your problem?” and “Our problem is creative and does not use the words 'multiply', 'add', 'subtract', or 'divide.'” Each criterion connects to the learning goals, so by creating a problem with reference to the criteria, the students are progressing toward the goals. In the second phase, students peer edited another pair's word problem, where both pairs had the same given equation. This process was also scaffolded and structured for the students to give their peers feedback in certain areas (see Appendix B for some), including the clarity of the wording, precision of the wording, and how well the problem matches the given equation. Again, this requires students to understand both the structure of word problems and how certain words or phrases indicate specific mathematical operations. This portion of the task did not require students to apply knowledge of solving single-variable equations, so I designed the third phase of the task to accomplish the remaining learning goals. In this final phase (see Appendix C), students were asked to work with a partner to find the equation which corresponded to a student-created word problem, explain how they did so, solve the equation, and provide suggestions to the problem creators. This final portion required the students to understand concepts connected to each of the learning goals and gave me a concrete means to assess these goals.
In addition to providing students with scaffolding to engage in the task, I also designed the activity's structure to give me a means to elicit student thinking and assess student understanding. Specifically, I asked questions which directly addressed the learning goals and provided students with the necessary scaffolding for the task. Related to this scaffolding, I wanted to provide students with enough structure that they could be independent enough so as to work without constant guidance from me, enabling me to monitor student progress and address and clarify any confusions or misconceptions held by the students. At the same time, I had also structured the activity so that students could help to clarify others' confusions and misconceptions in the editing portion to build interdependence and enable the students to build their own understanding without me needing to direct their thought. In this way, I raised the cognitive demand of the task while also providing a means to maintain it by reducing the need for me to guide them during the task. This being said, in the enactment of the task, some pairs did require more guidance than others. The most common form of guidance was that students decided on a context and identified what they would have their variable represent, but were unsure of how to word their problem to discriminate between constants and the coefficients of variables (see Appendix D: these students knew that they wanted students to use bouquets of flowers in their problem, and had written the first sentence and decided on their question, but needed guidance to figure out how to represent constants, so I guided them toward individual flowers). Along these lines, some students struggled to attach the coefficient to the variable in their wording of the problem, but these issues were resolved through the editing portion of the task with help from their peer editors (see Appendices E and F). The resolution of these issues primarily by students demonstrated to me that my students have, at least for the most part, a solid conceptual base that they can apply to unfamiliar situations with connections to known scenarios. Overall, the success of the peer editing process (all but one pair (see Appendix G, in which the problem creators accidentally added what appeared to be a second variable into the equation – the number of candies per bag) between two classes created problems that could be matched with their given equation, several pairs arriving at this point as a result of the editing portion) meant that the enactment of the activity aligned very well with the learning goals and the planning of the task. Students were also, for the most part, successful with using problem solving strategies that we had learned in class to find the equation which corresponded with the student-created word problem they had been given (see Appendices H, I, and J). Some students did struggle with explaining how they were able to generate the equation that corresponded to their given student-created word problem (see Appendix K), which leads me to believe that these students have some initial level of understanding which allows them to write the equation, but that they either are lacking the conceptual understanding that connects the words to mathematical operations or that they have not developed the metacognitive skills to understand how they know what they know. Still, these students engaged in the task and demonstrated achievement of many of the learning goals and progress toward the remainder. This activity very much demonstrated to me the importance of thoughtful formative assessment. The way I scaffolded the activity and deconstructed the thought process for the students gave me a greater window into student understanding and provided me with a means to assess that understanding with concrete examples that did not require me to be conveniently located in the classroom to overhear it. It also gave me a greater appreciation for building students' metacognitive abilities; if built in a general sense, students can apply these abilities across content areas to in turn build their conceptual understanding within any topic. In the future, I hope to incorporate more formative assessment activities with both a scaffolded structure that breaks student work into parseable units and that ask students to practice metacognitive skills, eventually to the point that it is built into the daily classroom routine. If I were to implement this activity again, I would like to ask the problem creators to justify why their word problem matched their given equation in order to further incorporate a metacognitive element into this task and to reinforce the connections between language and mathematics. I would also emphasize the importance of being able to explain how they approached and solved a problem, in the hopes that the students would consciously think about how they are thinking about the problem and build the skills to express this thought verbally or on paper. In the coming lesson, students will be completing a similar activity which connects the ideas of the investigation in an IB assessment task format. This task will require students to create their own problem while simultaneously using a problem-solving structure which we learned in this unit to explain how the problem which they are creating connects to their equation (as I would like my task to do were I to enact it again). To prepare students for the assessment task, I will be giving the students written feedback, specifically relating to the language that they used to write their problem and thinking about connections to other contexts that could be represented by the same equation. As I saw in this task, my students were building this understanding of how to represent variable coefficients and constants in their story, so I hope to use this assessment task to solidify this understanding with a second opportunity to utilize such creative skills.
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