The 4 elements of an effective set-up of a challenging math problem

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Not all kids love a challenge in math class. But more kids become engaged in solving the math problem if you do a good job setting it up, says a study in the Journal for Research in Mathematics Education.

After reviewing video recordings of 165 math teachers in their classrooms, researchers identified 4 key elements of an effective set-up. The classroom videos were gathered as part of a 4-year research project on improving middle school math in large urban U.S. math districts, The impact of a a good set-up on student participation and response was fairly dramatic in the videos, according to the researchers. Researchers used rubrics to score teachers on their practices, including the practice of setting up a problem.

With more students now studying ambitious math and with students all having different levels of math understanding, a good set-up provides support to many more students, according to the study. If more students understand what they need to do earlier in the process, teachers don’t spend a lot of time reintroducing the task. Students can begin solving the task earlier in the class, leaving more time at the end of the lesson for a whole-class discussion, a major priority in today’s math classroom.

“A central goal of this article is to elaborate on the “how” of ambitious mathematics teaching by identifying high-leverage practices that teachers can develop,” the researchers write. “Such practices have the potential to increase student participation and learning as they engage in mathematical activity aimed at rigorous learning goals.”

To outline the key steps of a high-quality set-up, researchers focused on the approach of one exemplary teacher, “Mr. Smith”, as he introduced a “Dancing for Dollars” problem. This problem asked students to calculate how many dollars would be raised in a dance marathon. The students had to use tables, graphs and equations to choose the best of 3 different proposals for pledges: $5 plus $1 for every hour danced; $3 per hour; or $8 per hour plus $0.50 for every hour danced.

Here are the 4 elements of a good set up as illustrated by Mr. Smith:

Explicitly discuss key contextual factors Make sure all students understand the situation/scenario in the problem. Some students might be familiar with dance marathons, but make sure all of them know what it is and how it works. Mr. Smith showed students Internet pictures of dance marathons to elicit their knowledge and then asked his students to propose different reasons for having a marathon. In this problem, the purpose of the fundraiser was to hire a deejay for the school’s Valentine’s Day dance. (The teachers in the study were more likely to provide background on math concepts than on context, report the researchers).

Point out key mathematical ideas and relationships. In the Dancing for Dollars problem, students had to represent the accumulation of money over time for 3 different plans. Students had to understand that money accumulates as a participant continues to dance for a greater number of hours. They also had to know that there were 2 different ways people could donate. One way was to give a simple upfront pledge just for the student’s participation and the other was to an hourly pledge with the total pledge based on the total number of hours a student danced. Mr. Smith asked students to explain and restate the distinction and often adopted the language they used. One danger in the set-up is reducing the cognitive demand of the problem by pointing out a solution path. Mr. Smith knew his students were comfortable making tables, graphs and equations to represent linear equations with y-intercepts of 0. Representing equations with y-intercepts not equal to 0 was a central challenge of this task. But Mr. Smith avoided the trap of giving too much away.

Develop a common language In a good set-up, teachers do not simply talk to students about the key features of the task, but solicit input and ask questions that require more than a yes or no answer. With broad participation, teachers can assess students’ understanding of the key features of the task and determine what support they need. “The use of common language is an indicator that students have developed taken-as-shared understanding of the key features of the task,” the authors write. While it is difficult to know how many of your students understand, common language is an indication that they have enough knowledge to proceed. At several points in the set-up process, Mr. Smith asked his students to describe the problem, restate ideas in their own words, add to their peers’ ideas and mark particular ideas as important. These talk moves helped him check for student understanding along the way.

Maintain the cognitive demand. In a good set-up, teachers identify the boundaries between background info and the material that must be left for the students to unlock. The teacher must take into account the students’ prior instruction, their present conceptions and the instructional goal. For this task, students had to use mathematical justifications to ultimately compare the merits of the particular plans. Mr. Smith was effective at the delicate task of making the problem accessible to students without compromising their opportunities to learn significant mathematics .

“Exploring Relationships Between Setting Up Complex Tasks and Opportunities to Learn in Concluding Whole-Class Discussions in Middle-Grades Mathematical Instruction,” by Kara Jackson et al., Journal for Research in Mathematics Education, 2013, Volume 44, Number 4, pp. 646-682.

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