# Engaging Middle School Mathematics Students in Problem-Based Learning: Placement of Cell Phone Towers

by Joe Savrock (October 2008)

UNIVERSITY PARK, Pa. – Teachers of mathematics are fully aware that the subject is challenging for some middle school students. By implementing a problem-based learning experience, mathematics teachers can help stretch students’ reasoning and writing skills, as three Penn State researchers demonstrated.

Rose Mary Zbiek, Shari Ann Reed, and Tracy Boone, of Penn State’s Mathematics Education program, developed and implemented a geometry lesson that incorporated problem-based learning stations, enabling the students to apply mathematics to real-world situations. Zbiek, professor of mathematics education, was the project’s lead investigator. Her scholarly interests focus on teachers’ and students’ mathematical processes, particularly as they unfold in technology-intensive environments at the secondary and college levels.

In their work with students in grades 6 through 8 at several classes in a middle school near Harrisburg, Pa., the researchers were looking to help students better understand the concept of area and its relationship to linear measurement. Their lesson motivated the students by presenting them with a challenging real-life problem—identify the best geographic placement of cell phone towers throughout their home region of Dauphin County, located in south-central Pennsylvania.

The students’ strategy was to allow cell phone signals to reach as many communities as possible without overlapping service, and to use as few towers as possible. Zbiek, Reed, and Boone describe the lesson in a published article, “Helping Students Achieve in Mathematics,” in the journal *Mathematics Teaching in the Middle School* (vol. 12, no. 6, 2007).

The problem-based lesson called for students to work in groups to collaborate on the best solution, and later to hold a whole-class discussion to reflect on the overall results. The class was broken into groups at three stations—(a) the “Area of Dauphin County” station, (b) the “Tower Placement” station, and (c) the “Who’s Correct?” station. The groups spent 20 minutes at each station and then rotated to the other stations.

While all three stations encouraged the students to reason about mathematics, each station involved a different aspect of linear measure or area measure. At each station, the students carried carrying out specific tasks aimed at finding the best solution while utilizing a large county map and a small county map marked to scale with 1-inch blocks. Among the tools they used were metersticks, overhead-projector pens, calculators, string, and “tower tools.” (The tower tools, which had been created by the researchers especially for this project, are a precursor to compasses that emphasize center and radius as defining features of a circle.)

The “Area of Dauphin County” station involved work with side lengths of geometric figures. The students were asked to find the square mileage of their county—a challenging task since Dauphin County is a nonpolygonal shape. The students quickly came to realize that basic formulas to determine the area of simpler shapes (for example, rectangles) would not work in figuring the area of an irregularly shaped county. They came up with different approaches. Some students drew a large rectangle that enclosed the outline of the county and then subtracted sections of blocks that straddled the county lines; others chose to count only the number of complete 1-inch blocks within the county’s outline and then factor in partial blocks. This task helped students develop new methods of measure and see how some methods of measurement are more accurate than others.

The “Tower Placement” station required work with radius and center of circles. Here, the students looked for ways to place each individual cell phone tower strategically so that the signal would reach multiple towns. As they worked, they learned to go beyond their intuitive notion that towers should be placed directly between two towns. They found that, in some cases, placing a tower outside a direct line might provide signal coverage for three, four, or more towns. This exercise encouraged the students to think regionally as well as linearly.

At the “Who’s Correct?” station, students located towns and potential cell tower positions and measured distances to determine which proposed solutions to real-world problems were most believable. Students used a combination of mathematical skills—including constructing circles from centers and radii and making conversions between different scales or units of measure. The students exercised logical reasoning and articulated their strategies in writing. They developed an understanding of how solving real-world problems requires attending to how everyday conversation differs from mathematical conversation.

The lesson’s final phase was a whole-group discussion in which the students reflected on their project. In this open discussion, they came to better understand the mathematical concepts that they employed and learned why some solutions were more beneficial than others.

“The teacher's orchestration of the discussion was crucial to the lesson's success,” stated Zbiek. “The discussion reflected several things—the teacher's purposeful attention to the mathematical ideas as the groups worked, careful selection of student work to include in the whole-class discussion, and explicit use of questions that focused students’ attention to the main points of the lesson.”

One important outcome of the lesson was the students’ ability to analyze two- and three-dimensional space and figures by measuring distance, angles, and radii. “The success of this lesson follows from some basic principles that make a classroom an enjoyable and productive place,” said Zbiek. “The tasks are meaningful in terms of students' real-world experiences. Responding to the real-world questions requires the mathematical ideas that are the target of the lesson. Students of various mathematical levels can make mathematical progress with each task, and whole-class discussions center on the mathematical ideas that students need to develop.”