Without question, the problem of finding the most economical location for a new stadium applies to the real-world, and such practical problems are necessary in order for students to appreciate the significance of mathematics. Two questions, then, warrant deeper inspection:
- What mathematical topics does this problem address, and where does it fit in curricula?
- How does the technology used to solve this problem effectively strengthen student understanding such that this problem deserves a spotlight in the classroom?
To answer the first question regarding mathematical topics integrated into this problem, the conceptual domains of Modeling and Geometry are integral to reaching the solution, and these are two of the conceptual categories outlined for high school (9-12th grade) mathematics in the 2015 Revised Alabama Course of Study: Mathematics (Alabama State Department of Education (ASDOE), 2015). Specifically, the steps implemented in solving this problem parallel to the modeling cycle delineated in the 2015 Revised Alabama Course of Study: Mathematics, represented below:
The solving strategy specific to the new stadium problem reflects this modeling cycle:
- Identify essential information in the problem: Constructing/locating the various centers in the given triangle; recognizing cost disparity between paving and repaving roads; understanding need to minimize length of roads (re)paved
- Create a geometric model to contextualize the relationship between variables in the problem: Constructing possible routes with GeoGebra
- Perform operations on evidenced relationships: Calculating cost to (re)pave given routes; organizing calculations in a spreadsheet; totaling costs for respective stadium locations
- Interpret results: Analyzing spreadsheet to determine most economical option
- Validate solution: Examining conclusion in light of given problem; considering reasonableness of answer
- Sharing results
According to the 2015 Revised Alabama Course of Study: Mathematics, “Modeling is best interpreted…in relation to other standards” (ASDOE, 2015, p. 75). In the new stadium problem, the modeling standard relates to the Geometry standard of making geometric constructions, as solving the problem requires a student to construct a triangle’s circumcenter, centroid, incenter, and orthocenter (ASDOE, 2015, p. 78). Additionally, this problem requires students to meet several of the Standards for Mathematical Practice outlined by the ASDOE (2015, p. 78): make sense of the problem and persevere to solve it; model with mathematics; use appropriate tools strategically; and attend to precision.
With a clear idea of how the new stadium problem fits into the mathematical curriculum, next comes determining the effectiveness of the technology implemented in solving this problem. Returning to the criteria for effective technology proposed by Dick and Hollebrands (2015), it has been shown that a student must (a) choose appropriate technology (GeoGebra and Numbers); (b) purposefully use the technology (make geometric constructions, create and organize a data spreadsheet); (c) make connections (see properties of triangle centers, understand concept of shortest direct and indirect distance); (d) reflect on the results (realize that most economical choice should maximize repaved and common roads). In this way, the technology used in this problem is justified and beneficial for building a student’s relational understanding. As explained by Skemp (1978), relational understanding involves comprehension of why a given concept is true, unlike instrumental understanding, which focuses on memorizing rules and formulas. Because GeoGebra enables students to see properties of triangle centers and distance, students relationally comprehend why the circumcenter is the most economical location. In this way, providing avenues by which students can gain relational understanding is important because it is relational understanding that enables students to think critically and solve new problems.
In regard to actually incorporating this problem into the classroom, it would most appropriately fit with the curriculum of a geometry class. Given the many components and amount of time necessary to solve this problem, this problem could serve as a group project placed at the end of a unit involving triangle centers. The grade for this project could potentially serve in place of a test grade on this material, since the students will likely learn more from interacting with the triangle centers in the project than they would learn by simply memorizing attributes of the centers, and to arrive at the solution they must demonstrate mastery of the material.
In conclusion, the new stadium problem presented in these pages exemplifies the types of problems all teachers should be posing to their students: problems with real-world applications that meaningfully incorporate technology in order to build students’ reasoning and sense-making.