Developing Authentic Mathematics Tasks
from Local Events
Kimberly Corum and Dr. Joe Garofalo
Mathematically proficient students are expected be able to apply their mathematical knowledge to contexts that extend beyond the mathematics classroom (NCTM, 2002; Virginia Board of Educators, 2009). Teachers can support their students’ ability to make connections by incorporating tasks that require applications of mathematics and encourage students to engage in authentic problem solving (NCTM, 2002; Virginia Board of Educators, 2009). However, such rich tasks are often absent from traditional curriculum materials. The inauthenticity of school mathematics is a concern that has been raised by mathematics education researchers for years. As Schoenfeld (1989) explains, “In the context of school mathematics, word problems always have answers. Those answers are almost always integers, which are obtained by manipulating the integers that appear in the problem statements. Thus, in classroom practice, one learns to combine the numbers, whether or not doing so makes sense in other contexts” (p. 85-86).
Despite this call for reform, mathematics textbooks continue to present content in isolation and application tasks are presented as a culminating activity. In a recent review of traditional and reform textbooks, Sherman, Walkington, and Howell (2016) found that most content was organized based on the symbol precedence view (SPV). The assumption of SPV is that “symbolic equations are easier to solve and should be taught prior to verbal problems” (Sherman, Walkington, & Howell, 2016, p. 135). But, formalizing mathematical concepts without providing students with tangible experiences to ground their thinking can hinder students’ depth of understanding (Sherman, Walkington, & Howell, 2016). One way that teachers can make mathematics more authentic and applicable is to supplement their curriculum with tasks developed from actual events and news stories. In this article, we share two tasks that were developed from local events that teachers can use in their classrooms. The first task uses earthquake data to explore applications of logarithms and the second task uses an actual pricing error from a Charlottesville gas station to explore decimal operations and place value. Both tasks address Virginia Standards of Learning (SOL) for Mathematics ranging from middle school through high school.
Figure 1: The historic portion of Louisa High School after the earthquake .The Louisa Earthquake Task
The Louisa Earthquake
On August 23, 2011, at 1:51 p.m., the small town of Mineral, in Louisa County, Virginia, experienced an earthquake. It was estimated that approximately one-third of the U.S. population could have felt the earthquake, more than any earthquake in U.S. history. The earthquake, which destroyed Louisa County High School, was first rated as a 5.8 on the Richter scale, then it was upgraded to 5.9, then it was revised again, back down to 5.8 (CNN, 2011).
· How different are earthquakes of 5.8 and 5.9 in terms of seismic wave amplitude?
· What Richter rating represents twice the amplitude of a 5.8 rating?
The purpose of the Louisa Earthquake task is to provide students with a real-world application of logarithms and to encourage students to engage in non-linear thinking. When thinking linearly, the difference between a 5.8 and a 5.9 rating appears minimal. However, since the Richter scale is logarithmic, an increase of 0.1 on the scale represents a 26% (100.1 ≈ 1.258) increase in the amplitude of the seismic wave. And, since 100.3 ≈ 1.995, an earthquake with magnitude 6.1 would have double the seismic wave amplitude (100.3 * 105.8 = 106.1). The Louisa Earthquake task could be posed as either an introductory investigation for students or as an exploration of the application of logarithms after students have developed more familiarity with the concept. Teachers can also collaboratively plan lessons with their colleagues so that students explore the mathematics behind the Richter scale and the science behind what causes earthquakes to occur. Teachers can further extend this task by having students compare other earthquakes or exploring other applications of logarithmic scales, such as decibels or pH.
The Charlottesville Gasoline Task
Beginning in the middle grades, students are expected to be able to solve multistep problems involving decimals (6.7) and proportional reasoning (7.4) (Virginia Board of Educators, 2009). This task was developed using an actual receipt from a local gas station where a decimal error in assigning the unit cost of gasoline resulted in huge savings for customers. The receipt was submitted to us by an undergraduate student.
Cheap Gas in Charlottesville
Figure 2: Receipt from Charlottesville BP gas station.
On February 16, 2013, the BP gas station on Fontaine Avenue in Charlottesville, VA was selling its premium gasoline, BP Gold, for $0.409 per gallon. It appears that when changing the price at the pump, a mistake was made with the placement of the decimal point. BP Gold should have been priced at $4.099 per gallon. Below is a receipt for gas purchased at the station on that day. The driver took the opportunity to fill up her gas tank at the surprising low price.
· How much money did this driver save by filling up her tank that day?
· How much money would a 22-year old female, who drives a 2013 Mini Cooper with a 4-cylinder engine and automatic transmission, save if she were able to buy gas at this rate for an entire year?
· How much would a 40-year old male, who drives a 2013 Lamborghini Aventador Roadster with a 12-cylinder engine and automatic transmission, save if he were able to buy gas at this rate for an entire year?
The Cheap Gas in Charlottesville task can be modified to be appropriate for a range of grade levels. For example, if used in upper elementary grades, the purpose of this task could be to give students an authentic use of decimals. The task can be limited to only the first question and students can use the information from the receipt to compare the cost of the number of gallons purchased using the two prices. They could either use multiplication or repeated addition, or just realize that all is needed is to multiply the lower cost by nine. In the secondary grades, the purpose of the task could be developing problem-solving strategies (e.g., analyze the problem, interpret given information, generate a plan). The second and third questions require students to identify what information is missing and then conduct independent research to find what is needed (e.g., average miles driven per year for different genders and ages, gas mileage for different vehicles). Teachers can further extend this task by having students explore the savings that would result from a similar decimal error using today’s gasoline prices. With the rise of vehicles that run on alternative fuel, another related task is to have students analyze under which conditions hybrid cars and diesel cars are more economical than gasoline cars.
The National Council of Teachers of Mathematics (2002) recommends, “School mathematics experiences at all levels should include opportunities to learn about mathematics by working on problems arising in contexts outside of mathematics. These connections can be to other subject areas and disciplines as well as to students’ daily lives,” (p. 66). Considering the limitations of traditional mathematics textbooks and curriculum materials (e.g. Schoenfeld, 1989; Sherman, Walkington, & Howell, 2016), teachers are encouraged to supplement their mathematics teaching with authentic application tasks. Using tasks developed from actual events can support students’ ability to make these important connections between their classroom mathematics experiences and the world around them. We hope that you are able to incorporate the tasks we shared into your own classroom instruction and that you have been inspired to develop your own authentic tasks.
CNN Wire Staff. (2011, August 24). Virginia earthquake rattles East Coast. CNN. Retrieved from http://www.cnn.com/2011/US/08/23/virginia.quake/index.html
National Council of Teachers of Mathematics (2002). Principles and standards for school mathematics (Vol. 1). Reston, VA: NCTM.
Schoenfeld, A. H. (1989). Problem solving in context(s). In R. Charles & E. Silver (Eds.), The teaching and assessing of mathematical problem solving (Vol. 3, pp. 82-92). Reston, VA: NCTM.
Sherman, M. F., Walkington, C., & Howell, E. (2016). A comparison of symbol-precedence view in investigative and conventional textbooks used in algebra courses. Journal for Research in Mathematics Education, 47, 134-146.
Virginia Board of Education. (2009). Mathematics standards of learning for Virginia public schools. Richmond, VA: Commonwealth of Virginia, Department of Education. Retrieved from http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/ 2009/stds_math.pdf
University of Virginia
Dr. Joe Garofalo
Associate Dean, Professor
University of Virginia