A novel approach to teaching Algebra

Photo by Kathy F. Atkinson | Illustration courtesy of Teo Paoletti

Unlike other elementary and middle school mathematics courses, algebra begins to introduce abstract relationships and ideas. While textbooks attempt to give students real-world illustrations of these new concepts, many students struggle to match what they’re learning in the classroom with concrete experiences.

For students who struggle to master these foundational concepts, the challenges of algebra can have long-term economic and social impacts. Algebra often serves as a gatekeeper for future mathematics coursework and careers in science, technology, engineering and mathematics (STEM). How can K-12 educators support their students in developing this critical algebraic knowledge?

The University of Delaware’s Teo Paoletti, who is an assistant professor in the College of Education and Human Development’s (CEHD) School of Education (SOE), said he thinks that the answer might lie in helping students develop covariational reasoning or the ability to reason about relationships as quantities change together. He has received a considerable funding award from the National Science Foundation (NSF) that may help further his investigation. Through the Covariational and Algebraic Reasoning Project: A New Path to Algebra (Project CARe), Paoletti will create and test digital instructional tasks designed to support middle schoolers’ covariational reasoning using a free educational technology platform. He hopes to integrate education and research to create a new path into learning algebra for middle school students.

In support of this novel approach to teaching and learning algebra, Paoletti received a nearly $885,000 NSF CAREER award, one of the organization’s most prestigious awards. These awards support early-career faculty who have the potential to serve as academic role models in research and education and to lead advances in the mission of their organization.

“Part of what makes school algebra so difficult for students is that it can feel really abstract and different from what they experience in their day-to-day life,” Paoletti said. “When learning how to add, subtract, multiply and divide numbers, students have concrete experiences that they can draw on. However, algebra is more complex than arithmetic. By designing opportunities for students to see and experience quantities changing in context, they can begin to represent relationships between these changing quantities. My goal is for students to see how they can use graphs and algebraic expressions as representations of the concrete relationships they conceive of in a situation. This approach can not only provide a more accessible entry way into algebra, but also better connects with how algebra is used in science and engineering fields.”

What is Project CARe?

In Project CARe, Paoletti will explore how middle schoolers’ ability to reason covariationally might help them develop an understanding of ideas that are critical to learning algebra. Specifically, he will investigate how students’ covariational reasoning may serve as a foundation for the development of algebraic reasoning and identify the instructional paths — or what teachers can do in the classroom — for supporting their reasoning.

A series of novel, digital instructional tasks, designed and piloted by Paoletti, serves as the basis of this project. Using the free, publicly available Desmos platform, Paoletti will create a research-based sequence of instructional activities, including teacher support materials, that have been effective in supporting students’ algebraic and covariational reasoning.

For example, the Faucet Task, which asks students to make predictions about water flow and temperature changes as they adjust the hot and cold knobs on a digital faucet, is designed to help students develop understandings for graphing. After playing with the knobs and observing animated changes in flow and temperature (indicated by color changes), students are asked to make predictions about how specific actions — like turning on the cold knob slowly — will affect the water flow and temperature. Students then practice graphing these relationships.


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