Steven Jones recently published an article titled “Learning Integrals Based on Adding Up Pieces Across a Unit on Integration.” in the International Journal of Research in Undergraduate Mathematics Education. Steven has answered a few questions about this article below:
Who were your co-authors on this article?
Brinley Stevens, a former BYU masters student
Who would you say is the target audience for this article?
Undergraduate mathematics education researchers and calculus instructors.
What is the big problem you hoped to address with this article?
Definite integrals are a crucial STEM concept that are widely used across many disciplines. A lot of calculus education research has examined student understanding, thinking, and reasoning about definite integrals, but the research literature has few empirical examinations of students learning integrals across an entire integration unit. This study supplied this missing information.
What are some of the key ideas in the article?
Teaching definite integrals through quantities-based meanings is essential for their productive usage across STEM. It requires a shift away from the usual "area under the curve" meaning to a "sum of tiny bits of a quantity" meaning (which actually matches the mathematical definition better, anyway). This study used a small-scale teaching experiment to examine pairs of students learning definite integrals and integral functions within this quantities-based paradigm. It examined how they first learned integrals during the introductory lesson, and how their understanding developing as they worked up to the point of being ready to learn the Fundamental Theorem of Calculus.
Abstract:
Recent research on integration has shown the importance of quantities-based meanings for integrals. However, this research body is still in need of detailed empirical accounts of how students develop such understandings across an entire unit on integration. This paper contributes by providing one such account, based on a quantities-based orientation called adding up pieces (AUP). Our study examines how three separate pairs of students learned definite integrals and integral functions within interview settings over four consecutive interview-lessons, meant to correspond to four consecutive in-class sessions. The first interview developed the partition, target quantity, and sum structure for definite integral, which were solidified in the second interview. The third and fourth interviews extended this AUP structure to a variable upper bound, output, and function structure for integral functions in preparation for the Fundamental Theorem of Calculus (FTC).