Steven Jones recently had a paper titled “Approaches to Integration based on Quantitative Reasoning: Adding Up Pieces and Accumulation from Rate” published in the conference proceedings for the Research in Undergraduate Mathematics Education conference. Steven has answered a few questions about this paper below:
Who were your co-authors on this paper?
Robert Ely, University of Idaho
Who would you say is the target audience for this paper?
Math Education researchers, calculus instructors, and anyone new to the field of calculus education.
What is the big problem you hoped to address with this paper?
There’s been substantial recognition of the need for integral instruction to move past the stereotypical “area-under-a-curve” meaning to one based on quantities and quantitative reasoning. However, research on these types of approaches are scattered across many journals, conference proceedings, books, and other venues. This paper completed a systematic review of definite integral instruction and learning to pull all these results together into a single reference work for researchers, instructors, and anyone wanting to learn more about the learning and learning of definite integrals.
What are some of the key ideas in the article?
There’s been substantial recognition of the need for integral instruction to move past the stereotypical “area-under-a-curve” meaning to one based on quantities and quantitative reasoning. Doing so is crucial for students to successfully use integrals to model and reason within STEM disciplines. This paper reviewed efforts on quantitatively-grounded approaches to definite integrals to examine meanings, formalizations, foci, reasoning, and modeling within these approaches. While some approaches differ in some of these dimensions, they all contain the benefit of allowing students to flexibly and productively use, apply, model with, and understand integrals in real-world contexts.
Calculus education research on integration is coalescing around the theme that teaching integration based on quantitative reasoning is crucial for robust understanding of and usage of integrals. This paper contributes to this International Journal for Research in Undergraduate Education special issue on the teaching and learning of definite integrals by reviewing the research literature on quantitatively-based approaches to integration, in order to pull together the ideas that are spread across many papers in many outlets. We note that the literature in this area has largely developed along two distinct paradigms, which we call adding up pieces and accumulation from rate. While they are both based in quantitative reasoning, there are critical differences between them that have important ramifications for teaching and learning integration. We use these two paradigms to organize our review and we use the literature to discuss the meanings, formalizations, foci, reasoning, and modeling in these approaches.