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Pat Thompson, PI Fabio Milner, co-PI Mark Ashbrook, co-PI

"DIRACC" is an acronym for the *Developing and Investigating a
Rigorous Approach to Conceptual Calculus*. Project DIRACC gained support from the National Science Foundation in Fall 2016.

Project DIRACC grew out of Pat Thompson's research on calculus learning and teaching (see his 1994, 2008, 2013, and 2016 publications, or here for a full list).

The project began in the Fall of 2010 with Thompson's decision to conduct a series of three design experiments to see whether the standard calculus curriculum used at ASU could support an attempt to help students overcome a number of difficulties uncovered by research on students' calculus learning.

Some of these established facts from prior research are that, in students thinking:

- Calculus, like school mathematics, is about rules and procedures. Students think that calculus is difficult primarily because there are
*so many*rules and procedures. - Variables do not vary. Therefore rate of change is not about change and integrals are not about accumulation.
- Integrals are areas under a curve. Students wonder, "How can an area represent a distance or an amount of work?"
- Average rate of change has little to do with rate of change. It is about the direction of a line that passes through two points on a graph.
- A tangent is a line that "just touches" a curve.
- Derivative is a slope of a tangent. The net result is that, in students' understandings, derivatives are not about rates of change.
- With integrals being areas and derivatives being slopes of tangents, and with the Fundamental Theorem of Calculus mentioning neither area nor slope, students' understandings of derivatives and integrals stand in splendid isolation of each other.

The result of the design experiments was the decision that the calculus curriculum needed to be fundamentally restructured so that it was about ideas that build upon and connect with each other. The restructured curriculum is reported in this article. The textbook you are viewing is the embodiment of those ideas.

Thompson's design experiments also led to a proposal to the National Science Foundation (NSF) to extend DIRACC to Calculus 2 and to investigate student learning in restructured calculus courses. The proposal was funded in October, 2016.

Calculus is an essential tool, and provides a conceptual foundation, in each of the STEM disciplines. Acquiring deep conceptual understandings when first learning calculus poses many difficulties for students and creates many challenges for teachers. As a consequence, many concerted efforts in calculus reform appear to have failed to make educationally significant differences in student understanding that their proponents predicted. This project will build on the documented success of ASU’s already-redesigned and already-deployed Calculus I to redesign and implement Calculus II, and will research students' learning in both the redesigned and traditional calculus sequences. In addition, the project will develop conceptual inventories for Calculus I and Calculus II that other institutions can use to assess students’ progress on central ideas of the calculus. Students taking this redesigned sequence will be better prepared for using calculus in their science courses and for learning in subsequent mathematics courses. The textbook produced by this project will be made available as an open resource for others to use or build upon. Finally, our results on students’ calculus learning, and the calculus concept inventories built to investigate students’ learning, will inform future research in these areas. Results from the project will be disseminated widely at regional, national and international conferences. Finally, the project's research results will be published in *The
Monthly* and in peer reviewed mathematics education journals and conference proceedings. Two Ph.D. students will conduct their dissertations in the context of Project DIRACC.

One reason for the lack of effectiveness of calculus reform is that the fundamental structure of the underlying curriculum has remained unchanged. For example, reform projects have not focused on students’ development of richly connected meanings for rate-of-change functions and accumulation functions, which are essential to any introductory calculus. The DIRACC courses will be highly conceptual because they will be based upon students' development of coherent meanings for ideas applicable throughout calculus and that also facilitate learning of mathematical ideas beyond calculus. The first course addresses this challenge by making the fundamental theorem of calculus central to every aspect of students' experience of calculus in the course. At the same time, the sequence will be explicitly computational because, as in the existing first course, students will use computers to represent processes that define functions as models of situations, which then become objects of study themselves. The courses will also acknowledge and address known weaknesses in students' preparation for calculus. The calculus concept inventories will be developed using standard instrument-development techniques, given at the beginning and end of their respective courses, and their psychometric properties will be established with approximately 600 students per course.

Re-Imagining the Calculus Curriculum, I

Re-Imagining the Calculus Curriculum, II

Click here for our report of activities in the first year of NSF funding.

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