4183 Views (as of 05/2023)
  1. Fabio Milner
  2. http://math.asu.edu/~milner
  3. Professor and Director of Mathematics for STEM Education
  5. Arizona State University ASU
  1. Hedvig Mohacsy
  2. https://math.asu.edu/node/930
  3. Lecturer
  5. Arizona State University ASU
  1. Pat Thompson
  2. http://pat-thompson.net
  3. Prof.
  5. Arizona State University ASU

Developing & Investigating Rigorous Approach to Conceptual Calculus

NSF Awards: 1625678

2019 (see original presentation & discussion)

Grades 9-12, Undergraduate

We describe the philosophy and core principles that guided the design of DIRACC, Developing and Investigating a Rigorous Approach to Conceptual Calculus. We provide examples of non-standard problems that are posed in DIRACC calculus and how students approach their solutions. We discuss some high and some low points in the development and provide some evidence that this philosophy works for the benefit of students.

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Discussion from the 2019 STEM for All Video Showcase (9 posts)
  • Icon for: Fabio Milner

    Fabio Milner

    Lead Presenter
    Professor and Director of Mathematics for STEM Education
    May 12, 2019 | 05:18 p.m.

    The DIRACC (Developing and Investigating a Rigorous Approachmto Conceptual Calculus) team has worked for the past 4 years in the development of a new curriculum, materials, assessments, and training for faculty and graduate teaching assistants for first and second semesters of calculus for scientists and mathematics majors. The desire to develop a new approach to conceptual calculus arose from two sources: a desire that students learn calculus meaningfully and coherently and past research that revealed calculus students’ misunderstandings and faulty meanings both before and after studying calculus.

    DIRACC calculus frames the entirety of calculus in one variable with two foundational problems:

    • You know how fast a quantity varies at every moment; you want to know how much of it there is at every moment.
    • You know how much of a quantity there is at every moment; you want to know how fast it varies at every moment.

    The experience of DIRACC at Arizona State University produced significant results, both in terms of gains measured by pre- and post-tests and performance in "downstream" reasoning courses such as multivariate calculus and the first proof course.

    • The same pre-test and post-test taken by 320 students in the pre-test (140 DIRACC and 180 traditional) and 307 in the post-test (106 DIRACC and 201 traditional), showed indistinguishable scores from students in the two groups for the pre-test (2.98 for DIRACC vs. 3.18 for traditional) but an effect of 1.23 standard deviations in favor of the DIRACC students in the post-test (7.90 for DIRACC vs. 4.89 for traditional). Similar results appeared the following semester when comparing DIRACC students with engineering students: pre-test averages were 2.97 for engineering and 3.72 for DIRACC, while post-test averages were 4.38 for engineering students and 7.13 for DIRACC students.
    • Students who took Mathematical Structures, the first proof course at ASU, immediately after calculus 2 and who completed it in an academic-year semester during the last 5 years failed at five times the rate when coming from traditional calculus compared with DIRACC calculus (16% vs. 3%). Perhaps more telling about students’ understandings is the fact that 5% of students who earned a grade of A in traditional calculus 2 failed the proof course as compared with none who earned an A in DIRACC calculus 2.

    We invite you to get a taste of DIRACC calculus at http://patthompson.net/ThompsonCalc/. We hope you see potential value, whether you are a student trying to understand the main concepts of single-variable calculus or a teacher trying to help your students do that.


  • Icon for: Becca Schillaci

    Becca Schillaci

    Research Associate
    May 13, 2019 | 11:44 a.m.

    Hi Fabio. It's sounds like DIRACC Calculus is showing promise. What do you think are some of the course features that contribute to it's success with students?

  • Icon for: Fabio Milner

    Fabio Milner

    Lead Presenter
    Professor and Director of Mathematics for STEM Education
    May 13, 2019 | 12:06 p.m.

    Hi, Becca. One of the main contributors is laying responsibility on the students for expressing correctly what they are thinking. They need to do that both orally and in writing. They use the computer program Graphing Calculator and quickly discover that, in order that the program do what they want it to do, they need to "tell" it in writing, accurately and unequivocally what that is.

    Discussion is closed. Upvoting is no longer available

    Becca Schillaci
  • Icon for: Gregory Rushton

    Gregory Rushton

    Director, TN STEM Education Center
    May 14, 2019 | 07:06 a.m.

    Hi, thanks for sharing your work with us....I'm not clear on how studying the accumulation function helps students get to a fundamental understanding of calculus better than traditional approaches...can you say more about the connection between these two ideas theoretically? thanks again!

  • Icon for: Hedvig Mohacsy

    Hedvig Mohacsy

    May 14, 2019 | 03:25 p.m.

    Hello Gregory,

    Studying the accumulation function helps students answer to the question "You know how fast a quantity varies at every moment; you want to know how much of it there is at every moment."

    Recreating a linear function from an initial value and from the constant rate is an easy task for a calculus students. Can we recreate any differentiable function from an initial value and from the rate of change function as well? 

    Yes, it is quite simple. Every differential function, locally is essentially linear.  Thus, we can compute dy=mdx many times and adding the bits of changes together to find the value of the function.  The accumulation function, beyond its own mathematical beauty, gives the meaning of integration and can be applied in real life scenarios, instead of introducing integrals as a tool to calculate areas, volumes etc.  

    As we constructed the accumulation function in class and understood the meaning of it, the Fundamental Theorem of Calculus was proved effortlessly and elegantly. The structure of the course made us see the big picture after few weeks, only using simple concepts such as constant rate of change and linearity.

    In the online book, in chapter 5, you can find application problems: how can we find the the high of the rocket from the its velocity; what is the volume of the oil leaked, if we know the rate at which the volume of the oil changes with respect to time.   http://patthompson.net/ThompsonCalc/section_5_1.html

  • Patrick Thompson

    Higher Ed Faculty
    May 16, 2019 | 11:25 a.m.

    Hi Gregory,

    I'll add to Fabio's reply.

    Accumulation from rate of change is easier for students to conceptualize than is rate of change from accumulation. Studying accumulation functions first also makes the idea of deriving rate of change functions more natural: If you take any function that gives an amount of a quantity, you can re-conceive it as each value of that quantity as having accumulated. But if it accumulated, it accumulated at some rate and therefore must be of the form f(x)=int(a,x,r(t)dt). The problem then becomes recovering r(x) from f(x).

    Another reason DIRACC develops accumulation functions first is that, using the graphing program Graphing Calculator, students can define accumulation functions in open form and they become "first-class" functions. Students can evaluate them, graph them, or anything they could do with a function defined in closed form.

    As Fabio points out, chapters 4 and 5 lay this out in detail.



  • Icon for: Monae Verbeke

    Monae Verbeke

    Senior Research Associate
    May 15, 2019 | 08:34 p.m.

    After watching this, I'm wondering what you how student's take to other areas of their studies/lives and if you've tried to study this transfer? 

  • Patrick Thompson

    Higher Ed Faculty
    May 16, 2019 | 11:31 a.m.

    Hello, Monae.

    We haven't studied this question yet, but it will be important to do so.


    Pat Thompson

  • Icon for: Amy Hackenberg

    Amy Hackenberg

    Higher Ed Faculty
    May 20, 2019 | 03:07 p.m.

    Thanks for the video about this important project. I wonder what you might be doing next to spread the impact... for example, is there professional development in the works about this approach so that instructors/professors outside of ASU might get involved?

  • Further posting is closed as the event has ended.