By David Bressoud @dbressoud

As of 2024, new Launchings columns appear on the third Tuesday of the month.

If you do not know about the Calculus Videos Project (calcvids.org), you should check it out. This NSF-sponsored project [1] led by Aaron Weinberg, Jason Martin, and Michael Tallman produced high quality videos built around 30 topics from first semester calculus. For each topic, their team created up to three short videos designed to engage students with the motivating issues and essential understandings.

One of the most thoughtfully designed sets of videos is built around the concept of instantaneous rate of change. This, after all, is what we mean by the derivative. It sits at the heart of differential calculus, and yet I doubt many of our calculus students ever really understand it.

The treatment in the video on Approximating Instantaneous Rate of Change, as with each of the thirty topics, begins with an unfamiliar problem whose solution will get at the heart of the concept. Students are shown the picture in Figure 1 and asked how fast the ball was traveling when the picture was taken. The first of the three videos that deal with this topic shows two students trying to make sense of the problem. They observe that to determine velocity one needs a distance that is travelled and a time interval over which the travel takes place. The picture seems to show just a frozen instant of time. But they observe that the picture of the ball is blurry. The camera would have had a shutter speed, a time interval over which the picture was taken. Knowing shutter speed and using the blur to determine distance traveled should give us a velocity.

The second video provides numbers. The shutter was open for 1/2000th of a second. Measuring the blur indicates that the ball moved 0.572 inches, resulting in a speed of 1144 inches per second, or a suspiciously precise 65 mph. Clearly, the speed of the baseball is not going to change very much over that 1/2000th of a second. For all practical purposes, 65 mph can be given as the instantaneous rates of change.

The third video approaches a precise definition of what we mean by the limit of the average rate of change. Imagine we have a camera that can make two consecutive exposures of precisely 1/4000th of a second, splitting the first image into two. Assume we want to know the exact speed of the baseball at the precise moment the first picture ends and the second begins. Because of air resistance, the speed must be ever so slightly slower in the second 1/4000th of a second. This means that the actual speed at this instant, if it means anything, must be a value less than the average speed over the first 1/4000th of a second and greater than the average speed over the second 1/4000th of a second.

We can continue to squeeze the value at a particular moment in time between every closer bounds. While the video does not say this, the limit of the average rate of change is simply that value that lies at or below every upper bound and at or above every lower bound. It is worth noting that a limit exists if and only if there is exactly one such value. Observe that defining the limit as that unique value that is less than or equal to every upper bound and greater than or equal to every lower bound is fully equivalent to the epsilon-delta definition of the limit.[2]

The Calculus Videos Project and Intellectual Need

In the recently published paper “Observing Intellectual Need and its Relationship with Undergraduate Students’ Learning of Calculus (Weinberg et al., 2024), the authors explored the effectiveness of these videos in instilling “intellectual need”.

Guershon Harel has defined intellectual need as the need to resolve “a perturbational state resulting from an individual’s encounter with a situation that is incompatible with, or presents a problem that is unsolvable by, his or her current knowledge” (Harel 2013, p. 122). He has asserted that “for students to learn what we intend to teach them, they must have a need for it, where ‘need’ refers to intellectual need” (Harel 1998, p. 501). The authors of this paper boil intellectual need down to something that is easy to query: Did a given set of videos create a sense of wonder and curiosity? The specific question that they asked was

“When you were working on this task, were there any parts where you genuinely were curious or were left wondering about something? If so, please state them in the box below; if not, please leave the box empty.”(Weinberg et al 2024, p. 115)

The paper is well worth reading for the care and detail with which the authors explore this question. With results from roughly 1550 students and 25 instructors spread over 14 institutions, the answer was: Not so much. I find it interesting that there was tremendous variation by instructor. Within one standard deviation of the mean, the percentage of students reporting wonderment and curiosity varied from 2.4% to 9.4%.

I see a couple of positive take-aways from this. First, the fact that this project has created videos that can instill intellectual need in any students is encouraging. It is a proof of concept that calls for a better understanding of what works and why. Second, online videos are still a poor substitute for an instructor who can motivate students. The instruction that is wrapped around the videos is more important than the videos themselves. We are a long way from putting flesh and blood teachers out of work.

Footnote

[1] Project funded under NSF grants DUE #1712312, DUE #1711837, and DUE #1710377 

[2] Note that this definition of the limit works beautifully for the limit of Riemann sums (taking upper and lower bounds over all partitions of the interval), is precise in a way that talk of “approaching” never can be, and is more intuitive than the epsilon-delta definition.

References

Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa). The American Mathematical Monthly, 105(6), 497–507.

Harel, G. (2013). Intellectual need. In K. R. Leatham (Ed.), Vital directions for mathematics education research. New York, NY: Springer. https://doi.org/10.1007/978-1-4614-6977-3 

Weinberg, A., Corey, D.L., Tallman, M., Jones, S.R., Martin, J. (2024). Observing Intellectual Need and its Relationship with Undergraduate Students’ Learning of Calculus. International Journal of Research in Undergraduate Mathematics Education 10:107–137. https://doi.org/10.1007/s40753-022-00192-x

David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. Information about him and his publications can be found at davidbressoud.org

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