Making the Abstract Concrete
Bringing Clarity to Calculus
Objective: To clearly convey an abstract concept that is difficult to grasp without visual aids.
Topic: “What is a derivative, exactly?
Exploration of abstract concepts
The ideas I came up with for the topic for this video included:
- Why is the sky blue? (Why is the sky not violet?)
- How does GPS navigation work?
- The doppler effect
- What is a derivative?
I settled on calculus because it posed an interesting design challenge — the concept is always communicated using jargon and abstract symbols, but the math is rather beautiful and elegant when explained visually through graphs. The other topics also proved to be fairly easy to explain using only text.
What is a derviative?
A function is a mathematical relationship between an input value (the independent variable) and converts it to a single output value (the dependent variable). Functions can be mapped on a graph. For example, when you map speed the speed of an object at different points in time, you’ve mapped speed as a function of time.
But what about acceleration? Acceleration is the change in the speed, relative to the passage time, or in other words, it’s the change is the function, with respect to an independent variable. Whenever you want to know how quickly some quantity is changing, you are basically talking about a derivative. This is easy enough to find out when you’re comparing two different points in time, but slightly more complex when you what to find the acceleration at a single given instant. This is when derivatives are most helpful. They can be represented on a graph too, as the steepness of the function. This steepness is called the slope of function that point, and is shown as a tangent at that point.
Refining the Script
The initial feedback on the script was that it still heavily used jargon. Words like “function” and “slope” are not universal concepts. One suggestion was to illustrate this through practical examples.
A function is a mathematical relationship between an input value (the independent variable) and converts it to a single output value (the dependent variable)Functions can be mapped on a graph. For example, the cost of production incurred by a manufacturer can be mapped to the number of units he is producing.
But what if you want to find out how much money you’d have to spend in order to produce one more unit? Whenever you want to know how quickly some quantity is changing with respect to another, you are basically talking about a derivative. In our example, you need to find the rate of change in the cost function, relative to the increase in an independent variable — the number of units produced. This is easy enough to find out when you’re comparing the cost at two levels of production, but slightly more complex when for a single given instant.
This is when derivatives are most helpful. They can be represented on a graph too, as the steepness of the function. This steepness is called the slope of function that point, and is shown as a tangent at that point. The derivative formula is simply a mathematical calculation for the slope of this point.
Moyer’s “Napkin Sketch Workbook”
We read the chapter of “Napkin Sketch Workbook” that outlines drawing structures. I tried to use the idea of Swim Lanes to talk about costs of production in a lemonade or cookie stand.
Of the other structures, the ones that seemed most useful were graphs (for obvious reasons) or to “just show it”.
Searching for References
In order to get a better grip on the subject, I decided to poke around the internet to see how others had tackled the subject. Unfortunately, most of the content I found was “classroom” content — videos of lectures anywhere between 20 to 60 minutes long. Most of them also assumed that the viewer has a grasp on simpler concepts like functions, slopes, and graphing equations.
Mapping the Concept
To aid the script-writing process, which proved to be more complicated than I expected.
After getting some feedback on the mapping as well as the latest iteration of the script, I also realised that my script, much like the reference videos, was trying to teach calculus, rather then explaining it (that is, focusing on the calculation more than the high level concept).
This brought to attention an issue I had been struggling with—the audience of the piece. Starting out, I had imagined the audience to be students of mathematics, those who have a grasp on basic concepts of algebra and linear equations, but dislike calculus as it is taught in classes. (Here, I’d like to reference my own experience as a class 11 students in a large school in New Delhi. I was good at math, but only realised what the general derivative formulae meant when my mother drew them out for me on a graph.)
A major issue is attached to assumptions like this, especially in the context of this project. Because mathematical aptitude is extremely varied, any assumption I make narrows down the audience to a specialised chunk, each with their own ability to comprehend the subject, not to mention their interest in it.
Bogged down by time constraints, I quickly to decided instead to touch upon applications of calculus in real life, and focus on why calculus is important, rather than how to calculate the derivative.
Because writing was still proving to be a complex task, I decided to attack the problem via storyboard, thinking through the script visually first.
It was far easier to write a script that I felt explained calculus adequately:
Say you’re driving to a friend’s house. It’s 15 miles away, and the trip there takes half an hour. It’s quite easy to figure out that the average speed will be 30 miles per hour.
In reality, the speed of the car is constantly changing for a number of reasons, (…) at times going faster (…) or slowing down. The average speed doesn’t give us all the information.
For example (…) Where did you gain the most speed? How far were you from home when you realized you forgot to switch the lights off?
This is where calculus comes in. It deals with values that are always changing, and is mainly concerned with two kinds of questions.
The first is how quickly an object is moving at any given instant. This is called differentiation. The steepness of a curve at a given point represents the derivative.
The other is how much it has already moved. This is called integration. The integral is represented by the area under the curve.
These concepts are used everywhere, even though you may not realize it — to build bridges, develop vaccines, and in thermostats.
But why should you care about calculus?
Even without calculating the limit of x as it approaches infinity, calculus can help us model complex situations which are affected by constantly changing forces. We use these models all the time — from figuring out the best time to leave the house to beat traffic, or how to invest money to get the most bang for your buck .
With a clearer understanding of the subject, we can develop more accurate models, and therefore a better understanding of the world around us as it evolves over time — and that is definitely a valuable skill to have.
Some explorations of the visual style:
Digital Story Board
Further edits to the story board were made digitally, while working in illustrator and creating assets.
The video doesn’t communicate the concept as well as I would have liked. The major issue with the final piece is a disconnect between the two objectives is was struggling to accomplish simultaneously: “teaching” the mathematics and “explaining” the concept.
The feedback I got on the video was extremely varied. While some thought the bit with the graph that touches upon the math was interesting and started to clear up a lot of things, others found that the were not able to grasp what was happening in the graph at all, and felt the illustrations were much more helpful. The mathematical ability of the viewers affected how they responded to the video (as expected).
Clearly defining the audience of the piece, and setting the expectations at the beginning of the video might have helped. Though the objective of the video was defined in the brief of the project, I should have defined the context in which it will exist before getting into the script at all. As always, context is everything.