Derivatives in the Dorm -- By Kathy Jaqua

Living in a dorm is one of the hallmarks of college life.  That’s where you learn to share a small space, share a bathroom, and share the world of your roommate.  In reality, the dorm becomes a center point in the landscape of daily life.  So I wasn’t surprised when my Calculus I students found mathematical selfies of the general concept of a derivative at the dorm. While looking at the submissions, I saw three basic types of photos submitted to represent the idea of a derivative:  1) illustration of slope of a tangent line; 2) illustration of a graph of a function (in an algebraic sense) along with the graph of its derivative; 3) rate of change.  As one student wrote, “The next category Definition of a derivative’s pictures mostly illustrate tangent points on a graph with people finding objects that look like the graph. One student … chose to represent the idea of a derivative with the change in leaves over time.”  Come take a quick walk with me through the dorm and see what students photographed and how they and their classmates described the general concept or definition of a derivative.

Come in, come in…

Beginning at the entrance to the dorm, we see the first type, illustration of slope of a tangent line, which is a common beginning point for students in calculus.  It puts together the idea of slope of a line (something that they first encountered in middle grades) with tangents (something from precalculus) to explain change on curved graphs. 

The notions of positive slope, negative slope, zero slope, and no slope are fundamental to the idea of derivative and are often the easiest for students to understand visually.  The author of this selfie added line segments to the photo to show exactly what he saw in relation to slopes of tangent lines as he walked into his dorm. Students who chose this photo as best agreed that the addition of the line segments made clear his vision of derivatives as slope of the tangent line. “A commonality among many of the pictures was the slope of tangent lines that were drawn onto the pictures. Those who explained the slope of the tangent lines represented the category better than those who stated, “Using the derivative, we can find….” The best picture that illustrates the definition of a derivative would be the one using the arch ways and columns to describe derivatives because it explains all the concepts of what a derivative would look like for each tangent line slope. They even added what the derivative would be of a vertical line.”

I’m sure that I have sketched these same relationships during class.  Because I am not an artist, I was excited to see how the author could take my crude drawing from class and see it in the façade of his dorm.  

Don’t mind the mess…

Venturing inside the dorm, we come to a dorm room where clothes, shoes, books, and the occasional pizza box may end up on the floor.  The second type of derivative representation, the illustration of a graph of a function (in an algebraic sense) along with the graph of its derivative, included a classic equation f(x) = x² illustrated using the shoes that were dropped on the floor.

This photo was chosen by many students as the best representation of the general idea of a derivative, partly because they liked the shoes, but also because of the inclusion of one of the most basic derivative rules. 

“The best one in this category is probably the one with the tennis shoes, because even though these laces were positioned to match the description, they could have naturally land on the floor like that.”

“This is one of the better pictures for a definition for a derivative, the person who took this picture was smart enough to do so and it is easy to see the function and its derivative that he has made with the laces. This photo caught my eye right off the bat because I found it the easiest to understand out of all of the pictures. It is super easy to see the function and its derivative in the way that the laces are positioned.”

Being able to connect the algebraic form of a function and its derivative with a graphical form was the point of this photo.  I could also see the embodiment of a test question from early in the semester that had students draw relationships between graphical and algebraic forms of derivatives--just not with tennis shoe laces!

Let me see you out…

Finally, it is time to leave the dorm.  Outside of the dorm we find the third type of photo representation of the concept of derivative, rate of change.  Students were impressed with this photo and found it compelling. “The purpose of the pictures in this section is to show how something changes in term of something. In this case, the change of leaves in terms of time (DL/DT). This is my favorite picture because it is a great example of a derivative; how leaves change over time.”  

“The best picture that helps represent this category is the one with the different seasons of leaves. This is a good example for this category because it shows that the leaves depend on the time of season before they change and a derivative is relying on something else or has respect to something to help it change.

I thought this photo was very creative, while also showing a clear concept of rate of change.  The planning required for the author to take essentially the same mathematical selfie repeatedly over time (in this case a semester) clearly showed her understanding of the idea of change with respect to time.

Our tour of derivatives and dorms is now complete.  So until next time, in the words of one student, “The world is full of mathematics if you simply just open your mind to the things around you. The mathematical selfies project helped me to open my mind to the world around me and see it with more than just my eyes.”

Kathy