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
