The
idea of limits in mathematics seems to challenge many students when they first
encounter it. The mathematical notation
of a limit:
Outcome
1: Selfies
mimicking a graphical representation of limits
It
is not a surprise that because mathematical
selfies are visual, most students equated them with graphical
representations. Some students created
simulations that resembled typical graphs involving vertical or horizontal
asymptotes such as the cords positioned to mimic graphs with asymptotes as in
photo 1. Others found examples that mimicked graphs in objects around them.
Photo 2 shows several of those types of photos.
In
discussing their photos, students revealed a range of understanding of the
concept of limit. The author of the
photos of the cord (photo 1) revealed a somewhat muddled understanding of the
idea of limits. He describes the photo
on the left as “The next picture is of a paracord, and it represents a limit as
x goes to infinity. It represents that because it goes up which in return
creates a vertical asymptote meaning that the line (paracord) goes to infinity.” This description does not show that he distinguishes
the role of x and f(x) in the limit process as both could
be seen as going to infinity, but his description really inplies x going to a constant where the
asymptote occurs and y going to
infinity. Looking at his comment about
the photo on the right (photo 1) convinces me that he does not understand those
roles. “This next picture is also of a
paracord and it represents a limit as x goes to a constant. It represents that
because it levels off as it goes down making it a constant value.” In this description he clearly mixes the
roles of the x and y coordinates on his imagined
graph.
In
describing the copier shown above (photo 2), the author said, “This photo also
illustrates a limit as x approaches infinity, if you look at the part directly
above the sticker stating “questions or concerns”. A limit as x approaches
infinity means that the function will continually get close to a certain output
value (y-value), but will not actually reach that value. This illustrates that
because, on an XY graph, this is the shape that would be produced by graphing a
limit that approaches infinity.” He is
equating the shape he sees with a shape as an “XY graph”, and he understands
the role of the “X” and “Y” coordinates in his example as they relate to the
limit process. He does reveal a bit of a
misunderstanding when he implies that the function value can’t become constant
in the limit process, but it may just be in reference to the example he is
using. He also trips up on the language of limits when he states “a limit that
approaches infinity” when he means the limit as x approaches infinity. The author of “Subway Y” (photo 2), on the
other hand, confuses the idea of the limit as x goes to a constant with the
idea of a limit of a function that is a constant as did the first student with
his paracords. His photo shows the
function approaching a constant on the top edge of the curved section of the Y,
but he labeled this mathematical selfie
as “Limit at a constant”. The typical
language that mathematicians use with limits has not quite been perfected by
this student because he continues with the description: “The Subway “Y”
represents a limit at a constant because the slope gets closer to zero as it
approaches the “peak” or constant.” He
suggests the idea of the limit of the
function approaching a constant, but
he doesn’t see that as different from the limit at a constant.
The author
of “Christmas tree” and the author of “scissors” (photo 2) both describe the
idea of a limit at a constant using the definition of a limit based on left and
right hand limits. For Christmas tree,
the author says, “In my picture, the limit is the star, from both sides.” The author of scissors says “As x travels towards
a constant is when a point on a line is checked from both the left and the
right side to see if they both travel to the same point. If they do this shows
that x does in fact travel to this point. These scissors represent x traveling
to a constant by the blades being the line if you travel down both blades they
meet at a single point this point represents the constant.” There is some ambiguity (especially for the
scissors) in whether these authors distinguish between the constant that x approaches from each side with the
constant that the y value
approaches. Once again there appears to
be some confusion, at least in the discussion, of the distinction of a limit at a constant and the limit is a constant.
Outcome
2: Selfies
that don’t depict limits as graphs
One
student authored two mathematical selfies
that were more conceptual rather than mimicking a graph. In these mathematical
selfies, he reveals his knowledge of the distinction of the roles of the x and y components in a mathematical limit as well as the general concept
of a limit. The first, a photo of the
WCU clock tower (photo 3), reveals this distinction directly: “when the minute hand (or the x value)
approaches the twelve, the hour hand (or y value) approaches the next number on
the clock.” He does note “when a
function is shown on a graph, you can see where the points on the graph are
that the function intersect. As the line gets closer to a certain point that
means that the function is approaching that point.” So while his point of view is somewhat graphical,
he can translate that view into a more conceptual one when he discusses the
relationship between the minute hand and the hour hand. The same author also showed clear
understanding of a limit as x goes to infinity.
As part of a robotics project in an engineering course, he created a
robot powered by an Arduino board (photo 4).
He describes how this project illustrates the idea of a limit as x goes
to infinity in a truly nongraphical way.
“In this picture, you can see a robot that has a set function to drive
forward and avoid walls. This is achieved by looping the code on the Arduino
board, by doing this the board will constantly look at the code and infinitely
run it. At no point will the robot stop moving around unless an outside force
has tampered with it.” His focus on what
would happen to the robot if the code ran infinitely is a clear example of the
concept of a limit as the input (in this case code based) goes to
infinity.
Finally, what do all of these mathematical selfies reveal about students’ understanding of the
idea of limits? Based on typical test
questions completed during this same class, all of these students (and most of
the class) could calculate basic limits using a variety of techniques. They also could estimate limits based on
graphs and tables. But in the end, their
mathematical selfies revealed two
details not shown by typical testing.
First, as a class, they had a definite bias towards a graphical
understanding, which may reflect my own bias for introducing these ideas in
graph format. Second, I have some real questions on how well they truly
understood the concept of limits and the roles of inputs and outputs in the
process of a limit. Certainly their
tenuous knowledge of typical mathematical language of limits is shown. Mathematical language is very precise and
students often don’t see the crucial difference created by changing one small
word. In their descriptions of their mathematical selfies, students showed
confusion between the ideas of limit at
a constant and a limit that is a constant. They would say one and show the other, or
they would say them both about the same aspect of the photo. While I often note lack of precision in
students’ language during class discussions, these mathematical selfies allow me to more accurately assess their
confusion of these two distinct ideas involving limits. It also leads me to wonder if they can
distinguish among the six different potential limit problem types. In the end, I am left with more questions to
explore with my next class and with several areas to consider as I plan
activities, assignments, and assessments.
Kathy
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