We recently started in on sequences and the book opens the chapter with problems that require students to create a table and come up with a pattern. The first one was about multiplying bunnies. The problem stated that we start off with two bunnies and, each month, each pair of bunnies has two babies. The problem wanted students to figure out how many bunnies there would be at the end of 12 months, as well as to determine the pattern and what family of equations described the growth.
I had one student who fairly quickly completed the problem. One of the extension questions that the textbook suggested was for students to attempt to write an equation that represented the situation. This student, after he graphed the growth, recognized that it wasn’t a linear growth pattern. When asked if the situation could be represented by a straight line, he said no, it was something else. However, when he was set to the task of determining an equation for the situation, he always came up with a linear equation. When I asked him what the graph of his equation would look like, he knew it would be a line. When I showed him the basic exponential equation (y=b^x), he recognized that was the form he needed to use, but even by the end of class all of the equations he was creating were linear equations.
I didn’t expect him to come up with the correct equation, but it was interesting to me that he was constantly writing linear equations. If I had been able to sit with him, he probably could have been guided to finding the correct equation. It just wasn’t possible for me to do that. I know that it is a topic covered later on in the chapter, and in another chapter in the textbook, so I know that he’ll eventually have the opportunity to work with the concept more.
Something I’ve noticed for a while in both my math class and my chemistry classes is this aversion to graphing. In both cases, the graphs were simple linear relationships, but students will go out of their way to not graph.
I noticed it with my math students in a station activity that I had them work on. Of the nine stations, three of the stations were ones that required them to graph. When I reviewed the work papers later, almost all of the students had ignored the directions and simply solved algebraically. When we began sequences and were working to build rules for either arithmetic or geometric sequences, students still had no desire to graph. They would happily create a table, but were missing on what kind of relationship there was until they saw it graphically.
In my chemistry class, it became noticeable when there were two consecutive labs that had a graphing component. One student flat out asked me why we had to graph (though I think the underlying comment was “why do we have to do all this work?”). One of the questions on the second lab asked them to determine which value of the x-intercept was more accurate: the one from the calculations or the one from the graph. At least 3/4 of the students said that the graph provided the more accurate answer, and many of those responses came from the students who were vehemently against generating the graph in the first place.
I think what I find the most puzzling is why students have such an aversion to graphing. In math, of the many methods we discuss regarding solving a system of equations, graphing is certainly the easiest. In chemistry, graphing your data is often much easier than any of the required calculations, even when you have to create a best-fit line to go with the data. Perhaps students missed out on the discussion of why we graph data or given equations. I think it is important that students see the benefit of graphing and that it does tell you something, either about an equation or a given set of data.