The instructions seemed adequate: plot your data, draw a best-fit line, and extrapolate the line to find the y-intercept. Graph paper and rulers were available on a table at the front of the classroom. To me, the implication was crystal clear that the graph should be done accurately on graph paper, and the line placed carefully and drawn using a straightedge. Evidently, this is only clear to students who learned math during the pencil-and-paper era.
Education reforms of the 1990s required that math students be taught use of calculators. Educators thought that because students would always have access to a calculator, time spent drilling skills like memorization of multiplication tables and practicing pencil-and-paper arithmetic could be better spent educating students about the concepts of how and why the arithmetic works.
Now that these students are in high school and college, we are seeing the disastrous results: the ability to see the answers from simple arithmetic was what enabled students to easily think forward algebraically. Without this basic tool, students experience great difficulty in tackling problems that require them to choose between alternative pathways, because they have to follow each pathway step-by-step to its end, and then go back compare the answers. This cripples students’ ability to take a global approach, which is a crucial skill to develop for solving complex engineering problems, project management, and effective leadership. It also leaves students unable to make good estimates, which renders them unable to catch errors that result in ludicrous answers. This problem is beginning to receive some attention—at the speed of our educational system it will probably be addressed within ten or twenty years.
However, a related problem that appears not to be on the radar at all is the way students are taught to use graphs. In the pencil-and-paper era, graph paper was the way to get a good-enough answer to problems that could not easily be solved exactly with calculations. Graphs also provided a way to show what the data points and equations represented. The valuable skills of interpolation and extrapolation were the bread and butter of a lot of science and engineering calculations.
In the calculator-and-computer era, it is easy for students to get their calculators to exactly calculate the equation of a best-fit line, and use the equation to interpolate or extrapolate precisely, without their having any real concept of what is actually happening. To get around the pedagogical issues, math teachers have their students sketch graphs to show their understanding of what the data points and equations represent, but the “correct answer” comes out of the calculator.
The casualty of the answer-from-calculator, graph-as-illustration approach is that students simply don’t see graphs as a useful tool for performing calculations. To them, a graph is simply a picture of what the calculator is doing as a way to check their understanding. A sketch of the graph, with the axes, divisions, points and lines all drawn freehand, has always been sufficient for them to show their understanding. What I wasn’t prepared for is their assumption that if I ask them to produce a numerical result from their graph, estimating the result from the sketch is still good enough.
To be fair, I did allow students who had enough background in statistics to use their calculators to calculate the best-fit line and y-intercept, provided that they also gave the slope and correlation coëfficient that their calculator provided. I felt that this set the bar in an appropriate place—students who haven’t learned the statistics concepts are still reasonably likely to comprehend the graphical approach and why it works, but the statistical equations would be beyond what I could teach them in a few minutes out of a physics class. Unfortunately, the ability to point them toward a graphical solution also seems to be beyond what I can teach them in a few minutes out of a physics class.