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Should students learn how to graph functions by hand?

Software to create graphs of all different kinds electronically is ubiquitous. There is no question in my mind that we do students a great disservice if we do not give them opportunities to learn how to use at least a few of these programs. That being said, does the use of these programs potentially make learning concepts related to graphing, or through graphing, more difficult than it would be if the students used traditional paper and pencil graphing techniques?

It should be clear the skill of graphing is different when using technology. Some tasks, like choosing an appropriate scale, which are typically difficult for students are much easier using technology. One can replace time spent learning how to space lines correctly on paper with time spent learning how to choose the space between the lines in the software. In either case, time should be spent on visual design principles and why we might want horizontal lines in the first place.

One problem with using technology for graphing, especially when the purpose is to use the graph to determine a relationship between variables, is that the technology can potentially make the job of graphing too easy. A mind, recognizing that a task is easy, can potentially put insufficient energy into the task, and the mind’s ability to distinguish patterns is reduced. This born out by research on the effect of font type when people learn through presentations, and by Veritasium’s research on effective science videos. A mind insufficiently challenged, either by the task of character recognition, or on it’s misconceptions, is a mind that is less likely to learn.

On the other hand, some very useful learning tasks are so difficult to do when using paper and pencil techniques as to be pointless to do. These tasks can be much more manageable using technology. For example, the standard equation of a parabola, y = ax^2 + bx + c, can be explored through a graphing program. What effect does changing the values of a, b, and c have on the equation? Try this task with paper and pencil and then with technology to see why I ask students to do this task with technology. See the applet below for an example of this (requires Java).

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com

There are many graphing tasks which are typically learned very poorly by students. From my experience, there are many middle school students running around with muddled concepts of the equation of a line, wondering what the ‘x’ in the equation is for, and being asked to learn mechanical tasks related to this y = mx + b. I usually find that some exploration of the equation of a line in graphing software typically clears up at least some of these misconceptions.

Some tasks when done with a graphing program can disguise important concepts. There is something to be said for visually placing points on the coordinate plane for understanding coordinate systems, for example. I don’t think it matters if one uses a mouse for this activity or if one draws the point with a pencil, either way, the student has carefully chosen the location of the point. If one types the coordinates into a text box, and just sees the point magically appear, I suspect that one will find learning coordinate systems more difficult. This suggests that the choice of software matters, since some software will let you plot individual points "manually" and some software does not.

I suspect that these issues are less about which technology we use, since paper and pencil is itself a form of technology, and more about how we interact with the technology when learning graphing (or any other mathematical technique). We need to think carefully about what the technology allows us to do, and what underlying concepts we want students to learn. It may be that some concepts that used to be fundamental no longer are with the new technology, and other concepts become more important to learn. I suspect that insufficient research has been done on how pedagogy should change with the use of various technologies in mathematics, particularly ones that change so fundamentally the task of the student.

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