## 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.

• I think the immediate question should be “what is your objective?” If your goal is to learn the effect of the coefficients, a technological tool is a huge boon. If your goal is for a student to be able to plot points or graph without aid, then the tech will get in the way by removing the challenge.

The more central question is “what should the objectives be?” – especially given the existence of the technology.

I’m tempted to say the by hand stuff is now irrelevant, but I’m aware how mental numerical computation encourages understanding of number in a way that calculator computation does not, and think it will be similar for graphing. Graphing by hand fluently is a challenge, and challenge is often the start of learning.

That said, I feel the technology should definitely be taught, and the learning of the technology should be taught, because it so greatly enhances our problem solving power.

• David Wees wrote:

"What should the objective be?" is a question that drives much of my thinking around mathematics education. What indeed! I don’t think we have much agreement about the broad objectives in mathematics education, but I wonder if we have some agreement on specific topics within mathematics education.

• Hi David.

I am torn on this question as well, if access to technology is not an issue. If it is an issue, then I believe paper and pencil is the best choice for students outside of the classroom. If access is not an issue, I believe the important aspects of sketching any function are best introduced using some form of technology. I prefer GeoGebra as you use above, although there are may fine alternatives.

Afterwards, once students clearly understand the key concepts such as slope, y-intercept, x-intercepts, axis of symmetry, asymptotes, holes, extrema, amplitude, phase, periodicity, etc., depending upon the level of the mathematics, they should be able to apply their understanding manually (paper and pencil / pen, whiteboard and marker, etc) to simple functions, as that is still the most prevalent means of communicating in small groups. More complex functions are typically modeled using software due to their difficulty.

While writing this comment, my son asked me a question about finding the domain of a function such as y = sqrt(x^2+5x+6). While I explained it to him algebraically initially, I elected to show him graphically using GeoGebra to cement his understanding. I could have sketched it manually for him, but even this simple function would have taken too long.

Also, a couple of months ago, a new pre-calculus teacher asked me if she should spend time having students learn how to graph various functions manually. My advice was for her not to have students spend too much time using manual graphing methods to discover the key concepts. Rather for her to introduce, or review, them using software, then have students work with both manual methods and software. Unfortunately, access is a challenge still for most students, even if money is not an issue. The need to install the software and develop adequate proficiency with it still seems to challenge high school students, so students end up manually graphing on most problems anyways. Even calculus textbooks that incorporate uses of CAS or graphing calculators on occasion throughout still emphasize manual methods for curve sketching using sign charts, etc.

For a specific example of the power of technology to aid in teaching graphing, I sent the following link, and comments, to the pre-calculus teacher I mentioned earlier stating it is an excellent tool for exploring how the inverse of a function relates to a function, and vice versa

********

You must try this out.  It is an eye opener for students learning (and relearning) the inverse of a function.  Be patient when going to this url.  It takes a few moments to load in your browser.

The software running this tool is GeoGebra, which I have shown students several times.  This specific “app” allows you to input any function and simultaneously observe its graph and the graph of its inverse.  You can also observe how the inverse function becomes a reflection of the original function over the line y = x, and restrict the range of the output.  Lastly, it allows you to note how two points, one on each graph, are inverses of one another, where one is point A at (x,y) and the other is point B at (y,x). http://www.geogebratube.org/student/m3211

Also, note that you can change the function to be any f(x) you desire.  Start out with a constant such as f(x)=5, then try a negative slope line such as f(x) = -x, then quadratics, cubics, exponentials: f(x) = e^x, rationals: f(x) = 1/x, trigonometrics: f(x) = sinx, etc.

You will radically improve your students understanding of inverse functions with this tool!

*****

Dave

• David Wees wrote:

Your comment about access is important. I haven’t really considered access at all in this post, and I still need to consider it; at least for another 10 or so years (I’m being optimistic, I think).

Also, I agree with you Dave, there are discoveries about functions that you cannot easily make using paper and pencil techniques; it just takes too long to graph some functions and one never gets to see the fun stuff.

• I want my students to learn how to graph by hand because of the same reasons you mentioned (technology can potentially make the job of graphing too easy) and what John Golden said above (Graphing by hand fluently is a challenge). Also important is that I want my students to APPRECIATE the technology available and realize that it is humans who made these applets and softwares to do the same work — which took them 5 minutes by hand — in a split second. Once proficient by hand, we switch to using graphing calculators (and really never look back), but right before entering in the data, I ask the kids to predict first what they think the function will look like, this helps them slow down and be more thoughtful. Studying parent functions and shifts would be dreadful without use of technology! The slider example you provide is exactly what I mean.
Thanks, David!

• David Wees wrote:

I think we can also consider working memory when looking at this issue. Someone who is pain-stakingly working on a pencil and paper graph is using a lot of their working memory. This may impede their ability to see similarities between the function they are working on, and the functions they have previously worked on. They will have to be reflective and ensure that they take the time to look through the functions they have graphed for similarities. On the other hand, with technology, most of the working memory of the students is freed up, and provided they are looking for patterns, they will be more likely to notice these patterns.

I wonder, could we create a graphing app that required an estimation of the graph before showing the picture? Could we create an app which is somewhere between the "too easy" of many current graphing applications, and pencil and paper techniques?