Why don't we discard the traditional pencil and paper method of graph plotting in high schools and for freshers at colleges since there are many electronic devices doing the graphing? And please where do I get citation on studies that show students graphing ability

  • $\begingroup$ What information about students' graphing ability are you interested in? $\endgroup$ – Daniel Hast Aug 24 '16 at 19:41
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    $\begingroup$ Documented research on students' ability of choose reasonable scales, plot data correctly, determine slopes and intercepts correctly, connect information from graph with the theory behind the graph. $\endgroup$ – Mafuyai M.Yaks Aug 24 '16 at 20:32
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    $\begingroup$ There is almost nothing in undergraduate mathematics which can't be done better by computers. Why stop with graphing? Is there any point in teaching integration by hand when Maple does it better? Why bother to teach matrix multiplication? Even the ability to produce proofs in abstract algebra has been (to a large extent) automated. Why not discard the entire curriculum? My answer: the computer can't understand it for you. Hand-graphing is a tool for gaining understanding. The fact that it is an inferior technology isn't relevant. Running is slower than driving -- but it builds muscles. $\endgroup$ – John Coleman Aug 25 '16 at 13:19
  • $\begingroup$ I may form this into a proper answer later, but I don't have the documented research you ask for. Anyway, I do think that plotting individual points quickly becomes tedious, and if you want a precise graph then software does the job. But I think students need to know what graphs ought to look like and be able to sketch the general shape without plotting many points at all. This sort of thinking will help them solve problems that don't need the specifics that software might give them. They'll just KNOW what to expect. $\endgroup$ – DavidButlerUofA Aug 26 '16 at 2:32
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    $\begingroup$ Also, in my experience, students who have only ever used a calculator/computer to construct graphs just simply do not know what graphs ought to look like. I'm sure it's possible to develop this instinct even if you don't draw them by hand, but I think it will happen more naturally. $\endgroup$ – DavidButlerUofA Aug 26 '16 at 2:33

Knowing basic graphs is a simple way to geometrically understand a wealth of data about functions. If a student understands what a point on a graph means as it relates to a function then there is so much that can be remembered by a simple picture.

For example, $\sin \theta = \frac{1}{2}$. For the careless calculator user, they'll answer $\theta = \pi/6$ or $30$ degrees. But, if you think graphically, then you understand $\sin \theta = \frac{1}{2}$ is the intersection of two graphs. The sine function and the horizontal line $y=\frac{1}{2}$ intersect infinitely many times. The calculator produced zero percent of the total solution set. If the student trusts calculators and doesn't have a lot of mathematical common sense then they'll have a hard time telling when they have been tricked by the calculator.

Ultimately, calculators will improve. However, there will always be a need for students of math to understand math. Graphing by hand is in fact hands-on experience of math. When you create a graph it gives you a chance to get to know a function in some sense. Enough experience in this regard eventually brings the concept of a function to life. If all you do is punch formulas into a blackbox and like a robot sloppily copy a window for your "graph" it hardly allows you to internalize much.

Of course, there is a right way to use a graphing calculator and there is also a wrong way to graph by hand. I graphed by hand AND used a graphing calculator to explore possibilities. I don't see many students really doing this sort of exploration. I often see students in my classes who are graphically clueless. I assume this is in large part because they used graphing calculators too much in their youth.

Personally, the interplay between algebra and graphing is fascinating and I hate to see students deprived of their intuition for the sake of some supposed efficiency. Technology should be reserved for ugly graphs.

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    $\begingroup$ I do agree that data sets for experimental sciences are better left to technology. That sort of graphing is very different than what I am trying to get after in this answer. $\endgroup$ – James S. Cook Aug 25 '16 at 3:30

By replacing hand-graphing with computer graphing, we place a black box between equations and their graphs in the experience of our students.

Black boxes are bad for understanding. In fact, this is the entire point of a black box - to create a level of abstraction that frees a user from the gritty details of some particular task. Obviously there are times where this is appropriate. Lots of times. Who needs to know, while driving, how the fuel injection of their carburetor is working? Seriously considering this complicated mechanical process, while driving, would probably be dangerous (distracted driving, etc). Luckily all of our cars' complicated moving parts have been tucked away under the hood - a black box.

School mathematics is maybe the best possible example of a place where black boxes - the removal of complexity from the user's view - are not appropriate. (Or at least not universally appropriate). Here are some points to that effect, pertaining to the computer graphing vs manual graphing question.

  • An equation and a graph are different, but equivalent, representations of some relationship between quantities. Students who have had regular practice at sketching graphs will know and understand this, with or without being told. Students whose graph production has been dominated by computers may know this, but I promise that many will not. 'Graphs' will be one more abstract thing which must, for some reason, be produced for sacrifice at the math alter. And in any case, I believe that the students who do still know and understand this will have this understanding in a diminished way, as follows:

  • Manual graphing of unfamiliar functions or combinations of known functions reveals the particular 'forces' which inform the shape of the graph. When $ y = -5x + x^2 - 2^\frac{x}{5} $ is graphed manually, you can feel each term taking control of the action as you move along.

  • The above example will seem unreasonable to some - how should a student be prepared to graph a curve whose interesting bits occur near the origin and ~(50, 1300)? The answer is investigation, estimation, etc. Take some derivatives, find (even estimates for) maxima/minima/inflection points, do thought experiments on where different terms must overpower one another, etc. These 'equation debugging' skills are the product of cumulative experience, and, crucially, they are all highly transferable skills, and part of the general 'rigorous investigatory ethic' which in my mind is the entire point of mathematics education.

To put a question back to your question: why are your students producing graphs at all? If the purpose is to crank out graphs, then of course the computer is the way to go. But if the purpose is to produce something that enriches the mental model of the relationship, then the computer is lacking in some ways - the production itself lends a lot to the mental model of the relationship.

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I once worked in a factory that used "six sigma" to improve its production processes and yields. "Six sigma" involves a lot of graphs. In half of the factory, the manufacturing engineers used computers to prepare blank charts, and the line workers drew the graphs in the charts themselves. In the other half of the factory, the manufacturing engineers used computers to both format and prepare the graphs.

In the half of the factory where the workers drew the graphs:

  • The workers figured out which data collection was useful, and convinced the manufacturing engineers to simplify the data collection and graphing accordingly.
  • The workers noticed problems sooner.
  • More suggestions for process improvements were made.
  • More suggestions for process improvements were implemented.

The line workers' foremen were the ones who noticed that the teams that graphed their own data made more progress.

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    $\begingroup$ Could you please explain what the workers (in the first half of the factory) started with? You mention that the computers prepared blank charts, did the workers then also get functions they should then plot based on calculus, or did they just get data points, or was it anything else? Also, if you could provide a reference to read more about, that would be great. $\endgroup$ – yoniLavi Aug 26 '16 at 18:05

Students are expected to graph a quadratic equation, a parabola, and indicate on the graph the five points of interest -

  • vertex
  • 2 intercepts
  • y-intercept
  • symmetric point

If you graph this on a computer or smart phone, you are given the answers, same as plugging in the equation and asking a calculator to spit out the two solutions. I can't cite a study, but it seems to me the jumping to an answer with no work in between isn't as learning as it should be. (And I'm aware I just a assigned a different part of speech to the word learning than you'll see in the dictionary.)

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  • $\begingroup$ At the very beginning of learning graphs that may be true. But once a student has a grasp that a graph represents all possible solutions to an equation, why would it still be a benefit to graphing by hand as opposed to digitally? $\endgroup$ – Joey Kramer Aug 25 '16 at 1:33
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    $\begingroup$ @JoeyKramer Doing it by hand might help in understanding how to draw for example $f(x/2)$ or $f(x+2)$. $\endgroup$ – Tommi Aug 25 '16 at 6:11
  • $\begingroup$ @TommiBrander I'm not saying that it isn't beneficial to do it by hand at the beginning of learning a thing. But once they understand how to graph and what a graph is, why keep doing it? I would think that the underlying concept of what dividing the argument by 2 (f(x/2)) is doing to the graph is much quicker to reach (and is really the meat of the math in that situation) if we graph them on a computer/calculator. $\endgroup$ – Joey Kramer Aug 26 '16 at 13:11
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    $\begingroup$ @JoeyKramer Executing an algorithm can lead to further insight to its working, in my (unreliable) experience. It is the difference between seeing something happen and doing it yourself. (I don't think the comments here are a good place for having this discussion.) $\endgroup$ – Tommi Sep 2 '16 at 7:43

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