Is there any advantage of plotting graphs using traditional paper and pencil method?

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.)