Cactus: graphing software.

Hyde, Hartley

Students use complex educational software at many levels of comprehension. This article considers the ways students use graphing software and the expectations we may have of some commercial products.

At the most basic level students use the software as a tool to print a graph as quickly and easily as possible so that they can illustrate an important concept for a purpose that may be quite unrelated to mathematics.

If we are teaching mathematics and graphing skills we may wish to use software to generate a variety of related graphs so that the effect of changing a constant can be studied. At other times students will be asked to confirm what they have done as desk-work or we might want students to revise their learning by predicting what will appear on their screen before generating a graph. In all of these modes it is important that the graph produced on the screen reinforces the skills we have taught as desk-work.

For a parallel example, who would dream of asking their Australian students to use a word processor if the default speller was set to American spelling? How could we expect students to learn to spell if they are constantly being bombarded with inappropriate spelling 'corrections'?

Most of us are just as disturbed when graphing packages contradict concepts we are teaching. It is tempting to give up and teach students to use the same style the software uses because the students are going to follow the software's preset style anyway.

Unfortunately the graphing software available is often designed by programmers who have little memory of school mathematics. Autograph and FX-Graph are both designed by teachers and the better quality shows.

Let me describe what I expect when my students graph a function by hand. I expect students to use appropriate paper. In most cases this will be from a pad of squared paper but I may have to distribute paper that has been prepared for drawing log-log, semi-log, isomorphic or polar graphs. Such paper was once produced by photocopying pages from John Craver's must useful Graph Paper From Your Copier. Now it is easier to visit www.printfreegraphpaper.com. Software sufficiently flexible to draw this variety of graphs is rare.

Students should keep a sharp 2B pencil for drawing graphs. If they attempt to shape a curve using an HB pencil they will leave ruts in the paper and it is then impossible to draw a nice curve through the ruts. The equivalent requirement of graphing software is that it should be possible to print a smooth curve completely free of pixilation.

We used to expect that a well drawn curve could be read accurately to two-and-a-half significant figures (meaning that the third figure was unreliable). Computers should be more accurate than this, but often produce a curve too thick to read accurately.

Most graphing applications are now able to draw a reasonable quality, jaggy-free curve. The better packages let you control the thickness of the curve and allow you to export a much better quality graph that is good enough for publication. I was impressed to find that an app for the iPhone is capable of emailing a reasonably detailed graph.

I expect that the axes will be marked X and Y and have arrow heads on the positive ends of each axis. The origin should be clearly marked with a circle or the number "0". I have always been pedantic about the way students draw arrow heads. This shows students at an early stage how serious I am about getting the entire graph looking good. Even graphing software that does attempt to draw arrowheads settles for a ">" or a triangle on the end of each axis.

There is a tutorial that draws polar graphs at www.analyzemath.com/polarcoordinates/polar coordinates.html. It shows one polar axis as a ray pointing to the right from the pole. So--this is not just an Australian thing! Why then do foreign applications insert cartesian axes on polar graphs?

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How then can we blame students who think the software programmer knows more about graphing than their teacher? Version 4 of FX-Graph can be forced to draw a correct polar graph if we switch off both axes and draw the ray 8 = 0. Graphing applications seldom offer enough polar graph options.

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Scales on graphs should increment in multiples of ten of the integers 1, 2 and 5. It is a good idea to borrow some voltmeters from the physics lab and show students the types of scales used on instrumentation. The first graphing packages used typewriter characters on line printers. It was common to see scales like 6.82, 13.65, etc. The first generation of iPhone apps sometimes make the same error (see next page). Thankfully most packages now get this right but scales like 0, 3, 6, 9 still occur. We should always stress the importance of including the origin and the distortion that occurs if this is ignored. Excel graphs are prone to this error.

I expect students to write the function name somewhere in a clear space near the top of each graph. It is a tall ask that software can recognise a clear space, but most applications can now write formulas using the correct notation. It cannot be that hard to place a panel on each graph, with a correctly formatted, version of the function that the student can easily move to a clear space.

Very few graphing applications bother to identify asymptotes. In the worst cases (see next page) the software joins the last point calculated before a vertical asymptote to the first point calculated after the asymptote. Version 4 of FX-Graph does add asymptotes while Autograph expects that students will add their own vertical asymptotes.

We also expect that students can interpret what is happening at special points on their graph. When a graph cuts an axis at a point which is not labelled by the scale, the value on the axis should be added. If the graphing software is part of a CAS package it is reasonable that the labels should be in surd form when appropriate. We should expect similar labelling at asymptotes, turning points, points of inflection and points of symmetry. Such labels should be movable so that they can be moved to uncover the graph, the axes and the specific points described (just like the labels for Cabri). The PocketCAS application for iPhone overcomes the obvious limitations on real estate by inserting a small coloured square at each point of interest. When the user touches the point, the coordinates are displayed. Hopefully later versions will show the coordinates in surd form when appropriate. Students who use Autograph are expected to add their own labels.

Autograph has a scribble feature where students can anticipate what will appear on their screen when a graph is drawn. To appreciate the worth of this feature, take a piece of paper and graph the cube root function. Open WolframAlpha from your browser or iPhone and type y = x[conjunction](1/3).

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I wonder how many people who are capable of following the explanation at http://mathworld.wolfram.com/CubeRoot.html are ever likely to graph the cube root function? Educational software should try to anticipate the maturity of the likely users.

Above left you will see a graph generated by a free app for the iPhone. There was a time when graphing software on mainframe computers was worse than this. The curves were represented by patterns of asterisks that landed on the paper as close as possible to where a point should be. Students had then to draw a curve of best fit between the asterisks. The scales were just as bad and there was no attempt to manage asymptotes.

Above right you will see the same function graphed as I believe it should be graphed using modern computers. There is no package that I know off that can do all of this. This is a composite that should be possible.

Even then the graph falls short of what I expected my students to draw. A sketch graph had to fill half a page. If the graph was to be measured it had to fill the page with the best meaningful closeup. The curve should be thinner, but thin curves do not show up well in magazine illustrations. There was a time when students could gain half the marks needed to pass by getting the graph question correct. We all took graphing very seriously.

Progress has to be paid for. When we buy a software package we are also paying for the programmer to invest time improving the package. If we simply download free software that produces graphs that are close enough then that is what we will get back from our students: close enough is not good enough.

Such programming must cater for a variety of pedantic expectations. However, it is these expectations that set the tone for the effort we require from our students. The further away from home you roam the more variety you will find in those expectations. That is why the variety of settings and an ability to set defaults is so important if the software is to sell widely.

Setting up the software options to match your expectations may take time, but if you don't do that, it is a waste of time teaching your students any other way: over time, their graphing habits will be reinforced by the default software settings you have established for them to use.

Hartley Hyde

cactus.pages@internode.on.net