The phenomenon that is discussed in the paper is the tendency for people to switch off their brain when they use symbolic algebra programs (the paper specifically singles out Mathematica), but the problem is more widespread than this because it occurs with basic 4-function calculators (e.g. do I divide by 1.1 or multiply by 0.9?) or with advanced numerical software (e.g. why do the eigenvectors come out completely different for trivially different data?). This causes people to drop down into a calculational mode where they act merely as operators of the software/hardware, whilst not bothering to form a higher-level interpretation of what they are calculating (e.g. its physical interpretation).
This is like the difference between a worm's eye-view (e.g. low-level calculational mode) and a bird's eye-view (e.g. high-level physical interpretation mode). It is like the difference between having a local serialised view of each part of the problem that you are solving or a global parallelised view of the whole problem. It is like the difference between being a calculator or a visionary.
I know of people who are calculators but who can't see the grand picture, and who usually cannot communicate with anyone other than like-minded calculators. I know of people who are visionaries but who can't express their ideas in enough detail to carry them out, and who are highly articulate but whose apparent lack of rigour really annoys the ace calculators. I know of very few people who are both calculators and visionaries, but these people are really interesting to know.
The education system trains people to produce standard solutions to problems, so that everyone calculates using the same language. It is relatively easy to teach people to rote-learn standard procedures, and to then test them on this knowledge in exams. It is much less easy to teach people the skills that are needed to relate these calculations to the rest of the world, or so one would think.
My approach to counteracting the tendency to drop down into a calculational mode of thinking is to visualise what I am calculating; I try to avoid doing calculations that I can't visualise. When I draw a picture of what I want to calculate then the calculation itself follows almost automatically, and calculational subtleties (e.g. the epsilons and deltas) are easily resolved by referring back to the picture. I would go so far as to say that if I can't draw a picture then I don't understand what I am calculating.
I was amused to see that the principal example cited in the paper http://arxiv.org/abs/0712.1187 was one in which several students struggled to use Mathematica to evaluate the following integral (I have omitted various constants):
Integrate[x^2 Sin[x]^2, {x, -Infinity, Infinity}]
This is a clear example of students in calculational mode, who have adopted the worm's eye view of the problem as just being a calculation. They tried feeding the integral to Mathematica in various different ways, but without success. What they do not do was to diagnose their problem by simply visualising what they were calculating; it is not even necessary to know the physics behind this integral.
Using the same tool (i.e. Mathematica) that the students were using in their attempts to evaluate the integral, here is a plot of the integrand over a finite interval.
Normally, an integrand that is simple as this would not need to be plotted out explicitly because its behaviour is obvious from its structure, i.e. an x^2 factor that diverges times a Sin[x]^2 factor that oscillates between 0 and 1. Nevertheless, in this case I did plot it out as part of my ingrained habit of visualising calculations using Mathematica. The students should have been doing this as well, so I presume that they had not been very well tutored in their use of Mathematica. Had the students attempted even a rudimentary visualisation then they would have immediately realised that the limits of the integral they were trying to evaluate could not possibly be infinite.
As the visualisation habit becomes part of your way of working, you eventually reach a point where the solution of some problems comprises visualisation followed by calculation. There always remains a set of "difficult" problems for which your current set of visualisation techniques is inadequate, in which case you have to use pure calculation to get to a solution. But then you should be on the lookout for ways to capture the essence of your solution in a new visualisation technique.
Wouldn't it be nice if there was a standard set of visualisation techniques that you could use alongside the existing set of calculational techniques? If this set of visualisation techniques was carefully designed then it would be just as rigorous (and teachable) as standard calculational techniques. It would be a very interesting exercise to reformulate existing material using such a visual language; for instance, I made an attempt to do this sort of thing for the topology of the SO(3) rotation group here.
7 comments:
Please don't assume that visualization of mathematics is itself, necessarily, the correct approach. Using the same logic as you employ here, it could be argued that a conventional framework implies a lack of visualization in the same context that you use here.
For example, in reply to a comment attached to your blog entry of Sep 12 you express data compression of the "standard model" as if it were a *necessary* way forward on the road to achieving a more complete encapsulization of physical law. However, physical law might well be centred on a system that processes symmetries based on a more fundamental understanding of what "symmetries" and "processing" as we understand it actually are. The behaviour of fundamental processes and forces will then reduce to a collection of symmetries that are "processed" according to a much more fundamental (and possibly very simple AND counter-intuitive)mechanism.
The way that works needs a profoundly deeper level of "visualization" and intuition than the abdication of the "calculation" based approach that you describe. In fact if I were to be unkind, I could accuse you of attempting a purely calculational approach by using a standard framework of mathematics at all.
However, I am not unkind; I'm just pointing out that there are far deeper levels of abstraction and convention than implied by your example.
I agree with your comment about lack of visualisation in both conventional and symbolic algebra cases, but the tendency to focus on calculations alone when you have access to symbolic algebra means that visualisation gets less of a chance to be used unless you actively develop the habit of doing it.
As for "data compression", I am using this terminology in a very general way. I am taking the reductionist viewpoint that physics emerges from something deeper and simpler than our low energy measurements currently see. This imposes no prior constraint on the form of the underlying algorithms that we use in our fundamental physics theories.
This includes the possibility that physics turns out to be an accidental and contingent outcome of a simple underlying process (e.g. one point in a "landscape" of possibilities). I know that it may be the case that physics turns out to be a mess "all the way to the bottom" (there may be no "bottom"), and that physics is merely the way it is for purely anthropic reasons; I am just betting that this isn't the way things are.
As for the type of "visualisation" that this needs, I am using this terminology in a very general way. I don't limit myself to the types of visualisation that are immediately accessible using the visual intuition that we develop by living in our world of sensory experience. The example in the blog posting is artificially simple. That's only where we start. I add to that the various layers of more abstract visualisation that we can develop to extend our powers of visualisation. This is an approach that I have gradually built up over many years, and which is greatly helped by using Mathematica as a symbiotic extension of my brain.
Am I purely calculational because I use "standard mathematics"? I use whatever tools I find are productive to use, and I haven't yet discovered anything that I can use to replace standard mathematics, although I have found ways of using various existing mathematical techniques in new ways. Inevitably, it will be true that there are yet-to-be discover (Platonic, yes!) mathematical techniques that will make my life much easier, but I am realistic enough to see that I am not going to be the one to discover these, because am a user not a discoverer of mathematical techniques. That's why I did a theoretical physics rather than a maths degree!
Hmmm.. I struggled to understand how the integral could conere from reading your description, so I read the paper. Am I right in saying that the choice of constant to multiply by in sin ^ 2 is releant? Does multplying by pi n make the sin ^ 2 conerge faster than the ^ 2 dierges?
(sorry, broken '' and '' keys)
The integral does not converge when the integration limits are +/- infinity, but the students failed to recognise this fact which is obvious when you plot the function.
I wanted to point out in my blog posting that visualisation is a tool that you should use to get a feel for the inner workings of mathematics, including situations that are much more complicated than the simple integral here.
Perhaps this is my problem: the question asks for the integral from +/- infinity. Why is "the function doesn't got a finite integral over that range" not the correct answer?
The original point of my blog posting (and of the paper it was commenting on) was that the students failed to use simple visualisation to spot an error that they had made earlier on in their calculation, i.e. to use +/- infinity as the integration limits. The students had chosen these incorrect limits, not the examiner.
You are correct to observe that the function does not have a finite integral over the +/- infinity range, and this is what the students eventually realised after far too long. In detail, the actual problem the students were trying to solve is an expectation value in an infinitely deep square potential well problem in quantum mechanics, for which the range of integration is physically bounded by the edges of the potential well. Realising that the range of integration must be finite (i.e. bounded by the edges of the infinitely deep square potential well) was merely "debugging" the students' formulation of the problem.
Thanks for clearing up my confusion. I must have glossed over that section of the paper...
I have a new link for you, which you may find interesting.
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