Conceptual Mathematics: Exemplified by Nontraditional Lines&Planes Problems

A series of nontraditional "Lines&Planes" problems appeared in MaplePrimes sometime back. These problems required careful interpretation, visualization, and eventually, computation. (The author of this webinar frankly admits his struggles to think about and visualize the relationships between the objects being manipulated in these problems.)
In a reprise of these problems we will concentrate on the conceptualization required, not on the mechanics of the solution of each. Listed below, they are called "examples" of the kind of reasoning that would necessarily have to take place prior to their solutions. It is that reasoning, that conceptualization, this webinar calls "Conceptual Mathematics," and wishes to highlight.
Too often, teaching, learning, and doing mathematics is seen as progressing through a sequence of skills, with focus being on the acquisition of manipulative skills. This webinar calls that view of "math" into question. It will show how much more important and useful it would be to "resequence concepts and skills" so that mathematical thinking amounts to manipulating conceptual building blocks rather than to mere computation.
Before the advent of modern software tools found, for example in Maple, skill development necessarily came first. Manipulative skills were necessary for exploring concepts; the expectation seemed to be that from the accumulation of skills, conceptual understanding would emerge. The failure of this expectation might be signaled by the number of "new math" projects that have come and gone in recent years. But how could the emphasis be shifted to conceptual reasoning first? What would allow mathematics to be seen conceptually before being seen mechanically as a collection of manipulations? Clearly, this webinar is suggesting that Maple is the agent by which mathematics can be approached conceptually. And this
webinar uses some nontraditional problems involving lines and planes in space as a way of illustrating this perspective.
The computational tools in Maple allow the objects and processes in the "Lines&Planes" portion of multivariate calculus to be "conceptualized" rather than reduced to mere manipulations. Processes such as solving a set of equations, defining objects such as lines and planes in space, intersecting and projecting objects, finding distances between objects, etc., are concepts, not just manipulations. The examples below, gleaned from the MaplePrimes forum, will be used to illustrate what we mean by approaching mathematics, not computationally, but conceptually.
(If nothing else, the examples are challenging, interesting, and far removed from the types of
Lines&Planes exercises found in the standard calculus text.)
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