"There is no one way"

Thursday, August 17, 2017

More on Geometric Construction

(Previous posts on this topic.)

I suspect that by far the most common introduction to geometric construction in US classrooms is a presentation by the teacher (or textbook) on various compass and straightedge construction techniques. "This is how you construct a perpendicular bisector. This is how you construct an equilateral triangle." And so on. "Now memorize these techniques, because you will be tested on them." Fortunately, some teachers take this further. Best case scenario:
  1. students learn how to prove that the constructions are correct
  2. students are asked to apply the techniques they learned in increasingly challenging construction challenges (e.g. "construct a square with a given side" after learning how to construct a perpendicular bisector)
  3. students are asked to create beautiful symmetric designs using those techniques
Such a scenario is certainly preferable to stopping after introducing the basic techniques, and it is far preferable to completely avoiding the topic. But I believe it is not optimal. In this post, I will try to present mathematical and pedagogical arguments for a somewhat different approach.

First of all, I would like to discuss the underlying mathematics. The essential mathematical concept underlying geometric construction is not the use of straightedge and compass. Interesting versions of construction have been developed for straightedge and the collapsing compass, and for the compass alone, not to mention for pedagogical tools such as patty paper, Plexiglas mirrors, and of course interactive geometry software. There are even interesting challenges involving only a (two-sided) straightedge which allows one to readily create parallel lines.

The essential concept underlying geometric construction is that of intersecting loci.  The locus of a point is the set of all possible locations of that point, given the point’s properties. The locus can be a line, a circle, or some other curve. If one knows two loci for a certain point, the point must lie at their intersection. In other words, given a figure, an additional point can be added to it in a mathematically rigorous way by knowing the locus (location) of the point in two different ways. Geometric construction is the challenge of finding such points and, in some cases, using them to define additional parts of the figure.

For example, the standard straightedge-and-compass construction of the perpendicular bisector is based on this theorem: PA = PB if and only if P is on the perpendicular bisector of AB. A compass is used to find all points at a distance AB from A (a circle centered at A, with radius AB.) That is one locus. Similarly, the circle centered at B with radius AB is the locus of points at a distance AB from B. These circles intersect at two points. Each of these points is equidistant from A and B, and therefore must be on the perpendicular bisector of AB.

The essential construction question is: given this figure and these tools, construct these additions to the figure in a mathematically rigorous way. In other words, it is a puzzle to solve, not a recipe to execute. In this view, the student is not a programmable machine. The student is a thinking human being. The pedagogical question becomes: what tools are most effective if we want to use geometric construction to teach geometric concepts in part through student problem-solving? To think about this, you need to assess the "overhead" a given tool entails, in other words the learning curve it requires. And you need to weigh this against the educational benefits of using the tool. As is usually the case in math education, there is no one way. I will present an approach that has worked for me.

In my view, one should start with compass, straightedge, and patty paper. (Patty paper is inexpensive tracing paper. Its use in geometry class was pioneered by San Francisco teacher Michael Serra.) The reason for including patty paper in this initial phase is that it makes it easy to copy line segments and angles. Of course one can do that with compass and straightedge, but it is too laborious and complicated for beginners. Including patty paper makes it possible to do interesting things right away. This phase of the work is mostly intended to get at some basic ideas. The physical challenges in using real-world compasses suggest that one should quickly transition to interactive geometry software if that is at all possible.

One way to do this is to use GeoGebra, which is free and works on every platform, including smart phones and tablets. GeoGebra is a huge and powerful program, and can be intimidating. A good strategy for introducing it is to start with a few of its tools, and give students time to explore it without a particular goal in mind, other than developing familiarity with the application. I used to work at a school where every student had a laptop, and the initial interactive geometry homework was to create something interesting in Cabri, using any tools at all. (Cabri is the interactive geometry application we used before GeoGebra. The same would work with GeoGebra) When students shared their creations, there were invariably some stunning images, and students developed a positive attitude towards the software. After that, I introduced whichever tools were needed for the work at hand (in particular, the crucial compass tool.) I did not find it necessary to hide any tools. This can be done in Cabri and in GeoGebra, but in this context I believe it is counterproductive. If there was a particular activity where I wanted to restrict students to using certain tools, I just told them which ones were allowed for that activity. Not hiding any tools almost guarantees that some students will discover additional tools, develop some curiosity about them, and teach them to others. Nothing wrong with that!

Once students are using interactive geometry software, many powerful labs and lessons become possible: construction challenges on the one hand, and informal explorations on the other hand. In fact, the availability of geometric transformations tools in those applications opens up many other possibilities, but I will have to save those for a future post.

-- Henri

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Links

I share my construction unit here: 8th grade | 9th-10th grade.

In the approach to proof in a transformations-based geometry course which I am developing with Lew Douglas, an informal unit on construction is very helpful in laying the groundwork for many foundational theorems. The reason is that proving many of those requires the use of what we call the construction postulates. Read more about this here.

And once again, here are my previous posts on geometric construction.

2 comments:

  1. I think this is great but I can't imagine most school's fitting constructions back into the curriculum given the current time constraints. When you look through the modern geometry course, one sees a bit of trigonometry, 3D volumes, analytic geometry and sometimes an unrelated statistics unit tacked on as well.

    This has come at the cost of de-emphasizing Euclidean synthetic techniques including constructions and proofs. I've gone through some thought experiments and I think you could just as easily do a Geometry II course with all the material that can't be covered and in many ways its a more natural narrative arc than the topics in Algebra II.

    Another fun idea, what if we delayed computational math until late elementary school and focused on Geometry for the beginning years?











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    1. Alas, the decline in the amount of geometry in the US has been going on a long time... I wrote about this in my mega-article on Common Core. (http://www.mathedpage.org/teaching/common-core/) US pragmatism and anti-intellectualism may lead us to continue in this downward slide... Note that the wonderful geometric construction app Euclidea seems to originate in Russia.

      Still, a unit on construction does help teach some of the surviving topics, such as congruent triangles, and as you can see in my Transformational Geometry page (http://www.mathedpage.org/transformations/), it is one key to a strong transformational approach to proof. So I say: make room for it!

      As for Geometry 2, I'm all for it. See my Space course (http://www.mathedpage.org/space)

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