Why study quadrilaterals?

Actually, I have thought and read a lot about this topic. A brief disclaimer: I'm going to present two pictures below, both of which can be validly used and neither of which is an "objective" correct pictures. However, one has strong advantages and I won't be shying away from promoting it.

Quadrilaterals

Studying quadrilaterals is an excellent example of how definitions are used in mathematics. Most people don't appreciate this the first time around (I certainly didn't, in grade school.)

When we learn about squares, rhombi, trapezoids etc when we're young, the definitions often spring from the head of Zeus with very little connection. Most students would probably not notice what similarities connect these families.

But anyone who goes a little farther in mathematics can start seeing this "family tree":

What is to be learned here? The idea is that of a "hierarchy" of properties. Squares are special rectangles, are special parallelograms are special quadrilaterals, and etc through other paths of this picture.

Actually, you can make this picture completely symmetrical by adding in a node between kite and quadrilateral and adding a line to it from "parallelogram", but we don't have a good word for that particular shape. But that is a slight digression: the point is that all of these things fit into a coherent picture based on their properties.

Why thinking about hierarchies is useful

The main strength of exercising this kind of thinking is that it strengthens one's logical ability. For one thing, it teaches that one should view things with a lot of properties as being special cases of more general objects (like squares being special cases of parallelograms.) Another thing is that if you prove something for a quadrilateral high up, then it automatically holds for quadrilaterals lower, because they are just special cases.

Thinking in terms of hierarchies of properties like this can be found throughout mathematics. For example, when studying real analysis, I distinctly remember making diagrams of implications between pointwise convergence, a.e convergence, uniform convergence etc so that I could understand how they related to each other.

One can do the same with groups and rings in algebra. There is a little chain of implications between rings in the top of this wiki article, and I personally have an extensive one with about 90 ring types on it making a very interesting lattice.

Really, this example with quadrilaterals is one of the simplest and most accessible demonstrations of a hierarchy of properties.

An unfortunate situation at present

Unfortunately, many primary and secondary teachers still insist on teaching the "Frankenstein hierarchy" depicted below:

Thanks to presentations like this, the most important pedagogical point is pretty much obliterated. A tiny bit is retained in the parallelogram-rhombus-square-rectangle portion, but I bet many students perceive that as a "lucky break" in the organization of an otherwise terrible picture. The only way you could make it more terrible is if you had "quadrilateral" at the top, and put the other seven items in a row with individual lines going up to "quadrilateral." The result is a very shallow hierarchy with virtually no use.

So, what can you take away from the first picture?

1. Point out that tracing a line upward is "relaxing" requirements, and tracing a line downward is "adding" requirements. These things are interrelated by their properties, not just randomly named garbage to torture children with.

2. Notice that if you prove something for parallelograms, then you have automatically proven the same theorem for all rhombi, rectangles and squares. If you prove something for all kites, you've also proven the same thing for rhobmi and squares. If you prove something for trapezoids, you proved it for parallelograms as well.

3. Moreover, let's say you have proven something for rectangles. The next question might be "but wait, is rectangles the best I can do? Might I be able to prove this for parallelograms too?" And maybe you will be able to do it! This shows that knowing how things are interrelated can give you ideas for improving results.

There is a book about this

In fact, you might want to check this out from the library: The Classification of Quadrilaterals: a study in definition. I hope it doesn't sound too dry because it's really quite interesting. It is, if nothing else, a book with a well told story of math education.

Among other things it contains a historical overview of how the definitions evolved, and better written explanations about the pedagogical value than what I am writing here. Finally, they provide quite good analysis and evidence that supports the use of the first diagram above over the second.

Here is a brief list of works I've found useful investigating this topic. If you read anything, take a look at one of Keedy's papers and the book I mentioned above (which is also below.

• Aydin, N. and Halat, E. (2009) ”The Impacts of Undergraduate Mathematics Courses on College Students’ Geometric Reasoning Stages.” The Montana Mathematics Enthusiast, ISSN 1551-3440, Vol. 6, nos.1&2, pp.151- 164

• Bagni, G. T. (2001). “An investigation of some misconceptions in High School students’ mistakes”. Learning in Mathematics and Science and Educational Technology, I, Intercollege Press Cyprus, Nicosia: pp 3-24

• Craine, T.V. & Rubenstein, R.N., “A quadrilateral hierarchy to facilitate learning in geometry.” Mathematics Teacher 86 (January 1993): 30-36.

• DeVilliers, M. (1994). “The Role and Function of a Hierarchical Classification of Quadrilaterals.” For the Learning of Mathematics, Vol 14 No. 1. FLM Publishing Association: pp 11-18.

• Fujita, T & Keith Jones. (2007).“Learners’ understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing.” Research in Mathematics Education, 9 (1&2): pp 3-20.

• Fujita, T. (2012). “Learner’ level of understanding of the inclusion relations of quadrilaterals and the prototype phenomenon.” The Journal of Mathematical Behavior 31: pp 60-72.

• Keedy, M.L. “The Fuss about trapezoids.” Mathematics Teacher 73 (October 1980): 488-489.

• Keedy, M.L. “What is a trapezoid?” Mathematics Teacher 59 (7) (November 1966): 646.

• Leung, I.K.C. (2008) “Teaching and learning of inclusive and transitive properties among quarilaterals by deductive reasoning with the aid of SmartBoard.” ZDM Mathematics Education. Vol 40, #6: pp 1007-1021. DOI: 10.1007/s11858-008-0159-z

• Maraldo, S. “Properties of Quadrilaterals.” Mathematics Teacher 73 (1) (1980): 38-39.

• Usiskin, Z. Griffin, J, Witonsky, D, Willmore, E. “The classification of quadrilaterals: A study of definition.” Information Age Publishing. (December 2007)

Well, as a mathematician who just had to redo his kitchen floor due to water damage, I can tell you that my knowledge of quadrilaterals came in really handy. I had to measure and cut pieces of laminate flooring to fit quite exactly around corners which were generally not square.

I think that learning "by rote" is generally not such a good thing, although it can be useful in some contexts. For instance, I used the pythagorean theorem in this project without deriving it from scratch. I also used some facts like a parallelogram with equal diagonals is a rectangle.