“Proofs Without Words” are proofs based on a visual representation of a theorem which provides a convincing argument about its validity without the need for any accompanying text. The genre has been much enriched by the increased availability of computer animation. This is of course relevant to math education: many of the concepts we teach can be illustrated visually, including with clever and sometimes beautiful animations. In this post, I will discuss the place of such “visuals” in teaching.
First of all, I’d like to challenge the idea that those proofs should actually be “without words” in the educational context. In reality, proofs without words are mostly meaningful to people who already understand the result that is being illustrated. If we want them to support student learning, it is mostly through discussion, reflection, and writing. Our job as teachers is to help students come up with the words that explain these images.
For example, these images illustrate the “difference of squares” identity:
The point is clearer if you watch the GeoGebra animation on my website. Clearer for you, that is. For students, it only becomes clear if they think about, discuss, and/or write about the animation — perhaps as suggested by the prompts that accompany the figure.
This example illustrates a key idea about pedagogy. The fact that we (math teachers) appreciate the elegance of the visual representation in no way means that merely looking at it is helpful to students. It is a widely held belief that “if they can see it, they’ll understand it.” This verges on magical thinking. It is but one manifestation of one of the most harmful fallacies in math education, the illusion that students see the same thing as we do when they look at the same representation. In reality, what one sees is greatly influenced by what one already knows and understands.
This is not at all to deny the important role visual representations can play in teaching. Rather, it is an argument to think case by case about where those fit in our lesson plans. As a teacher and curriculum developer I had to learn that the hard way. Being enamored with my own creativity can make me forget the importance of actually listening to students. Here are some of the things I learned about “visuals” in my 42 years in the classroom.
Preparing to Watch
Often, work with physical materials before watching the animation helps prepare students to get the most out of the visual representation.
For example, the above illustration of the difference of squares could be preceded by a paper-scissors activity using grid paper. “Draw a square, different from your neighbors’. Draw a smaller square in one of its corners. Cut out the difference of the two squares. Make a single cut in the resulting figure, and rearrange the two pieces to make a rectangle.” Doing this will only take a few minutes, and it just about guarantees that students will have a better sense of what is happening in the animation. The discussion of the prompts should happen first about the paper model, and then about the computer version.
Along the same lines, paper-scissors activities could precede these animations: Area of a Circle and Sum of the Angles in a Triangle. An alternate preview of the latter is described in this post about tiling.
There are many, many visual proofs of the Pythagorean theorem. I share a few animations on my website, but again, I suggest that you precede them with work using concrete materials. One of the classic proofs can be previewed on the geoboard or dot paper. Another can be previewed with an activity on grid paper, and/or as a puzzle. (There are links to all this on my Pythagorean theorem launch page.) Once students understand those, you might send them to Steve Phelps’s amazing Pythagorean animations collection.
I have a number of applets that illustrate various algebraic concepts: Multiplying Binomials, Squaring a Binomial, Completing the Square, Three Representations of a Trinomial. Those are all based on the Lab Gear model, and should be used after students have worked with the manipulatives to make many rectangles and squares. Again: discussion and writing need to happen as part of the hands-on phase, and then again as commentary on the applets.
Just to be clear, I do not suggest the use of concrete materials because of a naive belief that those activities automatically confer understanding. Quite the opposite! The point is that actively doing something provides a better platform for discussion, reflection, and writing than watching something. There is a time for watching, but it comes later, and writing about what they see at that point provides a good way for students to assess how well they understand the concept.
Of course, concrete materials are not the only way to get students to engage actively. For example, there are worthwhile activities on the computer that do not require the sort of preparation I recommended above. Here are some examples.
Understanding rate of change more generally can be enhanced by visiting Doctor Dimension. In fact, those applets support the learning of many concepts about functions — 8th grade to calculus.
For most kids, learning does not happen merely by listening to teacher explanations, even if they are accompanied by beautiful images or clever animations. There are no shortcuts: if something is important, it is best to engage students intellectually in multiple activities that lead to the necessary reflection, discussion, and writing. Once that foundation is built, they can appreciate “proofs without words”, and put words to them.