Teaching Congruence, or Are You Happy, Professor Wu?

I first ran into the writings of Professor Hung-Hsi Wu in ed school, and never forgot him. How often do you see a math professor hyperventilating about elementary teachers as the abused children of math education:

For elementary teachers, there is at present a feeling that they have been so damaged by their K–12 experience…that we owe it to them to treat them with kid gloves…. Those that I have encountered are generally eager to learn and are willing to work hard. The kid-glove treatment would seem to be hardly necessary. …There is another school of thought arguing that for elementary teachers, one should teach them not only the mathematics of their classrooms, but at the same time also how children think about the mathematics. Again, I can only speak from my own experience. The teachers I observed usually had so much difficulty just coming to terms with the mathematics itself that any additional burden about children’s thinking would have crushed them.


There are ample reasons to believe that at present most teachers are operating at the outer edge of their mathematical knowledge. Now when one finds oneself in that situation, one is prone to being tense and inflexible, and is consequently not likely to create a friendly atmosphere for learning. There should be a study to look into how much of the so-called math phobia in this country can be traced to this fact (especially in elementary schools). The other simple reason is that no matter how elementary the topic, some students would bring up deep or at least non-elementary related questions.2 If the teacher fails to answer such questions too often, the students’ confidence in the teacher is eroded and, again, a non-productive learning atmosphere would result.

Abuse victims grow up to be abusers, you know?

He trains elementary school teachers in these special recovery workshops for abuse victims:

The main difficulty with the Geometry Institute, and the relative lack of success thereof, was the teachers’ unfamiliarity with anything geometric. With but mild exaggeration, some teachers literally trembled at the sight of ruler and compass or when they were handed a geometric solid. As mentioned in the preceding paragraph, we were prepared for teachers’ being ill-at-ease with geometric reasoning and lack of geometric intuition, but not for the degree to which both were true. School education in geometry is in deep trouble.

I know many who read Professor Wu and take his descriptions at face value, using him as evidence of teacher stupidity. Anyone who believes a math professor—a mathematician, that is—can be objective about elementary school teachers’ math knowledge has never met any mathematicians. Read the above link in which he describes the bare minimum of what fifth grade teachers need to know in order to prepare their students, and ask yourself if elementary school math teachers have ever known that much. How on earth did we all get to the moon and create the internet?

Still, his fulminations about geometry caught my attention over the summer, when I read this:

As to the subject of school geometry, the problem is that if universities do not teach it, or do not teach it well, then the only exposure to school geometry that geometry teachers ever have will be their own high school experience in geometry. The latter of course has been scandalously unsatisfactory for a long time, to the point where many school geometry courses cease to prove any theorems.

Well, hang on. As I’ve mentioned before, I don’t teach proofs and rarely prove theorems. I also typically dump transformations, most of construction, and solids. But I’m teaching non-honors geometry in a Title I school, where geometry is but a brief respite before the kids are dumped back into Algebra Hell. As a tutor, I’m very familiar with geometry as it is taught in the high-performance schools in the area, which are some of the highest performing schools in the country, and they are getting proofs a-plenty. Moreover, geometry hasn’t been “scandalously unsatisfactory” since we began forcing everyone into college prep math (an absurd notion in and of itself). What’s scandalously unsatisfactory is the idiocy of trying to teach proofs to low ability kids. But I digress. The point is, I dispute his notion that geometry is taught badly at all schools. A highly modified version of geometry is taught at some schools, for a very good reason, and given the kids involved there’s little evidence that it’s doing any damage, and regular “old school” geometry is routinely being taught to the top students.

But I was curious, nonetheless, as to what dire notions Prof Wu had about geometry as it is taught in schools, and so I googled. Teaching Geometry According to the Common Core Standards is the first article I read, but this interview with Rick Hess spells out the key point with the fewest words (would that I valued the same behavior in myself):

As another example, when state standards ask that the concept of congruence be taught in middle school, they do not realize that what students will end up getting is that congruence means same size and same shape. As a mathematical definition, the latter is completely unacceptable.

I sat up straight at this. Remember, I’m not a mathematician (which of course means that Prof Wu wouldn’t let me near fifth graders, much less high schoolers), and had never given much thought to congruence and similarity until I began teaching public school, as opposed to reviewing similarity for college admissions tests (congruence is not a tested subject). At that point, however, the book definitions seemed a tad circular. Check out the CPM section on similarity. It uses dilations, which (as you will see) is the right start, but at no point does the text explain the link between similarity and dilation. It’s all “cut out the figures, talk with your group, what’s the same, what’s different” crap. And when I taught geometry two years later using Holt, the official definition of congruence, straight from the book, was “Two polygons are congruent if and only if their corresponding sides and angles are congruent”. Really? Polygons are congruent if they are congruent?

I actually apologized to my class last year when we got to that point, because I hadn’t noticed this bizarre definition until the day before I taught congruence. I said yeah, a tad circular, and I hereby promise to investigate but for now, let’s go with it. (Sorry, Professor Wu, but most of them are never going to use congruence again.)

So here’s Wu saying that yes, this is a problem? How does he want it to be taught?

Holy Crap. Rigid motions? Isometries? (not isometrics. I make that mistake all the time) The sections I ignore? They have a purpose? I should have majored in math. No, not really.

This was a huge revelation, and incredibly easy to put into action. Most students got two “transparency triangles” and a white board. Some students used the graph paper with the transparency triangles. Three students (strong students) used the white boards with different colored pens and no manipulative. I wanted to see which methods worked best and if any problems came up with a particular method.

Day 1: Introduce translations and reflections, moving to increasingly complex reflection equations. Emphasize that reflections occur over a line; evaluate the change in coordinate points with pre-image and image, and then start calculating new coordinate values without using the manipulatives first.

Day 2: Rotations (by far the most difficult, in my mind). We focused on rotations of 90 degrees, and on reviewing the definition of perpendicular slopes, since that’s how the students found the new point—find the slope from the point of rotation (usually the origin) to the vertex to be rotated, convert to the perpendicular slope. Stressed that rotations were around a point, in contrast to reflections.

I foolishly didn’t take pictures in class that day—or if I did, I can’t find them. Here’s roughly what it would have looked like for a student using graph paper and the manipulatives, except they used colored pencils for the different slope connections. This is an example rotating a triangle 90 degrees clockwise.

First step (click on image) was to identify the slope from the point of rotation to each vertex. Then they identified the perpendicular slope for each of the rotation points—reinforcing perpendicular slope relationships being a big ol’ secondary point of the lesson—and sketched that line in as well. The students used different colors for each vertex, so they could easily see the before and after for each point, and recognize the 90 degree nature of the turn.

Then, with the points sketched, they did the actual rotation. Put the triangle on the original point, hold the manipulative at the point of rotation, and turn. Voila.


And then put the first manipulative in the original position to see what the rotation before and after looks like.


I’ve always had a difficult time teaching rotations, but the manipulative really helped.

I end the day pointing out that transformations preserve both degree and distance. They can see this because they are using identical manipulatives, but I have them calculate some side lengths and slopes to confirm.

Day 3: Congruence
And now, congruence. Instead of a circular definition, I have a clean syllogism:

If Polygon A is congruent to Polygon B, then A can be mapped onto B using a series of transformations. If the figures can be mapped into the same space, then their corresponding angles and sides are congruent, because the mapping preserves degree and distance. Therefore, congruent polygons’ corresponding angles and sides are also congruent.

From there, I go onto congruence shortcuts and proofs, blah blah blah. But it started much more cleanly. I taught transformations, reviewed perpendicular lines and other coordinate geometry formulas, and linked it all to congruence in a meaningful way.

A few weeks later, it was onto similar polygons. Again, instead of just saying “Similar polygons have congruent angles and proportional sides”, I can link it to dilation.


Day One: Review of Proportionality, then onto dilation

The kids did straight dilations as well as transformations and dilations in combination. I started with straight dilations, because I wanted the students to confirm the elements of similarity. The kids generally remember that parallel lines have the same slope, but I thought it would also be useful to see the transversal relationships with the parallel lines. We could prove, algebraically, that the lines of the dilated triangle were parallel to the original, and we could then extend those lines to prove that the corresponding angles on each triangle were congruent. Here’s an example (again, one I just sketched up) that shows how the kids determined the angles were congruent.


The kids colored the corresponding angles—there are three in each case (one of the green ones in my image is an error, you can see I xed it out, just too much hassle to draw again).

So again, the point was to algebraically and visually confirm the parallel relationship, and then follow the dual sets of parallel lines and transversals to confirm that the angles are congruent.

I had them do a combination transformation/dilation, confirming that order didn’t matter, and identifying which of the isometries had the parallel relationship.

Day 2: Review of Dilation, then onto Similarity.


Linking isometries to congruence and similarity was so much better, and whenever I tell math teachers about it they go oooh, ahhh and think about trying it themselves. And yet, I can’t point to why it’s so obviously superior. I can’t swear that my students learned congruence or similarity more thoroughly—in fact, I think they learned it as well but not any better than my students last year.

But it’s just more….cohesive, maybe? Not only am I finally linking in rigid transformations, which I never gave more than a quick review at test time (two of the three are intuitive, rotations are tough), to the rest of geometry, and creating an organic reason to review the relevance of perpendicular and parallel lines/transversals, but I am also linking both of these concepts to congruence and similarity, rather than just giving that annoying circular definition. While congruence doesn’t have much relevance past geometry, similarity runs through the next three years of math in a big way. So anything that makes the introduction more meaningful is probably a good thing. Moreover, transformations are easily grasped by even weak students, and their interest kept them going through the review of perpendicular and parallel lines.

None of this required complicated worksheets. I taught congruence using notes and Kuta worksheets; for similarity we used the book (Holt). I taught the transformations with boardwork–I really, really could have used a document camera, but that just came a couple weeks ago. Still, the kids got it well.

The really important thing, though, is that I have to feel mildly guilty about mocking Professor Wu. Next time I see him at Math Survivors Anonymous, I’ll grovel.

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