The mathematics education literature on student errors has documented how sometimes what students learn can be overgeneralized as they solve other problems and can even occasion errors. I was reminded of this as I puzzled over something I observed some Geometry for Teachers’ students doing as they worked on the problem of constructing a triangle congruent to a given triangle. 

Students were asked to create a triangle DEF whose sides would be congruent with those of a triangle ABC which was given. One student, “Angie,” produced a construction like the one below.

Angie’s construction was incorrect. She constructed DE to be congruent to AB and EF to be congruent to AC, which was promising. If by the notation (X, YZ) we mean the circle of center X and radius YZ, we can observe the following. Angie found Point F to be the intersection of the two circles (D, AB) and (E, AC), which meant that F belongs in circle (D, AB) and hence DF would be congruent to DE. Why Angie thought that DF would also be congruent to CB was not apparent to me. 

Then I realized that Angie and her classmates had just learned how to construct an equilateral triangle with straightedge and compass. In this construction, students had learned what Euclid does at the very beginning of Book 1. Euclid creates two circles, using the extremes of a given segment as centers and using the segment as radius. The third vertex GeT Activity of the equilateral triangle is found at the intersection of these two circles. Other than the fact that (E, AC) had a different radius, the procedure was very similar. There was also the alluring presence of a new point of intersection—two, in fact. If those points were not meant to be the points sought, what could one do with them?

Of course, the correct point F would need to be found by constructing circle (D, BC) and intersecting this circle with circle (E, AC). One could nudge students in that direction by, for example, asking them where to find all the points at a distance BC from D. Hopefully, that question would get them to think that there is a third circle that needs to be constructed.