I’m frustrating my students in ways that I don’t want to. I’m not sure exactly what to do about it. In geometry there’s proof. With proof comes a certain logical structure. Once you know this structure, it is terribly difficult to unknow.
Currently we are dealing with similarity, which involves using SSS, SAS, and AA postulates to prove whether or not two triangles are similar.
Suppose I gave this image to a student and asked them to find whether or not triangle FES and triangle GHS were similar.
Let’s suppose the student divides 54 by 24, and also 58.5 by 26. Both times the student gets 2.25 as a solution. The student assumes this is a scale factor and applies it to GH, finding that FE = 45. The student then divides 45 by 20 and gets 2.25 for a third time. That’s three pairs of proportional side lengths and BAM! Similarity proven by SSS.
Me, the teacher, is there is tell the student that he or she isn’t quite right. (You see the mistake, right?)
The student assumed similarity before it was proven. Then proceeded to use the assumed scale factor to find the missing side length, which ensured that the third quotient was going to be the same as the first two. This is circular reasoning. They are similar because FE = 45. FE = 45 because they are similar. I have seen this play out countless times.
I have addressed it with little success. I can’t seem to make sense to the students why that argument is weak. It sounds like “geometry teacher says we can’t, so do what geometry teacher says.” (especially when the very next question asks for FE, which is 45… because the triangles ARE similar…). I can’t stand using my authority as a teacher to enforce a math idea that the students are perfectly capable of actually learning.
I’m trying to decide how picky to be with this. I have a hard time allowing that circular reasoning argument to be called correct, although it is clear that the student has learned a lot about similarity, proportionality, and the structure of a proof.
But the more I push the point, the more frustrated I get and the students don’t seem to be getting any significant gains. I just continue to enforce that “math teachers are just picky like that.”
I am hoping for some help on this one. I’ve tried a lot of things, but that thing you like that works really well for you… I haven’t tried that one. Toss it my way. I want to see how well it works.