Proof and Consequences: Circular Reasoning

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.

Similarity4

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.

Except…

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.

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8 thoughts on “Proof and Consequences: Circular Reasoning

    • I agree with you, except the students take it upon themselves to find GH. It could be that I’m asking too many questions together. The fact that they know they will need to find GH might give the impression that it doesn’t matter what order they do the problems.

  1. I wish I had a better answer. We just finished the same similarity rules after spending quite some time on congruence. In any case, if they jumped to a conclusion without first establishing proof of congruence or similarity, I would usually ask something like “How did you know to do that?” or “What did you base that on?” I’d try to ask questions to get them to retrace their thought process and see what was already given or known. Not saying that always works!

    • Yeah, the students assume so much without paying attention to it. “Those two lines are parallel, so…” is a different thought than “those two lines look parallel, I should check that out.” I’m trying to enforce the latter. The tricky part, as I said in the post, is deciding how picky to be. I want to stay true to the spirit of the proof and the value of logical thought, but the erroneous, circular argument is filled with good math, and decent (although flawed) logic. I appreciate you chiming in, though.

  2. i have some suggestions, but first i have a few questions:

    1) i’ve never seen using SSS to describe anything but triangle congruence. is that an american (i teach in the Dutch and IB curricula) to also use that for similarity? to me it implies that the sides are congruent, not that they’re in proportion. i find the interchangeability of terminology there confusing.

    2) is there more text accompanying this? are the students given that FE & GH are parallel, or are they expected to measure the angles themselves?

    now ideas:

    1) do an in-class investigation with the research question “if we have the same scale factor for 2 corresponding sides of a triangle, can we ASSUME that it works for all 3?”

    step 1: instruct the students to draw a triangle (their choosing what kind and what measurements) and then construct a new triangle where the same scale factor works for all 3 sides.
    step 2: then instruct the students to construct another triangle based on the first triangle drawn, using the same scale factor as before, only this time make it so the scale factor only works for 2 of the sides (they may struggle with this initially).

    ask the students whether triangles 1&2 are similar and whether triangles 1&3 are similar and without referring to scale factor, how can you be SURE that your answers are correct. ask the students what conclusion they can draw from this.

    (i drew an example of this on the train home from work where i drew two isosceles triangles — the first had congruent legs 2 & 2 and the second had congruent legs 3 & 3, so scale factor of 1.5, but the first triangle was an isosceles right triangle and the second had an angle of a bit more than 90. i can send a jpeg if you want!)

    2) as a warm-up, give a problem that resembles the problem you gave, but with GH ever so slightly askew. show them a student’s working that uses the same logical reasoning they used to indicate similar triangles and thereby find FE. ask them to find the flaw in the student’s reasoning and what would need to be different about the problem and why for the student’s anser to be correct.

    3) vary how you are asking the question. for example, give it without the information that GH & FE are parallel and ask them what markings they would need to add to the diagram to ensure that they had similar triangles. another thing would be to give them GH & FE and then ask them what they can conclude about <F & <SGH. if you can get them coming and going, maybe they will see it better.

    if i think about other ways i'll post more! good luck – sounds fun either way! 🙂

    • There’s some good stuff here. Thanks. I would love to see a .jpg of that sketch you drew. Send it to andrew.shauver@gmail.com. In general, I love the practice of showing students work and asking students to analyze. My concern would be that the students don’t understand the limitations of the circular argument to call it out as a mistake. They don’t see anything wrong with it. However, I haven’t tried it. I do often ask as the students are looking at some student work, “What is the assumption that this student is making?” although that question isn’t really sinking in very well, either.

      Answers: 1. While I’m not sure if it is an American thing or not, SSS has been a part of every standards-based similarity unit I’ve seen, whether based on Michigan Merit and Common Core. SSS says that if you can show a common ratio among all three sides, then you can conclude the triangles are similar.

      2. There are not any texts that give additional information. Although, I would not expect the student to measure the angles, but to reason that the two side pairs give are proportional, but, regardless of the measure, both triangles make use of <A which is included by the two proportional side lengths. So, that would be concluding similarity by SAS, at least that was the idea.

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