The Language That We Use

I recently engaged in a spirited discussion prompted by Patrick Honner (@mrhonner) on twitter and on his blog. The original post that started this whole discussion can be found here and it is well worth your time. Engaging comments there an on the twitters and a friendly suggestion by Patrick himself has me writing here, thinking out loud.  To set the stage for this post, an image from Patrick’s post is important.

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A quick glance at this certainly suggests that these are congruent figures until you look more carefully at how the question is worded. This is a pretty classic example of the kind of question that makes students think that test writers are gaming the system to catch them in a mistake. We are looking at two figures that are equivalent to each other. A rigid transformation maps one onto the other. However, that mapping does not map them in the order suggested. A classic mistake that I lost points for as a student and one that, sadly, I admit that I have probably deducted points for when grading. The debate on the blog and on twitter raised some really challenging questions about our goals with this type of specificity. Yes, mathematics is a precise language and precision is a powerful habit to try to help develop. However, I keep thinking about my fun Geometry class from last year. When we were discussing how to determine whether  a triangle with given side lengths was acute, right, or obtuse we worked out a strategy where we assumed that the Pythagorean Theorem would hold and we decided what the consequence was when it did not. This led to my students saying things like this; “If the hypotenuse is bigger than we thought it would be, then the triangle is obtuse.” Now, I know that the largest side of an obtuse triangle is not called the hypotenuse. When pressed on the issue I suspect that almost all of my students knew this as well. Optimistically, I want to say that they know this as well, but is is early August… My concern here is that I was letting them down by letting them be a bit lazy with their language. What I did at the time was to gently remind them that hypotenuse was not the best word to use there but I understood what they meant when they said it. Should I have made a bigger deal about this at the time? Was I being understanding and flexible? Was I being undisciplined and imprecise? I suspect that there is a decent amount of both of these in my actions and I have to admit that I did not think too deeply about it at the time. In the wake of the conversation that Patrick moderated, I am thinking deeply about it. It is also early August (again, I note this) and it is the time of year that my brain reflexively starts dwelling on teaching again. I am also thinking about a distinction that I got dinged for as a student but this time it is one that I do not ding my students for. I remember losing points in proofs if I jumped from saying that if two segments were each the same length then they are congruent. This is, obviously, true but I was expected to take a pit stop by making two statements along the way instead of jumping straight to congruence. I know that equivalence of measure and congruence of segments (or the same argument with angles) are slightly different meanings. A nice explanation is here at the Math Forum. But I feel pretty strongly that my 9th and 10th grade Geometry students are not tuned in to the subtle differences and I think I am prepared to defend my point of view that they do not need to be. I want my students to be able to think out loud and I DO want them to be careful and precise in their use of language but I do not want them to think that this is some sort of ‘gotcha’ game where I am looking for mistakes and looking for reasons to penalize them.

I am thankful to Patrick for getting this conversation started and for gently nudging me to try and work out my thoughts more thoroughly on this issue. I am interested in hearing from other teachers – particularly Geometry teachers – on how they try to navigate these conversations. How precise should our high school students, especially freshmen and sophomores, be when discussing these issues?

As always, feel free to jump in on the comments section or reach out to me through twitter where I am @mrdardy

Thinking Out Loud

Super brief post here – class starts in 15 minutes. One of my Geometry colleagues asked me about a HW question I had written. I asked the students to find two cylinders that were not congruent but that had the same surface area AND the same volume. I thought it was a pretty interesting question, but I realized I did not have a coherent strategy for discussing it other than playing with numbers. I threw the question out to Twitter and engaged in a terrific conversation with Matt Enslow (@CmonMattTHINK), John Stevens (@Jstevens009), Dave Radcliffe (@daveinstpaul), and David Wees (@davidwees) Some good math was tossed around, but what really has my brain bubbling is an exchange with David Wees. He said he thought it was a great question but he would not have used it as a homework question. When I asked him why he said something that reminded me of a conversation with a former colleague. My former colleague once said that he sees a difference between exercises and problems and that he liked to keep problems for times together with the students where they could work together. I find that I feel (hope) that meaningful conversations can happen in class more readily about a problem like this one if the students have had time to think about it first. However, his words carry some weight with me as do David’s. I feel as if there is some conclusion I want to reach, but I also suspect that there is no right answer to this. I would love to hear some opinions about this in the comments or through Twitter where you find me at @mrdardy