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.

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

I do think that we need to do a better job as math teachers encouraging our students to use the precise mathematical language. I think we talked about this a little bit on the plane ride from Minneapolis to Charlotte coming back from TMC. I have to agree that the language of the congruence question is a bit tricky for high school students, so I am glad you showed this example. The best example I can think of is numerator and denominator. I frequently have to encourage students to use these words vs. top or bottom. I think that using the language of mathematics is something that needs to be reinforced from Kindergarten. Good post, Jim.

Jonathan

Thanks for dropping by and commenting. I do remember that conversation on the plane back from TMC16. I totally agree with you on the fraction language, but I will also defend my students using hypotenuse as short hand for longest side. I think that this might be one of those verbal shortcuts (dare I call it a ‘trick’?) that helped them to organize their thoughts. I do think I needed to be more firm in my correction of the language use, but it feels less sloppy to me than using ‘top’ and ‘bottom’ when referring to fractions.

I’m glad you put this post together. In your comments on my blog, and on Twitter, you took my post in an unexpected and interesting direction. I’m glad you unified your thoughts a bit here.

I definitely make the precise use of language an emphasis in my classes. Naturally it’s important in terms of communicating, but I think it’s equally important in terms of understanding. As I mentioned on my blog, I think segment congruence is a great example. On the one hand, distinguishing between segments being equal and being congruent can seem like a mere technicality. But on the other hand, it presents an opportunity to highlight fundamental features of different mathematical objects: numbers, which can be equal, are a different kind of mathematical object than, say, squares, which can be congruent.

I also like your “hypotenuse” example. In that situation, my students will say similar things to what your students say, and I’ll probably say such things occasionally, too. But I’ll consistently remind students that, techincally, only right triangles have hypotenuses, and so we are really talking about the longest side. Which is another good opportunity to reinforce understanding: the hypotenuse is interesting precisely because it is the longest side of the right triangle.

Like you, I try not to penalize students too often for such technicalities. Student thinking is paramount. But getting students to express their understanding with technical clarity is also one of our goals.

I think this could be an interesting blog series! Maybe you’ll see other examples of this phenomenon throughout the year.

Patrick

Thanks for the inspiration in this whole conversation and thanks for dropping by the comments here. I love your points here about comparing numbers – which we call equal – with something like geometric figures. I also like the point in the Math Forum link I provided where it was pointed out that ‘equal’ for geometric figures at some point meant ‘same area’ which we would never be happy about with our students. I also wish I had been clever enough to point out that hypotenuse is just a special word case for longest side of a triangle.

I actually love that the students came up with a genuine notion of type of triangle by comparing the “hypotenuse,” and using the language, “if the hypotenuse is longer than we thought…” I am in favor of letting them use the language to make a claim aloud, then being precise when writing and defining. I would then say, now that we have precisely defined what we mean, let’s use the correct language for it.

Tracy Zager is uncomfortable with my use of the word “massage” as in “massage the student’s thinking. Maybe I should have said, “guide.”

You are doing an amazing job and I personally am excited about the direction. Going over to read the math forum.

Amy

Thanks for dropping in on this conversation. I have to admit I kind of like the word massage in this context. It reminds me of my use of the word ‘marinate’ when I am discussing letting ideas slowly soak in over time.