Putting transformational approach into practice – notation

[JimOlsen]

Hi Bill (& others),
(I’m doing workshops with teachers this summer on the CCSSM.)
We are digesting the transformation approach to geometry (which, BTW, I like).

I think we (big ‘we’–teachers across the entire country) have a ways to go in terms of putting this into use in the classroom.

In particular, my question is about notation for the transformations.
On the one hand, we want students to have a conceptual understanding of, for example, of translation–and notation “shouldn’t matter.”
However, from a practical handout, in mathematics, ultimately we want to be able to write things down and also notation is important for thinking about, and communicating, mathematics.
In the case of translation, I’ve seen a few different notations. For example, (x+2, y+3) or T(2,3).

Is there a particular notation, for the various transformations, we want to make sure we expose our student to?
I ask this question, because it would be a shame if our students had a very good conceptual understanding of transformations, but on the PARCC (or Smarter Balanced) test a notation unfamiliar to the students is given and they cannot answer the question.

Thank you.

[Bill McCallum]

I think the notation should be delayed until the point that the students see the need for it. You can say “the translation that takes A to B” or “the rotation clockwise about O through 30 degrees” for quite a long time. So, for example, I would avoid notation in Grade 8. As for high school, I think it’s up to curriculum writers. I can imagine writing a curriculum which avoids it almost entirely (well, I can imagine trying, anyway). I can also imagine careful deployment of notation being very useful.

If you do introduce notation, I would be in favor of not having it tied to coordinates. For example, for translations, I would write something like $T_{A,B}$ for the translation that takes $A$ to $B$, rather than finding the coordinates of $A$ and $B$ and writing $T(2,5)$. For rotations, I would write something like $R_{\angle AOB}$ for the rotation about $O$ through the angle $\angle AOB$, rather than figuring out the coordinates of $O$ and the angle measure of $\angle AOB$ and writing $R_{(2,1),30^\circ}$. There are a couple of reasons for this. First, coordinates are unnecessary and might not be present. Second, knowing the position of $A$ and $B$ or the measure of $\angle AOB$ is unnecessary, and might be a distraction from the proof. I would want students to get used to the idea that the points, lines, and angles from which they construct the transformations should be points, lines, and angles already existing in the situation, not ones they have to come up with numbers for.

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