The first conversation deals with the equal sign. I love the way the author set it up so students can discuss what the equal sign represents and opens up the discussion to interpreting the problem.
The second conversation is on representations and the meaning that people put with the representation while the third deals with mathematical language.
I have been working on increasing mathematical conversation in my classroom but my teachers training never actually covered either conversation or writing in the math classroom. Since I work with ELL students, I've had to create my own activities with little or no help so when I see articles like this, I am thrilled.
I really enjoyed reading this article because it gives me a starting point for this fall when school begins again in August. It is my opinion that if students can verbally express their thinking process, they develop a better understanding of mathematics.
I would recommend you head over to check out the article, read it and include these three conversations into your math classes because it makes the students use higher level thought processes that are important and it helps pinpoint or eliminate misconceptions.