The analysis requires a couple of important points.
I was excited about this happening in the class because as teachers these students are going to need to feel comfortable as the authority on mathematics. Whether or not a teacher should be an authority figure in a mathematics classroom, they are seen that way by the students. Thus a math teacher must feel capable of figuring out the answer when put on the spot. Even when a teacher is not functioning as the authority figure in the class, the teacher needs to be comfortable with their role as an authority (Shulman, 1988). In fact, Shulman argues that it is only this comfort that makes it possible for the teacher to relinquish their authority.
In addition to seeing this seizing of authority as helpful to the teachers pedagogically, I also see it as mathematically important too. In this vignette, the students have taken control of the mathematics and are working together to understand why the numbers work in the way they do. Since the idea of transcendentality of numbers comes long after the notion of such addition of roots, the question of whether this number is algebraic is not particularly important for their teaching. The thinking that underlies the answer to this question, however, is.
As noted in the instructor log, the students found themselves working on a natural problem in the understanding of variable: x as an unknown number versus x as a variable. By confronting this confusion on their own and more importantly recognizing the difficulty that making a shift from one understanding of x to another understanding of x caused them allowed for the students to recognize a learning difficulty that their students will encounter. Moreover, it also allowed me to discuss an instance of transforming content knowledge to pedagogical content knowledge. The content knowledge is the different roles that symbols play, and the importance of those roles in doing mathematics. This knowledge becomes pedagogical knowledge when recognizing the difficulties incipient in these different roles of the same symbol, and the necessity of making clear what role the symbol is playing at any one time.
The last reason I saw this interaction as important to the goals of the class is that I saw this taking of authority was a sign that the students were beginning to take authority over the mathematics in their projects. The move to accepting responsibility for the correctness of the mathematics in their projects is one of the key issues for the students to navigate.
While these were my view, the students had a more general perspective on the class as a whole. In the interview, Alan said of the class as a whole:
So that's what it felt like to me, like we as a class decided where to take, where to take the next step...
Thus, he saw the students taking command supporting my earlier statements, since it was this particular class day that he listed as his main example. As the discussion about this class meeting continued, Alan then summed up the whole class again as:
...there was a lot of working as a group to going back and forth between our conjectures, we've been talking a lot about this in our TE class this idea of chucking back and forth between conjectures and the board and what we generally see in our class is "definition theorem proof" and in our class, in the 496 class it was very different from that a lot of times it was (Jill: Conjecture) it was like conjecture, start a proof, figure out it is not quite right, fix the conjecture, continue kind of stuff. So I think I got a better view of what math is.
Again, showing the idea of them taking command and having less of me leading the class.
Another interesting point is that while the students saw this class day as different from our typical class day, they didn't see the distinction between the two as great as I did. That is, they appear to have seen this class as a natural extension of the other classes that we had. I found this last development to be particularly striking since one of the great positives I saw in the class this year was the way the students notion of mathematical discussion expanded throughout the term.