Ed, Andy, here's my two-bits response to the practice of Ms Pena.
the unit of analysis i'm using here is each sequence, or turn, of the larger activity of the practice of doing math in Ms Pena's class.
She writes the problem on the board:
"Pedro has five dollars. He wants to buy 6 pens. If each pen costs 80 cents apiece and he buys 6, how much money will he have left?
"She asks Elvina to read the problem."
" Noticing that several students have their pencils out and are writing on paper, she admonishes them saying, “Put down your pencils and listen.”
After Elvina reads the problem, Ms. Peña points out that it is a two-step word problem. She asks the her students to say the word and
then, asking for a definition of two-step, calls on Jorge.
Jorge replies that two-step means “More than one step.”
Ms. Peña restates writing the word operation on the board, “A two-step problem is a word problem where there is more than one operation” and she gives possible arithmetic examples.
She then calls on her students to identify the operations in this word problem.
After some discussion, the students identify multiplication—that is, multiply the number of pens by eighty cents—and subtraction—that is, subtract that product from five dollars."
She tells her students they—herself and the class—will be solving two step problems; that is, problems that usually require more than one operation. Then, after checking to make sure that her students understand the term operation, ]s. In summary Ms. Peña writes and speaks
x 6 -$4.80
Ms. Peña then divides her class into four groups and giving each group a slightly different two-step problem sets them to work.
As students work, she circulates reminding each group that there are two-steps and that they need to identify the operations comprising these steps.
Towards the end of the period she has each group diagram their solutions on provided chart paper.
As class comes to a close, Tanil raises her hand. “Couldn’t you just subtract? Y’know, subtract six times.”
Ed, you now comment that:
Note my story has a number of basic characteristics which include for instance, managing those bodily senses relevant to a mathematics problem, producing meaning from a mathematics problem, and developing a view of a mathematics problem that is shared by a community.
what i think, Ed, is that the meaning and the view shared by the community isn't exactly managing bodily senses relevant to a mathematics problem, but rather that, for this math class community, a shared understanding is that it is the teacher who does the work.
out of the roughly 14 turns - actually i should have counted Elvina as well - the teacher had done about eleven of them. in other words, it's the teacher who is doing the actual activity of "learning".
and, Tanil is absolutely correct, and in the school i work in, the response of the teacher would have been to elicit how Tanil came up with that solution process, and then follow that it with the question, which solution was more efficient. (by the way, there is no right answer to the question about efficiency, except for the student's perspective.
(my guess is that the student is using a older process learned six or nine months ago.
Ed, you also wrote:
"Why do I care? The best I can say is that as a teacher of teachers it is very disconcerting to think, "Well, it strikes me what you are doing bears very little relationship to the teaching of mathematics" and be unable to create a context which is jointly meaningful within which my gut feelings as a teacher of elementary school mathematics can be respectfully, coherently, and helpfully expressed. perhaps to put it another way, the people I work with are serious and talented, but they haven't thought some things through nor have they been given the opportunity to do so."
i agree wholeheartedly here, and i'd look for opening within the system of activities of the school to begin to bring about this conversation.
in the school that i'm at, the teacher would have written the problem up on the board, _not_ told the students that it was a two part math lesson, and asked for what students thought the answer was.
possible answers would have been written up on the board.
then a student would have been asked to come up and show how they got the answer, or, would have told the teacher what steps to go through to get the answer. (personally i prefer to have the student tell me what to write on the board.
and the equations would have been written as:
.80 X 6 =
$5.00 - $4.80 =
not as an algorithm.
because, for Ms Pena, the question is not whether or not the students can do an algorithm, but rather can they parse out the part of the math problem.
(and yes, for many student, especially second language learners, this is also a literacy lesson.)
my apologies for going on at such a length - and really i've described the barest essentials of what a classroom teacher could do to get the students to take on the responsibility of learning how to solve math problems, rather than have the classroom teacher perseverate in the practice of mistaking telling students how to solve a math problem is the same as having students solve the problem of solving math problems.
finally, i think that breaking down a large activity into turns or sequences - who is doing what and how does the doing support the supposed goal of the activity - greatly helps in figuring out a unit of analysis.