I see.The text which is makes up the third point of the triangle with the teacher and student is in this case a mathematical text, yes? So for example, completely different problems arise than would arise in the case of reading a story or some other piece of writing. I can see the idea that that teacher-text-pupil relation is an archetype for a whole range of teaching. In itself, it certainly doesn't tell you anything distinctive about teaching mathematics in particular. I think you need to turn to other units specific to different topics being taught.
Different units give different insights. For example, Vygotsky used word meaning while Bakhtin used utterance. Utterance is a much larger unit than word meaning, but it proves useful for providing insights into communication and handling the framing and context, whilst word meaning is useful for understanding concepts and the development of conceptual thought. Davydov's germ cell in which two objects are compared in length is an elementary act of abstraction, and therefore captures the idea of quantity, which should take a student to the point of grasping the general idea of mathematical text and abstracting quantities from real situations. But that doesn't really do for the whole subject or tell you anything about the teacher-pupil relation.
I would not get obsessed on this phrase: "possessing all the basic characteristics of the whole." That phrase can lead you up a blind alley. I think it originates from Engestrom's 1987 book: “a viable root model of human activity ... [must be] the smallest unit that still preserves the essential unity and quality behind any complex activity," which is somewhat more precise than the phrase you have used, but can still lead to misconceptions. The interpretation "possessing all the basic characteristics of the whole," leads to a logical circle: which characteristics are essential, which characteristics are basic?
You need to form a concept of teaching mathematics.Perhaps you could elaborate a little, Ed, on your ideas for a unit of analysis for mathematics teaching? Why do you need a smaller unit?
Andy ------------------------------------------------------------------------ *Andy Blunden* http://home.pacific.net.au/~andy/ Ed Wall wrote:
AndyThe paper ("The Unit of Analysis in Mathematics Education") is about unifying branches of mathematical education research: nature and philosophy of mathematics, teaching of mathematics, learning of mathematics, and sociology of mathematics (this last something he has promoted for a number of years) under one unit of analysis (i.e. collaborative projects). Insofar as the section on mathematics teaching goes he just says the triad (and he fleshes it out a bit) isn't controversial so I wouldn't say he is always 'critically' reviewing, but that may be a matter of opinion. My question isn't directed at Ernest, but at you. I'm interested in the very idea of a unit of analysis possessing all the basic characteristics of the whole. The problem I am having with all varieties of the triad is that they seem yet too 'large'; i.e. in a sense the grain size is too large to, one might say, pick up the mathematical flavor that differentiates mathematics teaching from, say, reading teaching. So it would seem that the choice of the unit of analysis also needs to be done in a minimal fashion? Without such a unit of analysis, I find myself unable to talk usefully and coherently with my students about what I observe that is mathematically problematic (and I don't mean mistakes) in their planning and teaching of mathematics. There are times, unfortunately, when it appears I am viewing a well thought-out reading or grammar lesson.Anyway I doubt whether mathematics is unique in this regard and that teachers of all stripes aren't having similar problems with such units of analysis. Ed On Oct 23, 2014, at 9:41 PM, Andy Blunden wrote:Ed, Paul may quote me, but I actually know little about his work or mathematics education itself. But isn't he discussing a number of different proposals for a unit of analysis for mathematics teaching, one of which is the one you refer to. I take it that he is critically reviewing all such proposals before making his own proposal. Andy ------------------------------------------------------------------------ *Andy Blunden* http://home.pacific.net.au/~andy/ Ed Wall wrote:Andy The paper seems to be about unifying mathematics education research. Parts are a bit open to debate (especially arguments concerning the 'nature' of mathematics) and Ernest tends to somewhat gloss over this. However that is not relevant and you are correct Ernest does, among other things, put forth a unit of analysis for mathematics teaching which, as he admits is simplified for the purposes of the paper; i.e. the usual triad of teacher, student, and text (which is hardly unique to Ernest as he notes). At this point I have a question that I've been pondering about concerning such triads and their elaborations (and this goes back in a sense to things Schwab said elsewhere - the Schwab he quotes in the beginning of his paper) and, as he quotes you heavily, I will ask you: If this triad is indeed a prototype of mathematics teaching (i.e. posses all the basic characteristics of the whole), what makes this a prototype of mathematics teaching and not a prototype of, say, the teaching of reading? This is not a spurious question since, as a mathematics educator (of the type that Ernest wishes to unify - smile), I often find myself needing to help elementary school teachers realize there are actually substantial and observable differences (and substantial similarities) between teaching reading and teaching mathematics and, for sundry reasons, they tend to favor something like the former and cause their students some anguish in the learning of mathematics as time passes. Hmm, I guess I am asking whether the unit of the analysis can, in effect, be the 'world' or should it be, so to speak, among the 'minimum' relevant prototypes. It seems that it would be somewhat worthless otherwise (again similarities are important). Ed On Oct 23, 2014, at 3:57 PM, Andy Blunden wrote:Paul Ernest has a position on the unit of analysis for mathematics teaching: http://www.esri.mmu.ac.uk/mect/papers_11/Ernest.pdf Andy ------------------------------------------------------------------------ *Andy Blunden* http://home.pacific.net.au/~andy/ Julian Williams wrote:Andy: Now I feel we are nearly together, here. There is no 'final' form even of simple arithmetic, because it is (as social practices are) continually evolving. Just one more step then: our conversation with the 7 year old child about the truth of 7plus 4 equals 10 is a part of this social practice, and contributes to it....? The event involved in this Perezhivanie here involves a situation that is created by the joint activity of the child with us? Peg: Germ cell for the social practice of mathematics... I wonder if there is a problem with Davydov's approach, in that it requires a specification of the final form of the mathematics to be learnt (a closed curriculum). But let me try: One candidate might be the 'reasoned justification for a mathematical use/application to our project' ... Implies meaningful verbal thought/interaction, and collective mathematical activity with others. Not sure how this works to define your curriculum content etc. Julian On 23 Oct 2014, at 16:28, "Peg Griffin" <Peg.Griffin@att.net> wrote:And thus the importance of finding a good germ cell for mathematics pedagogy -- because a germ cell can "grow with" and "grow" the current "social practice of mathematics." Whether someone agrees with the choice of germ cell made by Davidov (or anyone else), a germ cell needs to be identified, justified and relied on to generate curriculum content and practice, right?PG