# [Xmca-l] Re: In defense of Vygotsky [[The fallacy of word-meaning]

```Andy

Apologies, I thought the triangle and its variations were somewhat common knowledge on xmca. Anyway, what you say is what Ernest is 'using' as a unit of analysis. He assumes that, as most, what is critical is the dialogical.

By the way that the text (written large or small) is mathematical doesn't make the teaching/learning mathematical.

Actually, I rather like the idea of basic characteristics (I tend to do hermeneutic phenomenology) and really find the idea of 'smallest' troubling.

I have, in a sense, been thinking about something like a unit of analysis for mathematics teaching for some time and to go into that in detail as it spans a number of papers would be a bit much. However, let me say a bit why I need a 'smaller' unit. I'll take this from a paper I'm working on so perhaps all this will make somewhat more sense. Let me tell a story (this story is more or less true, but it is intended as a case in point than data):

>

Ms. Peña has, in previous years, noticed that her fourth graders, at times, struggle to make sense of multi-step word problems. Many seem confused about both the nature of the required operations—most usually, addition, subtraction, multiplication, or division—or the order in which these operations are to be applied. Today she has put together an activity which she hopes will alleviate their difficulties. She begins by writing the following 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

80¢           \$5.00
x 6           -\$4.80
-------            -------
\$4.80               20¢

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.”

>

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.

Most of the teachers I work with and a number of people in mathematics education would agree that this is an exemplar for 'teaching' making sense (one could argue that it is actually impossible to teach a student how to make sense although a student can learn how to do so) in an elementary school mathematical classroom (I happen to think it is a thoughtful lesson, but). The problem is that this is more of a literacy lesson than a mathematics lesson and, on purpose, I picked Tanil to indicate this. Mathematics is, in a sense, the science of patterns and this is the very thing this lesson is not about. The pattern, by the way, is 'profit' minus 'loss' not a series of two-, three-, or four-steps. However, the triad seems to be too coarse for to pick this 'omission' up and in my next story (in the paper) I show 'word' and 'phrase' seem to be too coarse also (somewhat against Halliday, one might say). I could perhaps use something to do with patterns like Davydov uses comparisons in length, but this for various reasons doesn't seem quite right (although that may be because I am still not sure about a 'unit of analysis'). In the third section of the paper you might say I make a suggestion for a unit of analysis. I have some ideas, but none sufficiently argued that I wish to publicly put in print (smile).
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.

Ed

On Oct 24, 2014, at  1:24 AM, Andy Blunden wrote:

> 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?
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> 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:
>> Andy
>>
>>      The 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:
>>
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>>> 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
>>>>
>>>
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