[Xmca-l] Re: units of mathematics education

[Ed Wall]

Andy

      Tanil's comment is at the end of the lesson after she together with her  peers have, in effect, given their explanations and after the teacher has used multiplication in her initial demonstration.  While your interpretation is possible, an reasonable alternate interpretation would be Tanil is indeed aware that multiplication is a possibility, but so is repeated subtraction (however, she may not realize the mathematical equivalence: i.e. -.80 - .80 - .80 -.80 -.80 -.80 = -4.80 for obvious reasons). Multiplication, by the way, is not necessarily part of the pattern I was pointing at. 

      I think the teacher has both the processes you mention in mind. However, there is a sense in which the work of abstracting (although they are not 'abstracting', but making 'concrete')  is presented as decoding and the work of manipulating is presented as filling in a template. I don't necessarily have a problem with 'making sense' being 'do what I do' (i.e. there is a place for mimesis in teaching/learning), but again I would want to complicate such a lesson for a teacher who so argued. I think you said something about the social practice of mathematics evolving. It might be difficult to participate 'socially' in such evolution in such a classroom. Perhaps a story will help convey what I am pointing at: A number of years ago I was teaching a somewhat advanced mathematics class. One of my students was failing and he complained, "You never give the same problems for homework! In my other classes we always had a lot of practice and once I learned how to do a problem I got good grades." I explained to him that while each one of a particular set of problems looked textually different, they all referenced a pattern we had recently discussed in class. Yes, seeing patterns is difficult for a novice, but that is true in any field. However with the appropriate experiences there is the possibility of one 'evolving' to later stages of expertise.

      If you are asking about my focus. I have both of your processes in mind (although conceptualized a bit differently - see 'abstraction' above) and quite a bit more. However, I want to emphasize I am not engaging in the paper in a critique of this teacher (I did that only to try to answer your question "why" ), but, you might say, engaged in trying to iterate a 'smallest'  unit of analysis. Making sense is worked on in a certain way in this classroom and I find that interesting. My gut feeling (as a mathematics teacher and doer of mathematics) is there is the possibility of 'more' to such work. The question I have for myself is can I characterize (and I have chosen to do the characterization hermeneutically) this 'more' in a respectful, coherent, and useful fashion. So the important point is, for this teacher, teaching 'making sense' seems to include teaching students how to manage those bodily senses relevant to a mathematics problem,  how to produce meaning from a mathematics problem, and how to develop a view of a mathematics problem that is shared by a community (these somewhat reflect the processes you mention, which are not unimportant, but neither are those processes necessarily 'mathematical'). In the next section of the paper, I question this viewpoint of mine so as to iterate.

Ed

On Oct 25, 2014, at  1:54 AM, Andy Blunden wrote:

> But isn't it the case that Tanil hasn't grasped that repeated subtraction (or addition) is multiplication? S/he didn't see the pattern which she should have recognised as an instance of multiplication?
> 
> Aside from that: speaking as a novice in this area, isn't it the case that there are two distinct processes involved in doing mathematics, being able to abstract the "mathematical problem" from a situation, and being able to correctly manipulate the symbols to solve the problem. I gather it is the first which is the focus here. Yes?
> 
> Andy
> ------------------------------------------------------------------------
> *Andy Blunden*
> http://home.pacific.net.au/~andy/
> 
> 
> Ed Wall wrote:
>> Andy
>> 
>>      It wasn't, in a sense, a mathematics lesson. Mathematics is, in a sense, the science of patterns. Part of its usefulness is in reproducing generalized patterns 'accurately'. A child might after experiencing this lesson begin to think that one masters one-step problems, then two-step problems, three-step and so on. That, I think, is a rather depressing and inaccurate image of mathematics and one that many children experience and buy into (I can think of a *lot* of examples). Also, you might take Tanil's comment as indicating that, in fact, the proposed taxonomy is ambiguous as this could also be a one-step  or a six-step problem. Categorizing it as two-step significantly impacts the possibly of alternate techniques of solution.       Making sense mathematically would entail recognizing a pattern and generalizing it. Tanil has an inkling of this and has produced a possibly productive 'counter-example' to the proposed taxonomy. There are a few other problems here that I take up in the second part of the paper (this one I don't actually take up until the third); this is one of the easiest to explain.
>> 
>> Ed
>> 
>> On Oct 24, 2014, at  11:43 PM, Andy Blunden wrote:
>> 
>>  
>>> Thanks for this, Ed.
>>> Could make more explicit for me the point you are making in this example? What happened or failed to happen?
>>> Andy
>>> ------------------------------------------------------------------------
>>> *Andy Blunden*
>>> http://home.pacific.net.au/~andy/
>>> 
>>> 
>>> Ed Wall wrote:
>>>    
>>>> 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:
>>>> 
>>>>       
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