RE: [xmca] Numbers - Natural or Real?

[anna sfard <annasfar@math.msu.edu>]

Hello Andy, Larry, David, Bill, Tony, Mike and all the mathematically-minded
lurkers,

 

I feel personally invited to this conversation (think you, Andy, David, .)
and have been aching to join. And if I'm doing this late and for just a few
moments, I hope to be forgiven. The time couldn't be worse: I'm about to
travel and the unfinished jobs are taking my sleep away. While finally
succumbing to the temptation I feel as I felt years ago, when I was running
from my homework to play with other kids (do not interpret this metaphor as
reflecting my perception of the present audience!). My apologies if in what
follows I sound breathless. The topic deserves much better than that.  

 

The main thing I wish to do now is to disambiguate the conversation.
According to my reading, Andy's opening shot consisted of two separate, if
not entirely independent, problems: 

-         first, the issue of developmental relation between natural and
rational number; 

-         and second, the question of what would serve us better as the
pivotal concept (unit of analysis?) in our discourse as researchers. 

Both these queries implied that there is some kind of opposition between the
compared options.  My disambiguating effort aims at showing that there is
none. For the sake of clarity, I will do it in two separate posts, devoting
he present one to the first question and discussing the second one in the
next mailing.

 

Here, I wish to speak about numbers. I must be forgiven for not trying to
follow the (admittedly fascinating) meandering path of the former
contributions (but I do wish to acknowledge Larry's nice summary of some of
my writing - thank you, Larry). Let me make it quick and clear: I think that
in this conversation, we have confused discourses on quantities with
discourses on numbers. Of course, there is a tight relation between the two
and most of the time they appear to be the same thing. But if you think
about it, you realize the possibility of discourse on quantities that
features no numbers. Such numberless discourse on quantities is recognizable
by the nature of tasks for which it is used: quantitative comparisons. You
can compare objects with respect to their length, weight, volume without
ever using numbers. You can also compare discrete sets, thus discrete
quantities, but I agree with Davydov that this comparison is more tricky:
while comparing lengths of two sticks you simply align one stick with the
other and see which "sticks out". To compare two sets of discrete objects
you need special arrangement, called one-to-one correspondence, and this is
difficult for the children to make; but more importantly, the kids cannot
possibly figure out, at this point in time, why they need such arrangement
(recall that in Pigetean conservation tasks, when asked "where is more
marbles" they often compared lengths, density or areas covered by the set
rather than building 1-1 correspondences).  

 

Both historically and ontogenetically, the discourse on quantities is
present before the discourse on numbers. At this point, quantities come only
in the context of comparisons, and objects may be labeled with relative
adjectives such as "the longest", "longer", "the lightest", but there is no
point or even possibility to talk about "length" as an object  or as an
absolute property of another object.  

 

The transition to quantities as absolute properties of objects takes place
when the need arises to compare things that cannot be simultaneously present
in one's field of vision. How can I compare my height to yours, Andy, if you
insist to stay at the antipodes? (or maybe it is me who walks upside down?)
Well, I need some  agreed mediating entity to which both your and my heights
can be compared. Thus, I need a unit of length and. a number! The number
will tell me how many times the unit goes into each of the heights. And this
is, indeed, where numbers originate: in the need to compare quantities that
are not simultaneously present. The number that states your height becomes
your permanent label, reflecting your absolute property.

 

As to which of the discourses - the one on natural numbers on the one on
rational (or better still, real) should/can come first, well, you cannot say
what 2/3 (two thirds) is without being if only slightly acquainted with
creatures such as 2 and 3, right? So, yes, the discourse on natural numbers
has, developmentally  (and I speak about development of discourses, not of
kids) come first. And it does come first even in Davydov's teaching
sequence, which builds on the children's ability to count. After all, they
can already count when they come to school - some better some much less so,
but the idea is there (do take a look at Walkerdine's fascinating book, as
suggested by Tony). See, the discourse on natural numbers is subsumed within
the discourse on rational (and real, some time later) numbers, and it
constitutes this latter discourse's "generative" core.

 

The bottom line of all  this may be put as follows: Davydov does begin with
the discourse on continuous quantities, as opposed to the common practice of
beginning with the discourse on discrete quantities. In this way, he
provides a solid unifying basis on which both numerical discourses, the one
on natural numbers, and the larger one - the one on rational numbers can be
developed almost simultaneously. But it is a mistake to claim that he
proposes to begin with the discourse on rational numbers, that is, that he
tries to develop the latter in the absence of even the basics of the
discourse on natural numbers (once again, continuous quantity is not the
same as rational number). 

 

And one last remark before I rush to the other post: There is no
contradiction between Davydov's work and my own, including this last
statement, according to which one can hardly start developing a discourse on
rational numbers without already having at least basics of the discourse on
natural numbers. Davydov proposed a particular pedagogy - a specific, well
structured curriculum grounded in carefully argued basic assumptions. I am
not creating curricula - I'm doing research in which I am asking what is
possible and what is actually happening under different circumstances in
math classroom. Of course, whatever I have learned from research has
pedagogical implications, but there is nothing in it that would contradict
Davydov's pedagogy. On the contrary, there is much that can be said in its
support (see above). 

 

Enough for now. Hope to find time to write the other post as well :-)

anna

 

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