Re: [xmca] Numbers - Natural or Real?

[Bill Kerr <billkerr@gmail.com>]

hi anna,

Appreciate that you are busy. If you get a chance to read the Devlin
article http://www.maa.org/devlin/devlin_01_09.html  then I'd be interested
in your thoughts about it.

Agree with you that quantity is prior to counting. Once kids can count then
should a maths curriculum focus more on natural numbers or real numbers?

Natural numbers arise from the desire / need for peoples to keep track of
their possessions and trade with one another.

Systems of measurement (length and area) arise in order to measure land,
plant crops and eventually to design and erect buildings.

Western curriculum starts with positive whole numbers and addition and
proceeds through negative numbers and rationals. This may give rise to the
belief that natural numbers are somehow more basic or fundamental than the
reals.

But both natural numbers and reals arise directly from our real world
experience in parallel out of different cognitive processes used for
different purposes with neither dependent on the other.

This is the background context which to me makes the Soviet / Davydov
alternative a very interesting one to consider. Devlin briefly discusses the
implications of the different approach for teaching of concepts such as
multiplication, etc.

I still haven't read the Schmittau article, which is referenced by Devlin
and was posted by Larry and which appears to discuss the implications in
more detail, eg. for the teaching of algebra


On Sun, Jul 3, 2011 at 2:55 PM, anna sfard <annasfar@math.msu.edu> wrote:

> 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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