Re: education, technology & chat (The Mathematics of it)

[Peter Moxhay <moxhap@portlandschools.org>]

Peg,

What you write here makes perfect sense to me and resonates with my
experience of the Davydov math curriculum.

But are you arguing for adding measurement to a curriculum that
develops the concept of number in the traditional way (through the
counting up of discrete objects)? Or for placing measurement at the
center of the curriculum (as Davydov does) -- as the genetic source
of the theoretical concept of number?

Also, I would be interested to know if, like Davydov, you have 
expeuience
with children *modeling* the action of measurement, say with a formula 
like
A = nE, where A is a quantity, E is a unit, and n the the number. 
Davydov
argues that it is only via the model that a generalization is formed.

Peter

>     Measuring sets a precedent that units can be ever further 
> partitioned,
> breaking ground in which rational numbers can be planted in later 
> school
> years.  Measurement lessons provide a foil to developing 
> misconceptions that
> all numbers are whole and that number itself is a countable entity.  
> Later
> lessons on fractions may take advantage of a measurement curriculum
> essentially about a whole and its partitioned equal sized units.  
> Children
> with such an introduction to measurement may encounter fractions in 
> fourth
> grade more prepared to grapple with the idea that 2/3 is 1/3 plus 1/3 
> or
> that 3/4 is the same amount as 6/8 or that four halves is the same as 
> two.
> It is, after all, a matter of picking your unit and partitioning the 
> whole.
>
>     A measurement curriculum can enrich children's mathematics 
> development.
> A useful curriculum goes beyond direct object comparisons and seriation
> activities.  It does more than provide opportunities to cover space 
> with
> non-conventional units.  It does not stop at teaching techniques for
> mechanically applying rulers or balance scales and reading numbers from
> them.  The curriculum gives value to measurement activities by 
> mathematizing
> them: engaging students to focus on whole-part relations, thinking 
> about
> what they are counting, recognizing what makes a unit sensible to 
> count,
> improving specific skills that serve the essential ideas.  The 
> curriculum
> provides a context for cultural tools like rulers and scales to be 
> welcomed
> as ways to take a shortcut through the iteration of measurement units 
> and
> the counting of them.  It provides a context for estimated 
> measurements as a
> part of checking to see when a measurement result should be doubted 
> and the
> procedures should be executed again so that the goal of measurement is 
> met:
> The quantity is described with precision.