There is a dissertation on this topic that was done fairly recently.  If you request it, you can probably make good use of the reference list.  The dissertation is:

Marnich, Mark A (2009) A Knowledge Structure for the Arithmetic Mean: Relationships between Statistical Conceptualizations and Mathematical Concepts. Doctoral Dissertation, University of Pittsburgh.

Their website is http://d-scholarship.pitt.edu/9109/

Couldn't get it there but here:

http://core.ac.uk/download/pdf/12208932.pdf

There is good work by Kevin Miller with preschool children. He is now at the University of Michigan. I assume that you can find the reference on his website. 

Gerard VERGNAUD, the theory conceptual fields, specially, he worked the analysis of additive structures.

Vergnaud wrote:

There are two prototypical situations for addition: the binary combination of

two parts into a whole (‘4 boys and 5 girls for Kath’s birthday party; how many children altogether?’), and the increase of an initial state (‘Richard had 4 marbles, he wins 5; how many marbles does he have now?’), which can be better modelled by a unary operation, a function from the set of possible initial states into the set of final states.

The distinction between these two prototypes is made clear when one considers

the variety of problems that can be generated. In the first case, binary combination of two parts into a whole, only two classes of problems can be generated: knowing the two parts find the whole, and knowing the whole and one part find the other part.

In the second case, six classes of problems can be created: knowing the initial

state and the transformation find the final state (by increasing or decreasing the initial  amount), knowing the initial and the final states find the transformation, when

the final state is bigger or smaller than the initial one, knowing the final state and

the transformation find the initial state by increasing or decreasing the final state.

Among these six classes of situations, four require a subtraction and only two an addition. Addition and subtraction are not merely inverse of each other.

Regards,

I actually thought you were asking a different question, one about the teaching of verbal skills imbedded in math problems.  I'm attaching a document I created with a committee of math teachers when I was the Curriculum Director for Conejo Valley Unified School District in Thousand Oaks California.  It may or may not help you.

Thanks for your answer, Martha. Actually, I'm more interested in this particular type of mathematical problem that in the language involved. I have proposed it in Spain for children from 5 to 12 years to study how they solve it at different ages. I am studying transitions in numerical thinking. For example, how children move from manipulating and counting objects to solve the problem (5 years) to the combined use of addition and subtraction (6-7 years) or how they use division (with 10 years) for the same problem. However, your contribution is valuable for me to properly understand how to formulate these problems in English language.

Dear Soledad,

Thank you for your detailed contribution. I know the work of Gerard Vergnaud. The problem I pose to children has not an additive structure, but involves at least one multiplication or division. Equitable redistribution problems have not been studied by Vegnaud, although his vision of multiplicative structures is very important in my work.

regards,

Carlos

Thanks to Julia and Christian. I have downloaded the thesis of Mark A. Marnich. It is very interesting to me, because this thesis deals with the same concepts in my problem. Perhaps the thesis is very focused on the concept of arithmetic mean and its understanding for adult students. In my case, I study these problems with students from kindergarten (5 years). I use these problems because they are not common in school, in the early years, and that allows me to study children mathematical thinking.

Good luck, Carlos!