Cultural Historical Activity Theory extends Vygotsky's Activity Theory.
I am interested in garnering quantitative data relating to the varying degrees of individual participation group during collaborative digital games-based learning.
It looks to be a very complicated task. In fact I can't realize how division of labor could be measured.
As far, as I know, Historical Cultural Approach was the result of Vigotsky researches. Later, as a derivated theory, several of his followers developed Activity Theory (Leontiev, Rubinshtein, Luria, and many others).
Later, those theories have continued their evolution, mainly in Russia, but there are several approaches in other countries, like Scandinavian Activity Theory. Currently, both in Russia and outside, have taken place some process of convergence, as the result we got CHAT.
Regarding measuring division labor, I think the best approach could be representing the structure of individual activity, because It could help in representing differences among activities, and then you could try to measure those differences.
Now, finding the structure representation of activity is not an easy task. The best representation I know, was created by Russian professor, at Moscow State University, Tatiana Gabay, but the two main documents, containing the explanation of activity structure, her doctoral dissertation and the her book of Psychological Pedagogy, are written in Russian language.
Maybe you could if you analyse individual actions. However, the question is why do you need to quantify division of labour? What will that tell you? what for? how would you connect that to the activity?
I agree with Rey and Malba. I am not sure a) whether and b) why you would measure the quantitatively "division of labour". In my understanding this relates more to the roles various subjects assume within an Activity or an Activity System (following Engestrom) rather than the amount of labour that they complete.
Also I was interested in developing a quantitative research design to examine CHAT components with quantitative indicators. In my opinion, division of labour can be quantified by means of "social network analysis" of the "instrumental ties" such as closed ties and clique analyses. In other words if we could pose the right question to participants, - I guess- it is sensible to delineate division of labour by examining social interaction structures in which participants are already embedded...
I shall investigate "social network analysis" of the "instrumental ties" such as closed ties and clique analyses.
I am currently using R to perform textual analysis on student's writing in response to collaboration work that they undertook using a multi-player Mars simulation. I am interested in the correlation between pronoun usage - I and WE - and lexical richness, attainment, etc. I wonder if I and WE occurrences in writing about collaborative activities could help us provide clues about 'division of labour'.
I have developed a general approach to quantifying the division of labor using the example of the emergence of multicellularity. I believe that this approach can be applied to your case as well.
Check out the attached articles. And if you have any questions - write to me.