cK¢ takes up the challenge of modeling learners understanding (a response to Guershon Harel - continued)

Several of the questions Guershon Harel [*] asked after my talk at the PME-NA conference in Chicago concern the scope and objectives of the cK¢ modeling framework. In this post, I take each of these questions and give a short answer leaving for specific posts more elaborated answers when needed. So, here it is:

4. Is the cK¢ a model of learning processes or learning states?

The answer is very simple: cK¢ provides a framework for modeling learning states. Indeed learning processes are of a paramount importance, but they are in my opinion more an object of study for psychology than for mathematics education. Indeed, I don't confuse "learning processes" which are of a mental and intellectual nature, and "problem solving processes" which correspond (to make it simple) to the activity the learner engage when he or she has to solve a problem in a given situation. We need to understand and model these processes, but even if they may inform us about learning processes they are only a dimension of them.

5. Is the cK¢ a model of a learner (period), a model of a learner learning mathematics, or a model of a learner in a mathematics classroom setting?

cK¢ provides a framework to model learners' understanding (learning states, as just said) in mathematics from a situated perspective; situations may be set up within a classroom or in an other context. Actually, the objective is slightly larger, I would claim that cK¢ is a framework to model mathematical understanding taking into account the situational characteristics, not being restricted to learners. A key idea when I started the project was to find a way to model mathematical conceptions with the same tools, be it they conceptions of novices or experts, wrong or correct from whatever knowledgeable perspective.

6. What exactly are the challenging aspects of modeling learning relative to modeling content and pedagogy? 

Anyone will expect the content to be in some sense "correct" and explicit enough to be defined precisely as a content to be taught. Still, there are challenging aspects related to its nature; for example, to model algebra or geometry from an epistemic and teaching perspective is not of the same level of difficulty.
Concerning pedagogy, which is in the first place the product of a practice, the related knowledge is largely implicit. Practitioners can share and discuss their expertise within their community or with teacher students in an apprenticeship approach, but the communication register is largely based on a pragmatic co-construction of meaning referring to a shared practice; as a result it is difficult to model. The challenge is to make explicit what is largely knowledge in action. However, there are progresses as witnessed by research on tutoring and adaptive learning systems.
In line with what I wrote before, I will not consider learning but "the best conditions for learning" (best or optimal). So, I would consider the question: what are the challenging aspects of modeling (determining) the best conditions for learning compared to those of modeling content and pedagogy? I would say that these challenges are very close the one to the other. First, part of the challenge comes from the nature of the content at stake, in particular the role of representations and the complexity of validating the related mathematical statements. Arithmetic, algebra, geometry, calculus, probability, discrete mathematics raise their own specific learning challenges within mathematics. Then, determining the best conditions for learning requires knowing some critical things about the initial state of knowledge of the learners (as a matter of fact, we can only teach people who know); in other words it depends on our knowledge of the conceptions which have to evolve or to be rejected. Eventually, one can say that the challenge is of knowing the knowledge at stake from a learning perspective, understanding learners initial understanding, and being able to design situations likely to stimulate, support and validate the construction of new knowledge. Eventually, modeling pedagogy consists in stating principles for designing situations which implements the "best conditions for learning" under multiple constraints: curricula, institutional standard, cultural and economical environment, time and all material means to make the class working properly.

7. What are the interdependent relationships among these three models? 

Indeed, as the above answers suggests it, the three models are tightly related. In particular, from an educational perspective modeling knowledge is under learning constraints because what we need is not a "knowledge model" for itself but a model of the intended learning outcomes. This knowledge (intended learning outcomes) must be learnable (accessible to learners) and teachable (manageable by teachers); actually, the objective of the didactical transposition is exactly to produce this knowledge which is always at a distance from a knowledge of reference which to some extend justify it. 

8. What is the efficacy of such models if they are constructed independently from each other? In particular, can models of content and pedagogy be viable without the presence of a learning model?

Be it explicitly the case or not, any pedagogical model includes a learning model; I mean a model of the (claimed) best conditions for learning. 

Then, a more philosophically oriented, yet critical, question is

9. Are cognitive models of thinking possible?

Once we have agreed on what means "cognitive", "model" and "thinking" my answer would be: yes... but a discussion of this answer may go far beyond my field of expertise and beyond the scope of this blog as well.