The first question Guershon Harel [*] asked about cK¢ is
My response is very simple and direct:
Based on the evidence we can get from the learner's activity, the objective is to characterize it in terms intelligible from a mathematical perspective and which can serve as inputs to take teaching decisions. Two types of evidence are easy to get: representations manipulated by the learner and operators he or she uses in order to achieve a task or to solve a problem. Actually, these operators are not always explicit but it is not impossible to have an interpretation of the learner's behaviours which makes sense from a mathematical perspective (this corresponds to the Vergnaud coup de force when he coined the concept of "theorem in action"). It is then reasonable to claim that we have a picture of the learner understanding when these representations and operators are stable within a problem-space. This has to be completed by a description of the means the learner uses to take a decision about the validity of his or her activity and the related outcomes. It is the idea of the control structure. Once we have a characterization along these four dimensions, we can conjecture a mathematical meaning, but this does not tell what are the related mental activities or cognitive structures as psychology or neuroscience would understand them. It is very likely that different learning theories would shade different lights on these characterizations. However, my claim is that such characterizations are sufficient to assess the so-called mathematical understanding, and to take teaching decisions or to design learning environments.
For the rest, cK¢ shares many of the scientific characteristics of "cognitive models": it is based on "rigorous methods", it is "capable of generating testable predictions" and of generating descriptions in "formal, mathematical or computer, languages". It does not describe processes but nothing prevents it a priori to contribute to such descriptions, this is something to explore.
Eventually, it is important to realize that cK¢ does not ambition to construct models to respond to the question "How does a child transition from additive reasoning to multiplicative reasoning?" but to the question "What are the optimal conditions to initiate and support the child transition from additive reasoning to multiplicative reasoning?"
3. To what extent is the cK¢ a cognitive model?Actually, this question comes after a more general one: (1) "What is a cognitive model and what are its purposes?" and a more direct one (2) "Is the cK¢ a cognitive model?
My response is very simple and direct:
cK¢ does not propose a framework to construct cognitive models. It does not pretend to model an "approximation to processes of humans’ mental activities" and do not ambition to be "capable of explaining mental processes or interactions among them", eventually it does not aim at answering a specific question such as "how do we learn to categorize perceptual objects?"Yet, cK¢ has a very strong relation to the learner by being focused on his or her interaction with a learning environment (more precisely the "milieu"). Indeed, cK¢ could contribute to a cognitive approach, but it is not its objective in the first place.
Based on the evidence we can get from the learner's activity, the objective is to characterize it in terms intelligible from a mathematical perspective and which can serve as inputs to take teaching decisions. Two types of evidence are easy to get: representations manipulated by the learner and operators he or she uses in order to achieve a task or to solve a problem. Actually, these operators are not always explicit but it is not impossible to have an interpretation of the learner's behaviours which makes sense from a mathematical perspective (this corresponds to the Vergnaud coup de force when he coined the concept of "theorem in action"). It is then reasonable to claim that we have a picture of the learner understanding when these representations and operators are stable within a problem-space. This has to be completed by a description of the means the learner uses to take a decision about the validity of his or her activity and the related outcomes. It is the idea of the control structure. Once we have a characterization along these four dimensions, we can conjecture a mathematical meaning, but this does not tell what are the related mental activities or cognitive structures as psychology or neuroscience would understand them. It is very likely that different learning theories would shade different lights on these characterizations. However, my claim is that such characterizations are sufficient to assess the so-called mathematical understanding, and to take teaching decisions or to design learning environments.
For the rest, cK¢ shares many of the scientific characteristics of "cognitive models": it is based on "rigorous methods", it is "capable of generating testable predictions" and of generating descriptions in "formal, mathematical or computer, languages". It does not describe processes but nothing prevents it a priori to contribute to such descriptions, this is something to explore.
Eventually, it is important to realize that cK¢ does not ambition to construct models to respond to the question "How does a child transition from additive reasoning to multiplicative reasoning?" but to the question "What are the optimal conditions to initiate and support the child transition from additive reasoning to multiplicative reasoning?"
