Engaging in problem posing activities in a dynamic geometry setting and the development of prospective teachers’ mathematical knowledge

In the present study we explore changes in perceptions of our class of prospective mathematics teachers (PTs) regarding their mathematical knowledge. The PTs engaged in problem posing activities in geometry, using the “What If Not?” (WIN) strategy, as part of their work on computerized inquiry-based activities. Data received from the PTs’ portfolios reveals that they believe that engaging in the inquiry-based activity enhanced both their mathematical and meta-mathematical knowledge. As to the mathematical knowledge, they deepened their knowledge regarding the geometrical concepts and shapes involved, and during the process of creating the problem and checking its validity and its solution, they deepened their understanding of the interconnections among the concepts and shapes involved. As to meta-mathematical knowledge, the PTs refer to aspects such as the meaning of the givens and their relations, validity of an argument, the importance and usefulness of the definitions of concepts and objects, and the importance of providing a formal proof.     

What Is “System”: The Information-Theoretic Arguments

The problem of “what is ‘system’?” is in the very foundations of modern quantum mechanics. Here, we point out the interest in this topic in the information-theoretic context. E.g., we point out the possibility to manipulate a pair of mutually non-interacting, non-entangled systems to employ entanglement of the newly defined “(sub)systems” consisting the one and the same composite system. Given the different divisions of a composite system into “subsystems”, the Hamiltonian of the system may generate in general non-equivalent quantum computations. Redefinition of “subsystems” of a composite system may be regarded as a method for avoiding decoherence in the quantum hardware. In principle, all the notions refer to a composite system as simple as the hydrogen atom.     

Problem posing via “what if not?” strategy in solid geometry — a case study

The study describes the kinds of problems posed by pre-service teachers on the basis of complex solid geometry tasks using the “what if not?” strategy and the educational value of such an activity. Twenty-eight pre-service teachers participated in two workshops in which they had to pose problems on the basis of given problems. Analysis of the posed problems revealed a wide range of problems including those containing a change of one of the numerical data to another specific one, to a proof problem. Different kinds of posed problems enlightened some phenomena such as a bigger frequency of posed problems with another numerical value and a lack of posed problems including formal generalization. We also discuss the educational strengths of problem posing in solid geometry using the “what if not?” strategy, which could make the learner rethink the geometrical concepts he uses while creating new problems, make connections between the given and the new concepts and as a result deepen his understanding of them.