Cristian Voica

Is Problem Posing a Tool for Identifying and Developing Mathematical Creativity

The mathematical creativity of fourth to sixth graders, high achievers in mathematics, is studied in relation to their problem-posing abilities. The study reveals that in problem-posing situations, mathematically high achievers develop cognitive frames that make them cautious in changing the parameters of their posed problems, even when they make interesting generalizations. These students display a kind of cognitive flexibility that seems mathematically specialized, which emerges from gradual and controlled changes in cognitive framing. More precisely, in a problem-posing context, students’ mathematical creativity manifests itself through a process of abstraction-generalization based on small, incremental changes of parameters, in order to achieve synthesis and simplification. This approach results from a tension between the students’ tendency to maintain a built-in cognitive frame, and the possibility to overcome it, which is constrained by their need to devise mathematical problems that are coherent and consistent.

Cognitive Variety in Rich-Challenging Tasks

Cognitive flexibility—a parameter that characterizes creativity—results from the interaction of various factors, among which is cognitive variety. Based on an empirical study, we analyze students’ and experts’ cognitive variety in a problem-posing context. Groups of students of different ages and studies (from primary to university) were asked to start from an image rich in mathematical properties, and generate as many problem statements related to the given input as possible. The students’ products were compared in-between, and to the problems posed by a group of experts (teachers of mathematics and researchers) who received the same images as input. The study revolves around the question: “In what ways does cognitive variety depend on age or training in mathematically promising individuals?” We found that cognitive variety seems randomly distributed among the groups we tested, contradicting the intuitive idea that this is age (and training) related, except at the expert level. In addition, when talking about mathematical creativity, more sophisticated parameters, such as validity, complexity and topic variety, as well as the potential of respondents’ products to break a well-internalized frame have to be taken into account. All those are to be balanced against the person’s level of expertise in the specified domain.

When Mathematics Meets Real Objects: How Does Creativity Interact with Expertise in Problem Solving and Posing?

The paper analyzes the results of activities undertaken by Mathematics students enrolled in a pre-service teacher-training program. Students were given the task to describe the way of building a figure from which one could get a box, to identify the geometric properties that allow producing the box, and to propose new models from which new boxes can be obtained. For creativity analysis, a cognitive flexibility framework has been used, within which students’ cognitive variety, cognitive novelty, and their capacity to make changes in cognitive framing are analyzed. The analysis of some specific cases led to the conclusion that creativity manifestation is conditioned by a certain level of expertise. In the process of building a solution for a nonstandard problem, expertise and creativity support and mutually develop each other, enabling bridges to the unknown. This interaction leads also to an increase in expertise. Moreover, in order to get individual relevant data, the identification of creativity should take place based on tasks situated in the proximal range of the person’s expertise but exceeding his/her actual level of expertise at a time.

A Knowledge Discovery Platform for Spatial Education: Applications to Spatial Decomposition and Packing

The workshop aims to introduce the XColony Knowledge Discovery Kit -a new teaching platform based on geometric manipulatives and designed in the STEM education context for training creativity and spatial intelligence in primary and middle school students. After a brief introduction of the basic concepts, participants are invited to evaluate the platform from students’ perspective by actively participating in hands-on mini projects, with the goal of constructing 3D structures that allow them to discover new geometric properties. A relevant case study on how the platform can be utilized in class is presented and the participants discuss and identify other mathematical concepts and educational activities conducted in a similar manner. The workshop concludes with a test that participants can voluntarily take, or they can take it home and use it as a selfevaluation tool for spatial intelligence.