Cristian Voica

University of Bucharest | Unibuc · Department of Mathematics

While patterning was commonly seen as evidence of mathematical thinking, interdisciplinary interest has recently increased due to pattern-recognition applications in artificial intelligence. Within two empirical studies, we analyze the analogical-transfer capability of primary school students when completing three types of bi-dimensional patterns,...

Affects are intuitively accepted as having a role in the key stages that determine success in problem-solving (PS) and problem-posing (PP). Two disjoint groups of prospective mathematics teachers with similar background and competences have been exposed to PS and PP activities, respectively, and they had to describe their affective states during th...

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 s...

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 ca...

While a wide range of approaches and tools have been used to study children’s creativity in school contexts, less emphasis has been placed on revealing students’ creativity at university level. The present paper is focused on defining a tool that provides information about mathematical creativity of prospective mathematics teachers in problem-posin...

We use statistical data to identify the problems that appear to be difficult with students in a problem-solving contest counting 9,580 participants from grades 2 and 3. Our analysis considers the level of complexity of the reading and problem-solving processes, as well as the diversity of the forms the information is conveyed by. We found the stude...

Effective communication in the classroom is a key element for learning, yet when and how future teachers should acquire such competence is not clear. In this article we explore students-prospective teachers’ written productions of a set of instructions in a learning situation. Through three emblematic cases we illustrate how a communication task fo...

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 i...

The paper investigates the use of a 3D modular paper game within educational programs. The experiment involved students in grades 5 to 9, engaged in a guided play workshop, during a summer camp, in 2012, Romania. We report the conclusions on the use of this educational resource regarding its effectiveness in enhancing creativity and spatial intuiti...

We look at dynamic thinking and static thinking in relation to mathematical problem solving. We examine the distribution of answers chosen by large samples of students to multiple-choice problems. Our empirical data suggest that static thinking activated by students in problem solving is likely to be responsible for a certain pattern of students’ r...

The links between the mathematical and cognitive models that interact during problem solving are explored with the purpose of developing a reference framework for designing problem-posing tasks. When the process of solving is a successful one, a solver successively changes his/her cognitive stances related to the problem via transformations that al...

We analyse a student's creative expression in problem-posing situations. The findings suggest that a small but significant difference in creative behaviour at an interval of one year (from 11 to 12 years old) indicates a passage from cognitive variety to small incremental changes, under the constraints of a strong cognitive frame. We found a specia...

This paper presents the results of an experiment in which fourth to sixth graders with above-average mathematical abilities modified a given problem. The experiment found evidence of links between problem posing and cognitive flexibility. Emerging from organizational theory, cognitive flexibility is conceptualized through three primary constructs:...

We study the relationship between creative tasks and the quality of learning. We found that when confronted with problem posing contexts, a high achiever in mathematics displays cognitive flexibility, and reaches a number of new understandings that allow deep learning. Consequently, efficient learning can be promoted in a context that combines prob...

The paper presents a qualitative analysis of the impact of a conversion program on the teaching activities of the participants. We found changes in conceptions and attitudes towards the new domain, and we interpreted them as evidence that the participants became aware of the importance of the specialized content knowledge in effective teaching. It...

This study provides evidence that pre-service teachers who develop and implement educational projects during their training are more likely to better understand the complexity of the teaching profession. The students-prospective teachers of our sample had to design small scale projects and to apply them in school settings. In addition, they had to...

We analyze the statistical distribution of the answers given by 2nd to 10th graders to a set of number line problems. To structure our analysis of students’ misconceptions, we identified three clusters of problems related to the number line. Our analysis shows that neglecting one of the main features of the number line can be a potential cause for...

When reasoning about infinite sets, children seem to activate four categories of conceptual structures: geometric (g-structures), arithmetic (a-structures), fractal-type (f-structures), and density-type (d-structures). Students select different problem-solving strategies depending on the structure they recognize within the problem domain. They natu...

We explore children’s strategies in comparing infinite sets of numbers, based on an empirical study. We report four categories of structures that the children identified during this process: geometrical-based structures, topological-based structures, fractal-type structures, and arithmetical-type structures. Using the identified structure, students...

Based on an empirical study, we explore children’s primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endow Q with a...

In this paper, we obtain a complete classification of all rational surfaces embedded in P4 so that all their exceptional curves are lines. These surfaces are exactely the rational surfaces shown by I.Bauer to project isomorphicaly from P5 from one of their points, although no a priori reason is known why such a surface should be projectable in this...

We study rationally connected (projective) manifolds $X$ via the concept of a model $(X,Y)$ , where $Y$ is a smooth rational curve on $X$ having ample normal bundle. Models are regarded from the view point of Zariski equivalence, birational on $X$ and biregular around $Y$ . Several numerical invariants of these objects are introduced and a notion o...

In this paper we nd a bound of the degree of rational surfaces embedded in P4 with a linear system of type j L px0 x1 : : : xr j We determine all the possible (families of) rational surfaces embedded in P4 with a linear system as above, for the particular case p = 2.

The contractibility of a curve E on a projective surface X is equivalent to the existence of a semi-affine neighbourhood of E in X.

In this paper we obtain information about an algebraic set X using the equations of X. In a particular case, we can determine equations for the irreducible components of X.

We explore different types of behavior during the problem posing process by looking at the ways students value the problem data in solving and extending their own posed problems. Based on the outcomes of these analyses we explain the differences in students' success and failure in the problem posing approaches in relation to the level of understand...

A proof of the existence of a symmetry axis for a conic is given which is independent of the metric structure.

We determine the rational surfaces embedded in ℙ 4 using a 2-homogeneous linear system, i.e., the linear system of plane curves with base locus of multiplicity 2.