Singer, F. M. (2009). The Dynamic Infrastructure of Mind - a Hypothesis and Some of its Applications, New Ideas in Psychology, 27, 48–74

A mechanism underlying the computational properties of the cognitive architecture is construed based on a minimal list of operational clusters. This general processing mechanism constitutes the dynamic infrastructure of mind (DIM). DIM consists in categories of mental operations foundational for learning that contain inborn components called inner operations, which are self-developing in the interaction mind-environment. Within the DIM paradigm, the input cognitive systems are not domain specific or core-knowledge specific, they are operational specific and capable of further developments that become domain specific while experiencing the environment. Arguments for this construal come from three sources: literature review, data collected through classroom observations, and a four-year experimental study of teaching and learning mathematics in primary grades. The outcomes of that experiment led to a methodology of learning based on activating the operational infrastructure of mind, which enhances students’ flexibility of thinking and predicts the capacity to solve creatively a variety of problems.

Singer, F.M. (2007). Beyond Conceptual Change: Using Representations to Integrate Domain-Specific Structural Models in Learning Mathematics. Mind, Brain, and Education, 1(2), pp. 84-97, DOI: 10.1111/j.1751-228X.2007.00009.x, ISSN: 1751-2271.

Effective teaching should focus on representational change, which is fundamental to learning and education, rather than conceptual change, which involves transformation of theories in science rather than the gradual building of knowledge that occurs in students. This article addresses the question about how to develop more efficient strategies for promoting representational change across cognitive development. I provide an example of an integrated structural model that highlights the underlying cognitive structures that connect numbers, mathematical operations, and functions. The model emphasizes dynamic multiple representations that students can internalize within the number line and which lead to developing a dynamic mental structure. In teaching practice, the model focuses on a counting task format, which integrates a variety of activities, specifically addressing motor, visual, and verbal skills, as well as various types of learning transfer.

Singer, F. M., & Moscovici, H. (2008). Teaching and learning cycles in a constructivist approach to instruction. Teaching and Teacher Education, Vol 24/6 pp 1613-1634, DOI:10.1016/j.tate.2007.12.002., ISSN: 0742-051X.

This study attempts to analyze and synthesize the knowledge collected in the area of conceptual models used in teaching and learning during inquiry-based projects, and to propose a new frame for organizing the classroom interactions within a constructivist approach. The IMSTRA model consists in three general phases: Immersion, Structuring, Applying, each with two sub-phases that highlight specific roles for the teacher and the students. Two case studies, one for mathematics in grade 9 and another for science in grade 3, show how the model can be implemented in school, making inquiry realistic in regular classes. Beyond its initial purpose, the IMSTRA model proved to be a powerful tool in curriculum development, being used in producing mathematics textbooks, as well as in developing teaching courses for a long-distance teacher training program.

Singer, F.M., Voica, C. (2008). Between perception and intuition: thinking about infinity, Journal of Mathematical Behavior, 27(3), 188-205, PII: S0732-3123(08)00025-4, DOI: 10.1016/j.jmathb.2008.06.001, ISSN: 0732-3123.

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 discrete structure by making transfers from N to Q. In a continuous context, children are more likely to mobilize a topological perception. Evidence for a secondary perception of N arises from students’ propensities to develop infinite sequences of natural numbers, and from their ability to prove that N is infinite. Children’s perceptions on infinity change along the school years. In general, the perceptual dominance moves from sequential (processional) to topological across development. However, we found that around 11–13 years old, processional and topological perceptions interfere with each other, while before and after this age they seem to coexist and collaborate, one or the other being specifically activated by the nature of different tasks.

Singer, F.M. (2009). Recent research in cognitive science and neuroscience: is it relevant to mathematics learning? Proceedings of the 6th International Congress of Romanian Mathematicians, pp. 587-597, Bucharest: Romanian Academy of Science P.H.

During the last four decades, a large body of research was devoted to analyzing number representation in humans. The findings in neuroscience and their adequate interpretation in relation with cognition may reshape the traditional ways of  teaching and learning. This paper synthesizes recent research on human cognition, emphasizing that: there is a complex relationship mind-and-brain; numerical tasks activate some cortex zones that are mathematically specific; human beings possess inborn numerical predispositions independent of language; language plays a scaffolding role in developing the computational capacities; and advanced mathematics is based on cultural tools developed by humans at the intersection between innate predispositions and environmental interactions. Educational consequences derived from here might put an emphasis on some capacities traditionally neglected, such as  estimations and approximations, which facilitate access to inborn
predispositions and connect to real life. Moreover, these findings confirm the hypothesis that a dynamic training that stimulates connections, relations, and semantic associations is more effective for mathematics learning.

Singer, F. M., Voica, C. (2009). When the infinite sets uncover structures: an analysis of students’ reasoning on infinity. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, C. (eds.). In Search for Theories in Mathematics Education. Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, vol. 5, pp. 121-128. Thessaloniki, Greece: PME.

Singer, F.M. (2007). Balancing Globalisation and Local Identity in the reform of Education in Romania. In B. Atweh, M. Borba, A. Barton, D. Clark, N. Gough, C. Keitel, C. Vistro-Yu, and R. Vithal (Eds), Internalisation and Globalisation in Mathematics and Science Education, Dordrecht: Springer Science, pp.365-382, 2007, ISBN: 978-1-4020-5907-0.

This chapter reports on the mechanisms activated by the reform of education at the confluence of economic, political and sociocultural factors in a system in transition to democracy and a functional market economy. The paper analyses the fluctuating balance between ideological and professional positions in the decision making process based on a case study of the curriculum reform in Romania, which has been developed within the Education Reform Project co-financed by the Romanian government and the World Bank. The tensions among customs, traditions, mentalities, the will to integrate, and the desire to keep a cultural specificity in a global world, are not yet resolved. Such tensions bring new issues into the contemporary debates regarding globalisation and the new polarization of power. In this context, the reform in mathematics and science education is approached from the perspective of the relationship between knowledge and power during the industrial époque in Eastern Europe, stressing some of its implications for the post-industrial era and sketching some future developments in the framework of the knowledge society.

Singer, M., Sarivan, L. (2009). Curriculum Reframed. MI and New Routes to Teaching and Learning in Romanian Universities. In J.Q. Chen, S. Moran, H. Gardner (eds.), Multiple intelligences around the world. Pp. 230-244. 408 pages New York: Jossey-Bass Inc Pub.

Multiple Intelligences story in Romania started on the shattered foundations of a school system overwhelmingly burdened by its communist heritage.  The still very uniform school could not adopt the individualized approach, yet got attracted by the radical innovation and hastily turned MI into glamorous fashion. Could we try any better?  In our attempt to promote learning beyond the facts we have seen MI as a good inspiration to restructure the teacher training curriculum. We thus promote a teacher training program that encourage students to identify their clichés, provide contexts for them to structure and restructure domain-related and teaching competence, and challenge them to transfer their acquisition into similarly planned sessions for their own students. For methodological purposes we structured a multirepresentational model that supports deep understanding and metacognition. The model basically uses a transfer frame in which a concept turns into a procedure to promote another concept learning via various contexts.

Singer, F. M. (2007). Modelling both complexity and abstraction: a paradox? In W. Blum, P. Galbraith, H. W. Henn and N. Mogens (Eds.), Applications and Modelling in Mathematics Education, New York: Springer, Chapter III.3.2.: pp. 233-240, 2007, ISBN-13: 978-0-387-29820-7, Library of Congress Control Number: 2006932713.

Do we need models in explaining the outer world and the self? What types of models might be helpful in school to explain both complexity and abstraction? What level of representation is appropriate? What dimensions of training should be focused on in constructing an inquiry-based learning? How could these dimensions be reflected in developing students’ competencies? Analysing the dual relationship between complexity and abstraction, the study pro-poses some strategies to enhance learning in a model-building environment.

Singer, F.M. (2001). Structuring the information – a new way of perceiving the content of learning, Zentralblatt für Didaktik der Mathematik (ZDM)/International Reviews on Mathematical Education, MATHDI, 6/2001, p. 204-217.

Dominating information by formation seems to be the only realistic solution to overcome the crisis generated by the higher and higher accumulation of information in each domain. The paper is proposing a model of building structured knowledge able to generate in the child’s mind strategies for efficient processing of information.

A Cognitive Model for Developing a Competence-based Curriculum in Secondary Education

Singer, F.M. (2006). A Cognitive Model for Developing a Competence-based Curriculum in Secondary Education. In: Al. Crisan (Ed.), Current and Future Challenges in Curriculum Development: Policies, Practices and Networking for Change. Bucureşti: Education 2000+ Publishers. Humanitas Educational, pp. 121-141, ISBN-(13) 978-973-689-104-5

The last decade brought about a vivid discussion concerning a competence-based curriculum in order to better train the students for the knowledge society. The present paper describes a cognitive model that aims at designing competences for secondary education. It consists in six operational categories which are combined based on epistemological and pedagogical constraints and allows structuring knowledge from a domain expert’s perspective. The model was applied in developing the new curriculum for secondary education in Romania. A few examples from different subject matters are meant to show how the model is functioning in concrete situations.

Pelczer, I., Singer, F. M., Voica, C. (2009). Patterns of change in solving dynamic and static problems. “Quaderni di Ricerca in Didattica (Matematica)”, Supplemento n. 2, 2009., pp. 288-292. G.R.I.M. (Department of Mathematics, University of Palermo, Italy). Proceedings CIEAEM 61 – Montréal, Quebéc, Canada, July 26-31, 2009.

Dans cet article, nous analysons l’évolution dans le temps du nombre de réponses correctes et incorrectes ainsi que du pourcentage de l’absence de réponses à des problèmes provenant d’un test international à choix. Le principal intérêt à ce travail est de relier certaines caractéristiques de problèmes à certains changements. Pour catégoriser les problèmes, nous avons retenu les catégories « dynamique » et « statique ». Les problèmes dynamiques impliquent un changement dans leur configuration, une transformation du contexte ou une analyse des variantes ce que n’impliquent pas les problèmes statiques. Ces deux catégories de problèmes semblent définir différents patterns de réponses et ce, à différents niveaux scolaires. La connaissance de tels comportements pourrait aider les enseignants à mieux structurer les activités de résolution de problèmes tout au long de la scolarité.

Singer, F. M. (2007). Language across the mathematics curriculum: some aspects related to cognition. In S. Ongstad (ed.), Language in Mathematics? A comparative study of four national curricula, Strasbourg: Council of Europe, Language Policy Division, www.coe.int/lang, pp. 71-82.

Singer, F.M. (2004). The dynamic structural learning: from theory to practice. In M. Niss & E. Emborg (Eds.) 10th International Congress on Mathematical Education (ICME 10) Proceedings, Copenhagen, Denmark: IMFUFA, ISBN: 978-87-7349-733-3.

Singer, F.M. & Voica, C. (2010). In Search of Structures: How Does the Mind Explore Infinity? Mind, Brain and Education, 4(2), p. 81-93.

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 naturally search for structures in challenging learning contexts. This tendency to search for structure might be a characteristic of human cognition and a necessary condition for human knowledge development. For example, specific fractal structures are intrinsic to concepts such as the numerical system that have been developed by the human race over a long period of time. When these structures are emphasized within teaching, they can facilitate the deep understanding of several basic concepts, in mathematics and beyond.

Singer, F.M. (2002). Developing mental abilities through structured teaching methodology, in: A. Rogerson (Ed.), The Humanistic Renaissance in Mathematics Education, Palermo: The Mathematics Education into the 21st Century Project, p. 353-357