Layers of generality and types of generalization in pattern activities

Pattern generalization is considered one of the prominent routes for in-troducing students to algebra. However, not all generalizations are al-gebraic. In the use of pattern generalization as a route to algebra, we —teachers and educators— thus have to remain vigilant in order not to confound algebraic generalizations with other forms of dealing with the general. But how to distinguish between algebraic and non-algebraic generalizations? On epistemological and semiotic grounds, in this arti-cle I suggest a characterization of algebraic generalizations. This char-acterization helps to bring about a typology of algebraic and arithmetic generalizations. The typology is illustrated with classroom examples

The Role of Context-Related Parameters in Adults’ Mental Computational Acts

Researchers who have carried out studies pertaining to mental computation and everyday mathematics point out that adults and children reason intuitively based upon experiences within specific contexts; they use invented strategies of their own to solve real-life problems. We draw upon research areas of mental computation and everyday mathematics to report on a study that investigated adults’ use of mental mathematics in everyday settings. In this paper, we report on one adult’s use of mental computation at work and highlight the role of context and context related parameters in his mental mathematical activities

Identifying Quantum Structures in the Ellsberg Paradox

Empirical evidence has confirmed that quantum effects occur frequently also outside the microscopic domain, while quantum structures satisfactorily model various situations in several areas of science, including biological, cognitive and social processes. In this paper, we elaborate a quantum mechanical model which faithfully describes the 'Ellsberg paradox' in economics, showing that the mathematical formalism of quantum mechanics is capable to represent the 'ambiguity' present in this kind of situations, because of the presence of 'contextuality'. Then, we analyze the data collected in a concrete experiment we performed on the Ellsberg paradox and work out a complete representation of them in complex Hilbert space. We prove that the presence of quantum structure is genuine, that is, 'interference' and 'superposition' in a complex Hilbert space are really necessary to describe the conceptual situation presented by Ellsberg. Moreover, our approach sheds light on 'ambiguity laden' decision processes in economics and decision theory, and allows to deal with different Ellsberg-type generalizations, e.g., the 'Machina paradox'.Comment: 16 pages, no figures. arXiv admin note: substantial text overlap with arXiv:1208.235

The challenge of representative design in psychology and economics

The demands of representative design, as formulated by Egon Brunswik (1956), set a high methodological standard. Both experimental participants and the situations with which they are faced should be representative of the populations to which researchers claim to generalize results. Failure to observe the latter has led to notable experimental failures in psychology from which economics could learn. It also raises questions about the meaning of testing economic theories in “abstract” environments. Logically, abstract tests can only be generalized to “abstract realities” and these may or may not have anything to do with the “empirical realities” experienced by economic actors.Experiments, representative design, sampling, Leex

On the Development of Early Algebraic Thinking

This article deals with the question of the development of algebraic thinking in young students. In contrast to mental approaches to cognition, we argue that thinking is made up of material and ideational components such as (inner and outer) speech, forms of sensuous imagination, gestures, tactility, and actual actions with signs and cultural artifacts. Drawing on data from a longitudinal classroom-based research program where 8-year old students were followed as they moved from Grade 2 to Grade 3 to Grade 4, our developmental research question is investigated in terms of the manner in which new relationships between embodiment, perception, and symbol-use emerge and evolve as students engage in patterning activities