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

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

Multi-modal meaning – An empirically-founded process algebra approach

Humans communicate with different modalities. We offer an account of multi-modal meaning coordination, taking speech-gesture meaning coordination as a prototypical case. We argue that temporal synchrony (plus prosody) does not determine how to coordinate speech meaning and gesture meaning. Challenging cases are asynchrony and broadcasting cases, which are illustrated with empirical data. We propose that a process algebra account satisfies the desiderata. It models gesture and speech as independent but concurrent processes that can communicate flexibly with each other and exchange the same information more than once. The account utilizes the psi-calculus, allowing for agents, input-output-channels, concurrent processes, and data transport of typed lambda-terms. A multi-modal meaning is produced integrating speech meaning and gesture meaning into one semantic package. Two cases of meaning coordination are handled in some detail: the asynchrony between gesture and speech, and the broadcasting of gesture meaning across several dialogue contributions. This account can be generalized to other cases of multi-modal meaning

How Do Gestures Influence Thinking and Speaking? The Gesture-for-Conceptualization Hypothesis.

Peer reviewedPostprin

Directional adposition use in English, Swedish and Finnish

Directional adpositions such as to the left of describe where a Figure is in relation to a Ground. English and Swedish directional adpositions refer to the location of a Figure in relation to a Ground, whether both are static or in motion. In contrast, the Finnish directional adpositions edellä (in front of) and jäljessä (behind) solely describe the location of a moving Figure in relation to a moving Ground (Nikanne, 2003). When using directional adpositions, a frame of reference must be assumed for interpreting the meaning of directional adpositions. For example, the meaning of to the left of in English can be based on a relative (speaker or listener based) reference frame or an intrinsic (object based) reference frame (Levinson, 1996). When a Figure and a Ground are both in motion, it is possible for a Figure to be described as being behind or in front of the Ground, even if neither have intrinsic features. As shown by Walker (in preparation), there are good reasons to assume that in the latter case a motion based reference frame is involved. This means that if Finnish speakers would use edellä (in front of) and jäljessä (behind) more frequently in situations where both the Figure and Ground are in motion, a difference in reference frame use between Finnish on one hand and English and Swedish on the other could be expected. We asked native English, Swedish and Finnish speakers’ to select adpositions from a language specific list to describe the location of a Figure relative to a Ground when both were shown to be moving on a computer screen. We were interested in any differences between Finnish, English and Swedish speakers. All languages showed a predominant use of directional spatial adpositions referring to the lexical concepts TO THE LEFT OF, TO THE RIGHT OF, ABOVE and BELOW. There were no differences between the languages in directional adpositions use or reference frame use, including reference frame use based on motion. We conclude that despite differences in the grammars of the languages involved, and potential differences in reference frame system use, the three languages investigated encode Figure location in relation to Ground location in a similar way when both are in motion. Levinson, S. C. (1996). Frames of reference and Molyneux’s question: Crosslingiuistic evidence. In P. Bloom, M.A. Peterson, L. Nadel & M.F. Garrett (Eds.) Language and Space (pp.109-170). Massachusetts: MIT Press. Nikanne, U. (2003). How Finnish postpositions see the axis system. In E. van der Zee & J. Slack (Eds.), Representing direction in language and space. Oxford, UK: Oxford University Press. Walker, C. (in preparation). Motion encoding in language, the use of spatial locatives in a motion context. Unpublished doctoral dissertation, University of Lincoln, Lincoln. United Kingdo

Sobre el desarrollo de pensamiento algebraico temprano

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.Este artículo aborda la cuestión del desarrollo del pensamiento algebraico en estudiantes jóvenes. En contraste con los enfoques mentales de la cognición, sostenemos que el pensamiento está compuesto por componentes materiales y del mundo de las ideas tales como el discurso (interior y exterior), formas de imaginación sensitiva, gestos, tacto y acciones reales con signos y artefactos culturales. Con base en datos obtenidos de un programa de investigación longitudinal basado en el aula en el que se siguió el paso de estudiantes de 8 años de segundo grado a tercero y a cuarto, nuestra pregunta de investigación acerca del desarrollo es investigada en términos de la forma en que surgen y evolucionan nuevas relaciones entre el cuerpo, la percepción y el inicio del uso de símbolos a medida que los estudiantes participan en actividades sobre patrones.This article is a result of research programs funded by the Social Sciences and Humanities Research Council of Canada (SSHRC/CRSH)

Gesture meaning needs speech meaning to denote – A case of speech-gesture meaning interaction

We deal with a yet untreated issue in debates about linguistic interaction, namely a particular multi-modal dimension of meaning-dependence. We argue that the shape interpretation of speech-accompanying iconic gestures is dependent on its co-occurrent speech. Since there is no prototypical solution for modeling such a dependence, we offer an approach to compute a gesture’s meaning as a function of its speech context

Artificial Intelligence And Big Data Technologies To Close The Achievement Gap.

We observe achievement gaps even in rich western countries, such as the UK, which in principle have the resources as well as the social and technical infrastructure to provide a better deal for all learners. The reasons for such gaps are complex and include the social and material poverty of some learners with their resulting other deficits, as well as failure by government to allocate sufficient resources to remedy the situation. On the supply side of the equation, a single teacher or university lecturer, even helped by a classroom assistant or tutorial assistant, cannot give each learner the kind of one-to-one attention that would really help to boost both their motivation and their attainment in ways that might mitigate the achievement gap. In this chapter Benedict du Boulay, Alexandra Poulovassilis, Wayne Holmes, and Manolis Mavrikis argue that we now have the technologies to assist both educators and learners, most commonly in science, technology, engineering and mathematics subjects (STEM), at least some of the time. We present case studies from the fields of Artificial Intelligence in Education (AIED) and Big Data. We look at how they can be used to provide personalised support for students and demonstrate that they are not designed to replace the teacher. In addition, we also describe tools for teachers to increase their awareness and, ultimately, free up time for them to provide nuanced, individualised support even in large cohorts