Promoting student reasoning through careful task design: a comparison of three studies

Researchers have found that students as young as elementary school can engage in mathematical reasoning. Specifically, particular tasks tend to encourage this reasoning. This paper provides insight into some general characteristics of tasks that may lead to arguments that represent varied forms of reasoning. In this paper we report on arguments built by diverse student groups, of different ages, that were used to justify their solutions to problems from the fraction and counting strands of longitudinal and cross-sectional studies. We compare the characteristics of the two tasks and suggest how the implementation of tasks such as these can help elicit varied student reasoning

Integration of Action and Language Knowledge: A Roadmap for Developmental Robotics

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Improving Perception to Make Distant Connections Closer

One of the challenges for perceptually grounded accounts of high-level cognition is to explain how people make connections and draw inferences between situations that superficially have little in common. Evidence suggests that people draw these connections even without having explicit, verbalizable knowledge of their bases. Instead, the connections are based on sub-symbolic representations that are grounded in perception, action, and space. One reason why people are able to spontaneously see relations between situations that initially appear to be unrelated is that their eventual perceptions are not restricted to initial appearances. Training and strategic deployment allow our perceptual processes to deliver outputs that would have otherwise required abstract or formal reasoning. Even without people having any privileged access to the internal operations of perceptual modules, these modules can be systematically altered so as to better serve our high-level reasoning needs. Moreover, perceptually based processes can be altered in a number of ways to closely approximate formally sanctioned computations. To be concrete about mechanisms of perceptual change, we present 21 illustrations of ways in which we alter, adjust, and augment our perceptual systems with the intention of having them better satisfy our needs

Natural Language Syntax Complies with the Free-Energy Principle

Natural language syntax yields an unbounded array of hierarchically structured expressions. We claim that these are used in the service of active inference in accord with the free-energy principle (FEP). While conceptual advances alongside modelling and simulation work have attempted to connect speech segmentation and linguistic communication with the FEP, we extend this program to the underlying computations responsible for generating syntactic objects. We argue that recently proposed principles of economy in language design - such as "minimal search" criteria from theoretical syntax - adhere to the FEP. This affords a greater degree of explanatory power to the FEP - with respect to higher language functions - and offers linguistics a grounding in first principles with respect to computability. We show how both tree-geometric depth and a Kolmogorov complexity estimate (recruiting a Lempel-Ziv compression algorithm) can be used to accurately predict legal operations on syntactic workspaces, directly in line with formulations of variational free energy minimization. This is used to motivate a general principle of language design that we term Turing-Chomsky Compression (TCC). We use TCC to align concerns of linguists with the normative account of self-organization furnished by the FEP, by marshalling evidence from theoretical linguistics and psycholinguistics to ground core principles of efficient syntactic computation within active inference

The Topological Field Theory of Data: a program towards a novel strategy for data mining through data language

This paper aims to challenge the current thinking in IT for the 'Big Data' question, proposing - almost verbatim, with no formulas - a program aiming to construct an innovative methodology to perform data analytics in a way that returns an automaton as a recognizer of the data language: a Field Theory of Data. We suggest to build, directly out of probing data space, a theoretical framework enabling us to extract the manifold hidden relations (patterns) that exist among data, as correlations depending on the semantics generated by the mining context. The program, that is grounded in the recent innovative ways of integrating data into a topological setting, proposes the realization of a Topological Field Theory of Data, transferring and generalizing to the space of data notions inspired by physical (topological) field theories and harnesses the theory of formal languages to define the potential semantics necessary to understand the emerging patterns

On the Convergence of Evolutionary and Behavioral Theories of Organizations: A Tentative Roadmap

The behavioral theory of the firm has been acknowledged as one of the most fundamental pillars on which evolutionary theorizing in economics has been built. Nelson and Winter’s 1982 book is pervaded by the philosophy and concepts previously developed by Cyert, March and Simon. On the other hand, some behavioral notions, such as bounded rationality, though isolated from the context, are also at the heart of some economic theories of institutions such as transaction costs economics. In this paper, after briefly reviewing the basic concepts of evolutionary economics, we discuss its implications for the theory of organizations (and business firms in particular), and we suggest that evolutionary theory should coherently embrace an “embeddedness” view of organizations, whereby the latter are not simply efficient solutions to informational problems arising from contract incompleteness and uncertainty, but also shape the “visions of the world”, interaction networks, behavioral patterns and, ultimately, the very identity of the agents. After outlining the basic features of this perspective we analyze its consequences and empirical relevance.

Conceptualizing routines of practice that support algebraic reasoning in elementary schools: a constructivist grounded theory

There is ample literature documenting that, for many decades, high school students view algebra as difficult and do not demonstrate understanding of algebraic concepts. Algebraic reasoning in elementary school aims at meaningfully introducing algebra to elementary school students in preparation for higher-level mathematics. While there is research on elementary school students' algebraic reasoning, there is a scarcity of research on how elementary school teachers implement algebraic reasoning curriculum and how their practices support algebraic reasoning. The purpose of this study therefore was to discover practices that promote algebraic reasoning in elementary classrooms by studying elementary school teachers' practices and algebraic reasoning that the practices co-constructed. Specifically, the questions that guided the study included (a) what were the teachers' routines of practice, and (b) in what ways did the routines of practice support algebraic reasoning. I sampled On Track Learn Math project and worked with six teachers to explore their routines of practice and students' algebraic reasoning. As a participant observer, I analyzed video data of the classroom activities, memos, field notes, students' written transcripts and interview data using constructivist grounded theory approach and descriptive statistics. Member checking, data triangulation, and data coding by multiple raters ensured consistency and trustworthiness of the results. Descriptive analysis of students' written generalizations showed that about 74% of the generalizations were explicit and about 55% of the generalizations included names of variables indicating that students were learning how to reason algebraically. Data analysis also revealed five routines of practice. These routines are; (a) maintaining open-endedness of the tasks, (b) nurturing co-construction of ideas, (c) fostering understanding of variable, (d) creating a context for mathematical connections and (e) promoting understanding of generalizations. Teachers maintained open-endedness by giving minimal instructions when launching the tasks and providing students with workspaces. They nurtured co-construction of ideas by creating opportunities for students to collaborate, fostering collaboration, and balancing the support of discourse and content. They fostered understanding of variable as a changing quantity and as a relationship. Teachers created a context for mathematical connections between On Track tasks and students' everyday experiences, between student strategies, between different tasks, between On Track tasks and other curriculum ideas, and between different representations. Teachers promoted understanding of generalizations by encouraging students to justify their conjectures, to apply and evaluate peers' generalizations among other practices. These practices were dependent and informed each other