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Computer-supported Cooperative Work
Published in Giuseppe Mantovani, New Communication Environments, 2021
One particularly effective and widespread form of incorporation in artifacts of control functions which compete for power and authority is what Bowers (1992) calls ‘the politics of formalism’. Let us designate formalism as ‘a representational system of a certain sort. A formalism generates representations through the operation of rules over some vocabulary. The elements which make up the vocabulary and the terms which constitute the rules may represent human or machine action, computational procedures or operations, etc.’ (ibid., p.234). This allows us to separate distinct, countable elements, to see complex operations as a composition of simple components, to give temporal order to actions to be performed and, finally, ‘to distinguish between the legal and the illegal. Only some representations are allowable, only some orders or compositions can be realized or recognized in a particular formalism’ (ibid.). So formalism defines an order, priorities, a principle of discrimination between what is legal and what is not. In this sense, the assumption in a cooperative system of a formal model coincides with the establishment of a control function.
The Early Shaping of Cognitive Science by Artificial Intelligence
Published in Alessio Plebe, Pietro Perconti, The Future of the Artificial Mind, 2021
Alessio Plebe, Pietro Perconti
The attribute ‘formal’ can have different meanings when applied to ‘languages’. Within the domain of logic, Dutilh Novaes (2012) identified at least two main different notions: formal as computable and formal as de-semantification. The latter corresponds to the use of symbols and constructs abstracting from all meaning whatsoever. We met in §2.1.1 the constructions of formal languages for expressing reasoning, from Boole to Frege and Russell. The variables in these formalisms abstract from most semantic aspects of propositions, but still preserve a fundamental semantic property of linguistic expressions: their truth value. For this reason, logic is often considered a formalism that captures the semantics of linguistic expressions. What is definitely irrelevant in logic is the form of the words that constitute a proposition. The formality we are going to describe in this section is even more de-semantified that in Novaes’ category. It is the formal description of sentences introduced by Noam Chomsky, that reduces to a computational form the words of a sentence, taking care of their syntactic roles only. As we will see, in fact, Chomsky expanded on a previous mathematical framework for describing combinations of words, independent not only from their lexical meaning, but from syntax also. The figure of Chomsky has been central to cognitive science since the 1960s, but his work was soon borrowed by computer science and AI, despite the indifference of Chomsky towards these disciplines.
QBism: Subjective Probabilistic Interpretation of Quantum Mechanics
Published in Andrei Khrennikov, Social Laser, 2020
In applications any mathematical formalism has to be endowed with the corresponding interpretation of its entities. Here we present the original Kolmogorov interpretation of probability (see section 11.8.2 for another interpretation of the same mathematical formalism).
First-year engineering students’ mathematics task performance and its relation to their motivational values and views about mathematics
Published in European Journal of Engineering Education, 2021
Timo Tossavainen, Ragnhild Johanne Rensaa, Pentti Haukkanen, Mika Mattila, Monica Johansson
The present study builds on two theoretical perspectives: (1) students’ epistemological beliefs on the nature of mathematics (Hofer and Pintrich 1997) and (2) their motivation values in the framework of the Expectancy–value theory (Eccles et al. 1983; Wigfield and Eccles 2000). The epistemological beliefs about the nature of mathematics concern the structure, quality, certainty, and the source of mathematical knowledge. There is evidence that such beliefs have an influence on a learner’s actual competence and performance in mathematics, since they affect the perception of the mathematical task and, thus, also the learner’s choice of actions in handling the task (Felbrich, Müller, and Blömeke 2008). Further, these beliefs have an effect on individuals’ beliefs about themselves as learners of mathematics. Grigutsch, Raatz, and Törner (1998) studied German mathematics teachers’ beliefs about the nature of mathematics, and they identified four different categories of such beliefs. Their categorisation has been used for studying student teachers’ beliefs. For that purpose, Felbrich, Müller, and Blömeke (2008, 764) formulated and named the categories as follows: The formalism-related orientation: mathematics is viewed as an exact science that has an axiomatic basis and is developed by deduction (e.g. ‘Mathematical thinking is determined by abstraction and logic’).The scheme-related orientation: mathematics is regarded as a collection of terms, rules, and formulae (e.g. ‘Mathematics is a collection of procedures and rules which precisely determine how a task is solved’ [‘a toolbox’]).The process-related orientation: mathematics can be understood as a science which mainly consists of problem-solving processes and the discovery of structure and regularities (e.g. ‘If one comes to grips with mathematical problems, one can often discover something new (connections, rules, and terms)’).The application-related orientation: mathematics can be seen as a science which is relevant to society and life (e.g. ‘Mathematics helps to solve daily tasks and problems’).
Measuring changes in mathematics teachers' belief systems
Published in International Journal of Mathematical Education in Science and Technology, 2023
Andreas Eichler, Ralf Erens, Günter Törner
Teachers’ beliefs have several different definitions (Fives & Buehl, 2012; Leder, 2019) but we use that of Philipp (2007), who suggests beliefs to be individual propositions that have a truth value. Also, beliefs are understood to be organized in belief systems, in which some beliefs are central whereas others are peripheral (Green, 1971; Philipp, 2007). Strongly held beliefs are called central beliefs, while peripheral beliefs are of lesser importance. The aspect of centrality is crucial since teachers’ central beliefs seem to be particularly relevant for teachers’ classroom practice (Eichler, 2011; cf. also Furinghetti & Morselli, 2011). A further distinction of beliefs is between primary beliefs and derivative (subordinate) beliefs (Green, 1971). Finally, a belief system comprises clusters of beliefs (Green, 1971). To differentiate the term ‘cluster of beliefs’ from the statistical cluster which results from cluster analysis, we use the term belief-cluster. Fives and Buehl (2012, p. 472) identified six topics of teachers’ beliefs that may be understood as possible general belief clusters: ‘(a) self, (b) context or environment, (c) content or knowledge, (d) specific teaching practices, (e) teaching approach, and (f) students’. In a similar way, Hutchins and Friedrichsen (2012) modelled a teacher’s belief system as consisting of beliefs clusters concerning mathematics, mathematics learning, mathematics teaching, beliefs about self, social norms and environmental constraints. In this paper, we focus on belief-cluster (c) according to Fives and Buehl (2012), which is concerned with the nature of school mathematics and mathematics in general. This belief-cluster is viewed as a part of the whole belief system of a mathematics teacher. Grigutsch et al. (1998) called a teacher’s belief-cluster referring to the nature of school mathematics and mathematics in general as a teacher’s ‘mathematical world view’. This world view includes four aspects: The application-oriented aspect represents beliefs implying that the utility of mathematics for real-world problems is the main aspect of the nature of (school) mathematics.The process-oriented aspect represents beliefs implying that (school) mathematics is a creative activity consisting of problem-solving using different and individual ways.The formalist aspect represents beliefs implying that (school) mathematics is characterized by a strongly logical and formal approach.The schema aspect represents beliefs implying that (school) mathematics is a set of calculation rules and procedures to apply to routine tasks.