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Jul 27, 2009 · In this part of the book, I illustrate the workings of the commognitive approach by applying it to the special case of mathematical thinking. In so doing, my intention is to show what difference commognitive analysis makes in our interpretation of observed phenomena and in our practical decisions about teaching and learning.
For the mathematical theory of communication (MTC), information is only a selection of one symbol from a set of possible symbols, so a simple way of grasping how MTC quantifies information is by considering the number of yes/no questions required to determine what the source is communicating.
- Mathematics as A Mediator Between Theory and Observation
- Data-Driven Research and The Use of Big Data
- Tuning Models Mathematically
- Using Alternative Routes to Solving Equations
- Non-Representational Idealizations
- Synthesis
Mathematical theories are indispensable for forging links between theory and observation. This applies to measurement, i.e., to connecting theoretical quantities with data as well as to technology. The use of mathematics is essential for registering phenomena and for generating and shaping phenomena. In the former case, mathematics is employed as o...
Generalizing these considerations leads to the phenomenon of “data-driven research” (cf. J. Jost’s contribution to this volume). Data-driven research can be contrasted with “model-driven research,” in which theoretical expectations or a micro-causal model distinguishes certain patterns in the data as significant. Data-driven research is purportedly...
Theoretical mathematical models are rarely perfect. Even if a model is very good, it regularly provides a kind of mathematical skeleton that includes a number of parameters whose quantitative assignment has to be read off from empirical data. A case in point is the gravitational constant that does not follow from mathematical arguments though it do...
Solving differential equations by a computer simulation proceeds by not literally solving these equations but rather by calculating values of its discretized proxy. Solutions are calculated point by point at a grid and for specific parameter values. As a result, using computational models does not simply mean to enhance the performance of mathemati...
Idealizations are intimately connected to the role of mathematics in the sciences. All mathematical operations inevitably deal with objects or models that are in some sense idealized. The point is what sense of idealization is the relevant one in our context. “According to a straightforward view, we can think of idealization as a departure from com...
We have discussed five modes of using mathematics as a tool which partly overlap and are partly heterogeneous in kind. The instrumental use of mathematics is not governed by a single scheme. Rather, the heterogeneity of tools and of tool usages teaches lessons about mathematization, its dynamics and its limits. The examples also suggest that the mo...
- Johannes Lenhard, Martin Carrier
- 2017
Jul 1, 2019 · At the most fundamental level, the issue is not algorithms versus strategies; it’s about approaching math as the provision of a toolkit (OUTDATED), as opposed to developing a way of thinking (CRUCIAL). The former, toolkit approach was defensible, and arguably unavoidable, in the millennia before we had tools for procedural math.
Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public.
Sep 1, 2018 · Digital tools allow researchers to explore individually or collaboratively, both within and across disciplines, students’ and teachers’ interactional processes in new ways; including tracing interactions on video records, excavating massive amounts of data, and capturing classroom learning processes.
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ABSTRACT: Mathematical communication skills are the ability of students to understand mathematical definitions, symbols or notations appropriately, then able to express mathematical ideas through oral, written and drawing.
