Levels of Sophistication in Elementary Students’ Understanding of Polygon Concept and Polygons Classes

This paper reports sophistication levels in third grade children’s understanding of polygon concept and polygon classes. We consider how children endow mathematical meaning to parts of figures and reason to identify relationships between polygons. We describe four levels of sophistication in children’s thinking as they consider a figure as an example of a polygon class through spatial structuring (the mental operation of building an organization for a set of figures). These levels are: (i) partial structuring of polygon concept; (ii) global structuring of polygon concept; (iii) partial structuring of polygon classes; and (iv) global structuring of polygon classes. These levels detail how cognitive apprehensions, dimensional deconstruction, and the use of mathematical language intervene in the mental process of spatial structuring in the understanding of the classes of polygons.

Retrieval of Photometric Parameters of Minerals Using a Self-Made Multi-Angle Spectrometer Based on the Hapke Radiative Transfer Model

In this paper, a self-made, mineral, multi-angle, spectrum measurement device is employed to measure the multi-angle spectra of olivine and plagioclase; the multi-angle spectra of ilmenite in the Reflectance Experiment Laboratory (RELAB) Spectral Library are collected; and the optimized retrieval of the photometric parameters of the Hapke model is realized. Importantly, the derived result of the single-scattering albedo (SSA) is stable and has both mathematical meaning and physical meaning. The derived Legendre polynomial coefficients of the phase function can better simulate the variation in the mineral spectra with angle. This paper compares the effects of multi-angle and single-angle spectral data on the photometric parameter derived results. The setting of the Legendre polynomial coefficient of the scattering phase function mainly affects the simulation accuracy of the mineral spectra as a function of angle. Using this coefficient to optimize the retrieval, the simulation accuracy is moderately improved compared with the single-angle simulation. The estimation of photometric parameters based on multi-angle spectral data can provide a basis for setting the empirical values of the phase function parameters from single-angle spectral calculations, which can more truly reflect the law of reflectance spectra changing with angle than Lucey’s traditional empirical value of the phase function (b = −0.4 and c = 0.25). The results of multi-angle spectra retrieval in this paper show that the Legendre polynomial coefficients of the phase function vary with wavelength rather than being constant and that different minerals differ greatly.

Rational Interpretation of Numerical Quantity in Argumentative Contexts

Numerical descriptions furnish us with an apparently precise and objective way of summarising complex datasets. In practice, the issue is less clear-cut, partly because the use of numerical expressions in natural language invites inferences that go beyond their mathematical meaning, and consequently quantitative descriptions can be true but misleading. This raises important practical questions for the hearer: how should they interpret a quantitative description that is being used to further a particular argumentative agenda, and to what extent should they treat it as a good argument for a particular conclusion? In this paper, we discuss this issue with reference to notions of argumentative strength, and consider the strategy that a rational hearer should adopt in interpreting quantitative information that is being used argumentatively by the speaker. We exemplify this with reference to United Kingdom universities’ reporting of their REF 2014 evaluations. We argue that this reporting is typical of argumentative discourse involving quantitative information in two important respects. Firstly, a hearer must take into account the speaker’s agenda in order not to be misled by the information provided; but secondly, the speaker’s choice of utterance is typically suboptimal in its argumentative strength, and this creates a considerable challenge for accurate interpretation.

Constructionist Learning in School Mathematics: Implications for Education in the Fourth Industrial Revolution

Purpose: Amid rapid technological development in the Fourth Industrial Revolution, this article engages with an important question, especially in the context of science, technology, engineering, and mathematics (STEM) education: Can technology itself transform teaching and learning? Design/Approach/Methods: Constructionist learning responds to the current “maker movement,” which draws upon the innate human desire to make things with our hands. Two important elements of constructionist learning—technology literacy and engineering design—have implications for meeting the global need for expertise in the STEM disciplines. To date, the practice of teaching and learning mathematics remains to be dominated by manipulation of symbols with the paper-and-pencil medium. In response, this article discusses how constructionist learning can play an important role in teaching and learning school mathematics via a transdisciplinary approach for STEM education. Findings: Two examples of the authors’ empirical research on constructionist learning in school mathematics classrooms with 3D printing are illustrated. Findings suggest that the 3D Printing Pens played an active role in the construction of artifact (physically) and mathematical meaning (cognitively). Originality/Value: The empirical results as discussed in this article warrant more design-based classroom interventions to further investigate students’ constructionist learning in technological, hands-on, and innovation-oriented environments.

Happy Numbers in China and Japan

The paper outlines the functioning of cultural practices concerning numbers in Chinese and Japanese mundanity. The formation and use of such symbolic non-mathematical meaning of numbers is a distinctive aspect of linguistic, cultural and axiological systems in the countries of the Far East. The topic seems to be of particular interest due to high attention drawn by number-containing words and idioms in Chinese and Japanese linguistic studies in combination with cultural studies. Such an analysis seeks to develop the approaches to clarifying nation-specific mental representations and cultural aspects of using numeral vocabulary. Non-mathematical meaning of numeral vocabulary should be considered in a differentiated manner depending on factors that shape particular meanings. Religious and cosmogonic mythology as well as oriental philosophy serves as major origins of number-related meanings. Graphic interpretations also produce new associated meanings. The paper mostly delves into the effects of how the phonetics of number-containing words influences their meaning. Homonymy and homophony that are typical of the Chinese language considerably facilitate the process of mounting additional meanings. The axiological and cultural perspective embraces numbers as classifying factors that can be used to stratify the objects. The study of non-traditional meanings in number vocabulary not only allows to reveal the link between culture and language but highlights how cognitive processes operate in linguistics.