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Mathematical understanding continues to be one of the major goals of mathematics education. However, what is meant by “mathematical understanding” is underspecified. How can we operationalize the idea of mathematical understanding in research? In this article, I propose particular specifications of the terms mathematical concept and mathematical conception so that they may serve as useful constructs for mathematics education research. I discuss the theoretical basis of the constructs, and I examine the usefulness of these constructs in research and instruction, challenges involved in their use, and ideas derived from our experience using them in research projects. Finally, I provide several examples of articulated mathematical concepts.

The purpose of the current study was to explore pre-service middle school mathematics teachers’ personal concept definitions of a trapezoid, parallelogram, rectangle, rhombus, square, and kite. The data were collected by a self-report instrument through which the pre-service teachers provided their answers in written form. The participants’ definitions were coded by using Zazkis and Leikin’s (Educational Studies in Mathematics, 69(2), 131–148, 2008) framework, which includes the following four main categories for determining mathematical correctness: necessary and sufficient, necessary but not sufficient, sufficient but not necessary, and neither necessary nor sufficient. The findings revealed that about half of the all definitions were correct. More specifically, the participants generated considerably higher proportion of correct definitions for a parallelogram and rhombus, while they displayed a very low performance in defining a kite. The possible reasons of the participants’ varied performance levels in defining the six basic quadrilaterals are discussed based on the linguistic (syntactic, semantic, and lexical) structure of the Turkish names given to the these quadrilaterals. The current study may provide some feedback to teacher education programs regarding the extent of knowledge that should be possessed by the pre-service teachers about definitions of special quadrilaterals before they leave their programs. Such feedback may also help mathematics teacher educators ponder on more fruitful approaches that may promote the development of pre-service teachers’ knowledge and understanding of definitions of special quadrilaterals.

This study examined the effect of a professional development training programme on 20 second-year preservice mathematics teachers’ knowledge in foundational mathematical concepts at a rural university in South Africa. The training programme aimed to enhance preservice teachers’ mathematical knowledge for teaching. An embedded mixed-methods case study design was employed. Baseline and endline assessments were administered before and after the training. A participant feedback survey was also administered after the training. Results showed that the training significantly improved the preservice teachers’ understanding and confidence in the selected concepts, despite their low baseline scores. The participants also expressed satisfaction with the knowledge they gained and appreciated the integration of theory and practice in the training. These findings suggest the need for teacher training institutions to ensure that preservice teachers are well versed in both university-level and school-level mathematics. They also support the need for collaboration with other stakeholders to provide preservice teachers with relevant and engaging professional development opportunities that can enhance their mathematical knowledge for teaching.ContributionFindings of this study point to a renewed emphasis on the creation of greater collaborations between institutions of higher learning and other key stakeholders to promote the development of prospective teachers’ knowledge of what they will be expected to teach.

Tzur and Simon (2004) postulated 2 stages of development in learning a mathematical concept: participatory and anticipatory. The authors discuss the affordances for research of this stage distinction related to data analysis, task design, and assessment as demonstrated in a 2-year teaching experiment.

Meaning and understanding are didactic notions appropriate to work on concept comprehension, curricular design, and knowledge assessment. This document aims to delve into the meaning of school mathematical concepts through their semantic analysis. This analysis is used to identify and establish the basic meaning of a mathematical concept and to value its understanding. To illustrate the study, we have chosen the trigonometric notions of sine and cosine of an angle. The work exemplifies some findings of an exploratory study carried out with high school students between 16 and 17 years of age; it collects the variety of emergent notions and elements related to the trigonometric concepts involved when answering on the categories of meaning which have been asked for. We gather the study data through a semantic questionnaire and analyze the responses using an established framework. The subjects provide a diversity of meanings, interpreted and structured by semantic categories. These meanings underline different understandings of the sine and cosine, according to the inferred themes, such as length, ratio, angle and the calculation of a magnitude.

We introduce the concept of metamaterial analog computing, based on suitably designed metamaterial blocks that can perform mathematical operations (such as spatial differentiation, integration, or convolution) on the profile of an impinging wave as it propagates through these blocks. Two approaches are presented to achieve such functionality: (i) subwavelength structured metascreens combined with graded-index waveguides and (ii) multilayered slabs designed to achieve a desired spatial Green's function. Both techniques offer the possibility of miniaturized, potentially integrable, wave-based computing systems that are thinner than conventional lens-based optical signal and data processors by several orders of magnitude.

The reduced interest of students in learning mathematics and considermathematics to be a difficult subject to become a serious problem in Indonesia,especially in one of the provinces in Indonesia, Lampung. The purposeof this study was to determine the effectiveness of the demonstration methodassisted by Multiplication Board props to the understanding of the third grademathematical concept of Bandar Lampung.This type of research is quantitative research with the type of QuasyExperimental Design. The design used is the Pretest-Postest Control GroupDesign. Data collection techniques in this study consisted of interviews, testsand documentation. The population of this study was class III Bandar Lampungstudents. The sample in this study is class III A as the experimental classwith the demonstration method, class III B as the control class usingconventional methods. Data analysis techniques used the normality test withLilifors test and homogeneity test with Bartlett test. Followed by testing thehypothesis that is using independent tests.Based on the results of the analysis and discussion of the researchdata obtained the results of hypothesis testing manually with t count = 4.265and t (0.025; 38) = 1.960, so that tcount> t (0.025; 38) then HO isrejected. Based on these results, there is an understanding of the concept ofmathematical multiplication between students who are taught using thedemonstration method compared to using conventional methods.

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.

One of the goals of a pilot study is to identify unforeseen problems, such as ambiguous inclusion or exclusion criteria or misinterpretations of questionnaire items. Although sample size calculation methods for pilot studies have been proposed, none of them are directed at the goal of problem detection. In this article, we present a simple formula to calculate the sample size needed to be able to identify, with a chosen level of confidence, problems that may arise with a given probability. If a problem exists with 5% probability in a potential study participant, the problem will almost certainly be identified (with 95% confidence) in a pilot study including 59 participants.

Understanding the impact of collective social phenomena in epidemic dynamics is a crucial task to effectively contain the disease spread. In this work, we build a mathematical description for assessing the interplay between opinion polarization and the evolution of a disease. The proposed kinetic approach describes the evolution of aggregate quantities characterizing the agents belonging to epidemiologically relevant states and will show that the spread of the disease is closely related to consensus dynamics distribution in which opinion polarization may emerge. In the present modelling framework, microscopic consensus formation dynamics can be linked to macroscopic epidemic trends to trigger the collective adherence to protective measures. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.