An Analysis of Errors on Mathematical Symbol as a Metaphor in Linear Programming

Symbol sense is crucial in the understanding of mathematical problems comprising various symbols. The misuses of symbols happen due to misinterpretation, which is considered the constraint to learn algebra more comprehensively, including in linear programming. The term ‘metaphor’ is defined as a means to carry over symbol sense, and is used to improve mathematical understanding. This present research was aimed at analyzing errors on mathematical symbol as a metaphor in linear programming. This research was conducted by means of descriptive qualitative design, with a test and interview as the instruments. The test was made essay, and its results were analyzed qualitatively. The test, further, was administered to five eleventh graders selected according to highest rates of errors committed. This research has shown that the students committed a number of errors in some cases, such as representing symbols as variables, representing numbers, and interpreting symbols as relational operators. In addition, errors which the students committed in constructing mathematical models covered defining the final value, representing numbers, applying inequality system, and interpreting symbols as operation counts. This present research has provided some ways for symbol sense, and thus the errors on mathematical symbol as a metaphor could be lessened. Next, this research can be further followed up by reviewing the effectiveness of remedial instruction according to the committed errors on mathematical symbols.

Students’ Metacognitive Process in Mathematical Problem Solving Based on Cognitive Style

This study is a qualitative literature review aimed to describe junior high school students’ metacognitive process in mathematical problem solving based on field independent and field dependent cognitive style. The research was done based on these following steps: 1) Data library relevant to variable and in accordance to the data source criteria was collected; 2) the data library was classified according to the grade and the subjects; 3) the data was analyzed. The instruments used in this study were the researchers and interview. The interview was conducted to confirm the classified data. Based e this study, it can be concluded that students using field independent style, competently can employ metacognitive process in planning, monitoring, and evaluating because they can write down the known fact and the question using mathematical symbol, choose the appropriate strategy, and answer the question thoroughly. Students using field dependent style were having difficulties in writing down the known fact and the question using mathematical symbol, choosing an appropriate strategy, and answering the question thoroughly.

Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism

Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of real numbers as a continuous sequence of points on a number line. Moreover, as he understands mathematics to be an anthropological phenomenon, he rejects set theory due to its lack of application. As I argue here, it is possible to defend Georg Cantor’s theory by taking a standpoint I call quietistic conventionalism. The standpoint broadly resembles Wittgenstein’s formalist middle period and allows us to view transfinite set theory as a result of a series of definitions established by arbitrary decisions that have no ontological consequences. I point to the fact that we are inclined to accept such definitions because of certain psycho‐ logical mechanisms such as the hypothetical Basic Metaphor of Infinity proposed by George Lakoff and Rafael E. Núñez. Regarding Wittgenstein’s criterion of applicability, I argue that it presupposes a static view of science. Therefore, we should not rely on it because we are unable to foresee what will turn out to be useful in the future.

Sustainability Development in Mathematics Education—A Case Study of What Kind of Meanings Do Prospective Class Teachers Find for the Mathematical Symbol “23”?

In this article, our focus is on sustainable development in mathematics education from the point of view of teacher training. The aim was to develop prospective teachers’ content knowledge and pedagogical content knowledge of school mathematics. As a case study, we chose the mathematical symbol “a/b”, and examined how prospective class teachers in Finland connect it to the concepts of fraction, ratio, division, rational number or probability. Mathematics textbooks often have a central role in lessons, and they affect strongly how pupils understand concepts and the relationships between them. We chose languaging as a multi-semiotic approach to interpreting what kind of meanings the prospective class teachers gave the mathematical symbol “a/b”. The results show that some of these concepts are difficult to see at the same time from the given mathematical symbol. The concept of ratio is particularly difficult for prospective class teachers to interpret. Pictorial presentation supported the interpretations. Mathematics learning materials and teacher education should develop in accordance with the results of the study.

Binary fractions and a whole unit 'measure of value' in the seal-texts of the Indus civilization

Short sequences of symbols on tiny stone seals, miniature tablets and assorted media are the only evidence of writing in the Indus Civilization (flourished ca. 2600--1900 BCE). About five thousand specimens bearing such brief 'texts' have been cataloged so far, and despite nearly a hundred years of study, the texts cannot be read. This study proposes based on a consilience of evidences that a small number of symbols that occur prominently at the beginning of several texts in the corpus are pictorial expressions of binary fractions from the 'one-sixteenth' to the 'half', a 'whole' unit, and an 'equivalence' indicator whose form may have emerged from the same cognitive basis as the modern mathematical symbol for 'equality'. The correspondence of the fractions to the system of tiny weights in binary ratios found in the Indus archaeological record suggests the 'fractions' were not used in a mathematical sense, but were representations of small quantities of objects of value such as precious metal that likely functioned as measures of economic 'worth'. Parallels from contemporaneous Egypt and Mesopotamia, as well as specific Indus texts investigated in this study, suggest that the texts which begin with these symbols encoded 'worth equivalences', in which the worth of commodities of trade and articles of value were expressed against a standard measure of value. The use of such standardized expressions of economic worth over a vast geographical area offers concrete evidence of a high degree of economic integration among the different regions of the Indus Civilization.