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"Mathematical Languages."
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Eleven Grade 1 teachers’ understandings of mathematical language in a South African context
BackgroundFluency in mathematical language is essential for learning mathematics. Teachers must understand and use their diverse mathematical knowledge, including language and communication difficulties inherent to mathematics instruction. According to recent South African research, Grade 1 teachers are not equipped to utilise learners’ linguistic skills for efficient learning of mathematics.ObjectivesThis research investigates South African Grade 1 teachers’ mathematical language perceptions, experiences, and feelings. These Grade 1 teachers’ transcripts were analysed to discover their understanding of the language of mathematics.MethodExploratory, descriptive, and contextual research designs were used in conjunction with an adapted interactive qualitative analysis technique. Focus group interviews, individual interviews, and lesson observations, together with a purposive sampling technique, were used to gather the data from both public and private primary schools.ResultsThe results showed that Grade 1 teachers view mathematics as a separate language with its own vocabulary and register. The findings highlighted the need to simplify the language of mathematics to enhance understanding.ConclusionThis research concluded that language is essential to mathematics learning and that mathematics has its own register, which is acquired like any other additional language. To help isiXhosa learners understand mathematics in English, scaffolding strategies must be aligned with their linguistic demands.ContributionThis article provides important recommendations for teachers who need to recognise the reality that English is the lingua franca and ensure isiXhosa home language-speaking learners receive the necessary support to acquire actual proficiency in the academic register of English for mathematical language learning.
Journal Article
Language and the rise of the algorithm
\"A wide-ranging history of the intellectual developments that produced the modern idea of the algorithm. Bringing together the histories of mathematics, computer science, and linguistic thought, Language and the Rise of the Algorithm reveals how recent developments in artificial intelligence are reopening an issue that troubled mathematicians long before the computer age. How do you draw the line between computational rules and the complexities of making systems comprehensible to people? Here Jeffrey M. Binder offers a compelling tour of four visions of universal computation that addressed this issue in very different ways: G. W. Leibniz's calculus ratiocinator; a universal algebra scheme Nicolas de Condorcet designed during the French Revolution; George Boole's nineteenth-century logic system; and the early programming language ALGOL, whose name is short for algorithmic language. These episodes show that symbolic computation has repeatedly become entangled in debates about the nature of communication. To what extent can meaning be controlled by individuals, like the values of a and b in algebra, and to what extent is meaning inevitably social? By attending to this long-neglected question, we come to see that the modern idea of the algorithm is implicated in a long history of attempts to maintain a disciplinary boundary separating technical knowledge from the languages people speak day to day. Machine learning, in its increasing dependence on words, now places this boundary in jeopardy, making its stakes all the more urgent to understand. The idea of the algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction. \"-- Provided by publisher.
BOOK
Formal and informal mathematical discourses: Bakhtin and Vygotsky, dialogue and dialectic
The importance of the role of language/discourse in the learning and teaching of mathematics is noted in many mathematics curricula and standards documents. In the research literature, this role has been widely theorised from a Vygotskian perspective. This perspective is limited by some of its underlying assumptions, including an instrumental and systemic view of language as tool and its basis in dialectic. In this paper, I propose a Bakhtinian, dialogic perspective as an alternative. I focus my discussion on the long-standing issue of the relationship between formal and informal mathematical language in the learning and teaching of mathematics. I illustrate this discussion with an examination of interaction in an elementary school mathematics classroom in Québec, Canada. Based on Bakhtin's ideas, I argue that mathematical meaning emerges through locally produced, situated dialogic relations between multiple discourses, voices and languages in mathematics classroom interaction. From this perspective, students do not follow a linear path from informal to formal mathematical discourse; rather, they work with the teacher to expand the repertoire of possible ways to make meaning in mathematics.
Journal Article
Facing and challenging language ideologies towards a more inclusive understanding of language in mathematics education research—the case of sign languages
Research on language in mathematics education is largely dominated by a ‘normalcy’ of spoken languages. This modal hegemony does not only affect a whole group of learners in failing to provide access that is epistemologically equitable—those using sign language as their preferred mode for mathematical discourse—it also obscures our view on the roles language can play in mathematical thinking and learning. As a field, we can only win from seeking to understand Deaf learners of mathematics beyond a disability, as learners of mathematics with a specific linguistic background that influences mathematical thinking and learning in peculiar ways. In this contribution, I suggest a shift in mindset towards a more inclusive view on language in mathematics education research and practice. I propose basic principles to inform a perspective for reconsidering the role of language in mathematics thinking and learning, inspired by work of philosopher Francois Jullien. This perspective counters a perspective that merely integrates sign language into existing research and instead searches for dialogue between linguistic modalities in learning mathematics, looking beyond language as spoken or written. This approach will be illustrated by the case of the modal affordance of iconicity foregrounded in signed mathematical discourse, its role in Deaf students’ mathematics thinking and learning, and how this can inform existing research and practice dealing with language in mathematics education.
Journal Article
Language Barriers in Mathematics: Learning Challenges When Global Languages Replace Native Instruction
The teaching and learning process is a social practice that involves interactions between learners, teachers, and resources. These interactions are facilitated by various tools, especially tools for communication. The language of instruction is at the centre of this interaction and is the focus of this article. The article is conceptualised from the Tanzania context, where English is the medium of instruction, yet it is a second or third language to many learners. This article explores the phenomenon of English being the language of instruction in multilingual classrooms, posing opportunities and challenges in learning. Whereas fluency in English may aid the learning of mathematics, limited competence in English may hinder the learning of mathematics since learners must then simultaneously learn both English (the language of instruction) and mathematics. Additionally, the article discusses the differences in the meanings of various English terms when used in everyday English versus formal mathematical language. Suggestions for pedagogical practices to help learners develop English language competences without deferring the development of mathematical competences are provided.
Journal Article
Probabilistic Language in Spanish Secondary Textbooks
Probabilistic language is a main component in the teaching and learning of probability; however, research analyzing probabilistic language in textbooks, which are fundamental didactic tools, is scarce. Consequently, in this research, we studied the various probabilistic languages used in Spanish secondary school textbooks. We performed a detailed content analysis of two complete series (grades 1 to 4; the last with two options) of Spanish prestigious editorials published after the last curricular guidelines in 2022; 10 books in total. We researched the verbal, symbolic, tabular, and graphical language in each textbook. Results suggest differences in the way each editorial introduces its everyday and probabilistic language. Although the number of new symbols is small, some of them are complex or used inconsistently. There is scarce use of tables and graphs, except for tree diagrams and two-way tables, in the study of conditional and compound probability. We conclude with recommendations to improve probabilistic language in textbooks and facilitate the learning of probability in secondary education in this way.
Journal Article
Comparative study on the gastrointestinal- and immune- regulation functions of Hedysari Radix Paeparata Cum Melle and Astragali Radix Praeparata cum Melle in rats with spleen-qi deficiency, based on fuzzy matter-element analysis
Hedysari Radix Praeparata Cum Melle (HRPCM) and Astragali Radix Praeparata Cum Melle (ARPCM) are used interchangeably in clinics to treat spleen-qi deficiency (SQD) symptom mainly including gastrointestinal dysfunction and decreased immunity, which has unknown differences in efficacy.
To investigate the differences between HRPCM and ARPCM on intervening gastrointestinal- and immune-function with SQD syndrome.
After the SQD model was established, the Sprague-Dawley (SD) rats were randomly divided into nine groups (n = 10): normal; model; Bu-Zhong-Yi-Qi Pills; 18.9, 12.6 and 6.3 g/kg dose groups of HRPCM and ARPCM. Gastrointestinal function including d-xylose, gastrin, amylase vasoactive intestinal peptide, motilin, pepsin, H
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/K
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-ATPase, Na
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/K
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-ATPase, sodium-glucose cotransporter 1 (SGLT1), glucose transporter 2 (GLUT2) and immune function including spleen and thymus index, blood routine, interleukin (IL)-2, IL-6, interferon-γ (IFN-γ), tumour necrosis factor-α (TNF-α), immunoglobulin (Ig) M, IgA, IgG and delayed-type hypersensitivity (DTH) were detected. Finally, the efficacy differences were analysed comprehensively by the fuzzy matter-element method.
In regulating immune, the doses differences in efficacy between HRPCM and ARPCM showed in the high-dose (18.9 g/kg), but there were no differences in the middle- and low- dose (12.6 and 6.3 g/kg); the efficacy differences were primarily reflected in levels of IL-6, IFN-γ, TNF-α and IgM in serum, and the mRNA expression of IL-6 and IFN-γ in the spleen. In regulating gastrointestinal, the efficacy differences were primarily reflected in the levels of D-xylose, MTL, and GAS in serum, and the mRNA and protein expression of SGLT1 and GLUT2 in jejunum and ileum.
HRPCM is more effective than ARPCM on regulating gastrointestinal function and immune function with SQD syndrome. Therefore, we propose that HRPCM should be mainly used to treat SQD syndrome in the future.
Journal Article
Linguistic Conventions of Mathematical Proof Writing at the Undergraduate Level: Mathematicians' and Students' Perspectives
This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing.
Journal Article
Mathematics performance and the relations to the cognitive reflection, fluid intelligence, conditional reasoning, and logical-mathematical language in the university students
Recent research on university mathematics education has emphasized the need to understand how cognitive factors and specific mathematical skills contribute to students’ mathematical performance in advanced mathematical contexts. This paper examines the predictive value of fluid intelligence (Gf), cognitive reflection (CR), mastery of logical-mathematical language, and conditional reasoning on the academic achievement of first-year mathematics undergraduates. Combining theoretical perspectives from cognitive science and mathematics education, this study analyzes the extent to which (individually and jointly) these cognitive and disciplinary variables explain differences in mathematical performance at the university level. Logistic and multiple linear regression models are used to examine the relationships existing between these variables, paying special attention to their role in predicting students’ academic outcomes. The findings reveal that both general cognitive abilities and discipline-specific abilities, such as formal language use and conditional reasoning, are significant and independent predictors of academic success in university mathematics. This research contributes to the growing body of knowledge on the factors that shape students’ progression and performance in higher mathematics. Moreover, it highlights the complex interplay between cognitive resources and formal mathematical reasoning at the tertiary level.
Journal Article