Putting Technology in Its Place

2020 ◽  
Vol 113 (2) ◽  
pp. 140-146
Author(s):  
Jason Knight Belnap ◽  
Amy Parrott
Keyword(s):  
Mathematical Tasks ◽  
Mathematical Activity ◽  
Mathematical Practices

Technology is used effectively when it enables students to engage in authentic mathematical activity. Using four mathematical tasks, we discuss how technology, carefully designed tasks, and orchestrated discussions can both reveal our students' mathematical practices and provide opportunities to shape those practices.

scholarly journals Algebraization Levels in the Study of Probability

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 91
Author(s):  
María Burgos ◽  
Carmen Batanero ◽  
Juan D. Godino
Keyword(s):  
Mathematical Activity ◽  
Mathematical Practices ◽  
Educational Levels ◽  
Semiotic Approach ◽  
Mathematical Formalization ◽  
Selection Of

The paper aims to analyze how the different degrees of mathematical formalization can be worked in the study of probability at non-university educational levels. The model of algebraization levels for mathematical practices based on the onto-semiotic approach is applied to identify the different objects and processes involved in the resolution of a selection of probabilistic problems. As a result, we describe the possible progression from arithmetic and proto-algebraic levels of mathematical activity to higher levels of algebraization and formalization in the study of probability. The method of analysis developed can help to establish connections between intuitive/informal and progressively more formal approaches in the study of mathematics.

scholarly journals The Implementation of an Integrative Approach to Learning with the Use of Integrated Images

2021 ◽  
Vol 13 (1) ◽  
pp. 281-297
Author(s):  
Renat Rizhniak ◽  
Natalia Pasichnyk ◽  
Dolores Zavitrenko ◽  
Kateryna Akbash ◽  
Artem Zavitrenko
Keyword(s):  
Mathematical Tasks ◽  
Integrative Approach ◽  
Mathematical Activity ◽  
Approach To Learning ◽  
Knowledge And Skills ◽  
Definition Of ◽  
Integrated Knowledge

The article is dedicated to the definition of methodological conditions under which the use of the solution and research of the “task series” generated by a given task topic will acquire methodological expediency in the context of developing students' abilities to solve mathematical tasks of a productive nature. The authors suggested a possible way to solve it through the synthesis of integrated knowledge and the formation of the integrated images of the task topics that allowed students to develop knowledge and skills of integrative mathematical activity.

scholarly journals How To Learn To Understand Mathematics?

2017 ◽  
Vol 10 (2) ◽  
pp. 114
Author(s):  
Raymond Duval
Keyword(s):  
Mathematics Education ◽  
Cognitive Factors ◽  
Mathematical Tasks ◽  
Education For All ◽  
Teaching Mathematics ◽  
Mathematical Activity ◽  
Crucial Issue ◽  
Semiotic Representations ◽  
Learning Mathematics ◽  
Insight Into

The cognitive core process of mathematical activity is the recognition of a same object in two semiotic representations whose respective contents have nothing in common with each other. It is also the recurrent and insuperable difficulty of comprehension in learning mathematics and the main impediment to solving problems for most students. The theory of registers provides a cognitive analysis of the way of working and thinking in mathematics. It highlights the key cognitive factors to be taken into account in Mathematics Education for all students up to the age of 16. To give an insight into the theory this paper focuses on two topics. How to introduce letters and elementary algebra? How to learn to solve problems in mathematics? And to avoid the confusion of words arising in Mathematics Education whenever we talk about « theories », we shallshow how to analyze in terms of registers the mathematical tasks related to these two topics. This allow us to identify the cognitive thresholds to be crossed to understand and to solve problems in mathematics. Analyzing mathematical activity in terms of registers is quite different from the prevailing mathematical view. This concerns the hidden face of mathematical activity and not its exposed face. We are broaching here the crucial issue about teaching mathematics to all students up to the age of 16. What should be its objectives and priority areas?Keywords: Register. Transformation of Semiotic Representation. Conversion. Treatment. Discursive operation.

The Web of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Current Trend ◽  
Mathematical Practice ◽  
Mathematical Activity ◽  
Scientific Practices ◽  
General Introduction ◽  
Philosophical Work ◽  
Mathematical Practices ◽  
David Hilbert ◽  
Different Levels ◽  
The Web

This chapter is a general introduction to the current trend of studies of mathematical practice, with particular emphasis on historical and philosophical work. It offers a preliminary explanation of the notion of mathematical practice, first by considering the work of historians and philosophers on mathematical practices, from Archimedes and David Hilbert to Jens Høyrup, Penelope Maddy, Marcus Giaquinto, and Philip S. Kitcher. The chapter then characterizes the notion of mathematical practice by successively proposing several constraints. It argues that several different levels of practice and knowledge are coexistent and that their interrelationships are crucial to mathematical knowledge. It shows how the scheme of a web of interrelated practices—counting practices, measuring practices, technical practices, scientific practices—with their systematic links acting as constraint and guide, can be applied in the analysis of very different levels of mathematical activity.

scholarly journals Types of mathematical tasks used in secondary classroom instruction

2020 ◽  
Vol 9 (3) ◽  
pp. 751
Author(s):  
Valbona Berisha ◽  
Ruzhdi Bytyqi
Keyword(s):  
Classroom Instruction ◽  
Analytical Framework ◽  
Mathematical Tasks ◽  
Mathematical Activity ◽  
Symbolic Form ◽  
First Semester ◽  
School Year ◽  
State Education ◽  
Secondary Classroom ◽  
The Individual

This study examined the quality and types of mathematical tasks used for classroom instruction in an upper secondary school – gymnasium. All the mathematical tasks presented in nine different school classrooms during the first semester of the school year 2018/2019 were analysed against a 5D analytical framework. The dimensions of the individual task analysis were contextual features; the answer forms required; forms of presentation; types of required mathematical activity and cognitive demands involved. Performed analysis gived perspective on the learning opportunities offered in classroom instruction for building mathematical competencies specified in the current state education curriculum. The results indicate that the selection of tasks was not in accordance with the curriculum requirements. Mostly, teachers used close-ended, non-applicative, lower-level cognitive tasks presented in symbolic form, promoting operations and calculations as a central activity. These types of tasks are usually associated with knowledge and skills of lower orders. In this case, classroom instruction had low potential and very little room left to build a significant portion of competencies and learning outcomes of higher orders.

scholarly journals Connecting expectations of Te Whāriki and The New Zealand Curriculum: Examples of mathematical practices

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Jane McChesney ◽  
Margaret Carr
Keyword(s):  
Early Childhood ◽  
New Zealand ◽  
Mathematics Curriculum ◽  
First Year ◽  
Mathematical Activity ◽  
Early Childhood Mathematics ◽  
Conceptual Tool ◽  
Mathematical Practices ◽  
Childhood Education ◽  
Te Whāriki

The first year of primary school aims to be closely connected with early childhood education, yet this is often invisible in the curriculum of specific subjects. This paper sets out an approach that uses mathematical practices as a curriculum tool that reconceptualises school mathematics. Using the early childhood mathematics framework of Te Kākano, the strands of mathematical practices are important descriptors of mathematical activity for children. We describe examples of mathematical learning from both early childhood and the first year of school, and make a case for using mathematical practices as a conceptual tool for designing a mathematics curriculum in the first years of school.

scholarly journals FORMATION OF BILINGUAL MATHEMATICAL COMPETENCE IN PRIMARY SCHOOL PUPILS IN THE CONDITIONS OF NATIONAL RUSSIAN BILINGUALISM

2020 ◽  
pp. 27-38
Author(s):  
Наталья Ивановна Спиридонова
Keyword(s):  
Primary School ◽  
Bilingual Education ◽  
Language Proficiency ◽  
Mathematical Tasks ◽  
Mathematical Activity ◽  
Mathematical Competence ◽  
The Subject ◽  
High Level ◽  
Primary School Pupils ◽  
Special Language

Введение. В процессе билингвального обучения школьники должны не только качественно усвоить содержание учебного предмета, но также преодолеть языковые затруднения. Учитывая взаимосвязь между речевой и математической деятельностью, раскрываются сущность и структура билингвальной математической компетенции, сформированность которой позволяет обучающимся-билингвам успешно усваивать программу основной школы в условиях национально-русского двуязычия. Также предложены некоторые пути формирования билингвальной математической компетенции, сосредоточенные на развитии культуры математической речи, а также обучении школьников применению поликультурных знаний. Цель – описать методику формирования билингвальной математической компетенции у обучающихся основной школы в условиях национально-русского двуязычия. Материал и методы. Были использованы теоретические методы сравнительного анализа, синтеза и обобщения содержания отечественной и зарубежной научно-методической литературы по теме исследования. Результаты и обсуждение. В ходе работы проанализированы исследования, которые указывают на тесную взаимосвязь между языком обучения и предметным математическим содержанием. Было установлено, что в условиях билингвального образования необходимо учитывать родной язык обучающихся. Кроме того, выявлено, что уровень владения родным и русским языками влияет на математические достижения обучающихся-билингвов. Данный анализ показал, что результатом билингвального обучения должен стать синтез определенных компетенций, обеспечивающий высокий уровень владения языком и глубокое освоение предметного содержания. Заключение. В ходе исследования уточнено понятие «билингвальная математическая компетенция», синтезирующее в себе предметный, специальные языковые (на родном и русском языках) и межкультурный компоненты. Была представлена система математических задач, которая направлена на развитие математической речи обучающихся, что обеспечивает формирование предметного и специальных языковых компонентов, а также уточнены понятия, которые должны быть заключены в сюжете текстовых математических задач для формирования межкультурного компонента билингвальной математической компетенции. Результаты данного исследования в дальнейшем послужат основой для проведения экспериментального исследования по проверке эффективности предложенных средств обучения, которые разработаны для формирования билингвальной математической компетенции. Introduction. In the process of bilingual education, pupils must not only master the content of the subject, but also overcome language difficulties. Taking into account the relationship between speech and mathematical activity, this article reveals the essence and structure of bilingual mathematical competence, the formation of which allows bilingual pupils to successfully learn the program of main school in the conditions of national Russian bilingualism. Some ways of forming bilingual mathematical competence focused on the development of the culture of mathematical speech, as well as teaching students to apply multicultural knowledge are also proposed. The aim of the article is to describe the methodology for the formation of bilingual mathematical competence in primary school pupils in the conditions of national Russian bilingualism. Material and methods. In this study, we used theoretical methods of comparative analysis, synthesis and generalization of the content of domestic and foreign scientific, pedagogical and methodological literature on the problem of research. Results and discussion. In the course of the work, we analyzed the research that indicates a close relationship between the language of instruction and the subject mathematical content. It was found that in the conditions of bilingual education, it is necessary to take into account students’ native language. In addition, it was found that the level of native and the Russian languages proficiency affects the mathematical achievements of bilingual pupils. This analysis led to the conclusion that the result of bilingual education should be a synthesis of certain competencies that provide a high level of language proficiency and deep development of the subject content. Conclusion. As a result of the research, the concept of bilingual mathematical competence is clarified, which synthesizes the subject, special language (in native and the Russian languages) and intercultural components. The system of mathematical tasks was presented, which is aimed at the development of mathematical speech of pupils, which ensures the formation of subject and special language components, and also clarified the concepts that should be included in the plot of text-based mathematical tasks for the formation of an intercultural component of bilingual mathematical competence. The results of this research will later serve as the basis for conducting an experimental study to test the effectiveness of the proposed training tools, which are designed to form a bilingual mathematical competence.

Students' Mathematical Noticing

2013 ◽  
Vol 44 (5) ◽  
pp. 809-850 ◽  
Author(s):  
Joanne Lobato ◽  
Charles Hohensee ◽  
Bohdan Rhodehamel
Keyword(s):  
Social Interactions ◽  
Middle Grades ◽  
Applied Linguistics ◽  
Mathematical Tasks ◽  
Mathematical Activity ◽  
Student Reasoning ◽  
Material Resources ◽  
Discourse Practices ◽  
Subsequent Interview ◽  
Individual Cognition

Even in simple mathematical situations, there is an array of different mathematical features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noticing as a complex phenomenon that is distributed across individual cognition, social interactions, material resources, and normed practices. Specifically, this research demonstrates that different centers of focus emerged in two middle grades mathematics classes addressing the same content goals, which, in turn, were related conceptually to differences in student reasoning on subsequent interview tasks. Furthermore, differences in the discourse practices, features of the mathematical tasks, and the nature of the mathematical activity in the two classrooms were related to the different mathematical features that students appeared to notice.

scholarly journals Intellectual humility in mathematics

Synthese ◽  
2021 ◽  
Author(s):  
Colin Jakob Rittberg
Keyword(s):  
Case Studies ◽  
Virtue Theory ◽  
Philosophical Analysis ◽  
Mathematical Activity ◽  
Intellectual Virtues ◽  
Intellectual Humility ◽  
Mathematical Practices ◽  
The Individual ◽  
Individual Accounts

AbstractIn this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections.

Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning

1997 ◽  
Vol 28 (5) ◽  
pp. 524-549 ◽  
Author(s):  
Marjorie Henningsen ◽  
Mary Kay Stein
Keyword(s):  
Cognitive Processes ◽  
Mathematical Thinking ◽  
Mathematical Tasks ◽  
Mathematical Activity ◽  
Mathematics Classrooms ◽  
Student Cognition ◽  
Set Up ◽  
High Level

In order to develop students' capacities to “do mathematics,” classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.

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