scholarly journals Enhancing Number System Knowledge to Promote Number Sense and Adaptive Expertise: A Case Study of a Second-Grade Mathematics Student

2018 ◽  
Vol 15 (3) ◽  
Author(s):  
Cami Player ◽  
Jessica Shumway
Keyword(s):  
Mathematics Education ◽  
Mathematics Instruction ◽  
Number Sense ◽  
Number System ◽  
Second Grade ◽  
Adaptive Expertise ◽  
Instructional Treatment ◽  
Mathematics Problems ◽  
System Knowledge

Instruction for developing students’ number sense is a critical area of research in mathematics education due to the role number sense plays in early mathematics learning. Specifically, number system knowledge—systematic relations among numerals and the use of number relations to solve arithmetic problems—has been identified as a key cognitive mechanism in number sense development. Number system knowledge is a component of number sense, and the researchers of this study hypothesize that it plays a critical role in second-grade students’ understanding of relationships among numbers and adaptive expertise with mathematics problems. The purpose of this exploratory case study was to investigate the variations of an eight-year-old student’s number system knowledge learning as she participated in an instructional treatment over nine weeks. The main research question of this study was: In what ways does a student struggling in mathematics develop number system knowledge during a nine-week period in her second-grade classroom as she engages in a number system knowledge instructional treatment? The case in this study was selected based on her low pretest score combined with her desire for making sense of mathematics. The data sources for this study were a number system knowledge assessment and student interviews. The analysis involved a multiple-cycle coding process that resulted in themes of adaptive expertise and the union of procedural and conceptual knowledge in mathematics instruction. The results suggest that this number system knowledge instructional treatment provided this case-study student to develop more pronounced adaptive expertise in solving mathematics problems. An in-depth analysis of how and why one struggling student develops number system knowledge during a nine-week instructional treatment within the context of her mathematics class provides exploratory evidence to help researchers and teachers develop and implement similar practices in elementary mathematics instruction. KEYWORDS: Number Sense; Number System Knowledge; Mathematics Education; Whole Numbers and Operations; Elementary Education; Teaching and Learning; Case Study Research

Beliefs and Their Influence on Mathematical Performance

1989 ◽  
Vol 82 (7) ◽  
pp. 502-505 ◽  
Author(s):  
Joe Garofalo
Keyword(s):  
Mathematics Education ◽  
Mathematics Instruction ◽  
Mathematical Tasks ◽  
Mathematical Task ◽  
Mathematical Content ◽  
Mathematical Performance ◽  
Mathematics Problems ◽  
Course Of Action ◽  
Mathematical Behavior ◽  
Research In Mathematics Education

Recent research in mathematics education has shown that success or failure in solving mathematics problems often depends on much more than the knowledge of requisite mathematical content. Knowing appropriate facts, algorithms, and procedures is not sufficient to guarantee success. Other factors, such as the decisions one makes and the strategies one uses in connect ion with the control and regulation of one's actions (e.g., deciding to analyze the conditions of a problem, planning a course of action, assessing progress), the emotions one fee ls while working on a mathematical task (e.g., anxiety, frustration, enjoyment), and the beliefs one holds relevant to performance on mathematical tasks, influence the direction and outcome of one's performance (Garofalo and Lester 1985; Schoenfe ld 1985; McLeod 1988). These other factors, although not explicitly addressed in typical mathematics instruction, are nonetheless important aspects of mathematical behavior.

scholarly journals The Relationship of Reading Abilities With the Underlying Cognitive Skills of Math: A Dimensional Approach

2021 ◽  
Vol 12 ◽  
Author(s):  
Luca Bernabini ◽  
Paola Bonifacci ◽  
Peter F. de Jong
Keyword(s):  
Phonological Awareness ◽  
Cognitive Abilities ◽  
Cognitive Skills ◽  
Number Sense ◽  
Number System ◽  
Dimensional Approach ◽  
Reading And Math ◽  
Relationship Of ◽  
The Relationship ◽  
System Knowledge

Math and reading are related, and math problems are often accompanied by problems in reading. In the present study, we used a dimensional approach and we aimed to assess the relationship of reading and math with the cognitive skills assumed to underlie the development of math. The sample included 97 children from 4th and 5th grades of a primary school. Children were administered measures of reading and math, non-verbal IQ, and various underlying cognitive abilities of math (counting, number sense, and number system knowledge). We also included measures of phonological awareness and working memory (WM). Two approaches were undertaken to elucidate the relations of the cognitive skills with math and reading. In the first approach, we examined the unique contributions of math and reading ability, as well as their interaction, to each cognitive ability. In the second approach, the cognitive abilities were taken to predict math and reading. Results from the first set of analyses showed specific effects of math on number sense and number system knowledge, whereas counting was affected by both math and reading. No math-by-reading interactions were observed. In contrast, for phonological awareness, an interaction of math and reading was found. Lower performing children on both math and reading performed disproportionately lower. Results with respect to the second approach confirmed the specific relation of counting, number sense, and number system knowledge to math and the relation of counting to reading but added that each math-related marker contributed independently to math. Following this approach, no unique effects of phonological awareness on math and reading were found. In all, the results show that math is specifically related to counting, number sense, and number system knowledge. The results also highlight what each approach can contribute to an understanding of the relations of the various cognitive correlates with reading and math.

Visualizing Number: Instruction for Number System Knowledge in Second-Grade Classrooms

2020 ◽  
Vol 12 (2) ◽  
pp. 142-161
Author(s):  
Jessica F. Shumway ◽  
Kaitlin Bundock ◽  
Jessica King ◽  
Monika Burnside ◽  
Heather Gardner ◽  
...  
Keyword(s):  
Number System ◽  
Second Grade ◽  
System Knowledge

scholarly journals Developing Number Sense: An Approach to Initiate Algebraic Thinking in Primary Education

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 518
Author(s):  
Natividad Adamuz-Povedano ◽  
Elvira Fernández-Ahumada ◽  
M. Teresa García-Pérez ◽  
Jesús Montejo-Gámez
Keyword(s):  
Teaching And Learning ◽  
Number Sense ◽  
Methodological Approach ◽  
Number System ◽  
Algebraic Thinking ◽  
First Year ◽  
Symbolic Language ◽  
Decimal Number ◽  
Language Characteristic ◽  
Theoretical Foundations

Traditionally, the teaching and learning of algebra has been addressed at the beginning of secondary education with a methodological approach that broke traumatically into a mathematical universe until now represented by numbers, with bad consequences. It is important, then, to find methodological alternatives that allow the parallel development of arithmetical and algebraic thinking from the first years of learning. This article begins with a review of a series of theoretical foundations that support a methodological proposal based on the use of specific manipulative materials that foster a deep knowledge of the decimal number system, while verbalizing and representing quantitative situations that underline numerical relationships and properties and patterns of numbers. Developing and illustrating this approach is the main purpose of this paper. The proposal has been implemented in a group of 25 pupils in the first year of primary school. Some observed milestones are presented and analyzed. In the light of the results, this well-planned early intervention contains key elements to initiate algebraic thinking through the development of number sense, naturally enhancing the translation of purely arithmetical situations into the symbolic language characteristic of algebraic thinking.

scholarly journals Instructional improv to analyze inquiry-based science teaching: Zed’s dead and the missing flower

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Maggie Dahn ◽  
Christine Lee ◽  
Noel Enyedy ◽  
Joshua Danish
Keyword(s):  
Mixed Reality ◽  
School Science ◽  
Second Grade ◽  
Science Lesson ◽  
Instructional Decisions ◽  
Case Study Analysis ◽  
Second Grade Students ◽  
Lesson Context ◽  
Power And Agency

AbstractIn inquiry-based science lessons teachers face the challenge of adhering to curricular goals while simultaneously following students’ intuitive understandings. Improvisation (improv) provides a useful frame for understanding teaching in these inquiry-based contexts. This paper builds from prior work that uses improv as a metaphor for teaching to present a translated model for analysis of teaching in an inquiry-based, elementary school science lesson context. We call our model instructional improv, which shows how a teacher spontaneously synthesizes rules of improv with teaching practices to support student learning, engagement, and agency. We illustrate instructional improv through case study analysis of video recorded classroom interactions with one teacher and 26 first and second grade students learning about the complex system of honey bee pollination in a mixed reality environment. Our model includes the following defining features to describe how teaching happens in this context: the teacher 1) tells a story; 2) reframes mistakes as opportunities; 3) agrees; 4) yes ands; 5) makes statements (or asks questions that elicit statements); and 6) puts the needs of the classroom ensemble over individuals. Overall, we show how instructional improv helps explain how teachers can support science discourse and collective storytelling as a teacher (a) shifts power and agency to students; (b) balances learning and agency; and (c) makes purposeful instructional decisions. Findings have immediate implications for researchers analyzing interactions in inquiry-based learning environments and potential future implications for teachers to support inquiry learning.

scholarly journals Keterampilan Berfikir Siswa SMP dalam Menyelesaikan Soal Matematika Ditinjau dari Kemampuan Number Sense

2019 ◽  
Vol 6 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Lilik Setyaningsih ◽  
Arta Ekayanti
Keyword(s):  
Junior High School ◽  
Number Sense ◽  
Higher Order Thinking ◽  
Thinking Skills ◽  
Thinking Skill ◽  
Ability Test ◽  
Mathematics Problems ◽  
Qualitative Descriptive ◽  
And Mathematics ◽  
Mathematics Problem

This research aimed to describe the students’ thinking skills of each number sense category in solving mathematics problems. This study used a qualitative descriptive approach and involved one class of Year 7 students in one of junior high school in Ponorogo, Indonesia. Data collection involved test and non-test. The instruments were number sense ability test and mathematics problems including six cognitive categories. Data analysis included collecting data, reducing data, analyzing data and drawing conclusions. The results showed that students who had low number sense ability were classified as Lower Order Thinking Skill (LOTS) level. In this category, students can only solve mathematics problem involving remembering and understanding categories. While the students with medium number sense ability also identified at LOTS level. In this category, students can only solve the problem involving applying category. Furthermore, the students who had a high number sense ability were classified as Higher Order Thinking Skill (HOTS) level. In this category, students can solve the mathematics problem involving analyzing) and evaluating categories.

scholarly journals O ENSINO DE SITUAÇÕES MULTIPLICATIVAS: constatações a partir dos atos de mediação docente

2017 ◽  
Vol 24 (2) ◽  
pp. 74
Author(s):  
Eliziane Rocha Castro ◽  
Marcília Chagas Barreto ◽  
Antonio Luiz De Oliveira Barreto ◽  
Francisco Jeovane do Nascimento
Keyword(s):  
Elementary School ◽  
Mathematics Education ◽  
Teaching Practice ◽  
Field Research ◽  
Technical Skills ◽  
Theoretical Construct ◽  
Case Study Method ◽  
History Of ◽  
Professional History

ElResumo: Inserida no campo da Educação Matemática, esta investigação tem como objetivo central analisar os atos de mediação docente no ensino de situações multiplicativas no 5º ano do Ensino Fundamental, tendo como suporte referencial a Teoria dos Campos Conceituais. O constructo teórico prevê a estruturação dos conceitos de multiplicação e divisão em um único campo conceitual – o das Estruturas Multiplicativas. A pesquisa é de natureza qualitativa, ancorada no método do Estudo de Caso recaindo sobre os atos de mediação de uma docente do 5º ano do Ensino Fundamental de uma escola da rede pública do município de São Luís, Maranhão. A pesquisa de campo foi realizada nos meses de outubro e novembro de 2015. Os dados empíricos foram coletados por observação de três aulas previamente planejadas pela docente observada. Os achados dessa incursão investigativa apontam a carência do trabalho voltado para os aspectos conceituais das operações de multiplicação e divisão, bem como revelam a proeminência da simbolização em detrimento da conceitualização. As conclusões que se derivam dessa incursão investigativa entrelaçam aspectos inerentes à formação e à prática docente, na medida em que englobam o amplo repertório de eskemas concernentes à interação, comunicação, linguagem e afetividade, além do conjunto de competências técnicas e conhecimentos propagados nos espaços de formação que também modelam os atos de mediação docente no decurso da história individual e profissional dos professores.Palavras-chave: Situações multiplicativas. Mediação docente. Teoria dos Campos Conceituais.TEACHING SITUATIONS MULTIPLICATIVE: findings from the mediation acts of teachers Abstract: Inserted in the field of mathematics education, this research had as main objective to analyze the acts of teacher mediation in teaching multiplicative situations in the 5th year of elementary school, supported by the Theory of Conceptual Fields. The theoretical construct provides the structure of multiplication and division concepts into a single conceptual field - that of multiplicative structures. The research is qualitative in nature, anchored in the Case Study method falling on the acts of mediation of a teacher of the 5th year of elementary school in a public school in São Luís, Maranhão. The field research was conducted in the months of October and November 2015. The data were collected by observation of three classes previously planned by the teacher observed. The findings of this investigative foray point to the lack of focused work for the conceptual aspects of the multiplication and division operations , as well as reveal the prominence of symbolization at the expense of conceptualisation. The conclusions derived from this investigative foray intertwine aspects of training and teaching practice, in that it encompasses the broad repertoire  concerning the interaction, communication, language and affection, beyond the range of technical skills and propagate knowledge in the areas of training also model the acts of teaching mediation during personal and professional history of teachers.Keywords: Situations multiplicative. Mediation acts of teachers. Theory of Conceptual Fields.LA ENSEÑANZA DE SITUACIONES MULTIPLICATIVAS: resultados a partir de los actos de mediación docente Resumen: Insertado en el campo de la educación matemática, esta investigación tiene como objetivo principal analizar los actos de mediación docente en la enseñanza de las situaciones multiplicativas en el 5º año de la escuela primaria, utilizando como soporte de referencia la teoría de los campos conceptuales. La construcción teórica proporciona la estructura de los conceptos de multiplicación y división en un solo campo conceptual – el de las estructuras multiplicativas. La investigación es de naturaleza cualitativa, anclada en el método de estudio de caso que recae sobre los actos de la mediación de una docente de 5º año de primaria en una escuela pública en São Luís, Maranhão. La investigación de campo fue realizada en los meses de octubre y noviembre de 2015. Los datos empíricos fueron recogidos mediante la observación de tres clases previamente programadas por la profesora observada. Las conclusiones de este punto de incursión señalan la carencia de trabajo dirigido a los aspectos conceptuales de las operaciones matemáticas de multiplicación y división, así como revelan la prominencia de la simbolización en detrimento de la conceptualización. Las conclusiones derivadas de esa investigación entrelazan aspectos de la formación y la enseñanza práctica, ya que abarca el amplio repertorio de eskemas relativos a la interacción, comunicación, lenguaje y afectividad, además del conjunto de competencias técnicas y conocimientos propagados en los espacios de formación que también modelan los actos de mediación docente en el decurso de la historia personal y profesional de los profesores.Palabras clave: Situaciones multiplicativas. Mediación docente. Teoría de los Campos Conceptuales.       

scholarly journals USING ANALOGY IN SOLVING PROBLEMS: A CASE STUDY OF TEACHING THE RADICAL INEQUALITIES

2021 ◽  
Vol 8 (5) ◽  
Author(s):  
Bui Phuong Uyen
Keyword(s):  
High School ◽  
Mathematics Education ◽  
Group Work ◽  
Analogical Reasoning ◽  
Considerable Improvement ◽  
Problem Solving Skills ◽  
Three Phase ◽  
Phase Group ◽  
Can Tho City

In mathematics education, teachers can use several reasoning methods to find solutions such as inductive, deductive and analogy. This study was intended to guide students to find solutions to problems of radical inequalities through analogical reasoning. The experiment was conducted on 36 grade 10 students at a high school in Can Tho city of Vietnam. The instrument used was a problem of radical inequalities. A three-phase teaching process had been organized with this class comprising individual work phase, group work phase and institutionalization phase. The data collected included student worksheets and was qualitatively analyzed. As a result, many students discovered how to solve the above inequality by using the analogy, and they had a considerable improvement in their problem-solving skills. Additionally, a few ideas were discussed about the use of analogy in mathematics education. <p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/edu_01/0769/a.php" alt="Hit counter" /></p>

scholarly journals Inconsistency Among Beliefs, Knowledge, and Teaching Practice in Mathematical Problem Solving: A Case Study of a Primary Teacher

2017 ◽  
Vol 7 (2) ◽  
pp. 27-40
Author(s):  
Tatag Yuli Eko Siswono ◽  
Ahmad Wachidul Kohar ◽  
Ika Kurniasari ◽  
Sugi Hartono
Keyword(s):  
Problem Solving ◽  
Teaching Practice ◽  
Mathematics Learning ◽  
Mathematical Problem ◽  
Primary Teacher ◽  
Mathematical Problem Solving ◽  
Mathematics Problems ◽  
And Mathematics ◽  
Teacher's Beliefs

This is a case study investigating a primary teacher’s beliefs, knowledge, and teaching practice in mathematical problem solving. Data was collected through interview of one primary teacher regarding his beliefs on the nature of mathematics, mathematics teaching, and mathematics learning as well as knowledge about content and pedagogy of problem solving. His teaching practice was also observed which focused on the way he helped his students solve several different mathematics problems in class based on Polya’s problemsolving process: understand the problem, devising a plan, carrying out the plan, and looking back. Findings of this study point out that while the teacher’s beliefs, which are closely related to his problem solving view, are consistent with his knowledge of problem solving, there is a gap between such beliefs and knowledge around his teaching practice. The gap appeared primarily around the directive teaching which corresponds to instrumental view he held in most of Polya’s process during his teaching practice, which is not consistent with beliefs and knowledge he professed during the interview. Some possible causes related to several associate factors such as immediate classroom situation and teaching practice experience are discussed to explain such inconsistency. The results of this study are encouraging, however, further studies still need to be conducted.

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