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This research focuses on the development, validity, and practicality of booklet-based teaching materials for mathematics subjects, specifically on equivalent fractions. Following the Research and Development (R&D) model, which incorporates the ADDIE (Analysis, Design, Development, Implementation, and Evaluation) design stages, this research aims to create effective learning aids. The assessment of the developed Booklet showed high validity and practicality. The material expert assessment rated it "very valid" at 93.33%, while the media design and language experts rated it high at 98.57% and 95%, respectively. In addition, both teachers and students considered the Booklet practical, with response percentages of 87% and 94%, respectively. These findings underscore booklet-based learning materials' effectiveness and ease of use in mathematics education. The Booklet is a valuable tool for teachers to deliver curriculum content effectively and for students to improve their understanding of mathematical concepts. Learning outcomes can be significantly enhanced through such media, facilitating understanding and practical application of mathematical principles.

This research focuses on the development, validity, and practicality of booklet-based teaching materials for mathematics subjects, specifically on equivalent fractions. Following the Research and Development (R&D) model, which incorporates the ADDIE (Analysis, Design, Development, Implementation, and Evaluation) design stages, this research aims to create effective learning aids. The assessment of the developed Booklet showed high validity and practicality. The material expert assessment rated it "very valid" at 93.33%, while the media design and language experts rated it high at 98.57% and 95%, respectively. In addition, both teachers and students considered the Booklet practical, with response percentages of 87% and 94%, respectively. These findings underscore booklet-based learning materials' effectiveness and ease of use in mathematics education. The Booklet is a valuable tool for teachers to deliver curriculum content effectively and for students to improve their understanding of mathematical concepts. Learning outcomes can be significantly enhanced through such media, facilitating understanding and practical application of mathematical principles.

This research focuses on the development, validity, and practicality of booklet-based teaching materials for mathematics subjects, specifically on equivalent fractions. Following the Research and Development (R&D) model, which incorporates the ADDIE (Analysis, Design, Development, Implementation, and Evaluation) design stages, this research aims to create effective learning aids. The assessment of the developed Booklet showed high validity and practicality. The material expert assessment rated it "very valid" at 93.33%, while the media design and language experts rated it high at 98.57% and 95%, respectively. In addition, both teachers and students considered the Booklet practical, with response percentages of 87% and 94%, respectively. These findings underscore booklet-based learning materials' effectiveness and ease of use in mathematics education. The Booklet is a valuable tool for teachers to deliver curriculum content effectively and for students to improve their understanding of mathematical concepts. Learning outcomes can be significantly enhanced through such media, facilitating understanding and practical application of mathematical principles.

This research focuses on the development, validity, and practicality of booklet-based teaching materials for mathematics subjects, specifically on equivalent fractions. Following the Research and Development (R&D) model, which incorporates the ADDIE (Analysis, Design, Development, Implementation, and Evaluation) design stages, this research aims to create effective learning aids. The assessment of the developed Booklet showed high validity and practicality. The material expert assessment rated it "very valid" at 93.33%, while the media design and language experts rated it high at 98.57% and 95%, respectively. In addition, both teachers and students considered the Booklet practical, with response percentages of 87% and 94%, respectively. These findings underscore booklet-based learning materials' effectiveness and ease of use in mathematics education. The Booklet is a valuable tool for teachers to deliver curriculum content effectively and for students to improve their understanding of mathematical concepts. Learning outcomes can be significantly enhanced through such media, facilitating understanding and practical application of mathematical principles.

This article covers the methodology of teaching mathematics. In addition his theory was considered also about the content of the introduction of understanding about this.

This article covers the methodology of teaching mathematics. In addition his theory was considered also about the content of the introduction of understanding about this.

The purpose of this transcendental phenomenological study was to describe elementary teachers’ lived experiences of implementing constructivist strategies, specifically the concrete, representational, and abstract approach (CRA) when teaching mathematics in a midwestern school district by asking: What are elementary teachers’ lived experiences implementing constructivist strategies? The theory that guided this study is Bruner’s constructivist learning theory, as the enactive-iconic-symbolic learning he described remarkably parallels the concrete, representational, abstract (CRA) framework for teaching and learning mathematics. This qualitative study consisted of 10 elementary math teachers who instructed kindergarten through fifth grades in Montgomery City Schools in suburban Ohio. Data from interviews, focus groups, and journal entries were coded and placed into emerging themes. A detailed descriptive analysis of the data was included, and a complete description of the participants’ lived experiences were integrated. Four themes developed through data analysis using Moustakas’s modified method of analysis in this phenomenological study and include (a) mathematical understanding with concrete objects, (b) mathematical concepts with representations, (c) providing inquiry with abstract problems, and (d) recognizing needs with abstract concepts. These themes corresponded to the theoretical framework of the study. This study confirmed Bruner’s learning theory through participants shared lived experiences teaching math with constructivist strategies. Teachers experienced success using CRA to help teach students conceptual understanding of mathematics.

The article considers the development of the online educational model "Creative Thinking through Learning Elementary Maths" and the results of the efficiency to implement this course for students of the specialization “Mathematics” while learning Elementary Mathematics. The research represents the development of learning materials for the online course posted in free access on the educational platform "Higher School Mathematics Teacher". The course components (theoretical, practical, and feedback) are focused on the organization of motivated and consistent students’ activity; open and integrative Rich tasks of three levels of complexity. The course participants’ activity, focused on the recognition, classification, solution, creation of Rich tasks, encourages the formation of the main components of creative thinking that include a problem statement, fluency, flexibility, originality, and elaboration. The first version of the course materials was improved following course users’ wishes, and then the efficiency of online course education was checked using an experiment. The analysis of results of the experiment allows concluding about the efficiency of the developed model of learning an online course.

The article considers the development of the online educational model "Creative Thinking through Learning Elementary Maths" and the results of the efficiency to implement this course for students of the specialization “Mathematics” while learning Elementary Mathematics. The research represents the development of learning materials for the online course posted in free access on the educational platform "Higher School Mathematics Teacher". The course components (theoretical, practical, and feedback) are focused on the organization of motivated and consistent students’ activity; open and integrative Rich tasks of three levels of complexity. The course participants’ activity, focused on the recognition, classification, solution, creation of Rich tasks, encourages the formation of the main components of creative thinking that include a problem statement, fluency, flexibility, originality, and elaboration. The first version of the course materials was improved following course users’ wishes, and then the efficiency of online course education was checked using an experiment. The analysis of results of the experiment allows concluding about the efficiency of the developed model of learning an online course.

The purpose of this quantitative, quasi-experimental study was to examine the relationship between college students’ virtual math mentorships and instruction on third grade students’ math academic success. College students in teacher preparation programs investigated deficiencies in third grade students’ competency of math concepts as specified by state standards, and they designed lessons to address these challenges through weekly virtual meetings. Third grade teachers wanted to provide additional math supports for their struggling students, and this partnership between the college students and the third graders was the foundation of this study at four Title 1 rural elementary schools. This research is an extension of a previous investigation where only students’ attitudes towards math were documented, showing improvements after receiving targeted interventions from college mentors. For this study, an analysis of covariance (ANCOVA) was used, comparing six third grade classes, three with virtual math mentoring interventions and three without interventions. Pre and post standardized tests were used in the control and experimental groups, and from the data, it was determined that there was a slight difference in students’ math achievement scores as a result of the interventions by college students. Future research on elementary students’ math success should include: (a) the influence of parents, guardians, and teachers on students’ engagement in and focus on math skills, (b) the attitudes and instructional style of teachers, (c) the experience level of the teachers, (d) the gender and socio-economic status of students, (e) the strength of the relationship between a preservice teacher and elementary students, and (6) the social and emotional status of the students being tested in a non-pandemic year.

The purpose of this study was to determine if a 15-week mindfulness curriculum, MindUP, impacted participants’ performance on a standardized mathematics word problem measure. The study was conducted in two elementary schools in a large public school district in North Carolina during the 2020-2021 school year. Participants included 74 third- and fourth-grade students who received a supplemental mindfulness curriculum and 77 students who received only business as usual instruction (N = 151). The research question and null hypothesis were tested using a quantitative, quasi-experimental, pretest posttest, nonequivalent, control group design. The TOMA-3 Word Problems subtest was administered by school personnel to all participants and was used as the pretest and posttest instrument. An ANCOVA was used to determine if there was statistically significant difference TOMA-3 performance, based on group membership, while controlling for the covariate pretest score. The results of this study reveal no statistically significant difference in word problem performance following participation in the mindfulness curriculum. These findings reveal discrepancies with empirical evidence in favor of mindfulness practices in schools and raise questions for future examination. More work is needed to understand the specific components of mindfulness curricula that improve executive function skills and how these improvements transfer to mathematics performance.

Teacher noticing requires listening to student sense-making, interpreting the mathematical understandings teachers hear, and deciding how to respond based on what they notice in a specific interaction. Teachers manage what they notice about students in instructional interactions to both make key teaching decisions and adjust their interactions with specific learners. Elementary teachers narrate their procedural and declarative knowledge in the moment between listening and deciding how to interact. The data is generated through qualitative interviewing, observation, and documentation. There is a cyclical process of listening to student sense-making, then recalling the student criteria that triggered an interaction, as well as a balance between impulsivity and overthinking (in which the tension between a teacher’s best knowledge and the gathering of more student information is considered), before a teacher commits to a specific response. Teacher listening processes, context, and ability to narrate interactions are considered to describe how student sense-making can drive the opportunities to learn that teachers provide.

Teacher noticing requires listening to student sense-making, interpreting the mathematical understandings teachers hear, and deciding how to respond based on what they notice in a specific interaction. Teachers manage what they notice about students in instructional interactions to both make key teaching decisions and adjust their interactions with specific learners. Elementary teachers narrate their procedural and declarative knowledge in the moment between listening and deciding how to interact. The data is generated through qualitative interviewing, observation, and documentation. There is a cyclical process of listening to student sense-making, then recalling the student criteria that triggered an interaction, as well as a balance between impulsivity and overthinking (in which the tension between a teacher’s best knowledge and the gathering of more student information is considered), before a teacher commits to a specific response. Teacher listening processes, context, and ability to narrate interactions are considered to describe how student sense-making can drive the opportunities to learn that teachers provide.

Teacher noticing requires listening to student sense-making, interpreting the mathematical understandings teachers hear, and deciding how to respond based on what they notice in a specific interaction. Teachers manage what they notice about students in instructional interactions to both make key teaching decisions and adjust their interactions with specific learners. Elementary teachers narrate their procedural and declarative knowledge in the moment between listening and deciding how to interact. The data is generated through qualitative interviewing, observation, and documentation. There is a cyclical process of listening to student sense-making, then recalling the student criteria that triggered an interaction, as well as a balance between impulsivity and overthinking (in which the tension between a teacher’s best knowledge and the gathering of more student information is considered), before a teacher commits to a specific response. Teacher listening processes, context, and ability to narrate interactions are considered to describe how student sense-making can drive the opportunities to learn that teachers provide.

The article presents the results of the study of the impact of knowledge sections of elementary mathematics on the study of higher mathematics by students of Pskov State University. The analysis is based on the knowledge of the elementary mathematics sections tested on a unified state exam (USE). The most important sections of USE's monitoring materials have been identified for the study. The results of these assignments are given to school leavers who have enrolled in 2018 and 2019 at Pskov State University. The regression analysis method examined the impact of USE results on higher mathematics by students studying at the university's technical areas. The regression rate, which indicates the strength of the link between the results of higher and elementary mathematics, is about 40%. Recommendations are made for the organization of the educational process of studying higher mathematics, taking into account the low level of knowledge of school mathematics entering the first year of the university

Teacher noticing requires listening to student sense-making, interpreting the mathematical understandings teachers hear, and deciding how to respond based on what they notice in a specific interaction. Teachers manage what they notice about students in instructional interactions to both make key teaching decisions and adjust their interactions with specific learners. Elementary teachers narrate their procedural and declarative knowledge in the moment between listening and deciding how to interact. The data is generated through qualitative interviewing, observation, and documentation. There is a cyclical process of listening to student sense-making, then recalling the student criteria that triggered an interaction, as well as a balance between impulsivity and overthinking (in which the tension between a teacher’s best knowledge and the gathering of more student information is considered), before a teacher commits to a specific response. Teacher listening processes, context, and ability to narrate interactions are considered to describe how student sense-making can drive the opportunities to learn that teachers provide.

The article presents the results of the study of the impact of knowledge sections of elementary mathematics on the study of higher mathematics by students of Pskov State University. The analysis is based on the knowledge of the elementary mathematics sections tested on a unified state exam (USE). The most important sections of USE's monitoring materials have been identified for the study. The results of these assignments are given to school leavers who have enrolled in 2018 and 2019 at Pskov State University. The regression analysis method examined the impact of USE results on higher mathematics by students studying at the university's technical areas. The regression rate, which indicates the strength of the link between the results of higher and elementary mathematics, is about 40%. Recommendations are made for the organization of the educational process of studying higher mathematics, taking into account the low level of knowledge of school mathematics entering the first year of the university

Education is the responsibility of the school family and the environment. The success of the three component education is crucial. In addition to these three components, learning methods or strategies, learning tools are also absolutely necessary. The problem of this research is how to use concrete objects can increase the ability of students of Class I in SD Negeri No. 16 Sungai Apit Kecamatan Sungai Siak Kabupaten Siak in operationalizing addition and subtraction in mathematics. This study aims to improve students' ability in operationalizing the addition and subtraction with the help of concrete objects, also expected to benefit students, researchers, and parents. To achieve the purpose of the research, the researcher used classroom action research against the first grade students of SD Negeri 16 Apit Sungai Siak Kecamatan Sungai Apit Kabupaten Siak as many as 37 students who performed in 2 cycles. The method used in this study is a qualitative approach that describes the actual problems that exist in the field and then reflected and analyzed based on the theory that support followed by the implementation of research in the field.Data obtained in this study through observation discussion and evaluation. The results of this study indicate an increase of priminaliry activities, cycle I and cycle II.In priminaliry study, students who experience mastery learning as much as 35% after the action with the tools of concrete objects.Student learning completeness in the first cycle rose to 97% followed by the next cycle all students experience complete mastery learning. The conclusion that can be taken is that concrete objects can help students in operationalizing the addition and reduction of numbers on learning Class I Mathematics, so that learning achievement progress. Other findings children become happy, confident in doing the learning process. Thus it can be suggested that the implementation of education should be environmentally friendly because the environment prov

Children who experience difficulties with learning mathematics should be taught by teachers who focus on the child’s best way of learning. Analyses of the mathematical difficulties are necessary for fine-tuning mathematics education to the needs of these children. For this reason, an instrument for Observing and Analyzing children’s Mathematical Development (OAMD), based on action theory, has been developed. The use of levels of action is a new insight in the diagnostic process. Using the OAMD makes it possible to explore and analyze a child’s knowledge, proficiency and possible difficulties on four levels of acting in the domain of Number (counting, addition, subtraction, multiplication and division). The research concerns children from kindergarten up to grade three. In this article, we will discuss the purpose and the construction of the instrument with the focus on the usability of the OAMD. The study examines the quality and the diagnostic value of the instrument by means of the internal consistency, the test–retest reliability and the construct validity. The analyses show positive results. The conclusion of the research is that the OAMD is a suitable instrument for the analysis of numerical development of young children and their possible difficulties in this domain.

In this article we analyze the research studies carried out by northeast Portuguese about the existing Ethnomathematics knowledge in traditional professions and artefacts. We refer to craft trades such as cooperage, tinsmithing, locksmithing and carpentry, and artefacts built by these crasmen, such as barrels and vats, oil cans and other canned vessels, wrought-iron balconies, and yokes. We reflect on these studies and we disclose their potential in educational terms. We verified that it is possible to apply ethnomathematical knowledge and know-how in the context of the mathematics classroom in the 2nd and 3rd cycles of Portuguese education, and that this has its advantages. e mathematical contents are contextualized and the students discover uses of mathematics that they did not suspect, although being very close to them. In short, artisans feel their work valued and students become more aware of the culture of their region, in addition to learning math in an experimental and more engaging way., ResumoNeste artigo damos a conhecer os estudos de investigação que têm sido desenvolvidos no nordeste português sobre os saberes etnomatemáticos existentes em profissões e artefactos tradicionais. Referimo-nos a profissões artesanais como a tanoaria, a latoaria, a serralharia e a carpintaria e a artefactos construídos por esses artesãos, como barris, pipas, tonéis, almotolias e outras vasilhas em latão, grades das varandas feitas em ferro forjado e jugos. Refletimos sobre esses estudos e divulgamos o seu potencial em termos didáticos. Verificamos que é possível aplicar saberes e saberes-fazer etnomatemáticos em contexto de sala de aula de matemática nos 2.º e 3.º ciclos de ensino português, com vantagens. Os conteúdos matemáticos são contextualizados e os alunos descobrem utilizações da matemática que não suspeitavam, ainda que estando bem próximas deles. Em suma, os artesãos sentem o seu trabalho valorizado e os alunos ficam mais sensibilizados para a cultura da sua região, para além de aprenderem matemática de modo experimental e mais envolvente.

In this article we analyze the research studies carried out by northeast Portuguese about the existing Ethnomathematics knowledge in traditional professions and artefacts. We refer to craft trades such as cooperage, tinsmithing, locksmithing and carpentry, and artefacts built by these crasmen, such as barrels and vats, oil cans and other canned vessels, wrought-iron balconies, and yokes. We reflect on these studies and we disclose their potential in educational terms. We verified that it is possible to apply ethnomathematical knowledge and know-how in the context of the mathematics classroom in the 2nd and 3rd cycles of Portuguese education, and that this has its advantages. e mathematical contents are contextualized and the students discover uses of mathematics that they did not suspect, although being very close to them. In short, artisans feel their work valued and students become more aware of the culture of their region, in addition to learning math in an experimental and more engaging way., ResumoNeste artigo damos a conhecer os estudos de investigação que têm sido desenvolvidos no nordeste português sobre os saberes etnomatemáticos existentes em profissões e artefactos tradicionais. Referimo-nos a profissões artesanais como a tanoaria, a latoaria, a serralharia e a carpintaria e a artefactos construídos por esses artesãos, como barris, pipas, tonéis, almotolias e outras vasilhas em latão, grades das varandas feitas em ferro forjado e jugos. Refletimos sobre esses estudos e divulgamos o seu potencial em termos didáticos. Verificamos que é possível aplicar saberes e saberes-fazer etnomatemáticos em contexto de sala de aula de matemática nos 2.º e 3.º ciclos de ensino português, com vantagens. Os conteúdos matemáticos são contextualizados e os alunos descobrem utilizações da matemática que não suspeitavam, ainda que estando bem próximas deles. Em suma, os artesãos sentem o seu trabalho valorizado e os alunos ficam mais sensibilizados para a cultura da sua região, para além de aprenderem matemática de modo experimental e mais envolvente.

In this article we analyze the research studies carried out by northeast Portuguese about the existing Ethnomathematics knowledge in traditional professions and artefacts. We refer to craft trades such as cooperage, tinsmithing, locksmithing and carpentry, and artefacts built by these crasmen, such as barrels and vats, oil cans and other canned vessels, wrought-iron balconies, and yokes. We reflect on these studies and we disclose their potential in educational terms. We verified that it is possible to apply ethnomathematical knowledge and know-how in the context of the mathematics classroom in the 2nd and 3rd cycles of Portuguese education, and that this has its advantages. e mathematical contents are contextualized and the students discover uses of mathematics that they did not suspect, although being very close to them. In short, artisans feel their work valued and students become more aware of the culture of their region, in addition to learning math in an experimental and more engaging way., ResumoNeste artigo damos a conhecer os estudos de investigação que têm sido desenvolvidos no nordeste português sobre os saberes etnomatemáticos existentes em profissões e artefactos tradicionais. Referimo-nos a profissões artesanais como a tanoaria, a latoaria, a serralharia e a carpintaria e a artefactos construídos por esses artesãos, como barris, pipas, tonéis, almotolias e outras vasilhas em latão, grades das varandas feitas em ferro forjado e jugos. Refletimos sobre esses estudos e divulgamos o seu potencial em termos didáticos. Verificamos que é possível aplicar saberes e saberes-fazer etnomatemáticos em contexto de sala de aula de matemática nos 2.º e 3.º ciclos de ensino português, com vantagens. Os conteúdos matemáticos são contextualizados e os alunos descobrem utilizações da matemática que não suspeitavam, ainda que estando bem próximas deles. Em suma, os artesãos sentem o seu trabalho valorizado e os alunos ficam mais sensibilizados para a cultura da sua região, para além de aprenderem matemática de modo experimental e mais envolvente.

Learning progressions are one of the most important curriculum and assessment design ideas to be introduced in the past decade. A well-constructed learning progression can incorporate the knowledge needed to define the “track” that students may or should be on. This can inform teachers about when to teach what to whom. For the development of sound learning progressions, researchers investigate how learning typically unfolds in a particular area of study, and should empirically test and validate it. A learning progression of Number Sense was developed in a research project, Special Education Learning Progressions in Math (SELPM). The progression includes hypothesized levels of achievement for grades K to 3 for four sub-domains of Number Sense: Place Value, Addition, Magnitude Comparison, and Transcoding. This research was concerned with the validation of the learning progression. The study used three student assessment data sets to investigate the validity of the proposed learning progressions using item response theory, specifically the Rasch model and its extension. This research consists of three validation studies – Phase I: Preliminary Study, Phase II: Testing Validity of Learning Progression, and Phase III: Validation of Alternative Learning Progression. Through the iterative validation process, this research aimed to provide an empirically-validated and theoretically-based Number Sense learning progression and a set of assessments for the education community

Elementary school teachers in the United States are typically provided curricular materials by their school districts. However, little is known about how teachers may replace or supplement these provided materials with outside materials—materials found in their own curricular libraries or from external sources. This dissertation proposes a research framework on teachers’ use of outside materials that leads to the following research questions: What motivates teachers to consider using outside curricular materials? What approaches do they use to discover outside materials? What criteria do they use to evaluate outside materials? What steps do they take to prepare or adapt outside materials? And what external factors relate to teachers’ decisions to use outside materials? These questions were addressed through an online survey of 98 elementary teachers across two districts in Western Washington state, interviews with nine survey respondents who frequently considered using outside materials, and in-depth case studies of the decisions of three interviewees. The survey revealed that a large majority of teachers considered using outside curricular materials at least once in the 2015-2016 school year, that many of them did so frequently, and that their reasons for considering outside materials were linked to areas of dissatisfaction with the provided materials. A multiple linear regression found that higher teacher-reported feelings of curricular autonomy and more years of experience with their provided curriculum predicted higher frequency of considering outside materials. The survey also revealed that teachers used a wide variety of approaches to discover outside materials, including both online and offline methods. Interviews provided detail to these findings, revealing that teachers’ evaluation criteria were aligned with their initial motivation to consider outside materials, and that teachers tended to engage in minimal preparation of outside materials. Case studies of

Learning progressions are one of the most important curriculum and assessment design ideas to be introduced in the past decade. A well-constructed learning progression can incorporate the knowledge needed to define the “track” that students may or should be on. This can inform teachers about when to teach what to whom. For the development of sound learning progressions, researchers investigate how learning typically unfolds in a particular area of study, and should empirically test and validate it. A learning progression of Number Sense was developed in a research project, Special Education Learning Progressions in Math (SELPM). The progression includes hypothesized levels of achievement for grades K to 3 for four sub-domains of Number Sense: Place Value, Addition, Magnitude Comparison, and Transcoding. This research was concerned with the validation of the learning progression. The study used three student assessment data sets to investigate the validity of the proposed learning progressions using item response theory, specifically the Rasch model and its extension. This research consists of three validation studies – Phase I: Preliminary Study, Phase II: Testing Validity of Learning Progression, and Phase III: Validation of Alternative Learning Progression. Through the iterative validation process, this research aimed to provide an empirically-validated and theoretically-based Number Sense learning progression and a set of assessments for the education community

Elementary school teachers in the United States are typically provided curricular materials by their school districts. However, little is known about how teachers may replace or supplement these provided materials with outside materials—materials found in their own curricular libraries or from external sources. This dissertation proposes a research framework on teachers’ use of outside materials that leads to the following research questions: What motivates teachers to consider using outside curricular materials? What approaches do they use to discover outside materials? What criteria do they use to evaluate outside materials? What steps do they take to prepare or adapt outside materials? And what external factors relate to teachers’ decisions to use outside materials? These questions were addressed through an online survey of 98 elementary teachers across two districts in Western Washington state, interviews with nine survey respondents who frequently considered using outside materials, and in-depth case studies of the decisions of three interviewees. The survey revealed that a large majority of teachers considered using outside curricular materials at least once in the 2015-2016 school year, that many of them did so frequently, and that their reasons for considering outside materials were linked to areas of dissatisfaction with the provided materials. A multiple linear regression found that higher teacher-reported feelings of curricular autonomy and more years of experience with their provided curriculum predicted higher frequency of considering outside materials. The survey also revealed that teachers used a wide variety of approaches to discover outside materials, including both online and offline methods. Interviews provided detail to these findings, revealing that teachers’ evaluation criteria were aligned with their initial motivation to consider outside materials, and that teachers tended to engage in minimal preparation of outside materials. Case studies of

Procedural learning has dominated the study of K-12 mathematics in the United States. A lack of emphasis on the development of mathematical thinking among educators has undermined American students’ achievement in mathematics, thereby impacting students’ readiness for college and careers in the 21st century. Elementary teachers, in particular, are often under-equipped to tackle teaching practices that promote mathematical thinking and the development of conceptual mathematical knowledge, both emphasized in the Common Core Standards for Mathematical Practice. They are frequently restricted by their own content knowledge and confidence in understanding mathematics, and are themselves products of rote learning and an overemphasis on procedures. This study was designed to support development of mathematical teaching practices among a group of upper elementary (third through fifth grade) teachers. The study utilized a qualitative action research design in order to investigate teachers’ perceptions of the development of their practice over time. Ten upper elementary teachers participated in a bi-monthly mathematics study group over a period of 6 months. The study group initially focused on using warm-up tasks to promote mathematical discourse. Data was collected in the forms of transcriptions of the study group sessions, written and verbal reflections, and pre and post-questionnaires. Throughout the study, teachers regularly experimented with practices they had rarely or never tried before. Teachers reported that the warm-ups they learned in the professional development were the driving force for changes in their mathematics teaching practices. Additionally, teachers reported that there were substantial differences in the student discourse that occurred during math time between the previous school year and this year and that facilitation of student discourse during math time became a regular part of their practice.

This paper presents data collected as part of an intervention research project to develop exploratory talk and collaborative group work with six and seven year-old students in mathematics. A discursive approach was used to analyse and interpret variations in the way that three case-study teachers, involved in the project, managed the group work and developed the students’ talk. Analysis suggested that the intervention required, not just a change in practice, also but a shift in ideologies for these teachers. Ideologies are seen as discourses that legitimate and maintain a social pattern, and the paper explores how a research-based intervention can create conflicts between researchers’ and teachers’ ideologies.

Procedural learning has dominated the study of K-12 mathematics in the United States. A lack of emphasis on the development of mathematical thinking among educators has undermined American students’ achievement in mathematics, thereby impacting students’ readiness for college and careers in the 21st century. Elementary teachers, in particular, are often under-equipped to tackle teaching practices that promote mathematical thinking and the development of conceptual mathematical knowledge, both emphasized in the Common Core Standards for Mathematical Practice. They are frequently restricted by their own content knowledge and confidence in understanding mathematics, and are themselves products of rote learning and an overemphasis on procedures. This study was designed to support development of mathematical teaching practices among a group of upper elementary (third through fifth grade) teachers. The study utilized a qualitative action research design in order to investigate teachers’ perceptions of the development of their practice over time. Ten upper elementary teachers participated in a bi-monthly mathematics study group over a period of 6 months. The study group initially focused on using warm-up tasks to promote mathematical discourse. Data was collected in the forms of transcriptions of the study group sessions, written and verbal reflections, and pre and post-questionnaires. Throughout the study, teachers regularly experimented with practices they had rarely or never tried before. Teachers reported that the warm-ups they learned in the professional development were the driving force for changes in their mathematics teaching practices. Additionally, teachers reported that there were substantial differences in the student discourse that occurred during math time between the previous school year and this year and that facilitation of student discourse during math time became a regular part of their practice.

This paper presents data collected as part of an intervention research project to develop exploratory talk and collaborative group work with six and seven year-old students in mathematics. A discursive approach was used to analyse and interpret variations in the way that three case-study teachers, involved in the project, managed the group work and developed the students’ talk. Analysis suggested that the intervention required, not just a change in practice, also but a shift in ideologies for these teachers. Ideologies are seen as discourses that legitimate and maintain a social pattern, and the paper explores how a research-based intervention can create conflicts between researchers’ and teachers’ ideologies.

The process of generalization in mathematics was identified by mathematics education and educational psychology research, out of many mental actions or operations, as a cognitive function fundamentally required in the thinking process. Moreover, the current changes in education in the United States bring forward the dual goal of mathematics teaching and learning: students should have strong and rigorous mathematical content knowledge and students should be involved in practices that define the status of doing mathematical work. This dual role is totally dependent on the process of generalization. This study uses theories and research findings from the field of algebraic thinking, teaching, and learning to understand how the third grade teacher’s edition textbooks of three mathematics curricula portray the process of generalization. I started my study with the development of a theoretical coding system obtained by combining Kaput’s theory about algebra (Kaput, 2008), Krutetskii’s two way of generalization (Krutetskii, 1976), and the five mathematical representations identified by Lesh, Post, and Behr (1987). Then, I used the coding system to identify tasks that have the potential to involve students in the process of generalization. The findings from my study provide evidence that following a well-structured theory, such as Kaput’s theory about algebra, allows us to identify tasks that support algebraic thinking and to create new ones with higher potential to involve children in the process of generalization. Such tasks may support the development of algebraic thinking as a continuous process that should start from early grades of elementary school.

Fifty-two low performing schools were randomly assigned to receive Spatial-Temporal (ST) Math, a supplemental mathematics software and instructional program, in second/third or fourth/fifth grades or to a business-as-usual control. Analyses reveal a negligible effect of ST Math on mathematics scores, which did not differ significantly across subgroups defined by prior math proficiency and English Language Learner status. Two years of program treatment produced a nonsignificant effect. Publication of evaluation results from large-scale real-world supplemental mathematics instructional implementations such as this one can provide a realistic view of the possibilities, costs, and limitations of this and other computer aided instruction supplemental interventions.

Fifty-two low performing schools were randomly assigned to receive Spatial-Temporal (ST) Math, a supplemental mathematics software and instructional program, in second/third or fourth/fifth grades or to a business-as-usual control. Analyses reveal a negligible effect of ST Math on mathematics scores, which did not differ significantly across subgroups defined by prior math proficiency and English Language Learner status. Two years of program treatment produced a nonsignificant effect. Publication of evaluation results from large-scale real-world supplemental mathematics instructional implementations such as this one can provide a realistic view of the possibilities, costs, and limitations of this and other computer aided instruction supplemental interventions.

Fifty-two low performing schools were randomly assigned to receive Spatial-Temporal (ST) Math, a supplemental mathematics software and instructional program, in second/third or fourth/fifth grades or to a business-as-usual control. Analyses reveal a negligible effect of ST Math on mathematics scores, which did not differ significantly across subgroups defined by prior math proficiency and English Language Learner status. Two years of program treatment produced a nonsignificant effect. Publication of evaluation results from large-scale real-world supplemental mathematics instructional implementations such as this one can provide a realistic view of the possibilities, costs, and limitations of this and other computer aided instruction supplemental interventions.

Although there is wide consensus in the field of mathematics education that the teaching and learning of mathematics is a social, interactive, and relational practice, less attention has explicitly examined the role of the student-teacher relationship in the classroom or why this relationship matters for student learning. A central goal of this dissertation was to understand how teachers build productive working relationships with their students through their mathematics instruction and investigate how this relationship influences mathematics learning, with particular respect to student participation and mathematical dispositions. Using practitioner inquiry and design-based research methods, I took up the role of teacher-researcher to detail and surface the work involved in establishing pedagogical relationships that aim to support ambitious and equitable student learning outcomes. I designed an organized pedagogical approach to mathematics practice called an integrated caring approach (ICA) and implemented this approach in one fourth-grade classroom during a 12-week classroom-based intervention. Drawing from the theoretical lens of care, the framework of ICA conceptualizes the work of building relationships along the three dimensions of personal, mathematical, and political care. Primary data sources include a teacher-reflective journal, transcripts of audio-recorded lessons, and student surveys and interviews, which were contextually supplemented by lesson plans and student artifacts. Findings reveal that pedagogical relationships served as an instructional resource that enabled me to make opportunities to learn more accessible for students and attend to students' mathematical experiences in the classroom. Analysis indicates the instructional practices that emerged from ICA supported students' willingness to participate in mathematical discussions and fostered the development of positive student dispositions. Findings also suggest that students' experiences with I

Division, one of the most difficult topics to teach in elementary mathematics, is rarely examined from the perspective of student teachers engaging with peers and making use of mathematical or pedagogical knowledge in teacher-pupil interaction. In this paper, we examine prospective elementary teachers preparing, realizing and discussing a role-play on division with remainder. Using the concept of “structure of attention”, we offer a positive understanding of prospective teachers’ performances in terms of navigating the richness and complexity of teaching division.

Division, one of the most difficult topics to teach in elementary mathematics, is rarely examined from the perspective of student teachers engaging with peers and making use of mathematical or pedagogical knowledge in teacher-pupil interaction. In this paper, we examine prospective elementary teachers preparing, realizing and discussing a role-play on division with remainder. Using the concept of “structure of attention”, we offer a positive understanding of prospective teachers’ performances in terms of navigating the richness and complexity of teaching division.

The Year 1-10 Mathematics Syllabus, recently implemented in Queensland’s schools introduces mental computation as the main form of computation with written computation emphasising students’ self developed strategies (QSA, 2004). To facilitate the incorporation of mental computation into the curriculum a teaching experiment that adopted a case study design was conducted. A Year 2 teacher was provided with a series of professional development (PD) sessions that incorporated the mathematics of mental computation and the use of support materials. She then used this knowledge to develop (with the assistance of the researcher) a series of 8 half hour lessons delivered over an eight week period. This paper reports on the pre- and post-study interview results of two of her students. Their selection was based on responses to questions that probed for accuracy and flexibility of strategy. During the pre-study interview both students demonstrated inflexible and inaccurate mental computation. On post-interviewing the students remained inaccurate; however, their repertoire of strategies had developed such that they were categorised as flexible mental calculators. Close examination of teacher actions, including engaging students in mathematical discussions, the use of representations and teacher consideration of the mental computation process of proficient mental computers all appear to have supported student growth. Further growth in accuracy as well as flexibility, although seen in other students in the class was not evident with these two students. Close examination of error patterns will further support teacher acquisition of content and pedagogical knowledge essential for the teaching of mental computation.

This paper examines teacher actions during a teaching experiment aimed at enhancing Year 2 students’ mental computational strategies. Specific teaching instruction was conducted by the classroom teacher while the first author acted as participant observer. The teacher was provided with a theoretical background for mental computation and support materials for the development of the instructional program. The lessons were designed to enable students to access a range of representations to build mental models in order to calculate efficiently. The results indicate that these elementary students are not only capable of participating in class discussions on computation methods; they are also able to develop a range of computational methods of increasing sophistication.

This paper reports on a study of five, Year 2 students’ strategy choice, flexibility and accuracy when answering 20 addition and subtraction mental computation questions. All five students were identified as being inaccurate. However, two students employed a range of calculation strategies while the other three students remained inflexible in their strategy choice, choosing low order strategies. Individual interviews were conducted to identify these aspects of calculation. Two conceptual flowcharts developed by Heirdsfield (2001b) were utilised to identify factors and relationships between factors that impact on mental computation. Use of these flowcharts provides an avenue for identification of the breakdown in the structures thereby providing an understanding of where the child is operating and how they may be moved forward.