The Eurasia Proceedings of Educational and Social Sciences, Sep 1, 2015
The demand of interventions in daily lessons is high in the classroom, and curriculum programs ma... more The demand of interventions in daily lessons is high in the classroom, and curriculum programs make an effort to include resources for such interventions. Yet, there is no clear theoretical and practical guidance on daily interventions for both teacher and curriculum. This study examines interventions that are offered in written lessons from a range of elementary mathematics curriculum programs and those that teachers actually incorporate into instruction, aiming at understanding the nature of interventions embedded in daily lessons and the role of teacher and curriculum in classroom interventions. The results of the study highlight the importance of intervention resources in the curriculum and teacher role in recognizing the affordances of resources to provide appropriate interventions toward the mathematical point of the lesson.
International Group for the Psychology of Mathematics Education, Nov 1, 2015
The interaction between types of communication and socio-mathematical norms on students' learning... more The interaction between types of communication and socio-mathematical norms on students' learning autonomy in five curriculum programs was investigated. Uni-directional, contributive, reflective, and instructive types of communications were present in curriculum programs studied. Investigations in Number Data and Space and Math Trailblazers provided opportunities for students to explain, justify, and compare solution strategies. Math in Focus and Scott Foresman Addison Wesley-Mathematics required explanations, justifications, and comparisons of solution strategies, but those were mainly provided by the teacher. The former and latter programs potentially foster intellectual autonomous and intellectual heteronomous learning in students, respectively, while Everyday Mathematics almost equally supports both.
How are the results of the Third International Mathematics and Science Study (TIMSS) connected wi... more How are the results of the Third International Mathematics and Science Study (TIMSS) connected with your classroom? How would your students perform on these questions? How can you help your students perform well on questions like these? Do your students measure up metrically?
In this chapter, we examine mathematical treatment and emphasis in the five curriculum programs—w... more In this chapter, we examine mathematical treatment and emphasis in the five curriculum programs—what aspects of mathematics are emphasized and the way in which the mathematics is represented and organized for student learning. In doing so, we attend to mathematics content presented for daily instruction, in particular, the scope and sequence of number and operations. In addition, the cognitive demand of instructional activities, the nature of ongoing practice, and the visual and physical representations used are examined in order to discern the mathematical emphasis in the programs. The results show that there are substantial variations in the scope and sequence of number and operations, cognitive demand of tasks and problems, ongoing practice, and representations used across the five programs. Such variations reveal different mathematical treatments and emphasis in the programs. We summarize important findings of the study and discuss implications for teachers and teaching for each of the findings.
This chapter introduces our approach to analyzing how curriculum authors communicated with teache... more This chapter introduces our approach to analyzing how curriculum authors communicated with teachers. Built on recommendations by Ball and Cohen (Educational researcher 25:6–14, 1996), Davis and Krajcik (Educational Researcher 34:3–14, 2005), and Remillard (Curriculum Inquiry 29:315–342, 1999, Elementary School Journal 100:331–350, 2000), we explored whether and the extent to which curriculum authors provided guidance intended to support teachers in their roles as curriculum enactors. The chapter reports on the coding framework used by the ICUBiT team to categorize different approaches of communicating with teachers in mathematics lesson guides and presents findings from quantitative analysis of these data. We found differences in the quantity of communication and the authors’ tendencies to direct teachers’ actions versus communicate to them about mathematics, student thinking, or design rationale (which we considered potentially educative). We also found that one program tended to combine directive and educative communication much more extensively than others. When looking across findings from previous chapters, we found alignment between mathematical emphasis, pedagogical approach, and approach to communicating with the teacher in the lesson guide. The findings in this chapter set up Chaps. 6, 7, and 8, which offer an in-depth analysis of communication within each type of support for teachers.
Mathematics Teaching in the Middle School, May 1, 1999
The use of benchmarks, such as 0, 1/2, and 1, for comparing the size of fractions is not emphasiz... more The use of benchmarks, such as 0, 1/2, and 1, for comparing the size of fractions is not emphasized very often in instruction. Comparing fractions with 0, 1/2, or 1 by comparing the numerator with the denominator or by mentally modeling the fraction is a powerful tool for gauging the size of fractions, making quick estimates, and judging the reasonableness of computed results (Bezuk and Bieck 1992). If students are asked to compare 5/8 with 4/9, the traditional technique is to find the common denominator, convert both fractions to equivalent forms using this common denominator, then compare numerators. However, this problem can be solved more efficiently by comparing each fraction with 1/2: 5/8 is larger than 1/2, since it is larger than 4/8; and 4/9 is less than 1/2, since 4/9 is less than 4.5/9, or 1/2.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
One of the most pressing issues for classroom teachers is how to facilitate learning opportunitie... more One of the most pressing issues for classroom teachers is how to facilitate learning opportunities for all their students, especially at the beginning of each school year. Teachers are faced with new children, who have unique backgrounds and who bring different understandings to the classroom. Ms. Lee, a first-grade teacher, faced this challenge head on as she began implementing material on number sense in the first month of the school year with her twenty-one students.
This chapter synthesizes and builds on findings from the previous chapters to consider what we ha... more This chapter synthesizes and builds on findings from the previous chapters to consider what we have learned about mathematics curriculum materials as tools for teachers. Looking back at our analysis in Chaps. 2, 3, 4, 5, 6, 7, and 8, we briefly summarize our overall key findings and consider how these findings were influenced by key methodological decisions. We then discuss the complexity of curriculum materials as designed artifacts and the challenges associated with analyzing them. Using Hiebert and colleagues’ notion of a constellation of features that are needed to characterize classroom teaching, we argue that examining curriculum materials also involves considering a cluster of features. Using existing literature and our analysis in this volume, we also conceptualize curriculum materials as artifacts of design decisions comprised of multiple layers. We distinguish two main components, i.e., objectively given structures (what teachers see physically) and authors’ ideas and values, and discuss the relationship between them from curriculum authors’, the researchers’, and teachers’ perspectives. Finally, we provide some implications for teachers, curriculum designers, and researchers based on our findings and reflections on our work.
This chapter explores the pedagogical approach incorporated into each of the five curriculum prog... more This chapter explores the pedagogical approach incorporated into each of the five curriculum programs we analyzed (The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for more details about the programs.). Specifically, we consider explicit and implicit messages about how students should interact with mathematics, one another, the teacher, and the textbook around these mathematical ideas. The analysis considers the role the teacher is expected to play and the corresponding roles the textbooks and students play in shaping these interactions. Following Hiebert and colleagues’ notion that teaching is an interactive system, involving a constellation of interrelated features, we present the approach of each program in a holistic profile, which overviews five dimensions: participant structures, nature of student work, role of the teacher, sources of knowledge, and cognitive demand of tasks. We also consider ways that these features bolster or moderate one another. Using a dialogic-to-direct instruction continuum characterized by Munter and colleagues, we highlight several key differences of the pedagogical approaches across the five programs. We also discuss ways that a single continuum is insufficient for characterizing the way pedagogical approach is represented in teacher’s guides.
The 'Resource' Approach to Mathematics Education, 2019
Teachers use resources in order to support their teaching, to support student learning, and to ad... more Teachers use resources in order to support their teaching, to support student learning, and to advance their own pedagogical and content knowledge. Using resources is intrinsically linked to particular knowledge and skills. These are conceptualized within different theoretical frames as competencies, aspects of design capacity, teacher expertise, professional knowledge, or utilization schemes within the instrumentation process. We discuss four different conceptualizations of teachers’ work with resources, problems they aim to address, and exemplars of empirical studies in which such conceptualizations are used. We then discuss the affordances, constraints, and blind spots of these frameworks and indicate how they overlap and complement each other.
International Group for the Psychology of Mathematics Education, 2015
The interaction between types of communication and socio-mathematical norms on students’ learning... more The interaction between types of communication and socio-mathematical norms on students’ learning autonomy in five curriculum programs was investigated. Uni-directional, contributive, reflective, and instructive types of communications were present in curriculum programs studied. Investigations in Number Data and Space and Math Trailblazers provided opportunities for students to explain, justify, and compare solution strategies. Math in Focus and Scott Foresman Addison Wesley-Mathematics required explanations, justifications, and comparisons of solution strategies, but those were mainly provided by the teacher. The former and latter programs potentially foster intellectual autonomous and intellectual heteronomous learning in students, respectively, while Everyday Mathematics almost equally supports both.
This chapter serves as an introduction to this volume. It introduces the idea of curriculum analy... more This chapter serves as an introduction to this volume. It introduces the idea of curriculum analysis and situates the work of this volume in the Improving Curriculum Use for Better Teaching (ICUBiT) project. After providing a brief historical overview of the evolution of mathematics curriculum materials in the USA, beginning in 1990, we detail the framework guiding our analysis of five elementary mathematics programs, along with a set of assumptions that undergird our approach. We then locate our approach to curriculum analysis in the wider body of such work, identifying several key methodological questions that researchers consider when undertaking analysis of curriculum materials. Finally, we introduce the five curriculum programs that are the focus of our analysis and provide an overview of key aspects of our methodological approach. We conclude with an overview of the remaining chapters in the volume.
Mathematics teachers are always looking for innovative exercises or projects that can be used in ... more Mathematics teachers are always looking for innovative exercises or projects that can be used in the classroom to enhance students' opportunities to learn. One exercise that allows for mathematical thinking is exploring flags that have geometric features embedded in them. Having students search for patterns in these geometric features can develop algebraic thinking (for more information, see Selkirk [1992] and Hellman and Long [1993]). In addition, many flags can be used as contexts to develop conceptual knowledge of fractions and proportions.
Professional noticing of children’s mathematical thinking (PNCMT) is the ability for teachers to ... more Professional noticing of children’s mathematical thinking (PNCMT) is the ability for teachers to attend to students’ mathematical thinking, analyze it within a larger framework, and use that analysis to respond differentially to students, based on the strategies that they choose or mis/understandings that they demonstrate. This chapter analyzes whether and how teacher’s guides in five elementary mathematics curriculum programs offer support to teachers related to noticing student thinking. As reported in Chap. 5, we found that all five curriculum programs provided some supports for attending to and analyzing students’ thinking and work. Our analysis in this chapter reveals that only two of the programs frequently modeled the explicit connections to conceptual understanding necessary for PNCMT. Two other programs frequently modeled evaluating students based upon correctness or general characteristics, without considering the strategies they used. The final program used a combination of the two approaches. As teachers do not tend to use PNCMT without focused education and support, we hypothesize that curriculum programs that provide models for PNCMT are more likely to support teachers in learning to attend to student thinking. Our findings provide examples of ways that curriculum authors might support teachers in developing PNCMT.
Mathematics Teacher Education and Development, Mar 23, 2015
Developing and writing assessment items that measure teachers' knowledge is an intricate and comp... more Developing and writing assessment items that measure teachers' knowledge is an intricate and complex undertaking. In this paper, we begin with an overview of what is known about measuring teacher knowledge. We then highlight the challenges inherent in creating assessment items that focus specifically on measuring teachers' specialised knowledge for teaching. We offer insights into three practices we have found valuable towards overcoming challenges in our own cross-disciplinary work to create assessment items for measuring teachers' knowledge for teaching.
The Eurasia Proceedings of Educational and Social Sciences, Sep 1, 2015
The demand of interventions in daily lessons is high in the classroom, and curriculum programs ma... more The demand of interventions in daily lessons is high in the classroom, and curriculum programs make an effort to include resources for such interventions. Yet, there is no clear theoretical and practical guidance on daily interventions for both teacher and curriculum. This study examines interventions that are offered in written lessons from a range of elementary mathematics curriculum programs and those that teachers actually incorporate into instruction, aiming at understanding the nature of interventions embedded in daily lessons and the role of teacher and curriculum in classroom interventions. The results of the study highlight the importance of intervention resources in the curriculum and teacher role in recognizing the affordances of resources to provide appropriate interventions toward the mathematical point of the lesson.
International Group for the Psychology of Mathematics Education, Nov 1, 2015
The interaction between types of communication and socio-mathematical norms on students' learning... more The interaction between types of communication and socio-mathematical norms on students' learning autonomy in five curriculum programs was investigated. Uni-directional, contributive, reflective, and instructive types of communications were present in curriculum programs studied. Investigations in Number Data and Space and Math Trailblazers provided opportunities for students to explain, justify, and compare solution strategies. Math in Focus and Scott Foresman Addison Wesley-Mathematics required explanations, justifications, and comparisons of solution strategies, but those were mainly provided by the teacher. The former and latter programs potentially foster intellectual autonomous and intellectual heteronomous learning in students, respectively, while Everyday Mathematics almost equally supports both.
How are the results of the Third International Mathematics and Science Study (TIMSS) connected wi... more How are the results of the Third International Mathematics and Science Study (TIMSS) connected with your classroom? How would your students perform on these questions? How can you help your students perform well on questions like these? Do your students measure up metrically?
In this chapter, we examine mathematical treatment and emphasis in the five curriculum programs—w... more In this chapter, we examine mathematical treatment and emphasis in the five curriculum programs—what aspects of mathematics are emphasized and the way in which the mathematics is represented and organized for student learning. In doing so, we attend to mathematics content presented for daily instruction, in particular, the scope and sequence of number and operations. In addition, the cognitive demand of instructional activities, the nature of ongoing practice, and the visual and physical representations used are examined in order to discern the mathematical emphasis in the programs. The results show that there are substantial variations in the scope and sequence of number and operations, cognitive demand of tasks and problems, ongoing practice, and representations used across the five programs. Such variations reveal different mathematical treatments and emphasis in the programs. We summarize important findings of the study and discuss implications for teachers and teaching for each of the findings.
This chapter introduces our approach to analyzing how curriculum authors communicated with teache... more This chapter introduces our approach to analyzing how curriculum authors communicated with teachers. Built on recommendations by Ball and Cohen (Educational researcher 25:6–14, 1996), Davis and Krajcik (Educational Researcher 34:3–14, 2005), and Remillard (Curriculum Inquiry 29:315–342, 1999, Elementary School Journal 100:331–350, 2000), we explored whether and the extent to which curriculum authors provided guidance intended to support teachers in their roles as curriculum enactors. The chapter reports on the coding framework used by the ICUBiT team to categorize different approaches of communicating with teachers in mathematics lesson guides and presents findings from quantitative analysis of these data. We found differences in the quantity of communication and the authors’ tendencies to direct teachers’ actions versus communicate to them about mathematics, student thinking, or design rationale (which we considered potentially educative). We also found that one program tended to combine directive and educative communication much more extensively than others. When looking across findings from previous chapters, we found alignment between mathematical emphasis, pedagogical approach, and approach to communicating with the teacher in the lesson guide. The findings in this chapter set up Chaps. 6, 7, and 8, which offer an in-depth analysis of communication within each type of support for teachers.
Mathematics Teaching in the Middle School, May 1, 1999
The use of benchmarks, such as 0, 1/2, and 1, for comparing the size of fractions is not emphasiz... more The use of benchmarks, such as 0, 1/2, and 1, for comparing the size of fractions is not emphasized very often in instruction. Comparing fractions with 0, 1/2, or 1 by comparing the numerator with the denominator or by mentally modeling the fraction is a powerful tool for gauging the size of fractions, making quick estimates, and judging the reasonableness of computed results (Bezuk and Bieck 1992). If students are asked to compare 5/8 with 4/9, the traditional technique is to find the common denominator, convert both fractions to equivalent forms using this common denominator, then compare numerators. However, this problem can be solved more efficiently by comparing each fraction with 1/2: 5/8 is larger than 1/2, since it is larger than 4/8; and 4/9 is less than 1/2, since 4/9 is less than 4.5/9, or 1/2.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
One of the most pressing issues for classroom teachers is how to facilitate learning opportunitie... more One of the most pressing issues for classroom teachers is how to facilitate learning opportunities for all their students, especially at the beginning of each school year. Teachers are faced with new children, who have unique backgrounds and who bring different understandings to the classroom. Ms. Lee, a first-grade teacher, faced this challenge head on as she began implementing material on number sense in the first month of the school year with her twenty-one students.
This chapter synthesizes and builds on findings from the previous chapters to consider what we ha... more This chapter synthesizes and builds on findings from the previous chapters to consider what we have learned about mathematics curriculum materials as tools for teachers. Looking back at our analysis in Chaps. 2, 3, 4, 5, 6, 7, and 8, we briefly summarize our overall key findings and consider how these findings were influenced by key methodological decisions. We then discuss the complexity of curriculum materials as designed artifacts and the challenges associated with analyzing them. Using Hiebert and colleagues’ notion of a constellation of features that are needed to characterize classroom teaching, we argue that examining curriculum materials also involves considering a cluster of features. Using existing literature and our analysis in this volume, we also conceptualize curriculum materials as artifacts of design decisions comprised of multiple layers. We distinguish two main components, i.e., objectively given structures (what teachers see physically) and authors’ ideas and values, and discuss the relationship between them from curriculum authors’, the researchers’, and teachers’ perspectives. Finally, we provide some implications for teachers, curriculum designers, and researchers based on our findings and reflections on our work.
This chapter explores the pedagogical approach incorporated into each of the five curriculum prog... more This chapter explores the pedagogical approach incorporated into each of the five curriculum programs we analyzed (The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for more details about the programs.). Specifically, we consider explicit and implicit messages about how students should interact with mathematics, one another, the teacher, and the textbook around these mathematical ideas. The analysis considers the role the teacher is expected to play and the corresponding roles the textbooks and students play in shaping these interactions. Following Hiebert and colleagues’ notion that teaching is an interactive system, involving a constellation of interrelated features, we present the approach of each program in a holistic profile, which overviews five dimensions: participant structures, nature of student work, role of the teacher, sources of knowledge, and cognitive demand of tasks. We also consider ways that these features bolster or moderate one another. Using a dialogic-to-direct instruction continuum characterized by Munter and colleagues, we highlight several key differences of the pedagogical approaches across the five programs. We also discuss ways that a single continuum is insufficient for characterizing the way pedagogical approach is represented in teacher’s guides.
The 'Resource' Approach to Mathematics Education, 2019
Teachers use resources in order to support their teaching, to support student learning, and to ad... more Teachers use resources in order to support their teaching, to support student learning, and to advance their own pedagogical and content knowledge. Using resources is intrinsically linked to particular knowledge and skills. These are conceptualized within different theoretical frames as competencies, aspects of design capacity, teacher expertise, professional knowledge, or utilization schemes within the instrumentation process. We discuss four different conceptualizations of teachers’ work with resources, problems they aim to address, and exemplars of empirical studies in which such conceptualizations are used. We then discuss the affordances, constraints, and blind spots of these frameworks and indicate how they overlap and complement each other.
International Group for the Psychology of Mathematics Education, 2015
The interaction between types of communication and socio-mathematical norms on students’ learning... more The interaction between types of communication and socio-mathematical norms on students’ learning autonomy in five curriculum programs was investigated. Uni-directional, contributive, reflective, and instructive types of communications were present in curriculum programs studied. Investigations in Number Data and Space and Math Trailblazers provided opportunities for students to explain, justify, and compare solution strategies. Math in Focus and Scott Foresman Addison Wesley-Mathematics required explanations, justifications, and comparisons of solution strategies, but those were mainly provided by the teacher. The former and latter programs potentially foster intellectual autonomous and intellectual heteronomous learning in students, respectively, while Everyday Mathematics almost equally supports both.
This chapter serves as an introduction to this volume. It introduces the idea of curriculum analy... more This chapter serves as an introduction to this volume. It introduces the idea of curriculum analysis and situates the work of this volume in the Improving Curriculum Use for Better Teaching (ICUBiT) project. After providing a brief historical overview of the evolution of mathematics curriculum materials in the USA, beginning in 1990, we detail the framework guiding our analysis of five elementary mathematics programs, along with a set of assumptions that undergird our approach. We then locate our approach to curriculum analysis in the wider body of such work, identifying several key methodological questions that researchers consider when undertaking analysis of curriculum materials. Finally, we introduce the five curriculum programs that are the focus of our analysis and provide an overview of key aspects of our methodological approach. We conclude with an overview of the remaining chapters in the volume.
Mathematics teachers are always looking for innovative exercises or projects that can be used in ... more Mathematics teachers are always looking for innovative exercises or projects that can be used in the classroom to enhance students' opportunities to learn. One exercise that allows for mathematical thinking is exploring flags that have geometric features embedded in them. Having students search for patterns in these geometric features can develop algebraic thinking (for more information, see Selkirk [1992] and Hellman and Long [1993]). In addition, many flags can be used as contexts to develop conceptual knowledge of fractions and proportions.
Professional noticing of children’s mathematical thinking (PNCMT) is the ability for teachers to ... more Professional noticing of children’s mathematical thinking (PNCMT) is the ability for teachers to attend to students’ mathematical thinking, analyze it within a larger framework, and use that analysis to respond differentially to students, based on the strategies that they choose or mis/understandings that they demonstrate. This chapter analyzes whether and how teacher’s guides in five elementary mathematics curriculum programs offer support to teachers related to noticing student thinking. As reported in Chap. 5, we found that all five curriculum programs provided some supports for attending to and analyzing students’ thinking and work. Our analysis in this chapter reveals that only two of the programs frequently modeled the explicit connections to conceptual understanding necessary for PNCMT. Two other programs frequently modeled evaluating students based upon correctness or general characteristics, without considering the strategies they used. The final program used a combination of the two approaches. As teachers do not tend to use PNCMT without focused education and support, we hypothesize that curriculum programs that provide models for PNCMT are more likely to support teachers in learning to attend to student thinking. Our findings provide examples of ways that curriculum authors might support teachers in developing PNCMT.
Mathematics Teacher Education and Development, Mar 23, 2015
Developing and writing assessment items that measure teachers' knowledge is an intricate and comp... more Developing and writing assessment items that measure teachers' knowledge is an intricate and complex undertaking. In this paper, we begin with an overview of what is known about measuring teacher knowledge. We then highlight the challenges inherent in creating assessment items that focus specifically on measuring teachers' specialised knowledge for teaching. We offer insights into three practices we have found valuable towards overcoming challenges in our own cross-disciplinary work to create assessment items for measuring teachers' knowledge for teaching.
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