A 2D version of Cavalieri's Principle is productive for the teaching of area. In this manuscript, we consider an area-preserving transformation, “segment-skewing,” which provides alternative justification methods for area formulas, conceptual insights into statements about area, and foreshadows transitions about area in calculus via the Riemann integral.

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### Nicholas H. Wasserman, Keith Weber, Timothy Fukawa-Connelly and Juan Pablo Mejía-Ramos

The Asked & Answered department shares excerpts from discussion threads on the online MyNCTM community. In this issue, featured threads highlight responses to members' questions regarding 1st grade number sense, multiplication and division of fractions, issues of definition and precision related to circles, and the value of rationalizing denominators.

### John K. Lannin, Delinda van Garderen and Jessica Kamuru

This manuscript discusses two important ideas for developing student foundational understanding of the number line: (a) student views of the number sequence, and (b) recognizing units on the number line. Various student strategies and activities are included.

## Abstract

Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations recognizes that the strengths and needs of young children must be considered when addressing the continuity and alignment of mathematics education for this student group. It also identifies and addresses the critical conversations necessary to meet the following critical challenges:

- Broadening the purpose of school mathematics to prioritize development of deep conceptual understanding so that children experience joy and confidence in themselves as emerging mathematicians
- Dismantling structural obstacles that stand in the way of mathematics working for each and every student
- Implementing equitable instructional practices to cultivate students’ positive mathematical identities and a strong sense of agency
- Organizing mathematics along a common shared pathway grounded in the use of mathematical practices and processes to coherently develop a strong foundation of deep mathematical understanding for each and every child

## Abstract

*Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations* recognizes that the needs of young adolescents are different from elementary and high school–age students and that policies, practices, and issues must consider the unique needs of this student group. Students undergo significant developmental changes from elementary school to middle school. These changes contribute to how they see and understand the world as well as how they see and understand their place in the world. Critical conversations that middle school teachers need to initiate should center on the following serious challenges:

- Broadening the purpose of school mathematics’ focus to include the development of positive mathematical identities so that students can make purposeful decisions about their future endeavors
- Dismantling structural obstacles that stand in the way of mathematics working for each and every student
- Implementing equitable instructional practices to cultivate students’ positive mathematical identities and strong sense of agency
- Organizing middle school mathematics along a common shared pathway grounded in the use of mathematical practices and processes to coherently develop deep mathematical understanding

## Abstract

Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations recognizes that the strengths and needs of young children must be considered when addressing the continuity and alignment of mathematics education for this student group. It also identifies and addresses the critical conversations necessary to meet the following critical challenges:

- Broadening the purpose of school mathematics to prioritize development of deep conceptual understanding so that children experience joy and confidence in themselves as emerging mathematicians
- Dismantling structural obstacles that stand in the way of mathematics working for each and every student
- Implementing equitable instructional practices to cultivate students’ positive mathematical identities and a strong sense of agency
- Organizing mathematics along a common shared pathway grounded in the use of mathematical practices and processes to coherently develop a strong foundation of deep mathematical understanding for each and every child

## Abstract

*Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations* recognizes that the needs of young adolescents are different from elementary and high school–age students and that policies, practices, and issues must consider the unique needs of this student group. Students undergo significant developmental changes from elementary school to middle school. These changes contribute to how they see and understand the world as well as how they see and understand their place in the world. Critical conversations that middle school teachers need to initiate should center on the following serious challenges:

- Broadening the purpose of school mathematics’ focus to include the development of positive mathematical identities so that students can make purposeful decisions about their future endeavors
- Implementing equitable instructional practices to cultivate students’ positive mathematical identities and strong sense of agency
- Organizing middle school mathematics along a common shared pathway grounded in the use of mathematical practices and processes to coherently develop deep mathematical understanding

## Abstract

Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations recognizes that the strengths and needs of young children must be considered when addressing the continuity and alignment of mathematics education for this student group. It also identifies and addresses the critical conversations necessary to meet the following critical challenges:

- Broadening the purpose of school mathematics to prioritize development of deep conceptual understanding so that children experience joy and confidence in themselves as emerging mathematicians
- Implementing equitable instructional practices to cultivate students’ positive mathematical identities and a strong sense of agency
- Organizing mathematics along a common shared pathway grounded in the use of mathematical practices and processes to coherently develop a strong foundation of deep mathematical understanding for each and every child

## Abstract

*Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations* recognizes that the needs of young adolescents are different from elementary and high school–age students and that policies, practices, and issues must consider the unique needs of this student group. Students undergo significant developmental changes from elementary school to middle school. These changes contribute to how they see and understand the world as well as how they see and understand their place in the world. Critical conversations that middle school teachers need to initiate should center on the following serious challenges:

- Broadening the purpose of school mathematics’ focus to include the development of positive mathematical identities so that students can make purposeful decisions about their future endeavors
- Implementing equitable instructional practices to cultivate students’ positive mathematical identities and strong sense of agency
- Organizing middle school mathematics along a common shared pathway grounded in the use of mathematical practices and processes to coherently develop deep mathematical understanding

## Abstract

*Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations* recognizes that the needs of young adolescents are different from elementary and high school–age students and that policies, practices, and issues must consider the unique needs of this student group. Students undergo significant developmental changes from elementary school to middle school. These changes contribute to how they see and understand the world as well as how they see and understand their place in the world. Critical conversations that middle school teachers need to initiate should center on the following serious challenges: