This book gives teachers of students in grades 3-5 the deep understanding of geometry and measurement that they need to identify and dispel students common misconceptions, correct erroneous claims that they make, and move them forward mathematically. It prepares teachers to introduce terms from geometry and measurement in ways that not only bolster students vocabulary but also enhance their understanding of concepts and definitions. By helping teachers understand what changes or stays the same when shapes are transformed, the book equips teachers to support and extend students understanding of shapes and the space that they occupy.

Developing Essential Understanding of Geometry and Measurement for Teaching Mathematics in Grades 3–5

Barbara J. Dougherty

Hannah Slovin

Rose Mary Zbiek

Richard Lehrer

NCTM

Do your students think that the vertical line test is indispensable and foolproof for determining whether a relationship is a function? Do they believe that every function can be modeled by an equation? Do they interpret the graph of a function as the function itself? What tasks can you offer what questions can you ask to determine what they know or don t know and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach functions effectively in grades 9-12. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with functions not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of functions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. You have essential understanding. It s time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM's Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader reflection open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at nctm.org/more4U. Maximize the potential of student-centered learning and teaching by putting essential understanding into practice.

Putting Essential Understanding of Functions into Practice in Grades 9–12

Terry Crites

Robert Ronau

Dan Meyer

Explore rich, researched-based strategies and tasks that show how students are reasoning about and making sense of mulitplication and division. This book focuses on the specialized pedagogical content knowledge that you need to teach multiplication and division effectively in grades 3-5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these computations -- not only in their current work, but also in higher-level math and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of multipliacation and division. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. About the Series: You have essential understanding. It s time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM s Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader refl¬ection open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, understanding into practice, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at nctm.org/more4U.

Putting Essential Understanding of Multiplication and Division into Practice in Grades 3–5

John Lannin

Kathryn Chval

Dustin Jones

Do your students say that one object is "bigger" than another without attending more precisely to the attribute they are comparing? Do they have difficulty naming numbers from 11 to 19? Do some students still count all while their classmates are counting on or counting by tens? What tasks can you offer-what questions can you ask-to determine what your students know or don't know-and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach number and numeration effectively in prekindergarten-grade 2. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these computations-not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of number and numeration. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. You have essential understanding. It's time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM's Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics-topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader reaction open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at nctm.org/more4U. Maximize the potential of student-centered learning and teaching by putting essential understanding into practice.

Putting Essential Understanding into Practice: Number and Numeration PK–2

Science & Math

Karen S. Karp

Mathematics

Do your students think they can model ratios with sets of discrete objects and combine them to show the addition of ratios? Do they believe that equivalent ratios are based on additive relationships rather than multiplicative ones? What tasks can you offer what questions can you ask to determine what your students know or don't know and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach ratios and proportions effectively in grades 6-8. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with ratios and proportions--not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of ratios and proportions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning.

Putting Essential Understanding of Ratios and Proportions into Practice in Grades 6–8

Melfried Olson

Travis A. Olson

Do your students have the incorrect idea that addition “makes numbers bigger” and subtraction “makes numbers smaller”? Do they believe that subtraction is always “taking away”? What tasks can you offer - what questions can you ask - to determine what your students know or don’t know - and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach addition and subtraction effectively in prekindergarten–grade 2. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these computations - not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of addition and subtraction. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. You have essential understanding. It’s time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM’s Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics - topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader reaction open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at nctm.org/more4U.

Putting Essential Understanding of Addition and Subtraction

Janet H. Caldwell

Karen Karp

Beth Kobett

Nature & Ecology

Do your students recognize a square if the base is oriented horizontally but call it a diamond if it is turned 45 degrees? Do they claim that a shape is a triangle because it looks like one? Do they say that one object is bigger than another without attending more precisely to the attribute they are comparing? What tasks can you offer what questions can you ask to determine what they know or don t know and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach geometry and measurement effectively in prekindergarten to grade 2. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these concepts not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of fractions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. You have essential understanding. It s time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM s Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader reflection open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at More4U. Maximize the potential of student-centered learning and teaching by putting essential understanding into practice.

Putting Essential Understanding of Geometry and Measurement into Practice in Grades Pre-K–2

Juanita Copley

Do your students suppose that 1/3 is greater than 1/2, since 3 is greater than 2? Do they believe that having halves means having two, and only two, congruent pieces of a whole? What tasks can you offer what questions can you ask to determine what your students know or don t know and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach fractions effectively in grades 3 5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with fractions not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of fractions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. About the Series: You have essential understanding. It s time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM s Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader refl¬ection open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, understanding into practice, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at nctm.org/more4U. Maximize the potential of student-centered learning and teaching by putting essential understanding into practice

Putting Essential Understanding of Fractions into Practice in Grades 3–5

Dusty Jones

Do your students have concept images that limit their ideas of shapes to specific examples, oriented in particular ways? Do they confuse the size of an angle with the length of the rays in a drawing of an angle? When solving problems, do they mix up the concepts of perimeter the distance around a shape, measured in linear units and area the region within the shape, measured in square units? What tasks can you offer what questions can you ask to determine what they know or don t know and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach geometry and measurement effectively in grades 3-5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with geometry and measurement not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of geometry and measurement. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. Putting Essential Understanding of Geometry and Measurement into Practice in Grades 3-5 is the eighth title in NCTM s highly useful and very readable Putting Essential Understanding into Practice Series, edited by Barbara J. Dougherty. Each volume in the series builds on the companion volume in the earlier Essential Understanding Series to show teachers how to implement their understanding of mathematics in the classroom.

Putting Essential Understanding of Geometry and Measurement Into Practice in Grades 3–5

Do your students suppose that if the perimeter of a shape increases, the area also increases? Do they think that area is the product of base and height, without understanding that this applies only to parallelograms? Do they believe that a transformation happens only on or to a particular object and not the entire plane? What tasks can you offer-what questions can you ask-to determine what they know or don't know-and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach geometry effectively in grades 6-8. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with geometry-not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of geometry. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning.

Putting Essential Understanding into Practice: Geometry, 6–8

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Books by Barbara J. Dougherty