Like algebra at any level, early algebra is a way to explore, analyze, represent, and generalize mathematical ideas and relationships. This book shows that children can and do engage in generalizing about numbers and operations as their mathematical experiences expand. The authors identify and examine five big ideas and associated essential understandings for developing algebraic thinking in grades 3 5. The big ideas relate to the fundamental properties of number and operations, the use of the equals sign to represent equivalence, variables as efficient tools for representing mathematical ideas, quantitative reasoning as a way to understand mathematical relationships, and functional thinking to generalize relationships between covarying quantities. The book examines challenges in teaching, learning, and assessment and is interspersed with questions for teachers reflection.

Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3–5

Maria Blanton

Linda Levi

Terry Crites

Barbara Dougherty

Rose Mary Zbiek

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

Barbara J. Dougherty

Robert Ronau

Dan Meyer

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

Hannah Slovin

Karen Karp

Do your students think a triangle can be constructed from any three given line segments? Do they believe that a transformation affects only the pre-image, not the whole plane? Do they understand that examples, no matter how many they find, cannot prove a conjecture but one counterexample is sufficient to disprove it? Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of diagrams and definitions, transformations, and proof. Use the opportunities that these and similar tasks provide to build on their understanding and guide them in taking the next steps in learning.

Putting Essential Understanding into Practice: Geometry, 9–12

Do your students think that a sample, regardless of size, always has the same characteristics as the population from which it is drawn? Do they know the difference between the distribution of a sample and the sampling distribution of a statistic? What tasks can you offer, and 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 statistics 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 statistics--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 statistics. 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: Statistics, 9–12

Roy St. Laurent