Planting the Seeds of Algebra, PreK-2: Explorations for the Early Grades

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Monica Neagoy & Jennifer Bay-Williams

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    Foreword: Designing a Garden where Seeds can Grow

    When I initially read the title to this book, Planting the Seeds of Algebra PreK–2: Explorations in the Early Grades, I saw the word algebra and anticipated that the pages inside would provide types of algebraic experiences that are appropriate in the early grades (which it does). But, the more I read, the more the opening words planting the seeds took on meaning. I am a novice gardener, but my father is a farmer and my parents are both avid gardeners, so I draw from these experiences to illuminate the unique and powerful opportunities this book can provide for teachers and for students in cultivating young students to become good thinkers, strong in number sense and algebraic reasoning.

    Landscaping

    In landscaping, it is important to select the right plants for the setting and to be sure that the plants complement each other in ways that lead to a garden that effectively can grow and will provide overall beauty. In Planting the Seeds, four tasks are selected for in-depth development; they address addition, subtraction, patterns, and special numbers, respectively. These are foundational concepts in K–2. The Common Core State Standards (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010), in an effort to press for a focused curriculum, identify two to four critical areas at each grade level; these include the following:

    • Representing and comparing whole numbers, initially with sets of objects (K)
    • Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20 (Grade 1)
    • Building fluency with addition and subtraction (Grade 2)

    These significant concepts in K–2 are then connected to foundational and central algebraic thinking concepts: properties of numbers and operations (looking for structure); meaning of symbols/notations, in particular variables (that may vary or may represent an unknown value) and the equals sign; and connecting patterns to functions to algebra. What these concepts mean and why they are important are explicitly discussed in the chapters following the classroom implementation of the selected tasks (Chapters 2, 5, 8), providing important insights to the broader landscape that is our K–12 mathematics curriculum.

    Fertile Soil

    In order for our classrooms to be fertile environments for our students to learn, we must have our eye on the mathematical horizon, seeing what concepts relate to or build on each other, and then focusing student thinking on those ideas. Planting the Seeds is designed to support teacher learning in a way parallel to student learning. When we teach children mathematics, we begin with concrete explorations that set up the opportunity to follow up with discussions about the important mathematical concepts; we connect back to the concrete experience to develop an understanding for the more abstract mathematical ideas, and then provide opportunities for students to apply, connect, and extend their new knowledge. Planting the Seeds is organized in this way through the organization of its chapters.

    The first chapter in each set (Chapters 1, 4, 7) provides the actual concrete experience—an exemplar task or set of tasks taught in a classroom, including visuals, student work, and vignettes. For example, in Chapter 1, we can “see” how finding “partners of 7” can enhance students' understanding of number combinations, part-part-whole, missing addends, symbols, the commutative property, and making generalizations. In Chapter 4, we experience a lesson in which algebraic thinking is used to expand students' understanding of what subtraction means in order to help them be better at subtracting (what turns out to be a very effective intervention). Finally, Chapter 7 illustrates how repeating patterns can be analyzed and developed to connect to functions, while engaging students in important number relationships. The teacher moves in these chapters illustrate the critical importance of knowing the mathematics in our own garden and in the broader landscape.

    As mentioned above, the chapters following each of these chapters (2, 5, 8) discuss the key mathematical ideas, with a particular focus on the connections between number and algebra. Because these discussions relate back to the lesson itself, it is comprehensible, regardless of the reader's previous understanding of algebra. With a deeper understanding of mathematics (related to the focus task), we are ready to extend our mathematical thinking (or our students' thinking), which is the focus of the third chapters in each set (Chapters 3, 6, 9). Chapter 10 models this same approach of using concrete experiences to illuminate mathematical concepts and extend understanding in a shortened version intended for parents.

    This book structure—from concrete to abstract to application—is excellent for individual teacher learning, collaborative book study (e.g., through a professional learning community), or modified lesson study.

    Growing the Seeds

    I have learned one thing about gardening. If a plant is dying, it is probably for one of two reasons: too much water or too little water. So it is with student learning of mathematics. If we tell the students too much, we lower the level of cognitive demand in the lesson, and they are not able to see the connections that we are telling them to see. If we give a rich task, but do not provide enough scaffolding or the right tools, then the students will not be able to engage in the task and learn the intended mathematics. The transcripts of the lessons in Chapters 1, 4, 7, and 10 provide valuable insights into striking a balance that maximizes student opportunities to learn. For example, the first lesson is a differentiated lesson where one group uses dice to explore sums that equal 7, while a second group explores how seven birds can land on a tree and bush. The “set up” part of each lesson illustrates how to ensure students are ready to engage in the problem without taking away their thinking. In Chapter 4, we see how students (who understand subtraction only as “take-away”) can benefit from dropping back to simpler problems to expand their understanding. (In this case, that subtraction is also the distance or difference between two numbers). This additional “watering” allows the students to return to the initial problem they were trying to solve with an enhanced understanding that allows them to successfully solve the problem.

    In addition to differentiation, scaffolding, questioning, and use of tools, there are many other teacher moves in these chapters that can provide for rich discussion in a professional development setting—considering how the task selection (design) and teacher moves (implementation) of the tasks support or inhibit student understanding of number and of the related algebraic concepts (and mathematical practices).

    In her introduction, Monica Neagoy tells us that one of her purposes for writing the book is so that teachers may be able to, “Experience algebraic acculturation: that is, cultivate new thought and behavior patterns that naturally weave into algebra's cognitive fabric” (p. 2). Through the use of classroom vignettes, explicit connections to content, and opportunities to extend our knowledge, Planting the Seeds provides an opportunity for us as teachers, elementary mathematics specialists, coaches, and teacher educators to cultivate our own understanding of the mathematics we teach and how to teach it in ways that maximize students' opportunity to learn and connect foundational concepts in number and algebra. If PreK–2 children are nurtured and challenged in these ways, they will blossom into children who have sound mathematical foundations and the capacity (and disposition) to reason and think algebraically.

    Jennifer Bay-WilliamsMathematics Educator and Department Chair, University of Louisville, Louisville, KY Past-President of the Association of Mathematics Teacher Education (AMTE)

    Acknowledgments

    Writing this book has been an edifying journey. The process took me to private places of the mind and spirit and simultaneously connected me with others who directly or indirectly inspired me, enriched me, and enlightened me. Collectively, they helped to shape the goals and content of Planting the Seeds of Algebra. I thank my schoolteachers who nurtured my love for mathematics, my professors who led me down interesting paths, and my favorite authors and researchers who shed light on new domains. I especially thank the myriad teachers and administrators of the elementary schools—public and private—I have worked with over the past 20 years. Their questions, comments, and insights always prompted me to go deeper and search further. Their desire to excite children about mathematics nurtured my own. I will not risk listing all their names for fear of leaving out an important one. For this reason, I hope they will understand—as they know well who they are—that my gratitude to them is endless.

    That being said, I would be remiss not to specifically acknowledge a few individuals who assisted me, on the production end, in bringing this book to life: I thank the acquisition editor Cathy Hernandez who first contacted me to write a book on mathematics for Corwin, and Carol Collins, who took over the project halfway through. Carol unfailingly supported my work, and me, and sagaciously guided it to production. I thank the reviewers who made perspicacious comments on the manuscript draft and the talented Corwin production team!

    On a more personal note, I have been blessed with a supportive and thoughtful husband, who has lived this book with me, through its countless metamorphoses and iterations. I am profoundly grateful to Didier, a writer and artist himself, my partner on stage and in life, my intellectual companion and loving mate, and first editor of everything I write. He created a home environment conducive to creativity and writing, and graciously read every draft manuscript of this book, providing invaluable comments and insights. Merci infiniment!

    Lastly, I would like to dedicate this book to the memory of my parents, who held high education standards for their children.

    Publisher's Acknowledgments

    Corwin gratefully acknowledges the contributions of the following reviewers:

    Lenisera L. Bodison, Coordinator Mathematics Teaching and Learning DeKalb County School District Decatur, GA

    Tania Dymkowski, Instructional Strategist Science Hall Elementary School Fischer, TX

    Stacey Ferguson, Fifth Grade Teacher North Bay Elementary School Bay St. Louis School District Bay St. Louis, MS

    Ruth Heaton, Professor, Math Education University of Nebraska–Lincoln Lincoln, NE

    Charyl Kerns Hills, Elementary Math/Technology Integration Specialist Goodnoe & M. M. Welch Schools Council Rock School District Newton, PA

    Rosamaria Murillo, Principal and Former Elementary Math Coach Eastman Avenue Elementary Los Angeles Unified School District La Mirada, CA

    Pamela Opel, Fourth Grade Teacher Pass Road Elementary Biloxi, MS

    Renee Peoples, Teacher Leader Swain West Elementary Bryson City, NC

    H. Caroline Piangerelli, PreK–12 Mathematics Specialist Office of Curriculum, Instruction, and School Support Los Angeles Unified School District Los Angeles, CA

    Nichole Ponzer, Fifth Grade Teacher Read Elementary School Oshkosh, WI

    Gail Underwood, Teacher Leader (Curriculum) Grant Elementary School Columbia, MO

    About the Author

    Photo by Tim Coburn

    Monica Neagoy, international consultant and popular keynote speaker, has a passion for mathematics. The facets of her 25-year mathematics career include teacher professional development, math specialist training, national/international conference speeches, live television courses, video creation and hosting, and live shows including her popular MathMagic show. Whether here or abroad, whether speaking English, French, or Spanish, Monica's life-long goal is to infuse people with a fascination for the wonder of mathematics.

    Dr. Neagoy began her career teaching mathematics at Georgetown University. Later, she was a program director at the National Science Foundation (NSF), where she oversaw national projects. As an independent national mathematics consultant, she has served numerous school districts across the country and notable organizations including the Carnegie Institution of Washington, PBS TeacherLine, MATHCOUNTS, and the American Association for the Advancement of Science. Since 2005, she broadened her mathematics consulting to serve Europe, the Middle East, and Africa, in addition to the United States.

    Dr. Neagoy has had a parallel career in theatre as actor, choreographer, and stage-director, particularly with the professional LE NEON Theatre in the Washington, D.C. area. Her exposure to many cultures, mastery of several languages, double career in the arts and the sciences, and mindfulness training through yoga practice and teaching provide her with a unique perspective on the learning and teaching of mathematics. Her passions for mathematics and the arts converge in her popular presentation, The Mathematics of Beauty and the Beauty of Mathematics.

    Dr. Neagoy combined her love for the arts and the sciences in the creation of innovative mathematics videos series for teachers and students, including Discovering Algebra with Graphing Calculators (Discovery Education), Algebra Nspirations (Media4Math), and Video #7 of Mathematics: What's the Big Idea? titled “Algebra: It Begins in Kindergarten” (Annenberg Learner).

    Monica was educated in the French school system, grades 1–12, in Asia and the United States. She has a BS in mathematics and philosophy from Georgetown University, an MA in pure mathematics from Catholic University, and a PhD in mathematics education from the University of Maryland.

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