DOUBLE TAP TO ZOOM WITH PHONE OR TABLET MATHEMATiZiNG An Emergent Math Curriculum Approach for Young Children ALLEN C. ROSALES COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET MATHEMATiZiNG COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET MATHEMATiZiNG An Emergent Math Curriculum Approach for Young Children ALLEN C. ROSALES COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET Published by Redleaf Press 10 Yorkton Court St. Paul, MN 55117 www.redleafpress.org © 2015 by Allen C. Rosales All rights reserved. Unless otherwise noted on a specific page, no portion of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or capturing on any information storage and retrieval system, without permission in writing from the publisher, except by a reviewer, who may quote brief passages in a critical article or review to be printed in a magazine or news­paper, or electronically transmitted on radio, television, or the Internet. First edition 2015 Cover design by Percolator Cover illustration by Allen David Rosales Interior design by Percolator Typeset in ITC Stone Informal Interior photos by Allen C. Rosales Printed in the United States of America 22  21  20  19  18  17  16  15     1  2  3  4  5  6  7  8 Library of Congress Cataloging-in-Publication Data Rosales, Allen C. Mathematizing : an emergent math curriculum approach for young children / Allen C. Rosales. — First edition.        pages cm   Includes bibliographical references and index.   ISBN 978-1-60554-395-6 (pbk.)   ISBN 978-1-60554-396-3 (ebook) 1. Mathematics—Study and teaching (Elementary)  I. Title.   QA135.6.R65 2015   372.7' 043—dc23                                                             2015006439 Printed on acid-free paper COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET To my family, for their love and support that has made me   a better educator, husband, and father. To the teachers in our field who are the children’s champions.   Your love and perseverance for the work with children   is inspirational. To my Creator, for providing such a masterfully designed   and mathematically patterned creation. Thank you. COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET CONTENTS Acknowledgments ix Introduction: What Is Mathematizing? 1 CHAPTER 1: The Mathematizing for Learning Process Approach Theoretical Base 11 CHAPTER 2: Observation Component 17 CHAPTER 3: Exploration with Materials Component 25 CHAPTER 4: Language Modeling Component 43 CHAPTER 5: Inquiry Component 69 CHAPTER 6: Conclusion 115 Glossary 119 Bibliography 123 Index 135 COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET ACKNOWLEDGMENTS I would like to express my sincere gratitude to the many early childhood schools and educators that have been a part of my personal and professional life and who supported me throughout the writing of this book. The col- laboration and friendship that you demonstrated were fundamental to my work, and your dedication to the children and families you serve is inspira- tional. The following schools opened their hearts and resources to me and played an important part in my ability to apply the Mathematizing for Learn- ing Process approach in their settings. They include: Belmont-Cragin Early Childhood Center, Albany Park Community Center, Chinese American Ser- vice League, Erie Neighborhood House, Josiah L. Pickard Elementary School, Carole Robert­son Center for Learning, Family Home Daycare, and Laurance Armour Day School. A special thanks to Reggio Emilia approach practitioners in Italy and the United States for their ability to create spectacular learning environments for children and for sharing their perspectives with the field. I became a Reggio Emilia-inspired practioner in the mid ’90s after watching To Make a Portrait of a Lion and have followed their work ever since. Thank you for your com- mitment to the field. I am also grateful to Erikson Institute’s Early Math Collaborative team for their excellent work with early childhood schools in Chicago and for allowing me to participate in the collaborative as a coach and consultant. The math professional development, research, and dissemination practices I experi- enced in the collaborative strengthened my math practices and inspired me to share those perspectives with the field. A special thanks to my editors Todd R. Berger, David Heath, Douglas Schmitz, and everyone at Redleaf Press for their patience, collaboration, and professionalism. Lastly, I give a heartwarming thank-you to all of the teachers, children, and families that allowed me to observe and document their learning prac- tices and processes. I hope my documentation work in this book captured the essence of the mathematizing interactions you experienced at your schools. ix COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET INTRODUCTION WHAT IS MATHEMATIZING? My earliest recollection of mathematizing—​although I didn’t realize what was happening at the time—​occurred at a grocery store when I was five. My mother engaged me in a fun learning experience while shopping for fruit. “I wonder how many bananas we will need for the week?” she asked me. “Should we buy the big cantaloupe or the small one? Which fruit do you think is heavier, the cantaloupe or the orange? Why do you think the cantaloupe is heavier?” Using words like “how many,” “big,” “small,” and “heavier” as we shopped was my mother’s way of introducing mathematizing into my young life. The continual math play we engaged in helped me develop the ability to see and think about math (mathematize) as part of my everyday experiences. According to Merriam-​Webster’s Collegiate Dictionary, 11th edition, to math- ematize is “to reduce to mathematical form.” In their National Council for the Teaching of Mathematics News Bulletin article, Jacqueline Leonard and Nora Ramirez define mathematizing as “the ability to identify the relationships and quantities that exist in specific contexts” (Leonard and Ramirez 2009, 1). As an early childhood educator, I define mathematizing as “the process of understanding math within the contexts of children’s daily lives.” The intention with my definition of mathematizing is to direct attention to the words “process,” “understanding,” and “context,” which are key elements of respectful early childhood curriculum practices. This process can be cognitive or linguistic or both. The grocery store shopping experience I had with my mother is a good example of how a common situation can be reduced to pure mathematical form, in order to teach critical math concepts within a child’s context. Teach- ers can also develop young children’s mathematical thinking and provide meaningful math experiences by establishing micro-​math cultures in their classrooms. A micro-​math culture is a group of classrooms or individuals that share common thoughts, values, and behaviors. Micro-​math cultures can 1 COPYRIGHTED MATERIAL 2   Introduction DOUBLE TAP TO ZOOM WITH PHONE OR TABLET be seen as environments, both physical and intellectual, that are created to bring mathematics alive for children. Teachers who establish micro-​math cultures in their classrooms mathematize daily routines, activities, play, ex- plorations, and investigations that children encounter through relationships with their friends, families, and environments. Within any chosen task or activity, mathematizing teachers can help chil- dren make sense of the math concepts they are learning. Take for example infant and toddler children who love to fill and dump water into cups, bowls, and other containers at the water table. A teacher with a mathematizing eye observes and identifies the mathematics that the children are investigating, in this instance the concept of conservation and its accompanying variables (volume, capacity, adding, subtracting). Adults can promote critical thinking by having children explore water with cups, sponges, and many other tools and materials. Teachers can also support children’s linguistic development by incorporating language-​modeling techniques such as labeling, expand- ing/extending language, and parallel and spiral talking (see chapter 4) to help children develop language and conversation skills. A teacher’s math language at the water table could include statements such as these: • “You are pouring the water into the cup!” • “You are squeezing the sponge and filling the cup!” • “The cup is full!” • “You are dumping the water from your cup!” • “Your cup is empty now. The water is all gone!” At the preschool level, the same mathematical conversations could con- tinue, with the opportunity of including more advanced math vocabulary and inquiry to the exchanges. This will help children develop and make meaning of the math concepts they are investigating. Preschool teachers can use language such as the following: • “Your cup is half full!” • “The water is reaching the top of your cup!” • “I wonder how many cups of water it will take to fill the tall container?” • “Why do you think it took the same number of cups to fill the tall con- tainer and the short, wide container?” The Mathematizing for Learning Process (MLP) approach (see chapter 1) is an effective framework that can support teachers in their interpretation, COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET What Is Mathematizing?   3 Figure i.1 and Figure i.2. Teachers Elvira Mata and Andrea Richard mathematizing children’s water explorations at the Laurance Armour Day School in Chicago, Illinois design, and application of rich mathematical experiences in the classroom. It develops students’ math understanding in a natural, yet rigorous process. To get a sense of how teachers can mathematize experiences within a school setting, let’s take a look at two examples from the Belmont-​Cragin Early Childhood Center, a public school in Chicago. Teachers engaged chil- dren and parents in meaningful math learning experiences as a way to en- courage mathematizing at school and at home. In teacher Lourdes Molina’s preschool classroom, children became curi- ous about a rainbow that appeared in the sky one rainy day. The class had a long discussion about rainbows, and Mrs. Molina decided to have the chil- dren investigate the topic more in-​depth through a long-​term study. At first, the children observed and focused on the colors and shapes of a rainbow. This interest led to a recycling project where the children were encouraged to collect items at home based on the colors they identified on the rainbow. Once the materials were collected, children grouped the materials based on color, counted the number of items per set, and created a color-​pattern rain- bow. Sometime later, as children revisited the rainbow project, Mrs. Molina challenged them to think about the length of the rainbow lines. She asked children to estimate the length of each line and to predict which lines would COPYRIGHTED MATERIAL 4   Introduction DOUBLE TAP TO ZOOM WITH PHONE OR TABLET be shorter or longer. To prove and verify children’s esti- mations and predictions, the children were provided with clay in order to recreate the rainbow with a new mate- rial. Once completed the students used a tape measure to count and find the total length of each rainbow line. The children compared the results to their initial estimations and shared the outcomes. In a magnificent display of parent engagement, the Belmont-​Cragin Early Childhood Center incorporates a fifteen-​minute parent/child learning experience every day at the beginning of class. In all seven preschool classrooms, children and parents enter the rooms ready to engage in lesson activities prepared by the classroom teachers. During this time, the parents direct and guide the learn- ing of their children, while the teachers take on facilita- tor roles. Figures i.4–i.6 show how parents mathematized their children’s experiences by helping them identify the Figure i.3. Teacher Lourdes Molina and three children counting and measuring properties and attributes of the natural materials, create the lines of a clay rainbow shapes and letters, group the items, count the quantities of the materials, and finally chart the items on a graph. The classroom teachers end the fifteen-​minute session by modeling how to mathematize a read-​aloud with books that parents can check out through their classroom lending library. Figure 1.4, Figure i.5, and Figure i.6. Parents engage their children in mathematizing experiences at the Belmont-​Cragin Early Childhood Center in Chicago. COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET What Is Mathematizing?   5 MATHEMATIZING OPPORTUNITIES IN OUR ENVIRONMENTS Before delving into the MLP approach and how this framework can help you, I want to look at some addi- tional mathematizing opportunities within our envi- ronment. This first example can take place in any city with tall buildings. A mathematizing teacher can ob- serve a skyscraper (see fig. i.7) and interpret the kinds of math concepts that could be taught or investigated if this building were to be incorporated within a study or unit lesson. In your mind you can create a list (fig. i.8) of math concepts that can be seen in the skyscraper’s structure. Next you find creative and constructive materials to support the learning of the math concepts you have identified and children’s meaning-​making process. As the students begin to explore the properties of the mate- rials and commence their construction of the structure, you can observe the children’s interactions and look for clues about which math concepts they are interested Figure i.7. The Willis Tower in Chicago in investigating. Once you identify children’s chosen math concepts, you MATHEMATIZING can generate a second list of key math words you PROBABILITIES can use as the students create with materials and Height engage with their classmates (fig. i.9). Your goal is to ensure that the math learning Width that transpires through your interactions with Length students is connected to the concepts the chil- Perimeter dren have chosen to investigate. For example, one child’s representation of the Willis Tower Surface Area with Lego blocks (fig. i.10) was a culminating Patterns project based on an interest the child followed for Counting a couple months. As he observed the photo of the skyscraper and revisited the Lego structure, he Addition would add structural details and focus on differ- Subtraction ent math concepts throughout the process. This Shapes and Form boy’s sequential math focus and construction with Lego blocks during the two-​month study included these concepts: Figure i.8. COPYRIGHTED MATERIAL 6   Introduction DOUBLE TAP TO ZOOM WITH PHONE OR TABLET • Height: Child builds a narrow, tall skyscraper. • Length and width: Next skyscraper construction becomes longer and wider. • Height and number sense: Structure becomes taller and child counts the levels of the building. • Patterns: Child begins to create a structure with a focus on vertical and horizontal lines. • Area: Child notices and creates surface areas on the building. • Height and number sense: Antennas are added and counted. • Multiple math concepts: All math concepts investigated are incorpo- rated in the final Willis Tower representation. MATHEMATIZING PROBABILITIES Height (Child Focus) Tall* Height Short* Width Length Perimeter Surface Area Patterns Top Bottom Distance Between More Counting Less Addition Inches Subtraction Feet Shapes and Form Teacher Math Concept Development Words Figure i.9. Throughout the Willis Tower study, the teacher nurtured mathematical learning by fostering the interests and thinking that emerged from the chil- dren’s perspectives. This is what differentiates an emergent math curriculum from a traditional skills-​based approach to learning mathematics. Emergent COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET What Is Mathematizing?   7 mathematics promotes a construction-​of-​meaning pro- cess. Children are provided with real-​life contexts and the materials to make sense of the mathematics found within those experiences. The skills-​based approach to mathematics, on the other hand, focuses on providing children with an accumulation of factual knowledge with the goal of having them repeat the memorized information. Had the Willis Tower study been incorpo- rated using the skills-​based approach, the teacher would have shown the child a photo of the skyscraper and told him everything there was to know about the building. This teacher-​directed approach may help in naming parts of the skyscraper, but it does little to develop chil- dren’s understanding of the mathematics that were in- corporated in the construction of the building. In fact, Douglas H. Clements and Julie Sarama, two well-​known math-​education researchers, state that students may not Figure i.10. A five-​year-​old’s representa- reach the level of understanding needed for future aca- tion of the Willis Tower (twenty-​eight demic success “if children are not helped to mathematize inches tall) with Legos (reflect on, give language to)” (Clements and Sarama 2014). Mathematizing a tall building and pro- viding the students with many opportunities to create a model of the skyscraper with different MATHEMATIZING PROBABILITIES materials can help develop a student’s under- FOR TREE STUDY standing of how math is integrated within our environment. Angles Great mathematizing opportunities also Tall/Short occur when teachers use children’s environ- Wide/Narrow ments to promote mathematical learning. Take for example an exuberant tree I photo- Far/Near graphed in California (fig. i.12). The natural More/Less beauty of this tree embodies the essence of rich Shapes/Forms mathematics existing within living things. At first glance, some of the mathematical Patterns concepts that are visible in the photograph Quantities include angles, length, height, and quantity Proportions (fig. i.11). A mathematizing teacher interprets the kinds of math concepts that can be investi- Etc. gated with this photograph and plans learning Figure i.11. COPYRIGHTED MATERIAL 8   Introduction DOUBLE TAP TO ZOOM WITH PHONE OR TABLET experiences to help children understand the mathematical relationships be- tween the concepts. After providing meaningful experiences related to the first photograph presented, a teacher can then revisit the tree with photographs taken from different angles. Incorporating this strategy benefits the children in that the new photographs reveal other mathematical concepts that the chil- dren can investigate and represent with materials. By supporting children’s experiences with this tree study, teachers can ensure the students’ work is • mathematically rich, • cognitively demanding, and • promotes worthwhile tasks that “build connections they have not already learned” (Featherstone et. al. 2011, 58). Figure i.12. First perspective Figure i.13. Second perspective Figure i.14. Third perspective Figure i.15. A five-​year-​old’s drawing of the exuberant tree with pastels COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET What Is Mathematizing?   9 The function, purpose, and goal of mathematizing have evolved over the years. Today it is an in-​depth process that integrates the wisdom of parents, teachers, mathematicians, artists, architects, engineers, neuroscientists, and linguists to develop innovative practices. The MLP approach functions as a curriculum and instruction design framework that helps teachers create meaningful learning experiences for children. Its main purpose is to develop students’ higher-​level thinking and linguistic skills through intentional and purposeful interactions. As teachers mathematize children’s daily learning experiences, students learn to see and think about the mathematics that exists within children’s contexts. This process helps children understand the role of math in their lives. It begins to instill in them a love of mathematics, thereby reaching the goal of developing mathematically confident and pro- ficient students. The process of teaching mathematics within real-​life contexts has proven to be quite difficult for many teachers who may feel a lack of confidence or a certain type of “math fear.” This anxiety towards math can be expected as teachers sometimes relate math knowledge to secondary-​or college-​level pro- cessing, which can be daunting. But, for the early childhood teacher, becom- ing a proficient mathematizing teacher simply means having the knowledge of the math content and processing that occurs at the age level of the children they are working with and then having the ability to implement quality math experiences within the context of children’s lives. The MLP approach frames the kinds of skills, knowledge, and processing a teacher needs to implement successful math experiences. The real-​life teaching experiences presented in this book can serve as models for learning. The resources provided can help lower the anxiety some teachers may feel when the topic of math arises. The book can promote the development of positive dispositions toward the teach- ing of mathematics in preschool settings. I often hear teachers say “math is everywhere” or “math is in everything,” and although these statements are true, to understand the patterns, relation- ships, and mathematical systems of the concepts inside the “everywhere” or “everything” is the challenge. I wrote this book to provide teachers with a systematic approach to creating and engaging children in rich mathemat- ical experiences. By rich I mean experiences that integrate great environ- mental settings, creative learning materials, and meaningful teacher/child interaction. Rich experiences focus on the mathematics that exists within the topics being explored or investigated by the children. The MLP approach components can help teachers see and interpret the math inside the “every- where” or “everything,” and support teachers as they develop curricula for their students. COPYRIGHTED MATERIAL 10   Introduction DOUBLE TAP TO ZOOM WITH PHONE OR TABLET This book is intended to • provide a practical mathematizing approach for teachers working with very young children; • present vivid examples, processes, and illustrations of great math pro- cessing in early childhood settings; • present creative strategies and techniques for teachers to develop their pedagogical skills; and • promote and develop teachers’ and children’s mathematical and multi- dimensional thinking and processing skills. This is the kind of mathematical thinking and processing that is in high demand worldwide and “that develops students’ abilities to deal with com- plex systems” (English 2011b, 1). The methods, strategies, and techniques in this book can be used by teach- ers to develop their teaching skills. The four components in the MLP approach framework (observation, exploration, language modeling, and inquiry) con- stitute fundamental skills teachers need in order to implement high-​level in- teractions with children. In my work with early childhood programs, I’ve found that centers that incorporated this framework within long-​term profes- sional learning communities saw the greatest increases in the quality of their curriculum and instructional practices. The teachers were able to change their “math fear” into “math confidence” and learned to provide intentional and purposeful math experiences to the children in their care. –––––– Now that we have seen some mathematizing possibilities in our environ- ments and settings, let’s take a look at the MLP approach. COPYRIGHTED MATERIAL DOUBLE TAP TO ZOOM WITH PHONE OR TABLET MATH / CURRICULUM / STRATEGIES See and support the math already happening in children’s everyday lives Mathematizing provides teachers a proven, effective way of building children’s math and inquiry skills. The mathematizing approach is different from traditional math curriculums because it immerses children in a process that develops their understanding of math concepts in real-life, everyday contexts. Using child- centered learning, you will discover how to interpret children’s interests and use them as a catalyst for creating meaningful mathematical learning. This book is full of practical lessons, case studies, illustrations, documentation, and charts not available in other math-related books. It provides a framework and the tools for designing and implementing rich mathematical experiences. Recognize the math occurring in children’s lives every day and promote children’s math, language, and inquiry skills. Allen C. Rosales has over twenty years of experience as an early childhood educator. His work with teachers and professional-learning communities has won national and state recognition, awards, and grants that support the field in implementing and sustaining quality early childhood practices. Rosales is currently an early childhood and bilingual education instructor at Roosevelt University in Chicago, where he helps prepare the next generation of educators. ISBN 978-1-60554-395-6 COPYRIGHTED MATERIAL $29.95