We wrote this book in response to teachers’ requests. Teachers wanted better insight into children’s mathematical development. They wanted to know: “What does this child’s thinking tell me about what he or she understands?” “Where does this child’s mathematical thinking lie on a framework of primary numeracy development?” “What questions can I ask, or what activities can I do, to help this child move to the next phase?”
This resource is a sample video record of the mathematical development of 60 children over the course of two years. The teachers in the videos began with the children’s initial understandings and worked with their ideas to help the children develop increasingly sophisticated strategies and deepen their mathematical understanding. The development you see in these children is a result of this instruction.
Each chapter is organized around a series of video clips of interviews with children solving problems in a particular area of mathematics. These interviews were conducted over two and, occasionally, three years with the same children. The video icon, as shown in the margin, indicates that there is a video to be viewed. In chapters 2 to 5, the video is a snapshot of a time in a child’s mathematical development. In Chapter 6, it comprises several interviews with the same child, following a learning path from initial struggle through the development of increasingly sophisticated strategies.
This book is provided in print and eText format. To access the eText, see the information provided on the inside front cover of this book. Video clips can be viewed by clicking on the video icons in the eText.
Chapters 2 to 5 begin with a brief explanation of the strategies you will see children using in the videos, and the key ideas they are constructing. This introduction is followed by a brief discussion of the mathematical topics addressed in the chapter and children’s developing understandings of the topic. For each video, a chart describes what the child does during the interview and what this demonstrates about the child’s mathematical understanding. There are suggestions for assessing children’s development, sharing strategies in class discussions, and supporting children who are ready to try the strategy demonstrated or to construct the key idea implicit in the video. There may also be suggestions for what to do next with the children. Each chapter concludes with a chart that provides a brief explanation of the key ideas constructed by the children featured in the chapter.
Chapter 6 focuses on learning paths of particular children over a two-year period. Each child is developing a particular mathematical skill or key understanding. These children demonstrate learning paths that are representative of many we saw during our research.
The Teacher’s Math Kit, which follows Chapter 6, contains a selection of everyday living activities, mini-lessons, and games that can be used to support children as they develop the mathematical skills and understandings needed for number sense and early computation. These activities are referenced in the earlier chapters to indicate when they might best be used. Line masters, such as game boards, are provided for some activities. The line master icon, as shown in the margin, indicates when there is an activity master available for use.
View and download the line masters by clicking on the line master icons in the eText. You can access black line masters for the continua by clicking on the icons for MK22 and MK23.
Basic definitions of mathematical and instructional terms are provided in the margin the first time the terms are used in a chapter. The book also includes a reference section that lists the relevant scholarly articles and teacher resources referenced in the chapters.
How you use and interact with the book will depend on your goals and situation. Although the text builds chronologically from early addition through to beginning multiplication and division strategies, it is not expected that you will read it in order from cover to cover. Here are just a few of the ways you may use this resource:
All of these uses are possible, and they are aided by the detailed Chapter Overview of Videos charts. These charts, found in chapters 2 to 6, describe the problem being solved in each videotaped interview and identify the strategies and key ideas that can be seen or inferred from careful observation of the child as he or she works.
We expect the vocabulary to be new to you, and that expectation is part of the reason for writing the book. Deborah Ball, one of the most influential living mathematics educators today, makes the case that, as a profession, we do not achieve what we should with children, in part, because we do not have a common vocabulary of teaching. The most effective way to learn the new vocabulary is to approach it as you would when learning a new language: Use it. We encourage you to use the vocabulary in discussion with colleagues, administrators, and parents alike, as a way to solidify and refine your own understanding and capacity as a teacher of children’s mathematics.
Although most of our research team members were already familiar with the terminology, we still refined and solidified our thinking as we discussed the videos in order to write this book. Knowing the vocabulary provides you with a strong foundation, and maximizes the value of viewing the videos, reading this book, and discussing what you learn with others.
You could read this book and learn a great deal without looking at any of the videos. The description in each of the interview charts is a written record of what you would see in a given video. In this way, it is possible to use the book without viewing the videos. There may be times when you want to quickly access a given problem or strategy and do not have time to view the video. Having said that, there are aspects of the videos that we cannot capture in writing, so, at some point in your reading of the book, it would be worth viewing the videos.
The videos will help you recognize, understand, and appreciate children’s thinking in your classroom. Alternatively, if you don’t see evidence of similar thinking in your class, the videos offer illustrations of children demonstrating this thinking. They will help you refine your understanding of a given strategy, and possibly the key ideas, because you see the first in action and may be able to infer the second. Finally, the videos will help you locate the strategies on the continuum we reference, supporting your ability to assess your learners’ mathematical knowledge and to plan for next steps.
When I prepare to watch any of these videos with teachers in a professional development group, I first ask them to think about what they predict children might do to solve the given problem. They discuss their predictions with a partner, then watch the video. This helps prepare teachers to anticipate the wide range of strategies they may observe; further, they develop a common language and discuss what the video shows with a more critical and informed eye. Later, teachers often try out the problem with children in their own class to observe how they respond and to establish a basis for our next round of professional development.
Choose a problem from one of the chapters and develop a context that reflects the interests of the children in your class; for this first try, keep the numbers the same as in the sample problem. Use this problem as a “baseline.” At this point, resist the temptation to given the children too much support; look at what they know on their own. Compare their responses to what you see in the book. What strategies do you want to support your children in developing? Try out one of the games or activities suggested to foster a chosen strategy. Alternatively, try a new version of the same problem, this time changing the numbers, and have children discuss their strategies in a class debrief as the final part of the lesson. We turn now to the heart of the book: looking at children’s mathematics.