Learning and Teaching Early Math

The Learning Trajectories Approach

Douglas H. Clements, Julie Sarama

• Publisher: Routledge
• Year: 2020
• Print ISBN: 9780367521974
• eBook ISBN: 9781000288544
• Edition: 3
• Topic: Education
• Subtopic: Early Childhood Education

The third edition of this significant and groundbreaking book summarizes current research into how young children learn mathematics and how best to develop foundational knowledge to realize more effective teaching.

Using straightforward, practical language, early math experts Douglas Clements and Julie Sarama show how learning trajectories help teachers understand children's level of mathematical understanding and lead to better teaching. By focusing on the inherent delight and curiosity behind young children's mathematical reasoning, learning trajectories ultimately make teaching more joyous: helping teachers understand the varying levels of knowledge exhibited by individual students, it allows them to better meet the learning needs of all children.

This thoroughly revised and contemporary third edition of Learning and Teaching Early Math remains the definitive, research-based resource to help teachers understand the learning trajectories of early mathematics and become confident, credible professionals. The new edition draws on numerous new research studies, offers expanded international examples, and includes updated illustrations throughout.

This new edition is closely linked with Learning and Teaching with Learning Trajectories –[LT]²–an open-access, web-based tool for early childhood educators to learn about how children think and learn about mathematics. Head to LearningTrajectories.org for ongoing updates, interactive games, and practical tools that support classroom learning.

1 Young Children and Mathematics Learning

Snow was falling in Boston and preschool teacher Sarah Gardner’s children were coming in slowly, one bus at a time. She had been doing high-quality math all year, but was still amazed at her children’s ability to keep track of the situation: The children kept saying, “Now, 11 are here and 7 absent. Now, 13 are here and 5 absent. Now ….”

Why have so many people become interested in math for very young children lately? Because early math is surprisingly important.

First, math is increasingly important in a modern global economy, but math achievement in many countries has not kept up. Our own country, the USA, has fewer high-performing and more low-performing students than many other countries, especially in math (http://ncee.org/pisa-2018-lessons/). These differences appear as early as first grade, kindergarten … and even preschool (Gerofsky, 2015b; Mullis, Martin, Foy, & Arora, 2012b; OECD, 2014). Although some high-performing countries are showing improvements, many like the USA are not (Mullis et al., 2012b). This is one reason interest in improving early childhood math education has emerged from around the globe, such as in Africa, South and Latin America, and Asia. These increased interests are often paired with a special focus on children who have not been provided opportunities to learn (McCoy et al., 2018b).

Many young children do not even get the chance to learn the more advanced math taught in many other countries. If each child is given such opportunities, all people in each country benefit, economically and socially, because everyone contributes more to social and technological advancements.

During most of the 20th century, the United States possessed peerless mathematical prowess—not just as measured by the depth and number of the mathematical specialists who practiced here but also by the scale and quality of its engineering, science, and financial leadership, and even by the extent of mathematical education in its broad population. But, without substantial and sustained changes to its educational system, the United States will relinquish its leadership in the 21st century.

The National Mathematics Advisory Panel¹ (NMP, 2008, p. xi)

Second, these early childhood years have been found to be surprisingly important for development through life. That is, what math children know when they enter kindergarten predicts their math achievement for years to come (Duncan et al., 2007). Math also predicts later success in reading (Duncan et al., 2007; Duncan & Magnuson, 2011), so, math appears to be a core component of cognition. Further, knowledge of math in the early years is the best predictor of graduating high school (McCoy et al., 2017; Watts, Duncan, Siegler, & Davis-Kean, 2014). One more argument for early childhood math is that Number and arithmetic knowledge at age 7 years predicts socioeconomic status at age 42 (even controlling for all other variables, Ritchie & Bates, 2013).

These predictions may show that math concepts and skills are important to all of school and life. Math provides a new way to see the world, the beauty of it, and the way you can solve problems that arise within it. However, math is much more: Math is critical thinking and problem solving, and high-quality math experiences also promote social, emotional, literacy, and general brain development (Aydogan et al., 2005b; Clements, Sarama, Layzer, Unlu, & Fesler, 2020a; Dumas, McNeish, Sarama, & Clements, 2019; Sarama & Clements, 2019b; Sarama, Lange, Clements, & Wolfe, 2012b)! No wonder they predict later success.

Third, although the math-achievements gap between countries is troubling, an even larger and more damaging gap lies between children growing up in higher- and lower-resource communities. Both the income gap and the achievement gap have been increasing for decades (Bachman, Votruba-Drzal, El Nokali, & Castle Heatly, 2015; Reardon, 2011). Children shouldn’t be at a disadvantage just because their communities lack resources to provide charging stations for learning math—and they do not have to be. They would think and learn just as well if they have the same opportunities to learn math early. That’s why we are working to make good early math learning resources available to children in all communities.

Fourth, if our country’s children have limited math knowledge initially and achieve less later in school compared to children in other countries, can there possibly be bright spots? Yes. From their first years, children have boundless interest and curiosity in math … and the ability to learn to think like mathematicians. In high-quality early childhood education programs, young children can engage in surprisingly deep investigations of math ideas. They can learn skills, problem solving, and concepts in ways that are natural and motivating to them. This brings us to the main reason to engage young children in math: Young children love to think mathematically. They become exhilarated by their own ideas (like the 6-year-old quoted at the beginning of the preface) and the ideas of others. To develop the whole child, we must develop the mathematical child.

Fifth, teachers enjoy the reasoning and learning that high-quality math education brings forth from their children. High-quality math throughout early childhood does not involve pushing elementary arithmetic onto younger children. Instead, good education allows children to experience math as they play in and explore their world. A higher proportion of children are in early care and education programs every year. We teachers are responsible for bringing the knowledge and intellectual delight of math to all children, especially those who have not yet had many high-quality educational experiences. Good teachers can meet this challenge with research-based “tools.”

Most children acquire considerable knowledge of numbers and other aspects of mathematics before they enter kindergarten. This is important, because the mathematical knowledge that kindergartners bring to school is related to their mathematics learning for years thereafter—in elementary school, middle school, and even high school. Unfortunately, most children from low-income backgrounds enter school with far less knowledge than peers from middle-income backgrounds, and the achievement gap in mathematical knowledge progressively widens throughout their pre-K-12 years.

The National Math Advisory Panel (NMP, 2008, p. xvii)

Fortunately, encouraging results have been obtained for a variety of instructional programs developed to improve the mathematical knowledge of preschoolers and kindergartners, especially those from low-income backgrounds. There are effective techniques—derived from scientific research on learning—that could be put to work in the classroom today to improve children’s mathematical knowledge.

The National Math Advisory Panel (NMP, 2008, p. xvii)

These tools include specific guidance on how to help children learn in ways that are both appropriate and effective. In this book, we pull that knowledge together to provide a core tool: “learning trajectories” for each major topic in early math.

What are Learning Trajectories?

Children follow natural developmental progressions in learning and development. As a simple example, they learn to crawl, then walk, then run, skip, and jump with increasing speed and dexterity. These are levels in the developmental progression of movement. Children follow natural developmental progressions in learning math, too, by learning math ideas and skills in their own way.

Teachers who understand these developmental progressions for each major domain or topic of math, and base their instruction on them, build math learning environments that are particularly developmentally appropriate, effective, and meaningful (Figure 1.1). These developmental paths are the basis for this book’s learning trajectories. Learning trajectories help us answer several questions: What goals or objectives should we hold? Where do we start? How do we know where to go next? How do we get there?

Figure 1.1 Carmen Brown encourages a preschooler to “mathematize”

Learning trajectories have three parts: (a) a math goal, (b) a developmental path along which children progress to reach that goal, and (c) teaching practices, including the educational environment, interactions, and activities, matched to each of the levels of thinking in that path, that help children develop ever-higher levels of thinking. Let’s examine each of these three parts.

Goals: The Big Ideas of Math

The first part of a learning trajectory is a math goal. Our goals include the “big ideas of math”: clusters of concepts and skills that are mathematically central and coherent, consistent with children’s thinking, and generative of future learning. These big ideas come from mathematicians, researchers, and teachers (CCSSO/NGA, 2010; Clements, 2004; NCTM, 2006; NMP, 2008). They include math content but also research on students’ thinking about and learning of math. As an example, one big idea is that counting can be used to find out how many in a collection.

Development Progressions: The Paths of Learning

The second part of a learning trajectory consists of levels of thinking, each more sophisticated than the last, through which children develop on their way to achieving the math goal. That is, the developmental progression describes a typical path that children follow in developing an understanding and skill about that math topic.

Humans are born with a fundamental sense of quantity.

(Geary, 1994, p. 1)

This development of math abilities begins when life begins. As we will see, young children have certain math-like competencies in number, spatial sense, and patterns from birth. However, young children’s ideas and their interpretations of situations are uniquely different from those of adults. For this reason, good early childhood teachers are careful not to assume that children “see” situations, problems, or sol...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Acknowledgments
• What is Learning and Teaching with Learning Trajectories—[LT]2?
• 1. Young Children and Mathematics Learning
• 2. Quantity, Number, and Subitizing
• 3. Verbal and Object Counting
• 4. Comparing, Ordering, and Estimating
• 5. Arithmetic: Early Addition and Subtraction and Counting Strategies
• 6. Arithmetic: Composition of Number, Place Value, Multidigit Addition and Subtraction, Multiplication and Division, and Fractions
• 7. Spatial Thinking
• 8. Shape
• 9. Composition and Decomposition of Shapes
• 10. Geometric Measurement: Length
• 11. Geometric Measurement: Area, Volume, and Angle
• 12. Other Content Domains: Patterns, Structure, and Algebraic Thinking: Classification and Data
• 13. Mathematical Processes and Practices
• 14. Cognition, Affect, and Equity
• 15. Early Childhood Mathematics Education: Contexts and Curricula
• 16. Instructional Practices and Pedagogical Issues
• References
• Index