A Focus on Addition and Subtraction

Bringing Mathematics Education Research to the Classroom

Caroline Ebby, Elizabeth Hulbert, Rachel Broadhead

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781000220933
• Edition: 1
• Topic: Education
• Subtopic: Education General

This innovative text offers a unique approach to making mathematics education research on addition, subtraction, and number concepts readily accessible and understandable to pre-service and in-service teachers of grades K–3.

Revealing students' thought processes with extensive annotated samples of student work and vignettes characteristic of teachers' experiences, this book provides educators with the knowledge and tools needed to modify their lessons and improve student learning of additive reasoning in the primary grades. Based on research gathered in the Ongoing Assessment Project (OGAP), this engaging, easy-to-use resource features practical resources such as:

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• A close focus on student work, including 150+ annotated pieces of student work, to help teachers improve their ability to recognize, assess, and monitor their students' errors and misconceptions, as well as their developing conceptual understanding;

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• A focus on the OGAP Addition, Subtraction, and Base Ten Number Progressions, based on research conducted with hundreds of teachers and thousands of pieces of student work;

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• In-chapter sections on how Common Core State Standards for Math (CCSSM) are supported by math education research;

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• End-of-chapter questions to allow teachers to analyze student thinking and consider instructional strategies for their own students;

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• Instructional links to help teachers relate concepts from each chapter to their own instructional materials and programs;
  
• An accompanying eResource, available online, offers an answer key to Looking Back questions, as well as a copy of the OGAP Additive Framework and the OGAP Number Line Continuum.

A Focus on Addition and Subtraction marks the fourth installment of the popular A Focus on … collection, designed to aid the professional development of pre-service and in-service mathematics teachers. Following from previous volumes on ratios and proportions, multiplication and division, and fractions, this newest addition is designed to bridge the gap between what math education researchers know and what teachers need to know in order to better understand evidence in student work and make effective instructional decisions.

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Additive Reasoning and Number Sense

Big Ideas

• Additive reasoning includes various mathematical skills, concepts, and abilities that contribute to number sense.
• Additive reasoning is built on concepts of early number, including part–whole relationships, commutativity, and the inverse relationship between addition and subtraction.
• Additive reasoning both depends upon, and contributes to, the development of base-ten number understanding.
• The OGAP Additive Framework contains learning progressions that provide instructional guidance for teachers so that all students can access the important concepts and strategies that lead to additive reasoning.

Additive reasoning involves much more than being able to add and subtract. It involves knowing when to use addition and subtraction in a variety of situations, choosing flexibly among different models and strategies, using reasoning to explain and justify one’s approach, having a variety of strategies and algorithms for multi-digit addition and subtraction, and knowing if an answer or result is reasonable. Moreover, it is important for fluency with addition and subtraction procedures to be built upon conceptual understanding and reasoning. As the National Council of Teachers of Mathematics (NCTM) states:

Procedural fluency is a critical component of mathematical proficiency. Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.

(2014a, p. 1)

Additive reasoning is the focus of K–2 mathematics and provides a foundation for multiplicative reasoning in the intermediate grades. According to Ching and Nunes (2017), additive reasoning is one of the crucial components of mathematical competence and is built on conceptual understanding of number and part–whole relationships. As students learn to work with larger quantities, additive reasoning also involves understanding of the base-ten number system and relative magnitude.

Additive Reasoning: The Mathematical Foundations

Additive reasoning centers around part–whole relationships. As Van de Walle and colleagues (2014) state, “to conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers” (p. 136). At first students will use their counting skills to construct an understanding of the relationships between quantities, but over time they will develop strategies and concepts that move them towards reasoning additively. Understanding quantities in terms of part–whole relationships is a significant achievement which allows children to compose and decompose numbers and use those relationships to make sense of and solve problems in a range of situations. The part–whole relationship between two quantities involves understanding both the commutative property and the inverse relationship between addition and subtraction (Ching & Nunes, 2017).

Commutativity

Students often learn about commutativity when they are developing fluency with single digit addition facts. This property of addition has its roots in understanding that a quantity can be separated into two or more smaller quantities and that the order in which they are added does not change the value. Any number c can be made up of part a and part b (c = a + b) or part b and part a (c = b + a). Figure 1.1 is an illustration of both part–whole relationships and commutativity.

Figure 1.1 Nine apples broken into two groups can be thought of as 6 and 3 or 3 and 6 but in both cases, there are 9 apples

Modeling commutativity in concrete situations allows for students to be exposed to this idea while they are developing part–whole understanding. Using models to observe and generalize commutativity can be significantly more powerful for students than learning it abstractly as a rule to remember (e.g., 3 + 6 is the same as 6 + 3 because they are “turn-around” facts).

Chapter 5 Visual Models to Support Additive Reasoning for more on the importance of visual models and Chapter 6 Developing Whole Number Addition for more on the commutative property.

The Inverse Relationship between Addition and Subtraction

The second property of addition that is central to part–whole relations, and therefore additive reasoning, is the inverse relationship between addition and subtraction. Instruction about the relationship between addition and subtraction often includes a focus on students generating a set of related equations (sometimes called “fact families”), but the concept is much more dynamic. Let’s consider the 9 apples again. The inverse relationship means that taking away one part from the whole leaves the other part, so if you remove the 3 you are left with 6 apples. Furthermore, since the two parts, 3 and 6, are interchangeable parts, if you remove the 6 you are left with 3 apples. There are therefore four related equations that can represent the part–whole relationship as shown in Figure 1.2.

Figure 1.2 Four related equations

The equations in Figure 1.2 are the way we numerically represent the relationships, but having a strong understanding of the inverse relationship between addition and subtraction is the result of working with concrete objects, visual models, and various problem situations. In other words, understanding the relationships between the quantities should be central to instruction rather than simply teaching students how to write the related equations.

The integration of part–whole, commutativity, and the inverse relationship between addition and subtraction leads to the development of additive reasoning, characterized by the ability to think about the relations between the quantities when solving problems. For example, children with an understanding of the inverse relationship between addition and subtraction can solve problems modeled by equations such as 7 + x = 10 or x − 5 = 8 by using the inverse operation.

Connecting Additive Reasoning and Base-Ten Understanding

According to several researchers (Krebs, Squire, & Bryant, 2003; Martins-Mourão & Cowan, 1998; Nunes & Bryant, 1996), concepts of additive reasoning must be in place in order to develop an understanding of base ten. Since our number system is composed of place value parts in varying unit sizes that combine to make the whole, flexibly working with multi-digit numbers involves concepts of part–whole, commutativity, and the inverse relationship between addition and subtraction. For example, 68 can be thought of as additively composed of 60 and 8 or 8 and 60, and if 8 is taken from 68 then 60 remains. These ideas are foundational to flexible use of the base-ten number system and number sense. At the same time, as students develop base-ten understanding they are able to develop more sophisticated strategies for addition and subtraction.

When students truly understand and can meaningfully combine these ideas, they can apply them to construct a relational understanding of numbers and operations, which in turn leads to strong number sense. The OGAP Additive Framework, discussed in more detail in Chapter 2, includes progressions for base ten, addition, and subtraction, as these concepts develop concurrently throughout the early elementary years.

What Is Number Sense?

Number sense is a widely used term encompassing a range of skills and concepts across all levels of mathematics. Broadly, number sense can be thought of as a flexible understanding of numbers and their relationships. According to NCTM (2000):

As students work with numbers, they gradually develop flexibility in thinking about numbers, which is a hallmark of number sense… Number sense develops as students understand the size of numbers, develop multiple ways of thinking about and representing numbers, use numbers as referents, and develop accurate perceptions about the effects of operating on numbers.

(p. 80)

Number sense does not boil down to a single skill or concept. Many of the important components that make up number sense have their origins in the earliest grades: equality, base-ten understanding, relative magnitude, operations, number relationships, and estimation, to name a few. Building students’ number sense involves making connections between these concepts with a focus on understanding and flexible use to solve problems.

Additive Reasoning and Number Sense: From a Teaching and Learning Perspective

Teachers face many challenges in supporting the development of students’ additive reasoning and number sense. Most teachers did not experience math instruction that was focused on developing number sense, either as a learner through their K–12 instruction or in their teacher preparation programs. An important goal is to view the learning of math as a more dynamic process, one that involves curiosit...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Acknowledgments
• 1. Additive Reasoning and Number Sense
• 2. The OGAP Additive Framework
• 3. The Development of Counting and Early Number Concepts
• 4. Unitizing, Number Composition, and Base-Ten Understanding
• 5. Visual Models to Support Additive Reasoning
• 6. Developing Whole Number Addition
• 7. Developing Whole Number Subtraction
• 8. Additive Situations and Problem Solving
• 9. Developing Math Fact Fluency
• References
• About the Authors
• Index