Making Sense of Mathematics for Teaching Grades 35: (Learn and Teach Concepts and Operations with Depth: How Mathematics Progresses Within and Across
Develop a deep understanding of mathematics. This userfriendly resource presents grades 3–5 teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Focus on an understanding of and procedural fluency with multiplication and division. Address how to learn and teach fraction concepts and operations with depth. Thoroughly teach plane and solid geometry. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success.
Benefits
 Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics.
 Explore how to develop, select, and modify mathematics tasks in order to balance cognitive demand and engage students.
 Discover the three important norms to uphold in all mathematics classrooms.
 Learn to apply the tasks, questioning, and evidence (TQE) process to ensure mathematics instruction is focused, coherent, and rigorous.
 Use charts and diagrams for classifying shapes, which can engage students in important mathematical practices.
 Access short videos that show what classrooms that are developing mathematical understanding should look like.
Contents
Introduction
1 Place Value, Addition, and Subtraction
2 Multiplication and Division
3 Fraction Concepts
4 Fraction Operations
5 Geometry
6 Measurement
Epilogue Next Steps
Appendix A Completed Classification of Triangles Chart
Appendix B Completed Diagram for Classifying Quadrilaterals
10 Chapters 
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Introduction 
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The only way to learn mathematics is to do mathematics. —Paul Halmos When teaching, much of the day is spent supporting students to engage in learning new content. In mathematics, that often means planning for instruction, delivering the planned lessons, and engaging in the formative assessment process. There are opportunities to attend conferences and other professional development events, but those are typically focused on teaching strategies or on administrative tasks like learning the new gradebook program. Opportunities to take on the role of learner of the subject you teach are often neglected. As you read Making Sense of Mathematics for Teaching Grades 3–5, you will have the chance to become the learner once again. You will learn about the mathematics you teach by doing the mathematics you teach. There is a strong call to build teachers’ content knowledge for teaching mathematics. A lack of a “deep understanding of the content that [teachers] are expected to teach may inhibit their ability to teach meaningful, effective, and connected lesson sequences, regardless of the materials that they have available” (National Council of Teachers of Mathematics [NCTM], 2014, p. 71). This lack of deep understanding may have more to do with lack of exposure than anything else. 

Chapter 1 Place Value, Addition, and Subtraction 
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Place value and the operations of addition and subtraction on whole numbers, with a connection to decimals, are important topics in grades 3–5 mathematics instruction. In this chapter, we focus on creating a deep understanding of the base ten number system and place value to assist in developing an understanding of invented and standard algorithms. You will explore how to use manipulatives such as base ten blocks to build students’ understanding of how to compose and decompose numbers in order to help with the process of regrouping. We also include multiple strategies for developing fluency to add and subtract within 1,000, which will lead to success with the standard algorithm. In the first task (see figure 1.1), consider how to add two threedigit numbers using an invented algorithm. Imagine that you don’t yet know the standard algorithm of lining up the addends vertically, adding the ones and regrouping if necessary, then adding the tens, and so on. As with all tasks in this book, take the time to work the task out before reading on. The time you take with the task makes the text that follows more meaningful. 

Chapter 2 Multiplication and Division 
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Exploring tasks with and without context allows you to consider different reasoning strategies that lead to solutions. The progression of learning with multiplication and division begins with singledigit factors and builds to multidigit facts, including knowing and applying the standard algorithms for multiplication and division. Throughout this chapter, you will apply a deep understanding of meanings of multiplication and division to strategies for determining basic facts, using approaches for invented and alternative algorithms for multidigit multiplication and division, and making sense of standard algorithms. The initial task (see figure 2.1) brings attention to varied strategies for multiplying multidigit numbers. Multiply 12 × 15 in three different ways. Figure 2.1: Task to multiply multidigit numbers in many ways. How did you respond to this task? It is likely you first used the standard algorithm to see that the product is 180 and then you paused as you tried to think of other ways to multiply. As an adult, you may find it difficult to think beyond the standard algorithm because that is what you were taught and what you remember. Now the expectation is that students should be able to use invented algorithms to solve problems. That means you need to become comfortable with them as well. Thinking of multiplication as groups of objects might help you reason with this problem more flexibly. How might this thinking help you make sense of 12 × 15? Note that the focus is not just on getting a correct product through a procedure but also on understanding the problem conceptually. 

Chapter 3 Fraction Concepts 
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When learning and teaching fraction concepts, the ideas of partitioning, unitizing, equivalence, and comparison are introduced in first grade and continue through fourth grade to prepare students for conceptualizing operations with fractions. Word problems with the aid of visual models can provide the foundation for understanding fraction concepts in depth. Use drawings to determine which student is correct in figure 3.1 before reading the discussion of the task. A group of students was asked to solve the following problem: Share four cookies equally among five people. How much of a cookie will each person receive? Analyze the following student responses to determine who is correct and incorrect and why. •Student A: Each person will receive ^{1}⁄_{5} of each cookie. •Student B: Each person will receive ^{4}⁄_{20} of all the cookies. •Student C: Each person will receive ^{4}⁄_{5} of a cookie. 

Chapter 4 Fraction Operations 
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This chapter focuses on mathematics for teaching addition, subtraction, multiplication, and division of fractions with depth. You can develop students’ fraction operation sense by embedding operations in context through word problems. To do so, first use visual models to solve the problems, then represent the contexts and solutions with equations, and finally, make sense of procedures for solving the equations more efficiently. Check the results through estimation to be sure solutions are reasonable. The initial task in this chapter (see figure 4.1) begins this process by providing word problems to be solved with visual models. These three problems may be challenging if you have not previously explored representing fraction operations with drawings. The key is to act out the context of each problem with pictures. The discussion that follows will be much more meaningful if you make an attempt to solve each problem using a picture and then write the situation equation before proceeding. 

Chapter 5 Geometry 
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The focus of this chapter is the mathematics for teaching plane and solid geometry with depth so that you and your students develop a strong foundation for the study of geometry. What you need to know about the study of geometry is sometimes beyond what you will address with students in grades 3–5. This is in large part to be sure that you do not teach rules that will expire as students learn geometry in later grades. Geometry is the study of space, objects in space, and the movement of objects in space. School geometry includes a focus on objects with zero, one, two, and three dimensions. Consider the images in figure 5.1. Figure 5.1: Dimensions of plane and solid geometry. An example of an object with zero dimensions is a point. A point does not have dimensions such as length, width, and height. Although the geometric object of a point seems very simple, the point is quite vital to the subsequent dimensions in geometry. For instance, it takes two distinct points to create a line segment. A line segment is an example of an object with one dimension; it has length. By connecting line segments that do not exist on the same line, you can create objects with two dimensions, such as a rectangle. A rectangle has the dimensions of length and width. Zero, one, and twodimensional geometry (commonly described as plane geometry) includes abstract representations of the real world. When you hold up an attribute block that is the shape of a rectangle and say, “This twodimensional shape is a rectangle,” you are not actually correct. The shape you are holding is actually threedimensional because it has length, width, and height to it. Even when you draw a representation of a rectangle on a sheet of paper, the drawn lines have a thickness, even though it is quite small. It is understood, by most adults, that you are ignoring the third dimension, the height, in order to represent the rectangle in a way that makes sense in the real world. 

Chapter 6 Measurement 
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In grades 3–5, topics central to measurement include concepts related to perimeter, area, volume, elapsed time, angles, and conversion within systems. Contexts related to measurement offer opportunities for students to engage in problem solving. The initial task in this chapter (figure 6.1) provides an opportunity for you to engage in problem solving related to measurement. This task involves the use of the geoboard and geoboard dot paper. If you do not have access to a geoboard, the activity can be completed using virtual geoboard manipulatives or the geoboard dot paper provided. Make nine different (simple) polygons, each with an area of four square units. Record them on geoboard dot paper so that each polygon is on its own board. Figure 6.1: Geoboard polygons task. Visit go.solutiontree.com/mathematics for a free reproducible version of this figure. In order to solve this task, you also need to define different in terms of the problem. In this problem, different means not congruent. Therefore, the polygons in figure 6.2 would not qualify as different because they are transformations of one another. In figure 6.2, you can see that the second figure was just a translation of the first, and the third figure was a rotation of the first figure. Polygons are the same if they are congruent, regardless of their location or orientation on the geoboard (see chapter 5 for more on polygons). 

Epilogue Next Steps 
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An important role of mathematics teachers is to help students understand mathematics as a focused, coherent, and rigorous area of study, regardless of the specific content standards used. To teach mathematics with such depth, you must have a strong understanding of the mathematics yourself as well as a myriad of teaching strategies and tools with which to engage students. Hopefully, by providing the necessary knowledge, tools, and opportunities for you to become a learner of mathematics once more, this book has empowered you to fill this role. Now what? How do you take what you learned from doing mathematics and make good use of it as the teacher of mathematics? Our position is that you first need to apply what you learned to your lesson planning. Are you planning for instruction that focuses on teaching concepts before procedures? How is your planning aligned to developing learning progressions? How will you ensure that your lessons do not end up as a collection of activities? What follows are strategies that will help you use what you experienced as learners and apply it to what you do as teachers. 

Appendix A: Completed Classification of Triangles Chart 
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Appendix B: Completed Diagram for Classifying Quadrilaterals 
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Details
 Title
 Making Sense of Mathematics for Teaching Grades 35: (Learn and Teach Concepts and Operations with Depth: How Mathematics Progresses Within and Across
 Authors
 Juli K. Dixon, Juli K Dixon, Edward C Nolan
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 9781942496427
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 Solution Tree Press
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 April 11, 2016
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