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Making Sense of Mathematics for Teaching Grades K-2: (Communicate the Context Behind High-Cognitive-Demand Tasks for Purposeful, Productive Learning)

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Develop a deep understanding of mathematics. This user-friendly resource presents grades K–2 teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. 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. Clarify math essentials with figures and tables that facilitate understanding through visualization.

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              Number Concepts and Place Value

2              Word Problem Structures

3              Addition and Subtraction Using Counting Strategies

4              Addition and Subtraction Using Grouping Strategies

5              Geometry

6              Measurement

Epilogue               Next Steps

Appendix A         Completed Classification of Triangles Chart

Appendix B         Completed Diagram for Classifying Quadrilaterals

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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 not available. As you read Making Sense of Mathematics for Teaching Grades K–2, 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 Number Concepts and Place Value

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Number concepts and place value provide the mathematical foundation that all students need for future success in mathematics. Number concepts describe the meaning of numbers and are prerequisite to making sense of operations (addition and subtraction and eventually multiplication and division). Place value involves how numbers are grouped in ones, tens, hundreds, and so on. As you consider these two pillars of mathematics, you will have the opportunity to think about ways in which students develop number sense and build an understanding of place value relationships. You will also learn about student misconceptions, how to facilitate student engagement through meaningful tasks, and ways to address common student errors related to number and place value.

There are several tools that help engage students with meaningful contexts to support their learning of these two concepts. We share these tools throughout the chapter. As you read, it is important to remember that students require time to make sense of number and place value. They benefit from being exposed to a variety of tasks that are conceptually based, grounded in everyday life experiences, and challenging to their present notions about number and place value.

 

Chapter 2 Word Problem Structures

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Word problems are extremely important when an emphasis of instruction is on teaching for understanding. Students make sense of mathematics by exploring it in real-world contexts. This is especially true as young learners develop number sense with adding and subtracting. You must be intentional about providing students with word problems to solve and even having them write their own word problems.

Prior to reading further, write your own word problems according to the directions in figure 2.1. Once you write these word problems, set them aside. You will refer to them later after exploring word problems more deeply.

Write four word (or story) problems: one addition, one subtraction, one multiplication, and one division.

Figure 2.1: Word problem task.

In this chapter, you will make sense of word problem structures. The focus is primarily on addition and subtraction problems. However, since so much of what is addressed in the primary grades leads to readiness for multiplication and division, we will briefly address word problems supportive of these operations as well. Research demonstrates that kindergarten students can solve a variety of problem types, including multiplication and division, as long as the students can act out the action of the problem (Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993). To explore this concept, unpack the set of word problems provided in the word problem sorting task (see figure 2.2, page 32). We’ve adapted these problems from Children’s Mathematics: Cognitively Guided Instruction (Carpenter, Fennema, Franke, Levi, & Empson, 2015), which provides an in-depth look at word problems and how they relate to Cognitively Guided Instruction (CGI). Rather than attempting to replicate their excellent work, we use it to support the continued development of your knowledge of the mathematics you teach with respect to problem structures. In order to fully appreciate the experience of unpacking these problems, copy the problems provided in figure 2.2, and cut them out along the dotted lines.

 

Chapter 3 Addition and Subtraction Using Counting Strategies

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This chapter connects your understanding of number concepts and word problem structures with the operations of addition and subtraction using counting strategies. In the following pages, you will consider how to develop understanding of the concepts of addition and subtraction. The addition and subtraction learning progression begins with learners using strategies to solve real-world problems. We then blend the use of these strategies with the learning of facts to develop a deep meaning for fluency.

Our initial task for you provides a context for the addition of two two-digit addends (see figure 3.1).

Jamila has a total of 28 playing cards in her set. Jocelyn has 37 playing cards in her set. If the girls combine their two sets, how many playing cards would be in the combined set?

Figure 3.1: Adding two two-digit addends task.

What strategies can be used to solve the problem in figure 3.1? Make a list of different ways to solve the problem, and provide a brief description for each strategy. What was the first strategy you considered? Did you begin by using the standard algorithm? Other strategies demonstrate how to make sense of the task in meaningful ways, including the modeling of place value. The use of concrete materials such as base ten blocks can aid in representing the place value connection (see figure 3.2).

 

Chapter 4 Addition and Subtraction Using Grouping Strategies

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The key to obtaining procedural fluency is to begin with building conceptual understanding of addition and subtraction. Addition and subtraction contexts, along with visual models, aid in understanding the operations, which enables students to create self-invented strategies. In this chapter, you will explore invented strategies and a variety of standard algorithms with the purpose of supporting the development of computational fluency with multidigit addition and subtraction using grouping strategies. An important aspect of fluency is the ability to choose strategies that are most efficient for a given task. You will examine the progression of the standards related to obtaining this fluency alongside potential misconceptions by engaging with meaningful tasks. The initial task in this chapter begins with a candy shop context referred to in chapter 1. This context will assist in making sense of addition and subtraction procedures based on place value and properties of operations.

Complete the task in figure 4.1 using your knowledge that there are 10 pieces of candy in a roll and 10 rolls in a box.

 

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 well beyond what you will address with K–2 students. This is in large part to ensure 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 two-dimensional geometry (commonly described as plane geometry) are abstract representations of the real world. When you hold up an attribute block that is the shape of a rectangle and say, “This two-dimensional shape is a rectangle,” you are not actually correct. The shape you are holding is actually three-dimensional 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 kindergarten through grade 2, topics in measurement are primarily related to linear measurement, time, and money. Contexts related to measurement offer opportunities for students to engage in problem solving in real-world contexts.

The initial task in this chapter (see figure 6.1) provides an opportunity for you to engage in problem solving related to linear measurement, time, and money, respectively—each of the primary measurement topics in K–2. Be sure to solve each problem before proceeding. While solving the task, think about how you might use the problems, or modify the problems, with your students.

1. Sergei and Lois each measure the length of the same object. Sergei says the length is 5 units. Lois says the length is 15 units. How can they both be correct?

2. What time is it when the hour hand is on the 24th minute mark?

3. Jackie had 45 cents in her purse, but she has no dimes. What coins could she have in her purse? What process did you use to determine the coins?

 

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 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

ePub

 

Appendix B: Completed Diagram for Classifying Quadrilaterals

ePub

 

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