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Making Sense of Mathematics for Teaching High School

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Develop a deep understanding of mathematics by grasping the context and purpose behind various strategies. This user-friendly resource presents high school 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. Combine student understanding of functions and algebraic concepts so that they can better decipher the world.

 

Benefits

 

  • Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics.
  • Explore how to develop, select, or 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.
  • Gain clarity about the most productive progression of mathematical teaching and learning for high school.
  • Watch short videos that show what classrooms that are developing mathematical understanding should look like.

 

Contents

 

Introduction

  1. Equations and Functions
  2. Structure of Equations
  3. Geometry
  4. Types of Functions
  5. Function Modeling
  6. Statistics and Probability

Epilogue: Next Steps

Appendix: Weight Loss Study Data

References

Index

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Introduction

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Introduction

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 High School, 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: Equations and Functions

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CHAPTER 1

Equations and Functions

What is the purpose of teaching functions in secondary mathematics? The concept of functions is one of the most important concepts in mathematics; it enables students to make connections within and across mathematical topics. Students need to understand patterns and the link to functional relationships, learning that mathematics is more than just manipulating algebraic expressions. As students use functions to represent relationships in multiple ways, they begin to recognize mathematical ideas in real-world situations and form notions of mathematical modeling (see chapters 5 and 6 for discussions on modeling). In order to emphasize the importance of exploring and connecting different types of functional relationships, different function types, such as linear and quadratic functions, and systems of linear equations are explored through real-world situations in this chapter and in chapter 2. Connections are made among verbal, numerical, graphical, and algebraic representations of mathematical concepts to illustrate the role and significance of multiple representations in understanding functions.

 

Chapter 2: Structure of Equations

ePub

CHAPTER 2

Structure of Equations

The initial focus of this chapter is solving systems of two linear equations. The discussion will then focus on exploring different forms of linear and quadratic functions and linking expressions and equations to their visual representations. This exploration sets the stage for later work with other types of functions.

The Challenge

The initial task in this chapter provides an application of linear functions (see figure 2.1). This application is an extension from the context presented in chapter 1 (see figure 1.1, page 16).

Figure 2.1: A race between a father and son.

In this task, the two linear graphs representing distance over time intersect at a single point, indicating that the system of equations has a unique solution. In high school, there has traditionally been an emphasis on two methods for finding the solution of a system of linear equations—substitution and elimination. These methods don’t always address the meaning of the solution to a system of equations. What does the solution mean within a given context? This chapter explores various methods to make sense of solving systems of equations to illustrate the mathematical reasoning behind why different methods work and how they are interconnected graphically and algebraically. What does the solution mean graphically? What does a solution to a system of equations mean algebraically? You may wish to consider these questions before reading the discussion on graphical and algebraic solutions of the initial task in the following paragraphs.

 

Chapter 3: Geometry

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CHAPTER 3

Geometry

Geometry often provides the first opportunity for a formal exploration of proof. Students make sense of and apply both inductive and deductive reasoning in this exploration. Inductive proof could be thought of as connecting to recursive thinking in the study of arithmetic sequences in algebra, while deductive thinking links more closely to determining explicit rules for those sequences. Thinking back to the bridges problem from chapter 2 (see figure 2.5, page 42), the recursive rule would be that the number of beams in the next bridge is equal to the number of beams in the current bridge with four additional beams added. The number of beams in each subsequent bridge is based on the previous example. An explicit rule for the number of beams in any bridge is not based on the number of beams in a prior bridge but is rather determined using the length of the bridge.

Specifically, inductive reasoning is using observations and examples to reach a conclusion. For example, when you recognize a pattern in a number of different isosceles triangles that the base angles are congruent, you may inductively conclude that base angles of isosceles triangles are congruent. However, this type of reasoning is always subject to finding an example that does not follow the pattern. For example, in the bridges problem if you only examine a bridge of length 4 (see figure 2.6, page 43), you might conclude that a rule for the number of beams in a bridge of any length is 3 + n(n − 1). However, upon further examination, you would see that this rule cannot be generalized to bridges of other lengths.

 

Chapter 4: Types of Functions

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CHAPTER 4

Types of Functions

How does the understanding of functions evolve as students experience different aspects of high school mathematics? As students transition from early coursework in high school mathematics to more advanced courses, they use their understanding of linear and quadratic functions (see chapters 1 and 2) to make sense of exponential, logarithmic, and rational functions, among others. As with earlier work with functions, these more complex functions are explored contextually, numerically, graphically, and algebraically. This chapter will focus on rate of change and other characteristics of various function types. Furthermore, the role of parameters in function transformations and the connections between functions and their inverses are examined, while highlighting the role of symmetry and geometric relationships. In chapter 5, tasks that use these mathematical concepts will be explored with a focus on modeling.

The Challenge

 

Chapter 5: Modeling With Functions

ePub

CHAPTER 5

Modeling With Functions

How does the understanding of various function types enable you to make sense of and mathematize the world around you? This chapter explores ways to integrate the understanding of functions and algebraic concepts to make sense of the world around you. Real-world situations provide the setting to investigate functions and to purposefully use mathematics to explain, interpret, and make predictions through the use of modeling with functions.

The Challenge

A critical part of learning mathematics involves connecting mathematical content to relevant contexts in order to explore and analyze situations. Mathematical Practice 4, “Model with mathematics,” focuses on this aspect of mathematizing the world by using mathematics to examine, predict, and gain greater insight into everyday phenomena. What are some different ways that you can make sense of data using the mathematics studied in high school? Consider the task in figure 5.1.

 

Chapter 6: Statistics and Probability

ePub

CHAPTER 6

Statistics and Probability

This chapter connects statistics and probability concepts to the reasoning and sense making you have explored from algebra and geometry. Some tasks will connect with the reasoning and justifying aspects of the previous chapters, such as reasoning how to make predictions using information from a real-world situation, justifying decisions made from the analysis of a given data set, or examining how sample surveys, experiments, and observational studies can be used as mathematical models. Other tasks will look at how the analysis of a data set connects to the shape of the data display and measures how well a function models the pattern of a data set.

By definition, statistics is the collection, analysis, and interpretation of data. Probability is the ratio of successful outcomes to the total number of outcomes. These two topics, along with the use of randomness, are important concepts in mathematical modeling and simulating real-world events.

 

Epilogue: Next Steps

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EPILOGUE

Next Steps

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: Hypothetical Weight Loss Study Data

ePub

Appendix: Hypothetical Weight Loss Study Data

 

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