37 Chapters
Medium 9781942496489

Appendix: Hypothetical Weight Loss Study Data

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

Appendix: Hypothetical Weight Loss Study Data

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Chapter 3 Fraction Concepts

Juli K. Dixon Solution Tree Press ePub

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 15 of each cookie.

•Student B: Each person will receive 420 of all the cookies.

•Student C: Each person will receive 45 of a cookie.

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Introduction

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

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.

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Epilogue Next Steps

Juli K. Dixon Solution Tree Press ePub

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.

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Chapter 5 Geometry

Juli K. Dixon Solution Tree Press ePub

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.

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