LITMIR.BIZ

      Pause 2: The exploration done in the mathematizing sandbox leads students to the “a-ha moment” when they can match what they see happening in the problem to a known problem situation (Figures 1.2 and 1.3). Understanding the most appropriate problem situation informs which operation(s) to use, but it also does so much more. It builds a solid foundation of operation sense.

      Step 3 (Express): Here students leave the sandbox and are ready to express the story either symbolically or even in words, graphs, or pictures, having found a solution they are prepared to discuss and justify.

      A Note About Negative Values

      Negative rational number values represent multiple challenges for students. The shortcuts and rules that are often taught can feel nonsensical or random, and students may have internalized ideas about computation that are now challenged. For example, students may still believe that addition and multiplication always make things bigger. This is not necessarily true, and that realization is a big cognitive transition for students to make.

      We know that integer computation is a challenging skill for many students to develop. It remains, even for some adults, a mystery of mathematics that equations like this one (⁻6 − ⁻8) with so many signs expressing a negative value, still yields a positive 2. After all, how can subtraction and two negative numbers possibly yield a positive result? For that matter, why does a negative multiplied by a negative give a positive product? However, our focus in this book is not on computation strategies but, rather, on making sense of problem situations.

      We firmly believe that if students reason about the problem situation, they can not only find a solution pathway, but they are more likely to understand where the answer comes from and why it’s correct. Further, a deeper understanding of the structure of the problem situations and operations better prepares them to engage in mathematical modeling now as well as in future mathematics classes and into adulthood.

      In each chapter, we will explore the problem situation first with fractions, decimals, and whole numbers. In the second half of each chapter, we introduce problem situations that include negative values. We also explore the symbols used in mathematics to describe a negative value. The negative symbol (−) actually has three different meanings (Stephan & Akyuz, 2012):

      Distinguishing among these three different uses of the negative symbol may help students recognize them in context and help them be more deliberate in their own use. Conventions about the use of negative numbers are not intuitive for students (Whitacre et al., 2014). They may initially use values and signs (magnitude and direction) in ways that make sense to them but that may or may not correspond to standard conventions (Kidd, 2007). The flashlight problem at the beginning of this chapter is typical. The student’s solution relied entirely on positive numbers and a subtraction operator to find the correct answer (10 − 2 = 8). This worked for the student likely because she recognized that the explorer never reached 0 to leave the cave. However, a more accurate equation for a problem situation that describes a descent and a climb out of a cave needs to include negative values to be accurate, as in ⁻10 + x = ⁻2. Does this matter? In this book we will make the case that it does matter. The incorrect equation given by this student may not be so much a “mistake” as it is a mistranslation of her understanding of the problem situation to a more accurate notation. We will return many more times to this idea of connecting the meaning of a problem situation to the various representations used to describe it.

      Final Words Before You Dive In

      We understand that your real life in a school and in your classroom puts innumerable demands on your time and energy as you work to address ambitious mathematics standards. Who has time to use manipulatives, draw pictures, and spend time writing about mathematics? Your students do! This is what meeting the new ambitious standards actually requires. It may feel like pressure to speed up and do more, but paradoxically, the way to build the knowledge and concepts that are currently described in the standards is by slowing down. Evidence gathered over the past 30 years indicates that an integrated and connected understanding of a wide variety of representations of mathematical ideas is one of the best tools in a student’s toolbox (or sandbox!) for a deep and lasting understanding of mathematics (Leinwand et al., 2014). We hope that this book will be a valuable tool as you make or renew your commitment to teaching for greater understanding.

      Descriptions of Images and Figures

       Back to Figure

      The figure shows the following five representations:

      Double-headed arrows indicate the interconnections of each representation with all the other representations individually.

      FIGURE 1.1 FIVE REPRESENTATIONS: A TRANSLATION MODEL

      Source: Adapted from Lesh, Post, and Behr (1987).

      DescriptionBack to Figure

      The figure shows “The Mathematizing Sandbox” which consists of the following steps: