Читать книгу Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong - Страница 24

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):

1 Subtraction: This symbol (−x), which children learn in elementary school, functions like a verb, an operator between the two values that come before and after the symbol.

2 Less than zero: In the middle grades, we introduce a symbol (−x) that distinguishes a negative from a positive number. In this case, the symbol functions more like an adjective. For example, the symbol in front of −5 describes a value that is 5 units less than zero. In contrast the symbol in front of +5 describes a value 5 units greater than zero.

3 The opposite: This use of the negative symbol (−x) conveys the idea of “the opposite,” or the additive inverse. In this respect, it toggles back and forth between positive and negative values. Reading −x as “the opposite of x” instead of as “negative x” communicates that −x represents the additive inverse of x. If the value of x is already negative, students are often confused by the outcome. For example, when x is −5, −x is the additive inverse of −5, or +5. How can a number that appears negative (−x) have a positive value (+5)?

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.