![Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong Corwin Mathematics Series](/cover_863768.jpg "Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong")
style="font-size:15px;"> 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.
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:
Pictorial
Symbolic
Verbal
Contextual
Concrete
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).
The figure shows “The Mathematizing Sandbox” which consists of the following steps:
1 EnterUnderstand the words: Students focus on reading comprehension of words and quantities.Light bulb icon with a question mark inside: Students pause and answer, “What is the story in this problem?”
2 ExploreAn image shows three cogwheels labeled as Engage Operation Sense, Represent, and Translate. An arrow over each cogwheel indicates that Engage Operation Sense rotates anti-clockwise and is connected to Represent, Represent rotates clockwise and is connected to Translate, and Translate rotates anti-clockwise. Under, it says “Students focus on mathematical comprehension.”
3 ExpressLight bulb icon with an exclamation mark inside: Students pause to identify a problem structure that fits the story.Show a solution: Students show and justify a solution.
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