![Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong Corwin Mathematics Series](/cover_863768.jpg "Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong")
1.3 MULTIPLICATION AND DIVISION PROBLEM SITUATIONS
Note: These representations for the problem situations reflect our understanding based on a number of resources. These include the tables in the Common Core State Standards for mathematics (Common Core Standards Initiative, 2010); the problem situations as described in the Cognitively Guided Instruction research (Carpenter, Hiebert, & Moser, 1981), in Heller and Greeno (1979), and in Riley, Greeno, and Heller (1984); and other tools. See the Appendix and the companion website for a more detailed summary of the documents that informed our development of these tables.
These problem structures are seldom if ever identified in middle-grades standards. They are typically addressed in the early elementary grades as students master basic whole number operations, and taken as known from there. Many of the challenges middle-grades students have with word problems may be rooted in a lack of familiarity with the problem structures, so it is helpful for middle school math teachers to understand them and recognize them within a word problem. We open each chapter in this book with a look at the new problem situation structure with positive rational numbers (whole numbers, fractions, and decimals); the second part of each chapter examines the same structure when the full range of values (positive and negative) are included.
In the chapters—each of which corresponds to a particular problem situation and a row on one of the tables—we walk you through a problem-solving process that enhances your understanding of the operation and its relationship to the problem situation while modeling the kinds of questions and explorations that can be adapted to your instruction and used with your students. Our goal is not to have students memorize each of these problem types or learn specific procedures for each one. Rather, our goal is to help you enhance your understanding of the structures and make sure your students are exposed to and become familiar with them. This will support their efforts to solve word problems with understanding—through mathematizing.
In each chapter, you will have opportunities to stop and engage in your own problem solving in the workspace provided. We end each chapter with a summary of the key ideas for that problem situation and some additional practice that can also be translated to your instruction.
Exploring in the Mathematizing Sandbox: A Problem-Solving Model
To guide your instruction and even enhance your own capacities for problem solving, we have developed a model for solving word problems that puts the emphasis squarely on learning to mathematize (Figure 1.4). The centerpiece of this model is what we call the “mathematizing sandbox,” and we call it this for a reason. The sandbox is where children explore and learn through play. Exploring, experiencing, and experimenting by using different representations is vital not only to developing a strong operation sense but also to building comfort with the problem-solving process. Sometimes it is messy and slow, and we as teachers need to make room for it. We hope that this model will be your guide.
FIGURE 1.4 A MODEL FOR MATHEMATIZING WORD PROBLEMS
The mathematizing sandbox involves three steps and two pauses:
Step 1 (Enter): Students’ first step is one of reading comprehension. Students must understand the words and context involved in the problem before they can really dive into mathematical understanding of the situation, context, quantities, or relationships between quantities in the problem.
Pause 1: This is a crucial moment when, rather than diving into an approach strategy, students make a conscious choice to look at the problem a different way, with a mind toward reasoning and sense-making about the mathematical story told by the problem or context. You will notice that we often suggest putting the problem in your own words as a way of making sense. This stage is critical for moving away from the “plucking and plugging” of numbers with no attention to meaning that we so often see (SanGiovanni & Milou, 2018).
Step 2 (Explore): We call this phase of problem solving “stepping into the mathematizing sandbox.” This is the space in which students engage their operation sense and play with some of the different representations mentioned earlier, making translations between them to truly understand what is going on in the problem situation. What story is being told? What are we comparing, or what action is happening? What information do we have, and what are we trying to find out? This step is sometimes reflected in mnemonics-based strategies such as STAR (stop, think, act, review) or KWS (What do you know? What do you want to know? Solve it.) or Pólya’s (1945) four steps to problem solving (understand, devise a plan, carry out a plan, look back) or even CUBES. But it can’t be rushed or treated superficially. Giving adequate space to the Explore phase is essential to the understanding part of any strategic approach. This is where the cognitive sweet spot can be found, and this step is what the bulk of this book is about.
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):
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
