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Mathematize It! [Grades 6-8]. Kimberly Morrow-Leong
Читать онлайн.Название Mathematize It! [Grades 6-8]
Год выпуска 0
isbn 9781071819876
Автор произведения Kimberly Morrow-Leong
Жанр Учебная литература
Серия Corwin Mathematics Series
Издательство Ingram
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Teaching Students to Mathematize
As we discussed earlier, learning to mathematize word problems to arrive at solutions requires time devoted to exploration of different representations with a focus on developing and drawing on a deep understanding of the operations. We recognize that this isn’t always easy to achieve in a busy classroom, hence, the appeal of the strategies mentioned at the beginning of the chapter. But what we know from our work with teachers and our review of the research is that, although there are no shortcuts, structuring exploration to focus on actions and relationships is both essential and possible. Doing so requires three things:
1 Teachers draw on their own deep understanding of the operations and their relationship to different word problem situations to plan instruction.
2 Teachers use a model of problem solving that allows for deep exploration.
3 Teachers use a variety of word problems throughout their units and lessons, to introduce a topic and to give examples during instruction, not just as the “challenge” students complete at the end of the chapter.
In this book we address all three.
Building Your Understanding of the Operations and Related Problem Situations
The chapters that follow explore the different operations and the various kinds of word problems—or problem situations—that arise within each. To be sure that all the problems and situational contexts your students encounter are addressed, we drew on a number of sources, including the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010), the work done by the Cognitively Guided Instruction projects (Carpenter, Fennema, & Franke, 1996), earlier research, and our own work with teachers to create tables, one for addition and subtraction situations (Figure 1.2) and another for multiplication and division situations (Figure 1.3). Our versions of the problem situation tables represent the language we have found to resonate the most with teachers and students as they make sense of the various problem types, while still accommodating the most comprehensive list of categories. These tables also appear in the Appendix at the end of the book.
NOTES
FIGURE 1.2 ADDITION AND SUBTRACTION PROBLEM SITUATIONS
Situation charts are available for download at http://resources.corwin.com/problemsolving6-8
FIGURE 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.