![Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong Corwin Mathematics Series](/cover_863768.jpg "Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong")
representation: A depiction of a mathematical situation using one or more of these modes or tools: concrete objects, pictures, mathematical symbols, context, or language.
Verbal
A problem may start with any mode of representation, but a word problem is first presented verbally, typically in written form. After that, verbal representations can serve many uses as students work to understand the actions and relationships in the problem situation. Some examples are restating the problem; thinking aloud; describing the math operations in words rather than symbols; and augmenting and explaining visual and physical representations including graphs, drawings, base 10 blocks, fraction bars, or other concrete items.
Contextual
The contextual representation is simply the real-life situation that the problem describes. Prepackaged word problems are based on real life, as is the earlier flashlight problem, but alone they are not contextual. Asking students to create their own word problems based on real-life contexts will bring more meaning to the process and will reflect the purposes of mathematics in real life, such as when scientists, business analysts, and meteorologists mathematize contextual information in order to make predictions that benefit us all. This is a process called mathematical modeling, which Garfunkel and Montgomery (2019) define as the use of “mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena.”
Mathematical modeling: A process that uses mathematics to represent, analyze, make predictions, or otherwise provide insight into real- world phenomena.
Concrete
Using physical representations such as blocks, concrete objects, and real-world items (for example, money, measuring tools, or items to be measured such as beans, sand, or water), or acting out the problem in various ways, is called modeling. Such models often offer the closest and truest representation of the actions and relationships in a problem situation. Even problem situations where negative quantities are referenced can rely on physical models when a feature such as color or position of an object shows that the quantity should be interpreted as negative.
Modeling: Creating a physical representation of a problem situation.
Pictorial
Pictures and diagrams can illustrate and clarify the details of the actions and relationships in ways that words and even physical representations cannot. Using dots and sticks, bar models, arrows to show action, number lines, and various graphic organizers helps students see and conceptualize the nature of the actions and relationships.
Symbolic
Symbols can be operation signs (+, −, ×, ÷), relational signs (=, <, >), variables (typically expressed as x, y, a, b, etc.), or a wide variety of symbols used in middle school and in later mathematics (k, ∞, ϕ, π, etc.). Even though numerals are more familiar, they are also symbols representing values (2, 0.9,
There are two things to know about representations that may be surprising. First, mathematics can be shared only through representations. As a matter of fact, it is impossible to share a mathematical idea with someone else without sharing it through a representation! If you write an equation, you have produced a symbolic representation. If you describe the idea, you have shared a verbal representation. Representations are not solely the manipulatives, graphs, pictures, and drawings of a mathematical idea: They are any mode that communicates a mathematical idea between people.
Second, the strength and value of learning to manipulate representations to explore and solve problems is rooted in their relationship to one another. In other words, the more students can learn to move deftly from one representation to another, translating and/or combining them to fully illustrate their understanding of a problem, the deeper will be their understanding of the operations. Figure 1.1 reveals this interdependence. The five modes of representation are all equally important and deeply interconnected, and they work synergistically. In the chapters that follow, you will see how bringing multiple and synergistic representations to the task of problem solving deepens understanding.
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
