![Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong Corwin Mathematics Series](/cover_863768.jpg "Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong")
the question
Eliminate unnecessary information
Solve and check
She tells you that her teachers walked students through the CUBES protocol using a “think-aloud” for word problems, sharing how they used the process to figure out what is important in the problem. That evening, as you settle down to plan, you decide to walk through some problems like the history assignment problem using CUBES. Circling the numbers is easy enough. You circle
Then you tackle “important information.” What is important here in this problem? Maybe the fact that there are two different assignments. Certainly it’s important to recognize that students do one of two kinds of history assignment. You box the question, but unfortunately the question doesn’t help students connect
If you think this procedure has promise as a way to guide students through an initial reading of the problem, but leaves out how to help students develop a genuine understanding of the problem, you would be correct.
What is missing from procedural strategies such as CUBES and strategies such as key words, is—in a word—mathematics and the understanding of where it lives within the situation the problem is presenting. Rather than helping students learn and practice quick ways to enter a problem, we need to focus our instruction on helping them develop a deep understanding of the mathematical principles behind the operations and how they are expressed in the problem. They need to learn to mathematize.
What Is Mathematizing? Why Is It Important?
Mathematizing is the uniquely human process of constructing meaning in mathematics (from Freudenthal, as cited in Fosnot & Dolk, 2002). Meaning is constructed and expressed by a process of noticing, exploring, explaining, modeling, and convincing others of a mathematical argument. When we teach students to mathematize, we are essentially teaching them to take their initial focus off specific numbers and computations and put their focus squarely on the actions and relationships expressed in the problem, what we will refer to throughout this book as the problem situation. At the same time, we are helping students see how these various actions and relationships can be described mathematically and the different operations that can be used to express them. If students understand, for example, that equal-groups multiplication problems, as in the history assignment problem, may include knowing the whole or figuring out the whole from a portion, then they can learn where and how to apply an operator to numbers in the problem, in order to develop an appropriate equation and understand the context. If we look at problems this way, then finding a solution involves connecting the problem’s context to its general kind of problem situation and to the operations that go with it. The rest of the road to the answer is computation.
Mathematizing: The uniquely human act of modeling reality with the use of mathematical tools and representations.
Problem situation: The underlying mathematical action or relationship found in a variety of contexts. Often called “problem type” for short.
Solution: A description of the underlying problem situation along with the computational approach (or approaches) to finding an answer to the question.
Making accurate and meaningful connections between different problem situations and the operations that can fully express them requires operation sense. Students with a strong operation sense
Operation sense: Knowing and applying the full range of work for mathematical operations (for example, addition, subtraction, multiplication, and division).
Understand and use a wide variety of models of operations beyond the basic and intuitive models of operations (Fischbein, Deri, Nello, & Marino, 1985)
Use appropriate representations of actions or relationships strategically
Apply their understanding of operations to any quantity, regardless of the class of number
Can mathematize a situation, translating a contextual understanding into a variety of other mathematical representations
Intuitive model of an operation: An intuitive model is “primitive,” meaning that it is the earliest and strongest interpretation of what an operation, such as multiplication, can do. An intuitive model may not include all the ways that an operation can be used mathematically.
Focusing on Operation Sense
Many of us may assume that we have a strong operation sense. After all, the four operations are the backbone of the mathematics we were taught from day one in elementary school. We know how to add, subtract, multiply, and divide, don’t we? Of course we do. But a closer look at current standards reveals nuances and relationships within these operations that many of us may not be aware of, may not fully understand, or may have internalized so well that we don’t recognize we are applying an understanding of them every day when we ourselves mathematize problems both in real life and in the context of solving word problems. For example, current standards ask that students develop conceptual understanding and build procedural fluency in four kinds of addition/subtraction problems, including Add-To, Take-From, Compare, and what some call Put Together/Take Apart (we will refer to this category throughout the book as Part-Part-Whole). Multiplication and division have their own unique set of problem types as well. On the surface, the differences between such categories may not seem critical. But we argue that they are. Only by exploring these differences and the relationships they represent can students develop the solid operation sense that will allow them to understand and mathematize word problems and any other problems they are solving, whatever their grade level or the complexity of the problem. It does not mean that students should simply memorize the problem types. Instead they should have experience exploring all the different problem types through word problems and other situations. Operation sense is not simply a means to an end. It has value in helping students naturally come to see the world through a mathematical lens.
Using Mathematical Representations
What would such instruction—instruction aimed at developing operation sense and learning how to mathematize word problems—look like? It would have a number of features. First, it would require that we give students time to focus and explore by doing fewer problems, making the ones they do count. Next, it would facilitate students becoming familiar with various ways to represent actions and relationships presented in a problem context. We tend to think of solving word problems as beginning with words and moving toward the use of variables and equations in a neat linear progression. But as most of us know, this isn’t how problem solving works. It is an iterative and circular process, where students might try out different representations, including going back and rewording the problem, a process we call telling “the story” of the problem. The model that we offer in this book is based on this kind of active and expanded exploration using a full range of mathematical representations. Scholars who study mathematical modeling and problem solving identify five modes of representation: verbal, contextual, concrete, pictorial, and symbolic representations (Lesh, Post, & Behr, 1987).
Problem context: The specific setting for a word problem.
