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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.

      Mathematical 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, , 1,000).

      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

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