Читать книгу Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong - Страница 12

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.