Читать книгу Mathematize It! [Grades 6-8] - Kimberly Morrow-Leong - Страница 19

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 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.