Математическая грамотность. Минск: рикз, 2020. 252 с
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| Understanding quantity, number systems and their algebraic properties
49. The basic notion of quantity may be the most pervasive and essential mathematical aspect of engaging with, and functioning in, the world (OECD, 2017, p. 18 [22] ). At the most basic level it deals with the useful ability to compare cardinalities of sets of objects. The ability to count usually involves rather small sets – in most languages, only a small subset of numbers have names. When we assess larger sets, we engage in more complex operations of estimating, rounding and applying orders of magnitude. Counting is very closely related to another fundamental operation of classifying things, where the ordinal aspect of numbers emerges. Quantification of attributes of objects (measurement), relationships, situations and entities in the world is one of the most basic ways of conceptualising the surrounding world (OECD, 2017 [22] ). 50. Understanding quantity, number systems and their algebraic properties includes the basic concept of number, nested number systems (e.g., whole numbers to integers to rationals to reals), 181 the arithmetic of numbers, and the algebraic properties that the systems enjoy. In particular, it is useful to understand how progressively more expansive systems of numbers enable the solution of progressively more complicated equations. This lays the foundation for enabling students to see more evidence of mathematics in the real world in as they learn more mathematics. 51. To use quantification efficiently, one has to be able to apply not just numbers, but the number systems. Numbers themselves are of limited relevance; what makes them into a powerful tool are the operations that we can perform with them. As such, a good understanding of the operations of numbers is the foundation of mathematical reasoning. 52. It is also important to understand matters of representation (as symbols involving numerals, as points on a number line, as geometric quantities, and by special symbols such as π) and how to move between them; the ways in which these representations are affected by number systems; the ways in which algebraic properties of these systems are relevant and matter for operating within the systems; and the significance of the additive and multiplicative identities, associativity, commutativity, and the distributive property of multiplication over addition. Algebraic principles undergird the place value system, allowing for economical expression of numbers and efficient approaches to operations on them. They are also central to number-line based operations with numbers, including work with additive inverses that are central to addition and subtraction of first integers, then rationals and finally reals. 53. The centrality of number as a key concept in all the other mathematical areas under consideration here and to mathematical reasoning itself, is undeniable. Students’ grasp of the algebraic principles and properties first experienced through work with numbers is fundamental to their understanding of the concepts of secondary school algebra, along with their ability to become fluent in the manipulations of algebraic expressions necessary for solving equations, setting up models, graphing functions, and programing and making spreadsheet formulas. And in today’s data-intensive world, facility with interpretation of patterns of numbers, comparison of patterns, and other numerical skills are evolving in importance. 54. A broad understanding of quantity and number systems supports reasoning in the real-world applications of mathematics envisaged by this framework. жүктеу/скачать 7,1 Mb. Достарыңызбен бөлісу:
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