Математическая грамотность. Минск: рикз, 2020. 252 с
жүктеу/скачать 7,1 Mb. Pdf көрінісі
|
2-ex pisa
| Appreciating the power of abstraction and symbolic representation
55. The fundamental ideas of mathematics have arisen from human experience in the world and the need to provide coherence, order, and predictability to that experience. Many mathematical objects model reality, or at least reflect aspects of reality in some way. However, the essence of abstraction in mathematics is that it is a self-contained system, and mathematical objects derive their meaning from within that system. Abstraction involves deliberately and selectively attending to structural similarities between mathematical objects, and constructing relationships between those objects based on these similarities. In school mathematics, abstraction forms relationships between concrete objects, symbolic representations and operations including algorithms and mental models. This ability also plays a role in working with computational devices. The ability to create, manipulate, and draw meaning in working with abstractions in technological contexts in an important computational thinking skill. 56. For example, children begin to develop the concept of “circle” by experiencing specific objects that lead them to an informal understanding of circles as being “roundish”. They might draw circles to represent these objects, noticing similarities between the drawings to generalise about “roundness” even though the circles are of different sizes. “Circle” becomes an abstract mathem atical object when students start to “use” circles as objects in their work and more formally when it is defined as the locus of points equidistant from a fixed point in a two-dimensional plane. 182 57. Students use representations – whether text-based, symbolic, graphical, numerical, geometric or in programming code – to organise and communicate their mathematical thinking. Representations enable us to present mathematical ideas in a succinct way which, in turn, lead to efficient algorithms. Representations are also a core element of mathematical modelling, allowing students to abstract a simplified or idealised formulation of a real world problem. Such structures are also important for interpreting and defining the behaviour of computational devices. 58. Having an appreciation of abstraction and symbolic representation supports reasoning in the real-world applications of mathematics envisaged by this framework by allowing students to move from the specific details of a situation to the more general features and to describe these in an efficient way. Seeing mathematical structures and their regularities 59. When elementary students see: 5 + (3 + 8) some see a string of symbols indicating a computation to be performed in a certain order according to the rules of order of operations; others see a number added to the sum of two other numbers. The latter group are seeing structure; and because of that they don’t need to be told about the order of operation, because if you want to add a number to a sum you first have to compute the sum. 60. Seeing structure continues to be important as students move to higher grades. A student who sees f ( 𝑥 ) = 5 + ( 𝑥 − 3)2 as saying that f ( x ) is the sum of 5 and a square which is zero when x = 3 understands that the minimum of f is 5. This lays the foundation for functional thinking discussed in the next section. 61. Structure is intimately related to symbolic representation. The use of symbols is powerful, but only if they retain meaning for the symboliser, rather than becoming meaningless objects to be rearranged on a page. Seeing structure is a way of finding and remembering the meaning of an abstract representation. Such structures are also important for interpreting and defining the behaviour of computational devices. Being able to see structure is an important conceptual aid to procedural knowledge. 62. The examples above illustrate how seeing structure in abstract mathematical objects is a way of replacing parsing rules, which can be performed by a computer, with conceptual images of those objects that make their properties clear. An object held in the mind in such a way is subject to reasoning at a level that is higher than simple symbolic manipulation. 63. A robust sense of mathematical structure also supports modelling. When the objects under study are not abstract mathematical objects, but rather objects from the real world to be modelled by mathematics, then mathematical structure can guide the modelling. Students can also impose structure on non-mathematical objects in order to make them subject to mathematical analysis. An irregular shape can be approximated by simpler shapes whose area is known. A geometric pattern can be understood by hypothesising translational, rotational, or reflectional transformations and symmetry and abstractly extending the pattern into all of space. Statistical analysis is often a matter of imposing a structure on a set of data, for example by assuming it comes from a normal distribution or supposing that one variable is a linear function of another, but measured with normally distributed error. 64. жүктеу/скачать 7,1 Mb. Достарыңызбен бөлісу:
|