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Mathematics makes up that part of the human conceptual system that is special in the following way: It is precise, consistent, stable across time and human communities, symbolizable, calculable, generalizable, universally available, consistent within each of its subject matters, and effective as a general tool for description, explanation, and prediction in a vast number of everyday activities ...
[20] The Panel effectively called for an end to the Math Wars, concluding that research showed "conceptual understanding, computational and procedural fluency, and problem-solving skills are equally important and mutually reinforce each other. Debates regarding the relative importance of each of these components of mathematics are misguided."
The general consensus of large-scale studies that compare traditional mathematics with reform mathematics is that students in both curricula learn basic skills to about the same level as measured by traditional standardized tests, but the reform mathematics students do better on tasks requiring conceptual understanding and problem solving. [3]
The CESSM de-emphasised manual arithmetic in favor of students developing their own conceptual thinking and problem solving. The PSSM presents a more balanced view, but still has the same emphases. Mathematics instruction in this style has been labeled standards-based mathematics [1] or reform mathematics. [2]
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality. [1]
A concept definition is similar to the usual notion of a definition in mathematics, with the distinction that it is personal to an individual: "a personal concept definition can differ from a formal concept definition, the latter being a concept definition which is accepted by the mathematical community at large." [1]
Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines.
The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities.