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Reasoning: Mathematical Reasoning
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Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts and from the earliest grades. At all levels, students reason inductively from patterns and specific cases. For example, even a first grader can use an informal proof by contradiction to argue that the number 0 is even: "If 0 were odd, then 0 and 1 would be two odd numbers in a row. But even and odd numbers alternate. So 0 must be even."
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Mathematical Reasoning™ helps your child devise strategies to solve a wide variety of math problems. These books emphasize problem solving and computation to build the math reasoning skills necessary for success in higher level math and math assessments. All books are written to the standards of the National Council of Teachers of Mathematics. Core or Supplemental.
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Mathematics teachers often claim that all types of critical thinking and problem solving are really examples of mathematical reasoning. But employers have a different view, rooted in a paradox: graduates with degrees in mathematics or computer science are often less successful than other graduates in solving the kinds of problems that arise in real work settings. Often students trained in mathematics tend to seek precise or rigorous solutions regardless of whether the context warrants such an approach. For employers, this distinctively "mathematical" approach is frequently not the preferred means of solving most problems arising in authentic contexts. Critical thinking and problem solving about the kinds of problems arising in real work situations is often better learned in other subjects or in integrative contexts [Brown, 1995].
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Without an explicit mathematical model that abstractly specifies the state of a List object and the behavior of List operations, reasoning about List objects is reduced to speculation and guesswork. How should objects and their operations be explained, given that a basic objective of software engineering is to be able to reason about and understand the software? The rest of this article illustrates an answer to this fundamental question using the List example. The issue at hand is one that you must address no matter which programming language or paradigm you use. But it is especially important for component-based software development, where source code for the components used often is not available to the client programmer.
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Experienced teachers know that knowledge and performance are not reliable indicators of either reasoning or understanding. Deep understanding must be well-connected. In contrast, superficial understanding is inert, useful primarily in carefully prescribed contexts such as those found in typical mathematics classrooms [Glaser, 1992]. Persons with well-connected understanding attach importance to different patterns and are better able to engage in mathematical reasoning. Moreover, students with different levels of skills may be equally able to address tasks requiring more sophisticated mathematical reasoning [Cai, 1996].
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Component-based software development aggravates the reasoning problem because it significantly widens the "semantic gap" between the kinds of real-world information you can write programs to process, and the bits that computer hardware ultimately is able to process. Appropriate mathematical models have long since been adopted for the built-in types provided by programming languages. But in component-based software development you use not only these built-in types -- which are one or two levels removed from the hardware -- but ... much higher-level types defined by off-the-shelf software components with powerful operations whose exact behavior can be complex and even mysterious. What are appropriate mathematical models for these types?
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Mathematical Reasoning