Blackboard for Problem Solving with C++

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report : adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

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This approach defines five problem solving steps you can use for most problems...
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However, in order to be useful, reductions must be . For example, it's quite possible to reduce a difficult-to-solve problem like the to a trivial problem, like determining if a number equals zero, by having the reduction machine solve the problem in exponential time and output zero only if there is a solution. However, this does not achieve much, because even though we can solve the new problem, performing the reduction is just as hard as solving the old problem. Likewise, a reduction computing a can reduce an to a decidable one. As Michael Sipser points out in : "The reduction must be easy, relative to the complexity of typical problems in the class [...] If the reduction itself were difficult to compute, an easy solution to the complete problem wouldn't necessarily yield an easy solution to the problems reducing to it."

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Therefore, the appropriate notion of reduction depends on the complexity class being studied. When studying the complexity class and harder classes such as the , are used. When studying classes within P such as and , are used. Reductions are also used in to show whether problems are or are not solvable by machines at all; in this case, reductions are restricted only to .

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Student Value Edition for Problem Solving with C++

The current model ignores several important features, including communication and learning. Our perspective-heuristic framework could be used to provide microfoundations for communication costs. Problem solvers with nearly identical perspectives but diverse heuristics should communicate with one another easily. But problem solvers with diverse perspectives may have trouble understanding solutions identified by other agents. Firms then may want to hire people with similar perspectives yet maintain a diversity of heuristics. In this way, the firm can exploit diversity while minimizing communication costs. Finally, our model also does not allow problem solvers to learn. Learning could be modeled as the acquisition of new perspectives and heuristics. Clearly, in a learning model, problem solvers would have incentives to acquire diverse perspectives and heuristics.

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Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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The main result of this paper provides conditions under which, in the limit, a random group of intelligent problem solvers will outperform a group of the best problem solvers. Our result provides insights into the trade-off between diversity and ability. An ideal group would contain high-ability problem solvers who are diverse. But, as we see in the proof of the result, as the pool of problem solvers grows larger, the very best problem solvers must become similar. In the limit, the highest-ability problem solvers cannot be diverse. The result also relies on the size of the random group becoming large. If not, the individual members of the random group may still have substantial overlap in their local optima and not perform well. At the same time, the group size cannot be so large as to prevent the group of the best problem solvers from becoming similar. This effect can also be seen by comparing . As the group size becomes larger, the group of the best problem solvers becomes more diverse and, not surprisingly, the group performs relatively better.