A question about formulation of problems in mathematics

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In summary, the conversation discusses the ability of mathematics professors to formulate challenging problems. While any professor should be able to come up with difficult examples that can be solved with the methods shown in textbooks, creating puzzles or famous unsolved problems may require a special skill that not all professors possess. It is suggested that the ability to formulate such problems may come from a deeper understanding of the field or simply by combining regular problems to create a multi-step problem.
  • #1
flamengo
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I have a simple question. Is a mathematics professor able to formulate more difficult problems than the problems that are in the books ? If so, how difficult is it to create such problems ? Is this a difficult task ? Or is this something easy that comes naturally for a professor ? Could someone give me a detailed explanation ?
 
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Problems in textbooks should be examples that can be solved with the methods shown in the textbook. Any professor should be able to think up difficult examples like that if he wanted to.
Problems in puzzle books may be a lot harder, but should be solvable. The method of solution may not be easy to discover. Puzzles take a special skill that even a very good professor might not have.
Problems in other math books may be famous unsolved problems. It would be extremely difficult for any professor the make up a truly new unsolvable problem. They would become famous.
 
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How do professors develop this ability of formulating problems that are more challenging ?
 
  • #4
Enybody can create a problem that he/she cannot solve or must work hard to solve it. When this is done by professor, then it is a challenging problem almost by definition.
 
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flamengo said:
Could someone give me a detailed explanation ?
Just try to make up a problem that you cannot solve. The things you do during this try, exactly the same is done by the professor (except at a higher level).
 
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  • #6
I think that, given a certain field in mathematics, it sometimes requires a deeper understanding of said field to formulate problems that are both challenging and interesting (this is not the same thing, i.m.o.) than to solve them.

Probably this is less valid at the true research level, but I think it is certainly true at the (elementary, intermediate as well as advanced) textbook level.
 
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  • #7
Take some regular problems and combine them to make one that takes two or three steps to solve. It's easy.
 
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What is the importance of formulating problems in mathematics?

Formulating problems in mathematics is crucial because it helps to clearly define the problem at hand, identify the necessary information and variables, and develop a systematic approach to finding a solution. It also allows for a more organized and efficient problem-solving process.

How do you formulate a problem in mathematics?

To formulate a problem in mathematics, you need to first identify the question or issue that needs to be solved. Then, define the problem by clearly stating what is known and what is unknown. Next, determine the relevant variables and relationships between them. Finally, develop a strategy or plan for solving the problem.

What are some common challenges in formulating problems in mathematics?

One common challenge in formulating problems in mathematics is identifying the underlying concepts and principles that are relevant to the problem. Another challenge is determining the appropriate level of complexity for the problem, as too simple or too complex problems can hinder the problem-solving process.

How can formulating problems in mathematics help in other areas of study?

The process of formulating problems in mathematics involves critical thinking and problem-solving skills that can be applied to other areas of study. It also helps to develop logical reasoning and analytical skills, which are valuable in various fields such as science, engineering, and computer science.

Can problem formulation in mathematics lead to multiple solutions?

Yes, problem formulation in mathematics can lead to multiple solutions. Depending on the complexity of the problem, there can be more than one approach or method to finding a solution. It is important to carefully consider all possible solutions and evaluate their validity and efficiency.

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