Mathematical problem
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Mathematical problem may mean two slightly different things, both closely related to mathematical games:
- general meaning
- a question that can be answered with the help of mathematics
- formal meaning
- any tuple (S, C( ), r), where S is a set of objects, C(s) is a set of conditions such that for every s∈S (every element s of S), C(s) is either true or false (in other words, that C(s) is well formed). The solution to the problem is a series of steps (an algorithm) that leads to finding r∈S that satisfies C(r).
In the formal meaning, a problem is a mathematical object, unlike in the informal meaning.
Informal mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 - 3", even if one knows the mathematics required to solve the problem. They are used on all levels of mathematics education to teach students to connect real world situations to the abstract language of mathematics, that is, to think logically.
Solving a mathematical problem involves three basic steps that are common to all types:
- Comprehend what the problem is asking.
- Abstract the problem, find a mathematical expression that represents the problem, and solve it.
- Understand what the solution means and implicates.
More theoretical mathematical problems are questions about mathematical objects and structures, requiring explanation and proof. They include specific questions like "What are the real number solutions of x2 − 1 = 0?" and more general tasks such as "Formulate and prove a conjecture about why these numbers appear in apparently different situations."
