What's the problem?

Each investigation begins with a problem. In inquiry-based teaching the problem is sometimes called a launch problem.

A problem usually consists of two parts: a situation, and a question about that situation.

Where do problems originate?

They don't start in textbooks or web sites. Someone has to come up with them. How?

Some problems come from tradition. Many problems in algebra textbooks, for example, are found (with the words altered) in books 800 years old.

Many problems come from outside mathematics, from what's sometimes known as the real world.

Some problems arise out of curiosity about mathematical ideas. For example, you might ask What can I figure out about this new mathematical idea? Or, after adding lots of numbers together, a six-year-old might ask, How many ways can I add numbers to get 17? Or, after you've solved a particular mathematical problem, you might ask In general, how can I solve all problems like this? What other problems use this idea? What if  . . . ?

A very important kind of problem in mathematics arises from conjectures. A conjecture is a tentative claim, usually that an observed pattern will always hold true. A long tradition in mathematics requires that the truth of conjectures be established by deductive reasoning rather than by large amounts of data. So a common problem in mathematics begins How can I prove or disprove. . . ? in reference to a particular conjecture.