A framework for research into students' problem posing in mathematics classrooms could be developed around the classification shown in Table 1, and around the three principles listed in the section on structured problem-posing situations. However, the author believes that there is much to be gained by invoking the ideas of Krutetskii (1976), and in particular, by extending Krutetskii's problem-solving categories into the realm of problem posing. Note that, although Krutetskii (1976) did draw attention to the value of problem posing, his emphasis was on linking students' mathematical abilities with their problem-solving processes.

Classification of Problem-posing Situations

Structure	Problem-posing Situations
Free (The situation is not defined)	Every-day-life situations; Free problem posing; "Problems I like"; Problems for a maths competition; Problems written for a friend.
Semi-structured (An open situation is given)	Open-ended problems (ie mathematical investigations); Problems similar to a given problem; Problems with similar solutions; Problems related to specific theorems; Problems derived from given pictures; Word problems.
Structured (The design is based on a specific problem )	Problem-posing situations for understanding the formal structures of the problem; Problem-posing situations for understanding the formal structures of the solution.

In fact, although Krutetskii's (1976) major focus was problem solving, his insights into the relationship between problem solving and problem posing has inspired the author to reflect on how his ideas could be extended to embrace both problem solving and problem posing. This would mean that Krutetskii's problem-solving categories could be more readily applied by educators wishing to develop quality structured problem-posing situations in mathematics classrooms.

Stimuli for Potentially Rich Problem-posing Situations

A problem-posing situation on the basis of an unstated question

Students can create problems by omitting the question from given problems. Questions such as the following can be asked: "How can we finish the problem?", "Can we pose another question?" Or "Write down all problems you can pose related to this situation."

Example: The prime decomposition of a number is 2³ x 3⁵.

Questions posed by the students might be: How many factors does the number have? Which are they? Is this number a factor of 2⁴ x 3⁴?

A problem-posing situation on the basis of incomplete information being provided for a given problem

Open-ended questions (Clarke & Sullivan, 1991) can be regarded as belonging to this category. By omitting a numerical fact or a mathematical relationship from the statement of a problem, a new problem-posing situation can be created. Helpful directions for the students to start to work might be "How many solutions can you find?", "Write down everything you know from the facts provided," or "Write down some problems you can pose by adding some additional facts."

A problem-posing situation which incorporates surplus information with respect to a given problem

Students can be invited to modify well-defined problems so that the newly-posed problems incorporate surplus "irrelevant" information ("noise"). This process can help students to analyse the formal structure of a problem. Problem-posing situations can be created by asking students to change a conclusion or to add numerical information or mathematical relationships.

Example: Write the smallest number divisible by 35 and 55, which has exactly eight factors and its last digit is five.

A Year 8 student immediately noticed that if the number has to have exactly eight factors, then it is the smallest one and its last digit will be five.

A problem-posing situation which incorporates elements that can be interpreted in different ways

Problem-posing situations can be generated by combining two or more geometrical figures (for example) and then looking at the elements from different perspectives.

A problem-posing situation with varying vocabulary

Students can be encouraged to modify the wording of given problems so that equivalent problems are formed. This can help students to develop their understanding of the meanings of mathematical terms and symbols, and of interrelationships between mathematical synonyms.

Example: Find a number, which is 2 larger than the smallest number divisible by 3, 4 and 5. A variant of this problem proposed by one of the students was: "Which number is 2 larger the least common multiple of 3, 4 and 5 ?"

A problem-posing situation with varying semantic structure

A problem-posing situation can be designed by changing the semantic structure of a given problem.

Example: A Year 3 class was asked to create a problem which had the same numbers and the same answer as: "Mary had 7 mice and 3 of them ran away. How many did she have then?" One student suggested: "Mary had 7 mice. She had 3 more than Gina. How many mice did Gina have?"

Framework Conclusion