The phrase ‘problem solving’ has different meanings in mathematics education. Even its role in mathematics teaching and learning is not clear cut. Some view problem solving as an end in itself. Others see it as starting point for learning. Nevertheless, here are some of the types of problem solving tasks we would see in textbooks and teaching. They are arranged according to cognitive demand. Remember that learners will only consider something a problem if they experience some sort of barrier in a situation they are curious about. There’s a difference between an exercise and a problem.
1. Word problems with arithmetical steps. These are mainly use in teaching mathematics to elementary school learners. The purpose is either to introduce elementary concepts using learners informal knowledge in contextual situations or to provide a situation where learners can apply their knowledge of operation. You might find The Little Green Math Book: 30 Powerful Principles for Building Math and Numeracy Skills (2nd Edition)
2. Worded contexts which require the learner to decide to use standard techniques. Majority of textbooks problems are of this type. For example:
The area of a triangular lawn is 20 square metres, and one side is 5 metres long. If I walk in straight line from the corner opposite this side, to meet it at right angles, how far have I walked?
The standard technique here is of course the recognition of the base and height of the triangle in the context used in the problem and knowledge of the area of a triangle. You can find additional problems in K-12 Math Problems.
3. Worded context in which there is no standard relationship to apply, or algorithm to use, but an answer is expected. These kind of problems usually requires setting up an equation or thinking of using specific cases or perhaps representing the problem using other representation such as graph. For example:
One side of the rectangle is reduced in length by 20%, the other side is increased by 20%; what change take place in the area?
These type of problems are best use for teaching a concept or new procedure. Here are two samples: Teaching the properties of equality via problem solving and Teaching Trigonometry through problem solving. They usually have multiple solutions or multiple correct answers.
4. Exploratory situations in which there is an ill-defined problem. Because the problem was set-up to be ill-defined, it requires the learner to mathematise by identifying variables and conjecturing relationships, choosing representations and techniques. Knowledge of a range of function is useful. Note that sometimes there may be no solution to the problem. For example:
Describe the advantages and disadvantages of raising the price of cheese rolls at the school tuck shop by 5 cents, given that cheese prices have gone down by 5% but rolls have gone up by 6 cents each.
I would include math investigations in this category.
5. Mathematical problems in which a situation is presented and a question posed for this has no obvious method. Problems in this category would be what mathematicians would call ‘problem’ since the expected line of attack is to use the forms of inquiry and mathematical thinking specific to mathematics. For example:
What happens to the relationship between the sum of squares of the two shorter sides of triangles and the square on the longer side if we allow the angle between them to vary?
As teachers, it would be good to be aware also of the features of good problem solving task and levels of problem solving skills. There is also a study that describes teachers conceptions of problem solving.
Reference: Paper 7: Modelling, problem solving and integrating concepts by Anne Watson.
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I like how you state that math exercises should only be considered “problems” if they pose a barrier. Too often these are called problems when in fact there is little to no challenge involved, especially if working through the exercises as repetitive practice challenges, and the student has mastered the concept. Your breakdown of the different types of problems is a good idea to keep in mind when talking about math practice!