Characteristics of a Good Math Problem
1. The problem is accessible. It minimizes vocabulary and notation (and vocabulary and notation that does exist should simplify, not complicate). It should only be as precise as necessary. The problem should have multiple entry points, and include ways to collect data of some sort. It should have multiple methods that promote different learning styles and celebrate different ways of being smart. It may have multiple valid solutions.
2. The problem is deep. It is rich enough to spend hours, days, weeks, months, or years working on variations, generalizations, and extensions. It leads to and connects different aspects of mathematics. The problem motivates developing procedures, vocabulary, notation, and mathematical concepts.
3. The problem is captivating. This does not mean that it has to be a “real world” problem and pseudocontext (from Jo Boaler's What's Math Got To Do With It?) should be avoided. A captivating problem may lead to a surprising result. It may feel like a puzzle waiting to be solved. It may be necessary to solve a different, interesting problem (which is not the same as “you’ll need to know this next year”). The problem may be posed by students. The problem consists of benchmarks along the way where one is re-energized by the feeling of success.
4. The problem scales sideways. This characteristic may be more applicable to school mathematics, but sideways scaling allows for practice and quality assessment (beyond just solving exercises measuring the ability to follow procedures).
5. The problem is mathematical. Problem solving skills and/or the language of mathematics help make progress in defining, simplifying, quantifying, dividing and/or solving the problem. Exploring the problem promotes mathematical habits of mind.