If you hang around long enough in the world of math education you’ll hear people refer to “rich problems”. What exactly are rich problems?
One definition is: “A problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”
My definition differs a bit: “One-off, open-ended, ill-posed problems that supposedly lead students to apply/transfer prior knowledge to new or novel problems that don’t generalize.”
For example: “What are the dimensions of a rectangle with a perimeter of 24 units?”
A student who may know how to find the perimeter of a rectangle but cannot provide some of the infinitely many possibilities is viewed as not having “deep understanding”.
Rather, the student’s understanding is viewed as “inauthentic” and “algorithmic” because the practice, repetition and imitation of procedures is merely “imitation of thinking”.
I teach 8th grade algebra and use a 1962 textbook by Dolciani. Here’s a problem from that book that I gave to my students:
“If a+1 = b, then which is true? a>b, a<2b, or a<b?”
This is my idea of a “rich problem” but don’t tell anyone. I might get fired.