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.
traditional math 
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.”
They always leave out the next bit of that definition, which is “The child won’t know where they made any mistakes, unless the teacher kills themselves trying to mark every answer as it is generated.”
I set my students the task of designing Aztec pyramids with 1,500 m^3 volume yesterday, as a bit of end of week fun. Am I meant to take in each of the 25 answers and mark them for the next lesson? Because they won’t know if they were successful otherwise,nor where they made mistakes. But if I do that regularly, then I’m marking two hours for every lesson I teach. That’s simply not sustainable.
Instead I modelled how I would approach the task, and explained my thinking at each step.
I can only do it that way because the class have already been taught, explicitly, how to do volumes of the requisite shapes. Thus I can focus on higher level thinking, knowing safely that the key skills have been drilled in. But if they made any mistakes in their calculations, then they won’t be corrected, and you can’t teach Maths usefully like that unless they are already near expert level.
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Good example of an open-ended problem that is well-posed. That is, the question does not say “what are the dimensions” which gives an impression of their being only one answer. Your question directs students to design a pyramid with a specific volume. And you modeled it rather than leaving them to a “productive struggle” or in current parlance, a “desirable difficulty”.
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Reblogged this on Nonpartisan Education Group.
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“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.”
This is the classic ed-speak I heard about differentiated instruction (learning) when my son was young. This wasn’t just for math. The onus is on the child (parents at home) for anything more than NON-STEM CCSS proficiency. I’ve always thought that this was just a K-8 excuse for full inclusion academic age tracking. Somehow, those things magically disappear in high school due to the coming reality of college and real life.
Has the “math zombie” fad passed, or do they claim some specific math curriculum that works better? Have the Phillips Exeter Harkness Table wannabes realized the difference between fuzzy class talk and hard homework problems that have one correct answer?
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