Withholding comes from my struggle with figuring out how to teach mathematical modeling. There are an enormous number of resources out there for modeling lessons (see here, here, here, and here for a ton of examples), but what I’ve struggled with more specifically is teaching the skills that allow students to be successful in modeling. I don’t have any good solutions for this, but one piece that I think is critical, not only for mathematical modeling (MP.4), but also for reasoning abstractly and quantitatively (MP.2) and algebraic thinking in general.
Withholding is taking a problem, then asking what information is needed to solve it. Withholding is absolutely useful at a basic level–like giving students one side of a rectangle and asking what they need to know in order to find the area, or which angles are needed to identify the third angle in a triangle. However. it has much more value as a problem solving question asking students to analyze a number of possibilities and determine the easiest route to a solution.
A function goes through the points (2, 5) and (3, 7). What do you need to know to determine the y-coordinate where x = 4?
Withholding is a pretty simple idea that is mathematically rich when it is applied to an open middle problem. Being able to identify necessary information, in particular in divergent problem-solving situations, is a critical skill in modeling with mathematics. However, I’m much more pessimistic on the value of withholding relative to other problem types I’ve talked about. Mathematical modeling is a complex skill that requires sustained attention and practice, and withholding is only one very small piece of that.