While many of the elements of problem design are useful tools in formative assessment or pushing students to think deeper about a concept they have had some experience with, splitting is most useful when a concept is first introduced. Splitting a concept takes students whose understanding is immature and gives them two (or more) options to argue about, with the goal of provoking 1) rigorous, engaged mathematical discussion and 2) a student-centered understanding of the concept. Splitting is an incredibly useful tool for application of CC.MP.3 Construct viable arguments and critique the reasoning of others.
I once saw a master teacher ask a group of elementary students who had just been introduced to decimals which was greater: .9, or 1.1. She was an absolute master — half the students thought .9 was greater because the first digit was larger, and an excellent discussion ensued where all of the students became convinced that 1.1 was actually greater and had the chance to articulate their understanding of place value.
When introducing the idea of association in scatter plots, students catch on quickly to the idea that negative association corresponds with a negative slope, and positive association with a positive slope. Then, throw this at them:
A scatter plot has negative association. Which statement is true?
As one variable decreases, the second variable decreases.
As one variable decreases, the second variable increases.
Splitting is a key step in CC.MP.3: Construct viable arguments and critique the reasoning of others. Students can best construct and critique arguments if the mathematics are worth arguing about. This requires well-chosen arguments that provoke genuine disagreement–and disagreements that can be facilitated so that as many students as possible come out with a deeper and more flexible understanding of the topic.