When analyzing CCSS-aligned tasks and student work and considering pedagogy to support student learning with teachers from a variety of states and school districts, the question we were most frequently asked was:
There is no simple answer to this question. No single approach will guarantee student success. We know that teaching and learning is a complex process with a lot of back and forth—trying something, observing how students respond, paying attention to what students understand and misunderstand, and then adjusting instruction accordingly.
That said, there is guidance for the perplexed teacher. It comes from research, the Common Core State Standards themselves, and what we practitioners know about good teaching.
We begin with the research. Learning is an active endeavor on the part of the learner (Baroody, 1987; Cobb, 1988; Fosnot, 1996; von Glaserfield, 1990, 1996). To understand a new idea and construct knowledge, learners must actively think about learning and ask themselves, “How does this fit with what I already know?” “How can I understand this in the context of my current understanding of this idea?” “Knowledge cannot be 'poured into' a learner." (Van de Walle, Karp, Bay-Williams, 2010.)
Given this body of research, it is not surprising that the CCSS writers insist that quality instructional materials include problems “worth doing.” They tell us that, “In solving problems, students learn new mathematics. It is in the 'figuring out' that students learn to do mathematics. Students develop ownership of the concepts they discover by 'doing the figuring out.'"
So where might direct instruction, such as teaching the procedures, fit in?
Let's try to answer this question by thinking through the lens of an example about proportional reasoning. Children have an intuitive understanding of the concepts behind ratio and proportional reasoning. Consider the example of children who sell candies on the streets in Mexico—3 for twenty-five cents, 6 for fifty cents, etc. These young entrepreneurs intuitively calculate the price for different quantities of candy. What is not intuitive is the knowledge that they are working with ratios or the ability to express a ratio using correct mathematical notation. Our students need experiences with similar, simple ratio problems—"How much will 30 candies cost?" "How many candies can we buy for $4.00?" After they find ways to solve such problems on their own (perhaps with a table of values, perhaps using simple multiplication or a guess-and-check method), we can introduce them to ratio notation using direct instruction. It might look and sound something like this:
This instruction is direct, in the sense that the teacher is specifically giving students a notational system because we neither expect nor want students to invent one. You may be surprised to see so many questions linked to direct instruction but without additional probing and pressing questions, students are not likely to internalize the idea. Nor, without additional questions, is the teacher likely to determine what students understand. Even with direct instruction, questioning is still important.
So when is the best time to use direct instruction: after student discovery and exploration or initially when a new concept or procedure is introduced? The IFL tends to encourage direct instruction after student discovery and exploration. We believe that for direct instruction teachers should capture the student reasoning during problem solving and then mathematize and extend it. Regardless of when teachers provide direct instruction, we know it is just “good practice” for students to be actively engaged in the process and to be pressed to respond to probing questions.