reflecting on their activity, for example, kindergartners can “prove” theorems about sums of even and odd numbers.34 Through a carefully constructed sequence of activities about adding and removing marbles from a bag containing many marbles,35 second graders can reason that 5+(–6)=–1. In the context of cutting short bows from a 12-meter package of ribbon and using physical models to calculate that 12 divided by is 36, fifth graders can reason that 12 divided by cannot be 72 because that would mean getting more bows from a package when the individual bow is larger, which does not make sense.36 Research suggests that students are able to display reasoning ability when three conditions are met: They have a sufficient knowledge base, the task is understandable and motivating, and the context is familiar and comfortable.37

One manifestation of adaptive reasoning is the ability to justify one’s work. We use justify in the sense of “provide sufficient reason for.” Proof is a form of justification, but not all justifications are proofs. Proofs (both formal and informal) must be logically complete, but a justification may be more telegraphic, merely suggesting the source of the reasoning. Justification and proof are a hallmark of formal mathematics, often seen as the province of older students. However, as pointed out above, students can start learning to justify their mathematical ideas in the earliest grades in elementary school.38 Kindergarten and first-grade students can be given regular opportunities to talk about the concepts and procedures they are using and to provide good reasons for what they are doing. Classroom norms can be established in which students are expected to justify their mathematical claims and make them clear to others. Students need to be able to justify and explain ideas in order to make their reasoning clear, hone their reasoning skills, and improve their conceptual understanding.39

It is not sufficient to justify a procedure just once. As we discuss below, the development of proficiency occurs over an extended period of time. Students need to use new concepts and procedures for some time and to explain and justify them by relating them to concepts and procedures that they already understand. For example, it is not sufficient for students to do only practice problems on adding fractions after the procedure has been developed. If students are to understand the algorithm, they also need experience in explaining and justifying it themselves with many different problems.

Adaptive reasoning interacts with the other strands of proficiency, particularly during problem solving. Learners draw on their strategic competence to formulate and represent a problem, using heuristic approaches that may provide a solution strategy, but adaptive reasoning must take over when

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