88 ◆
Adaptive reasoning is the capacity for logical thought, reflection, expla-
nation, and justification.
(Kilpatrick, Swafford & Findell,
2001, p. 116)Reasoning is one of the essential elements of mathematical proficiency. It
means that students will “think logically about the relationships among
concepts and situations” (Kilpatrick et al., 2001, p. 129). Students need to
be able to think about different ways to approach the problem, about
whether or not their answers make sense, and about the ways they can
verify and justify their answers. Students in elementary school should have
many opportunities for “explanation and justification but also intuitive
and inductive reasoning based on pattern, analogy and metaphor (Kilpat-
rick et al., 2001).” Students should be encouraged to use both “mental and
physical representations” as “tools to think with” (Kilpatrick et al., 2001).
Kilpatrick et al. (2001) note that “with the help of representation-building
experiences, children can demonstrate sophisticated reasoning abilities
(p.130).” When children work in pairs and groups to prove the math by
acting it out, working with manipulatives, drawings, diagrams, tables and
other models, and they are encouraged to discuss their thinking and reflect
on the thinking of others, they get good at it.
Contextualizing
Students should not only solve problems but also pose them. To get stu-
dents to make up word problems, they need to be scaffolded by acting
them out and using concrete objects, drawings and diagrams. This activity
takes time and practice. It is important to give students the context at first.
For example, tell the students that the answer is 5 marbles and ask them,
“What is the question?” Sometimes, I tell the students what type of prob-
lem it is. This is different from just saying the answer is 5. At first, you