Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Defining Reasoning and Proof
|Introduction | Investigating Conjectures | Reasoning and Justification | Additional Methods | Your Journal|
"Proofs (both formal and informal) must be logically complete, but a justification may be more telegraphic, merely suggesting the source of the reasoning" (Adding It Up, National Research Council, NAP 2001, p. 130).
When students provide justifications, they are explaining and defending their reasoning for a solution or a solution method. In the elementary grades, justification is less formal and less detailed than proof, but it is still a valuable means of communication.
When justifying a problem-solving procedure, it is helpful to first reflect on one's understanding of the procedure and its underlying concepts. For example, when asked to justify a method that involves regrouping during subtraction, a student is likely to go beyond shallow reasons, such as "Because this is what my teacher told us to do," to reexamine his or her conceptual understanding of place value, of regrouping, and of the operation of subtraction. The NCTM Process Standards point out that upper elementary students "should increasingly base their arguments on an analysis of properties, structures, and relationships" (NCTM, 2000, p. 191).
Working on rich tasks that foster efforts for justification presents valuable challenges for students. Students are likely to reexamine their comprehension of the problem and to reconsider the appropriateness of their solution steps. They may also think more generally about their methods and consider analogies or comparisons with other problems where the strategies would also work.
Justifications are frequently stated verbally and are sometimes recorded in writing. They may also be given visually through a demonstration or a diagram. For example, students who know the formula for finding the area of any rectangle may be able to make the beginnings of a deductive proof of the formula for the area of any parallelogram by using a paper rectangle and a pair of scissors. By physically exploring the properties of the new and old figures, they can show that any rectangle can be cut into two parts by making a cut between any two points on two parallel sides. These parts can always be rearranged into a new parallelogram with the same base length and same height:
This will work for any rectangle and parallelogram.
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