Double negative: The necessity principle, commognitive conflict, and negative number operations

• Arithmetic rules for negative integers are conventions, not theorems.
• Effective pedagogies for teaching conventions and theorems differ.
• Pattern-based justifications may not address students’ intellectual needs.
• Analysis of students’ intellectual needs leads to new pedagogical strategies.

We use the DNR framework to analyze a classroom episode introducing negative integer exponents, comparing and contrasting our analysis with Sfard's recent commognitive analysis of a similar episode concerning multiplication of signed numbers. Students in both episodes objected to the standard rules for integer products or exponents, and they persisted in preferring their own rules even after the teacher justified the standard ones. We examine how pattern-based justifications may not address students’ intellectual needs, and we suggest other pedagogical strategies that promote student reasoning.