Which arithmetic knowledge and reasoning facilitate students’ entry into algebra?
Our research comes under the Early Algebra current, upholding that the arithmetic knowledge and reasoning developed by early middle-school students in France (11–12 years old) can facilitate, or conversely raise obstacles to, the transition from arithmetic to algebra. We define criteria for analyzing beginning cinquième (Canadian secondary two, 12 year old) students’ personal relationships to arithmetic before they start algebra. Drawing on anthropological theory, we then define praxeologies (Chevallard, 1999) for identifying the meaning and role that students assign to equality, numeric expressions, properties of operations including the distributive property, and analytical reasoning. We then present the beginning of cinquième evaluation designed in this context and examine the students’ answers, before exploring the instructional conditions that could facilitate students’ entry into algebra.