“You're going to want to find out which and prove it”: collective argumentation in a mathematics classroom

One of the proposals of the North American educational reform movement is that teachers should stress scientific argumentation more than the manipulation of symbols and algorithms in their mathematics instruction. The aim of this article is to apply some theoretical concepts, drawn from the fields of sociolinguistics and rhetoric, to the analysis of argumentation in a lesson conducted in an urban middle school classroom. Our analysis focuses on the implementation of the classroom teacher's instructional goals during a lesson on area measurement. As a result of our analyses, we found that she achieved her instructional goals of being nondirective in her teaching and getting students actively involved in arguing about mathematical concepts. The teacher was able to orchestrate discussion by recruiting attention and participation from her class, aligning students with argumentative positions through reported speech, highlighting positions through repetition, and pointing out important aspects of their arguments through expansion. In addition, we also found that her students differed in the way they framed the mathematical content of the lesson in terms of the facts or grounds, algorithms or warrants, premises or backings, as well as solutions or claims. Their arguments also varied in terms of explicitness and ability to integrate their classmates' arguments. In conclusion, we feel that discourse analysis, based on sociolinguistic and rhetorical theoretical frameworks, can be a valuable tool for the evaluation of educational reform in mathematics.