To browse Academia. This longitudinal case study presents a sequence of episodes that document the mathematical thinking of one child, Stephanie, over a 5-year period. Her development of the idea of mathematical justification spans grades Stephanie worked on several combinatorics tasks in smallgroup, whole-class, and individual interview settings.
The documented events indicate Stephanie's progress in classifying, organizing, and reorganizing data. The study provides some significant insight into the process by which Stephanie learned to make proofs, within a setting that encouraged the development of her ideas. In a first-grade classroom 6-year-old Stephanie is observed seated around a table with three boys working on a story problem and objecting to her classmate's attempt to provide an answer without a reason.
Wait a second, Buddy, you can't just say six! I think we should read this word. Wait for us, Buddy! Remember, you can't just jump to conclusions like "I know this" Stephanie, Grade 1, 23 March In a fourth-grade classroom, 9-year-old Stephanie is observed seated with three other children in a conference-like setting trying to convince one of her classmates, Jeff, that she has found all possible towers three cubes tall that could be built by selecting from plastic cubes in two colors, one blue and the other red.
Stephanie: So I've convinced you that there are only eight [towers three cubes tall when selecting from two colors]? Jeff: Yes. Her more sophisticated use of justification surfaced again in grade 4, when she used a form of "proof by cases" to present her solution to Jeff and other classmates.
In grade 5, in a classroom written assessment, year-old Stephanie provided an elegant version of a proof by cases to justify having found all towers three cubes tall when selecting from two colors. In this work we explored proof schemes used by 41 middle school students when confronted with four mathematical propositions that demanded verification of accuracy of statements. The students' perception of mathematically complete vs. Lastly, we considered whether the students recognized and identified advantages associated with using justification models different from their own in order to offer a theoretical account for how individuals' proof scheme choice might be impacted by such an exposure.
