Abstract
In this paper, we propose an enriched and extended application of Harel and Sowder’s proof schemes taxonomy that can be used as a diagnostic tool for characterizing secondary students’ emergent learning of proof and proving. We illustrate this application in the analysis of data collected from 85 Year 9 (age 14–15) secondary students. We capture these students’ first encounters with proof and proving in an educational context (mixed ability, state schools in Greece) where mathematical proof is explicitly present in algebra and geometry lessons and where proving skills are typically expected, and rewarded, in key national examinations. We analyze student written responses to six questions, soon after the students had been introduced to proof and we identify evidence of six of the seven proof schemes proposed by Harel and Sowder as well as a further eight combinations of the six. We observed these combinations often within the response of the same student and to the same item. Here, we illustrate the eight combinations and we claim that a dynamic use of the proof schemes taxonomy that encompasses sole and combined proof schemes is a potent theoretical and pedagogical tool for mapping students’ multifaceted and evolving competence in, and appreciation for, proof and proving.
Original language  English 

Pages (fromto)  277294 
Number of pages  18 
Journal  Mathematical Thinking and Learning 
Volume  20 
Issue number  4 
Early online date  20 Sep 2018 
DOIs  
Publication status  Published  Sep 2018 
Keywords
 proof schemes
 proof and proving
 first encounter with proof
 secondary mathematics
Profiles

Irene Biza
 School of Education & Lifelong Learning  Associate Professor in Mathematics Education
 Research in Mathematics Education  Member
Person: Research Group Member, Academic, Teaching & Research

Elena Nardi
 School of Education & Lifelong Learning  Professor of Mathematics Education
 Research in Mathematics Education  Member
Person: Research Group Member, Academic, Teaching & Research