Proof schemes combined: mapping secondary students’ multi-faceted and evolving first encounters with mathematical proof

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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’ multi-faceted and evolving competence in, and appreciation for, proof and proving.
Original languageEnglish
Pages (from-to)277-294
Number of pages18
JournalMathematical Thinking and Learning
Issue number4
Early online date20 Sep 2018
Publication statusPublished - Sep 2018


  • proof schemes
  • proof and proving
  • first encounter with proof
  • secondary mathematics

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