At the University of Paderborn, the course “Introduction into the culture of mathematics” is required for all first-year students who enter the study program for future mathematics teachers at lower secondary level (grade 5–10). In this inquiry-based transition-to-proof course, we use four different kinds of proofs (the generic proof with numbers, the generic proof with figurate numbers, the proof with figurate numbers using geometric variables, and the so-called “formal proof”) to engage students in exploration, reasoning, and proving. In this paper, we report findings from an empirical study in winter term 2014/15 (pre- and posttest) concerning proof validation and acceptance. We used different kinds of ‘reasoning’ taken from Healy and Hoyles (2000) to assess students’ proof validation. At the beginning of the course, about a third of the students judged the purely empirical verifications and wrong algebraic operations as correct proof. These forms of reasoning being judged as correct proofs decreased greatly in the posttest. To investigate proof acceptance, the students had to rate different aspects – such as “conviction”, “explanatory power”, or “validity” - of the four kinds of proofs. “Proof acceptance scales” with very high reliabilities (Cronbach’s α> .864) were constructed using factor analysis. While in the pretest most of the students did not accept the generic proofs and the proof with geometric variables as general valid verifications, their acceptance increased during the course. However, in the posttest, the ratings of the different aspects vary greatly concerning the four kinds of proofs.

中文翻译：

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职前教师从以探究为基础的过渡到证明课程（以通用证明为重点）的收益

在帕德博恩大学，所有进入大学学习课程的一年级学生都需要“数学文化入门”课程，以供未来的初中数学老师（5-10年级）使用。在此基于查询的证明过渡过程中，我们使用四种不同的证明（带数字的通用证明，带数字的通用证明，带几何变量的带数字的证明以及所谓的“形式证明” ”），让学生参与探索，推理和证明。在本文中，我们报告了2014/15冬季学期（测试前和测试后）关于证据验证和接受的实证研究结果。我们使用了来自Healy和Hoyles（2000）的不同类型的“推理”来评估学生的证明有效性。在课程开始时，大约三分之一的学生认为纯粹的经验证明和错误的代数运算是正确的证明。在后期测试中，被认为是正确证据的这些推理形式大大减少了。为了调查证据的接受程度，学生必须对四种证据的不同方面进行评分，例如“定罪”，“解释权”或“有效性”。使用因子分析构建了具有很高可靠性（“克伦巴赫α> .864”）的“证明接受量表”。尽管在预测中，大多数学生不接受通用证明和带有几何变量的证明作为一般有效证明，但在课程中，他们的接受程度有所提高。但是，在后测中，关于这四种证明，不同方面的等级差异很大。