Unpacking the logic of mathematical statements
Unpacking the logic of mathematical statements
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解开数学陈述的逻辑
DOI：
10.1007/bf01274210
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发表时间：
1995
影响因子：
3.2
通讯作者：
Annie Selden
中科院分区：
文献类型：
--
作者：
J. Selden;Annie Selden
这项研究关注的是本科生将非正式书写的数学陈述转化为谓词演算语言的能力。1989年至1993年间，他们从一门旨在介绍证明和数学推理的“桥”课程的六个小节中的61名学生那里收集了数据。我们从将概念映象的概念扩展到陈述映象的角度来讨论这些数据，并引入证明框架的概念来指示定理的映象中与证明的顶层逻辑结构相对应的部分。对于简化的非正式微积分语句，只有8.5%的解包尝试成功；对于微积分文本中的实际语句，这一比例下降到5%。我们推断，这些学生将不能可靠地将非正式陈述的定理与他们证明的顶层逻辑结构联系起来，因此不能期望他们构造证明或验证它们，即确定它们的正确性。
This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a “bridge” course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion ofproof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness.