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The Use of Novel Items to Uncover Lapses in Mathematical Reasoning

Reasoning is integral to mathematics. However it is often difficult, in a written assessment, to judge if a student had possessed the correct mathematical reasoning behind a page of right answers. This paper reports the use of novel items and interviews to probe into the mathematical thinking behind the mathematical solutions of Grade 12 (pre-university) students in Singapore in an attempt to gain an insight into the lapses in mathematical reasoning. Of particular interest in this study was an examination into what students accepted as valid mathematical explanations and justifications and how they justified their mathematical arguments. Some novel items were developed and administered to a group of 69 Grade 12 students and the errors recorded. Further information on the students’, reasoning processes was obtained through semi-structured interviews of a selected group of students from the sample. The responses to the items and the interviews were analyzed in the light of existing literature. The key finding was that many mathematical errors arose because students had overly relied on instrumental understanding, which limited their transfer of knowledge to new environments. Another important finding was that inductive strategies from previous experiences in deriving mathematical generalizations were mistakenly accepted as valid mathematical proofs.

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