TEACHER KNOWLEDGE OF STUDENTS' MATHEMATICAL ERRORS

mathematical errors, the first regularity refers to certain errors made by different
students that are extremely common, and the second kind of regularity refers to the
wrong answers given by one person, in response to a sequence of questions.
Brousseau (1981) used historical elements in order to explain pupils’ errors in
decimal factions, and he found that pupils make the same errors independently of the
teaching methods used and, thus, concluded that there are errors that can be attributed
to the pupils’ epistemological foundations. And there is the widely recognized
conceptual change framework, within which errors initially conceptualized
negatively are now seen as a natural stage in knowledge construction and thus
inevitable (Vosniadow & Verschaffel, 2004)
At the same time, in the past two decades, there has been a significant and developing
research focus on teacher knowledge since Shulman (1986) introduced the notion of
‘pedagogical content knowledge’ (PCK), which emphasized knowledge of students’
thinking about particular topic, typical difficulties that students have, and
representations that make mathematical ideas accessible to students. Research on
teacher knowledge has expanded from studies of teachers’ subject-matter knowledge
of various content areas to the organization of teachers’ knowledge for teaching
particular content to students (Ball, 1990; Even, 1993; Peng, 2007; Izsak, 2008). This
expansion follows a generation of research that emphasizes knowledge of content and
students, include the ability to anticipate student errors, to interpret incomplete
student thinking, to predict how students will handle specific tasks, and what students
will find interesting and challenging. In this aspect, Hill et al. (2008) identified that
responding to students inappropriately—the degree to which teacher either
misinterprets or, in the case of student misunderstanding, fails to respond to student
utterance as a key aspect of the mathematical quality of instruction. Peng and Luo
(2009) developed a framework to investigate mathematics teacher knowledge as used
in error analysis.
From the literatures, it can be observed that there are insights from studies in both
analysing students’ mathematical errors and mathematics teacher knowledge, but it
lacks of how mathematics teachers are knowledgeable of students’ mathematical
errors, especially absence of empirical evidences. This study aims to fill this gap
within the existed framework.
THEORETICAL FRAMEWORK
In Peng and Luo (2009), the framework below (shown in Table 1) is introduced in
order to analysis teacher knowledge of students’ mathematical errors. The framework
includes two separate dimensions, namely, the nature of mathematical error and the
phrases of error analysis, which are closely linked together in a complex way. There
are four keys for the nature of mathematical error, namely, mathematical, logical,
strategical and psychological, and the four keys for the phrases of error analysis,
namely, identify, interpret, evaluate, and remediate.
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