TEACHER KNOWLEDGE OF STUDENTS' MATHEMATICAL ERRORS

TEACHER KNOWLEDGE OF STUDENTS’ MATHEMATICAL
ERRORS
Aihui Peng
Umeå Mathematics Education Research Centre, Umeå University, Sweden
Analysing students’ mathematical errors is a fundamental aspect of teaching for
mathematics teachers, and it is challenging which demands teachers to have specific
knowledge. This study aims to investigate how mathematics teachers are
knowledgeable about their students’ mathematical errors. Within the theoretical
framework for examining mathematics teacher knowledge as used in error analysis, it
was conducted with a questionnaire and in-depth interviews, in which 25 middle
school teachers participated. The results show that although teachers’ knowledge of
students’ mathematical errors differs in different tasks, there are emerging patterns
on the extent of knowledgeable of students’ mathematical errors for mathematics
teachers.
INTRODUCTION
Mathematical errors are a common phenomenon in students’ learning of
mathematics. Students of any age irrespective of their performance in mathematics
have experienced getting mathematics wrong. It is natural that analyzing students’
mathematical errors is a fundamental aspect of teaching for mathematics teachers.
Due to the challenge from the variety and complexity of students’ mathematical
errors, it is important that mathematics teachers require specific knowledge for
analyzing students’ mathematical errors. However, although there is increasing
interest in mathematics knowledge for teaching, there is still a lack of detailed
understanding regarding how mathematics teachers are knowledgeable of students’
mathematical errors. This study will give a nuanced understanding of it through
empirical investigation.
LITERATURE REVIEW
Students’ errors in mathematics learning are a world-wide phenomenon, and there is
a long history for error analysis in mathematics education (Radatz, 1979). Due to the
variety and significance of students’ mathematical errors, it attracts a number of
researchers’ interests, which leads to the formation of many theories about the nature
of mathematical errors, their interpretation and the ways of overcoming them
(Gagatsis & Kyriakides, 2000; Luo, 2004). For example, focusing on the student’s
cognitive process, Davis (1989) proposed two kinds of regularity about students’
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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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Peng
Dimension
Nature
of
mathematical
errors
Phrases
of
error
analysis
Analytical
categorization
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Description
Mathematical Confusion of concept and characteristics,
neglect the condition of formulas and
theorem
Logical False argument, rearrange concept,
improper classification, argue in a circle,
equivalent transform
Strategical
Couldn’t distinct from pattern, lack of
integral concept, not good at reverse
thinking, couldn’t transform the problem
Psychological Mentality deficiency, lack proper mental
state
Identify Knowing the existence of mathematical
error
Interpret Interpretting the underlying rationality of
mathematical error
Evaluate Evaluating students’ levels of performance
according to mathematical error
Remediate Presenting teaching strategy to eliminate
mathematical error
Table 1: A framework for examining mathematics teacher knowledge as used in error
analysis
This framework is used in this study for research design and data analysis.
METHODOLOGY
Participants
25 middle school mathematics teachers in their short in-service training courses
participated in this study. There are 10 males and 15 females. And there were 2, 16, 7,
respectively for the degree of master, bachelor, associate degree. Their years of
teaching ranged from 1 to 38. And all of them expressed that they’d like to provide
the data relevant to the reliability and validity of this study.

Instruments
Data were collected with an author-constructed Mathematical Error Analysis
Questionnaire and in-depth interviews followed the questionnaire. The questionnaire
consisted of four tasks that were designed to examine how mathematics teachers
analyse student’s mathematical errors in typical algebraic and geometric topics of
polynomial, equation and triangle. Each of the four tasks focused on mathematics
teacher knowledge of identifying, addressing, diagnosing, and correcting student’s
mathematical errors, preferably, every task mainly focused on one aspect of logic,
mathematical knowledge, psychology and strategy. These tasks were examined from
middle school students in a pilot study. After reviewing and analysing the responses
to the questionnaire, every teacher was invited to an in-depth interview to further
explore how teachers understand and handle student’s mathematical errors. Every
interview occurred within 1 hr. The interviewer posed the initial question and then
followed the teacher’s lead, asking follow-up questions based on the teacher’s
responses. The interviewer cycled back to topics to elicit more detail. Each interview
was audio taped and transcribed.
Data analysis
Qualitative analysis method was used in the analysis of the questionnaire and the
transcriptions of the interviews. Firstly, 4 different categories consisted of identifying,
addressing, diagnosing, and correcting students’ mathematical errors was identified
which included the responses to every task. Next, different perspectives from logic,
mathematical knowledge, strategy and psychology were identified. The responses
were categorized into groups and assigned a descriptive code. Two researchers used
the resulting codes to analyse the responses independently. Both sets of codes were
compared, and then, through discussion with the third researcher, the disparities were
reconciled to reach valuable agreements on the responses. The coding of the four
tasks of questionnaire was used by T1, T2, T3, and T4. And the investigated teacher
was used by six codes ABCDEF, which represented age, gender, degree, teaching
years, school district and confidence about analysing students’ mathematical errors
respectively.
RESULTS
According to teachers’ responses to questionnaire and interview, their knowledge of
students’ mathematical errors manifested differently in every dimensions of phrases
of error analysis and perspectives for sources of errors. But when they were mixed in
one picture through focusing on every specific task, the distinctive different levels of
knowledgeable of students’ mathematical errors were shown. The results were
presented including the clarification of the levels (see table 2) and exemplification of
the typical features of the levels supported by some interview excerpts (see table 3).
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Level 1 ● Couldn’t identify the students’ mathematical errors.

Peng
●Couldn’t find the reasonable causes of students’ mathematical
errors; couldn’t find the reasons for students’ mathematical errors or simply
attribute to students’ mathematical errors to such as ‘non-seriousness of
learning’ and ‘poor basic knowledge and basic skills’.
● Couldn’t evaluate the influence of students’ mathematical errors
on their latter learning, couldn’t evaluate students’ levels of performance
according to their mathematical errors.
● Couldn’t present teaching strategies for correcting students’
mathematical errors.
Level 2 ● Could identify the students’ mathematical errors but couldn’t
find the underlying reasons.
● Couldn’t find the reasonable causes of students’ mathematical
errors, could explain the reasons for students’ mathematical errors, but only
consider them as they stand, couldn’t analyse them for advanced viewpoint.
● Could simply evaluate the influence of students’ mathematical
errors on their latter learning, can evaluate students’ levels of performance
according to their mathematical errors to a certain degree.
● Could present teaching strategies for correcting students’
mathematical errors, but not so suit for the specific cases.
Level 3 ● Could identify the mathematical errors of students and find the
underlying reasons.
● Could evaluate students’ mathematical errors from a
reasonable point of view, could analyse students’ mathematical errors for a
singly relative high level theoretical point of view.
● Could properly evaluate the influence of students’
mathematical errors on their latter learning, could evaluate students’ levels
of performance according to their mathematical errors.
● Could present singly teaching strategies for preventing
students’ mathematical errors.
Level 4 ● Could identify the students’ mathematical errors and the
underlying reasons in a right and quick way.
● Could understand the reasonability of students’ mathematical
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