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Teachers’ knowledge about domain specific student errors
Janosch M. Türling, Jürgen Seifried & Eveline Wuttke
Abstract
A teachers ability to diagnose student errors and use them constructively
in the classroom is a key aspect of teacher professionalism. In the field of
business education and accountancy in particular, very little is known about
student errors and which competences teachers need to handle them
constructively. In this article we report the findings of a study concerning the
ability of (prospective) teachers to diagnose student errors in the domain of
bookkeeping.
Keywords
Student errors measurement of
competences, video vignettes
1. Professional Error Competence (PEC)
It is now commonly held that it is possible to develop professional
competence by learning from errors at school and in the workplace (q.v.
Baumgartner/Seifried in this edition). However, the idea that errors can bear
a potential for learning was rarely supported at first (e.g. Weimer 1925).
Recently, the focus has shifted to whether a negative evaluation of and
response to errors is the most effective approach in pedagogical contexts
(Fischer et al. 2006; Oser/Spychiger 2005; Yerushalmi/Pollingher 2006). A
key aspect is seen - Here the
about making errors (emotional
component) and enable reflection as well as support learning processes
2
through feedback (cognitive component). Several disciplines study how
teachers should best errors; namely, Pedagogy, Psychology,
Medical Science, Neurology or Engineering Sciences (e.g. Bauer 2008;
Graber 2009; Mehl/Wehner 2008; Oser/Spychiger 2005; Weingardt 2004).
Consequently, in the field of teaching-learning-research, increasing effort has
been directed towards identifying error types and the possibility of learning
from errors as well as analysing how teacher behaviour influences
chances of learning from errors (e.g. Baumert et al. 2010; Heinze 2004;
Seidel/Prenzel 2007).
These questions can be dealt with against the background of the current
discussion about teacher competences. Generally these are considered as
distinctly different categories of professional knowledge, and three are seen
as crucial: general pedagogical knowledge, pedagogical content knowledge
and content knowledge (Graeber/Tirosh 2008; Hill/Ball/Schilling 2008;
Shulman 1987). The broadest and most common definition of professional
teacher competence portrays a complex construct which includes knowledge,
beliefs and motivational orientations (Baumert/Kunter 2006; Desimone
2009). Referring to Shulman, pedagogical content knowledge (PCK) can be
described as a specific type of knowledge on how to transform subject-matter
knowledge into teaching practice. This kind of knowledge enables features
such as the effective structuring of lessons, the use of specific representations
or analogies, and an awareness of possible misconceptions or content-related
learning difficulties (van Driel/Berry 2010). Despite rather broad research in
this domain, it is still quite uncertain exactly what competences teachers
should have in order to deal with errors constructively.
With this argumentation in mind, we suggest that in order to use
errors constructively (from a cognitive point of view)1, teachers
need to be competent in three ways (three facets of Professional Error
Competence, PEC):
(1) Knowledge of possible error types: First, teachers have to actually recognise
the specific logical flaws and false assumptions made by students. To be able to
do this, teachers need domain-specific knowledge about possible learner errors.
(2) Available strategies of action/teachers reaction: After having recognised the
error, teachers must treat it adequately. For this they have to know about
various alternatives of action (e.g. providing adequate feedback or when it is
better to ignore errors).
(3) A constructive view on errors and their use in classroom interaction: Roughly
speaking, a so-called error-prevention-didactic (errors are to be prevented so that
false trains of thought do not become habitual) can be set against a constructive
management of errors. In the latter approach teachers are prepared to become
involved in students errors even if there are time constraints.
1 For a more climatic or emotional point of view see also Spychiger et al. (1998) or
Seifried/Wuttke (2010a).
3
Our current project focuses on how teachers develop competence in the
areas of error diagnosis and dealing with learner errors in the domain of
accounting (Wuttke/Seifried 2009; Seifried/Wuttke 2010b). We assume that
teachers can develop these competences in the course of their training and
professional life. Because little is known about when teachers acquire error
knowledge and ways of dealing with errors, we are using a combined cross-
sectional longitudinal design to test teachers at several stages of their
professional development. As a specific characteristic of the German teacher
educational system, prospective teachers have to complete a Masters degree
at university, where they already have didactical and pedagogical courses,
and then successfully complete a practical training of about 1.5 to 2 years
before they begin teaching. Therefore, corresponding development processes
during professionalisation will be considered for four groups: bachelor and
master students, pre-service teachers and professional teachers.
In this article we will focus on the first facet of PEC, namely the
knowledge of possible error types. As our work is still in progress, the first
discussed. This type of knowledge can be seen as a prerequisite for enabling
teachers to diagnose typical student errors and handle these errors in an
adequate way.2 From a first view (cross-section of the above mentioned
stages) the following objectives will be considered:
1) What domain-specific knowledge about student errors do the participants
have and are there differences due to the process of professionalisation?
2) How do they perceive their own knowledge and does this perception relate
to actual performance?
The second section of this article gives an overview of the theoretical
foundation of learning from errors; in particular in the specific field of
interest: accounting lessons in business education. How PEC could be
measured and a description of our sample will be specified in chapter three.
Finally, empirical findings concerning knowledge and the ability to diagnose
typical student errors (section 4) will be presented and discussed before
concluding in section 5.
2 As current research studies like COACTIV (
, Baumert
et al. 2010; Krauss et al. 2008
necessary but not sufficient condition for the quality and effectiveness of teaching. So the
PCK of a teacher (here to be aware of typical false assumptions of students, to have the
ability to get to the bottom of student errors and to handle errors in an adequate way)
apparently has a higher impact on the quality of teaching than the very existence of subject
knowledge. Nevertheless, a substantial correlation between CK and PCK can be found.
4
2. Student errors in the domain of bookkeeping
2.1 Learning from errors in school settings
Analysing the process of learning from errors is difficult both within
domains and a
(Rohe/Beyer/Gerlach 2005:15; Weingardt 2004:199). Reasons could be
found in a domain specific view as well as in linguistic barriers; in particular,
many relevant research activities can be identified in English speaking
ther conveys an intended differentiation
(Senders/Moray 1991). For our research interests in the field of business
education, and bookkeeping in particular, we refer to Heinze (2004:223) who
conceives student errors as domain specific and related to a specific setting of
instruction. According to Heinze (2004) a situation in a classroom where
errors occur is characterised by a triad of (1) a student
error identification and (3) respective handling of by the
teacher (225).
A possible basis for the modelling of error-learning-processes can be
found in the concept of negative knowledge or negative expertise. Recently
Minsky (1994) popularised this concept (see also Oser/Spychiger 2005 or
Gartmeier et al. 2008). Negative knowledge incorporates both procedural
(knowledge, how something does not work; Minsky 1994) and declarative
knowledge (knowledge, how something is not and what one does not know;
Parviainen/Eriksson 2006). The basic idea is that people recognise their
deficits when they make mistakes and as a consequence of this initiate
learning processes. Whether the potential connected with the acquisition of
negative knowledge can actually develop and result in knowledge acquisition
depends on whether deeper reasons for errors are analysed and reflected on
and if constructive feedback is given on how to improve in the future. But the
actual process of learning from errors, if it really happens, is still largely a
mystery.3 As a first step, a systematic conceptualisation of possible error
types is necessary for every domain. A look beyond the border of our own
discipline
3 Even examinations at a physiological level do not paint a homogeneous picture. On one
hand, results point to the fact that errors have a positive effect on subsequent learning
processes (Wills et al. 2007), on the other hand some people, because of an impaired
processing of Dopamine, hardly seem to learn from the negative consequences of their
actions (Klein et al. 2007).
5
mathematics (q.v. Seifried/Türling/Wuttke 2010). It is also remarkable that
many student errors apparently neither occur randomly nor are they caused
by a lack of concentration. Instead they can be recognised continuously and
across generations (Swan 2004). Examples in language learning could be
n incorrect use and exclusion of binomial
phrases like (a + b)2 = a2 + b2.
2.2 Domain specific considerations
Although the domain of bookkeeping can generally be seen as crucial for
the development of economic competence (Preiß/Tramm 1996;
Sembill/Seifried 2005; Sloane 1996), very little empirical evidence on its
learning and instruction exists. Subsequently, we conducted a preliminary
study interviewing experienced teachers (N = 51) about typical errors and
error situations in the domain of bookkeeping. Accounting, in the opinion of
(experienced) teachers and students, has a high rate of student error. The
results show that these errors can be classified using three different
perspectives (Türling et al. in press).
(1) Subject topics that are prone to errors. These are in particular: the transition
from asset to profit & loss accounts, value added tax, adjusting entries and
difficulties which refer to the logic and structure of this subject in general.
(2) Steps during a learning process, which have to be paced within solving a task
or a problem: the economic literacy, use of technical terms, formal operations
like allocating to an account as well as constituting a booking record and
mathematical operations.
(3) Possible causes for errors: studen
methodological issues, teacher-caused errors, and the abstractness of the subject
were most frequently mentioned.
Summarizing recent explanations of learning from errors and dealing
as having both a high
relevance and at the same time an insufficient evidence-base in the field of
teaching-learning-research.
6
3. Method
3.1 Measurement of PEC
Current discussions and research trends concerning the assessment of
competences are, among other things, characterised by a preference for
behavioural data collected in performance situations, even if this means a
higher test diagnostic effort than self-reports. However, a major disadvantage
of using self-reports can be seen in the bias caused by over- or
underestimation found in self-assessment (e.g. Leutner/Hartig/Jude 2008:
185f.). Simultaneously, to the increased impact of behavioural data,
methodological progress, namely models of Item-Response-Theory (IRT),
has enhanced the scope of new methods of test design and analysis (e.g.
Adams/Wu 2002; Hartig 2009; van der Linden/Hambleton 1996; Walter
2005; Wu/Adams/Wilson/Haldane 2007). All in all the current situation
regarding the design and assessment of teacher competences still covers new
territory and has not been sufficiently investigated using empirical evidence
(e.g. Desimone 2009; Kunter/Baumert 2010).
For our purposes of measuring PEC, we refer to a mixed methods
approach (Tashakkori/Teddlie 2008). Therefore, we use both performance
data and self-reports to consider various areas of competence. To analyse the
the basis of
performance data we used video vignettes and a paper-pencil-test. The
findings of the preliminary study (Interviews with experts, see chapter 2)
formed the background for the production of the video vignettes. These
vignettes present short error situations in the classroom and are used as
prompts to test whether teachers are able to identify errors and how they
respond (see also figure 1). The vignettes are interactive in such a way that a
second sequence builds on the first. In the first sequence an error situation is
shown to the participants. Afterwards, in a guided interview the participants
explain how they would react in the given situation, and especially how they
would handle the error(s). Furthermore, it is recorded which errors the
participants identify and what causes they assume as reason for the error(s).
Depending on the statement, one of four possible follow-up
sequences is then activated. The test administrator has to choose one of them.
The sequences vary systematically regarding two criteria: (1) The first aspect
focuses on the extent to which the participants would give students hints for
the correct solution and (2) the participants have to decide whether to take the
entire class or single students into consideration by dealing with the shown
7
problem/error. After the ending of the sequel the participants are again asked
to explain their reaction in the shown situation. This multiple confrontation
with a particular student error will show to what extent participants are able
to present and explain an identical learning objective to students from
different points of view. This ability is commonly seen as a reliable indicator
for the pedagogical content knowledge and competence of a teacher (Brunner
et al. 2006).
Using video vignettes to generate performance data has several
advantages (e.g. Barter/Renold 1999; Jüttner/Neuhaus 2010;
Oser/Salzmann/Heinzer 2009; Seguin/Ambrosio 2002; Veal 2002;
Wason/Polonsky/Hyman 2002). Firstly, to measure PEC as an adequate and
near to active treatment, a stimulus is needed that requests situative decisions
on action video-taping real
classroom situations, the production of vignettes with professional actors
ensures standardised conditions for the tests and the ability to vary and utilise
typical errors that actually should be investigated. The one weakness of this
instrument could be seen in social desirability, or that the vignettes in fact
only represent a
Figure 1: Assessment of Competences with Video vignettes
In addition we used a paper-pencil-test to investigate knowledge about
was designed as a fictive class
test including . The participants had to identify and correct
these errors within a given time. Post-hoc analysis with item response
8
modelling should present an appropriate way to assess how the participants
scored in these performance tests and to ascertain the level of difficulty of the
errors used. Here, due to the use of dichotomously scored responses such as
a one-
parameter logistic model (1PL, Rasch Model) was chosen (e.g. Hartig 2009;
van der Linden 2010). To obtain information on how the participants
questionnaire (adapted version of a scale from the COACTIV-study). On a
scale of 1 to 6 indi
(mean of about 4; 4 Items, Cronbach´s ).
3.2 Sample
In 2010, data was collected from 246 German (prospective) teachers. The
participants from stages 1 and 2 are in teacher training programmes at the
Universities of Constance and Frankfurt. The pre-service teachers were
attending their practical training at teacher education institutes, and the
professional teachers were employed at commercial schools. All institutions
considered are within the German federal states of Baden-Wuerttemberg and
Hesse. At the time this paper was submitted, the completion of the sample
was still in progress.
Table 1: Sample (n = 246)
Stage
N
sex
age
term/
professional
experience
female
male
M
SD
M
SD
1-Teacher Training (Bachelor)
76
43
33
23.29
3.05
3.70
1.62
2-Teacher Training (Master)
64
44
20
26.50
4.29
2.58
.88
3-Practical Training
73
32
41
28.92
3.80
.38
.25
4-Professional Teachers
33
20
13
32.18
4.63
3.43
.15
With the exception of stage 3, gender distribution is slightly unbalanced
in favour of female participants. The bachelor students have an average age
of about 23 and are in the middle of their undergraduate studies, the master
students are about 3 years older and are close to graduation. Participants from
stage 3 are about 29 years old and are mostly at the beginning of their
practical training, while the professional teachers have an average age of
about 32 and an average of three and a half years teaching experience. The
number of participants in the sample groups is slightly unbalanced because of
and data sampling, as stage 4 is still incomplete.
9
4. Findings
4.1 Test achievement and comparison across
professionalisation steps
The subsequent figure 2 shows in detail how the participants (sorted by
professionalisation steps) scored in the tests (relative frequencies of correct
item responses within their subgroup). To analyse possible differences
between the four groups a chi-square test (df = 3) was used. The two vignette
related tasks show that, apparently, the first task is easier to handle than the
second task (all groups had correct responses of about 80 %), except for the
professional teachers who scored high in both tasks. Thus, task V2 provides
2 = 70.585; p = .000). A look at the items related to
the paper-pencil-test shows that the first task (PPT1) can be considered as
being on a lower difficulty level with only a slight statistical difference
between groups 2 = 10.530; p = .015). The analysis of the other three items
revealed that and , as well as pre-service
teachers, achieved a rather low score. Here the professional teachers once
again scored (significantly) higher in the test (PPT2: 2 2 =
2 = 33.412; p = .000 for all three items). All in all, with the
exception of the two obviously easier tasks (V1 & PPT2), participants in the
three earlier professionalization steps do not differ substantially and achieve a
rather low score, thus indicating that they did not recognise numerous errors.
However, the professional teachers clearly outperformed the other groups.
Comparing the average achievement of the four groups across all tasks
showed significant differences and explained nearly 28 % of the variance (F
= 31.481; p = .000; 2 = .281). A linear increase of performance related to the
different stages could not be found.
10
Figure 2: Chi-square Test: Correct item responses within subgroup (n = 242)
Whereas the above mentioned results were based on behavioural data,
the self-perception of the participants regarding their ability to identify and
correct errors will now be considered (Table 2). Altogether, the participants
mostly perceive their own knowledge on a rather high level. A comparison of
the four professionalisation steps showed significant differences (F = 7.763; p
= .000) with a moderate effect size (explained variance 2 = 0.88). Analysis
of the relation between perception of knowledge and actual performance
generates no significant correlation on group level.4 Looking at the aggregate
level over all groups, only a low correlation (r = .14*) can be ascertained.
Considering the test scores and a rather high level of agreement within the
questionnaire, this is not surprising. However, the professional teachers
apparently have both a higher ability (performance tests) and a more realistic
self-perception (questionnaire).
4 For professional teachers, the relation of self-perception and performance is moderate but
not significant (r = .33). The sample size of this subgroup (N = 33) could explain this (e.g.
Fan/Konold 2010).
11
Table 2: ANOVA: Self-perception of error diagnosis (n = 246)
Instrument
Stage
N
M
SD
F
P
η2
Self report
1-Teacher Training (Bachelor’s)
76
4.12
.55
7.763
.000
.088
2-Teacher Training (Master’s)
64
4.00
.52
3-Practical Training
73
3.83
.63
4-Professional Teachers
33
4.38
.58
Note: Scale from 1 = full disagreement to 6 = full agreement.
In order to test content knowledge or the ability to identify and correct
domain-specific errors one has to include (domain-related) prior knowledge
into analysis. A comparison of the mean values (test performance) due to
socio-demographic variables of the bachelor and master students is given
in Table 3. To avoid a possible bias, e.g. due to in-school practical experience
of pre-service or professional teachers, only the students were taken into
consideration. Here the school type, i.e. the different ways of achieving
The several types of business related prior knowledge showed significant
differences (F = 2.795; p = .043) but only with a small effect size (explained
variance η2 = .058). Although prior knowledge is commonly held as a
meaningful predictor for learning effectiveness, here the explained relation is
rather low. This issue will be investigated more deeply in the ongoing
research, e.g. by considering further variables and aspects.
Table 3: ANOVA: Prior knowledge of Bachelor and Master Students (n = 140)
Prior knowledge
N
M
SD
F
P
η2
No prior knowledge
44
2.16
1.38
2.795
.043
.058
Commercial secondary school
37
3.00
1.05
Dual vocational training
28
2.61
1.35
Commercial sec. school & Dual vocational training
31
2.68
1.49
Note: sum scores (range of 0 to 6)
4.2 Item-fit and IRT parameters
Firstly, the Item-Fit must be proved. -Fit and therefore a
weighted MNSQ (weighted mean square) is commonly specified within a
range of .75 to 1.33 (Adams/Khoo 1996; Bond/Fox 2001; quoted in Winther
2010: 152). Table 4 shows that all 6 items used in the performance tests are
within the above mentioned range, and moreover, they mostly have an
approximately exact fit (1.0). T-values have to be lower than 1.96 (at 5 %
level of significance), which, in fact, applies for all 6 items. Furthermore, this
post-hoc analysis of item-difficulties tends to ascertain three levels of
difficulty (see also section 4.1). So the participants had to handle two rather
12
easy tasks (V1 & PPT 1), two more intermediate tasks (PPT 2 & PPT3) and
two more difficult tasks (V2 & PPT4).
Table 4: IRT - Item parameter estimates
For the estimation of personal parameters the WLE (weighted likelihood)
is commonly used and recommended (Hartig/Kühnbach 2006; Rost 2004;
Wu 2005). Comparing item and personal parameters showed that the ability
parameters are nearly equated to the sum score of the items.
Figure 3: IRT - Wright map
A Wright map illustrating the estimates for the Rasch model is shown in
Figure 3. The item difficulty and the personal ability are represented within
the same dimension, meaning a response e.g. of item 2 indicates the highest
level of ability, whereas, a response e.g. of item 3 indicates a lower level of
ability. Although the additional use of an IRT-approach shows that the items
fit well, because the set only has 6 items, the added value of this analysis here
Item1
Correct
response
(in %)
ESTIMATE
Weighted
MNSQ
T
CI
Discrimi-
natory
Power
1
V1
85
-2.312
0.99
-0.1
(0.76, 1.24)
.49
2
V2
26
1.429
0.93
-0.8
(0.84, 1.16)
.65
3
PPT1
78
-1.721
1.00
-0.0
(0.81, 1.19)
.56
4
PPT2
41
0.482
0.95
-0.8
(0.87, 1.13)
.69
5
PPT3
34
0.892
1.09
1.2
(0.86, 1.14)
.56
6
PPT4
29
1.230
1.04
0.5
(0.85, 1.15)
.58
1 Note: V = Vignette; PPT = Paper-pencil-test.
Separation Reliability = 0.995
Chi-square test of parameter equality = 756.84, df = 5, Sig. Level = 0.000
13
is low.
5. Conclusion
Coming back to the objectives presented in the beginning we can state:
(1) Knowledge about student errors: the ability to identify and to correct errors
(also in relation to the professionalisation step) of both students and pre-
service teachers can, all in all, be seen as quiet low. Obviously school
relevant content cannot be applied in an adequate way. This could be due to
the fact that students quite often acquire inert knowledge which they are not
able to use or apply in their professional life (Gruber/Renkl 2000). In
contrast, the professional teachers scored very high in the testsconsequently
generating significant differences. However, there is no linear positive
relation between the stage of professionalisation and achievement in the
tests.
(2) A comparison of self-perception and actual performance showed low
correlation. All in all students and pre-service teachers tend to over-estimate
their own ability to diagnose errors, whereas professional teachers
apparently see their ability in a more realistic way (high perception and also
high performance). According to section 3, this can be seen as a further
argument for the use of performance data. Furthermore, a substantial
relation (r = .51***) was generated by correlating the two single
performance tests (video vignette and paper-pencil-test). For us this is an
indication that the two different instruments measure similar competences,
but do not provide exactly the same information. Thus, the vignettes could
provide information which the paper-pencil-test did not.
However, due to the fact that the sample has not been completed yet, the
findings should be seen as an initial tendency. As previously mentioned, this
article only concerns the first facet of PEC: the diagnosis. In particular, only
the ability to identify and to correct student errors, which is strongly related
to content knowledge (CK), has been analysed. In ongoing research, further
aspects of PEC and in particular, the handling of errors by the teacher, such
as getting to the bottom of an error cause or giving adequate feedback to
enhance learning, will be analysed. Also the relation between single PEC-
facets and further personality traits like self-regulation or self-efficacy must
be taken into account.
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