OUT-OF-FIELD TEACHING MATH TEACHERS AND THE
AMBIVALENT ROLE OF BELIEFS - REPORT FROM
INTERVIEWS WITH THESE TEACHERS
Marc Bosse & Günter Törner
University of Duisburg-Essen (Germany)
This inquiry deals with out-of-field teaching math teachers and the affectivemotivational components of their professional competence model. With the help of a
tool, which was designed to describe beliefs and orientations, the crucial
characteristics of these teachers’ affective-motivational components (orientation
systems) will be shaped. We want to state first results from qualitative interviews with
this group and deduce consequences for designing in-service training programs.
INTRODUCTION AND MOTIVATION
It is an internationally widely reported phenomenon that mathematics is often taught
by teachers, which have not been educated as mathematics teachers at university and
thus might have deficits in their knowledge repertoire. In Germany – for example –
we know that more than 80 % of the primary teachers have not attended relevant
mathematics courses at university – for various reasons, although there is tendency to
change this situation in the future by study regulations.
We would like to point out that many of these teachers are often highly engaged in
teaching and in most of the cases they are willing to teach mathematics, although the
principal is permitted to request the teachers’ commitment. Since in Germany nearly
all teachers have to teach at least two subjects, these persons possess a solid basis in
methodology and general didactics with respect to the relevant groups of students of
their school type.
In different school types the percentage is not as high as in the primary grades,
however, by no means negligible. Since there is a shortage in mathematics teachers,
even in the lower secondary grades we estimate that about 15 % of the teachers are
not originally mathematics teachers. On the other hand, since a principal is
responsible for organizing the teaching resources in his school, ministerial
administrations are seldom involved, and so the problem is concealed. It is easy to
understand that a school will not discuss their own restrictions in their daily school
management in public.
Abroad, there exist several papers devoted to the problem field of out-of-field
teaching (Ingersoll 1998, Ingersoll 2001) and the issues are often discussed to the
topic of minor qualifications or underqualifications of teachers (Ingersoll 1999 and
Friedman 2000). However, these reports do address the school subjects in general,
but not exclusively mathematics. Recently, the second author took a first inventory in
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the German school system – with respect to mathematics (Törner & Törner 2012),
nevertheless there is a lack in various research publications.
In addition, there is also a lack in remedies to lower the problems which are induced
by this situation. We are aware of the fact that at some places, courses are offered to
qualify teachers. But what are customer-adequate, sustainable and successful
initiatives, which should be recommended for in-service-teacher training courses?
Thus, our talk will be divided into three parts.
First, we will give some description on possible grounded theories, which help to
model our virtual ‘workplace’. It is more than a hypothesis – and confirmed by first
interviews: we do not only have to focus the cognitive elements in a teacher’s
knowledge account in terms of the subject matter knowledge (Shulman 1985,
Shulman 1986, Bromme 1992, Weinert 1999) and so the missing elements and
deficits, but we also pay high attention to the affective-motivational components,
among which are convictions, beliefs and elements of motivations and selfregulations, last not least emotions. Second, we will present some results we have
gained by leading half-standardized qualitative interviews with this group of teachers.
As this paper is limited to 10 pages, we will only refer to crucial results.
Third, we will draw some consequences for designing in-service training courses for
this group. Actually, we believe that these courses should incorporate at least three
elements which may be regarded as our starting hypothesis. These components are all
intertwined with beliefs:
(i) Textbooks in the hand of these teachers are important and indispensable. How outof-field teachers should be instructed to use textbooks, this is a non-trivial question.
(ii) Next, the teachers under discussion often lack of direct approach to the ‘heart’ of
mathematics, the problem solving. Thus, active doing of mathematics is substantial to
change views on mathematics.
(iii) Finally, since mathematics is a highly emotional enterprise, we have to discuss
how to handle affects and emotions and thus propagate an open approach to
mathematics by which we promote pluralistic world views on mathematics.
GROUNDED THEORIES
Professional Competence of Math Teachers
According to Blömeke et al. (2012), teachers’ competences can be modelled with a
so called competence profile. They refer to Shulman (1985; 1986), Weinert (1999),
and Bromme (1992) as cognitive components (Content Knowledge (CK),
Pedagogical Content Knowledge (PCK), General Pedagogical Knowledge (PK)) are
included in the model on the one side. Moreover, beliefs, professional motivation,
and self-regulation are also part of the model and are considered as affectivemotivational components on the other side.
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Affective-Motivational Characteristics
Because beliefs have an orientating function in regard to the teachers’ perception of
math lessons as well as retrieving CK, PCK and PK (cf. Blömeke et al. 2012), we
want to analyse in particular the affective-motivational components of the out-offield teaching teachers’ competence profile. We assume that these competence-facets
differ from those of ‘regular’ math teachers and have to be considered particularly
with regard to in-service-training programs. Therefore, we think that training out-offield teaching math teachers’ in cognitive components related topics is not enough.
To be more precise, we need to build an appropriate terminology as follows.
Beliefs and Orientations
There is no need to refer to the various classical articles on beliefs in LederPehkonen-Törner (2003) and in handbooks, but we want to point out some
conceptual extension, the orientations, offered by Schoenfeld (2011).
According to Schoenfeld, we use the term “orientations” as an inclusive term,
having in mind we want to describe the teachers’ preferences, attitudes, and beliefs
on many different levels and to gain insights as wide as possible. That means if we
use the term orientation, we do not have to be as narrow as when speaking of ‘world
views’ or beliefs. It is the first time that we use this terminology within our
interviews and we are convinced that this term is flexible as well as precise.
“The main point is that how people see things (their ‘world views’ and their more
specific attitudes and beliefs about things they interact with) shapes the very way they
interpret and react to them. In terms of cognitive mechanism, people’s orientations to
various situations influence what they perceive in various situations and how they orient
to them.” (Schoenfeld 2011)
Orientation Systems
The modified tool used for analysing and describing the affective-motivational
components of the teachers’ professional competence is able to characterize the socalled Orientation System of a person, consisting of overall four components: the
Orientation Objects (OO), the Associated Orientations (AO), the Degree of Intensity
(DoI) and the Evaluative-Affective Responses (EARs).
Orientation Objects (OO): As we have pointed out above, the affective-emotional or
behavioural consequences of a person’s orientation depends on a context or more
exactly on objects related to that context (e.g. division, the definition of a square,
angles; or algebra, calculus, geometry; or more abstract/general: math lessons,
learning of mathematics, learning of a special mathematical content). We like to
name these objects Orientation Objects as teachers can be oriented to them.
Associated Orientations (AO): The way a person sees abstract or concrete people or
things, i.e., the manner this person thinks about, interprets, communicates with, reacts
to, and interacts with orientation objects, shall be called the Associated Orientation.
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Further, we use the verb ‘to orient’ when describing the process of thinking,
interpreting, communicating, reacting, and interacting based on specific orientations.
Degree of Intensity (DoI): We have to be aware that not every AO of an individual
has the same intensity regarding certitude and consciousness. Thompson (1992) point
out that “one feature of beliefs [and therefore orientations] is that they can be held
with varying degrees of conviction” (ibid., p. 129). In order to consider this, we will
study the level of certitude when analysing AO. Besides, we assume that a higher
degree of intensity concerning consciousness will lead to a greater integration of
orientations and practice (cf. Törner 2003). That means we expect that a high level of
consciousness of an AO will have a stronger impact on teaching and pedagogical
behaviour. On the other side, a low level of consciousness is a marker for “hidden”
parameters characterizing out-of-field teaching math teachers as these did not have
any chance to become professionally aware of such orientations.
Evaluative-Affective Responses (EARs): It is visible that a person’s orientation can
accompany affective-emotional or behavioural consequences (cf. Ellis 1994). Thus,
we want to clarify if there are negative affective-emotional or behavioural
consequences (like fear or anxiety) and if there are specific positive affectiveemotional or behavioural patterns of out-of-field teaching math teachers.
METHODOLOGICAL CONSIDERATIONS
In the very first phase of our research activities, we needed to choose a research
method which is capable to fulfil the following tasks:
First, it is to clarify if the choice of orientation objects to be analysed is useful to
characterize specifics of out-of-field teaching math teachers. Further, we have to
prove if there are other OO that should be in scope of our research. Clarifying
predefined and discovering new OO should be done in one step.
Second, the relevant AO, their DoI, and the EARs of both predefined OO and those
being discovered during our inquiry have to be identified. Thus, a high grade of
flexibility is needed when gathering data.
For these reasons, we decided to lead qualitative, half-standardized, video-based
face-to-face interviews with the aid of an interview-guideline (cf. Lamnek 2010). On
the one hand, the guideline provides the possibility to discuss different predefined
OO. If it becomes clear that one of these are not helpful when characterizing out-offield teaching math teachers, the interview item can be left out. An item can also be
neglected for individual reasons, e.g. when it becomes obvious that a specific OO
does not play any role in a certain type of school, a certain federal state, or just in the
specific professional life of the interviewee. On the other hand, the frame of the
interview-guideline can be left or can be added if an interesting new OO occurs while
leading the interview.
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By mainly asking open questions, the EARs of the interviewee can be noticed. In
order to get more data about EARs, which can also manifest themselves in mimic or
gestures, the interviews are recorded on video.
The interviews were led in three different federal states in Germany and took round
about thirty minutes each, owing to the limited time being available to the
interviewees. We have interviewed all in all nine out-of-field teaching math teachers
of different types of school: primary school (n=2), middle-school (n=5) and
comprehensive school (n=2). The interviewees took part voluntarily. Five of them
were asked because of attending an in-service training in their respective states. We
have to consider these circumstances as the teachers’ orientations may have been
affected by the training.
RESULTS
Concerning Orientations towards Mathematics and Mathematics Education
We must underline that in most cases the interviewed teachers mainly saw
mathematics in educational contexts: Their orientation towards mathematics is
exclusively restricted to the mathematical content they are confronted with in school,
lessons, or in-service training. Nobody could refer to mathematical applications or
topics out of curricular contexts or the everyday life of the students. Therefore, we
can proceed on the following assumption: When the interviewees referred to
mathematics, they primarily refer to its educational role in the teachers’ individualoccupational domain. When we asked for a kind of definition of mathematics, every
interviewee explained mathematics is something logical and refers to structural work
and thinking. In this regard we could observe three different types of this orientation:
Teachers with orientations of type “A” pessimistically think that the structural and
logical character of mathematics is the equivalent of being an abstract construct that
is not relevant for any field of everyday life (or possible applications). For example,
one of those teachers claimed that fractions and powers do not play a role in students’
life. Furthermore, they do not see any kind of sense in dealing with mathematics due
to missing relevance for everyday life.
Teachers with orientations of type “B” conclude that mathematics is a tool you can
use for solving problems and various applications (e.g. banking, paying when
shopping, traveling with train or bus, painting walls). Only one of those teachers
referred to the job-preparing function of mathematics education as one application.
Teachers with orientations of type “C” realize that mathematics is not only a
possibility to solve given problems as an application but it is an instrument of
discovering their surroundings and making the world accessible. They emphasize the
potential of doing research with mathematics and mention mathematical objects, e.g.
numbers, with which they could explore their living environment.
If the interviewees are specifically asked if mathematics plays a role in their students’
life today, they have the opinion that it is something which is meaningful in school
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but nothing which they really need for ‘surviving’ in society. Even types “B” and “C”
relativize their orientations (also admitting that some basics have to be learned). It
seems to be a difference if teachers refer to the role mathematics plays generally in a
normative perspective or if they refer to the role mathematics plays for students they
personally know.
Further, it became clear that out-of-field teaching math teachers may have different
orientations towards problem solving and modeling (approaches and results!) in
mathematics education:
Interviewee 1: “[…] Mathematics is very, very clearly structured. There is no grey,
there are mainly black and white. Right and wrong. It is relatively clear
which tool to use for a solving a specific problem.”
Interviewee 2: “[…] in mathematics education, if I […] don’t see that there are other
approaches than that one I had in mind and I insist on my approach, then I
can ruin a lot.”
Every interviewee refers to deficits in PCK. They primarily named missing
knowledge about the students’ mathematical learning processes and the connected
problems students can be confronted with. Furthermore, they complain about missing
diagnostic methods to recognize these problems and in addition, to anticipate possible
problems when preparing math lessons. One teacher explains that “it is difficult to
recognize what cannot be understood about my solution”. Another one adds that she
had problems with finding alternative approaches and with implementing students’
approaches she did not expect to be named in lessons.
Besides, they are aware of the (curricular) requirement to consider open problems,
applications, modeling and problem solving, whereupon orientations of type “B” and
“C” are attended by a higher level of consciousness. In this regard they refer to
missing knowledge about possible applications and furthermore, of student-friendly
ways to consider these applications in their math lessons:
Interviewee 1: “[…] I lack the expertise to find applications. […] Where can I find
illustrative and suitable examples in everyday life? […]”
Interviewee 2: “It is hard for me to implement applications in my lessons, although I
attach importance to it.”
Interviewee 3: “The children are quite good in solving 13 problems of the same type
with the same strategy. But I urgently need ideas how to implement [new]
applications.”
We can assume that missing knowledge about how to implement applications in
lessons compulsorily leads to standardized problem solving schemes and students not
being able to mathematically work in unknown and new contexts. Even if those
teachers’ orientation towards mathematics is of type “B” and “C”, they would surely
not be able to transfer those useful orientations for lack of knowledge about modeling
and applications.
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Additionally, the teachers compared out-of-field teaching in mathematics with out-offield teaching in other subjects by referring to their studied subject(s). One
interviewee underlined that she could not draw parallels to her studied subject,
German, when teaching math, as she could do when teaching English.
Concerning Orientations towards Teaching Material, Textbooks and other
Support
The theory of textbooks in the hand of out-of-field teaching teachers being important
and indispensable could be confirmed. Everyone, whose orientation towards
mathematics is not of type “A”, thinks that textbooks are helpful and supportive when
preparing for lessons. Some of the teachers fully adopt the content of their textbooks
and implement it in their lessons. One interviewee explained that he prepared about
70 % of his lessons with the help of the textbook. He added that he usually checked if
a subject matter was a content of the textbook and if it was not, he would not
implement this in his lessons. Some of the teachers cause the mainly use of the book
by referring to the prestructured subject matter, to the offering of methods which can
be used, and to the information about what is important for understanding a specific
subject matter according to the textbook.
Besides, some teachers refer to missing elements in their textbooks like activelearning material. More specific, there was “no textbook which can fully cope with
the challenges of everyday school life”. Some teachers mention the use of additional
material (e.g. SINUS material, PIK AS material, worksheets, and so forth).
A teacher with an orientation of type “A” has the opinion that textbooks are not logic
for non-mathematicians, that they are badly comprehensible, and that the presented
problems are too artificial. In this case, it should have to been clarified if a presumed
bad textbook has an impact on the teachers’ orientation as it enhances (or creates?)
the factual-pessimistic character or if the orientation let the teacher to think in this
manner.
In addition, almost every interviewee stated that they got information and support by
their (studied and regular-teaching) colleagues. Some explained that they would
compare notes with each other on demand, others reported of institutionalized coworker structures.
Concerning Evaluative-Affective Responses
As mentioned above we have observed evaluative-affective responses accompanied
by the teachers’ directly or indirectly articulated orientations. We can group those
responses in three clusters:
Positive Rational Evaluative-Affective Responses (RP-EARs): First, we could witness
EARs which are positive on the one hand but rational and helpful according to
mathematics education and didactical and pedagogical behaviour on the other hand.
The interviewed teachers articulated those PR-EARs by saying: “Doing math is fun.”;
“I have no anxiety when doing or teaching math.”; “I think mathematics is
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fascinating and exciting.”; “I can really get into math.”; “I’ve got a positive gut
feeling.”. They give reasons for these positive EARs by stating that mathematics is
helpful when solving problems (correlation to orientation type “B”), that mathematics
is an instrument for exploring and researching (correlation to orientation type “C”)
and that mathematics is something you can do with children (no correlation to
orientation type “A”). Therefore, we name this cluster rational as the positive EARs
are reasoned by education-supporting orientations. We have to underline that some
teachers’ PR-EARs can be observed more frequently when the interviewees were
talking about doing mathematics with children.
Positive Irrational Evaluative-Affective Responses (PI-EARs): In contrast, we could
observe positive EARs which are not helpful for mathematics education as they limit
the view on what mathematics actually is. Nevertheless, the observed EARs are
positive, e.g. fun. The teachers reason their positive EARs by cutting mathematics
down to “something being always clear”:
Interviewee: “It is so much fun to teach math. It’s because of … (pause) … Math is so
structured, you’ve always one way and one result and you know: that’s
right or that’s wrong. Well, therefore, I find it easy to do math and it is
fun.”
We name these positive EARs irrational as they do not contribute to a desirable
orientation of a math teacher towards mathematics and doing mathematics.
Negative Evaluative-Affective Responses (N-EARs): In addition, we observed
negative EARs which are (obviously) exclusively irrational and not helpful for
didactical and pedagogical behaviour in mathematics education. The interviewed
teachers articulated those N-EARs by saying: “Doing math is not fun.”; “I feel
uncomfortable”; “Math is not exciting at all.”; “I am overtaxed.”; “I am often
unconfident.”; “Doing math is a lot of stress and always connected to high amount of
work.”; “It is embarrassing to admit not to know how to teach correctly”; “I’ve got a
bad conscience and feel responsible for the children”; “Some colleagues fear others
saying they don’t know the curriculum of class 7.” It is interesting that nobody of the
interviewees mentioned having fear when teaching as we had expected. One teacher
expressively explained: “I’ve no fear, but I’m stressed.” Either they did not admit
having fear in the interview due to its public context or we have to abolish this
hypothesis. From gestures and mimic one can conclude that they really do not have
those negative emotional N-EARs when talking about affects before entering the
classroom or while teaching.
IMPLICATIONS
What do the results above now mean for designing in-service training programs? We
want to specify central implications which should be considered in the wake of our
first interview study:
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Mathematics – World View Aspects: If we want to enhance the out-of-field teaching
math teachers’ orientations with a view to better mathematics education, we have to
make clear that mathematics is more than a school subject, can be something relevant
for children’s everyday life and future life, and is not something you can only really
cope with when you have studied math. Further we have to explain that mathematics
is more than solving predefined tasks but a way to explore, discover and access the
world which means that you can do a kind of ‘research’ with mathematics. That
means we have to expand the teachers’ individual-occupational orientation to
mathematics (in school) to a more professional one. We can assume that you have to
consider orientations associated to mathematics as well as those associated to
mathematics education if you want to rectify the full orientation system.
Doing Mathematics: We have to show how mathematics can provide more than one
approach, one tool, or one solution for a certain problem. Furthermore, parallels to
other (studied) school subjects should be picked out as a central theme so that
mathematics is not seen as a closed subject without any relations to assumed nonmathematical problems anymore.
Deficits with Respect to Professional Competence: We could primarily observe
deficits in orientations associated to PCK-related orientation objects. In the wake of
our Interviews the interviewed teachers should be trained in
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
modelling and considering (student-friendly) applications as these enhance a
useful orientation towards mathematics,
mathematical learning-processes (as this training offers the possibilities to
get insights into what students possibly cannot understand, to get ideas about
various ways of teaching the same mathematical subject matter and to cope
with unexpected students’ approaches and solutions),
implementing didactical differentiation in mathematics education.
The Role of EARs: Concerning EARs we must be aware that positive EARs like fun
are not automatically helpful for mathematical education, for example, if fun is
caused by a limited orientation towards mathematics. PR-EARs could be observed in
context of orientations of type “B” and “C” which means, that in-service training
should foster such orientations. Besides, we recognized that the N-EARs fear and
anxiety are not that relevant as we had expected. Nevertheless, some interviewed
teachers reported that they feel uncomfortable and in the first instance overtaxed.
Overloading teachers must be supported by institutional-cooperative structures, e.g.
professional learning communities, tandems and continuous training.
The Role of Textbooks: Our considerations concerning the role of textbooks could be
confirmed. Textbooks added by other material are the out-of-field teaching teachers’
crucial source of planning and performing mathematics education. Therefore
textbooks play an important role concerning the teachers’ orientations as the books
are a dominant medium in defining orientations associated to specific mathematical
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and didactical contents. Nevertheless, we should not lose sight of the (studied and
‘regular’-teaching) colleagues’ role as they have influence on orientations, too.
CONCLUSIONS
Concluding we want to point out some open problems.
The Role of Anxiety: We could not observe any N-EARs like fear or anxiety.
Nevertheless, teachers we had interviewed in a previous inquiry mentioned these NEARs. It is our task to go into that matter in further studies. Therefore, we will have
to ask further questions concerning this: Do those N-EARs play a role in context of
out-of-field teaching or are they some kind of individual singularities? And if they
are not – how do they arise and how can we eliminate them with in-service training?
The Role of Domain Specificity: In this inquiry we have not considered the subject
profile of the teachers but respect this: Is there a difference between an out-of-field
teaching math teacher who studied physics and engineering and one who studied
social studies and German? We have to assume that there are differences and that the
domain specificity influences the orientations associated to mathematics.
Technical Terms and Systemisation: Our study is a first try to investigate a so far
unexplored problem. We have created, introduced, and proposed technical terms and
systemizations in order to notionally determine the shape and the quality of
orientations. This terminology and systemization have to prove of value in further
studies.
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