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Why use multiple representations in the mathematics classroom?
Views of English and German pre-service teachers
Running head: Why use multiple representations?
Abstract Dealing with multiple representations and their connections plays a key role for
learners to build up conceptual knowledge in the mathematics classroom. Hence, professional
knowledge and views of mathematics teachers regarding the use of multiple representations
certainly merit attention. In particular, investigating such views of pre-service teachers affords
identifying corresponding needs for teacher education. However, specific empirical research
is scarce. Taking into account the possible role of culture, this study consequently focuses on
views about using multiple representations held by more than 100 English and more than 200
German pre-service teachers. The results indicate that there are culture-dependent aspects of
pre-service teachers’ views, but also that there are common needs for professional develop-
ment.
Keywords: fractions, multiple representations, pre-service teachers, trans-national design,
views
INTRODUCTION
There may be many good reasons for using multiple representations for teaching in general,
such as for instance the possibility of taking into account the learners’ individual differences
and preferences. However, since representations play a special role in mathematics, there are
also discipline-specific reasons for using multiple representations. As mathematical concepts
can only be accessed through representations, they are crucial for the construction processes of
the learners’ conceptual understanding (Duval, 2006; Goldin & Shteingold, 2001). Awareness
of such discipline-specific reasons can clearly influence teachers’ abilities to design rich
learning opportunities. For instance, acknowledging that only the combination of different
2
representations affords the development of a rich concept image (Tall, 1988) may better sup-
port teachers in designing mathematical activities than seeing the main purpose of multiple
representations in keeping pupils’ interest. Hence, specific knowledge and views about using
multiple representations merit attention – in particular when it comes to professional devel-
opment. Exploring such views of pre-service teachers at the beginning of their teacher educa-
tion affords the identification of specific needs and prerequisites. Consequently, this study
focuses on pre-service teachers’ views on using multiple representations in the mathematics
classroom. We use a trans-national design with English and German pre-service teachers to
explore whether these views are strongly culture-bound. In line with a multi-layer model of
professional knowledge, such views are examined on different levels of globality to find out
how consistent general views on using multiple representations are with corresponding content
domain- and task-specific views. For the content domain-specific parts of this study we chose
the domain of fractions because of the high relevance of multiple representations specifically
in this content domain (e.g., Ball, 1993; Brenner, Herman, Ho, & Zimmer, 1999). Furthermore,
possible interrelations with specific content knowledge (CK) are explored. In the following
first section, we introduce the theoretical background of this study; the second and third sec-
tions present research questions and the research design. Results are reported in the fourth
section and discussed in the fifth section.
THREORTICAL BACKGROUND
The theoretical background of this study includes several aspects which constitute the structure
of this section. First, we focus on the special role that representations play for teaching and
learning mathematics. The second part is about (pre-service) teachers’ views on dealing with
multiple representations in the mathematics classroom as aspects of pedagogical content views
in the context of a model for teacher professional knowledge. Particular emphasis is put on
reasons why different levels of globality should be taken into account when exploring such
3
views. After giving some answers to the question as to why this study involves pre-service
teachers of two different countries, the last part of this section revolves around possible in-
terrelations between the views in the centre of this study and specific CK.
Multiple representations in the mathematics classroom
In mathematics and consequently also in mathematics classrooms representations play a spe-
cial role. According to Duval (2006) mathematical objects are not directly accessible and hence
experts as well as learners have no choice other than using representations when dealing with
those objects. We take the notion “representation” to mean something which stands for
something else – in this case for an “invisible” mathematical object (Duval, 2006; Goldin &
Shteingold, 2001). Figure 1 shows an example of some representations for a fraction in picto-
rial, symbolic or language-based formats.
Figure 1: Some representations for a fraction
The example illustrates that usually a single representation can only emphasise some properties
of a corresponding mathematical object. For instance, the string of pearls emphasises the ratio
aspect of the fraction, whereas the pie chart rather shows the fraction as being a part of a whole.
Hence, multiple representations which can complement each other are usually needed for the
development of an appropriate concept image (Ainsworth, 2006; Elia, Panaoura, Eracleous, &
Gagatsis, 2007; Even, 1990; Tall, 1988; Tripathi, 2008). Consequently, multiple representa-
“three
quarters“
×
+
“the proportion of goals
scored by the winners, if
the final score is 3 – 1”
“three out
of four“
−
“what’s left from the
whole after I’ve taken
away one quarter”
“the number in the
middle between 1
2 and
1”
4
tions appear to be important for the construction processes of mathematical understanding and
it is often emphasised that the ablilty to deal with them flexibly is essential for successful
mathematical problem solving (e.g., Acevedo Nistal, van Dooren, Clareboot, Elen, & Ver-
schaffel, 2009; Even, 1990). And indeed, there is substantial empirical evidence for the posi-
tive effects of learning with multiple representations on pupils’ conceptual understanding
(Ainsworth, 2006; Rau, Aleven, & Rummel, 2009; Schnotz & Bannert, 2003). Rau and col-
leagues (2009) for instance conducted a study with intelligent tutoring systems and found that
pupils learned more with multiple pictorial (i.e., graphical) representations of fractions than
with a single pictorial representation – but only when prompted to self-explain how the picto-
rial representations relate to the symbolic fraction representations. The fact that this positive
result comes with a certain restriction is not a coincidence: various studies have shown that
providing pupils with multiple representations does not per se foster pupils’ learning, since
integrating and connecting the different representations is usually difficult for pupils (Ains-
worth, 2006; Leikin, Leikin, Waisman, & Shaul, 2013; van der Meij & de Jong, 2006). It
should also be noted that a representation does not stand for a mathematical object in an ob-
vious way. This connection depends on interpretation and negotiation processes (Gravemeijer,
Lehrer, van Oers, & Verschaffel, 2002; Meira, 1998) and it is usually created in the interaction
of the participants in a learning environment (Steinbring, 2000). Therefore, learners need to be
supported in constructing meaning with respect to every single mathematical representation
and also in making connections between different representations. Findings from several
studies underpin this reasoning by showing that pupils need to be encouraged to actively create
connections between representations in order to benefit from using multiple representations
(Bodemer & Faust, 2006; Renkl, Berthold, Große, & Schwonke, 2013). To sum up, fostering
the learners’ competencies in dealing flexibly with multiple representations should be a central
goal in the mathematics classroom (Graham, Pfannkuch, & Thomas, 2009). Corresponding
5
objectives can be found in many national standards, where dealing with representations is
described as an important aspect of mathematical competence (e.g., KMK, 2003; NCTM,
2000; Qualifications and Curriculum Authority, 2007). In the English national curriculum for
mathematics “representing” is considered as one of the “essential skills and processes in
mathematics that pupils need to learn to make progress” (Qualifications and Curriculum Au-
thority, 2007, p. 142). Even more explicitly, the German national standards characterise “using
mathematical representations” as one out of six general mathematical competences, which
includes “applying, interpreting and distinguishing different representations for mathematical
objects and situations”, “recognising connections between representations” and “choosing
different representations depending on the situation and purpose and changing between them”
(KMK, 2003, p 8, translation by the authors).
Whereas “using multiple representations” is an overarching idea which is relevant for all parts
of mathematics (see Kuntze, Lerman, Murphy, Kurz-Milcke, Siller, & Winbourne, 2011),
there are content domains in which multiple representations are exceedingly significant for
pupils’ learning. “Fractions” – which is the focus of the domain specific parts of this study – is
considered as one of them (e.g., Ball, 1993; Brenner et al., 1999; Siegler et al., 2010). As
different representations can emphasise different core aspects of the concept of fraction (e.g.,
part-whole, ratio, operator, quotient, etc., see e.g., Charalambous & Pitta-Pantazi, 2007; Malle,
2004; cf. also Figure 1), the development of an appropriate multi-faceted concept image of
fractions requires integrating and connecting multiple representations. In particular, fostering
the pupils’ abilities to match symbolic-numerical representations with appropriate pictorial
(diagrams, sketches, illustrations) and content-related representations such as real world situ-
ations can play an important role for sustainable learning of fraction calculations (e.g., Ball,
1993; Malle, 2004). Taking the example shown in Figure 1, it may for instance support con-
6
ceptual understanding to establish a connection between the representation 1 2
⁄ + 1 4
⁄ and
the pie chart representation where a quarter pie is added to a half of a pie.
(Pre-service) teachers’ views on dealing with multiple representations
Since it is well-known that teachers’ views about teaching and learning mathematics influence
their instructional practice and what their pupils learn (e.g., Kunter, Baumert, Blum, Klus-
mann, Krauss, & Neubrand, 2011; McLeod & McLeod, 2002), it is also likely that in particular
their views about dealing with multiple representations play an important role. For exploring
such views this study uses the model of teacher professional knowledge which is shown in
Figure 2 (see Kuntze, 2012).
Figure 2: Overview model of components of professional knowledge (Kuntze, 2012, p. 275)
It integrates three dimensions according to which different components of mathematics
teachers’ professional knowledge can be structured. Considering the difficulty of distin-
guishing knowledge from beliefs/views with respect to mathematics instruction (Lerman,
2001; Pajares, 1992; Pepin, 1999), in a pragmatic approach such views are included as aspects
of professional knowledge and consequently the spectrum between knowledge and views
pedagogical knowledge
pedagogical content knowledge
content knowledge
curricular knowledge
generalised/global
content domain-specific
related to particular
content
related to a particular
instructional situation
knowledge
beliefs/views
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constitutes one of those dimensions. Hence, it should be noted that the notions knowledge and
views are not considered to be strictly separable. Both of the notions are used in this work, since
some components may be seen as being rather views or rather knowledge. As the term beliefs is
usually used for a very broad construct (e.g., Pehkonen & Pietilä, 2003), we prefer the notion of
views, which may be seen as a term which describes certain aspects of beliefs. For instance, a
cognitive constructivist view of teaching and learning (see e.g. Staub & Stern, 2002) may be
described as a component of professional knowledge which is rather on the side of views than
of knowledge, and it may be part of a teacher’s beliefs. A second dimension affords structuring
aspects of professional knowledge according to the domains by Shulman (1986) which form
the basis of many recent models of teacher professional knowledge (e.g., Ball, Thames, &
Phelps, 2008; Kunter et al., 2011). It is of course possible to refine these categories, for in-
stance by using the domains of Mathematical Knowledge for Teaching suggested by Ball,
Thames & Phelps (2008). Even if the cells in Figure 2 clearly have overlaps, an advantage of
the model (Kuntze, 2012) used here compared to others lies in the possibility to structure
components of professional knowledge also with respect to their globality (see Törner, 2002),
which constitutes its third dimension. Knowledge and views about teaching mathematics can
be very global – such as for instance views about the discipline of mathematics – but they can
also be specific to a certain content domain, a particular content or even to a particular in-
structional situation.
Regarding views on dealing with multiple representations in the mathematics classroom, the
distinction of different levels of globality is very useful: Firstly, there are general views about
the role that multiple representations play for pupils’ learning in mathematics and in particular
views on reasons for using multiple representations in the mathematics classroom. Perceptions
of such reasons have probably a significant impact on how teachers design learning opportu-
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nities using multiple representations. For instance, seeing the main purpose of multiple rep-
resentations in making mathematics instruction fun and diverse may serve the design of con-
ceptually rich mathematical activities less than being aware of the fact that usually the interplay
of different representations is needed for the development of an appropriate concept image.
Secondly, content domain-specific views about how to deal with representations when teach-
ing fractions merit further attention. As reasoned above, the development of a concept image of
fractions which is sufficiently multi-faceted requires integrating and connecting multiple
representations. Hence, focusing exclusively on one “standard” pictorial representation like
“the pizza” probably does not foster deep conceptual understanding of fractions. However,
such views about how to deal with multiple representations when teaching fractions might still
be different from perceptions of what role multiple representations should play in particular
tasks about fractions. So thirdly, we focus on views about how multiple representations can
foster pupils’ learning in specific tasks, which may be seen as views related to a particular
content. For instance, being aware of opportunities which can encourage pupils to actively
create connections between representations of fractions and their operations may better support
teachers in choosing and designing tasks with a high learning potential. In contrast, focusing on
using pictorial representations of fractions for the sole purpose of encouraging pupils in en-
gaging with the particular task, even if those representations are not useful for the solution,
may be less helpful for designing conceptually rich learning opportunities. Fourthly, there are
even more situated views about dealing with multiple representations, namely conceptions
about how to use representations in specific instructional situations. All these views about
dealing with multiple representations in the mathematics classroom on different levels of
globality are considered to be part of a teacher’s pedagogical content views. Those on the first
three levels of globality are in the focus of this study and therefore, the corresponding com-
ponents are highlighted in the model of professional knowledge shown in Figure 2. However,
9
of course the most situated level of teachers’ views should not be neglected. In a larger study
about teacher professional knowledge, which forms the framework of the research presented
here, we also address views about dealing with multiple representations of fractions in specific
classroom situations (see Dreher & Kuntze, 2015).
Assessing views on different levels of globality affords exploring to what degree these views
are interconnected consistently. This is particularly interesting, since there is some evidence
suggesting that the development of professional knowledge for teaching mathematics is ac-
companied by a growth in consistency across levels of globality: Investigating how practicing
teachers acquire professional knowledge, Doerr and Lerman (2009) have identified “the
teachers’ learning as a recurring flow between the procedural and the conceptual” (p. 439),
where specific responses to problem situations were seen as pedagogical procedures and more
general principles for actions as pedagogical concepts. “Procedural knowledge” is thus more
situated and less global than “conceptual knowledge”. Hence, the question as to what degree
teachers’ views on different levels of globality are consistent merits attention and might serve
as an indicator for the development of professional knowledge. Inconsistency across levels of
globality with respect to views on dealing with multiple representations could mean for in-
stance that a teacher acknowledges in general that multiple representations should be con-
nected for the pupils to develop an appropriate concept image, but nevertheless he or she thinks
that teaching fractions works best when concentrating on a single pictorial representation.
Consequently, research into teachers’ professional knowledge – in this case into pre-service
teachers’ views on dealing with multiple representations in the mathematics classroom –
should take into account different levels of globality and explore also to which degree these
levels are interconnected or consistent. Yet, research into views on dealing with multiple
representations in the mathematics classroom is scarce. There are to our knowledge only
10
studies regarding some selected aspects of views on dealing with multiple representations,
such as findings by Ball (1993) concerning global beliefs of teachers about pictorial repre-
sentations. These findings suggest that the interviewed (American) teachers attached great
importance to the motivational potential of pictorial representations, whereas they rather ne-
glected their role for conceptual learning. Another qualitative study has focused on teachers’
expectations of pupils being able to perform different conversions of representations (Bossé,
Adu-Gyamfi, & Cheetham, 2011). However, as reasoned above, there is a need for assessing
views on dealing with multiple representations in a more multi-faceted manner taking into
account global and more situated views as well as their interconnections in order to identify
needs for professional development. It may be assumed that pre-service teachers even at the
beginning of their teacher education already hold certain views about dealing with multiple
representations which may in particular be shaped by their experiences as pupils. Therefore,
analysing pre-service teachers’ prerequisites regarding such views and their degree of con-
sistency could support the design of appropriate learning opportunities for their teacher edu-
cation.
The possible role of culture
Exploring teachers’ views, one should bear in mind that some aspects may be cul-
ture-dependent (Pepin, 1999). The results of Pepin’s qualitative research into epistemologies,
beliefs and conceptions of mathematics teaching and learning in England, France and Germany
suggest that teachers’ beliefs were influenced by their cultural environment. Her findings
include for instance that for English teachers an individualistic and child-centred view was
dominant, whereas the conception of mathematics revealed by the investigated German
teachers was relatively formal. These global tendencies might also become evident in teachers’
perceptions of the role that multiple representations play in the mathematics classroom. Since
pre-service teachers’ views at the beginning of their teacher education may in particular be
11
influenced by their experiences as pupils, the investigation of their views should be seen in the
light of the characteristics of mathematics teaching in their countries. Kaiser (2002) pointed out
the strong influence of different educational philosophies in England and Germany on the
mathematics classroom. Based on her ethnographic study she described typical aspects of
mathematics teaching in England and Germany in a contrasting way, which are briefly sum-
marised in the following. According to Kaiser’s study, the most important principle of the
English education philosophy is the high priority of the individual. Hence, in the English
mathematics classroom long phases of individual work are typical, where great emphasis is put
on the pupils’ own ways of problem solving with openness towards individual versions of
notation and formulation. In Germany however, class discussion in which ideas are developed
collectively is usually a dominant teaching-and-learning style and thus a precise mathematical
language which is comprehensible by all learners and common notation is seen as being more
important. Following Kaiser (2002), distinct characteristics of mathematics teaching in Eng-
land and Germany can furthermore be put down to contrasting understandings of the role of
theory for teaching mathematics. The predominantly scientific understanding of theory in
Germany typically leads to great significance of rules, formulae, and arithmetic algorithms
which often have to be learned by heart, whereas a rather pragmatic understanding of theory for
teaching mathematics in England goes along with a focus on work with examples and minor
relevance of rules and standard algorithms.
Bearing in mind these findings, comparing views of pre-service teachers from England and
Germany on dealing with multiple representations may give some insight into which aspects of
such views and corresponding needs might be rather culture-dependent versus cul-
ture-independent. This approach can also give feedback about the research instrument used
with respect to its culture-sensitivity and validity.
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The possible role of specific CK
It may be assumed that specific CK is interrelated with views on dealing with multiple repre-
sentations, in particular when it comes to tasks-specific views, since evaluating the learning
potential of a task requires a content-specific understanding of the given representations and
their interplay. Especially the ability to see and reflect connections between different repre-
sentations and – particularly in the case of fractions – to match symbolic-numerical represen-
tations with appropriate pictorial and content-related representations appears thus to be rele-
vant. Reviewing the released items (Ball & Hill, 2008) of the survey instrument developed by
Hill, Schilling, and Ball (2004) in order to measure mathematical knowledge for teaching
shows that there are items included which assess this special kind of CK. However it is not
conceptualised separately in that study. Hill and colleagues described a way of categorising
their items based on different sorts of teachers’ tasks, where “choosing representations” is one
of them (Hill et al., 2004), but in the light of the reasoning above, “choosing representations”
does not capture the full spectrum of what is involved in dealing with multiple representations
in the mathematics classroom. Although the COACTIV study (Kunter et al., 2011), which also
focuses on teachers’ professional knowledge, included “explaining and representing” as one of
three components in its model for pedagogical content knowledge, as far as CK is concerned,
representations appear not to play any explicit role in the research design. Consequently, there
is a need for developing test instruments assessing specific CK regarding connections between
multiple representations.
RESEARCH QUESTIONS
According to the need for research pointed out in the previous section, the study presented here
aims to provide evidence for the following research questions:
13
1) What views do English and German pre-service teachers have on the role of multiple
representations for learning mathematics? In particular:
a. How much importance do they attach to different (global) reasons for using
multiple representations in the mathematics classroom?
b. What (content domain-specific) views about how to deal with representations
when teaching fractions do they have?
c. What (content-specific) views related to the learning potential of tasks focusing
on conversions of representations, in comparison with tasks including rather
unhelpful pictorial representations do the pre-service teachers have?
2) To what degree are their views on those different levels of globality interrelated? Is
there evidence of inconsistencies, which may point to needs for further professional
development?
3) Do inter-cultural comparisons reveal any differences regarding such views and corre-
sponding needs, i.e.: Are there any indications for which aspects might be rather cul-
ture-dependent than culture-independent?
4) What specific CK about dealing with multiple representations in the domain of frac-
tions do English and German pre-service teachers have? Is this CK interrelated with
their task-specific views on dealing with multiple representations?
SAMPLE AND METHODS
For answering these research questions, a corresponding questionnaire was designed in Ger-
man and was then translated into English. This translation was examined carefully by two
native speakers of English, one of whom is also fluent in German and has taught mathematics
both in the UK and in Germany. The questionnaire is based on a previous version which was
14
tested in a pilot study (see Kuntze & Dreher, 2014) and subsequently developed further. At the
beginning of the questionnaire there were explanations of the notions “representation” and
“pictorial representation” in a mathematical context given in order to reach a similar under-
standing of these key terms for the study.
The questionnaire was administered to 139 English (99 female, 22 male, 18 without data) and
219 German (183 female, 26 male, 10 without data) pre-service teachers before the beginning
of a course at their university. They completed the questionnaire in the presence of the first
author or a research assistant and they were given as much time as they needed. The English
subsample consists of pre-service teachers studying at London South Bank University,
whereas the German participants were students at Ludwigsburg University of Education. The
pre-service teachers of both subsamples were preparing to teach at primary level. The English
participants had a mean age of 27.9 years (SD = 6.9), while the German participants were on
the average 20.7 years old (SD = 2.5), but (with only a few exceptions in both samples) all the
participants were at the beginning of their first year of teacher education at university. The age
difference between the English and the German pre-service teachers in our study is due to the
different systems of teacher education in the two countries. The English pre-service teachers
have already finished some university degree before starting a one-year teacher training, which
specialises them in teaching mathematics at primary level. However, the broad majority of
these English pre-service teachers had not been studying mathematics since secondary school.
The German pre-service teachers enter their university studies directly after secondary school,
starting a 4- to 5-year teacher education program. These German participants studied mathe-
matics as one out of two school subjects.
Corresponding to the research questions for this study four parts of the questionnaire were
included in the evaluations: The first three parts were assessing views on using multiple rep-
resentations on different levels of globality and moreover there was a section about the par-
15
ticipants’ specific CK, namely their ability to match symbolic-numerical representations of
fractions with appropriate pictorial and content-related representations. In the following, these
four questionnaire sections are described in more detail.
In order to explore how much importance the English and German pre-service teachers attach
to different reasons for using multiple representations in the mathematics classroom, the par-
ticipants were asked to rate the significance of possible reasons on a five-point Likert scale
(from “not important” to “extremely important”). The selection of four different types of
reasons which are shown in Table 1 is not only drawn from theoretical considerations and
literature review, but is also based on the results of a pilot study in which the participants could
give reasons for using multiple representations in the mathematics classroom in an open format
(see Dreher, 2012). The appropriateness of the corresponding model encompassing four con-
structs was examined empirically by means of a confirmatory factor analysis (CFA) using
AMOS 21 software (Arbuckle, 2012), which is reported in the results section.
Construct (identifier) Sample item # items
Necessity for mathemat-
ical understanding Enhancing the ability to change from one representation
to another is essential for the development of mathe-
matical understanding.
4
Motivation and interest They make it easier to keep pupils’ attention and in-
terest. 3
Supporting remembering Pupils can use pictorial representations as mnemonics. 3
Learning types and input
channels Different learning types and input channels can be
addressed. 3
Table 1: Scales regarding reasons for using multiple representations in the mathematics
classroom
Based on the above reasoning about domain-specific views on the role of multiple represen-
tations for teaching fractions a corresponding questionnaire section was designed which fo-
cuses on the five constructs presented in Table 2. The participants were asked to imagine
16
preparing a teaching unit about fractions. With respect to each item the pre-service teachers
could express their approval or disagreement on a four-point Likert scale. The structure of our
proposed model of five constructs was again examined with respect to the empirical data by
means of a CFA.
Construct (identifier) Sample item
# items
Multiple representations
(MR) for understanding To understand fractions properly, it is necessary to use
many different representations in class. 3
Multiple representations
(MR) for individual
preferences
In order to give pupils the opportunity to choose their
preferred type of representation, which they most easily
understand, they should be provided with many dif-
ferent representations.
3
One standard representa-
tion It is best to use only one kind of pictorial representation
for fractions in lessons, so that you can always come
back to this as a ‘standard’ representation.
3
Fear of confusion by
multiple representations
(MR)
Several different pictorial representations for fractions
could confuse pupils, especially the weaker ones. 3
Multiple representations
(MR) impede learning
rules
If pupils pay too much attention to pictorial represen-
tations, their ability to confidently do calculations with
fractions is impeded.
3
Table 2: Scales regarding views on dealing with multiple representations for teaching fractions
To explore if English and German pre-service teachers are able to recognise the learning po-
tential of tasks focusing on conversions of representations, in comparison with tasks including
rather unhelpful pictorial representations, the participants were asked to evaluate the learning
potential of six fraction tasks by means of three multiple-choice items. A sample item is: “The
way in which representations are used in this problem aids students’ understanding.” The
pre-service teachers could express their approval or disagreement concerning these items
regarding each task on a four-point Likert scale. They were told that the tasks were designed
for an exercise about fractions in school year six. Three of these tasks are about carrying out a
conversion of representations, whereas solving the other three tasks means just calculating an
17
addition or a multiplication of fractions on a numerical-symbolical representational level. The
pictorial representations which are given in the tasks of this second type are rather not helpful
for the solution, since they can’t illustrate the operation needed to carry out the calculation.
Samples for both kinds of tasks are shown in Figure 3.
Figure 3: Samples for tasks of type 1 (left) and of type 2 (right)
Obviously, these two types of tasks are not representative of all tasks about fractions and
moreover there are of course other kinds of fraction tasks that have a high learning potential.
However the idea behind this rather plain, bipolar design is that contrasting these two types of
tasks against each other affords insight into whether and where the pre-service teachers see the
learning potential of multiple representations for fraction tasks. The assumption that the the-
oretical classification of the tasks underlying their creation manifests itself also in the
pre-service teachers’ evaluations of their learning potential suggests that two second order
factors can be empirically separated which represent the evaluations of the types of tasks. The
analysis of how well this theoretical model fits the empirical data was carried out by a CFA and
is reported in the results section.
The last questionnaire section which is included in this study focuses on specific CK about
dealing with multiple representations in the domain of fractions. In particular, this CK test was
designed to assess the participants’ ability to match symbolic-numerical representations of
fractions and their operations with appropriate pictorial and content-related representations. As
in the sample item shown in Figure 4, given (incorrect) conversions between such representa-
18
tions had to be checked and corrected or a conversion had to be carried out. In a top-down
approach, the answers to the tasks were scored dichotomously as being correct or incorrect.
Please change the diagram, if necessary, so
that of is shaded. Otherwise just tick
the box on the right-hand side.
Figure 4: Sample item of the CK test
Differences between the two subsamples according to the third research question of the present
study were addressed by conducting T-tests. In order to measure to what degree different
components of the pre-service teachers’ knowledge and views are interrelated in a quantitative
way, Pearson's correlation coefficients were calculated. For all of these computations SPSS 21
for Windows software (SPSS Inc., Chicago, IL) was used.
RESULTS
We start with the results concerning the pre-service teachers’ rating of the importance of rea-
sons for using multiple representations in the mathematics classroom. In order to examine the
theory-based structure of the corresponding questionnaire instrument (cf. Table 1) empirically,
a confirmatory factor analysis (CFA) was conducted on the items in this section. The appro-
priateness of the theoretical model was assessed by several measures of global model fit.
Firstly, measures of incremental fit were employed, namely the TLI (Tucker-Lewis index) and
the CFI (comparative fit index). In both cases an acceptable fit is indicated by values ≥ 0.90
(Kline, 2005). For the proposed model the TLI is 0.93 and the CFI is 0.95 and hence both
criteria are met. Moreover, we have examined the RMSEA (root mean square error of ap-
proximation), which can be interpreted as the amount of information within the empirical
covariance matrix not explained by the proposed model. The model may be classified as ac-
ceptable if at most 8% of the information are not accounted for by the model, i.e. RMSEA ≤
5
3
4
1
19
0.08 (Kline, 2005). The current model meets this criterion as RMSEA is 0.064. Thus, it was
concluded that the model fits the data reasonably well. Concerning the local model fit, the
analysis showed that all factor loadings are highly significant (p < .001) and that the factor
reliabilities range from .68 to .79. In order to determine whether the four constructs of the
proposed model are empirically separable, the discriminant validity of each construct with
respect to the others was assessed by means of chi-square difference tests (see e.g. Jöreskog,
1971). More precisely, it was examined for each pair of constructs in the model, whether re-
stricting their correlation to 1 leads to a significantly poorer data fit. Since in each case this
parameter restriction has caused a significantly poorer fit (p < .001), it was concluded that the
four constructs of the proposed model show sufficient discriminant validity. Therefore, for
each of the four constructs a corresponding scale could be formed.
Figure 5: Views on the importance of reasons for using multiple representations
Figure 5 shows the means and their standard errors for these scales for both subsamples. First,
it is noticeable that both subsamples rated the discipline-specific reasons as less important than
the other more general reasons. Furthermore there are no significant differences between the
ratings of the English and the German pre-service teachers, except for the last scale: the
German pre-service teachers attributed a higher significance to the contribution of multiple
representations to remembering mathematical facts than did their English counterparts (t(207)
= 6.0, p < .001, d = 0.73).
1.0 2.0 3.0 4.0 5.0
necessity for mathematical understanding
learning types and input channels
motivation and interest
supporting remembering
not important extremely important
German participants
British participants
20
Regarding the questionnaire section about views on the role of multiple representations for
teaching fractions we also started by examining whether our theoretical model fits the empir-
ical data. Conducting a CFA, the model exhibited a reasonably good data fit (RMSEA = 0.054,
TLI = 0.92, CFI = 0.95). All the factor loadings are highly significant and the factor reliabilities
range from .62 to .76. Furthermore, conducting chi-square difference tests as described above
suggested that the five constructs in the proposed model have sufficient discriminant validity.
Consequently, five scales corresponding to the five constructs could be formed.
Figure 6: Views on the role of multiple representations for teaching fractions
Comparing the means of the two subsamples shown in Figure 6, one discovers an interesting
pattern: the English pre-service teachers compared to the German pre-service teachers were
more in favour of using multiple representations for teaching fractions and less afraid of pos-
sible negative effects. Cohen’s d shows that the difference concerning the scale “multiple
representations for understanding” is rather negligible (d = 0.25), whereas the other significant
differences represent weak or medium effects (0.35 < d < 0.58). The scales which have de-
tected the biggest differences between the English and the German participants are “MR for
individual preferences” and “fear of confusion by MR”.
Going down another level of globality, we focus now on the pre-service teachers’ evaluation of
the learning potential of the six tasks about fractions given in the questionnaire. Since this
evaluation was carried out by means of three items regarding each task and since these tasks
1.0 1.5 2.0 2.5 3.0 3.5 4.0
MR for understanding
MR for individual preferences
one standard rep.
fear of confusion by MR
MR impede learning rules
strong disapproval strong approval
German
participants
British
participants
* p < .05
** p < .01
*** p < .001
*
***
**
***
***
21
were in turn designed to represent two different types of tasks (“conversions of representa-
tions” versus “unhelpful pictorial representation”) the proposed model for this questionnaire
section encompasses 6×3=18 indicators of six first order factors (the evaluations of the tasks)
and two second order factors (the evaluations of the types of tasks). A CFA yielded a rea-
sonably good data fit for this model (RMSEA = 0.054, TLI = 0.92, CFI = 0.93). As regards the
local model fit, the factor loadings are all highly significant and both second order factors are
reliable with α = 0.75 and α = 0.73. For assessing the discriminant validity of the two constructs
“evaluation of the learning potential first type of tasks” and “evaluation of the learning poten-
tial of the second type of tasks”, a chi-square difference test was carried out. The fact that this
test was highly significant (p < .001) indicates that the model which distinguishes between the
pre-service teachers’ evaluations of the two types of tasks predicts the empirical data better
than the corresponding one-dimensional model. Therefore, two scales corresponding to the
evaluations of the two types of tasks were formed. The means and their standard errors of these
scales in Figure 7 show that both the English and the German pre-service teachers tended to
rate the learning potential of the tasks of type 2 on average slightly higher than the learning
potential of those of type 1. However, the difference is merely significant for the English
subsample (t(138) = 2.7, p = .008, d = 0.24).
Figure 7: Evaluations of the learning potential regarding the two types of tasks
So far we have reported results concerning pre-service teachers’ views on dealing with mul-
tiple representations on three levels of globality. Addressing our second research question, we
1.0 1.5 2.0 2.5 3.0 3.5 4.0
German participants
British participants
strong disapproval
Type 1: conversions of
representations
Type 2: unhelpful pictorial
representations
strong approval
22
focus now on relationships between these levels in order to explore to what degree the corre-
sponding views are consistent. The idea behind the following analysis is basically to explore
whether the pre-service teachers’ general conceptions about the role that multiple representa-
tions play for pupils’ learning in mathematics ‘translate’ into corresponding content- and
task-specific views. Seen against the theoretical background of our study, a key idea is that
learning with multiple representations is essential for the development of appropriate mathe-
matical concept images and therefore for conceptual understanding of mathematics. This view
is reflected in the scale “necessity for mathematical understanding” which is listed in Table 1.
Under the assumption that this global conception transfers into corresponding content do-
main-specific views, one would particularly expect that it is interrelated with the view that
fractions should be taught using multiple representations for the sake of the pupils’ under-
standing. Furthermore, one would expect that a higher approval of such discipline-specific
reasons for using multiple representations is related to a higher rating of the learning potential
of the tasks encouraging pupils to create connections between different representations. Con-
sequently, we examine whether the corresponding scales correlate. Pearson’s correlation co-
efficients are presented in Table 4 for both subsamples separately.
English participants MR for understanding (fractions)
Type 1 Type 2
Necessity for understanding .41** .12 ns .38**
German participants MR for understanding (fractions) Type 1 Type 2
Necessity for understanding .39** .21** .14*
ns = not significant (p > .05), * p < .05, ** p < .01
Table 4: Pearson’s correlation coefficients for the scale “necessity for understanding”
First, it can be noted that for both subsamples the assigned significance to the reason “necessity
for mathematical understanding” on a global level is indeed positively correlated with the
content-specific view that for teaching fractions multiple representations support the pupils’
23
understanding. In both cases the corresponding correlation coefficients represent moderate
effect sizes. However, looking deeper into the data in order to find reasons why these correla-
tions are not higher still, we found that in both subsamples there are participants whose views
on these two levels of globality are not consistent at all. For instance, one German pre-service
teacher rated the significance of the general reason for using multiple representations “neces-
sity for mathematical understanding” very high (4.75), whereas he showed very little approval
of the corresponding content-specific view regarding fractions (1.33).
Focusing on interrelations with the perceived learning potentials of the two types of fraction
tasks, our subsamples appear to be more distinct (cf. Tab. 4). Merely for the German subsample
the assigned importance to the discipline-specific reasons correlates slightly positively with the
perceived learning potential of the type 1 tasks (r = .21**). With respect to the English par-
ticipants it correlates instead positively with the perceived learning potential of the calculation
tasks of type 2 with unhelpful pictorial representations (r = .37**). For the German subsample
there was also such a correlation found, which is however barely significant and represents a
weak effect (r = .14*). This raises the question whether the pre-service teachers have recog-
nised that those pictorial representations in the type 2 tasks cannot illustrate the operation
needed to carry out the calculations. Thus, we next address research question 4b and focus on
the results regarding the specific CK test included in the questionnaire.
Figure 8: Specific CK scores
Figure 8 presents the mean scores (and their standard errors) of the two subsamples for this test.
It appears obvious that the German participants have achieved significantly higher scores than
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
German pre-service teachers
British pre-service teachers
24
their English counterparts (t(355) = 15.4, p < .001, d = 1.5). However, the English as well as the
German pre-service teachers in our sample have on average solved less than half of the items
correctly, which indicates a common need for development of professional knowledge, in this
case CK. This might suggest that the participants in this study did not have enough specific CK
for evaluating the use of representations in tasks of type 2 appropriately. Yet, no significant
correlation between the pre-service teachers’ CK scores and their evaluation of type 2 tasks
was found.
DISCUSSION AND CONCLUSIONS
The findings of this study about aspects of English and German pre-service teachers’ views on
the role of multiple representations for learning mathematics affords identifying prerequisites
and specific needs for initial teacher education as well as insight into culture-dependent facets
of such views. Before discussing these results in more detail, we would like to recall the lim-
itations of this study which suggest interpreting the evidence with care. Despite the size of the
subsamples, the study is not representative for German or English pre-service teachers.
Moreover, although a spectrum of different facets of views on dealing with multiple repre-
sentations was included in the design, the constructs can only give an indicator-like insight and
are mostly restricted to the domain of fractions. Bearing this in mind, the findings allow
however to answer the research questions and indicate several aspects of theoretical and
practical relevance: Firstly, the results can provide insight into aspects of pre-service teachers'
knowledge and views regarding the role of multiple representations for learning mathematics
in the sense of identifying corresponding prerequisites and needs for their teacher education.
Secondly, they can contribute to a better understanding of interrelations between pre-service
teachers’ pedagogical content views on different levels of globality. And thirdly, the compar-
25
ison of English and German participants allows to identify culture-dependent aspects of such
views and to review corresponding assumptions within the scope of the study empirically.
Common prerequisites and needs of English and German pre-service teachers
We start by discussing the findings regarding the first two research questions with a focus on
common aspects of the views and prerequisites of the pre-service teachers from England and
Germany. Both subsamples saw on average the special role of multiple representations for
understanding mathematics as less important than the other reasons for using multiple repre-
sentations, which are not discipline-specific. Moreover, the English as well as the German
participants were mostly not able to recognise the learning potential of tasks focusing on
conversions of representations, in comparison with tasks including rather unhelpful pictorial
representations, to which the pre-service teachers tended to assign a higher learning potential.
This demonstrates that the pure global conviction of “using multiple representations is good” is
not enough for designing rich learning opportunities, but in addition answers to questions such
as “for what purpose and in which way should multiple representations be used” are central.
Regarding the second research question about connections between the pedagogical content
views, the evidence suggests some expected interrelations of the pre-service teachers’ views on
different levels of globality. However, those interrelations are not strong, so it may not be
assumed that global views simply “translate” into content-specific views, but that the views on
the different levels of globality represent constructs of their own right. Considering cases of
participants whose views on different levels of globality appear to be even contradictory,
reinforces the impression that for many pre-service teachers, global views on dealing with
multiple representations in the mathematics classroom were not (yet) very consistent with their
corresponding domain- and task-specific views.
26
It may not be surprising that beginning pre-service teachers show little awareness for the spe-
cial role that representations play for mathematical understanding and that their views are not
(yet) very consistent. Nevertheless these findings provide insight into the specific prerequisites
and needs of these pre-service teachers and afford to customise their professional development
to the end of fostering their professional knowledge with respect to dealing with multiple
representations in the mathematics classroom. Regarding such common needs for initial
teacher education, we can draw the following conclusions: Awareness of the crucial role of
multiple representations and their connections for conceptual understanding of mathematics
should be seen as a key element in the development of pedagogical content knowledge. Sec-
ondly, the work on specific content, tasks and also instructional situations should be in the
centre of professional learning under the perspective of overarching ideas such as using mul-
tiple representations (cf. Kuntze et al., 2011), in order to support pre-service teachers develop
specific pedagogical content knowledge which is consistent with respect to different levels of
globality.
Possibly culture-dependent aspects of pre-service teachers’ views
Besides these common prerequisites and needs for teacher education, some of the findings
yielded differences between the English and the German subsample which indicate cul-
ture-dependent aspects of views on dealing with multiple representations. In line with the third
research question, looking at such differences also affords designing opportunities for profes-
sional learning which may be more valid within the framework of the specific cultural settings.
With respect to reasons for using multiple representations in mathematical classrooms in
general the only significant difference we identified was the greater emphasis of the German
pre-service teachers on remembering facts. Regarding content-domain specific views related to
the use of multiple representations, however, more differences became apparent. For teaching
27
fractions, the English pre-service teachers attached significantly greater importance to multiple
representations than their German counterparts – at least when reasons not specific to math-
ematics were in the focus. The German pre-service teachers rather feared confusing their pupils
by multiple representations and favoured the use of one 'standard' representation on average
more than did their English counterparts. The German pre-service teachers may hence have put
a focus mainly on learning rules, whereas for the English participants taking into account
individual preferences was predominant. Interestingly, these differences in the views ex-
pressed by the English and German pre-service teachers in this study reflect very well the
differences in the teaching and learning styles in England and Germany as they were described
by Kaiser (2002). This can in particular be seen as a further validation of the questionnaire
instrument used in this study. Moreover, our results are also consistent with the findings by
Pepin (1999) regarding more general views of teachers in England and Germany and they may
in particular add PCK-specific aspects to these findings.
However, in order to draw conclusions from this culture-related evidence, the content-specific
views regarding the fraction tasks should be included in order to provide a more complete
picture. For instance, correlations of the investigated task-specific views with the perceived
significance of discipline-specific reasons for using multiple representations indicates incon-
sistencies across levels of globality, specifically for the English participants. Greater appreci-
ation of the role of using multiple representations for building up conceptual understanding
was on average associated with a more positive evaluation of the learning potential of calcu-
lation tasks with rather unhelpful pictorial representations (cf. Tab. 4). Moreover, the on av-
erage very low scores in the specific CK test of the English participants may indicate that most
of them didn’t realise that those pictorial representations were not helpful, but merely noticed
that there were different representations provided. Hence, these pre-service teachers may need
28
a strengthened CK background; teacher education should combine specific help in that area
with learning opportunities connected to content-specific PCK.
Need for further research
We could not observe any direct correlation between specific CK and the pre-service teachers’
task-specific views. Nevertheless, corresponding CK might be of increased significance for the
teachers’ ability to implement the global goal of fostering the learners’ understanding with
multiple representations, as representations and their interrelations must be analysed accurately
on the content level. Hence, this calls for deepening studies which explore the role that do-
main-specific CK plays regarding domain-specific views about how to deal with multiple
representations. In particular, further research should also include in-service teachers, since
this may give further evidence for the hypothesis that the development of professional
knowledge for teaching mathematics is accompanied with a growth in consistency across
levels of globality. Moreover, such extended research should focus on the most situated level
of globality as well and examine teachers’ professional knowledge and views on dealing with
multiple representations regarding specific classroom situations (see Dreher & Kuntze, 2015).
In addition, deepening studies should use qualitative methods in order to illustrate different
profiles of professional knowledge about dealing with multiple representations and analyse
them in greater depth.
Acknowledgements The data gathering phase of this study has been supported in the
framework of the project ABCmaths which was funded with support from the European
Commission (503215-LLP-1-2009-1-DE-COMENIUS-CMP). This publication reflects the
views only of the authors, and the Commission cannot be held responsible for any use which
may be made of the information contained therein.
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