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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

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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

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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

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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,

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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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