The role of reading comprehension in mathematical modelling: improving the construction of a real-world model and interest in Germany and Taiwan

Abstract

To solve mathematical modelling problems, students must translate real-world situations, which are typically presented in text form, into mathematical models. To complete the translation process, the problem-solver must first understand the real-world situation. Therefore, reading comprehension can be considered an essential part of solving modelling problems, and fostering reading comprehension might lead to better modelling competence. Further, ease of comprehension and involvement have been found to increase interest in the learning material, and thus, improving reading comprehension might also increase interest in modelling. The aims of this study were to (a) determine whether providing students with reading comprehension prompts would improve the modelling sub-competencies needed to construct a model of the real-world situation and their interest in modelling and (b) analyze the hypothesized effects in two different educational environments (Germany and Taiwan). We conducted an experimental study of 495 ninth graders (201 German and 294 Taiwanese students). The results unexpectedly revealed that providing reading comprehension prompts did not affect the construction of a real-world model. Further, providing reading comprehension prompts improved students’ situational interest. The effects of providing reading comprehension prompts on the construction of a real-world model were similar in Germany and Taiwan. Students’ interest in modelling improved more in Germany. An in-depth quantitative analysis of students’ responses to reading prompts, their solutions, and their interest in the experimental group confirmed the positive relation between reading comprehension and modelling and indicated that the reading comprehension prompts were not sufficient for improving reading comprehension. Implications for future research are discussed.

Full text

The role of reading comprehension in mathematical
modelling: improving the construction of a real-world
model and interest in Germany and Taiwan
Janina Krawitz
1
&Yu-Ping Chang
2
&Kai-Lin Yang
3
&Stanislaw Schukajlow
1
Accepted: 18 April 2021/
#The Author(s) 2021
Abstract
To solve mathematical modelling problems, students must translate real-world situations,
which are typically presented in text form, into mathematical models. To complete the
translation process, the problem-solver must first understand the real-world situation.
Therefore, reading comprehension can be considered an essential part of solving model-
ling problems, and fostering reading comprehension might lead to better modelling
competence. Further, ease of comprehension and involvement have been found to
increase interest in the learning material, and thus, improving reading comprehension
might also increase interest in modelling. The aims of this study were to (a) determine
whether providing students with reading comprehension prompts would improve the
modelling sub-competencies needed to construct a model of the real-world situation and
their interest in modelling and (b) analyze the hypothesized effects in two different
educational environments (Germany and Taiwan). We conducted an experimental study
of 495 ninth graders (201 German and 294 Taiwanese students). The results unexpectedly
revealed that providing reading comprehension prompts did not affect the construction of
a real-world model. Further, providing reading comprehension prompts improved stu-
dents’situational interest. The effects of providing reading comprehension prompts on
the construction of a real-world model were similar in Germany and Taiwan. Students’
interest in modelling improved more in Germany. An in-depth quantitative analysis of
students’responses to reading prompts, their solutions, and their interest in the experi-
mental group confirmed the positive relation between reading comprehension and model-
ling and indicated that the reading comprehension prompts were not sufficient for
improving reading comprehension. Implications for future research are discussed.
Keywords Modelling competence .Reading comprehension .Interest .Comprehension questions .
Country-specific differences .Word problems
https://doi.org/10.1007/s10649-021-10058-9
*Janina Krawitz
krawitz@uni–muenster.de
Extended author information available on the last page of the article
Published online: 20 May 2021
Educational Studies in Mathematics (2022) 109:337–359
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

1 Introduction
Mathematical modelling competence is an important part of mathematical literacy. However,
research on modelling has demonstrated that students encounter various difficulties when solving
modelling problems (Blum, 2015). Even at the beginning of the solution process, learners often
struggle to understand the real-world situation and to structure and simplify the given information
(Blum, 2015;Kintsch&Greeno,1985; Krawitz et al., 2017;Wijayaetal.,2014). In order to
overcome these barriers, they need modelling sub-competencies to construct a structured and
simplified mental representation of the real-world situation, here called the real-world model
(Kaiser & Brand, 2015). Consequently, teaching methods for modelling problems have often
included elements that are aimed at improving the construction of a real-world model (Greefrath
et al., 2018; Kaiser & Brand, 2015; Schukajlow et al., 2012). Reading comprehension plays a
decisive role in the construction of a real-world model (Leiss et al., 2010) because the modelling
problems encountered in the classroom are often presented in text form. Also, the process of
solving modelling problems in everyday life often includes gathering and interpreting information
presented in text form (e.g., newspapers, timetables, books). Hence, reading comprehension is
often required to understand the real-world situation, and consequently, interventions that address
students’reading comprehension seem to be promising for fostering the ability to construct a real-
world model and thereby improving one’s overall modelling competence. However, there has not
been much research that has focused on the effects of reading interventions on modelling
competence and modelling sub-competencies. In particular, there has been a lack of experimental
interventional studies in the field (Schukajlow et al., 2018).
Further, students’motivation plays a decisive role in the learning process in mathematics
(Middleton & Spanias, 1999; Schukajlow et al., 2017). One important motivational variable is
students’interest in the learning material. Interest has been found to enhance students’
learning, to predict academic decisions such as students’course choices in high school, and
to have a positive effect on mathematics achievement (Heinze et al., 2005;Hidi&
Harackiewicz, 2000). Hence, it is important to investigate students’interest in modelling
and to consider interventions for improving interest. One promising approach for triggering
situational interest that we examined in our study involves facilitating the reading comprehen-
sion of the given texts or problems (Schraw et al., 1995;Wadeetal.,1999).
The present article analyzes the effects of a reading intervention on students’modelling
sub-competencies to construct a real-world model and on interest in two different educational
environments. Prior research identified various perspectives on modelling, which differ in the
aims they pursue with modelling and which can be related to different cultural backgrounds
(Kaiser & Sriraman, 2006). Also, the value assigned to modelling has been found to differ
from country to country. We selected Germany and Taiwan because the educational environ-
ments in Germany and Taiwan are very different from each other. We targeted different
educational environments in this study in order to determine whether the findings held for the
different educational contexts that the students had been exposed to.
Following these considerations, the aims of the present study were (a) to test the effects of a
reading intervention on the construction of a real-world model and on interest in modelling and
(b) to examine whether the effects of the reading intervention on the construction of a real-
world model and interest in modelling were similar in the two educational environments.
Further, we conducted an in-depth analysis of students who participated in the reading
intervention. Thereby, we focused on students’reading comprehension ability and examined
its relations to the construction of a real-world model and students’interest in modelling.
338 Krawitz J. et al.
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2 Modelling competence, interest in modelling, and reading
comprehension in Germany and Taiwan
2.1 Modelling competence
The core of mathematical modelling is the translation of a real-world problem into a
mathematical model with the aim of solving the problem. The process of modelling is
typically depicted as a cyclic process that moves from the real world to the mathematical
world and back to the real world, passing through different phases that are required to
solve the problem (see, e.g., Blum & Leiss, 2007; Galbraith & Stillman, 2006;
Verschaffel et al., 2000). Demonstrating the willingness and ability to solve real-world
problems through mathematical modelling is referred to as mathematical modelling
competence (Kaiser, 2007). More specifically, we refer to an analytic understanding of
modelling competence that is based on different sub-competencies (a description of
different modelling strands can be found in Kaiser & Brand, 2015). Modelling sub-
competencies include—among metacognitive and social competencies—competencies
that are related to the different phases of the modelling cycle (Kaiser, 2007;Maaß,
2006; Niss et al., 2007), namely: the competencies to (1) understand the real-world
situation and construct an initial mental representation of the real-world situation (called
the situation model); (2) structure and simplify the situation model, the resulting mental
representation of which is referred to as the real-world model; (3) mathematize the real-
world model, resulting in a mathematical model; (4) apply mathematical procedures to
find a mathematical result; (5) interpret the mathematical result at the end of the solution
process; and (6) validate the result with regard to the real-world situation.
2.2 The modelling sub-competencies needed to construct a real-world model
In the present article, we focus on the modelling sub-competencies needed to construct a real-
world model. These sub-competencies are further explained and illustrated using the example
of the Parachuting modelling problem presented in Fig. 1.
First, students have to construct a situation model. Therefore, they have to understand the
information, presented here in the form of text that is accompanied by a table and a picture.
Fig. 1 The parachuting modelling problem adapted from Schukajlow and Krug (2014b,p.500)
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Second, the learner has to transform his or her situation model into a real-world model (Fig. 2).
This means the learner has to simplify the situation model by making an assumption about the
wind speed (e.g., assume that there is a strong wind blowing). He or she has to structure the
information by separating important from unimportant information (e.g., identifying the
important information that the horizontal shift per each thousand meters of descent in strong
wind conditions is 340 m during free fall and 3060 m while gliding) and construct relation-
ships between the pieces of important information (e.g., connecting the information that the
parachutist free falls about 3000 m to the information that the horizontal shift per each
thousand meters is 340 m).
Research on modelling has shown that solving modelling problems is demanding, and
students are often already struggling at the beginning of the modelling process in trying to
construct a real-world model (Blum, 2015; Kintsch & Greeno, 1985; Krawitz et al., 2017;
Leiss et al., 2010; Wijaya et al., 2014). In the study by Wijaya et al. (2014), more than one
third of students’errors in solving modelling problems were related to the construction of a
real-world model. These difficulties emphasize the need for research to examine interventions
that can improve students’modelling sub-competencies to construct a real-world model while
solving modelling problems.
2.3 Students’interest in modelling
Interest is considered a person-object relationship that refers to both the psychological state of
attention and affect toward a particular topic (situational interest) and an enduring predisposition
to reengage with the topic over time (individual interest) (Hidi & Renninger, 2006). Interest is a
domain- or content-specific motivational variable that combines affective and cognitive qualities
(Harackiewicz et al., 2016; Schiefele et al., 1992). Theories of the development of interest propose
that students pass through several phases as their interest develops: from unstable and triggered
situational interest to stable and well-developed individual interest (Hidi & Renninger, 2006). If a
person repeatedly experiences situational interest with respect to a particular topic, he or she may
also develop individual interest in the topic over time. Hence, the environment can contribute to
the development of individual interest by stimulating situational interest and building on prior
Fig. 2 Illustration of a real-world model for the parachuting problem under the assumption that a strong wind is
blowing
340 Krawitz J. et al.
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individual interest. For mathematical modelling, this means that if learners repeatedly experience
situational interest when solving modelling problems, they are likely to develop individual interest
in modelling. For modelling problems, different aspects can be sources of interest, namely,
students may be interested in the process of modelling, the content, or the intramathematical
problem. As affect and modelling competence are related to each other (Chamberlin, 2019;
Schukajlow & Krug, 2014a), enhancing interest in modelling is also beneficial for students’
modelling competence. The important role that affect plays in modelling has been acknowledged
in modelling research, and predictors such as authenticity, meaningfulness, and contexts have
been discussed (Di Martino, 2019; Goldin, 2019). Several studies have addressed the question of
how interest in solving mathematical problems can be enhanced. Building connections to reality is
one approach that can be used to increase students’interest in mathematics because problem
contexts can be a source of students’interest in working on the problems. However, the study
conducted by Rellensmann and Schukajlow et al. (2017) showed that students do not perceive
problems connected to reality per se as more interesting than intramathematical problems. In this
line of research, studies have investigated whether personalizing the problems increases students’
interest in working with the problems (Bates & Wiest, 2004; Høgheim & Reber, 2015). The
results have shown that tailoring the context to students’personal interest has benefits for students’
situational interest in working with the problems. However, personalized problems are often
constructed with the help of digital tools and therefore are not easy to implement in classrooms.
Our approach focuses on text-based interest (i.e., situational interest that comes from reading a
text). As the problem is described in a textual format, we considered factors that are claimed to
trigger text-based interest, such as meaningfulness, ease of comprehension, involvement, text
cohesion, novelty, and emotiveness (Mitchell, 1993;Palmer,2009;Schrawetal.,1995). Empir-
ical results from factor analysis and correlational analysis have supported the importance of these
sources of students’situational interest (Mitchell, 1993;Palmer,2009;Schrawetal.,1995). For
the present study, we consider involvement and ease of comprehension to be particularly
important. Involvement refers to the extent to which students feel they are active participants.
Ease of comprehension refers to how easy it is to understand a text. We discuss both sources in the
context of reading comprehension and modelling in the next section.
2.4 Reading comprehension and its impact on constructing a real-world model
and on interest in modelling
Reading comprehension is defined as the active process of building an adequate mental repre-
sentation of a text (Durkin, 1993;Kintsch,1986). Texts in mathematics often include discontin-
uous elements such as tables, figures, or formulae. If the text is accompanied by pictures, an
integrated mental representation is built on the basis of the text and pictures (Schnotz & Bannert,
2003). Reading comprehension can be claimed to be one of the sub-competencies needed to
understand the real-world situation because the situation is often presented in a textual format in
the classroom or in everyday contexts involving textual information, such as newspaper articles,
product information, reports, and many others. Hence, reading comprehension can also be
considered a sub-competency that is necessary for constructing a real-world model because
structuring and simplifying the given information directly depend on an adequate understanding.
The importance of reading comprehension for modelling has been acknowledged in research on
modelling (Leiss et al., 2010; Leiss et al., 2019), and theoretical descriptions of the modelling
process have been built on research on text comprehension (Kintsch & Greeno, 1985). Empirical
findings have supported the positive relation between reading comprehension and modelling
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competence (Krawitz et al., 2017; Leiss et al., 2010; Leiss et al., 2019; Vilenius-Tuohimaa et al.,
2008). Hence, interventions that address students’reading comprehension seem to offer a
promising approach for fostering modelling sub-competencies to construct a real-world model
and thereby improving overall modelling competence. However, hardly any intervention studies
have tried to enhance modelling competence by fostering reading comprehension, and the few
existing ones have not been successful (Hagena et al., 2017; Krawitz et al., 2017). Thus, further
investigations are necessary to identify the conditions under which reading interventions are
beneficial for modelling. One approach for enhancing reading comprehension is to present
questions that address important pieces of information and their relations given in the text,
referred to here as reading comprehension prompts. The impact of questions on reading compre-
hension is widely acknowledged in reading research, and answering questions is considered an
important strategy for boosting reading comprehension. In particular, reading research has shown
that reading comprehension prompts can guide readers’attention to important aspects of the text
(Ge & Land, 2003) and increase their engagement with the text because the contents of the text are
more actively processed when the reader has to answer questions about the contents (National
Reading Panel, 2000). However, questions are not as beneficial per se. The impact strongly
depends on factors such as the type of question (Cerdán et al., 2009) and readers’reading
proficiency (van den Broek et al., 2001). Working on high-level questions was found to be more
beneficial for reading comprehension than working on low-level questions (Cerdán et al., 2009).
The results from the study by van den Broek et al. (2001) suggest that more proficient readers
benefit more from questions, whereas less proficient readers can suffer from having to answer
questions because of an increase in cognitive demand from having to think about them. However,
these findings were based on scientific or narrative texts, and one question that remains unan-
swered is whether reading comprehension prompts are also beneficial for the text included in
modelling problems. Research on modelling has provided initial indications that reading com-
prehension prompts might foster the construction of a real-world model and thereby enhance
modelling competence. In the study conducted by Schukajlow et al. (2015) and similarly also in
the study by Hankeln and Greefrath (2020), students received a scaffolding instrument called a
solution plan to guide their modelling processes. The solution plan consisted of prompts referring
to the different phases of the modelling cycle, including prompts to trigger reading comprehension
(“Read the text precisely! Imagine the situation clearly!”). The results showed that using the
solution plan was beneficial for students’modelling competence, but the specific role of reading
comprehension prompts could not be derived from the data as it was not clear which prompts were
responsible for the positive effect on modelling competence.
Further, reading comprehension prompts might increase students’interest in modelling
because they address two important sources of situational interest: involvement and ease of
comprehension. First, reading comprehension prompts might trigger involvement because, by
working with the prompts, students become more actively involved in the reading process.
Second, reading comprehension prompts might affect ease of comprehension because they are
suggested to facilitate reading comprehension, and reading research has indicated that if texts
become easier to understand, they are perceived as more interesting (Schraw et al., 1995;
Wade et al., 1999). Modelling problems often place high demands on students’reading
comprehension, and thus, their ease of comprehension may be compromised. This is a
potential reason for the unexpected findings that students perceive modelling problems as
similar to (Schukajlow et al., 2012) or even less interesting (Rellensmann & Schukajlow,
2017) than problems with no connection to reality. Consequently, we expected that reading
comprehension prompts would increase students’interest in modelling.
342 Krawitz J. et al.
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2.5 Educational environments in Germany and Taiwan
One of the aims of the present study was to investigate the role of the educational environment
the students had been exposed to. Comparing students from educational environments that are
very different, such as Germany and Taiwan, can provide indications for the validity of the
theoretically assumed relationships between reading comprehension, modelling competence,
and interest in modelling. This section discusses differences in the educational environments of
the two countries that we analyzed in the present study.
Students from East Asian countries—among them Taiwanese students—have been found
to perform extremely well in international comparative studies of mathematics achievement
such as TIMSS and PISA. However, there are some indications that modelling and applica-
tions play only minor roles in Taiwanese compared with German mathematics education. In
Germany, modelling competence is embedded in the curriculum as one of six compulsory
competencies (KMK, 2004), whereas it is not explicitly mentioned in the Taiwanese curric-
ulum (Ministry of Education in Taiwan, 2003). Further, Taiwanese textbooks seem to focus on
intramathematical tasks, as Taiwanese textbooks were found to contain the lowest proportion
of real-world problems for geometry problems when compared with textbooks from Singa-
pore, Finland, and the USA (Yang et al., 2017). This result is reflected in students’reports, as
students from Taiwan reported that they encountered real-world problems in their math classes
less often than German students (OECD, 2014). The few comparative studies that have
analyzed students’achievement in modelling have pointed out that Western students are more
experienced in solving modelling problems. Chang et al. (2020) showed that German students
had higher modelling competence than Taiwanese students when the students from the two
countries were on the same level of intramathematical competence. This difference was
particularly remarkable for students with a low level of intramathematical competence.
Further, German and Taiwanese students’interest in modelling might also differ.
Because they have less experience with modelling problems, Taiwanese students may
find it more interesting to work on modelling problems than German students because
novelty is an important source of situational interest (Palmer, 2009). However, more
experience with modelling problems could also lead to greater interest in working on the
problems because they might perceive the problems as more meaningful, which is also
known as a source of situational interest (Mitchell, 1993). Little is known about
differences in students’interest in modelling problems in different countries. As interest
in modelling is related to interest in mathematics, the first indications of students’
interest in modelling in Germany and Taiwan can be derived from the results of PISA
2012, where students’interest in mathematics was assessed. The results indicate that
German students have a higher interest in mathematics compared with Taiwanese
students (OECD, 2013), which is surprising given the much higher mathematical per-
formance of Taiwanese learners. As cognitive and affective theories such as theories of
modelling competence and theories of interest do not depend on education environments,
we expected that reading comprehension prompts would have similar effects on the
construction of a real-world model and interest in modelling in Germany and Taiwan.
2.6 Hypotheses and path-analytical model
On the basis of theoretical considerations and the prior empirical findings described above, we
developed the following hypotheses:
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Hypothesis 1 (reading comprehension prompts)
The presentation of reading comprehension prompts will positively affect the modelling sub-
competencies needed to construct a real-world model and students’interest in solving model-
ling problems:
a) Presenting reading comprehension prompts will lead to higher scores on the sub-
competencies needed to construct a real-world model.
b) Presenting reading comprehension prompts will lead to a higher interest in solving
modelling problems.
Hypothesis 2 (educational environment)
The effects of presenting reading comprehension prompts on the construction of a real-world
model and on interest will be similar in both educational environments.
a) The effect of presenting reading comprehension prompts on the construction of a real-
world model will be similar in the two educational environments (Germany and Taiwan).
b) The effect of presenting reading comprehension prompts on interest in modelling will be
similar in the two educational environments (Germany and Taiwan).
The hypothesized path model (Fig. 3) links reading comprehension prompts—that were
operationalized by presenting questions (reading comprehension prompts group vs. control
group)—with the outcome measures (construction of a real-world model and interest in
modelling) while controlling for intramathematical competence. Educational environment
(Germany vs. Taiwan) was included as a moderator of the effects of reading comprehension
prompts on the outcome variables.
Fig. 3 Path-analytic model. Paths illustrate the direct effects of one construct (reading comprehension prompts)
on the other construct (e.g., construction of a real-world model) or the moderating effects of a construct
(educational environment) on the direct effects. The two paths from the control variable (intramathematical
competence) to the outcomes illustrate that the investigated effects were controlled for intramathematical
performance.
344 Krawitz J. et al.
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3 Method
3.1 Sample and procedure
The present sample involved 495 ninth graders, including 201 German students from nine classes
from high-track schools (German gymnasium; 50% female, mean age = 14.96 years) and 294
Taiwanese students from 12 classes in which all performance levels were taught (52% female, mean
age = 14.89 years). Prior studies (e.g., Chang et al., 2020;OECD,2019) have demonstrated that
Taiwanese students have much higher intramathematical competence than German students. In
order to balance these differences and improve the comparability of the groups regarding this
important background variable, we collected data from German high-track schools and regular
Taiwanese schools. We further compared the intramathematical competence of the two groups to
control for whether the sampling strategy led to the intended result (see Section 4).
In each of the 21 classes, students were randomly assigned to an experimental condition
(reading comprehension prompts group; RPG) or a control condition (control group; CG).
Students in both conditions worked on a paper-and-pencil modelling test. In the RPG, reading
comprehension prompts were used to trigger reading comprehension. Accordingly, students in the
RPG received reading comprehension prompts that referred to the textual descriptions of the real-
world situations (called situational descriptions). Students first read the situational description,
then worked on two corresponding reading comprehension prompts, and subsequently worked on
two modelling problems. Two sample pages from the RP test booklet from the context “Parachut-
ing”are presented in the Appendix (Fig. 8). This procedure took 60 min. After completing all
tasks, students worked on the intramathematical problems for 20 min. Students in the CG
followed the same procedure, but they did not receive any reading comprehension prompts.
Reading comprehension prompts were operationalized as questions that referred to the informa-
tion presented in the situational descriptions. The modelling problems were presented on a separate
page after the reading comprehension prompts in order to reduce the risk that students would work on
the modelling problems before answering the reading comprehension prompts. Responding to the
reading comprehension prompts was aimed at helping students focus on important objects and on
important relations between the elements given in the situational description. Consequently, one of
the two questions for each situational description targeted important information, and the other
question targeted relations between the given pieces of information. For example, for the parachuting
situation, the situational description was the textual description of the parachutist’s jump, including
how he or she was carried off target by the wind (text and table presented in Fig. 1). The first reading
comprehension prompt was “What is the horizontal shift per each thousand meters of descent while
gliding when a parachutist is carried by a light wind?”(correct answer: “540 m”). This question
referred to information provided in the table. The second reading comprehension prompt was “What
is the horizontal shift per each thousand meters of descent when a parachutist is carried by a strong
wind at about an altitude of 2,500 meters?”(correct answer: “340 m”). This question addressed the
relations between and the interpretation of the given pieces of information. To respond, learners have
to use the information given in the text to interpret the altitude of 2500 m as the free fall phase and
then use the table to read out the horizontal shift for strong wind conditions during free fall.
The reading comprehension prompts were tested in a pilot study (Krawitz et al., 2017)and
subsequently revised with a focus on the theoretically expected benefits of asking questions on
reading comprehension. These benefits include addressing important pieces of information and their
relations given in the text and thereby increasing students’engagement with the text (see
Section 2.4). In the reading comprehension prompts, we decided also to address information that
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is not needed to solve the modelling problems because the aim of the prompts was to enhance the
understanding of the situation and not to provide clues about which data should be used to solve the
modelling problems.
3.2 Measures
3.2.1 The modelling sub-competencies needed to construct a real-world model
and intramathematical competence
The modelling test included eight modelling problems that referred to the four situational descriptions
(two modelling problems for each situational description). All situational descriptions used in the study
were similar in length. Six modelling problems were adapted from previous studies (Blum, 2011;
Schukajlow & Krug, 2014b), and two modelling problems were developed in this study. The modelling
problems could be solved with methods, such as applying the Pythagorean Theorem or by drawing a
scaled diagram. The decision to limit the mathematical content area of the modelling problems was
made to improve the fit between the modelling test and the intramathematical test. One modelling
problem is presented in Fig. 1. Another modelling problem that referred to the same situational
description was: “For his last jump, a parachutist glided about 1,600 meters after he had opened his
parachute. Using the above clue, make reasonable assumptions about what kind of wind conditions
most likely prevailed during this jump. Find a solution and clearly provide reasons for your answer.”
In order to measure the modelling sub-competencies needed to construct a real-world
model, students’solutions to the eight modelling problems were analyzed for whether the
solution was based on a correct real-world model of the situation (scored 1) or not (scored 0).
The use of problems that require students to work on all phases of the modelling process
prevented us from asking questions that may have confused the students because they are not
used to describe their construction of the real-world model. For example, for the modelling
problem that went with the parachuting situation presented in Fig. 1(“What possible distance
might the parachutist move during the entire jump, including free fall and gliding?”), students
had to make an assumption about the wind conditions and link this assumption to the
information presented in the text and table (see Fig. 2). Figure 4presents a student’s solution
that was scored as correct real-world model because the student selected the important
information needed to solve the problem, made an assumption about the wind condition,
and correctly assigned the data to the respective objects. The accuracy of the real-model was
estimated based on students’written solutions (see Fig. 4).
In the solution presented in Fig. 5, the student assumed that a light wind was blowing but
interpreted the side deviation as the distance traveled. Hence, the data were incorrectly
assigned to the objects. Such a solution was scored as an incorrect real-world model.
Fig. 4 Example of a student’s solution that was scored as a correct real-world model
346 Krawitz J. et al.
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The scale reliability (Cronbach’s alpha) for measuring the construction of a real-world
model was 0.616. Two coders were involved in scoring the German test booklets, and six
coders coded the Taiwanese part of the sample. At least 20% of the test booklets in each
country were used to calculate intercoder reliability. Two coders scored the solutions for each
item. The intercoder agreement between two coders (Cohen’sκ) was 0.694 or higher,
indicating a substantial level of agreement. The coding was carried out by university students
who completed a training and received a coding manual. When differences occurred, the
coders discussed their judgments and made a consensual decision to choose one code.
Students’intramathematical competence was assessed as students’ability to solve
intramathematical problems on the topic of the Pythagorean Theorem. On the
intramathematical test, students were asked, for example, to calculate the length of the diagonal
of a rectangle with a length of 3 cm and a width of 4 cm or to judge whether a given figure
(nonright triangle) represents the Pythagorean Theorem. The scale consisted of 10 items, and
its reliability (Cronbach’s alpha) was 0.815.
3.2.2 Interest in solving modelling problems
We used task-specific questionnaires in the present study in order to take into account the state-
like nature of situational interest and the task-sensitivity of the construct (Knogler et al., 2015).
We adapted the task-specific scale used in prior studies (Rellensmann & Schukajlow, 2017;
Schukajlow et al., 2012). For each of the four situational descriptions, after working on two
modelling problems, students were asked whether they were interested in working on these
problems. Using a 5-point Likert scale (1 = not at all true,5=completely true), students’
responses indicated the extent to which they agreed with the following statement: “It was
interesting to work on the problems ‘[Name of the situational description, e.g., Parachuting]’.”
The scale consisted of 4 items, and its reliability (Cronbach’s alpha) was 0.835.
3.2.3 Translation of the material
The material that was adopted from prior studies was translated into the English language.
New material was directly developed in the English language. The English material was
translated into German and Chinese. The second author of the paper, who understands all
three languages, checked for the compatibility of the German and Chinese translations.
3.3 Data analysis
Means, standard deviations, and Pearson correlation coefficients, which are presented in Table 1,
were calculated using SPSS. All estimation and data fitting procedures for testing the hypothesized
Fig. 5 Example of a solution that was scored as an incorrect real-world model
347The role of reading comprehension in mathematical modelling: improving...
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path-analytic model (Fig. 3) were carried out with Mplus (Muthén & Muthén, 1998–2017). The
variance-covariance matrix was analyzed by using maximum-likelihood estimation with robust
standard errors. The reported pvalues for the effects of the reading intervention were one-tailed
because our expectations were directional, but they were two-tailed for the effects of educational
environment. To examine clustering effects produced by the nonindependence of students nested in
classes (n= 21), we calculated the intraclass correlation coefficient (ICC) for intramathematical
competence. The ICC (0.31) indicated that the intramathematical competence of students from the
same classes was more similar than that of students from different classes. Thus, we used the “TYPE
=COMPLEX”Mplus analytic option to account for the clustering effects (Stapleton, 2006). The
treatment variables were dummy coded (RPG = 1 and CG = 0). The model included 13 free
parameters and 495 participants. The ratio of participants to parameters was about 38 (495/13) and
hence above the critical value of 5 for obtaining solid results (Kline, 2005). The model was fully
saturated so that the fit indices were noninformative (i.e., CFI = 1; SRMR = 0).
4Results
4.1 Overall results
First, we conducted a preliminary analysis of the sample and tested it for differences between
German and Taiwanese students and between the conditions (RPG and CG) in order to
obtain some indication of the comparability of the groups. For intramathematical compe-
tence, the results indicated that there were no differences between the German and Taiwanese
students (Germany: M=0.520SD = 0.251; Taiwan: M=0.512SD =0.300),t(473.266) =
0.305, p= 0.760, nor were there differences between the students from the different
experimental conditions (RPG: M=0.518SD =0.284;CG:M=0.512SD = 0.279),
t(493) = −0.269, p= 0.788. These results justified the randomized assignment of students
to the reading comprehension prompts and control conditions in our sample. Further, this
preliminary analysis indicated that German and Taiwanese students were comparable
concerning an important cognitive prerequisite: students’intramathematical competence.
The estimates of the path model that we created to test our hypothesis were based on the
correlation matrix presented in Table 1.Figure6presents a graphical representation of the
estimates. The means, standard deviations, and correlations of the study variables are presented
separately for each experimental condition (RPG and CG) and educational environment
(Germany and Taiwan) in Table 3in the Appendix.
Table 1 Means, standard deviations, and correlations of all variables
Variable MSD12345
1 Construction of a real-world model 0.123 0.165 -
2 Intramathematical competence 0.515 0.281 0.470** -
3 Interest in modelling 2.781 0.950 0.079 0.191** -
4 Reading comprehension promptsa--−0.025 0.012 0.090*-
5 Educational environmenta--−0.256** −0.013 0.18 0** 0.001 -
aDummy coded (reading comprehension prompts: RPG = 1, CG = 0; educational environment: Taiwan = 1,
Germany = 0).
*p<.05,two-tailed.** p< .01, two-tailed
348 Krawitz J. et al.
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4.2 Effects of reading comprehension prompts on the construction of a real-world
model and on interest in modelling
Regarding our reading intervention, we expected positive effects of presenting reading
comprehension prompts on the construction of a real-world model (Hypothesis 1a) and on
students’interest in solving modelling problems (Hypothesis 1b). The analysis
partially supported our hypothesis. Presenting reading comprehension prompts did not
affect the construction of a real-world model (β=−0.064, p= 0.220, one-tailed), but
it positively affected students’interest in solving modelling problems (β= 0.344, p<0.01,
one-tailed).
4.3 Educational environment as a moderator of the effects of reading comprehension
prompts
We further expected that the hypothesized positive effects of presenting reading com-
prehension prompts on the construction of a real-world model (Hypothesis 2a) and on
interest in modelling (Hypothesis 2b) would be similar for the two educational environ-
ments of the students. There were no country-specific differences in the effect of
presenting reading comprehension prompts on the construction of a real-world model
(β=−0.001, p= 0.996). Further, contrary to our expectations, the path analysis revealed
an effect of the country on the effect of reading comprehension prompts on students’
Fig. 6 Path model for testing the effects of presenting reading comprehension prompts on the construction of a
real-world model and students’interest in modelling. Significant paths (p< .05) are presented as solid lines and
nonsignificant paths as broken lines (Estimates of the effects of binary covariates (reading prompts and
educational environment) are standardized with respect to the dependent variables (STDY), and estimates of
the effects of continuous covariates (intramathematical competence) are standardized with respect to both the
covariate and the dependent variable (STDXY). STDY standardized regression coefficients (β) can be
interpreted as the predicted change in the (residualized) criterion measures (in standard deviation units) when
the covariate (reading prompts or educational environment) changed by one unit. For example, if the value for
reading prompts changes from 0 to 1 (change in the reading treatment condition), interest increases by β×SD
Int
= 0.344×SDInt. STDXY is interpreted as the predicted change in standard deviation units when the covariate
changes by one standard deviation)
349The role of reading comprehension in mathematical modelling: improving...
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interest in modelling (β=−0.295, p< 0.05), indicating that presenting reading
comprehension prompts is more beneficial for interest in solving modelling problems
for German students than for Taiwanese students. An analysis of the effects of reading
comprehension prompts on interest in the respective educational environment revealed
significant positive effects in German but not in Taiwanese students (Germany: β=
0.352, p< .01; Taiwan: β=0.051,p= 0.309).
5 In-depth analysis of the reading comprehension prompts condition
We conducted an in-depth analysis to investigate the role of reading comprehension in
determining the modelling sub-competencies necessary to construct a real-world model
and interest in modelling in the group of students who participated in the reading
intervention. The aims were, first, to validate the positive relation between reading
comprehension and modelling competence that is assumed in modelling theories and,
second, to obtain an indication of why providing reading comprehension prompts only
partly supported our hypotheses on the positive effects of reading comprehension
prompts on the construction of a real-world model and interest in modelling. The
research questions for the in-depth analysis were:
1. Is reading comprehension positively related to the construction of a real-world model and
interest in modelling?
2. Are the effects of reading comprehension on the construction of a real-world model and
interest in modelling similar in different educational environments?
5.1 Method of the in-depth analysis
To conduct our analysis, we assessed students’reading comprehension in addition to
other measures. Reading comprehension was measured by scoring the answers to the
reading comprehension prompts in the experimental condition (N=245).Students
received a score of 1 if they responded correctly to a reading comprehension prompt
and a score of 0 if they responded incorrectly or did not responded at all (see the
examples in Section 3.1).Thisscalerangedfrom0to8becauseweprovidedeight
reading comprehension prompts in our study. The scale reliability (Cronbach’salpha)
was 0.759. Interrater agreement was calculated on a subset of at least 20% of the
participants with sufficient agreement (Cohen’sκ≥0.745). The path model included
13 free parameters and 245 participants. Hence, the ratio of participants to parameters
was above the critical value of 5. We followed the same statistical approach that we used
for the analysis of our primary hypotheses in the prior section.
5.2 Results of the in-depth analysis
The model parameter estimates were based on the correlation matrix presented in Table 2.
Means, standard deviations, and correlations of all variables are also presented in this table.
Figure 7presents a graphical representation of the estimates.
350 Krawitz J. et al.
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We found a positive impact of reading comprehension on the construction of a real-world model
(β= 0.425, p< 0.01, one-tailed). However, reading comprehension did not affect students’interest
in solving modelling problems (β= 0.060, p= 0.376, one-tailed). Further, contrary to our
expectations, educational environment was found to moderate the effect of reading comprehension
on the construction of a real-world model (β=−0.635, p< 0.05). The analysis of the effects of
reading comprehension on the construction of a real-world model and interest in modelling in the
respective educational environment revealed that reading comprehension had a positive impact on
the construction of a real-world model for German students (β= 0.235, p< 0.01, one-tailed) but not
for Taiwanese students (β=−0.002, p= 0.488, one-tailed). The effect of reading comprehension on
interest in modelling did not differ between German and Taiwanese students.
6Discussion
In this study, we hypothesized and tested the effects of reading comprehension in two different
educational environments on the modelling sub-competencies needed to construct a real-world
model and students’interest in solving modelling problems while controlling for
Table 2 Means, standard deviations, and correlations of all variables used in the in-depth analysis based on the
students’responses in the RPG
Variable MSD12345
1 Construction of a real-world model 0.118 0.164 -
2 Intramathematical competence 0.518 0.284 0.446** -
3 Interest in modelling 2.868 0.895 0.097 0.174** -
4 Reading comprehension 0.712 0.247 0.339** 0.460** 0.146*-
5 Educational environmenta--−0.283** −0.061 0.100 −0.269** -
aDummy coded (Taiwan = 1, Germany = 0)
*p<.05,two-tailed.** p< .01, two-tailed
Fig. 7 Path model for testing the effects of reading comprehension on the construction of a real-world model and
on students’interest in modelling. The analysis included only students from the experimental condition.
Estimates are standardized (STDXY). Thus, the regression coefficients can be interpreted as the predicted change
in the criterion measures in standard deviation units when the covariate changes by one standard deviation.
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intramathematical competence. Reading comprehension was experimentally manipulated by
providing students with reading comprehension prompts, operationalized as questions that
addressed the real-world situations in the situational descriptions of the modelling problems.
An in-depth analysis of the students who received reading comprehension prompts was
conducted to investigate the impact of students’reading comprehension on the construction
of a real-world model and interest in modelling in different educational environments and
while controlling for students’intramathematical competence.
6.1 Effects of reading comprehension on the construction of a real-world model
and interest in modelling
Contrary to our expectations, the construction of a real-world model was similar for
students who were provided with reading comprehension prompts when compared with
their peers who solved problems without reading comprehension prompts. The positive
impact of questions on reading comprehension found in the domain of reading research
(McKeown et al., 2009;Rickards,1976) did not hold for students’mathematical modelling
competence. An in-depth analysis of the relation between reading comprehension and
modelling showed that students who correctly answered the reading comprehension ques-
tions were better at constructing a real-world model, indicating the importance of reading
comprehension for modelling proposed in prior research. Apparently, providing reading
comprehension prompts is not enough by itself. Rather, the quality of a student’s engage-
ment with the reading comprehension prompts, indicated by accurate answers, seems to
enhance the modelling sub-competencies to construct a real-world model and thereby also
enhances overall modelling competence. But why did the reading comprehension prompts
fail to improve students’abilities to construct a real-world model in our study? A possible
explanation is that students might have answered the reading comprehension prompts
superficially without putting effort into reprocessing the text, and thereby, the benefits of
presenting reading comprehension prompts could not take effect (Bråten et al., 2014;
Pressley et al., 1989). Consequently, one implication from our study is that reading
comprehension instructions should prompt students’ability to process the description of
the real-world situation in the text (Pearson et al., 1992). We suggest that future studies
should expand the presentation of reading comprehension prompts by teaching students
how to use them in longer and more comprehensive interventions. Another reason could be
that the reading comprehension prompts guided learners’attention to specific information
and thereby did not enhance their understanding of the whole situation. It might be more
beneficial to use more general reading comprehension prompts, such as “What is the text
about? Write a short summary in your own words,”or specifically for the parachuting
situation, “Explain what horizontal shift means here and describe the factors that influence
horizontal shift.”Further, the cognitive cost of answering the prompts may have inhibited
the positive effect of the reading comprehension prompts (van den Broek et al., 2001).
These inhibiting effects might be particularly strong for readers with low proficiency
levels who were not able to answer the questions. In addition, time constraints might have
affected the results because students in the RPG had the same amount of test time as
students in the CG.
The positive relation between reading comprehension and the construction of a real-world
model found in our study adds to previous findings and indicates the importance of
reading comprehension for modelling activities. The correlation of 0.339 found in our study
352 Krawitz J. et al.
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is consistent with findings from previous studies (.198 in a study by Krawitz et al., 2017;
.282 by Plath & Leiss, 2018; and .486 by Leiss et al., 2010). Differences in the magnitudes of
correlations between different studies can be explained by the specifics of the analysis. In
the present study, we focused on the construction of a real-world model and not on
modelling competence as a whole. Our study expanded the prior results by indicating the
relevance of reading comprehension for the construction of a real-world model.However,
future studies are necessary to investigate whether reading comprehension affects modelling or
vice versa.
In line with our expectations, we found a positive effect of presenting reading
comprehension prompts on students’interest in modelling. Even if presenting
reading comprehension prompts did not directly improve modelling competence, it had
a positive impact on students’perceptions of modelling. The in-depth analysis provided
initial hints about the importance of different sources of interest in modelling. We
found no effect of the accuracy of reading comprehension on students’interest in
modelling. This finding indicates that ease of comprehension, which was found to be a
source of situational interest in prior studies (Mitchell, 1993;Schrawetal.,1995), did
not enhance interest in modelling in our studies. Consequently, other sources of situa-
tional interest such as students’level of involvement, which was triggered by the reading
comprehension prompts, were potentially responsible for the positive effect on interest in
modelling.
6.2 Educational environment as a moderator of the effects of reading comprehension
We collected data in Germany and Taiwan in order to validate the effects of a reading
intervention on modelling and interest in modelling in two educational environments that
are very different from each other. We expected that reading comprehension prompts would
enhance modelling competence and interest in modelling for students in both educational
environments.
There were no differences regarding the effect of presenting reading comprehension
prompts on the construction of a real-world model. However, the in-depth analysis showed
that reading comprehension is a significant predictor of the construction of a real-world
model for German but not for Taiwanese students. One explanation for this result is that
perhaps the Taiwanese students tended to fail to construct a real-world model even when
they managed to understand the situation. In order to successfully construct a real-world
model, students also need to structure and simplify the information, which also includes
making assumptions. Students seem to lack meta-knowledge about modelling, particular
the knowledge that solving modelling problems often requires learners to make assump-
tions (Krawitz et al., 2018), and it was found to be a particular strength of students educated
in Germany compared with students educated in other countries. This is presumably
because German students have more experience working with modelling problems
(Chang et al., 2020;Hankeln,2020).
The effect of presenting reading comprehension prompts on students’interest in modelling
differed between the two educational environments such that the German students benefitted
more from the reading comprehension prompts. A potential explanation is that different levels
of interest in mathematics from German and Taiwanese students (OECD, 2013) caused this
effect. Further studies should focus on the conditions in which fostering reading comprehen-
sion is beneficial for interest in modelling.
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7 Limitations
In the present study, reading comprehension was assessed by rating students’answers to the
reading comprehension prompts. This allowed us to measure reading comprehension in a
domain-specific way. For modelling, it is important that reading comprehension is measured in
a mathematics-specific way (e.g., reading numerical information presented in tables) (Leiss
et al., 2010), and hence, construct validity could be increased compared with the use of a
general reading comprehension test. However, because of this assessment, we did not collect
any information about reading comprehension in the control condition. Thus, we could not
determine whether the reading comprehension prompts led to better reading comprehension in
the experimental condition compared with the control condition.
Another limitation is that the scores for the construction of a real-world model were found
to be low, which might have resulted in floor effects. However, solving modelling problems is
known to be a demanding activity, and students’low scores on the construction of a real-world
model reflect the use of demanding modelling problems to measure modelling competence.
The construction of a real-world model was assessed by coding solutions to modelling
problems. Results may have been different if the problems had focused on only this sub-
competency. However, artificial tasks might have to be used if students are going to be asked
to construct a real-world model. Further, the reading comprehension prompts and the model-
ling problems were presented on separate pages, but we do not know whether the students in
the RP condition followed the given order and answered the reading comprehension prompts
before working on the modelling problems. Future studies should include a treatment check.
Another limitation addresses the use of a questionnaire for measuring students’interest in
modelling. Students were asked to rate how interesting it was to work on the respective task.
We do not know which aspects of the tasks they referred to when they made their judgments. It
is possible that different cultural and societal factors influenced the reports of the German and
Taiwanese students, and thus, the comparison of these measures should be treated with
caution. This measure also does not provide information about which aspects of the presented
modelling problems the students referred to. German students might have referred to their
interest in the real-world context, which is triggered by engagement with the reading compre-
hension prompts, whereas Taiwanese students might have referred to the task format of
modelling problems themselves, which they found interesting because of its novelty. Further
studies, particularly ones including qualitative approaches, are necessary to make more
elaborative statements.
8 Conclusion
Our study shows that students’interest in modelling but not their modelling competence
can be improved by presenting reading comprehension prompts. However, the findings
differ for learners in Germany and Taiwan, indicating the relevance of educational
environments for research in modelling. Consequently, reading comprehension is an
essential but not sufficient condition for modelling. Students’experience with modelling
seems to play a decisive role. Hence, we suggest that, in addition to students’reading
comprehension, students’meta-knowledge about modelling, particularly the knowledge
that modelling problems often require assumptions, should be addressed in modelling
research and practice.
354 Krawitz J. et al.
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Appendix
Fig 8 Sample pages from the test booklet for the RP condition, including the situational description, reading
comprehension prompts, and modelling problems for the real-world situation parachuting.
Table 3 Means, standard deviations, and correlations presented separately for each experimental condition (RPG
and CG) and educational environment (Germany and Taiwan)
Group Variable MSD1234
RP condition 1 Construction of a real-world model 0.118 0.164 -
2 Intramathematical competence 0.518 0.284 0.446** -
3 Interest in modelling 2.868 0.895 0.097 0.174** -
4 Educational environmenta--−0.283** −0.061 0.100 -
Control condition 1 Construction of a real-world model 0.127 0.166 -
2 Intramathematical competence 0.512 0.279 0.495** -
3 Interest in modelling 2.697 0.995 0.068 0.205** -
4 Educational environmenta--−0.231** 0.034 0.251** -
Germany 1 Construction of a real-world model 0.174 0.175 -
2 Intramathematical competence 0.520 0.251 0.351** -
3 Interest in modelling 2.576 0.949 0.086 0.144*-
4 Reading comprehension promptsa- - 0.004 0.078 0.193** -
Taiwan 1 Construction of a real-world model 0.088 0.148 -
2 Intramathematical competence 0.512 0.300 0.584** -
3 Interest in modelling 2.923 0.926 0.169** 0.227** -
4 Reading comprehension promptsa--−0.050 −0.026 0.021 -
aDummy coded (reading comprehension prompts: RPG = 1, CG = 0; educational environment: Taiwan = 1,
Germany = 0). *p< .05, two-tailed. **p< .01, two-tailed
355The role of reading comprehension in mathematical modelling: improving...
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Acknowledgments This study was conducted in the framework of the Taiwanese-German research program on
cultural-societal influences on mathematics education (TaiGer).
Funding Open Access funding enabled and organized by Projekt DEAL. The meetings of the German and Taiwanese
partners were funded by Deutsche Forschungsgemeinschaft (DFG) and the Ministry of Science and Technology Taiwan
(MOST), allocated to Aiso Heinze (IPN Kiel, Germany) and Kai-Lin Yang (NTNU Taipei, Taiwan), respectively.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and
indicate if changes were made. The images or other third party material in this article are included in the article's
Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included
in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or
exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy
of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Affiliations
Janina Krawitz
1
&Yu-Ping Chang
2
&Kai-Lin Yang
3
&Stanislaw Schukajlow
1
1
University of Münster, Münster, Germany
2
National Pingtung University, Pingtung, Taiwan
3
National Taiwan Normal University, Taipei, Taiwan
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