Culture-Specific Norms Regarding High-Quality Use of Task Potential for Mathematical Learning—Contrasting Researchers’ Perspectives from Germany and Taiwan

Abstract

Factors like the potential of tasks to support students’ mathematical learning and its use in instruction are consensually understood to be relevant for instructional quality across cultural contexts. Yet, research has also shown that perspectives on instructional quality may vary between cultural contexts. As an explanation, it is argued that such perspectives depend on instructional norms, which correspond to the expected behavior in instruction within a cultural context. Notably, research contrasting mathematics instruction from East Asian and Western cultures hints at potentially different instructional norms regarding high-quality use of task potential, but systematic evidence is lacking so far. This study addresses this gap and uses three text vignettes of instructional situations to systematically elicit and contrast instructional norms regarding the use of word problems for mathematical learning. Researchers from Germany (N = 17) and Taiwan (N = 19) evaluated the use of tasks in various instructional situations in an online survey, and their answers were qualitatively analyzed to determine possible culture-specific norms based on their reasoning. In two of the three cases, culture-specific norms in line with assumptions could be identified. In the third case, researchers in both countries referred to an interculturally shared instructional norm. Differences between the reasoning in answers from Germany and Taiwan indicate further cultural influences in line with assumptions based on prior research. We discuss the findings and their implications for the validity of intercultural research in mathematics education.

Full text

ORIGINALARBEIT/ORIGINAL ARTICLE
https://doi.org/10.1007/s13138-024-00237-5
Journal für Mathematik-Didaktik
Culture-Specific Norms Regarding High-Quality
Use of Task Potential for Mathematical
Learning—Contrasting Researchers’ Perspectives from
Germany and Taiwan
Josephine F. Paul · Anika Dreher · Ting-Ying Wang ·
Feng-Jui Hsieh · Anke Lindmeier
Received: 8 December 2022 / Accepted: 4 July 2024
© The Author(s) 2024
Abstract Factors like the potential of tasks to support students’ mathematical learn-
ing and its use in instruction are consensually understood to be relevant for instruc-
tional quality across cultural contexts. Yet, research has also shown that perspectives
on instructional quality may vary between cultural contexts. As an explanation, it
is argued that such perspectives depend on instructional norms, which correspond
to the expected behavior in instruction within a cultural context. Notably, research
contrasting mathematics instruction from East Asian and Western cultures hints at
potentially different instructional norms regarding high-quality use of task potential,
but systematic evidence is lacking so far. This study addresses this gap and uses
three text vignettes of instructional situations to systematically elicit and contrast
The first and the last author contributed equally to the publication.
Josephine F. Paul · Anke Lindmeier
Fakultät für Mathematik und Informatik, Abteilung Didaktik, Friedrich-Schiller-Universität Jena
(FSU Jena), Ernst-Abbe-Platz 2, 07743 Jena, Germany
E-Mail: josephine.paul@uni-jena.de
Anke Lindmeier
E-Mail: anke.lindmeier@uni-jena.de
Anika Dreher
Institut für Mathematische Bildung, Pädagogische Hochschule Freiburg (PH Freiburg),
Kunzenweg 21, 79117 Freiburg, Germany
E-Mail: anika.dreher@ph-freiburg.de
Ting-Ying Wang · Feng-Jui Hsieh
Department of Mathematics, National Taiwan Normal University (NTNU Taipei), Roosevelt
Road, Taipei 10617, Taiwan, Province of China
Ting-Ying Wang
E-Mail: tywang@gapps.ntnu.edu.tw
Feng-Jui Hsieh
E-Mail: hsiehfj@math.ntnu.edu.tw
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J. F. Paul et al.
instructional norms regarding the use of word problems for mathematical learn-
ing. Researchers from Germany (N= 17) and Taiwan (N=19) evaluated the use of
tasks in various instructional situations in an online survey, and their answers were
qualitatively analyzed to determine possible culture-specific norms based on their
reasoning. In two of the three cases, culture-specific norms in line with assump-
tions could be identified. In the third case, researchers in both countries referred to
an interculturally shared instructional norm. Differences between the reasoning in
answers from Germany and Taiwan indicate further cultural influences in line with
assumptions based on prior research. We discuss the findings and their implications
for the validity of intercultural research in mathematics education.
Keywords Cultural norms · Instructional norms · Cross-cultural research ·
Intercultural validity · Quality of mathematics instruction
Kulturspezifische Normen zur Nutzung von Aufgabenpotenzial im
Mathematikunterricht – Ein Vergleich von mathematikdidaktischen
Perspektiven aus Deutschland und Taiwan
Zusammenfassung Das Potenzial von Aufgaben zur Förderung des mathemati-
schen Lernens von Schülerinnen und Schülern sowie dessen Nutzung im Unterricht
werden allgemein als relevant für die Unterrichtsqualität in verschiedenen kultu-
rellen Kontexten angesehen. Die Forschung hat aber auch gezeigt, dass die Per-
spektiven auf Aspekte der Unterrichtsqualität je nach kulturellem Kontext variieren
können. Als Erklärung wird angeführt, dass solche Perspektiven von Unterrichtsnor-
men abhängen, die das im Unterricht erwartete Verhalten innerhalb eines kulturellen
Kontextes widerspiegeln. Untersuchungen, die den Mathematikunterricht in ostasia-
tischen und westlichen Kulturen kontrastieren, deuten insbesondere auf potenziell
unterschiedliche Unterrichtsnormen in Bezug auf eine qualitativ hochwertige Nut-
zung von Aufgabenpotenzial hin. Bislang fehlt es jedoch an systematischen Belegen.
Die vorliegende Studie adressiert diese Lücke und nutzt drei Textvignetten, die je
eine Unterrichtssituation repräsentieren, um systematisch Unterrichtsnormen bezüg-
lich der Nutzung von Textaufgaben für mathematisches Lernen zu erheben und
zu kontrastieren. Forschende aus Deutschland (N= 17) und Taiwan (N= 19) haben
den Umgang mit Aufgaben in verschiedenen Unterrichtssituationen in einer Online-
Befragung bewertet. Ihre Antworten wurden qualitativ analysiert, um auf der Grund-
lage ihrer Argumentationen mögliche kulturspezifische Normen zu bestimmen. In
zwei der drei Fälle konnten erwartungskonforme kulturspezifische Normen identifi-
ziert werden. Im dritten Fall wiesen die Antworten der Forschenden beider Länder
auf eine interkulturell geteilte Norm hin. Unterschiede in den Argumentationen der
deutschen und taiwanesischen Forschenden deuten außerdem auf weitere kulturelle
Einflüsse hin, die mit Annahmen basierend auf der bisherigen Forschung überein-
stimmen. Wir diskutieren die Ergebnisse und ihre Implikationen für die Validität
interkultureller mathematikdidaktischer Forschung.
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Culture-Specific Norms Regarding High-Quality Use of Task Potential for Mathematical...
Schlüsselwörter Kulturelle Normen · Unterrichtsnormen · Kulturübergreifende
Forschung · Interkulturelle Validität · Unterrichtsqualität
1 Introduction
The question of what constitutes instructional quality is crucial in mathematics
education. On the one hand, several aspects were described as relevant for high-
quality mathematics instruction across different cultural contexts, such as responding
to student thinking or using representations. Among those aspects, the potential of
mathematical tasks and its use in instruction is typically considered since tasks
are central for students when learning mathematics (e.g., Doyle 1988; Hsieh et al.
2017;LinandLi2009; Praetorius and Charalambous 2018; Stein and Lane 1996).
On the other hand, research indicates that perspectives on what constitutes high-
quality instruction may also vary between cultural contexts, like East Asian and
Western ones (Clarke 2013; Hsieh et al. 2017). Exemplarily, this can be seen in
the way tasks are used. For instance, one aim of East Asian instruction is to guide
students to memorize solution strategies and reflect on them to acquire efficient
strategies. In contrast, Western instruction aims at supporting the development of
various individual solution ideas (Leung 2001).
One explanation for perspectives being different across cultural contexts builds
on the notion of instructional norms, which characterize what is seen as appropriate
behavior in instruction. Moreover, these norms shape expectations about people’s
actions in instruction (Herbst and Chazan 2011). For example, when researchers are
asked to evaluate a task’s potential and its use in a specific instructional situation,
they evaluate the situation against their expectations. The researchers’ expectations,
in turn, correspond to the (often implicit) instructional norms, which may be specific
to the cultural context to which the person belongs.
Although there are good reasons to assume that perspectives on what constitutes
high-quality use of task potentials depend on culturally shaped instructional norms,
there has been little systematic research on the culture-specificity of instructional
norms. This is particularly critical since such instructional norms are typically used
as a frame of reference when designing assessment instruments for cross-cultural
comparative research (Dreher et al. 2021). Thus the lack of knowledge threatens the
validity of cross-cultural research (Clarke 2013).
Addressing this problem, we systematically investigate whether instructional
norms regarding high-quality use of task potential in mathematics instruction are
shared across cultures or culture-specific. We exemplarily compare the instructional
norms in Taiwan and Germany because we consider this contrast to be particularly
informative: Students in these countries perform very differently in comparative
studies like PISA or TIMSS (e.g., Mullis et al. 2020; Reiss et al. 2019), and thus,
differences regarding the instructional quality are plausible. Moreover, prior research
indicates differences between East Asian and Western countries from a broader per-
spective, which allows inferring expectations regarding specific aspects of mathe-
matics instruction against which new findings can be evaluated. To pursue our aim
to explicate and compare instructional norms across the two countries, we contrast
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the evaluations of mathematics education researchers from Taiwan and Germany re-
garding the high-quality use of word problems using vignettes of three instructional
situations.
2 Theoretical Background
To set the theoretical background, we introduce the concept of instructional norms
as specific social norms and discuss reasons for their potentially culture-specific
nature. According to our focus on instructional norms regarding the high-quality
use of tasks, we analyze the role of tasks, especially word problems, in mathematics
instruction and present expected differences regarding the high-quality use of word
problems in East Asian and Western mathematics instruction.
2.1 Instructional Norms
In their endeavor to understand what matters in instructional interactions, researchers
use sociological concepts like the notion of social norms. Social norms are generally
considered rules of behavior on which there is—potentially implicit—agreement
within a certain social context. They pre-structure action and shape expectations
regarding the behavior of members of the context (Coleman 1990). Thus, members
of a certain social context are largely familiar with the corresponding norms, even
if they are not necessarily conscious of them.
Classrooms have been described as constituting social contexts of their own
where social norms affect students’ and teachers’ actions. In particular, Herbst and
Chazan (2011) explicated the general concept of classroom social norms by defin-
ing instructional norms as “a collective sense of what is conceivable and perhaps
desirable to happen in classrooms” (Herbst and Chazan 2011, p. 406). Instructional
norms permeate the actions in an instructional situation, their temporal structure,
and the expectations of who should respond when and how (Herbst and Miyakawa
2008). Thus, instructional norms require specific situations to become evident.
Research on social norms specific to mathematics instruction typically incorpo-
rates the idea that the nature of the subject of instruction shapes the norms (e.g.,
Herbst and Chazan 2011; Yackel and Cobb 1996). However, despite this common-
ality, different norm concepts were considered in prior research with a focus on
mathematics instruction. For example, Herbst’s and Chazan’s (2011) perspective is
motivated by the question of how practical rationalities may be explained through in-
structional norms, aiming at explaining how specific situations are typically handled
in mathematics instruction, given the various obligations teachers face. Obligations
to the discipline of mathematics constitute one aspect, while obligations to students
and other components of the schooling process make up others (Herbst et al. 2011,
p. 223). Yackel and Cobb (1996)definedsociomathematical norms as what is consid-
ered mathematically normative in a mathematics classroom, emphasizing obligations
to mathematics. They aimed to clarify how teachers and students construct shared
mathematical concepts, like what counts as “an acceptable mathematical explanation
and justification” (Yackel and Cobb 1996, p. 461). Sociomathematical norms were
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also productive in explaining observed differences in instructional practices (Yackel
and Cobb 1996). However, in contrast to the broader concept of instructional norms,
they may be understood more narrowly as capturing the practical rationalities of the
inner mathematical structure of a situation. Thus, the concept of instructional norms
seems better suited to describe differences that may occur in contexts that differ in
terms of various obligations, as we explain, would be expected when contrasting
cultural contexts.
Finally, it will be briefly discussed how instructional norms originate. As with any
social norm, they are learned via socialization processes (Coleman 1990). During
teacher education, training, and further professional development, these norms are
subject to (implicit or explicit) social negotiation processes between (prospective)
teachers and teacher educators (Herbst 2003; Tatto 1998). In Germany and Taiwan,
mathematics education professors play an essential role in the socialization pro-
cesses: As teacher educators, the professors transmit norms by deciding on learning
opportunities and grading (future) teachers with consequences for teacher admission.
Due to their position as researchers, their associated enculturation and experiences,
which are typically not limited to single classes or schools, and their involvement in
policy decisions on teacher education, the professors also play an essential role in
shaping instructional norms within their cultural contexts (see Schwille et al. 2013,
especially p. 82 for Taiwan).
2.2 Cultural Influences on Instructional Norms
As the social context of education is part of an overarching cultural context, instruc-
tional norms regarding mathematics instruction are part of a broader cultural norm
system (Coleman 1990). With culture, we refer to the “shared motives, values, be-
liefs, identities, and interpretations or meanings of significant events that result from
common experiences of members of collectives that are transmitted across gener-
ations” (House et al. 2004, p. 15). Accordingly, countries with a shared history of
Confucianism are often called East Asian cultures. Countries in Western Europe that
share the experience of the Renaissance and Enlightenment, as well as the Anglo-
Saxon countries, are called Western1cultures (Leung 2001).
Prior research surfaced differences in mathematics instruction in East Asian and
Western cultures, spanning questions of the nature of mathematics to broader ques-
tions of the organization of instruction (e.g., Clarke 2013; see also Leung 2001 who
synthesized the differences along six dichotomies). For Taiwan as an East Asian
country, for example, mathematics instruction is characterized as exam-, content-
and product-oriented (Leung 2001;Yangetal.2022). The mastery of procedures is
accordingly valued highly. In Germany, as a Western country, mathematics instruc-
tion follows a competence-based curriculum emphasizing mathematics as a tool to
1It is essential to be aware that the term Weste r n expresses a Eurocentric perspective. It does not only
refer to countries within the geographical West, e.g., North America and Europe, but also to countries that
share historical roots with Europe, e.g., Australia. In the lack of a better alternative, we continue to use
the term as it is done by others, e.g., Leung (2001). We are aware that using these terms carries the risk of
over-generalizing, as both terms span many different countries whose traditions also vary.
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solve real-world problems, valuing mathematics as a process (Chang 2014; Leung
2001; Verschaffel et al. 2020). Meaningful learning is accordingly valued higher
than mastery of procedures. Furthermore, the instruction in Germany and Taiwan
may differ in the orientations towards the organization as whole class teaching ver-
sus individualized learning, which may stem from different perspectives on the role
of teachers: teachers are seen as subject-matter specialists in East Asia but are pri-
marily seen as pedagogues in Western countries (Leung 2001). The latter refers to
the observation that, in Western countries compared to East Asian countries, being
able to facilitate individual learning processes may be relatively more important for
teachers than being an exemplary mathematician.
These differences were reasoned to reflect cultural traditions and result in different
obligations of teachers. Accordingly, it may be assumed that (potentially implicit)
norms of mathematics instruction may also be culture-specific. At the same time,
there are reasons for some instructional norms being similar across different cul-
tural contexts. Mathematics is considered an internationally homogeneous discipline,
and there is an increasing international research discourse on instructional quality
in mathematics education (e.g., Praetorius and Charalambous 2018). To systemati-
cally investigate the open questions regarding the culture-specificity of instructional
norms, situations in which the practical rationalities can become visible must be
focused. Among the many possible focuses, situations that center on the use of
tasks may be considered particularly suitable as learners in mathematics instruction
engage most of the time with tasks2, and how tasks are used in instruction is in-
ternationally considered crucial for instructional quality (Mu et al. 2022; Stein and
Lane 1996).
2.3 Focusing on Word Problems to Investigate Instructional Norms
Word problems are specific types of tasks characterized by verbal descriptions of
problem situations, which raise questions that can be answered by applying math-
ematical operations (Verschaffel et al. 2020). Word problems require students to
understand the given problem situation and model it mathematically. This includes
transforming the verbal description into a mathematical model (making assump-
tions about the situation as necessary), working mathematically with the model, and
relating the mathematical solution (steps) to the problem context (repeatedly), for
instance, when validating results (e.g., modeling cycle, Blum and Leiss 2007; Chang
et al. 2020).
Word problems3can be used with different goals in instruction. In addition to
their potential to promote modeling skills, word problems can provide opportunities
2Doyle (1988) suggested using a broader definition of the notion of a task, including its implementation
in instruction. To emphasize the distinction between the potential of a task and its use by the teacher in
instruction, we refrain from using the broad definition of the term task, but instead, identify the term task
with its representation in the instructional materials.
3The delineation of the term word problems is subject to discourse (Verschaffel et al. 2020). For instance,
some researchers differentiate between realistic and standard word problems, depending on whether or not
real-world knowledge is needed to solve them (Fitzpatrick et al. 2020). Others distinguish between (stan-
dard) word problems and tasks that refer to real-world situations (e.g., Chang et al. 2020). The latter—so-
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to practice problem-solving strategies and support learners in building mathematical
concepts or thinking creatively (Sun-Lin and Chiou 2019; Verschaffel et al. 2020).
Despite the high potential of word problems for mathematical learning, it depends on
the way it is used by teachers and students in instruction whether such potential can
be used fruitfully for learning (Doyle 1988;Neubrandetal.2013; Stein and Lane
1996). Relevant aspects of the task’s use that may support or impede instructional
quality are, for instance, the alignment with the learning goal, the product a teacher
expects, the actions and resources used to create the product, and the importance
of the task in the accountability system (Doyle 1988). For example, how students
work mathematically may align with a task’s potential, available resources may
support or hinder mathematical thinking, and how a teacher implements a task may
affect what students perceive as relevant about mathematics (Doyle 1988;Herbst
2003). As practices of using word problems in mathematics instruction can greatly
vary, a focus on the use of word problems is suited to (exemplarily) investigate
instructional norms.
2.4 Word Problems in Western and East Asian Mathematics Instruction
Using the theoretical considerations of instructional norms, it can be expected that
what is considered normal practice regarding the use of tasks for mathematical
learning may vary between cultural contexts (Herbst and Chazan 2012). Particularly,
there are indications that the perspectives on the use of word problems may differ
in East Asian and Western contexts. To substantiate this argument, we will draw
again on the characterizations of East Asian and Western mathematics instruction
by Leung (2001) and infer specific assumptions for the contrast between Germany
and Taiwan.
The dominant process-oriented view on mathematics in Western countries (vs.
the product-oriented view in East Asian countries) manifests in the observation that
mathematical modeling is valued more in Western countries (e.g., Denmark, Aus-
tralia, the United States of America, Italy, or Sweden; Niss 2018)thaninEastAsian
countries (Leung 2001). Particularly in Germany, the role of modeling and real-
world contexts in mathematics instruction has grown strong since the realistic turn
of mathematics education in the 1970s (Sträßer 2019). For instance, the modeling
cycle (Blum and Leiss 2007), which characterizes mathematical modeling as a pro-
cess, is widely known and used in research and practice in Germany. Mathematical
modeling has been an explicit part of the competence-based curriculum in Germany
since 2003 (Chang 2014; Chang et al. 2020). In Taiwan, mathematical modeling
only recently gained curricular attention in teaching reforms (Chang et al. 2020;
called modeling tasks—may need students to make assumptions or consider multiple solution methods
(e.g., Krawitz et al. 2022; for the role of assumptions in modeling processes, see Galbraith and Stillman
2001). Mathematical modeling, in the sense of solving modeling tasks, received much attention in many
countries, especially in Western ones. In this Taiwanese-German study, we use the term word problem in
line with Verschaffel et al. (2020) as verbal descriptions of problem situations of any kind. This decision
implies that mathematical modeling, in our study, applies to both real-world and other problem situations
that all have in common that they demand a transformation of a real-world situation into a mathematical
model, even if they do not necessarily require students to make assumptions.
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see also for China, Cao and Leung 2018). Accordingly, one would assume that
instructional norms regarding the high-quality use of word problems in Germany
reflect a stronger emphasis on their use to engage students in mathematical modeling
processes in comparison to Taiwan.
The findings of comparative studies also resonate with these expected differences.
For instance, Chang et al. (2020) compared the modeling competences of German
and Taiwanese students. They found that despite the Taiwanese students being more
mathematically knowledgeable, German students were more capable of setting up
models than Taiwanese students. Similar observations were made for pre-service
teachers in Taiwan whose performance in abstraction and mathematical work was
better than in process-oriented tasks (Hsieh et al. 2012).
In contrast, acquiring mathematical knowledge as a product is more important
in East Asian mathematics instruction. Instructional content in general, and thus
also word problems in particular, would be expected to support the consolidation
or application of learned knowledge (Pratt et al. 1999). Memorizing content (often
misinterpreted as rote learning4without understanding) is valued highly and seen as
essential before applying it to problems since a broad knowledge base is understood
to be a key to performance (Leung 2001; Pratt et al. 1999). Hence, learning is first
oriented towards acquiring mastery or an ideal way of solving, often in a way it was
shown by the teacher (Pratt et al. 1999). Repetitive practice with variations is one
path to achieving this end (Wong 2006).
Studies indicate that this focus on optimal solution strategies in East Asian math-
ematics instruction is also relevant when engaging in problem-solving processes
like mathematical modeling (Cai 2006). Remarkably, Xu et al. (2022) found that
Chinese mathematics teachers prefer to teach modeling in a teacher-centered way
(representing the subject-matter expert). It was also reported that in East Asian in-
struction, the usefulness of solutions is valued over the novelty of solutions (Morris
and Leung 2010).
However, another essential aspect of East Asian instruction is the idea that content
learned by students should be reflected on because there is no real learning without
thinking and reflection (Leung 2001). Under this perspective, teachers are encour-
aged to reflect on solution strategies or to ask students to reflect on the strategies by
themselves so that students can understand why a strategy is considered ideal. Ac-
cordingly, one would assume that instructional norms regarding the high-quality use
of word problems in Taiwan show a stronger emphasis on reflecting and building
on existing, optimal, and efficient solution methods when applying mathematical
concepts compared to Germany (Morris and Leung 2010).
In contrast, Western traditions value abilities to flexibly solve mathematical tasks
by developing individual strategies according to the task’s characteristics (Morris
and Leung 2010). Comparing different solution strategies has indeed shown to be
helpful for flexibly solving tasks in studies in Germany and the USA (Silver et al.
2005; Schukajlow et al. 2015). Hence, exploratory working of students that sup-
ports them in finding and comparing individual solutions is recognized to support
4Leung (2001) originally named the corresponding dichotomy “rote learning versus meaningful learning”
(p. 39) which may have fostered misinterpretations.
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Culture-Specific Norms Regarding High-Quality Use of Task Potential for Mathematical...
meaningful learning in Western contexts (Dreher et al. 2021; Leung 2001). In partic-
ular, it is considered essential to scaffold students’ interactions with tasks and their
multiple solutions (for Germany: Borneleit et al. 2001). Hence, teachers should en-
able individual experiences, attend to solutions of the learners, compare strategies
and use students’ work to establish meaning. Consequently, one would assume that
instructional norms regarding the high-quality use of word problems in Germany
more strongly reflect the importance of meaningful learning through establishing
connections.
To sum up, Western and East Asian instruction, as discussed in this chapter, can
be expected to differ in three aspects referring to practicing modeling processes, to
reflecting and understanding the efficiency of solution strategies, and to comparing
different individual strategies.
3 Research Interest
Although our argumentation shows that there are good reasons that culturally shaped
instructional norms may impact what counts as high quality in mathematics in-
struction, there has been little systematic research on the culture-specificity of in-
structional norms. This may be a problem for cross-cultural comparative research,
where frameworks or instruments are used whose underlying norms often remain
implicit. Our study addresses this problem by explicating instructional norms regard-
ing a high-quality use of task potential based on German and Taiwanese researchers’
evaluations of the use of tasks in three exemplary instructional situations. Further-
more, we examine whether the norms are culturally shared or culture-specific.
Against the background of cultural differences and the resulting assumptions
about differences in instructional norms, which are in turn expected to influence
evaluations of teaching within the cultural contexts, we ask:
RQ1) Do mathematics education researchers’ evaluations indicate that there are, as
anticipated, culture-specific instructional norms regarding the high-quality use of
tasks for mathematical learning in Germany and Taiwan?
RQ2) Do the evaluations reflect further differences supporting the assumption of cul-
ture-specific influences on conceptions of high-quality use of tasks for mathematical
learning?
4 Context and Methods
This study is part of the binational project “Teacher noticing in Taiwan and Ger-
many” (TaiGer Noticing). The project aims to investigate the role of potentially
culture-specific norms regarding aspects of instructional quality (Dreher et al. 2021;
Lindmeier et al. 2024). To achieve this goal, the project focuses on three different
aspects of instructional quality, which are relevant in Taiwanese and German mathe-
matics instruction and may show cultural differences: The use of representations
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(Dreher et al. 2024), responding to students’ thinking (Dreher et al. 2021), and the
use of task potential. This contribution focuses on the latter aspect5.
4.1 Instruments
As instructional norms are often implicit but become evident in instructional sit-
uations, we follow a situated approach using representations of practice to elicit
instructional norms (Herbst and Chazan 2011). The approach is based on the ethno-
methodological notion of a breaching experiment (Mehan and Wood 1975)and
indirectly allows inferring instructional norms via a person’s reaction when con-
fronted with the breach of an anticipated norm. Specifically, we use descriptions of
instructional situations where teachers use tasks in a way not in line with the behav-
ior expected according to an anticipated instructional norm. If someone evaluates
a situation negatively and expresses the corresponding points of criticism, it can be
concluded that the person had expected the (breached) instructional norm to hold.
Following this approach, we designed text vignettes that all include breaches of
anticipated instructional norms regarding high-quality use of task potential in mathe-
matics instruction from the perspectives of the authors from Germany or Taiwan
(Dreher et al. 2021).
When designing the vignettes, a specific focus was on ensuring ecological valid-
ity (i.e., the vignettes represent instructional situations that are conceivable in both
countries’ secondary mathematics instruction). Hence, developing the vignettes fol-
lowed a sophisticated concurrent development process conducted symmetrically in
Germany and Taiwan. This process was documented in detail by Dreher et al. (2021).
It included the consolidation of a joint item development process, the independent
development of vignettes in each country, as well as methods of reviewing and
translating the vignettes for the intended binational use. As part of this process,
the research team also decided to focus on the topic of (linear and quadratic) func-
tions and equations since it is central in the secondary curriculum (grades 7–9) in
Germany and Taiwan.
Each vignette consists of 1) a picture of the task implemented in the instruc-
tional situation, which is assumed to have a high potential for mathematical learning
from the perspective of the authoring research team from Taiwan or Germany, and
2) a fictitious transcript of a classroom interaction (about 200 words). Since the
vignettes were designed following breaching experiments, they are supposed to il-
lustrate a non-optimal use of the task’s potential (breach of an instructional norm).
In total, we developed six vignettes focusing on high-quality use of tasks—three in
each country. During the development process, we did not specify which kind of task
the vignettes should focus. Only three vignettes finally focused on word problems
(see Sect. 2.3), so this report is based on these three vignettes (see Lindmeier et al.
2024, for another vignette with a focus on the use of tasks not built on a word
problem). Two of the three vignettes were developed in Germany (Task1, Task2),
5A working paper based on the same data was published in the proceedings of an international conference
in 2022 (Lindmeier et al. 2022).
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Culture-Specific Norms Regarding High-Quality Use of Task Potential for Mathematical...
Fig. 1 Vignette Task2
whereas one (Task46) was developed in Taiwan. Each vignette includes a breach of
an anticipated norm from the perspective of the authoring research team from one
country but not necessarily from the perspective of the research team from the other
country.
Vignette Task2 focuses on the task “cliff-jumping” (topic: quadratic functions,
Fig. 1). The task prompts students to engage with a real-world scenario presented
as graphically supported text. It requires students to understand the real-world sit-
uation, make an educated guess about the solution based on the real-world context
and its graphical visualization and then determine the immersion point by a given
mathematical model (mathematization and working mathematically). The quadratic
equation has two solutions, but only one can be the point of immersion. Hence, the
mathematical results are needed to be interpreted and validated. Thus, the learning
potential of this task refers to the known difficulty of students connecting real-world
situations and mathematical models. Considering that the highlighted steps corre-
spond to the modeling cycle (Blum and Leiss 2007), the vignette’s authors assume
that the task has a high potential to support students to engage in mathematical
modeling processes and expect teachers using this task to scaffold the students in
connecting their understanding of the situation to the mathematical model and in
using their understanding of the situation to validate the results.
6The vignettes are referred to with labels according to the labels in the overall project TaiGer Noticing as
we want to guarantee consistency between publications.
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Fig. 2 Vignette Task4
The instructional situation in vignette Task2 starts after a group working phase.
The teacher collects the students’ guesses and then shifts the focus toward using the
given function. In an interactive teaching style, two different ways of finding the
roots of the equation are presented verbally (solution formula, factoring). A student
mentions that one of the roots is irrelevant as a solution to the word problem.
The teacher confirms and redirects the interpretation of the remaining result to
a student. The instructional situation is closed with a remark on how to find the roots
of quadratic equations. The teacher does not emphasize the relationship between
the real-world situation and its mathematical model by using educated guesses or
the visualization of the situation. From the perspective of the vignette’s authors,
the teacher neglects the task’s potential to support students to engage in modeling
processes.
Vignette Task4 focuses on the task “student camp” (topic: systems of linear
equations, Fig. 2). The task requires students to understand a real-world scenario
presented as text and to set up a system of equations to determine the solution. The
potential of this task from the perspective of the Taiwan authoring team is that the
task can support the acquisition of efficient or ideal solution strategies, as it allows
for different variable assignments: a) The assignment of x and y as the number of
student groups or b) the assignment of x and y as the number of students in line with
the question formulated in the word problem. The first approach results in a simpler
calculation, while the second approach may be quicker to find for the students
because of the congruency with the word problem formulation. Hence, the Taiwanese
team members would expect a teacher to discuss the various characteristics of the
different variable assignments. A teacher should support students reflection of the
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Culture-Specific Norms Regarding High-Quality Use of Task Potential for Mathematical...
Fig. 3 Vignette Task1
pros and cons of each solution path to let them understand which one is the most
efficient to solve the task.
This expected behavior is breached in the instructional situation represented in
the vignette. The teacher presents how to set up the system of equations with the
variables as the number of groups (a). The students express confusion, and one asks
why the variables were not set to represent the number of students (b). The teacher
presents the corresponding approach, asks for preferences, and highlights the first
resulting in a simpler calculation. From the perspective of the Taiwanese authoring
team, the teacher does not use the potential of the task to discuss the pros and cons
of different variable assignments but merely states that the students should go with
the first approach because of the simpler calculation.
Vignette Task1 focuses on the task “medicine dosage” (topic: proportional rela-
tionship, Fig. 3). The task prompts students to engage with the real-world situation
presented by a text and an illustrative image. First, the task requires students to de-
termine a specific dosage for Paul. Further, the task prompts students to think about
different solution strategies and find at least two. The potential of this task from the
perspective of the vignette’s authors is that it may support students in learning to
solve word problems flexibly by actively prompting them to find different ways to
solve the problem. Accordingly, when using this task in instruction, the German au-
thors of the vignette would expect the teacher to discuss different solution strategies
of the students to support flexible problem-solving and to emphasize connections
between the solutions.
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In the instructional situation represented in vignette Task1, the teacher first col-
lects the solutions of two students verbally, praises them without specific feedback,
and marks them verbally as strategies without further elaboration. A third student’s
way of solving based on equations is elaborated on and noted on the blackboard by
the teacher. From the perspective of the vignette’s authors, the teacher misses the
opportunity to connect the different ways of solving the task. Thus, the teacher fails
to focus on meaningful learning because s/he just checks the result for correctness
and notes one solution, which is considered a non-optimal use of the task’s potential.
4.2 Sample and Data Collection
To examine whether there are, as anticipated, culture-specific instructional norms
regarding the high-quality use of tasks, we invited mathematics education professors
in Germany and Taiwan to evaluate the teacher’s use of the task in the exemplary in-
structional situations. The selection criteria were that they are active in mathematics
education research, have a background related to secondary mathematics teaching,
and are involved in educating future mathematics teachers. We aimed for a sample
size of 15 participants in each country. Because of an assumed completion rate of
50%, a random sample of 30 professors in Germany meeting the criteria was invited.
In Taiwan, only 32 persons met the criteria. Hence, all of them were invited. In total,
19 Taiwanese professors (6 female, 13 male) from 10 universities and 17 German
professors (7 female, 10 male) from 13 universities worked on the vignettes. Com-
pletion rates were similar in both countries (TW 59%, GER 56%). All participants
answered each of the vignettes.
On average, the German researchers spent 18.4 years (SD= 9.8) on research in
mathematics education and they were involved in teacher education for 15.9 years
(SD = 7.6). Eight German researchers also did research in mathematics. The Tai-
wanese researchers spent, on average, 16.9 years (SD= 7.5) on research in mathe-
matics education. They were involved in teacher education for 11.47 years (SD=
7.3). Five Taiwanese researchers also did research in mathematics.
To elicit the implicit instructional norms held by the researchers, the vignettes
were presented in the form of an online survey in the participant’s native language
(German or Chinese) with the following prompt: “Please evaluate the teacher’s use
of the task in this situation and give reasons for your answer.” The researchers
answered in their native language.
4.3 Translation and Analysis
Translation Subsequently, the national research teams translated the answers into
English as the common language within the research team. The other team members
reviewed the translations, and an external trilingual person checked each translation.
The original answers are attached in the digital supplementary material.
Development of the Coding Manual Afterward, a coding manual was coopera-
tively created by the German and Taiwanese members of the research team. The
coding follows a two-step process for each evaluation: Step 1) Did the researcher
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Tab l e 1 Deductively developed part of the coding manual regarding coding step 1
Step 1: Codes regarding researchers’ critique regarding the teacher’s use of the task
Did the researcher criticize the teachers’ use of the task as represented in the vignette?
(Codes used across vignettes)
Code Explanation
0=no The teacher’s use of the task was not criticized in any aspect
1 = yes The teacher’s use of the task was criticized
Tab l e 2 Deductively developed part of the coding manual regarding coding step 2
Step 2: Codes regarding the breaches of the anticipated norms
Did the researcher see the breach of a norm anticipated by the vignette’s authors?
Vignette Code
(Critique regarding the)
Explanation
Tas k 2 Breach of the anticipated
norm
It was mentioned that the modeling process/relation
between the situation and the mathematical model was
not emphasized sufficiently
Tas k 4 Breach of the anticipated
norm
It was mentioned that the teacher is not seizing the
opportunity to discuss the pros and cons of different
methods of assigning unknowns
Tas k 1 Breach of the anticipated
norm
It was mentioned that there is no connection/
comparison between the different solution strategies or
only one answer was highlighted
criticize the teachers’ use of the task represented in the vignette? (Code 0= no and
1 = yes; Table 1, codes used across vignettes) And, if so, step 2) Did the researcher
see the breach of a norm anticipated by the vignette’s authors, or did the researcher
give other reasons for critique? (Codes per reason per vignette).
In step 2, the codes are vignette-specific with a deductively determined code for
critique regarding the breaches of the potentially culture-specific anticipated norms
(RQ1). The corresponding part of the coding manual is shown in Table 2.
To analyze whether the researchers’ evaluations reflect further cultural differences
(RQ2), the coding manual for step 2 was enriched for each vignette via inductive
category formation (Mayring 2014). Codes were added for each discernible reason
for critique that repeatedly occurred in the answers. Finally, a code for other reasons
that occurred only singularly was added. These inductively generated codes are
showninTable3. The final coding manual contained all codes shown in Tables 1,
2and 3. All codes are explained in the findings section in detail.
Coding Process After the coding manual was finalized, independent parallel cod-
ing of all answers was conducted in Taiwan and Germany. In detail, every team
member (from Germany as well as Taiwan) coded each answer independently. Dur-
ing the coding process, the coders knew the nationality of the participant as it was
helpful to code answers where, for instance, critical aspects were not clear from the
English translation. If an answer contained more than one reason for a negative eval-
uation, multiple codes could be assigned to reflect the reasoning comprehensively.
Note that the inductively generated codes could also be assigned in addition to those
referring to the breaches of the anticipated norm.
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Tab l e 3 Inductively developed part of the coding manual regarding coding step 2
Step 2: Codes regarding other critiques
Did the researcher mention further critical aspects? If so, which?
Vignette Code
(Further critique regarding)
Explanation
Tas k 2 Algebraic explanations It was mentioned that the teacher does not deal suffi-
ciently with the student’s algebraic solutions
Heterogeneity It was mentioned that it is not sufficiently ensured that
weaker students understand the procedure/can follow
A lack of conclusion It was mentioned that the teaching sequence lacks
an appropriate conclusion on how to deal with such
problems
Use of the context It was mentioned that the context is not/only poorly
used
Treatment of educated
guesses
It was mentioned that the teacher did not talk enough
about the educated guesses
Other reasons (Singularly occurring other reasons, hence no defini-
tion possible)
Tas k 4 Connection of the two ap-
proaches
It was mentioned that the teacher is not connecting the
two approaches to show the equivalence of the ways of
solving
Teacher setting up the equa-
tion
It was mentioned that the students do not get the op-
portunity to assign equations/unknowns on their own
Teacher’s preference It was mentioned that the teachers’ conclusion is pre-
ferring his/her solution and exclusively focused this
one solution
Other reasons (Singularly occurring other reasons, hence no defini-
tion possible)
Tas k 1 Explanations It was mentioned that there is no further explanation/
elaboration of the solution strategies
Other reasons (Singularly occurring other reasons, hence no defini-
tion possible)
Differences in the coding were first discussed within the national teams, and
an agreement coding at the national level was synthesized. This resulted in two
independent codings of the complete data set. The interrater reliabilities (Cohen’s
Kappas) between the German and the Taiwanese coding regarding whether the
answers include critique (step 1) and on the level of the vignette-specific codes
(step2)aredisplayedinTable4. They were observed to be almost perfect (>0.80,
Landis and Koch 1977), except for agreement regarding step 2 for Task4 (moderate
agreement). The differences between the two national codings were then discussed
by the whole research team, and a final coding was generated.
Subsequently, we counted the number of answers per step 2 code. Since there
are no documented cutoff values for which quotas should be used to identify an
instructional norm, we resorted to the assumption that to be able to speak of a norm,
the members within one context should be largely familiar with it. So we decided
to apply the following criterion: If at least half of the researchers from one country
issued the same reason for critique, we concluded that this reason could be consid-
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Tab l e 4 Cohen’s Kappa to
assess interrater reliabilities
between the independent codings
from Germany and Taiwan
Task Cohen’s Kappa for step 1 Cohen’s Kappa for step 2
Tas k 2 1.0 0.85
Tas k 4 0.84 0.60
Tas k 1 1.0 0.93
Cohen’s Kappas regarding the agreement on step 2 were calculated
with a dummy coding accounting for differences in step 1 so that dif-
ferences in step 1 are also reflected in Cohen’s Kappas for step 2. The
coefficients for step 2 hence represent a conservative estimation of
interrater reliability
ered to reflect an instructional norm in their context (majority criterion, Dreher et al.
2021).
5 Findings
5.1 Overview and Organization of the Report
We will describe the findings in detail per vignette in the following way: First, we
report how many researchers criticized the teachers’ use of the task represented in the
vignette in each country (coding step 1). Second, to answer RQ1, we report whether
the majority of researchers per country mention critique regarding the breach of the
anticipated norm and illustrate with sample answers how the researchers addressed
the anticipated norms7. Third, we explain further points of critique as captured by the
inductively generated codes by means of sample answers and show their frequency
per country.This provides evidence of whetherthe evaluations reflect further culture-
specific differences (RQ2).
5.2 Task2 “cliff jumping”
Regarding the use of tasks in this instructional situation, 13 German (76.5%) and
17 Taiwanese (89.5%) researchers issued criticism.
As nine German (52.9%) and only three Taiwanese researchers (15.8%) men-
tioned critique regarding the breach of the anticipated norm, we infer that this an-
ticipated norm regarding the high-quality use of word problems to support students
to engage in modeling processes is a culture-specific norm in Germany (RQ1).
The sample answer of GER1 (Fig. 4) illustrates this: S/He stated that the teacher
largely misses addressing mathematical modeling, which is identified as a potential
of the task. To justify the critique, GER1 suggested improving the use of the task’s
potential through teacher questioning focused on mathematical modeling processes,
including a higher engagement regarding the context and the educated guesses.
7As with vignettes Task2 and Task4 culture-specific norms were identified, our report focuses on dif-
ferences between the countries. Regarding Task1, the report focuses on the commonalities between the
German and Taiwanese researchers’ answers to characterize the elicited interculturally shared norm.
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[GER1] The teacher obviously focuses on solving the quadratic equation, while the
modeling aspects contained in the task are hardly or not at all addressed. The
following questions are therefore not clarified: - Mark in the illustration what is to be
calculated. - How did you come up with your educated guesses? Can the illustration
be used to justify which educated guesses is particularly realistic? - Why is the
approach of S1 correct? - What is described by the solution -4? What is the
difference between the factual context and the function described?
Fig. 4 Example of a German researcher’s answer referring to the anticipated culture-specific norm in
vignette Task2
[GER2] […] What is negative is that the teacher does not really address the reference
to reality and particularly does not mention that and why one of the two
mathematically correct solutions is not a solution of the factual problem.
Fig. 5 Example of a German researcher’s answer mentioning further critique regarding vignette Task2
GER2’s answer (Fig. 5) illustrates another critical aspect. S/He was concerned
that the context is not used for ruling out the second algebraic solution. As there
is no indication that the researcher saw the teacher’s support to connect the un-
derstanding of the situation to the mathematical model as deficient, we inductively
coded a separate category that was assigned to four cases in total (critique regarding
the use of the context).
Similarly, we inductively extracted a code regarding the treatment of the educated
guesses used for answers that criticized how the teacher used the prompt presented
in the task without indicating that the teacher’s support of mathematical modeling
processes was considered deficient. Remarkably, this code occurred five times in the
Taiwanese subsample but only once in the German subsample.
The first sentence of the Taiwanese researcher TW1’s answer (Fig. 6)represents
a negative evaluation of the teacher’s use of the task regarding the algebraic ex-
planations: The researcher criticized that the mathematical concepts “equation” and
“function” are not used consistently by the teacher and that s/he did not consider ask-
ing the students to explain their solution strategies. Researchers from both countries
mentioned that the teacher should have paid more attention to the algebraic solution
paths, for example, by further elaborating on them or supporting their formulation
through written notes. Nine Taiwanese researchers pointed to this aspect, so it is
the main point of criticism issued by the Taiwanese participants. In the German
answers, such reasons were mainly presented along with reasons indicating that the
researchers saw the breach of the anticipated norm.
The Taiwanese researchers generally focused more on aspects of the mathematical
content than the German researchers when evaluating the teacher’s use of the task.
This can also be seen in the answer TW2 (Fig. 6). It indicates that the researcher
expected the teacher to propose a conclusion regarding using or not using the for-
mula to solve quadratic equations. This critique regarding a lack of conclusion was
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[TW1] There were serious flaws in this problem! Moreover, [the teacher talked
about] equations one moment and functions in the next, which was confusing! This
teacher had no intention of asking the students about their solution strategies and
solved the problem directly from the perspective of functions.
[TW2] […] [The teacher] did not let the students notice the meaning of using and not
using formulas.
Fig. 6 Examples of two Taiwanese researchers’ answers mentioning further critique regarding vignette
Task2
Tab l e 5 Distribution of codes regarding further critiques for vignette Task2
Further critique regarding the Other
reasons
Use of the
context
Treatment of
educated guesses
Algebraic
explanation
A lack of
conclusion
Hetero-
geneity
GER 2
(11.8%)
1
(5.9%)
7
(41.2%)
0
(0%)
4
(23.5%)
0
(0%)
TW 2
(10.5%)
5
(26.3%)
9
(47.4%)
5
(26.3%)
3
(15.8%)
1
(5.3%)
observed five times in the Taiwanese sample but was not mentioned by any German
researcher.
Finally, four German and three Taiwanese researchers evaluated the teachers’ use
of tasks negatively because of aspects related to heterogeneity, questioning whether
weaker students could follow the instruction.
Table 5summarizes the coding of the researchers’ answers for Task2.
5.3 Task4 “student camp”
The use of the task, as presented in this vignette, was criticized by 18 Taiwanese
(94.7%) and 15 German researchers (88.2%). In total, eleven Taiwanese (57.9%) and
four German researchers (23.5%) mentioned that the teacher should have discussed
the pros and cons of the strategies of assigning variables. Referring to the majority
criterion, we can infer that the anticipated norm regarding the use of the potential of
the task to discuss the pros and cons of different variable assignments is a culture-
specific norm in Taiwan (RQ1).
Exemplarily, TW3 (Fig. 7) pointed out that the students could have seen the
advantage (pro) of the first approach (being easier to calculate) better if they had
compared both sets of equations. S/He particularly emphasized discussing how to
set the variables in the different solution paths and criticized the teacher insisting
on a uniform method.
Further criticism is, for instance, illustrated by GER3’s answer (Fig. 8). GER3 also
saw a benefit for the students in solving the sets of equations. However, in contrast
to the Taiwanese researcher TW3, s/he requested that the teacher should have finally
considered the “contextual equivalence” of the solutions instead of discussing the
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[TW3] 1. The last line of the teacher’s statements ran too fast. It was obvious that
some students expressed their preference for the second method, [the teacher]
insisted that everyone uniformly learned the first method, and the lesson immediately
progressed to solving the problem without spending time discussing how to choose
“groups” to set the unknowns. 2. Some students preferred the second method; it may
be because they could only set the unknowns based on what the problem asked.
Although the first method was easy to solve, the students did not know how to
choose which appropriate variables in the problem to set the unknowns. The teacher
must spend time discussing with the students how to set the unknowns rather than
skipping and proceeding to solving the simultaneous equations.
Fig. 7 Example of a Taiwanese researcher’s answer referring to the anticipated culture-specific norm in
vignette Task4
[GER3] The teacher wants to show his/her students that the assignment of variables
in such tasks is not determined by the task. In principle, the variables can be defined
arbitrarily as long as this makes sense for the solution of the task and the variables
are used consistently. By notating the alternative, the teacher clarifies what another
solution approach might look like. However, it would make sense to solve both
approaches with the different variable assignments and then reflect on the contextual
equivalence of the solutions.
Fig. 8 Example of a German researcher’s answer mentioning further critique regarding vignette Task4
[GER4] […] Especially, [the teacher] does not further discuss the heuristics which
are used by him/her for the creation of a model (e.g., what is the rationale for
assigning variables). As it can be "seen" in the following, this is particularly
problematic since [the teacher] chooses a variable assignment that is seemingly not
clearly related to the task at first glimpse. The teacher seems to be unclear about
his/her goal: Discussing heuristics for assigning variables in factual contexts (sub-
part of a step within the modeling cycle) or discussing the whole modeling cycle. [...]
Fig. 9 Example of a German researcher’s answer mentioning another reason for critique regarding vi-
gnette Task4
pros and cons. So, the answer focuses on understanding the meaning of the results of
the calculations regarding the situation in the word problem rather than on effective
solution strategies. Seven German researchers mentioned such critique regarding the
connections between the different approaches assigning the variables. Focusing on
the equivalence of the two approaches appears to be a German perspective in this
situation since none of the Taiwanese researchers mentioned it.
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Tab l e 6 Distribution of codes regarding further critiques for vignette Task4
Further critique regarding the Other
reasons
Connection of the two
approaches
Teacher setting up the
equation
Teachers’ prefer-
ence
GER 7
(41.2%)
6
(35.3%)
5
(29.4%)
1
(5.9%)
TW 0
(0%)
6
(31.6%)
7
(36.8%)
3
(15.8%)
However, researchers from both countries criticized that the students had no
chance to set up the equations on their own (critique regarding the teacher setting
up the equation, 12 cases in total) and that the teacher was too focused on his or her
own solution strategy (critique regarding the teacher’s preference, 12 cases in total).
Both critiques were often mentioned in addition to other reasons but also occurred
as the only reason.
Compared to the other vignettes, the largest number of other reasons for criticism
was found regarding vignette Task4 (four cases). One German researcher (GER4,
Fig. 9) related the task to the modeling cycle and criticized an insufficient imple-
mentation. S/He also sees an issue in not naming the variable as the task suggests.
GER4 mentions that this leads to an insufficient relation between the mathematical
model and the real-world situation.
Table 6summarizes the coding of the researchers’ answers for Task4.
5.4 Task1 “medicine dosage”
The use of the task, as presented in this vignette, was evaluated negatively by
12 German (70.6%) and 13 Taiwanese researchers (68.4%). The majority of the
Taiwanese (11, 57.9%) and the majority of the German researchers (10, 58.8%)
noticed the breach of the anticipated norm. Hence, we infer that the anticipated norm
regarding the use of the task’s potential to support flexible problem-solving through
emphasizing the connections between the solutions illustrated in this vignette can
be considered an interculturally shared norm (RQ1). Figure 10 presents two sample
answers—one of a German and one of a Taiwanese researcher—that both addressed
insufficient support to connect the different solution strategies to learn to solve word
problems flexibly (breach of the anticipated norm). The German researcher criticized
that the teacher “does not go to the bottom” and elaborated on this as the missed
opportunity to relate the strategies in which the researcher saw different degrees of
sophistication. The Taiwanese researcher also discussed a missing integration of the
solution strategies, which are seen as different methods to deal with proportional
relations.
Further criticisms indicated that some researchers considered the explanation of
the solution paths insufficient (critique regarding the explanations). Six German and
three Taiwanese researchers mentioned this. In four of these cases in Germany, the
researchers also mentioned the missing connection between the solutions (breach of
the anticipated norm). The sample answer of the German researcher (GER5, Fig. 10)
exemplifies this case: After criticizing the teacher for “not going to the bottom”, this
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[GER5] The teacher collects student solutions and evaluates them ("Well done",
"Great") but does not get to the bottom of them. For example, [the teacher] does not
let S2 explain how he/she found the solution with the help of a table. The two simple
strategies of S1 and S2 (which are only possible with sufficiently simple numbers)
and the elaborated and far-reaching strategy of S3 are not related to each other. The
teacher is only doing "work-to-rule" regarding the textbook task and thereby misses
a learning opportunity.
[TW4] […] The teacher only explained the method of S3 but did not integrate and
synthesize that the three methods were all based on the same principle of
proportionality; just the methods of the three students of solving the proportional
expression were different. The teacher should have taken the opportunity to review
the principle of proportionality, transform this problem into a mathematical model of
proportionality, and then briefly explain the different methods with which the three
students applied this proportional expression.
Fig. 10 Examples of a German and a Taiwanese researcher’s answer referring to the interculturally valid
norm in vignette Task1
Tab l e 7 Distribution of codes
regarding further critiques for
vignette Task1
Further critique regarding the Other
reasons
Explanations
GER 6
(35.3%)
0
(0%)
TW 3
(15.8%)
1
(5.3%)
evaluation is warranted first by a lack of explanation of S2’s solution. The missing
connection of the solution paths is mentioned after that, indicating that the researcher
expected the teacher to use the explanation of the strategies to connect them. Finally,
one researcher’s answer could not be assigned to any category (other reasons). The
coding of the researchers’ answers to Task1 is summarized in Table 7.
6 Discussion and Conclusion
This contribution reports a study that systematically investigated whether anticipated
instructional norms regarding the use of tasks in mathematics instruction are cul-
ture-specific. Based on prior work on cultural differences in education, these norms
were expected to show certain differences across cultural contexts, despite a com-
mon understanding of the importance of high-quality use of tasks for mathematical
learning.
To explicate and compare the potentially implicit instructional norms, we used
a situated approach and contrasted the evaluations of mathematics education profes-
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sors from Germany and Taiwan regarding specific instructional situations represented
by vignettes. These vignettes were designed to include breaches of anticipated norms
regarding the use of word problems.
Qualitatively analyzing researchers’ written evaluations of the teacher’s use of
the task in the situations, differences and commonalities in the perspectives of the
researchers from Germany and Taiwan could be extracted by a sophisticated bi-
national process. Regarding RQ1, we found that the perspectives regarding two
situations differed between the cultural contexts in line with the assumptions of
the vignette’s authors. The answers indicate that using a task’s potential to support
student engagement in modeling processes is a culture-specific norm in Germany
(Task2), and discussing the pros and cons of different ways of assigning variables
describes a culture-specific norm in Taiwan (Task4). We could not find any cultural
specificity regarding the use of tasks that promote flexible solving by connecting
different solution paths. Instead, supporting flexible solving is a shared norm in both
countries (Task1).
To answer RQ2, we situate the findings regarding the instructional norms and
further critiques as mentioned in the evaluations from Germany and Taiwan in
previous research regarding cultural differences. Specifically, we revisit the assump-
tions regarding differences in the high-quality use of word problems. The German
instructional norm elicited with Task2 (supporting students’ engagement in model-
ing processes focusing on connecting the real-world situation and the mathematical
model) aligns with the important status of modeling in Germany (e.g., Sträßer 2019)
and the importance of a process orientation in Western countries (Leung 2001). The
Taiwanese researchers, in contrast, focused more on algebraic aspects like using
or not using a formula when evaluating vignette Task2, in line with the content
orientation valued in East Asian mathematics instruction (Leung 2001).
The Taiwanese instructional norm regarding discussing the pros and cons of
solution methods elicited with Task4 mirrors especially the expectation of reflection
of optimal solution methods (Cai 2006). Remarkably, a similar perspective became
visible in some evaluations of Task2, where only researchers from Taiwan criticized
the lack of conclusion.
Regarding Task1, the researchers’ perspectives across cultural contexts showed
agreement regarding the teacher missing the opportunity to connect the different
ways of solving a problem to develop flexibility. It is important to note that in-
terculturally shared norms may be observed for various reasons. Two plausible
explanations should be suggested.
First, developing flexible problem-solving may be considered an instructional aim
that can be associated with word problems under a lens of mathematical modeling,
but also with a lens of application of mathematical concepts. Connecting different
mathematical concepts explicitly is emphasized in Taiwan (Hsieh et al. 2017). It can
be seen as one way to reflect and learn to apply mathematical contents flexibly (Le-
ung 2001). Thus, the breach of the anticipated norm included in vignette Task1 that
refers to the missing connections between solutions using different mathematical
concepts can be seen as beneficial from a process-oriented perspective of mathe-
matical modeling and a content-oriented perspective on dealing with proportional
relations, as the sample answers also illustrate (Fig. 10).
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Second, the results may also reflect Western influences on Taiwanese instructional
norms. As Morris and Leung (2010) stated, flexible solving is especially emphasized
in Western educational traditions but has been widely discussed in the international
research community. With growing globalization and as one side-effect of interna-
tional comparative studies, certain trends in education were observed (Clarke 2013).
For example, Taiwanese curricula have been reformed several times, influenced by
global trends, while also recently, the own educational context has been explicitly re-
considered (Yang et al. 2022). Hence, conceptions of instructional quality regarding
mathematics in Taiwan today may reflect not only traditional perspectives but may
also be shaped by Western ideas of instruction (Hsieh et al. 2020).
Among the study’s limitations, we want to draw attention to the limited number
of vignettes focusing on word problems so that we could only use three instructional
situations to compare the researchers’ perspectives. However, given that the overar-
ching study took three different aspects of instructional quality under investigation,
it was not possible to incorporate more vignettes. The sample size is limited, yet
it can be considered an exceptionally large researcher, i.e., expert sample. Despite
the sampling strategy aiming at representativeness, it is, of course, not clear if the
participating mathematics education researchers entirely represent the perspectives
of the mathematics education researchers in Germany and Taiwan. The nature of the
study also allows no conclusion whether our findings can be generalized to other
Western or East Asian countries, even though the findings largely align with major
cultural characteristics as expected.
One may further argue that the applied majority criterion to determine norms
within a country is too lenient and that the observed percentages may indicate low
agreement among the researchers in one country. This is true to a certain extent. Still,
it should be considered that we did not ask researchers directly whether they agreed
with certain anticipated norms of instructional quality but instead analyzed their
written evaluations of instructional situations where a not entirely obvious breach
of the anticipated norm was included. Hence, we considered the majority criterion
to be appropriate.
Finally, in this study, we only consideredthe perspective of mathematics education
professors to infer whether the instructional norms regarding mathematics instruction
in the two contrasted countries apply as anticipated. This decision was based on
their important role in forming instructional norms because of their duties in teacher
education and research, but it would be important to investigate whether the findings
can be replicated with other relevant groups from the same cultural contexts (e.g.,
teachers).
Bearing in mind these limitations and the need for further research, our findings
indicate that despite mathematics being a relatively homogeneous discipline and
growing international consensus regarding important aspects of instructional quality
in mathematics education, instructional norms in different contexts with different ed-
ucational traditions may still differ substantially and in line with cultural differences
when looking at specific instructional situations. There is now a certain awareness of
the risk that (implicit) instructional norms, for example, as underlyingassessment in-
struments, could impair findings in comparative research (see, e.g., Lindmeier et al.
2024). Nevertheless, systematic investigations on the role of instructional norms are
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Culture-Specific Norms Regarding High-Quality Use of Task Potential for Mathematical...
so far rare. Talking about mathematics education in a common research language
may easily level differences at first sight. Thus, in order to draw attention to poten-
tially different instructional norms across cultures, we focused explicitly on making
them visible by means of instructional situations in our overarching binational re-
search project with equal participation of researchers from a Western and an East
Asian country. Beyond the evidence for influences of culture on norms regarding
high-quality use of tasks reported in this article, we made similar observations re-
garding other aspects of instructional quality, which are widely used in comparative
research (student thinking, use of representations: Dreher et al. 2021,2024;foran
overview: Lindmeier et al. 2024). In times of increasingly global mathematics ed-
ucation research endeavors, one of the biggest challenges will be finding ways to
address cultural specificities and use them to advance our mutual understanding of
conceptions of high-quality teaching.
Supplementary Information The online version of this article (https://doi.org/10.1007/s13138-024-
00237-5) contains supplementary material, which is available to authorized users.
Acknowledgements This study is part of the project TaiGer Noticing, funded by the DFG—German
Research Foundation (grant numbers DR 1098/1-1 and LI 2616/2-1) and the Ministry of Science and
Technology (MOST, grant number 106-2511-S-003-027-MY3). Some of the research was carried out at
the IPN—Leibniz Institute for Science and Mathematics Education (Kiel), where the last author used to
work in the past.
Funding Open Access funding enabled and organized by Projekt DEAL.
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 Com-
mons 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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