Students need language skills to participate in classroom instruction. There is empirical evidence that students with lower levels of proficiency in the language of instruction benefit less from mathematics instruction than students with higher levels of proficiency in the language of instruction (Barwell et al., 2016; Braswell et al., 2005; Ellerton & Clarkson, 1996; Haag et al., 2013; Organization for Economic Cooperation and Development [OECD], 2007; Secada, 1992). In studies that distinguished between procedural and conceptual mathematical knowledge, the development of conceptual knowledge was especially closely tied to learners’ proficiency in the language of instruction (Erath et al., 2021). Consequently, it seems crucial to develop and evaluate specific language-responsive instructional approaches to support students' acquisition of conceptual knowledge.

There is no commonly accepted definition of language-responsive instruction, and studies vary in how they operationalized it: More general approaches focus on teaching mathematics following different literacy strategies (Benjamin, 2011), making use of everyday language (Prediger et al., 2019; Setati & Adler, 2001) or students’ native language (Bermejo et al., 2021; Wagner et al., 2018). Content-specific approaches use learning materials with integrated systematic language learning paths (Coggins et al., 2007; Pöhler et al., 2017b; Prediger & Wessel, 2013). Teaching methods range from classroom discourse to adapting students’ learning materials. Overall, these different approaches are united by the idea of supporting learning by addressing learner's language requirements.

Although these language-responsive instructional approaches were initially developed for multilingual students with limited proficiency in the language of instruction, their implementation also seems promising in language-diverse mainstream classrooms because both multilingual and monolingual students can face language difficulties in mathematics classrooms (Barwell et al., 2016; Prediger & Wessel, 2018; Setati, 2005).

Nevertheless, studies that test the effectiveness of language-responsive instruction with broader groups of students are scarce (Prediger & Neugebauer, 2021). Therefore, the present study examines the effectiveness of a language-responsive learning approach based on materials with additional language support in linguistically diverse classrooms under controlled conditions. Given the great heterogeneity in classrooms (Nusser & Gehrer, 2020; Vock & Gronostaj, 2017), the key question concerns the extent to which students with different learning prerequisites benefit differently from learning materials with or without additional language support.

1 Theoretical and empirical background

1.1 Relationship between mathematics and language

Considering the learning process in mathematics classrooms, two functions of language come into play. First, language has a communicative function. It is used to describe, explain, and present mathematical content (Erath et al., 2018, 2021; Moschkovich, 2015). Language is needed to communicate, document thoughts and solutions, discuss mathematical problems with each other, and much more. Second, language has an epistemic function. It is a cognitive tool for thinking and learning, and is thus necessary to understand mathematics (Pimm, 1987; Prediger et al., 2018).

Language proficiency includes various receptive and productive skills (Oller, 1974; Pimm, 1987; Vollmer & Sang, 1983). Receptive skills include reading and listening, and learners need them to follow teacher explanations or extract text information. Productive skills include writing and speaking, and are needed to produce utterances, for example, to describe mathematical facts (Paetsch et al., 2016; Pimm, 1987). Learning mathematics encompasses various activities, including understanding concepts, solving problems, using mathematical language and notation, receiving instruction, collaborating with peers, and applying mathematical knowledge. Proficiency in the language of instruction is deeply intertwined with each of these components and is, therefore, essential to learning mathematics.

To learn mathematics in school, students need cognitive academic language proficiency (CALP) in the language of instruction, which encompasses the more academic and cognitive aspects of language needed for success in school (Cummins, 2000). CALP is essential for understanding cognitively demanding and abstract content (Bailey et al., 2004; Cummins, 2000). It is characterized by specialized vocabulary and complex grammatical structures and includes impersonal forms and decontextualized and abstract formulations (Bailey, 2007; Morek & Heller, 2012). In contrast, children use basic interpersonal communicative skills in their daily lives (e.g., when playing together, exercising at a sports club, or shopping). CALP is more related to written language, which many students need to become more experienced in as they are less familiar with it (Cummins, 2000). Accordingly, a lack of CALP in the language of instruction may impede overall mathematics performance (Haag et al., 2013; Ní Ríordáin & O’Donoghue, 2009; Prediger et al., 2018).

1.2 Proficiency in the language of instruction and its relevance to mathematics achievement

The large-scale Program for International Student Assessment (PISA) found positive and very high correlations between proficiency in the language of instruction and mathematical performance (average latent correlation of 0.85 between mathematics and reading; OECD, 2012), and the National Assessment of Educational Progress in the U. S. (NAEP) reported that English language learners in Grade 8 lagged far behind the mathematics achievement of non-English language learners (NAEP, 2017).

Much of the research on the impact of language proficiency on mathematics achievement has been framed by sociocultural theories (e.g., Cho et al., 2015; Forman, 2003; Takeuchi, 2015). However, when studying the impact of materials rather than classroom communication on students’ learning, as is the case in the present study, cognitive load theory is a more appropriate theoretical framework. Cognitive load theory (Sweller, 1988; Sweller et al., 2011) rests on the idea that working memory is limited to processing information from the environment. Hence, learning processes can only succeed when the total cognitive load does not exceed the working memory capacity. Given that standardized tests need to be completed in the language of instruction and contain language-dependent tasks, they compound cognitive demands, particularly for second language learners (Abedi & Lord, 2001; Campbell et al., 2007). Accordingly, second language learners perform less well when completing tasks in a non-native language due to increased cognitive demands (Campbell et al., 2007). Linguistic features, such as the lengths of words and sentences, passive constructions, comparative structures, or conditional clauses, are associated with morphological and syntactic complexity (Abedi & Lord, 2001). These features require little mental processing by students with higher proficiency in the language of instruction. However, they may lead to increased cognitive load for students with lower levels of proficiency in the language of instruction, which may lead to misinterpretations, slower reading, and task solving, and thus poorer test performance (Abedi & Lord, 2001; Abedi et al., 2020; Campbell et al., 2007; Haag et al., 2013). In line with these findings, mathematical test items can be linguistically adapted to reduce language load. For example, Abedi and Lord (2001) examined the effects of a linguistic modification of test items in a study with N = 1,031 Grade 8 students in Southern California. The findings show that linguistic simplifications of test items improved the test performance of English language learners (ELLs).

Although adapting test items allowed students to show higher achievement in mathematics, the research on the effect of adapting the linguistic demands of instructional material is scarce. Since instructional material plays a key role in mathematics classrooms, it seems crucial to investigate the effects of using linguistically adapted learning materials to support learners with lower levels of proficiency in the language of instruction of their mathematics learning.

1.3 Relationship between proficiency in the language of instruction and conceptual knowledge of mathematics

There is evidence that the development of conceptual knowledge is related to learners’ proficiency in the language of instruction (Erath et al., 2021). Conceptual knowledge comprises knowledge of concepts and principles that are fundamental in a particular domain, as well as knowledge of why a mathematical procedure works (Byrnes & Wasik, 1991; Crooks & Alibali, 2014; Hiebert & Lefevre, 1986; Kilpatrick et al., 2001). Specifically, students with lower levels of proficiency in the language of instruction performed less well on tasks that needed conceptual knowledge than on language-free computational tasks (Abedi & Leon, 1999; Heinze et al., 2007; Prediger et al., 2018; Ufer & Bochnik, 2020; Ufer et al., 2013). Such findings are not surprising, given that mathematical discourse practices, such as arguing and explaining meanings, are fundamental to the development of conceptual knowledge (Prediger, 2022; Prediger et al., 2018; Ufer et al., 2013).

In the example of learning fractions, students with lower language proficiency may not fully benefit from teachers' conceptual explanations. This may affect their ability to understand and internalize fraction concepts, ultimately affecting the depth of their conceptual understanding. In addition, these students may face challenges due to the specific linguistic demands of the learning content. For example, when multiplying fractions, the meaning of the term "of" can be an obstacle according to the specific linguistic demands (e.g., Prediger & Wessel, 2011; Purnomo et al., 2022). Explaining the meaning of 2/3 × 1/4 requires relating the two thirds to the correct whole, which is itself given as a part. This can be challenging due to the linguistic complexity, especially for students who are not proficient in the language of instruction (Prediger & Wessel, 2011). However, explanations based on a concept's underlying principles can help develop conceptual knowledge (Geller et al., 2017). In contrast, learners who are more proficient in the language of instruction are better able to express ideas and concepts, which can enhance their conceptual knowledge (Pimm, 1987; Wessel et al., 2020). Therefore, learners with lower proficiency in the language of instruction may benefit from additional language support to further develop their linguistic skills and engage in conceptual learning activities. Thus, instructional approaches are needed to support students in improving their use of mathematical language, especially for the development of conceptual knowledge (Erath et al., 2021; Moschkovich, 2015; Prediger & Neugebauer, 2021).

Qualitative studies that have focused on supporting students’ conceptual knowledge in various domains through language-responsive instruction are, for example, Moschkovich (2002), for the slope of functions, Pöhler et al. (2017a), for percentages, Prediger and Wessel (2013), for fractions, Prediger and Zindel (2017), for functional relationships, and Smit and van Eerde (2013), for line graphs. From these studies, six major design principles were identified: (1) engaging students in rich discourse practices, (2) establishing various mathematics language routines, (3) connecting language varieties and multimodal representations, (4) including students’ multilingual resources, (5) using macro-scaffolding to sequence and combine language and mathematics learning opportunities, and (6) comparing language pieces (form, function, etc.) to raise students’ language awareness (Erath et al., 2021). The results of quantitative studies provide empirical evidence for the effectiveness of language-responsive instruction based on the above design principles. For example, Prediger and Wessel (2013) designed a language-responsive intervention program for multilingual 7th-grade students with lower proficiency in the language of instruction to enhance their conceptual knowledge of fractions. They evaluated its effectiveness in a quasi-experimental pre-post intervention study in a small-group laboratory setting in the context of remedial instruction (six 90-min lessons with a student-teacher ratio of 2:1). The intervention group (n = 36), working under language-responsive conditions, scored significantly higher on the posttest than the control group (n = 36), which was taught by the regular teacher using the standard textbook. In a follow-up study with 7th graders performing low in mathematics (N = 343), Prediger and Wessel (2018) compared two language-responsive fraction interventions, one with and one without emphasis on vocabulary learning. Both intervention groups achieved significantly higher learning gains than the control group, but no significant differences were found between the two intervention groups. Monolingual and multilingual students had similar learning gains.

Previous research on language-responsive instruction on fractions has predominantly concentrated on specific student groups, namely, second-language learners with lower mathematics performance and proficiency in the language of instruction. As a result, the impact of language-responsive learning materials on broader groups of students with diverse learning prerequisites remains largely unexplored.

1.4 Learning fractions as a challenging field

The development of fraction knowledge is crucial to student success in more advanced mathematics (Booth & Newton, 2012; Siegler et al., 2011; Torbeyns et al., 2015) and predicts academic, occupational, and financial success (Siegler & Lortie-Forgues, 2015). At the same time, fractions are one of the most challenging topics in school mathematics (Lortie-Forgues et al., 2015). Typical errors are well explored and documented (Ashlock, 2010; Eichelmann et al., 2012). These difficulties may result from whole number bias (Ni & Zhou, 2005), which means that new knowledge about fractions interferes with prior knowledge regarding whole numbers. Consequently, children often consider the numerator and denominator of a fraction to be two separate numbers, leading to errors when performing operations with fractions or comparing them (Reinhold et al., 2020; Vamvakoussi et al., 2012). One way to overcome these problems is to foster students’ conceptual knowledge of fractions rather than focusing mainly on procedural knowledge (Ashlock, 2010; Geller et al., 2017; Siegler & Pyke, 2013).

Conceptual knowledge of fractions involves knowing specific characteristics of fractions (e.g., the relation of numerator and denominator and density of fractions) and different interpretations of fractions (e.g., part of a whole, quotient, and ratio) and is activated whenever two representations (concrete materials, visualizations, verbal representations, and symbolic notations) are used and related (Haapasalo & Kadijevich, 2000; Padberg & Wartha, 2023; Reinhold et al., 2020). It is assumed that integrating conceptual knowledge reduces errors when making calculations with fractions. For example, by utilizing visual representations, students can understand that fractions need to have the same denominators in order to add them. This is because only combinations of equal parts can be expressed symbolically with a fraction.

1.5 The role of mathematics anxiety

It is widely acknowledged that mathematical development depends on cognitive skills and learners’ attitudes, beliefs, emotions, and motivations (Batchelor et al., 2019; Zan et al., 2006). Specifically, mathematics anxiety, as a negative emotion about mathematics-related situations and tasks (Ashcraft & Moore, 2009; Foley et al., 2017; Halme et al., 2022), is associated with mathematics performance from young children to adults (Ashcraft & Krause, 2007; Miller & Bichsel, 2004; Ramirez et al., 2013; Vukovic et al., 2013). In particular, mathematics anxiety correlates negatively with mathematics performance (r = –0.28 in a meta-analysis by Barroso et al., 2021). This relationship between mathematics anxiety and mathematics performance is bidirectional (Ashcraft, 2002; Dowker, 2005). Children with higher mathematics anxiety experience negative emotions when working on mathematics tasks. They may devote less time and effort to studying mathematics to avoid triggering these negative emotions. This avoidance of mathematics leads to lower mathematics performance, further increasing mathematics anxiety and creating a vicious cycle (Ashcraft & Moore, 2009; Jansen et al., 2013).

Fractions are a topic that has been shown to elicit more negative attitudes in students than whole numbers (Mielicki et al., 2021; Sidney et al., 2021), and even educators with years of teaching experience report more anxiety about fractions than other mathematical content (Mielicki et al., 2021). In addition, studies have found that mathematics anxiety negatively affects fraction learning (Mielicki et al., 2023; Rayner et al., 2009; Sidney et al., 2018). Learning fractions is associated with more complex mathematical skills and may interfere with prior knowledge of whole numbers. Children may encounter difficulties with fractions, which can exacerbate mathematics anxiety (Starling-Alves et al., 2022). Therefore, considering students’ mathematics anxiety when supporting them in learning fractions seems essential. Although there is little research on the relationship between mathematics anxiety and language proficiency, it seems plausible that language difficulties in understanding classroom demands and mathematics concepts may increase anxiety about mathematics learning and, therefore, affect learning outcomes.

1.6 Differential effectiveness of language-responsive instruction: the framework of aptitude-treatment interaction

Research on the relationship between learning outcomes and heterogeneous learning conditions has been conducted since the 1970s using the aptitude-treatment interaction (ATI) approach (Cronbach & Snow, 1977; Kalyuga, 2007). ATI is based on the assumption that instructional approaches (treatments) are more or less effective for particular groups of learners, depending on their specific abilities or characteristics (aptitudes) (Corno et al., 2002; Cronbach & Snow, 1977; Snow, 1991).

Interactions between aptitudes and treatments generally affect cognitive load (Chen et al., 2017). More precisely, it depends on the student’s specific aptitude when the demands associated with information processing reach a critical threshold. For example, Yeung et al. (1998) found that integrating explanatory notes directly into a text improved the text comprehension of learners with lower levels of proficiency in the language of instruction. In contrast, for students with higher levels of proficiency in the language of instruction, the same format hampered their text comprehension. As the explanatory notes were difficult to ignore, learners with higher levels of proficiency in the language of instruction needed to process redundant units of information and match them with their existing knowledge schemas, which caused unnecessary cognitive load. This study shows that learning material that is effective with novice learners may be less effective or even harm more experienced learners (the so-called expertise reversal effect) (Kalyuga & Sweller, 2014).

Furthermore, noncognitive individual characteristics, such as mathematical anxiety, can mediate ATI effects. Specifically, anxiety may reduce working memory resources (Ashcraft & Kirk, 2001; Mammarella et al., 2015; Suárez-Pellicioni et al., 2016). Accordingly, mathematics anxiety could be a mediator when investigating the effectiveness of language-responsive instruction. In particular, learning materials with additional language support presumably impose an extraneous load, which can affect learners with higher levels of mathematics anxiety.

2 Research questions

As a result of previous research, it seems important to consider how the use of language-responsive learning material affects learning of fractions by students with varying levels of proficiency in the language of instruction. Therefore, we developed learning materials for fractions that did or did not include additional language support and explored the effectiveness of these learning materials. We addressed two research questions:

  • RQ1) Are there differences in students’ learning gains in fraction knowledge between students who use material with additional language support and students who use material without additional language support?

  • RQ2) To what extent do students with different learning prerequisites (regarding their proficiency in the language of instruction and their mathematics anxiety) benefit differently from learning materials with or without additional language support?

3 Methods

3.1 Sample

A total of 211 Grade 7 students from eleven secondary school classrooms (nine in the intervention condition and two in the control condition) of five schools in Germany participated in this study. The average age of the participants was 12.6 years (range 11–14 years). Approximately half of the students were male (53.1%), and half were female (46.9%). The sample encompassed a range of socioeconomic status backgrounds (assessed by the book-at-home index), including low (27.5%), middle (33.6%), and high (36.5%). Concerning language background (operationalized by family language), the study involved a diverse group of participants, including both monolingual (72%) and multilingual learners (28%).

The students attended either the middle or the lowest track of Germany’s tripartite secondary school system. These different types of schools ensured a broad range of heterogeneous aspects in the sample, particularly concerning mathematical competence and German language proficiency. In German curricula, most fraction instruction occurs in Grades 5 and 6, where the mathematics class focuses on fractions for a few weeks. Furthermore, the curriculum includes repetition units in the upper grades (especially Grades 7 and 8) as fractions of the necessary basis for rational numbers (Ministerium für Kultus, Jugend und Sport,  2016). Our study is situated within a repetition unit on fractions in Grade 7 that aims to reactivate and consolidate students’ knowledge of fractions.

The responsible Ministry of Education approved the study, and the students and their parents gave informed consent.

3.2 Research design and procedure

We ran a quasi-experimental intervention study under highly controlled conditions in typical classrooms, following a pretest and posttest design with two treatment groups and a waiting control group.

Both treatment groups worked for three lessons of 45 min each on self-learning materials with repetition units of fractions. The first intervention group (n = 79) received materials with additional language support, and the second intervention group (n = 87) received materials without additional language support (cf. section Intervention material). Between the pretest and posttest, the classes of the control group (n = 45) were taught by their regular teachers. They worked on a textbook unit unrelated to fractions. After the posttest, they received the same intervention on fractions as the intervention groups.

The two intervention groups were recruited from nine classes by randomly assigning the students (within classes) to one of the two groups with minor practical limitations. For example, students sitting at the same table worked on the same material to ensure that communication stimulated by the material was possible in partner work. The waiting control group (n = 45) comprised two complete classes recruited from the same schools as the intervention groups. Accordingly, the treatment and comparison groups shared similar sociodemographic characteristics, which contributed to minimizing systematic differences.

The intervention was conducted by the first author and a specially trained research assistant, both of whom had teaching experience at the secondary school level. This approach was used to ensure a standardized intervention procedure and a high implementation fidelity level, as other studies have suggested that even trained teachers may have varying levels of implementation fidelity (e.g., Cho et al., 2015). The regular teachers participated in the intervention to create a familiar classroom setting for the learners. They were responsible for classroom management and pedagogical issues (e.g., addressing ritualized procedures, motivating the learners, and avoiding and dealing with disturbances in class).

The teaching method of working with self-learning materials was familiar to the students, and most of them were motivated by the pretest to fill their knowledge gaps. Before working with the material, it was explicitly pointed out that oral and written explanations are part of the learning concept and promote an understanding of the mathematical content. As in regular lessons, students could ask the teachers for help. If needed, the teachers provided limited motivational support (“You can do that”), general strategic support (“Reread the task carefully”), or organizational support (making students aware of missing tasks) but no content-related support.

3.3 Intervention material

The intervention contained a repetition unit on fractions that focused on conceptual knowledge but also addressed procedural knowledge, as the two types of knowledge are closely intertwined. It comprised three lessons: (1) the concept of fractions, (2) the equivalence of fractions, and (3) the addition and subtraction of fractions. The intervention material was based on existing learning materials to promote conceptual knowledge, which were designed and evaluated in previous research (Prediger & Hußmann, 2014; Wessel et al., 2018) and are available as an open educational resource through the German Centre for Mathematics Teacher Education (DZLM).Footnote 1 The original material contains oral and written language production activities that require describing, explaining, justifying, and promoting language learning and conceptual knowledge development. There were two different versions of the learning material (Fig. 1). One version (M_LING) contained language support (see the blue box in Figs. 1 and 2) to support students’ oral and written language production specifically; the second version (M_NOLING) contained no additional language support. Except for the additional language support, both versions of the material contained the same tasks.

Fig. 1
figure 1

Intervention Material With and Without Additional Language Support

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Fig. 2
figure 2

Tasks Sample tasks with additional Language Support

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Overall, the following design principles for language-responsive learning (Erath et al., 2021) were considered in the material: the students were encouraged to engage in rich discourse practices through prompts (see partner tasks in Fig. 2). Both oral and written language production are stimulated through prompts (cf. Swain, 1995; the output hypothesis). The prompts focus primarily on explaining, describing, and arguing, which are particularly relevant to promoting conceptual knowledge in mathematics learning (Erath et al., 2021; Geller et al., 2017; Prediger et al., 2016). On the design level, engagement in discourse practices is accompanied by sentence modules (see the blue boxes in Fig. 2, e.g., sentence starters, word lists, sentence patterns, and language frames). These sentence modules make the needed vocabulary available and serve as scaffolds for students to solve tasks and become fluent in fraction-specific academic language without impairing conceptual explorations. When students document what they are exploring, they can concretize abstract ideas, internalize the new information, make relationships between concepts more explicit, and thus better understand the content (Countryman, 1992; Pugalee, 2004).

Language varieties (everyday, academic, and technical language) and multimodal mathematical representations (iconic, symbolic, and verbal) were connected to support conceptual learning (Duval, 2006; Lesh et al., 1987). For example, the symbolic notation of fractions is intertwined with the iconic representation in the rectangle model, which ultimately leads to a description of the explorations made (Task 1 in Fig. 2). Learners are initially supposed to communicate with their learning partners in everyday language. To document their mathematical discoveries, technical vocabulary and sentence frames were provided to support the students in expressing their explorations in a precise manner.

Macro-scaffolding was used to sequence and combine language and mathematics learning opportunities (Erath et al., 2021; Gibbons, 2002; Pimm, 1987; Pöhler & Prediger, 2015). The material started from students’ everyday experiences using everyday language and gradually introduced more formal concepts with academic and technical language.

Some tasks compared different explanations or arguments to raise students’ awareness of the specificities of language (Erath et al., 2021; García, 2017). Furthermore, in mathematics classrooms, when two near concepts or two slightly varied sentence structures are compared and contrasted (e.g., Task 2 in Fig. 1), students become sensitized to the subtleties of language (Erath et al., 2021).

The adaptation process was performed in close collaboration with in-service teachers and teacher trainers to ensure that all tasks were appropriate for students and could be implemented in real-life conditions.

3.4 Test instruments

The assessment was conducted in written form and consisted of six parts: (1) a test of conceptual and procedural fraction knowledge, (2) a test of German language proficiency, (3) a test of basic arithmetic competence, (4) a test of general cognitive abilities, (5) a mathematics anxiety rating scale, and (6) demographics (age, gender, socioeconomic status, and multilingual background).

3.4.1 Fraction knowledge

To assess fraction knowledge in both pretest and posttest, we used a validated test that measured conceptual and procedural fraction knowledge (Lenz et al., 2020; Table 1). The conceptual knowledge subscale contained three different task types: (1) verbalization of conceptual knowledge (e.g., writing explanations), (2) application of conceptual knowledge (e.g., number line tasks to assess the measurement aspect), and (3) visualization of conceptual knowledge (e.g., to draw a visualization or shade a figure to match a given fraction in symbolic form). The subscale for procedural knowledge comprised two task types: (1) verbalization of procedural knowledge (e.g., writing instructions concerning a calculation procedure) and (2) application of procedural knowledge (e.g., expanding and simplifying fractions or adding and subtracting fractions). These task types align with the basic fraction knowledge addressed in the intervention material (the concept of fractions, the equivalence of fractions, and the addition and subtraction of fractions).

Table 1 Descriptive Data and Comparability of the Intervention Groups and the Control Group
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The test instrument exists in two parallel versions with structurally identical items. We used one version in the pretest and the other in the posttest. For each of the two versions, we created two test booklets containing the same items but in a different order. The students were randomly assigned to one of the test booklets and worked at their own pace (approximately 30–45 min). The students’ responses were scored dichotomously (0: incorrect, fragmentary, or missing; 1: correct). The internal consistency was excellent, with Cronbach’s α = 0.92 for the pretest (50 items, N = 211) and α = 0.93 for the posttest (50 items, N = 211).

3.4.2 German language proficiency

German academic language proficiency was assessed in a time-saving but valid way using a C-Test, which is based on cloze texts (Grotjahn, 2002) and especially addresses skills related to reading and writing. Following previous studies (e.g., Pöhler & Prediger, 2015; Prediger & Neugebauer, 2021; Prediger & Wessel, 2018), we used a C-Test by Daller (1999), which consisted of three texts in everyday and academic formal language (Cronbach’s α = 0.76, three texts, N = 211).

3.4.3 Basic arithmetic competence

Before the intervention, basic arithmetic competence was measured with three subtests (addition, subtraction, and multiplication) of the standardized test “HRT 1–4” (Haffner et al., 2005) (Cronbach’s α = 0.80, 3 subtests, N = 211).

3.4.4 General cognitive abilities

General cognitive abilities were measured using the “Figural Analogies” subscale of “KFT 4–12 + R” (Heller & Perleth, 2000), a German version of the “Cognitive Abilities Test” (Thorndike & Hagen, 1971). This subscale consisted of 25 multiple-choice items requiring the accurate drawing of analogies to show an understanding of the relationships between the two presented figures and to transfer these to another pair of figures (Cronbach’s α = 0.88, N = 211).

3.4.5 Mathematics anxiety

Mathematics anxiety was measured using a scale from PISA 2012, comprising five items describing negative emotions in mathematical situations (OECD, 2012). The students had to indicate how they felt when they anticipated having to perform mathematical tasks, when they anticipated their performance in mathematics class, and when they were attempting to solve mathematics problems (Cronbach’s α = 0.80, N = 211) on a four-point Likert scale ranging from “strongly disagree” to “strongly agree”.

3.4.6 Demographics

Demographic data were collected using a self-report questionnaire. Socioeconomic status (SES), a relevant factor for achievement (OECD, 2007), was measured by the book-at-home index, an easy-to-use and reliable instrument. Previous research grouped the response categories into low, medium, and high SES (Prediger & Neugebauer, 2021; Prediger & Wessel, 2018).

Multilingual background was operationalized by family language. If students reported speaking multiple family languages or a non-German family language with parents and/or siblings, they were classified as having a multilingual background.

3.5 Data analysis

The Statistical Package for the Social Sciences (SPSS 28) was used for the data analysis. There were no missing data, as children who did not fully participate in the intervention were excluded from the data set.

To answer RQ1 (learning gains in fraction knowledge), repeated measures of analysis of variance (one-way ANOVA) were conducted to compare the mean values in the fraction knowledge of the pretest and the posttest. Group and time were the main factors in the ANOVA. Partial Eta squared was used as a measure of effect sizes.

A multiple linear regression analysis was conducted to investigate how the intervention effects with and without additional language support depended on learners’ proficiency in the language of instruction and mathematics anxiety (RQ2). The analysis included the two intervention groups but not the control group. According to Snow (1991), learning gains in fraction knowledge were used as the dependent variable, operationalized as the difference between pretest and posttest scores. All variables were mean-centered and z-standardized for the regression analysis to establish meaningful zero points. The intercept can be interpreted as the learning gains of one child with average language ability in the condition coded as “0” (Aiken & West, 1991; Enders & Tofighi, 2007). In our study, the condition without additional language support was coded as “0,” and the condition with additional linguistic support was coded as “1.”

Hierarchical regression analyses were conducted to examine whether interaction terms significantly contributed to predicting learning gains in fraction knowledge. In the first step (Model 1), we analyzed the main effects of the two independent variables, treatment type (with or without additional language support) and German language proficiency, on learning gains in fraction knowledge (dependent variable). Additionally, we controlled for general cognitive abilities, basic arithmetic competence, and mathematics anxiety because we wanted to analyze the associations beyond these potential third variables. All variables were entered simultaneously. The second step (Model 2) added interaction terms (treatment × German language proficiency and treatment × mathematics anxiety). Finally, in the third step (Model 3), the triple interaction term (treatment × German language proficiency × mathematics anxiety) was entered.

4 Results

4.1 Descriptive analysis

Table 1 shows the descriptive characteristics of the two intervention groups and the control group and the results of the variance tests (ANOVA for continuous variables and chi-square test for dichotomous variables). There were no significant differences between the two intervention groups and the control group in the pretest score for fraction knowledge, the control variables (German language proficiency, general cognitive abilities, basic arithmetic competence, and mathematics anxiety) or the distribution of gender, multilingual background, and SES (p > 0.05 for all variables). These results suggest that the three groups were comparable before the intervention and that there were no systematic differences.

Table 2 shows the intercorrelations, Cronbach’s alpha value means, and standard deviations of all variables. As expected, there was a strong correlation between basic arithmetic competence and fraction knowledge and moderate correlations between these mathematical competencies and German language proficiency. Moreover, in our sample, the moderate negative correlations related to mathematics anxiety show that anxiety is related to mathematics-related performance and linguistic competence.

Table 2 Summary of Intercorrelations, Cronbach’s Alpha, Means, and Standard Deviations
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4.2 RQ 1: Learning gains in fraction knowledge

There was a main effect of time, suggesting that regardless of group, scores increased significantly from pretest to posttest (Ftime (1, 211) = 168.97, p < 0.001, η2 = 0.45). No main effect of group was found (Fgroup (2, 211) = 1.576, p = 0.21). However, the interaction effect of time x group was significant (Ftime × group (2, 211) = 14.48, p < 0.001, η2 = 0.12), suggesting that the learning gains differed to some extent among the three groups. Post hoc analyses with Tukey’s HSD showed that both intervention groups had higher learning gains from the pretest to the posttest than the control group. The t-test estimates showed an increase in the test score of 11.65 from the pretest to the posttest (p < 0.001; d = –1.25) for the intervention group without additional language support and 10.35 (p < 0.001, d = –1.17) for the intervention group with additional language support. The learning gains were both significant, with high effect sizes. The learning gains in the control group were not statistically significant (mean difference = 1.84; p = 0.24) (post hoc results in the appendix).

Together, these results show that both materials effectively promoted fraction knowledge. The control group showed no significant change in the mean between the pretest and posttest, suggesting that the learning gains were due to the treatments and that the testing had no effect on learning.

4.3 RQ 2: Differential effectiveness

RQ2 asks whether there were differential effects of the treatments with and without additional language support for students with different levels of proficiency in the language of instruction. Table 3 presents the results of the linear regression analyses. The requirements for regression analyses were met in all models (Model 1: main effects, Model 2: main effects and interaction effects, and Model 3: triple interaction). The range of the variance inflation factor values (1.04–2.49) confirmed that there was no multicollinearity between the predictor variables. No model has autocorrelation; the values of the Durbin-Watson statistic are 1.61 for Model 1, 1.64 for Model 2, and 1.64 for Model 3, which indicates the independence of the residuals. Furthermore, the data show the normality of errors and homoscedasticity.

Table 3 Results of the Regression Analyses (Dependent Variable: Learning Gain in Fraction Knowledge)
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Fit values in terms of R2 were weak to moderate (Cohen, 1988). Model 1 shows that proficiency in the language of instruction and general cognitive abilities are significant predictors of learning gains. Treatment type, basic arithmetic competence, and mathematics anxiety do not significantly impact learning gains. In Model 2, interaction factors, including treatment type, basic arithmetic competence, general cognitive abilities, and mathematics anxiety, do not significantly impact the learning gains. However, Model 2 shows both a main effect of German language proficiency on learning gains in fraction knowledge and significant interaction effects between treatment type and German language proficiency, and between treatment type and mathematics anxiety. The significant R2 change shows that including the interactions increases the explained variance. The two interaction terms explained an additional 4% of the variance in fraction learning gains. Finally, Model 3 shows no significant triple interaction among treatment type, German language proficiency, and mathematics anxiety, which indicates that mathematics anxiety does not mediate the effect of German language proficiency.

The linear regression results provide evidence of the differential effectiveness of the treatment with and without additional language support. Specifically, they show that learning gains in fraction knowledge depend on German language proficiency in the treatment without additional language support. Concerning students’ achievement on the posttest, students with higher levels of German language proficiency performed significantly better on the fraction knowledge test when using the material without additional language support. Students with lower levels of proficiency in the language of instruction performed significantly better on the fraction knowledge test when using the material with additional language support (Fig. 3). Furthermore, the statistically significant interaction between mathematics anxiety and treatment suggests that students with higher levels of mathematics anxiety gained significantly less from the material with additional language support (Fig. 4). The nonsignificant three-way interaction indicates that the effect of proficiency in the language of instruction and learning material was not mediated by mathematics anxiety.

Fig. 3
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Associations Between Learning Gains in Fraction Knowledge and Proficiency in the Language of Instruction in Both Treatment Groups

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Fig. 4
figure 4

Associations Between Learning Gains in Fraction Knowledge and Mathematics Anxiety in Both Treatment Groups

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5 Discussion

5.1 Learning gains in fraction knowledge

Our study compared the effectiveness of two learning materials for fractions (with and without additional language support) in a controlled classroom setting using a quasi-experimental design. Both intervention groups achieved significant but comparable learning gains in fraction knowledge, whereas no significant learning gains were found in the control group. First, this confirms the quality of both intervention materials. Second, the results show that language-responsive learning material with additional language support can also be effective in a more heterogeneous group of students with varying levels of proficiency in the language of instruction. This finding extends previous research that has primarily focused on second language learners with lower levels of proficiency in the language of instruction.

The effect sizes in our study were comparable to other studies in the field of language-responsive instruction on fractions conducted in Germany, for example, Prediger and Wessel (2013) (η2 = 0.13 and d = 1.22), Prediger and Wessel (2018) (η2 = 0.12 and d = 1.03), and Schüler-Meyer et al. (2019) (η2 = 0.09 and d = 0.9). This result is remarkable because, compared to these studies, our intervention was relatively concise (three lessons) and based only on self-learning material (macro-scaffolding). As such, the results indicate an effect on the material itself, so our research can provide some guidance for designing instructional materials, such as textbooks. Furthermore, the learning gains might be even higher when teachers support the learning process in a communicative way (micro-scaffolding). However, teachers must be professionally trained to accompany mathematics and language learning.

Given that the present study was conducted under highly controlled conditions, it remains to be explored to what extent the effectiveness of language-responsive instruction with additional language support can manifest in ecologically valid learning situations. Therefore, further studies are needed to explore language-responsive instructional approaches under real-life conditions. Bridging the gap between controlled settings and practical contexts is crucial, as it is essential to consider the complexities of the real world to deepen our understanding of language-responsive learning.

5.2 Differential effectiveness

By empirically demonstrating the differential effects of language-responsive learning materials in heterogeneous classes, the present study shows that interactions between aptitudes and treatments appear not only in laboratory settings but also in a more noisy field setting. This aspect addresses a research gap in ATI research, in which interactions have been demonstrated mainly in laboratory experiments (Blumenthal et al., 2014; Walter, 2008). Moreover, we linked two research strands: research on ATI as a framework for research on individual differences in learning on the one hand, and research on language-responsive instruction on the other.

Although both types of instruction (with and without language support) resulted in similar learning gains from the pretest to the posttest across the whole sample, the linear regression analysis revealed differential effects with regard to language proficiency and mathematics anxiety. Learners with lower proficiency in the language of instruction benefit more from learning materials with additional language support than from materials without language support. In comparison, learners with higher proficiency in the language of instruction benefit more from learning materials without language support than from materials with additional language support. Additional language support may hamper learners with higher levels of proficiency in the language of instruction. As the sentence modules were directly integrated into the tasks, learners could not ignore them. Therefore, learners with higher levels of proficiency in the language of instruction had to process them and match them with their existing knowledge, although they may not have needed them to solve the tasks. For these learners, additional language support may lead to unnecessary cognitive load. Yeung et al. (1998) provided a silmilar explanation to their finding that integrating explanatory notes into a text improved the comprehension of learners with lower proficiency in the language of instruction, while it hampered the understanding of learners with higher proficiency in the language of instruction.

Furthermore, learners with higher mathematics anxiety benefit more from the material without additional language support than material with additional language support. This finding highlights the important role of anxiety in the learning process, which previous research has not yet addressed in such detail. For example, previous research has focused on group-level correlations between mathematics anxiety and mathematics performance (see the meta-studies by Ma, 1999 and Hembree, 1990). This approach provided important information about the overall relationship between mathematics anxiety and mathematics performance but did not convey information about the underlying learning process. Our study shows that mathematics anxiety has a significant impact on learning gains, even in short-term interventions. This suggests that mathematics anxiety may moderate the effectiveness of different learning materials. The cognitive load theory can explain this ATI effect, as learners with limited working memory capacity may struggle to regulate their anxiety levels (Hofmann et al., 2011). Anxiety can also deplete their working memory resources (Ashcraft & Kirk, 2001; Mammarella et al., 2015; Suárez-Pellicioni et al., 2016). New stimuli can create uncertainty, particularly for anxious learners, so additional linguistic support may increase their cognitive load to a critical threshold. Following this assumption, newly designed learning materials may initially challenge anxious learners, so they need to be well-designed and require introduction and guidance. This explanation remains speculative because we did not assess the learners’ cognitive load.

Our study has implications for classroom practice. Considering the heterogeneity of students in typical classrooms (Nusser & Gehrer, 2020; Vock & Gronostaj, 2017), teachers face the challenge of providing adaptive instruction to accommodate varying cognitive and emotional prerequisites among students. In our study, we focused on two aspects of heterogeneity, namely proficiency in the language of instruction and mathematics anxiety, demonstrating ATI effects based on these factors. The findings indicate that the principle of 'more is better' does not always apply to additional language support, and that identical learning materials may not be suitable for all students. This highlights the importance of adaptive teaching, while also demonstrating the challenges of tailoring instruction to diverse learners, given the complex interactions among various factors. For example, based on our findings, the following question arises: What constitutes appropriate learning materials for students with lower levels of language proficiency who also experience mathematics anxiety? This question also pertains to the practical implementation of assigning learning materials to students. Such assignments must be tailored to specific content and grounded in empirical evidence. Additionally, it is challenging to simultaneously address and monitor the specific needs of multiple learners.