Reality-based tasks (Blum and Leiss, 2007) involve authentic problem situations in a realistic setting. They are usually presented in a descriptive text (Verschaffel et al., 2020) and play a strongly growing role in mathematics education [e.g., Boaler, 2001; National Council of Teachers of Mathematics (NCTM), 2003]. How a student approaches such a task—and, subsequently, its degree of difficulty—is principally determined by the mathematic demands it contains, such as the mathematical attributes and competencies focused or the number and kind of mathematical operations required (Blomhøj and Jensen, 2003; Maaß, 2010; OECD., 2003; Schleicher et al., 2009).

All phases in a reality-based mathematical task's solution process contain potential difficulties for students (Galbraith and Stillman, 2006; Newman, 1977; Stillman et al., 2010). However, the first step is reading and comprehending the text prompt (Blum and Leiss, 2007; Leiss et al., 2019), where the reader seeks to establish the task situation's mental model that sets the foundation for a precise understanding (Heine et al., 2018). Following the established models of reading, this comprehension process can be understood as the construction of a situation model, an individual cognitive representation, with a combination of the bottom-up and top-down processes. The textual elements are decoded and provide situational data; however, they are simultaneously interpreted based on prior knowledge and integrated into the existing knowledge structures (Kintsch and van Dijk, 1978; McNamara et al., 2010; Caccamise et al., 2008). Comprehension can be considered successful when the readers' resulting situation model converges with the author's intended meaning (Reusser, 1989). In reality-based tasks, the text comprehension processes form the basis for any appropriate application of mathematical knowledge and skills, thus putting particular weight on this element (Cummins et al., 1988), especially since the presentation of an authentic problem situation from daily life often requires an extensive text, compared to word problems describing only a simple situation (“In a bowl, there are two apples and three pears...”). Therefore, reality-based tasks often require high language proficiency (Leiss et al., 2019). Several studies have shown that comprehension problems in this phase cause errors in the following solution steps (Leiss et al., 2010; Mayer and Hegarty, 1996; Wijaya et al., 2014). Thus, language proficiency is an essential student variable, comprising the ability to decode words and sentences, as well as strategies such as analyzing text structure, summarizing, or annotating texts (Artelt et al., 2009).

A general and extremely basic assumption in general linguistics is that language provides various alternatives for expressing ideas. Hence, the same content can be linguistically represented in diverse ways (Cummins, 2000, 2008; Schleppegrell, 2007, 2010, 2012). One can decide on the vocabulary level regarding how to present ideas verbally. One can use highly frequent verbs and nouns with a clear, straightforward meaning, with which even beginners will be familiar. Alternatively, one can utilize infrequent vocabulary poorly represented in everyday linguistic input. One can use short sentences with an evident, canonical structure or compress large amounts of information into one single, complex sentence. Most conceptual contents and ideas always have several possibilities of linguistic expression that can theoretically serve the same purpose. However, they could differ in readability and be more difficult to understand for students with low language competencies.

Accordingly, the same reality-based tasks can be presented in linguistically different versions. Imagine two different variants, A and B, of realistic tasks. Variant A presents a reality-based situation regarding a mathematical task with highly familiar vocabulary and simple sentence structures. Variant B denotes the same situation through infrequent vocabulary and long, complex syntactic structures. The mathematical demands for solving both tasks are the same; however, Variant B obviously requires higher text comprehension demands than Variant A. Since the former will put greater constraints on linguistic knowledge and processing, students with identical mathematical skills will presumably score lower on Variant B than A.

A comprehensive study on the influence of language factors on solving mathematical tasks was conducted by Walkington et al. (2018). Using data from 20 years of the National Assessment of Educational Progress (NAEP) and Trends in International Mathematics and Science Study (TIMSS), they examined the effect of readability factors such as word length, polysemous and technical vocabulary, vocabulary size, and syntactic complexity in mathematics word problems on students' solution rates. The results of the analyses showed that these factors influenced the difficulty of the mathematics word problems and that they also interacted to some extent with students' background characteristics, such as race/ethnicity, math achievement, and socioeconomic status.

Students' mathematical and language proficiency can be considered one of the most important characteristics influencing the likelihood of successfully solving a mathematical word problem (Prediger, 2019).

The level of students' mathematical knowledge and skills affects the identification of mathematical relationships and concepts when solving mathematical word problems, for instance, by identifying mathematical operation patterns. In several studies, pre-knowledge was the most significant predictor of achievement (Nguyen et al., 2016; Hailikari et al., 2008). Furthermore, mathematical problem-solving strategies (Schoenfeld, 2014; Polya, 1945) are also relevant. Regarding reality-based math tasks, researchers have emphasized the importance of problem-solving strategies as mental tools for solving mathematical tasks (Capraro et al., 2012; Galbraith and Stillman, 2006; Schukajlow and Leiss, 2011). Leiss et al. (2019) and Wienecke et al. (2023) showed that strategies such as note-taking are conducive to finding solutions to reality-based math problems.

A sufficient language proficiency level is essential to develop a proper situation model for reality-based tasks. Specifically, presenting an authentic situation from daily life is often provided by a considerably longer text than “traditional” word problems. Thus, such tasks often require high-level language proficiency (Leiss et al., 2019). Studies on linguistic tasks' attributes showed that students belonging to sub-groups, such as low-achieving (Wheeler and McNutt, 1983), bilingual (Moschkovich, 2002), and English as an additional language (Barwell, 2005), encounter specific challenges when solving mathematical word problems. Moschkovich and Scott (2021) summarized language issues in mathematics word problems, especially for English language learners. They summarized that the most relevant language features are on three levels: (a) the cultural background, (b) syntactic (sentences and paragraphs), and (c) lexical levels (words and phrases).

This section addresses mathematics teachers' perceptions of students' language difficulties when solving reality-based mathematical tasks. Given the limited research evidence in this area, we draw upon findings examining the broader role of language in educational processes. For example, Skinnari and Nikula (2017) showed that (Finnish) teachers possess some awareness of the subject-specific language within their discipline and the value of multiliteracy practices. However, the role of multilingualism, encompassing the diversity of students' languages and its impact on pedagogical practice, was only minor. Seah (2016) showed that teachers perceive a wide range of student difficulties related to language use, especially concerning the use of technical terms. Contrastingly, few studies to date have addressed the perception of difficulties in solving reality-based mathematical tasks.

Khoshaim's (2020) study shed light on teachers' reluctance to incorporate word problems into their instruction, citing perceived challenges faced by students. These include insufficient mathematical skills and a negative disposition toward mathematics when engaging with such tasks. Consequently, these difficulties in task engagement can exacerbate classroom stress and anxiety, potentially leading to math-related anxiety disorders among students.

Pearce et al. (2013) contributed insights into teachers' perceptions of their students' primary hurdle in solving word problems—namely, the ability to comprehend the text. Teachers in the study predominantly recommended keyword searching as a strategy for overcoming this perceived difficulty. Interestingly, the study highlighted a gap in teachers' pedagogical approaches, as none reported explicitly teaching problem-solving strategies to their students.

Basaraba et al.'s (2019) study analyzed teachers' perceptions and elucidated factors contributing to difficulty in math word problems. Particularly, teachers identified language presentation as a more formidable challenge for English language learners (ELLs) than for native English speakers. These findings underscore teachers' awareness of the linguistic demands inherent in mathematical tasks, especially for diverse learner populations.

Despite these valuable insights, the extent to which mathematics teachers' perceptions align with actual language-related difficulties in reality-based tasks remains ambiguous. Teachers should be aware that both linguistic and mathematical demands influence the solving of reality-based mathematical tasks. If linguistic demands are significantly overestimated or underestimated, teachers may not select the appropriate difficulty level for their students and thus overwhelm or underwhelm them. Particularly for students with low language skills, this can contribute to increasing the disadvantage of these students and widening social disparities in the education system. Addressing this discrepancy is therefore crucial for teacher training. By understanding and anticipating potential linguistic challenges in learning materials and assessments, teachers can better support student learning and adapt teaching strategies to effectively meet the needs of different learners.

In summary, an in-depth exploration of teachers' perceptions of students' difficulties with reality-based mathematical tasks offers valuable insights into the intersection of language and mathematics learning. By bridging the gap between perceived and actual challenges, educators can enhance pedagogical practices and promote equitable access to high-quality mathematics education for all learners.

A challenge in researching student characteristics' impact on reality-based tasks (or on mathematics word problems, too) is that mathematical and language task attributes are often confounded and, due to the methodological design, cannot be analyzed independently. For example, mathematically complex problems that require multiple solving steps often have longer problem descriptions with more and possibly longer sentences.

As mentioned, Walkington et al. (2018) showed that the influence of linguistic factors such as length, word difficulty, and pronouns makes processing mathematical word problems more difficult. However, this study was based on a cross-sectional design (observed data from large-scale assessments) and failed to separate the confounding factors between a task's linguistic and mathematical demands. Additionally, students' language proficiency could not be controlled, so no differentiated analyses for weak and strong students were possible. Therefore, although this research provided strong evidence that language factors are relevant, no causal relationships could be demonstrated.

Moreover, how the factors at the student and task levels work together or interact remains unclear. Many researchers assume that student characteristics moderate the influence of mathematical and linguistic task attributes on the successful solving of mathematical word problems (Abedi et al., 1997; Abedi and Lord, 2001; Haag et al., 2013; Daroczy et al., 2015; Martiniello, 2008; Prins and Ulijn, 1998).

An experimental research design is required to analyse the effects of the interaction of student characteristics and mathematical and language task characteristics. This is the only way to control or systematically vary linguistic and mathematical factors in a targeted manner. However, experimental studies that vary readability factors or the linguistic dimensions of test items in experimental designs under tight control of other factors are rare. Given these methodological challenges, we have summarized the state of research in the following section, considering only studies with a strict experimental design.

Abedi and Lord (2001) investigated the importance of language on student test performance in mathematical word problems. Overall, 1,174 students were given items released from the National Assessment of Educational Progress mathematics assessment and parallel ones modified to reduce their linguistic demands. The linguistic modification (seven factors on syntactic and lexical levels) of 20 test items caused significant differences in the mathematics performance; the linguistically modified version's scores were slightly higher (effect size: d = 0.09). Some student groups benefited more from the items' linguistic modification, particularly those in low-level and average mathematics classes, ELLs, and those with a low socio-economic status.

Johnson and Monroe (2004) examined the impact of simplified language on one state's mathematics performance assessment through 20 items. The simplified language was received by applying seven recommendations designed to improve the accessibility of text material from Kopriva (2000). Overall, 1,232 seventh-grade students had participated. The variance analysis indicated no benefit for those in general education (effect size: d = 0.05) and ELLs. Only those receiving special education services (N = 138) benefitted from the simplified language (effect size: d = 0.17); nonetheless, their scores remained significantly below those of their general education counterparts.

From a test accommodation perspective, Sato et al. (2010) evaluated whether ELLs significantly benefitted from a linguistic simplification (different modification strategies on syntactic and lexical levels) of 25 multiple-choice mathematics items. Four different item response theory-based (IRT) scoring approaches, commonly used by the states in analyzing the performance data from the United States state-wide testing, assessed their mathematics performance. Differences across sub-groups (ELLs, English language proficiency, and Non-English language proficiency) in the effects of linguistic modification on the students' mathematics performance depended on the scoring approach. Only when the scores were constructed based on the 1-PL item response model (Rasch model) a significant difference in the theta scores on the two item sets (original and linguistically modified) was detected across the student subgroups (effect size: d = 0.16). An analysis based on the raw scores or other IRT models revealed no significant effects of linguistic simplification.

Moreover, Haag et al. (2015) tested the impact of simplifying mathematics word problems' language (19 factors on syntactic and lexical levels). Using a randomized experimental design, they conducted a large-scale linguistic simplification study to test whether the performance gap between those who use a heritage language at home and monolinguals was smaller when assessed with linguistically simplified items in the schooling language. They utilized data from 17,738 fourth graders. Although the differences between the multilingual and monolingual students in mathematics achievement were related to those in their language proficiency and socioeconomic status, they found no significant primary effects of linguistic simplification. However, the multilingual students' differential effects emerged, indicating that some might profit from linguistic simplification during elementary school.

Pöhler et al. (2017) investigated whether students with low and high language proficiency score differently on mathematical word problems and whether this was due to the items' text format or conceptual mathematical challenges. A test with percentage problems of different types, in a purely mathematical, text, or visual format, was given to 308 seventh graders, with the scores analyzed statistically using a cognitive diagnosis model. The probability for those with a low language proficiency to solve the text format items was not lower than that for the pure format, indicating that conceptual challenges might have a stronger impact than language ones.

Walkington et al. (2019) systematically manipulated six different language features (number of sentences, pronouns, word concreteness, word hypernyms, consistency of sentences, and problem topic) of algebra story problems and analyzed the effects on student performance. They found limited evidence suggesting individual language features considerably affect the mathematics word problem-solving performance of the general student population. Among other differential results, language modifications were suggested to benefit or harm students depending on their familiarity with the computer-based assessment instrument employed.

Experimental studies so far have provided crucial empirical evidence regarding linguistic task characteristics' influence on task difficulty. Despite this research progress, a significant gap persists concerning the generalizability of the studies' findings across diverse student populations. While some studies have identified substantial effects of linguistic task characteristics, particularly for multilingual students or those with special educational needs (Haag et al., 2015), none have consistently demonstrated a larger effect across the entire student sample. This discrepancy highlights a critical gap in the literature, suggesting that the impact of language demands on task difficulty may vary significantly depending on individual student characteristics, particularly their language proficiency levels. Furthermore, existing research has often failed to adequately consider students' language proficiency levels in a nuanced manner, leading to potential confounding effects in the analyses. Consequently, the true relationship between linguistic task characteristics, student language proficiency, and task difficulty remains inadequately understood. This study aims to address these research gaps by adopting a comprehensive approach that considers students' language proficiency levels as a crucial factor influencing task difficulty. By measuring and controlling for students' language proficiency levels, we aim to provide a more nuanced understanding of the interplay between linguistic task characteristics and task difficulty across diverse student populations. We believe that elucidating the differential effects of language demands on task difficulty and considering the moderating role of students' language proficiency levels will help bridge the existing research gap and advance knowledge in this area. Through rigorous empirical investigation, we aim to elucidate the nuanced dynamics of task difficulty in reality-based mathematical tasks and inform more effective instructional practices considering students' diverse linguistic needs.

We constructed three linguistically different versions each (low, medium, and high linguistic demands) of five reality-based tasks. All items' mathematical content addressed reality-based problems in the linear functions' content area. In constructing these tasks, we accounted for various design criteria, which were intended to ensure that they were reality-based tasks in the sense of Depaepe et al. (2015). This means that the processing phases of Understanding, Structuring, Mathematizing, Working mathematically, Interpreting, and Validating had to be completed. Besides that, the students should not be deterred from seriously starting the solution process by elements that are too unfamiliar, such as an unusually long description of a real-life situation or the still unfamiliar making of assumptions of required quantities. In this respect, the tasks used in our study can be characterized by the following content-related features:

Considering the numerous aspects different researchers apply to “real” mathematical modeling tasks (Wess et al., 2021), our tasks are deliberately placed in a middle range on this continuum of normative specifications. They are neither clothed tasks whose context does not play a role nor authentic modeling tasks that leave a lot of freedom for the individual processing and answering of the central question. Rather, our reality-based tasks have the particular element that linear functions imply a fixed mathematical model for dealing with the real-life context. However, the students must translate worldly objects into mathematics, including the linguistic identification of constant starting elements (ordinate intercept) and changing quantities (variables and slope). Our preliminary studies have shown that many students have problems comprehending and translating functional relations even in these well-structured tasks (Leiss et al., 2019). Accordingly, the goal was for the students to process many tasks, the analysis of which yields findings whose transfer to more complex modeling tasks seems obvious but requires further studies.

Our study was conducted in Germany; therefore, the reality-based tasks were offered in German. Our approach to modifying these tasks' linguistic demands was described comprehensively by Heine et al. (2018).¹ Their model integrated several difficulties, inducing linguistic surface features.

Regarding the linguistic elements' frequency, the less frequent it is, presumably the less it is part of the language users' knowledge base and the less automated it is, thus creating gaps or vagueness in comprehension. These gaps must be filled from the linguistic co-text or world knowledge (Heine et al., 2018). However, on the syntax level, the frequency dimensions play an important role (Chen and Meurers, 2018). Thus far, the most frequent sentence structures were short main clauses with canonical word order in an active voice (Heine et al., 2018), while more complex sentences were generally rare and more typical for specific contexts (e.g., academic language use) and, thus, less familiar to the language users in general.

In our study, linguistic demand refers to the varying linguistic features present in reality-based mathematical tasks, which may influence task difficulty and student performance. We employed a systematic approach to vary the tasks' linguistic demands, as elaborately outlined in Heine et al. (2018). The specific linguistic features encompassed syntactic structures, verb forms, semantic transparency, stylistic elements, deixis, and impersonal expressions. These variations were meticulously designed and implemented by a team of linguists, ensuring consistency in mathematical demands in each version while systematically manipulating linguistic complexity. The aim was to ensure consistency in the mathematical requirements in each version while systematically varying the linguistic complexity. Starting from a task at the intermediate language level, simpler and more complex variants were then designed. Particular care was taken to ensure that the mathematical content and requirements remained consistent.

Heine et al.'s (2018) model serves as a basis for systematically manipulating texts into linguistically different versions on the word and sentence levels while keeping the mathematical task constant. A research team of linguists and mathematicians applied this approach to the five mathematical word problems and modified each text prompt systematically into linguistically simple, medium, and advanced language levels (LL1–LL3). The linguistic modifications only affected the task stimuli. The item questions and single-choice response options remained constant for all three versions of the five tasks. They were carefully formulated on a linguistically low demand level. The mathematics experts reviewed the linguistic modification process to ensure that all versions of the mathematically relevant information needed to solve the reality-based tasks remained constant. Table 1 illustrates an example task stimulus with low (LL1) and high (LL3) levels of linguistic demands.

In addition to the theoretically based model's systematic application, we conducted a simple modification examination and quantified the difference between LL1 and LL3 of the two German versions in Table 1 using the Flesch Reading Ease Readability Formula (Flesch, 1948; Schöll, 2021). After applying the index, the results revealed 86 (“easy”) and 57 (“fairly difficult”) score points for the low and high complex variants, respectively, validating that the texts should be relatively easily understood by students aged 13–15 years.

This study employed an experimental design with the within-factor items' linguistic demand level (low vs. high), controlling for language proficiency as a metric moderator variable. The sample comprised 1,346 students from 17 schools in northern Germany (female: 49.4%, male: 50.6%). The average age was 14.0 years. Approximately 34% of the students had an immigrant background (=parents were not born in Germany).

This research is part of a larger study that, besides mathematics, comprises other subject domains (physics, music, physical education, and German).² The elicitation duration was 90 min, with 60 and 30 min for the test and background variables, respectively (c-test and questionnaire).

It was decided post-hoc to exclude items with a medium language level and use only those with low and high linguistic demands for further analysis. Due to time constraints, only the two extreme variants could be employed within the teacher survey; nevertheless, no significant differences between the medium and low or high variants were found regarding the empirical item difficulty. Thus, we focused on the contrast between the low and high demand levels that could be more sensitive in detecting the effects of linguistic demands on the students' performance. The sample size remained unchanged by this decision because each student solved only one or two items with a medium linguistic demand.

The first research question's sample comprised 72 mathematics teachers from secondary schools in northern Germany (82.1% male, 17.9% female). It included a convenience sample drawn from the teacher professional development courses and was independent of the student sample. On average, the teachers had 13 years of teaching experience (SD = 10.1 years).

We requested the sample to review the reality-based tasks and estimate if they expected difficulties for the students. To establish a common reference base, we advised them to imagine a fictitious class characterized by information about the number of female and male students (14 and 10, respectively), the percentage of migration background students (33%), and the distribution of school grades within the class in mathematics and German (given by two tables). We instructed them regarding three different ratings: (1) “Guess what percentage of the students can solve the reality-based tasks correctly.” The response format was a percentage value between 0 and 100%. (2) “Evaluate the following statement regarding the class described at the beginning: the students will have linguistic difficulties solving the problem.” (3) “Judge the following statement regarding the class described at the beginning: the students will have mathematical difficulties solving the problem.” The response format of ratings (2) and (3) was as follows: 1 = “disagree,” 2 = “somewhat disagree,” 3 = “partly disagree/partly agree,” 4 = “somewhat agree,” and 5 = “fully agree.”

A condensed version of a c-test, which is a specific format of the cloze test, was administered to assess the students' language proficiency over a 10-min duration. This test is widely recognized for its reliability and validity in measuring language proficiency, offering insights into both receptive and productive language skills (Grotjahn, 2010; Grotjahn et al., 2002; Grotjahn and Drackert, 2020). Previous research has demonstrated the discriminative validity (DCLL+3) of texts utilized in this assessment. Each c-test comprised two texts, with 30 gaps in each test for students to complete. The total of 60 gaps were treated as individual items (Harsch and Hartig, 2010), resulting in a high internal consistency, as indicated by a Cronbach's alpha coefficient of 0.96. Item responses were calibrated using the Rasch model, and the resulting scale scores were standardized to a mean of 0 and a standard deviation of 1.

We calculated the descriptive statistics for research questions 1 and 2 and assessed significant group differences between the LL1 and LL3 task versions by conducting t-tests. For all analyses, we chose a significance level of α = 0.05.

Sato et al.'s (2010) study suggested that applying the IRT models for estimating students' achievement scores, compared to using the sum scores or the percentage correct, is more suitable for detecting the effects of the linguistic task attributes. Moreover, for analyzing the moderator effects between the personal characteristics (e.g., reading proficiency) and item attributes (e.g., language demands), explanatory item response models (EIRMs) seemed (Meulders and Xie, 2004) adequate and sensitive (De Boeck and Wilson, 2004). The EIRM is an approach to cognitive assessment that employs explanatory measurements utilizing item and person covariates to explain what is being measured, thus adding an explanatory value to the measurements (De Boeck et al., 2016).

Thus, for the third research question, we employed the EIRMs that provide a framework for modeling the item responses directly as a function of the item, as personal predictors, or as a combination of both (De Boeck and Wilson, 2004). This makes them especially suitable for the present analysis, as we expected the item responses to depend on an interaction between the item's linguistic level and the students' language proficiency. In our case, the item responses were binary, and we modeled the students' propensity to answer the items correctly, which were the logarithmised odds of a correct answer (e.g., Wilson and De Boeck, 2004). The models were specified as hierarchical generalized linear mixed models, with item responses nested within students and items (e.g., De Boeck et al., 2011).

The modification status of the reality-based tasks and the student characteristics (general language proficiency) and their interactions were defined as fixed effects. A possible primary effect of general language proficiency indicated an overall effect when solving reality-based tasks. The interaction effects of the general language proficiency (student characteristic) with the linguistic task level (item characteristic) demonstrated differences in the effect of linguistic demands related to the students' general language proficiency (Beretvas et al., 2012; Meulders and Xie, 2004). The models were estimated using the lmer function of the lme4 package in R (Bates et al., 2012).

Table 2 provides the results of how our sample's mathematics teachers rated the expected difficulties regarding the language and mathematical demands on a Likert scale. On average, the teachers expected exceedingly large differences (all five items' effect size: d = 1.68, p < 0.001) between the reality-based tasks with low and high linguistic demand levels (Table 2). This indicated that they anticipated a considerable effect of linguistic modification on the students, suggesting that they expected high linguistic demands to result in serious language difficulties. Table 2 demonstrates an example task of this language modification.

Furthermore, the teachers expected mathematical difficulties between the tasks with low linguistic demands (LL1) and high linguistic demands (LL3; Table 3). The mean values between them were statistically significantly different; the total scale's effect size was d = 0.51. Sullivan and Feinn (2012) rated an effect size of 0.2, 0.5, 0.8, and 1.3 as small, medium, large, and very large, respectively.

Finally, we requested the mathematics teachers to estimate how many students in a fictitious class would find each task version's correct solution (Table 4). On average, they expected that the LL1 and LL3 task versions would be solved by ~53% and only 36% of the students, respectively. The mean difference was statistically significant (p < 0.001). The effect size of the mean difference was approximately d = 0.73, which was a practically relevant effect. The results showed that the teachers anticipated a task's linguistic shape to strongly affect the students' success. As a manipulation check's form, the linguistic demand model's application (Heine et al., 2018) could be considered successful regarding face validity.

Regarding the second research question, 29 and 24% of the students successfully solved the mathematics tasks for the LL1 and LL3 versions, respectively (Table 5). The effect size of this mean difference was 0.12.

Noticeably, four out of five mean differences were not statistically significant. However, the LL1 versions were easier than the LL3 ones in all five tasks. Comparing the results (effect sizes) between the teachers' assessments (d = 0.73) and the students' performance (d = 0.12), the results show that teachers significantly overestimated the influence of linguistic requirements in real-life math problems. Their expected language difficulties were much higher than the students' empirical difficulties. This effect's overestimation was also accompanied by underestimating the tasks' absolute difficulty.

Regarding the third research question, we divided the student sample into four quartiles based on their language proficiency test results. Comparing the difficulties between the LL1 and LL3 task versions, we found the following effect sizes for the four quartiles: first (=25% of the students with the lowest language proficiency): d = 0.20 (p < 0.05); second quartile: d = 0.17 (p < 0.05); third quartile: d = 0.15 (ns); and fourth quartile (highest language proficiency): d = 0.03 (ns). These descriptive results indicated that the students' language proficiency influences the effect of a task's linguistic demands on its overall difficulty. Therefore, the language requirements of reality-based tasks are essential for students with low language proficiency.

To establish the moderation effect in a more advanced model, we applied an EIRM that allowed direct modeling of the interaction effect between personal abilities (language proficiency) and item characteristics (low vs. high linguistic demands) (De Boeck and Wilson, 2004). We investigated whether the effect of a task's linguistic demands was higher for the students with a low language proficiency than those with a high language proficiency.

Table 6 provides the three analyses' results. In Model 1, we found a statistically significant primary effect for a task's linguistic demands (β = −0.338, p < 0.01). This replicated the descriptive finding in Table 5: the students' performance is lower if the tasks' linguistic demands are high.

Model 2 additionally accounted for the students' language proficiency. The results showed that both student-level predictors, namely the students' language proficiency (β = 0.672, p < 0.001) and item-level predictors such as the task's linguistic demands (β = −0.345, p < 0.001), are predictive of students' performance on the reality-based tasks. This indicated that those with a high language proficiency have a higher propensity for solving a task than those with a low proficiency. Furthermore, the probability of correctly solving a task decreases as the task's linguistic demands increase.

Finally, in Model 3 (Model 2 plus the interaction effect), the students' language proficiency, along with the task's linguistic demands, was included in the analyses. The results demonstrated a significant interaction effect (β = 0.263, p < 0.05), indicating that the students' language proficiency moderates the effect of a task's linguistic demands regarding successfully solving an item.

We see implications for further research in the following areas in particular: