Is Formal Language Proficiency in the Home Language Required to Profit from a Bilingual Teaching Intervention in Mathematics? A Mixed Methods Study on Fostering Multilingual Students’ Conceptual Understanding
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Abstract
To what degree can multilingual students profit from bilingual teaching approaches, even when they lack experience in the academic or technical register in their home languages? This study explores this research question in a mixed methods design for a German/Turkish bilingual intervention aimed at fostering conceptual understanding of fractions (five sessions of 90 min each). The sample consisted of German/Turkish bilingual students (n = 128) in Grade 7 in German schools without prior formal mathematics education in Turkish. In a randomized control trial, the bilingual intervention was compared to the corresponding monolingual intervention and a control group. A repeated analysis of variance (ANOVA) showed that students in both interventions had significantly higher learning gains than in the control group, and in fact profited equally from both interventions, although some time and effort was required for overcoming initial barriers in the home language and especially in the academic register. A qualitative analysis of the videotaped bilingual learning processes revealed insights into specific obstacles and chances of connecting both languages in order to foster conceptual understanding. The students with some formal language proficiency in Turkish seemed to profit even more from the bilingual intervention, but a rigid technical register was not necessary.
Keywords
Bilingual teaching intervention in mathematics Formal language proficiency Learning of fractions Randomized control trialIntroduction
There is a persisting mathematics achievement gap between monolingual and multilingual learners, with research in some countries dating back more than 30 years (see Secada, 1992, for early summaries). For other countries, this gap has been acknowledged more recently and traced back to a limited language proficiency in the official language of instruction (Haag, Heppt, Stanat, Kuhl & Pant, 2013; Organisation for Economic Cooperation and Development [OECD], 2007).
One major approach for facilitating access to subjectmatter learning for multilingual learners is to build upon their home languages as a resource (Barwell, 2009; Gogolin, 2011; Grießhaber, Özel & Rehbein, 1996). When offering bilingual teaching programs to multilingual students in secondary mathematics classrooms, one typical obstacle is the limited familiarity with formal (i.e. academic or technical) registers in the students’ home language. This specifically applies to minoritylanguage students initially schooled only in the official language. This article theoretically and empirically explores this issue by investigating how much proficiency in the formal registers of the home language is required for participating in and profiting from a bilingual teaching intervention on fractions. This research was pursued in a German/Turkish bilingual teaching intervention in Grade 7 that aims at deepening the conceptual understanding of fractions.
In the first section, the theoretical and empirical background of bilingual teaching programs and the role of formal language registers are presented. The second section outlines the research context, namely the German language context and the teaching intervention investigated. The mixed methods design of the study with a randomized control trial and qualitative video analysis from the teachinglearning processes are reported in the third section and the quantitative results and first qualitative insights in the fourth section.
Background: Bilingual Teaching Programs and the Role of Formal Language Registers
Benefits and Obstacles of Activating Students’ Home Languages
In many European countries, bilingual teaching programs are discussed controversially (summarized in Meyer, Prediger, César, & Norén, 2016). Although the Council of Europe pleads for including students’ home languages in subjectmatter courses (Beacco, Byram, Cavalli, Coste, Egli Cuenat, Goullier & Panthier, 2010), most European school systems are still reluctant to open their monolingual classrooms to home languages (Meyer et al., 2016).
The state of research on the benefits of including home languages is not yet consistent. Several qualitative studies on bilingual mathematics learning processes have illustrated that students’ home languages can be a resource for mathematics learning in various ways, such as participating in mathematical discourses (Planas, 2014), activating everyday out of school experiences (Domínguez, 2011), upgrading resources for meaning making processes (Clarkson, 2006; Norén, 2015), and others (Barwell, 2009, for several possible effects). However, little quantitative evidence exists: Some randomized control trials have provided evidence for the higher efficacy for bilingual teaching programs for other school subjects (Slavin & Cheung, 2005, although many have been criticized for methodological biases; see Rossell & Kuder, 2005), but for mathematics learning, this evidence has not yet been convincing (Reljić, Ferring & Martin, 2015).
Two main arguments have often been raised against bilingual teaching programs: First, a simple timeontask argument proposes that home languages and official languages may compete for students’ scarce time resources, while maximum exposure to a language is crucial (Gathercole, 2002; Leseman, Scheele, Mayo & Messer, 2009); this argument, however, has been refuted by others (e.g. Cummins, 2000; MacSwan & Rolstad, 2010). The second argument has referred to multilingual students’ limited familiarity with the formal register in their home language (L1, for this paper Turkish): Students who have only been schooled in the official language of instruction (L2; German, in this paper) and not in L1, the formal register (encompassing the academic school register as well as the technical register of school subjects; see next section) in L1 has been developed with limited proficiency and is dysfunctional for school learning (Grosjean, 2001). Accordingly, home languages have been questioned in their role as means of participating in classroom mathematical discourse or a resource to build on to foster the formal registers in the official language (Barwell, 2009; Snow & Uccelli, 2009). Teaching in two languages can result in a tension between activating multilingual resources and fostering the appropriation of formal language (Adler, 2001). Hence, in spite of the qualitative indications for potential benefits of activating students’ home language in mathematics, the assumed need to develop some aspects of the academic and technical register in the home language may raise a more complex timeontask concern (adapted from Gathercole, 2002): Time for learning mathematics may be competing with time for learning formal registers in the home language. This must therefore be investigated more deeply.
Fostering Formal Language Learning in Mathematics Classrooms
The verbal representation is differentiated in two times three registers: the everyday register (with familiar vocabulary and simple sentence structures), the academic school register (which has been described by the more complex vocabulary and structures on the sentence and discourse level; see Schleppegrell, 2004; Snow & Uccelli, 2009), and the technical register of mathematics (comprising the specialized language means specific to mathematics), each in the home language L1 and the official language of instruction L2.
On a prescriptive level, the mathematics learning of language learners can be fostered by systematically relating all registers and representations in both languages (Moschkovich, 2013; Prediger et al., 2016). On a descriptive level, multilingual students’ limited familiarity with the formal registers in the home language may thus be a key to explaining why bilingual teaching programs may not be immediately beneficial for some multilingual students: Students with few experiences with the formal register in their home languages may have fewer resources for learning the formal language in the language of instruction, as the home language is less likely to be the takeoff point (Setati & Adler, 2001). However, no intervention study has compared learning gains for different home language proficiencies, and no differentiation between the academic and the technical register has been provided. On an empirical level, this study investigates the connection between learning gains and home language proficiency. With the presented research findings, we hypothesize that students with higher language proficiency may profit more, as they have more time to work mathematically and have more resources to meaningfully connect to formal language. While we acknowledge the high relevance of the home language everyday register (Moschkovich, 1999, 2002; Setati, 2005), we follow Adler (2001) in assuming it may not be sufficient for an immediate participation in classroom mathematical discourses. The adapted timeontask concern may be specifically significant in a shortterm teaching intervention, where—with L1 being dysfunctional for school learning—time is needed for overcoming L1 barriers. This leads to questions of how and to what degree adolescents who are familiar with monolingual classrooms can be encouraged to activate their home language resources and whether it is possible to overcome initial barriers (Meyer & Prediger, 2011).
Refined Research Questions
 (Q1)
How can bilingual teachinglearning situations be established in which multilingual students, who have never used their home languages in school, can activate these languages?
 (Q2)
How do the learning gains (for the conceptual understanding of fractions) differ between the bilingual intervention, the monolingual intervention, and the control group?
 (Q3)
How do the learning gains differ for students with high or low Turkish formal language proficiency?
The instruction design of the intervention, which will be presented in the next section, is a first answer for Q1; it was developed in previous design research cycles (see next section; Kuzu 2014, unpublished; Prediger & Wessel, 2013). In this paper, we provide results from a quantitative video analysis of students’ and teachers’ use of Turkish during the intervention as empirical evidence for the realizability of late start in bilingual instruction. For Q2 and Q3, a randomized control trial is conducted and triangulated, especially for Q2, using qualitative insights into the learning processes.
Research Context and Design of the Teaching Intervention
German Language Context
The research questions (among others) were pursued in the project MuMMulti (funded by the German ministry BMBF, grant 01JM1403A, held by Prediger, Redder, Rehbein).
More than 25% of the students in German schools are multilingual (Haag et al., 2013); however, so far, multilingual teaching programs are rare. Here, we explore a bilingual program for the largest language minority group: German/Turkish bilingual students. Most of these students were born in Germany and schooled in German, and only a minority learn Turkish formal language at home (Daller, 1999). In our sample, most may have no experience with mathematics in Turkish; even homework is often done in German (as revealed by the questionnaire data).
Design Principles and Realization of the Teaching Intervention
The research questions were pursued in a German/Turkish bilingual teaching intervention in Grade 7 aimed at deepening conceptual understanding of fractions (partofwhole concept, equivalence, and order of fractions). The intervention comprised five sessions of 90 min. each, which is a relatively short time to adapt to a new multilingual situation.
Design principles for adopting a bilingual intervention
Monolingual intervention  Added or adopted for bilingual intervention 

Principle of creating rich opportunities for communication and language production • following the output hypothesis (Swain, 1985), language production is encouraged in teachermoderated small groups • microscaffolding moves to foster students’ pathways along the language registers  Creating opportunities for bilingual communication and Turkish language production • doing small group work also crucial for engaging multilingual learners (Planas & Setati, 2009; Grießhaber et al., 1996) • due to missing Turkish math experience, encouraging Turkish language production systematically (Meyer & Prediger, 2011) 
Principle of macroscaffolding (Gibbons, 2002) • constructing meanings for concepts by starting the conceptual learning trajectory with everyday contexts and visual models (Freudenthal, 1991) • sequencing the conceptual learning trajectory from everyday resources to formal concepts and formal procedures (Freudenthal, 1991) • enriching the conceptual learning trajectory with languagelearning opportunities from everyday language to the technical register (Gibbons, 2002)  Developing the Turkish formal registers and taking into account culturally sensitive contexts • showing also the relevance of contexts for multilingual students, especially for contexts stemming from languagespecific, nondominant communities (Domínguez, 2011) • additionally, developing Turkish formal registers as far as necessary • in particular, establishing meaningrelated vocabulary, based on an analysis of Turkish mathematical vocabulary (Kuzu 2014, unpublished) 
Principle of relating registers and representations • systematically moving forward and backward between all registers and representations for constructing meanings (see Fig. 1) • initiating various activities for relating registers and representations (Prediger & Wessel, 2013)  Additionally, relating both languages • relating home and official language by encouraging codeswitching and strategies of relating languages in bilingual modes (Auer, 2010; Grosjean, 2001) • bilingual copresence of all material in German and Turkish (Setati, Molefe & Langa, 2008) • systematically comparing expressions for concepts in both languages to deepen conceptual understanding 
In order to help students overcome obstacles with the formal register in both languages, a macroscaffolding approach (Gibbons, 2002) was extended to provide phrases for discourse practices in both languages. For example, we introduced the word düşen pay in Turkish (share in English) when communicating about a “part of a whole.” In a fraction bar, a part can be shown as a gray area within a whole (fulllength bar), but students should also have the possibility of reifying the partofawhole concept in a single word.
Examples for important meaningrelated lexical means in German and Turkish
German  Turkish 

3/5: Spoken as “3 5tel” (three fifths)  3/5: Spoken as “5te 3,” specific Turkish conceptualization of the fraction as “5 therein 3” 
Anteil = share: A specific German word for thinking about “part of whole”  düşen pay: The Turkish expression (meaning “the part that one gets”) was newly created due to missing correspondents 
Teil and Ganzes = “part” and “whole”  parça and bütün 
kleiner/größer als = “smaller/bigger than”  daha büyük/daha küçük, daha büyük/küçük düşen pay 
3/5 von … ist … = “3/5 of … is …”  ’ün/nın/in 5de 3’u/ü/ı …: Turkish has a different order of sentence elements and suffixes (due to the suffixvocal harmony) (“of 5, there is 3”) 
The main principle of flexibly moving up and down between the registers and representations also refers to the connection between the two languages: Rather than only activating both languages separately and accepting codeswitching, it is the connection of languages that may be most promising for promoting deeper understanding (House & Rehbein, 2004). Although previous instruction on fractions was only in German, activating the Turkish language in order to promote conceptual understanding may help students to think mathematically by activating everyday experiences about fractions (see Domínguez, 2011). Furthermore, it acknowledges that multilingual resources are best addressed not as separate languages, but by translanguaging and flexibly connecting both languages (House & Rehbein, 2004). To overcome habitual monolingual practices, the teachers spoke a flexible mix of languages and revoiced students’ German utterances into Turkish or mixed language (see Setati & Adler, 2001, p. 265).
Furthermore, the bilingual teaching intervention comprised activities designed to help students reflect on diverging expressions for mathematical concepts, such as the reading direction of fractions (Bartolini Bussi, BaccagliniFrank & Ramploud, 2014): The Turkish way of expressing fractions such as \( \frac{3}{5} \) as “5te 3” corresponds to the Asian (e.g. Mandarin) reading direction from down to up in the numeric way of writing a fraction, literally translated “5 therein 3.” In comparing it to the German “3 5tel” (translated as “three fifths”), with the reading direction from up to down, the students reflect that the focus on the referent whole in “5 therein 3” is closer to relevant meanings than in German. This contrasting activity allows a deepened understanding of fractions (as shown in Wagner, Kuzu, Redder, & Prediger, 2017).
Summing up, the monolingual as well as the parallel bilingual teaching intervention aims at enhancing students’ conceptual understanding of fractions by providing a language and mathematicsintegrated learning opportunity in which the conceptual learning trajectory is supported by sequenced language learning opportunities along the registers. The bilingual contains all elements of the monolingual intervention, and additionally strengthens the activation and combination of both languages by extending the relating registers approach to the home language in all registers.
Research Design and Methodology
Overview of the Mixed Methods Design
The video data from the bilingual intervention groups offer the data corpus for the time measurement of Turkish use (Q1) and the qualitative analysis for pursuing Q2 and Q3 on the microlevel of the learning processes.
Preparation of Teachers and Video Data Corpus
The monolingual and bilingual teaching interventions encompassed five sessions of 90 min each. They were held in groups of 3 to 5 students by welltrained teachers in the master’s program of mathematics education. These teachers were specifically qualified in the linguistics and didactical concerns of the interventions in a twoday workshop on the principles of the teaching intervention (see above). Regular meetings between the interventions guaranteed the comparability of both intervention programs. The bilingual interventions were videotaped in all 11 groups, 5 of them with two cameras.
Instruments and Measures for Further Data Gathering

Measures for students’ conceptual understanding of fractions. Students’ mathematics achievement was operationalized as conceptual understanding of fractions (dependent variable) and measured by a fraction test (developed and standardized in a study involving 268 students in Prediger & Wessel, 2013). The test items covered the following: specifying and drawing fractions in partwhole and partgroup models, ordering fractions according to size and explaining order in a contextual situation (e.g. dividing a cake) or graphical representations (fraction bar, rectangle, or circle), finding equivalent fractions with a given numerator/denominator and explaining equivalence in a contextual situation or graphical representation, analyzing “part of” situations, finding locations on the number line, subtracting with proper and improper fractions, and solving word problems with fractions as operators (the last three topics were not treated in the intervention). The items in the pretest, posttest, and followuptest were structured in parallel versions with changing numbers. The test has a satisfactory internal consistency, with Cronbach’s α = 0.834 for the pretest with 28 items (in a sample of N = 1120 students), and α =0 .754 for the posttest (29 items, N = 417 students).

Measures for students’ socioeconomic status and general cognitive ability. Students’ socioeconomic status (known as a relevant factor for achievement; OECD, 2007) was measured by the book scale, an economic and reliable instrument (r = 0.80, Paulus, 2009). Students’ general cognitive ability, more precisely the subconstruct of fluid intelligence, was measured using a matrix test (BEFKI 7, adapted from Wilhelm, Schroeders & Schipolowski, 2014). Within our initial sample of 1124 students, the test reaches Cronbach’s α of 0.763. Further control variables were captured in the students’ selfreport questionnaire: age, gender, multilingual family socialization (operationalized by languages spoken with parents or grandparents), and immigrant status (operationalized by parents’ and own country of birth).

Measures for German and Turkish language proficiency. Students’ language proficiency in Turkish and German were measured with two Ctests, a widely used, economic, and valid measure based on cloze texts (Grotjahn, KleinBraley & Raatz, 2002). The German Ctest consisted of three texts by Daller (1999) in informal and formal language. The Turkish Ctest contained informal and formal language texts was developed for the project (by the linguistic project partners Meryem Çelikkol and Jochen Rehbein) because available Turkish Ctests for native Turkish speakers were too complex for the multilingual students who had grown up in Germany. Both tests proved to be highly reliable: the Turkish Ctest with α = 0.874 (N = 254) and the German Ctest with α = .774 (N = 1122). The maximum number of correctly filled gaps was 60. As all students were fluent in everyday Turkish but only 254 volunteered for reading and writing in Turkish, the Turkish Ctest can be considered as sufficient to capture the language proficiency in the formal (written) registers.
Initial Sample and Sample of Teaching Intervention
The initial full sample consisted of seventh grade students (N = 1124; 48% female and 52% male) in 12 secondary schools in an urban metropolitan region in Germany. Three hundred and twentytwo students selfreported that they speak (among other languages) everyday Turkish at home (at 27% of the total, this group is overrepresented in the sample due to the selections of schools), with 262 agreeing to take part in the written Turkish Ctest.
Descriptive data for the full sample and the intervention groups
m (SD) for different measures  Full sample (N = 1124, N = 262 for T Ctest)  M Monolingual intervention (n = 44)  B Bilingual intervention (n = 41)  C Control group no intervention (n = 43) 

Mathematics achievement (fraction pretest)  10.31 (4.73)  7.52 (3.41)  7.93 (2.60)  8.42 (2.95) 
General cognitive ability (BEFKI)  7.94 (3.41)  7.8 (2.69)  7.20 (2.92)  7.35 (2.43) 
Turkish language proficiency (T Ctest)  23.95 (13.17)  24.18 (12.44)  24.83 (12.60)  22.93 (11.61) 
German language proficiency (G Ctest)  35.27 (9.17)  31.55 (6.13)  32.32 (6.21)  31.07 (6.10) 
Age  12.76 (0.7)  12.71 (0.77)  12.88 (0.7)  12.81 (0.6) 
Socioeconomic status SES (percent of low/medium/high SES)  39/30/31  42/42/16  40/23/37  35/26/40 
Ftests comparing the three intervention groups show no significant differences with respect to mathematics achievements in the pretest (F [2, 125] = 0.962, p > .05), German language proficiency (F [2, 125] = 0.439, p > .05), Turkish language proficiency (F [2, 125] = 0.264, p > .05), and fluid intelligence (F [2, 125] = 0.580, p > .05). As Table 3 shows, the groups were also relatively comparable with respect to age and socioeconomic status.
Descriptive data for the subsamples of all intervention groups
Students with high Turkish formal language proficiency (n = 61)  Students with low Turkish formal language proficiency (n = 67)  

Mathematics achievement (fraction pretest)  m (SD)  8.25 (2.69)  7.69 (3.28) 
General cognitive ability (BEFKI)  m (SD)  7.59 (2.76)  7.33 (2.61) 
Turkish language proficiency (T Ctest)  m (SD)  34.51 (6.53)  14.37 (7.01) 
German language proficiency (G Ctest)  m (SD)  32.79 (5.44)  30.58 (6.54) 
Methods for the Quantitative Analysis
 (H1)
The five sessions of bilingual intervention are less effective for increasing students’ conceptual understanding of fractions than those of the monolingual intervention.
 (H2)
Students with lower Turkish formal language proficiency profit less from the bilingual intervention than students with higher Turkish formal language proficiency.
In order to test these hypotheses, repeated measure analyses of variance (oneway ANOVA) were conducted with the scores of the fraction test in the pretest, posttest, and followup test. Group and time were the main factors of the ANOVA and group by time was the interaction factor for the analysis, in the second case with subsamples with low and high Turkish formal language proficiency. Within these analyses of variance, the intergroup effect sizes were determined by partial eta squared. Additionally, the intragroup effect sizes d were captured by determining learning gains within each group.
In a quantitative video analysis of the videotaped interventions, teachers’ and students’ use of Turkish or mixed language in the third sessions was captured. For this, each sentence of students and teachers was tagged with a time code and classified as German, Turkish, or mixed (operationalized on a lexical level by both languages within a sentence) in the TRANSANA data corpus (TRANSANA is the video analysis software). The sum of time intervals between tags provided the total time of language uses for each participant.
Methods for the Qualitative Analysis
In step 1 of the qualitative analysis, the video data of students’ processes of conceptual development were inventoried using TRANSANA in order to select interesting episodes for transcription. In step 2, an interpretative sequential analysis was carried out on the transcripts using a systematic extensive interpretation (Beck & Maier, 1994) to prepare for extrapolating crucial moments and typical features in the individual and interactional processes and to identify the language use in different registers and languages. In step 3, the most conceptually dense moments in the transcripts were analyzed in depth with the analytical tools developed in Prediger and Wessel (2013), allowing identification of students’ processes of constructing mathematical meanings and developing conceptual understanding. In this paper, two episodes and their analyses are briefly presented. These episodes mainly serve to explain the quantitative results, without accounting for all analytic results.
Results from the Analyses of Videos and Test Scores
Results of the Quantitative Video Analysis: Home Language Resources Can Be Activated
Distribution of language production in German and Turkish by teachers and students (averages)
Share of German utterances  Share of Turkish utterances  Share of mixed utterances  Total time of language production  

Teachers’ language productions  32%  28%  39%  99% (1% unidentified) 
Students’ language productions  66%  16%  15%  97% (3% unidentified) 
These results can be interpreted as the first evidence for the feasibility of multilingual learning opportunities: In sum, 45% of the time, students were exposed to Turkish or mixed language (data reported in SchülerMeyer, Prediger, Wagner, & Weinert, submitted). Thus, even under difficult circumstances (starting very late in Grade 7 in a Germanonly school system and for only five sessions), students can be activated to use their home language for mathematics learning, when also taking into consideration the students’ language reception time (language production of the teacher and other students).
However, this does not yet show that the additional languages really provide a benefit for mathematics learning. In contrast, Hypothesis 1 assumes that the investment in Turkish may distract from learning mathematics, so the learning gains must be considered in a randomized control trial.
Learning Gains: Monolingual and Bilingual Intervention Equally Effective
Effects of two forms of intervention and control groups in pretest, posttest, and followup test
Scores in pretest m (SD)  Scores in posttest m (SD)  Scores in followup test m (SD)  Effect size d for pre to posttest  Effect size d for pretest to follow up  

Monolingual intervention (n = 44)  7.52 (3.41)  11.57 (3.46)  10.98 (3.24)  1.18  1.04 
Bilingual intervention (n = 41)  7.93 (2.60)  10.88 (3.38)  11.27 (4.72)  0.99  0.91 
Control group: No intervention (n = 43)  8.42 (2.95)  9.4 (3.57)  10.74 (3.67)  0.3  0.7 
Intergroup and time effect (for pre to posttest)  F_{(time)} = 93.13, p < .001, η^{2} = 0.41; F_{(group)} = 0.3 (ns), η^{2} = 0.005; F_{(group × time)} = 6.83, p < .01, η^{2} = 0.09  
Intergroup and time effect (for pre to followup test)  F_{(time)} = 63.66, p < .001, η^{2} = 0.34; F_{(group)} = 0.43 (ns), η^{2} = 0.007; F_{(group x time)} = 4.91, p < .01, η^{2} = 0.07 
In the followuptest, the learning gains of the bilingual intervention group slightly increased compared to the monolingual intervention, but not significantly. In the ANOVA, the significance of the main factor “time” with F_{(time)} = 93.13 and its high effect size of η^{2} = 0.41 suggests that, over the time of the teaching intervention, the students’ conceptual understanding of fractions in the three groups increased.
The significant interaction effect of “group × time” suggests that the three groups developed significantly unequally from pre to posttest: The effect size of η^{2} = 0.09 hints at medium intergroup differences. The posthoctest does not show a pairwise difference between the monolingual and bilingual teaching intervention, but between the mono or bilingual teaching interventions and the control group.
In summary, Hypothesis 1 is shown to be false, as the monolingual and bilingual teaching interventions were equally effective in fostering the students’ conceptual understanding of fractions.
Differential Learning Gains: Students with Higher Turkish Formal Language Proficiency Profit More
Comparison of students with high and low TFLP in pre and posttest
Scores in pretest m (SD)  Scores in posttest m (SD)  Scores in followup test m (SD)  Gains in average scores from pretest to followup  Intragroup effectsize d from pretest to followup  

Monolingual intervention for high TFLP (n = 22)  7.91 (3.05)  11.96 (3.96)  11.59 (2.92)  ∆ = 3.68  d = 1.23 
Monolingual intervention for low TFLP (n = 22)  7.14 (3.77)  11.18 (2.92)  10.36 (3.49)  ∆ = 3.22  d = 0.89 
Bilingual intervention for high TFLP (n = 20)  8.1 (2.47)  11.55 (3.35)  12.75 (5.14)  ∆ = 4.65  d = 1.22 
Bilingual intervention for low TFLP (n = 21)  7.76 (2.77)  10.24 (3.36)  9.86 (3.90)  ∆ = 2.10  d = 0.63 
Intergroup and time effect from pre to posttest  F_{(time)} = 91.63, p < .001, η^{2} = 0.43; F_{(group)} = 0.64, p = .67 (ns), η^{2} = 0.03; F_{(group x time)} = 4.49, p < .01, η^{2} = 0.16 
In summary, the quantitative data tend to confirm H2, i.e. students with higher TFLP profit substantially more in their conceptual understanding of fractions from the bilingual intervention than students with lower TFLP. The much smaller difference between students with high and low TFLP in the monolingual intervention underlines that the differential effects of the multilingual intervention are not connected to general cognitive factors or the German language proficiency gap.
These quantitative results call for a qualitative investigation into why students’ low TFLP limits the benefits of the specific shortterm bilingual learning opportunities.
Qualitative Insights into Bilingual Learning Processes: Obstacles Can Be Overcome
This section illuminates how a bilingual discourse can be established and explains why Hypothesis 1 must be rejected, although Turkish can also pose additional obstacles that can then be overcome by some students, depending on their formal language proficiencies.
Transcript of Episode 1: How to read fractions in Turkish
In this episode, students struggle with two different interpretations of the Turkish expression “5 therein 3,” namely, 5/3 and 3/5. The first interpretation seems to be viable to the students because they work within the reference context of the German way of expressing a fraction as “three fifths,” where the part is expressed first. In line with this German reference context, the students interpret “5 therein 3” as “five thirds.” Only by exchanging the German reference context for the fraction bar as new reference context can the students interpret “5 therein 3” using the Turkish everyday suffixes “thereof” and as a way of expressing the parts within the whole (introduced by Ilknur in Turns 15 and 23). In the later course of the intervention, this Turkish conceptualization is their solid resource for conceptual development.
This episode provides examples that back two crucial points of the argument in this paper. First, with respect to Q2 and Q3, this episode illustrates how some students can struggle with the formal Turkish register: In this case, it is Halim who also struggles in German. In the first section (Turns 1–14), we see how the students discuss 3/5 and 5/3 artificially, without relating to the meaning of these fractions. In these moments, it seems that the formal Turkish language for expressing fractions does not yet have a meaning for the students, and the discursive resources for discussing it in Turkish are too limited. However, when relating to the fraction bar and to gestures, as intended in the relating registers approach, the students can explain their thinking about how to read the formal Turkish expression “5 therein 3” in a meaningful way (Turns 23ff). Interestingly, relating registers occurs in a translanguaging mode, as both languages are used synthetically (Grosjean, 2001; House & Rehbein, 2004). This is one specific way of mixing languages; others have been found in other transcripts.
In addition, collaboratively working on how to read fractions in a bilingual teaching intervention takes time, so the beginning discussion leaves less time for the mathematical task than in a monolingual teaching intervention. This discussion (Turns 1–14) does not directly contribute to solving the underlying conceptual problem. However, having to distinguish different reading directions leads to a deeper understanding of, in this case, the meanings of numerator and denominator. On a more general note, it seems that this reflection on the reading direction seems to support students to flexibly change their perspective of fractions towards the partofwhole concept.
Transcript of Episode 2: Explaining with the Turkish fraction
When Halim evaluates Paul’s utterance in the new task as correct (Turn 1), the other students contradict. Although the discourse starts in German (Turns 1–4), Hakan switches to Turkish to explain why Paul and Halim are wrong: By speaking Turkish, he makes use of the Turkish reading direction of fractions. He first refers to the referent whole (5; Turn 5) and then to the part (3; Turn 7), showing that only 2 are left, not 5 (in Turn 9). For him, the partofwhole concept may be more accessible in Turkish. In his argument, Hakan interprets Paul’s utterance about 3/5 by activating the Turkish conceptualization of fractions (not only the order of speaking; Turns 7–9). Hakan activates Turkish without the teacher’s prompting and adopts the Turkish conceptualization of fractions to build his argument about the partofwhole concept. He does not, however, switch to Turkish because Halim might understand him better, as in this episode Halim speaks only German, indicating that this is his preferred language here.
In sum, the two episodes provide some insights into a pattern that we find in many more transcripts of the same and other students. They offer insights into a qualitative treatment for all three research questions: With respect to Q1 and Q2, we see in many transcripts that the students are able to get involved in mathematical discourses despite having no experience with discussing mathematics in Turkish. They are repeatedly able to solve mathematical tasks by activating their Turkish language in a “bilingual mode,” which is more than juxtaposing the languages (Grosjean, 2001), even within the first encounter in Session 1. With respect to Q3, it is interesting to relate the students’ repeated roles in the discourses to their Turkish Ctest scores (in the complete intervention sample for 128 students, the score was m = 24.2 with SD = 12.2 and max = 49). All four students have above average scores, but with differences of 1 or 1.5 SD: Hakan’s score is 44, Akasya’s and Ilknur’s are 31, and Halim’s is 26. As in many other episodes, the discourse in Episode 1 is dominated by Akasya and Ilknur (Turns 23ff), who have medium TFLP among the four students. Thus, there is no direct connection between TFLP and participation.
Hakan who has the highest TFLP among the four students is rarely as active as in Episode 2. For the most part, he participates less actively, as in Episode 1, but he is very attentive and involved at critical moments. His strong learning gains (while the full sample had m = 10.31 and SD = 4.73 in the pretest, Hakan increased from a pretest score of 7 to a followup score of 17) are also the best example for learning by language reception instead of active participation. Halim has the lowest TFLP among the four and also low German language proficiency. He seems to struggle most to express his ideas (Episode 1, Turns 19, 21), and this is also reflected by his very limited learning gains (from 8 to 9). In other episodes, these patterns are also relatively stable in the analyzed group.
Discussion
Central Results and Limitations
The presented study contributes to the research on fostering subjectmatterintegrated language learning in mathematics. While there are institutional arguments for both monolingual (e.g. lack of qualified multilingual teachers in Germany) and bilingual teaching interventions (Cummins, 2000; Planas, 2014; Reljić et al., 2015), the reported results suggest that both types of teaching interventions can be at least equally effective, even under difficult circumstances where Turkish is initially nonfunctional for mathematics.
With respect to Q1, an existence proof was provided that it is possible to establish a bilingual teachinglearning process for multilingual seventh graders on fractions, despite the students’ limited experience with the technical register in Turkish (Q1). The quantitative video analysis on the percentages of chosen languages shows that when the teachers invest in Turkish and a mixed language, the students also start using them, in spite of all barriers of language dominances and the late start in Grade 7 (Meyer et al., 2016). The quantitative investigation of language use has its limitations, as it gives no insights into the qualitative use of languages, for example, the phenomena of codeswitching within an utterance (e.g. Turns 15, 23; Table 8) or use of Turkish grammar in a German utterance. It is, however, a means to investigate whether a bilingual teaching intervention can successfully promote the use of Turkish. Further research is required to clarify these modes of using mixed languages and relate to different modes of translanguaging in more detail.
 (1)
Sample sizes too small: There is a difference with low effect size which may become more visible with larger sample sizes. [This hypothesis assumes that the monolingual intervention is more effective, for which there is almost no external evidence in the literature (see Reljić et al., 2015)].
 (2)
Too little individual use of Turkish: The average time of Turkish or mixed language use was quite high at 45%, but students used the Turkish language much less, and it is possible that this is a reason for limited learning gains. Thus, in future research, the correlation between individual Turkish language use and the individual learning gains must be investigated on the students’ individual level.
 (3)
Intervention too short: As Grade 7 is late to start bilingual mathematics learning, the short intervention time of 5 × 90 min may be a reason that the multilingual intervention did not provide higher learning gains; in light of the political nature of language (Setati, 2005), students may need time to accept and get adapted to Turkish as a language of learning.
 (4)
Little difference in mathematics learning gains: If other studies are able to replicate the results presented here, it may be sensible to hypothesize that it is not important for mathematical learning gains whether the intervention mobilizes multilingual resources or not. Bilingual learning can then be established for other reasons, such as identity building and motivation, without negative effects on conceptual learning.
The qualitative analysis in the last section provides some deeper insights into the processes of using Turkish and mixed language in a bilingual teaching intervention. It shows barriers where students initially need to establish means to coordinate languages and conceptualizations. Episode 1 may be an example of the concern about timeontask: Becoming adapted to a bilingual learning situation requires more time than a monolingual situation does. However, this example also provides more insights: More individual use of Turkish may help students to get more used to such “needs for conceptual coordination,” strengthening the second explanation hypothesis mentioned above. In light of the time needed to establish Turkish as a language of learning (Hypothesis 3), getting used to “needs for conceptual coordination” may take more time than five 90min sessions. Those students who participate regularly or have a higher TFLP may adapt to this need faster, which may explain why both the monolingual and bilingual intervention lead to comparable learning outcomes, especially for students with high TFLP (see Table 7; discussion below). Episode 2 and many other transcripts not analyzed in this paper illustrate that Turkish is not so much a barrier, but a learning opportunity for fostering involvement. This suggests that the hypothesis that the monolingual intervention is more effective has little substance on the level of learning processes. The use of Turkish or mixed language is then not so much an issue of limited timeontask but of providing students with an additional means for understanding fractions by coordinating and reflecting their conceptualizations in the different languages and of learning to flexibly use the most appropriate conceptualization. From this perspective, Turns 1–14 (Table 8) show the students searching for means to coordinate a conceptualization of fractions, which in the long run fosters conceptual understanding.
With respect to Q3, differential learning gains have been shown: Students with lower TFLP seemed to profit less from the bilingual teaching intervention than their peers with higher proficiency, so the title question of the article, “Is formal language proficiency in the home language required to profit from a bilingual teaching intervention in mathematics?” must be answered with “No, it is not required, but it helps.” The qualitative analysis suggests that the technical register is much less relevant than the academic register, and both may be only two among other factors, as some students (even those with low TFLP) engage more in the discourse while others engage less. The multifaceted learning processes in the two episodes indicate that the Ctest may also not be the adequate instrument to capture the students’ bilingual language proficiency, because it measures students’ written language proficiency in only one language without taking into account their oral bilingual discourse competence, which includes translanguaging for participating in bilingual teachinglearning processes.
Further Research Perspectives

First, it may be that the Turkish teaching material and the teacher favor students with high TFLP. The teacher, as an expert Turkish speaker, may engage more often with students with high TFLP, as they are more fluent. As a consequence, these students may have more slots to participate in the mathematical discourse and to engage in complex discourse practices such as negotiating meanings. Other studies have shown that English language learners need to be positioned as competent so that they can participate productively (Turner, Domínguez, Empson & Maldonado, 2013), and these positionings may be mediated by students’ home language proficiency. Thus, issues of proficiency and their relation to positionings and identities must be investigated in further studies.

Second, it may be that the test formats employed in this study do not capture students’ competencies, as these tests are monolingual and thus do not allow for translanguaging and a mix of Turkish and German. In the future, specific multilingual test formats may be required for testing the students’ bilingually represented conceptual understanding. In light of this, we intend to analyze the oral data in order to evaluate students’ translanguaging proficiency rather than assuming two singlelanguage proficiencies.

Third, in a bilingual intervention, students with low TFLP may not be able to follow bilingual conversations with the same depth that they can follow monolingual ones. They may also have fewer learning opportunities for more abstract and decontextualized discourses in outofschool contexts. Alternatively, they may not be able to exercise agency to change the classroom discourse towards their learning needs (LangerOsuna, Moschkovich, Norén, Powell & Vazquez, 2016). Thus, future interdisciplinary studies should investigate how low TFLP interferes with discourse competency and should also address in more detail the role of German language proficiency in its interplay with Turkish language proficiency. Furthermore, opportunities for agency in implementing a teaching intervention should be investigated.
These points illustrate that it is not yet clear whether changes in the teaching material, teaching practices, or students’ ways of participating in the discourse can overcome the issue that students with lower TFLP seem to profit less from a bilingual teaching intervention.
Beyond these open questions, the presented empirical study is methodologically limited by the small number of students in the bilingual and monolingual interventions. The resulting tendencies must therefore be interpreted and generalized with caution. Furthermore, the learning outcomes were only measured using German tests, so that the test may have artificially contributed to the effect that students with low TFLP profited less from the intervention. In addition, implementation control is required to investigate how the intended learning trajectory was put into praxis, and whether the teachers in the intervention differed in this regard.
However, even if this study is limited in different aspects, and the results invite further investigation, it is already encouraging to see that the widely claimed call for bilingual learning opportunities are also realizable in countries such as Germany that do not have traditions of bilingual teaching and learning. This may be enough reason to start the process of change towards multilingual classrooms, for which so many cultural and political reasons exist (Barwell, 2009).
Notes
Acknowledgements
The project “MuMMulti: Fostering Language in multilingual mathematics classrooms – effects and conditions of a content and language integrated intervention” is funded by the German minis try BMBF (grant 01JM1403A, grant holder S. Prediger, Jochen Rehbein and A. Redder). We thank our partners Jochen Rehbein, Jonas Wagner, and Meryem Çelikkol for the insightful interdisciplinary cooperation in the project.
References
 Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht, The Netherlands: Kluwer.Google Scholar
 Auer, P. (2010). Codeswitching/mixing. In R. Wodak, B. Johnstone, & P. E. Kerswill (Eds.), The SAGE handbook of sociolinguistics (pp. 460–478). London, England: Sage.Google Scholar
 Bartolini Bussi, M. G., BaccagliniFrank, A., & Ramploud, A. (2014). Intercultural dialogue and the history and geography of thought. For the Learning of Mathematics, 34(1), 31–33.Google Scholar
 Barwell, R. (Ed.). (2009). Multilingualism in mathematics classrooms. Bristol, England: Multilingual Matters.Google Scholar
 Beacco, J.C., Byram, M., Cavalli, M., Coste, D., Egli Cuenat, M., Goullier, F., & Panthier, J. (2010). Guide for the development and implementation of curricula for plurilingual and intercultural education. Strasbourg, France: Council of Europe. Retrieved from http://www.coe.int/lang.
 Beck, C., & Maier, H. (1994). Zu Methoden der Textinterpretation in der empirischen mathematikdidaktischen Forschung [About methods of text interpretation in empirical mathematics education research]. In H. Maier & C. Beck (Eds.), Verstehen und Verständigung (pp. 43–76). Köln, Germany: Deubner.Google Scholar
 Clarke, D. J. (2013). Contingent conceptions of accomplished practice: The cultural specificity of discourse in and about the mathematics classroom. ZDM—Mathematics Education, 45(1), 21–33.Google Scholar
 Clarkson, P. (2006). Australian Vietnamese students learning mathematics: High ability bilinguals and their use of their languages. Educational Studies in Mathematics, 64(2), 191–215.Google Scholar
 Cummins, J. (2000). Language, power and pedagogy. Clevedon, England: Multilingual Matters.Google Scholar
 Daller, H. (1999). Migration und Mehrsprachigkeit [Migration and multilinguality]. Frankfurt, Germany: Peter Lang.Google Scholar
 Domínguez, H. (2011). Using what matters to students in bilingual mathematics problems. Educational Studies in Mathematics, 76(3), 305–328.Google Scholar
 Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.Google Scholar
 Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht, The Netherlands: Kluwer.Google Scholar
 Gathercole, V. C. M. (2002). Monolingual and bilingual acquisition. In D. K. Oller & R. E. Eilers (Eds.), Language and literacy in bilingual children (pp. 220–252). Clevedon, England: Multilingual Matters.Google Scholar
 Gibbons, P. (2002). Scaffolding language, scaffolding learning. Portsmouth, England: Heinemann.Google Scholar
 Gogolin, I. (2011). Bilingual education. In J. Simpson (Ed.), The Routledge handbook of applied linguistics (pp. 229–242). London, England: Routledge, Taylor & Francis.Google Scholar
 Grießhaber, W., Özel, B., & Rehbein, J. (1996). Aspekte von Arbeits und Denksprache türkischer Schüler [Aspects of language of work and language of thinking of Turkish students]. Unterrichtswissenschaft, 24(1), 3–20.Google Scholar
 Grosjean, F. (2001). The bilingual's language modes. In J. Nicol (Ed.), One mind, two languages: Bilingual language processing (pp. 1–22). Malden, MA: Blackwell.Google Scholar
 Grotjahn, R., KleinBraley, C., & Raatz, U. (2002). Ctest: An overview. In J. A. Coleman, R. Grotjahn, & U. Raatz (Eds.), University language testing and the Ctest (pp. 93–114). Bochum, Germany: Finkenstaedt.Google Scholar
 Haag, N., Heppt, B., Stanat, P., Kuhl, P., & Pant, H. A. (2013). Second language learners’ performance in mathematics. Learning and Instruction, 28, 24–34.Google Scholar
 House, J., & Rehbein, J. (2004). What is ‘multilingual communication’? In J. House & J. Rehbein (Eds.), Multilingual communication (pp. 1–18). Amsterdam, The Netherlands: John Benjamins.Google Scholar
 LangerOsuna, J. M., Moschkovich, J., Norén, E., Powell, A. B., & Vazquez, S. (2016). Student agency and counternarratives in diverse multilingual mathematics classrooms: Challenging deficit perspectives. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J. Moschkovich, N. Planas, M. SetatiPhakeng, P. Valero, & M. Villavicencio Ubillús (Eds.), Mathematics education and language diversity. New ICMI study series (pp. 163–173). Cham, Switzerland: Springer.Google Scholar
 Leseman, P. M., Scheele, A. F., Mayo, A. Y., & Messer, M. (2009). Bilingual development in early childhood and the languages used at home: Competition for scarce resources? In I. Gogolin & U. Neumann (Eds.), The bilingualism controversy (pp. 289–316). Wiesbaden, Germany: VS.Google Scholar
 Lesh, R. (1979). Mathematical learning disabilities. In R. Lesh, D. Mierkiewicz, & M. Kantowski (Eds.), Applied mathematical problem solving (pp. 111–180). Columbus, OH: ERIC/SMEAC.Google Scholar
 MacSwan, J., & Rolstad, K. (2010). The role of language in theories of academic failure for linguistic minorities. In P. J. Petrovic (Ed.), International perspectives on bilingual education (pp. 173–195). Charlotte, NC: Information Age.Google Scholar
 Meyer, M. & Prediger, S. (2011). The use of first language Turkish as a resource – A German case study. In M. Setati, T. Nkambule, & L. Goosen (Eds.), Proceedings of ICMI Study 21 conference: Mathematics and language diversity (pp. 225–234). São Paulo, Brazil: ICMI.Google Scholar
 Meyer, M., Prediger, S., César, M., & Norén, E. (2016). Making use of multiple (nonshared) first languages: State of and need for research and development in the European language context. In R. Barwell, P. Clarkson, A. Halai et al. (Eds.). Mathematics education and language diversity. ICMI Study 21 (pp. 4766). Cham, Switzerland: Springer.Google Scholar
 Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11–19.Google Scholar
 Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning, 4(2–3), 189–212.Google Scholar
 Moschkovich, J. (2013). Principles and guidelines for equitable mathematics teaching practices and materials for English language learners. Journal of Urban Mathematics Education, 6(1), 45–57.Google Scholar
 Norén, E. (2015). Agency and positioning in a multilingual mathematics classroom. Educational Studies in Mathematics, 89(2), 167–184.Google Scholar
 Organisation for Economic Cooperation and Development (2007). Science competencies for tomorrow's world (PISA 2006) (Vol. 2). Paris, France: OECD.Google Scholar
 Paulus, C. (2009). Die Bücheraufgabe zur Bestimmung des kulturellen Kapitals bei Grundschülern [The booktask for determining the cultural capital of elementary school students]. Retrieved fromhttp://hdl.handle.net/20.500.11780/3344.
 Planas, N. (2014). One speaker, two languages: Learning opportunities in the mathematics classroom. Educational Studies in Mathematics, 87(1), 51–66.Google Scholar
 Planas, N., & Setati, M. (2009). Bilingual students using their languages in the learning of mathematics. Mathematics Education Research Journal, 21(3), 36–59.Google Scholar
 Prediger, S., & Wessel, L. (2013). Fostering German language learners’ constructions of meanings for fractions: Design and effects of a language and mathematicsintegrated intervention. Mathematics Education Research Journal, 25(3), 435–456.Google Scholar
 Prediger, S., Clarkson, P., & Boses, A. (2016). Purposefully relating multilingual registers: building theory and teaching strategies for bilingual learners based on an integration of three traditions. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J. Moschkovich, N. Planas, M. SetatiPhakeng, P. Valero, & M. Villavicencio Ubillús (Eds.), Mathematics education and language diversity. New ICMI Study Series (pp. 193215). Cham, Switzerland: Springer International Publishing. https://doi.org/10.1007/9783319145112
 Reljić, G., Ferring, D., & Martin, R. (2015). A metaanalysis on the effectiveness of bilingual programs in Europe. Review of Educational Research, 85(1), 92–128.Google Scholar
 Rossell, C.H. & Kuder, J. (2005). Metamurkey: A rebuttal to recent metaanalysis of bilingual education. In Arbeitsstelle interkulturelle Konflikte und gesellschaftliche Integration (AKI), The effectiveness of bilingual school programs for immigrant children (pp. 43–76). Berlin, Germany: Center for Social Scienes.Google Scholar
 Schleppegrell, M. J. (2004). The language of schooling. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
 Secada, W. G. (1992). Race, ethnicity, social class, language and achievement in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 623–660). New York, NY: Macmillan.Google Scholar
 Setati, M. (2005). Power and access in multilingual mathematics classrooms. In M. Goos, C. Kanes, & R. Brown (Eds.), Proceedings of the fourth international mathematics education and society conference (pp. 7–18). Gold Coast, Australia: Griffith University.Google Scholar
 Setati, M., & Adler, J. (2001). Between languages and discourses: Language practices in primary multilingual mathematics classrooms in South Africa. Educational Studies in Mathematics, 43(3), 243–269.Google Scholar
 Setati, M., Molefe, T., & Langa, M. (2008). Using language as a transparent resource in the teaching and learning of mathematics in a grade 11 multilingual classroom. Pythagoras, 67, 14–25.Google Scholar
 Slavin, R. E., & Cheung, A. (2005). A synthesis of research on language of reading instruction for English language learners. Review of Educational Research, 75(2), 247–284.Google Scholar
 Snow, C. E., & Uccelli, P. (2009). The challenge of academic language. In D. R. Olson & N. Torrance (Eds.), The Cambridge handbook of literacy (pp. 112–133). Cambridge, England: Cambridge University Press.Google Scholar
 Swain, M. (1985). Communicative competence: Some roles of comprehensible output in its development. In S. Gass & C. Madden (Eds.), Input in second language acquisition (pp. 235–256). Rowley, MA: Newbury.Google Scholar
 Turner, E. E., Domínguez, H., Empson, S., & Maldonado, L. A. (2013). Latino/a bilinguals and their teachers developing a shared communicative space. Educational Studies in Mathematics, 84(3), 349–370.Google Scholar
 Wagner, J., Kuzu, T., Redder, A. & Prediger, S. (2017). Vernetzung von Sprachen und Darstellungen in einer mehrsprachigen Matheförderung – linguistische und mathematikdidaktische Fallanalysen [Relating languages and representations in a multilingual mathematics intervention – case analyses]. Zeitschrift Fachsprache (in press).Google Scholar
 Wilhelm, O., Schroeders, U., & Schipolowski, S. (2014). Berliner Test zur Erfassung fluider und kristalliner Intelligenz (BEFKI 8–10) [Berlin test for determining fluid and crystallized intelligence (BEFKI 8–10)]. Göttingen, Germany: Hogrefe.Google Scholar
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