An empirical model of mixed-age teaching

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An empirical model of mixed-age teaching Robbert Smit *, Eva Engeli University of Teacher Education St. Gallen, Mu¨ller-Friedbergstr. 34, 9400 Rorschach, Switzerland

A R T I C L E I N F O

A B S T R A C T

Article history: Received 19 October 2014 Received in revised form 12 April 2015 Accepted 21 May 2015 Available online xxx

Many research studies analysing the effects of multi-graded classes have stated the need for a description of mixed-age teaching for pedagogical reasons. We tried to fill this gap by presenting a theoretical concept transferred onto a measurement model with seven typical elements of mixed-age teaching practices as a basis for future research. Within this structural-equation model, we were also able to trace the relationships between these practices. The empirical data derives from a teacher survey (N = 280) conducted as part of the Swiss-Austrian project, ‘‘Schools in Alpine Regions 2’’. ß 2015 Elsevier Ltd. All rights reserved.

Keywords: Mixed-age teaching Multi-grade schools Rural schools

1. Introduction Multi-grade classes (also called mixed-grade or composite classes) are a common phenomenon in schools worldwide (Hattie, 2002; Kalaoja & Pietarinen, 2009; Little, 2001). While some research has been conducted about the effects of multi-grade classes (Lindstro¨m & Lindahl, 2011; Sims, 2008; Veenman, 1995) or the context of such classes, often found in small and rural schools (Hargreaves, Kvalsund, & Galton, 2009; Hattie, 2002), empirical studies of teaching in multi-grade classes from a pedagogical point of view are rare (Little, 2001). Veenman (1995) suggested that future research should focus more on the processes by which learning effects are accomplished in multi-grade classes and not just address the organizational aspects; the quality of instruction is the key factor for successful learning. Most studies in Veenman’s often-cited synthesis, however, provide no information whatsoever about the micro-level practices employed in the classroom. The same conclusion of poor information on teaching practices was also reached by a German research group analysing the effects of multi-graded schools (Kuhl, Felbrich, Richter, Stanat, & Pant, 2013). In a related study, Hallam, Ireson, and Davies (2004) reported no consistent effects of ability-grouped or single-grade classes on achievement compared with mixed-ability groups within classes. They and others, like Quail and Smyth (2014) and Wilkinson and Hamilton (2003), concluded that effects are much more likely to be attributable to the quality of teaching than to the composition of classes. Teaching that targets learning across ages with the same instructional material for pedagogical reasons in multi-grade classes is called mixed- or multi-age teaching (Mason & Burns, 1997). This is different from multi-graded classes where the teacher splits his time between the different grades in the class and where only little learning activities are shared across the grades (Cornish, 2006a). Mixed-age teaching as a set of pedagogical practices is based on a more integrated curriculum, not one curriculum for each age group (Hoffman, 2003). Constructivist perspectives on learning and instruction emphasize independent learning, meta-cognition, ‘‘active learning’’ and learning from experts (cognitive

* Corresponding author. Tel.: +41 [7_TD$IF]71 [8_TD$IF]858 [9_TD$IF]71 [10_TD$IF]36. E-mail address: [email protected] (R. Smit). http://dx.doi.org/10.1016/j.ijer.2015.05.004 0883-0355/ß 2015 Elsevier Ltd. All rights reserved.

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apprenticeship). This goes hand-in-hand with the more student-centred approaches of mixed-age teaching, where instruction has to be more differentiated due to the age-related heterogeneity. From the literature (Hargreaves, 2001; Hoffman, 2002, 2003; Little, 2007; Stone, 1998) seven central elements of mixed-age teaching can be derived: (1) the role of the teacher as a facilitator; (2) differentiated and individual learning; (3) cooperative learning; (4) flexible and multiage grouping, where elder pupils may become tutors of the younger ones; (5) common learning topics for different levels; (6) open-ended, problem-oriented learning tasks; and (7) formative assessment. In this article, we present a measurement model of mixed-age teaching based on these seven suggested elements. This model could be a starting point for discussions of the advantages and difficulties of mixed-age teaching, and it could help to avoid simplistic comparisons of grouping concepts that differ only in organization. In addition, the model might lead teams in small – and other – schools to discuss the features and evaluate how these are implemented in their own school for reasons of professional development.

2. The concept of mixed-age teaching 2.1. Research on multi-grade classes Although multi-grade classes are seen by many as inferior to single-grade classes, student achievements and social skills are comparable in both types of classes (Gutie´rrez & Slavin, 1992; Hattie, 2002; Kuhl et al., 2013; Pratt, 1986; Quail & Smyth, 2014; Veenman, 1995). In their research, Ong, Allison, and Haladyna (2000) reported better results for grade 3 students in multi-grade classrooms for reading, writing and mathematics than for those in singlegrade classes. This favourable result is, however, limited to the group of better students and non-immigrants. It has to be mentioned that there are studies with negative results as well, e.g., a large study in Sweden, where approximately one-fourth of all children attend multi-grade classrooms during grades 4–6, by Lindstro¨m and Lindahl (2011). The authors did not distinguish between classes with mixed-age practices and those without special pedagogical foci. In Norway, a neighbour of Sweden, positive results for multi-graded schools have been reported (Leuven & Rønning, 2011). In a study by McClellan and Kinsey (1999), children in a mixed-age learning context showed less aggressive and more prosocial behaviour than their counterparts in same-age classrooms. Teachers in a small Catholic school, which is in transition from multi-grade to mixed-age teaching, noted more collaboration, shared responsibility in the class and at home and less disciplinary incidents (Proehl, Douglas, Elias, Johnson, & Westsmith, 2013). Parents are rather critical of mixed-age teaching (Cornish, 2006b), especially if they are ambitious ([1_TD$IF]Smit & Humpert, 2012a) or if their child is a high achiever (Anderson, & Pavan, 1993). They fear cross-age learning groups are less beneficial for their children’s learning because the teacher might have less time for each child. Parents are often not well informed about the intentions of the programme (Aina, 2001). If a school encourages parents’ participation, support for mixed-age teaching rises ([1_TD$IF]Smit & Humpert, 2012a). For example, at the Reggio Emilia schools, parents are seen as vital components of its philosophy; the parents even volunteer in the classrooms and are involved in aspects of the curriculum (Finegan, 2001). 2.2. Mixed-age teaching strategies While the principals for good teaching are the same as for age-grade classes, mixed-age classes contain a higher agerelated heterogeneity, which a teacher needs to consider when planning lessons (Miller, 1991). If we look at the excerpt from our interviews (below), it is obvious that a teacher in a mixed-age class is challenged by additional planning tasks compared to the teachers of single-grade classes. Let’s take as an example the topic ‘‘mountains’’. We talked about different vegetation zones and what animals live in each zone. One child in grade 3 was grouped with two children from grade 2; then they had to gather information about a typical animal of a zone, with the help of the Internet or an animal encyclopaedia. Then, they had to work together in groups and the child in grade 3 was the leader, quasi. (SG1 teacher) Teachers have to think how a certain topic can be used as a learning resource to achieve goals for different grade levels (Aina, 2001). They think of planning a learning environment instead of planning a lesson (Stone, 1996). A decision has to be made about how the children will work, either in homogeneous groups, heterogeneous groups, with peers or individually. Another question is how the support of the learning process will be organized. Teachers in a mixed-age setting are less often direct lecturers and more often facilitators, who guide students’ independent efforts to acquire and construct knowledge (Benveniste & McEwan, 2000). Therefore, instruction is more individualized (Miller, 1991). In the example above, older students have the function of tutors. These tutors relieve the teacher from having to be omnipresent in the classroom and needing to coach each group or student at the same time. It becomes clear that mixed-age teaching contains a variety of practices that are typical for such learning environments. In the following section, we will describe which practices of mixedage teaching can be found in other studies. The aim is to present a conglomeration of mixed-age practices that could be used as a theoretical underpinning of the construct ‘‘mixed-age teaching’’ and can serve as a basis for a measurement model. An overview will be presented in Table 1.

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Table 1 An overview of the elements of mixed-age teaching. Element

Description

Sources (selection)

The teacher’s role

Teacher is a facilitator and a coach, in addition to being an instructor Learners in each grade group engage in learning tasks appropriate to their level of learning Supportive classroom climate; students help each other and collaborate flexibly Learning is flexibly organized in the whole class and includes teacher-led groups, individuals within groups, collaborative groups and individuals The same general topic/theme in the same subject is covered for all learners The learning tasks are more open-ended, explorative and problem-oriented To observe and diagnose how a learner is learning and is intended to improve teaching and learning

Hoffman (2003), Miller (1991)

Differentiated instruction Socially collaborative classroom Flexible grouping

General learning topic The quality of the learning tasks Formative assessment

Hoffman (2003), Little (2007) Hoffman (2003), Stone (1998) Hoffman (2002)

Little (2007), Stone (1998) Stone (1998), Benveniste and McEwan (2000) Little (2007), Hargreaves (2001), Kappler and Roellke (2002)

2.2.1. Tasks for common topics Tasks in mixed-age learning settings should be chosen so that students of different ages can participate in activities that require different abilities within the same topic or theme (Broome, 2009; Hoffman, 2002). A more able student in the classroom can model a skill and perhaps provide scaffolded support to a less able student. This is what might be expected to happen from a Vygotskyan perspective, but is also related to Bruner’s idea of scaffolding (Wood, Bruner, & Ross, 1976). In 1960, Jerome Bruner described the idea of a spiral curriculum in which there is an iterative revisiting of topics, subjects or themes throughout a course or, in case of the mixed-age classroom, throughout the grades (Harden, 1999). In planning preparation, a teacher can look at the scope and the sequence of the subjects for all grades. The similarities and the topics that require sequential knowledge can be pulled out, and the differentiated tasks for the students can be defined (Aina, 2001). Often, the tasks are more open. Whole-class instruction should revolve around open-task activities if all students are to be engaged (Miller, 1991). For example, a teacher can introduce a writing assignment through topic development in which all students brainstorm for ideas. In this context, students from different grades can discuss and share their different perspectives. Some of the tasks require more time and take the form of small projects. Another way to plan a mixed-age learning environment with open-ended tasks is to use an ‘‘integrated curriculum’’ (Stone, 1996); the teacher selects a theme or topic, e.g. ‘‘animals in the forest’’, to focus on for a week, several weeks or even longer. Next, goals in different subjects or for different contents are chosen and aligned with the central theme. In Language, Arts and Science it will be easy to find stories and learning tasks for that topic. Even in Mathematics, figures, charts and statistical information about animals in the forest can be the start for several mathematical activities. Finegan (2001) reports the alternative schools in the Italian Reggio Emilia use progressive teaching methods, such as project work, within their mixed-aged learning groups. These projects are often teacher initiated, based on the teachers’ observations of the childrens’ interests, and allow mixed-age groups to explore a certain topic. There is a theoretical connection to the work of John Dewey and his project method (Dewey, [12_TD$IF]1938), a method often used in open education classrooms. Alternative or open education is not generally beneficial for students’ learning, and it is dependent on the quality of the instruction. Not every teacher in a multi-grade classroom using progressive teaching methods attains high student achievement or positive attitudes (Giaconia & Hedges, 1982). However, teachers who emphasized the role of the child in learning, use formative (rather than norm-referenced) evaluations, individualized instruction, and the presence of manipulative materials, generated higher student outcomes. While the reader will encounter the first three features in our description of[2_TD$IF] mixed-age teaching in the following sections, the last one, manipulative materials, comprises, amongst other items, materials for exploration and discovery, real-world materials or those which provide a rich learning environment. Such material is optimal for working on open-ended tasks. Multi-grade schools are often found in rural areas of developing countries (Miller, 1991). One of the reasons is that they are less cost intensive than single-grade schools because fewer teachers need to be employed. The Escuela Nueva programme in Columbia (Benveniste & McEwan, 2000) was introduced to raise the quality of educational practices in multi-grade schools. The programme consists of the following elements of mixed-age classroom practices: (1) individual problem solving by students, (2) exploration of spaces other than the classroom, (3) work with various objects and (4) group work. The first three elements indicate that in mixed-age learning environments differ and more open types of tasks seem to be preferred. The fourth underlines the importance of working on tasks collaboratively. This is in line with socio-constructivist learning theories (Vygotsky, 1978) in which the creation of knowledge is seen as a process that happens between learners in a social context.

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2.2.2. Differentiated and individualized instruction The expected standards of performance (in terms of outcomes) in the core areas of the curriculum are clearly defined, so that the subject content to be reached by the end of a designated period (e.g., from one to up to three years) is well known (Anderson, & Pavan, 1993). However, the time and path taken to reach that end is allowed to vary for the students; the diversity of the multi-grade group makes it impossible to view children as the same. Therefore, children must have access to varied and open-ended experiences that allow for differences in their learning rates and styles (Stone, 1996). Children can work in centres on different topics and subjects. There might be a reading centre, where children read or listen to stories, or a mathematics centre, where they can work with mirrors and explore axial reflections. These centres can easily be integrated into the current thematic unit. To differentiate their lessons, many teachers in mixed-age classes use progressive teaching methods, such as Jenaplan work ([1_TD$IF]Smit & Humpert, 2012b). Dutchwoman Suus Freudenthal, who introduced Jenaplan education to the Netherlands in 1962 (Braster, 2014), was further inspired by the work of two American pedagogues, John Goodlad and Robert H. Anderson (see Anderson, & Pavan, 1993). She used some elements of their research on non-graded elementary schools. Teachers in German-speaking countries often use weekly plans for the students, which show the subject areas and assignments that are due at the end of the week, including non-obligatory tasks for additional training or those that are especially challenging. These plans can be tailored for individual students or tiered for different achievement groups. Some teachers use the strategies of ‘‘planning up’’ and ‘‘planning down’’, a planning strategy that works especially well for two-grade classrooms (Berry & Little, 2007). ‘‘Planning up’’ means that teachers use the objectives for the older children and extend them downward with differentiated tasks. Planning down is the opposite. 2.2.3. Flexible group work Typical elements of mixed-age teaching mentioned by many authors are group work (Benveniste & McEwan, 2000; Kappler & Roellke, 2002) and peer learning. Hoffman (2002) writes that teachers in mixed-age classes often plan peer learning in collaborative groups. These groups are teacher or student led. They can change in configuration during the school day: the three most frequent are common interest groups, shared task groups and dyads. A productive group size is between 2–5 children (Gillies & Boyle, 2006). Anderson and Pavan (1993) indicate that sometimes even larger groups, between 12–15 pupils, might form a group when discussions, decision-making activities or planning events take place. Larger groups, however, are not feasible; ordinarily, it is too difficult for all 25 or so children to make a contribution to the discussion. Peer learning not only takes place in structured, purposely planned instruction but also in less-structured situations that occur in the classroom every day as students are flexibly grouped for instruction. However, structured collaborative situations seem to be more beneficial for learning than unstructured (Gillies & Ashman, 1998; Roberts & Eady, 2011). 2.2.4. Social collaborative classroom Collaborative learning settings require social competencies, but can also foster their development. As mentioned within the section about learning tasks, open-ended activities allow for fruitful discussions about how to move forward with the assignment and how to fairly divide the tasks. Children working in collaborative activities increase their competencies for self-regulated learning (Grau & Whitebread, 2012). To facilitate this development, teachers should instruct and model highquality group work. An advantage of multi-age classes is that the younger children learn appropriate social behaviours from older children, who are already familiar with the routines in class. Younger children acquire behaviours by observing and then imitating their social models (Bandura, 1977). Before and during collaborative group work, the students and teacher talk about the social skills needed to achieve the learning goals (Hoffman, 2002; Jordan & Me´tais, 1997). Students in mixedage groups need to learn how helping in collaborative situations can be effective; receiving only the answer in response to an error or a request for an explanation or receiving no reply to a request for information are negatively related to achievement (Webb, 1989). There are some tools, such as ‘‘reciprocal teaching’’ or ‘‘CTIR’’, that can structure peer and collaborative learning situations and help students acquire strategies for interacting in learning dialogues (Palincsar & Herrenkohl, 2002). 2.2.5. The teachers’ role As the students’ work is often self-directed or self-regulated and tasks are accomplished collaboratively, the teacher’s role shifts from direct instruction of procedures, concepts and skills in front of the whole class towards using more time to support the students’ individual learning processes (Miller, 1991). Nevertheless, teacher-led instructional sequences do not disappear, but become less dominant (Hoffman, 2002). When direct instruction is needed, working in one-on-one or small groups is a sensible venue. To understand students’ learning styles and unique personalities well, the teacher needs to observe many instances of students working together to accomplish tasks. In collaborative situations, the teacher monitors students’ interactions closely to make sure that the interactions are not only positive in nature but that the same students are not always taking on the role of the ‘‘teacher’’ (Hoffman, 2002). To support positive interaction, teachers provide direct instruction on how to ‘‘work together’’ before and during monitoring. 2.2.6. Observation and assessment Hargreaves (2001) suggests that mixed-age learning settings lend themselves to formative assessment or assessment for learning because they encourage the teachers to recognize individual differences in learning. Formative assessment is most

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valuable when it assists the teacher and the learner in the diagnosis of learning difficulties. One unique feature of the teaching philosophy of the Reggio Emilia teachers is the use of documentation of children’s experience as a standard classroom practice (Finegan, 2001). Teachers share the responsibility of observing, taking notes and recording conversations between children systematically. They use journals, transcriptions, tape recordings, photographs and representations of the children’s thinking and learning to document their learning progress. Because teaching is more individualized in mixed-age settings, teachers often find it necessary to change to more flexible assessment practices, based on individual student performance rather than performance in comparison to other children (Kappler & Roellke, 2002). Portfolios and narrative reports are helpful for documenting and describing the child’s individual progress. Assessment should also help the teacher learn each child’s level and where she or he needs to go next (Stone, 1996). Knowing this, the teacher can adjust and plan the next instructional steps. To learn to regulate their own learning, children conduct self-evaluations and reflect upon their learning successes and difficulties. While checklists equip a child with evaluation criteria, logbooks can be used by students to note reflections and observations. Individual responsibility for learning may be encouraged by teachers offering continuous checks for self-assessment of the child’s progress (Guskey, 1990). Of course, peer-feedback can be an assessment resource as well as a learning resource. The teacher can help to initiate a classroom culture of learning-enhancing feedback, from pupil to teacher or between pupils (Hargreaves, 2001). Specific training for teachers and students in giving and receiving of feedback will be necessary for this to be beneficial for learning (Webb, 1989). According to Hattie and Timperley (2007) feedback should be clearly directed to the task, processes and regulation, but not to the character level. A classroom climate that fosters peer and self-assessment and allows for learning from mistakes is also important. 2.3. Problems The practice of a high-quality mixed-age teaching has several obstacles. Miller (1991) reports on an UNESCO-conference on multi-grade instruction, during which five general problem areas emerged: (1) inadequately trained teachers, (2) scarcity of varied levels and types of materials, (3) lack of flexible and special types of curriculum organized for multi-grade classes, (4) inadequate school facilities and (5) lack of incentives for teachers of multiple classes. For the topic of this article, points 1 through 3 are of special concern. Suggestions to address points 2 and 3 are presented within the sections of mixed-age practices above. Point 1, teacher training, occurs as a main problem for raising the quality of teaching in multi-graded classes. In Iceland, teachers in small schools mainly prepare their lessons based on the individual needs of the students (Aðalsteinsdo´ttir, 2008). Instruction is done on a one-to-one interaction basis. The students are mostly assigned individual tasks, however, these tasks are generally intended for a specific age group. Group work is used rarely and the same is true for differentiated instruction. Teachers’ lack of skills or training in such teaching methods may explain this finding. Aðalsteinsdo´ttir (2008) especially criticizes a lack of co-operative methods because these can include peer support and joint learning experiences. That teacher training in mixed-age practices is essential for a high quality of instruction in multi-grade classes is also an issue in the Columbian programme Escuela Nueva; not all of the teachers who took part in the programme used the four practices mentioned in section ‘‘Tasks for common topics’’ to the same extent; those who underwent a teacher training were more likely to do so (Benveniste & McEwan, 2000). This starts with pre-service teacher training that is primarily designed for teaching in single-graded classrooms (Miller, 1991) and ends with in-service teaching that can be organized inside the school, as well as being a part of a more formalized training (Berry & Little, 2007; Moyer & Surbeck, 1992). Mulryan-Kyne (2007) defines the content of pre-service teacher training for mixed-age classes. Team teaching seems to be helpful for professional development inside the school (Hoffman, 2003). According to Broome (2009), staff development training for those involved in multi-age instruction is, however, generally not addressed enough in mixed-age literature. 2.4. Research questions The research questions for our study were as follows: (1) Is it possible to demonstrate the construct multi-age teaching empirically with the help of a structural equation model? and (2) How do the elements of mixed-age teaching relate to each other? Our hypothesis is that the elements derived from theory also relate to each other in the empirical model. 3. Method 3.1. Sample Our sample (N = 280) consisted of teachers and school principals from primary schools with multi-grade classes in three participating regions of eastern Switzerland and western Austria. The principals of all eligible schools were invited per email to participate. Participation was voluntary and we achieved a response rate between 30 and 50% (depending on the region) of the targeted schools. Most principals in the small schools were teachers who also acted as a principal. Of the participants, 73% were female and the mean age was 42. The length of service was between one-half year and 43 years, with a mean of 18 years. While half of the participating schools had less than 50 students, approximately 10% had more than 100 students. Although these 10% were not the typically small, rural schools, they were integrated in the study due to their pedagogical mixed-age orientation.

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3.2. Measurements Based on an input–output model (Scheerens, 2000), with the elements of multi-age teaching as process indicators, we measured teacher beliefs, school and team resources as interacting variables in mixed-age teaching. All participants filled out either an online or a paper and pencil questionnaire in March 2013. To gain further insight we also conducted interviews with a part of the sample. From a methodological point of view our mixed-method research design has a sequential timing (Creswell, & Plano Clark, 1993). 3.2.1. Teacher questionnaire A questionnaire for teachers with approximately 200 open and closed items was administered. Some of the items derived from the previous project, ‘‘Schools in Alpine Regions 1’’ ([1_TD$IF]Smit & Humpert, 2012b). The teachers functioning as school principals were asked to answer an additional 40 items. A large range of topics was covered, including school context, working conditions, school leadership, school teams, teacher’s attitudes and instruction practices. For the closed items, we mostly employed a 6-point Likert scale (6 = absolutely agree, 5 = agree, 4 = somewhat agree, 3 = somewhat disagree, 2 = disagree, and 1 = absolutely disagree); however, for frequencies of practices, a 5-point scale was used (5 = almost always, 4 = often, 3 = sometimes, 2 = seldom, and 1 = never). Some of the survey items were summarized in scales. For all 35 scales, a reliability index (Cronbach’s alpha) was computed. Scales with Cronbach’s alpha values below 0.7 were eliminated. In Table 2, the means and standard deviation for the scales are presented. They are consistent with the elements of mixed-age teaching derived from theory as shown in Fig. 1. 3.2.2. Interviews In all three regions, guided interviews with 31 school principals and one or two teachers in each school were conducted. This included a visit of the teacher’s class. Thus, a total of 75 interviews focusing the theme of teaching and learning in mixedage surroundings took place. For the large part, interviewees had also participated in the written survey. 3.3. Analysis To test the relationships of the latent variables, structural equation models (SEMs) using the software M-Plus 7 (Muthe´n & Muthe´n, 2012) were constructed. As a combination of factor analysis and path (respectively regression) analysis, the SEM is appropriate for testing proposed theoretical models with latent variables. The relations between indicators and latent variables and the relations between latent variables can be evaluated in a single model. The transcribed texts from the interviews were categorized in accordance with qualitative content analysis (Mayring, 2000). The software MAXQDA (VERBI, 2012) was employed for the coding. It has to remarked that in this study the interview transcripts only serve as an illustration. 4. Results Approximately 40% of the teachers in our sample worked with two grades, while each 25% had three or four grades in their classroom. A small group (5%) had five grades, and another 5% were teachers working in mixed-age surroundings, but not responsible for their own class. Only 12% stated that they never or seldom initiated mixed-age learning on a shared

Table 2 Pearson correlations of the scales. Scale

M

SD

1

1. Teacher as a facilitator

5.06

0.63

1

2

2. Individualized learning

3.72

0.86

0.30** **

3

4

5

6

1

3. Social competence

4.55

0.62

0.24

0.22**

1

4. Flexible grouping

3.37

0.65

0.28**

0.30**

0.13*

1

5. General learning topic

3.67

0.74

0.17**

0.19**

0.23**

0.22**

0.60

**

**

**

**

0.22**

**

**

6. Task quality 7. Formative assessment

3.79 3.16

0.86

0.33

**

0.37

0.18

**

0.51

0.24

**

0.26

0.22 0.21

1

0.17

1 0.23**

Example item It is important to me to listen to the student explaining his approach. How often do you offer individual learning material? How competent is your class in terms of assisting other students’ learning? How often do you let students lead small learning groups? How often do you use the same lesson topic for every grade? How often do you use tasks for creative problem solving? How often do you ask students for a demonstration of their level of learning?

Note: N = 280; ** p < 0.01; * p < 0.05; 1 & 2; Likert-scale 1–6 and 3–8: Likert scale 1–5 (highest value more).

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Fig. 1. Structural equation model of mixed-age teaching, n = 263. All of the paths are significant,[5_TD$IF] except for n.s., p < [6_TD$IF]0.01, and all of the effects are standardized.

theme; 24% used mixed-age learning settings occasionally. The remaining teachers who choose ‘‘often’’ or ‘‘always’’ created mixed-age learning environments. Not surprisingly, the subjects where mixed-age learning across ages took place most often were the arts, music and science. Language (German) and mathematics took place less often, a result that Bennett et al. (1983) also found in English schools and Mulryan-Kyne (2005) found in Irish schools. In the interviews, some of the teachers stated that these two subjects were less often used in mixed-age leaning environments because the progressive content, the curriculum and the schoolbooks were difficult to adapt for mixed-age teaching. During our visits in the schools, however, we found interesting didactical practices for the two subjects being used in some classrooms, which indicated that this may be merely a problem of ‘‘not knowing how’’ and, thus, should be a goal of teacher training. We found rather high agreement for most of the items that asked about teaching practices (see means in Table 2), which indicates high validity because the practices seem to be well known and applied in the classroom. Except for one scale (differentiated instruction), we were able to construct the suggested scales for mixed-age teaching (Table 1). Because the items of the scale ‘‘differentiated instruction’’ did not fit to one factor, we used only two of its items for a new scale, ‘‘individualized learning’’. The correlations of the scales are in Table 2. We found medium-to-high correlations (r > 0.30) between ‘‘teacher being a facilitator’’ and ‘‘individual learning’’, and ‘‘task quality’’ and ‘‘formative assessment’’. There was also a correlation of r = 0.30 between ‘‘individual learning’’ and ‘‘flexible grouping’’. Rather unexpectedly, the correlations between ‘‘general learning topic’’, a key theme in mixed-age teaching, and the other scales were significant, but below medium strength. 4.1.1. The structural equation model To ascertain whether we could group our seven elements of mixed-age teaching (see Table 1) in one analytical model, we used SEM. To study the relationship of these seven variables, we used latent indicators, that is, the items grouped in seven topics. Our model (Fig. 1) consisted of a measurement portion for each of the seven latent variables and a structural equation model. For the measurement portion, each latent variable was based on a set of items (the measures) similar to factor analysis. In the structural model, the theoretical causal relationships among the latent variables were described by a set of general linear equations. Dependencies were represented in diagrams by one-headed arrows (paths) representing regression relationships. We expected that our model, in accordance with the literature, would contain several paths between the latent variables. Guidance for establishing the first paths within the models was found in the correlation matrix (Table 2); high

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correlations indicate relationships or paths. The directions of the paths were chosen based on theoretical considerations. While a first SEM-model consisted of all seven supposed elements of mixed-age teaching, we finally arrived at a model with six elements due to insufficient fit measures (see Fig. 1). We could not fit the variable ‘‘teacher as a facilitator’’ in the[3_TD$IF] model. In addition, only three items from the scale ‘‘formative assessment’’ could be included in the model. Items referring to the use of portfolios or individual grading had to be omitted. Finally, an eight latent variable (differentiation) was introduced, which combined individualization and observation. The final model had the following fit indices: N = 263, X2 = 99.56, df = 80, p = 0.07, CFI = 0.98, TLI = 0.97 and RMSEA = 0.03. Although this model has good fit indices, other models with similar fit indices probably exist. However, from a theoretical and practical point of view presented here, the model is absolutely meaningful. 4.1.2. Characteristics of the model The model suggests that when a teacher chooses a new learning topic, he or she decides if and how the topic can be used to enhance learning for all grades. For example, developing a writing task in a mixed-age primary class with grades one to three requires consideration of the children in the first grade, who are still in the process of learning to read and write, and the children in grades two and three, who could work on the same writing task at their own level. For grade one, the teacher might assign a different task. As one can see in the example from the interviews (below), small modifications allow for the use of the same assignments in a mixed-grade setting. Basically, the whole class works on the same topic. This is also true for mathematics, where all children can do the same calculations but those in grade 3 use numbers up to 1000 and those in grade 4 up to 1,000,000. (SG2 teacher) Other learning arrangements, such as the observation and identification of the animal life around a pond, are less dependent on specific basic skills. In our final model, however, we could not include a significant path from ‘‘differentiation’’ to ‘‘general learning topic’’. It appears that the realization of differentiated lessons depends on the availability of appropriate assignments for different grade levels. Therefore, the path from ‘‘differentiation’’ goes first to ‘‘task quality’’—quality in relation to the learning potential of a task for a heterogeneous class—and then to the ‘‘general learning topic’’. If teachers differentiate their lessons for the individual needs of students or groups, e.g., of the same grade, they must consider the individual achievement levels of the students. In the model, the second order variable ‘‘differentiation’’ consists of two variables, ‘‘individualized learning’’ and goal-oriented ‘‘targeted observation’’. Observations are useful as a source of information upon which the teacher builds his or her further lesson planning. This can include information on students who work well together in collaborative or peer situations. A teacher can also learn, while observing his or her students working together, who might have the potential for being an effective tutor for others in small group work. The model shows that individualized learning includes collaborative working. There is a path from ‘‘differentiation’’ in lessons to ‘‘flexible grouping’’. Furthermore, there is also a relationship between ‘‘flexible grouping’’ and ‘‘general learning topic’’. When teachers in mixed-age settings use the same topic for different achievement groups, they rely on their students to be either good colearners or having some advanced, mostly older, students who can help the teacher support the other students during the working process. Surprisingly, we could not integrate a significant path between ‘‘flexible learning groups’’ and ‘‘social competence’’ of the students. Working in flexible groups could have been a favourable environment for the development of social competence. Instead, there is a path from ‘‘differentiation’’ to ‘‘social competence’’. Teachers who differentiate their lessons more often state that their children have more social competence. Children can develop social competence in more self-regulated learning sequences, but social competence is also a prerequisite for such learning. Finally, there is a path from ‘‘general learning topic’’ to ‘‘social competence’’. When children are working in mixed-grade situations on the same topic, social competencies can be fostered, as in the following example from the interviews. A situation where it is very helpful, when younger and older children are working together, is when we are writing texts, . . . then sometimes an older child writes a text together with a younger child or they mutually check their texts, e.g., within a writing conference, they have a partner, who is reading the text and correcting it before I will go through it. (V1 teacher) Other such situations could involve solving mathematical problems, experimenting in science, preparing a musical or collecting information for a file about the ancient Romans. Not every lesson must or can be a mixed-age lesson. As mentioned earlier, sometimes a specific competency, like addition and subtraction of fractions, requires single-grade instruction, while other grades are still working on understanding the concept of fractions. In the following chapter, we discuss the differences between the theoretical assumptions and the empirical model.

5. Discussion The structural model presented allows for new studies in which the effects of good mixed-age teaching, not just multigrade vs. single-grade, can be tested. It might also engender discussion about the didactical features of mixed-age teaching

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and learning. For the theoretical discussion we also refer to the introductory article of this special issue. We could not construct a model with all of the assumed elements; the scale ‘‘teacher’s role’’ and part of the assessment scale had to be dropped. In addition, some of the expected paths could not be added, which may be because of measurement problems (e.g. dependency of sample, choice of categories and residuals) or to theoretical underpinnings as discussed above. Therefore, models in other studies might present slightly different paths and relationships, but such differences will only enhance the discussion of how the elements of mixed-age teaching interact with each other. While Hoffman (2003) provided research on mixed-age teaching practices based on case studies, we aimed at generalizing such practices. In addition, we conducted interviews and integrated excerpts, but a profound analysis has yet to be conducted. What is still missing, however, is research that looks at the outcomes of students in mixed-age classes with regard to the features identified in the empirical model. Berry and Little (2007) suggested that research should investigate how well students from mixed-age classes are performing in subsequent schools to evaluate mixed-age teaching. This would allow the identification of the most effective mixed-age practices and the most beneficial combinations. In contrast to Hoffman (2003) and others, we added task quality as another feature of mixed-age teaching and learning. Although we outlined this element of mixed-age teaching, more studies are needed to describe effective learning tasks for specific contents of the curriculum, similar to the one by Nu¨hrenbo¨rger and Steinbring (2009). Those two authors showed how students in different aged dyads effectively worked together on numerical tasks, which included interaction and reflection from different points of view for each student. 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