Mathematics Instruction in Combination and Single-Grade Classes: An Exploratory Investigation
by DeWayne A. Mason & Thomas L. Good - 1996
An examination of curriculum, instruction, and organizational formats in multi-grade and single-grade mathematics classes
Combination classes (also termed split or multigrade classes) are a form of classroom grouping that typically occurs when school enrollments are imbalanced or inadequate, resulting in teachers managing students from two or more grades for most or all of the school day. These expediently formed classes, embedded within a graded system of schooling, therefore differ significantly from multiage or nongraded classes, formed deliberately because of pedagogical or philosophical interests in team teaching, flexible grouping, individualized instruction, continuous progress curriculum, and the elimination of all vestiges of gradedness. This exploratory study compared the curriculum, instructional strategies, and organizational formats used by six combination class teachers for mathematics with those used by eighteen single-grade teachers (six who used traditional whole-class teaching and twelve who used two within-class ability groups). Results showed that the instruction, classroom organization, and curriculum content and materials of combination class teachers differed in significant ways from those of both traditional whole-class and within-class ability-grouped (two-group) single-grade teachers. Observers ratings and low-inference measures indicated that combination classes included fewer instances of peer cooperation, innovative curriculum, and individualized instruction. Furthermore, teacher-directed and independent-group variables (e.g., meaningful presentations, use of manipulatives, higher-level thinking emphasis) varied significantly among these three grouping formats.
Combination classes (often called split or multigrade classes) are a form of classroom grouping in which one teacher manages students from two or more grades for most or all of the school day, usually as a result of unbalanced or inadequate enrollments (Knight, 1938; Miller, 1989; Veenman, 1995). These classrooms, where teachers tend to maintain two grade-level groups as well as two distinctly different curricula and instructional presentations (Mason & Burns, in press; Veenman, 1995), appear fundamentally different from the single-grade classrooms commonly found in most schools as well as from the often heralded yet quite uncommon multiage or nongraded classrooms that some schools form as a result of philosophical interests in team teaching, continuous progress curriculum, individualized instruction, and so forth (Anderson & Pavan, 1993; Goodlad & Anderson, 1987; Mason & Stimson, 1996). Interestingly, some argue that combination classes are no different from single-grade classes (Veenman, 1995) or that they encourage innovative curriculum and instruction (e.g., Grant, 1993), while others view these classes as difficult assignments in which instructional time and curriculum coverage are diminished (e.g., Brown & Martin, 1989; Mason & Burns, 1995; Mason & Doepner, in press; Pratt & Treacy, 1986).
Combination classes have long been a significant form of classroom grouping, especially in rural and international schools (Miller, 1989; Muse, Smith, & Barker, 1987; Veenman, 1995). Recently, however, as declining enrollments, class-size policies, and advocacy for year-round and multiage programs have increased the number of these classes in many areas (e.g., California Department of Education, 1992; Galluzzo et al., 1990; Miller, 1989; Quinlan, George, & Emmett, 1987; Veenman, 1995), their efficacy has been questioned. Unfortunately, programmatic research on combination classes is nonexistent, and reviewers of the extant literature have come to different conclusions about how these classes affect instruction and student outcomes. Veenman (1995), for example, concluded that combination class processes and effects are similar to those of single-grade classes. In contrast, Mason and Burns (in press) concluded that combination classes lead to lower-quality instruction and negative effects. Thus, current research provides practitioners few conclusive insights about the effects of combination classes, or about how to strategically organize, instruct, and manage curriculum in this alternative organizational structure.
Numerous scholars have called for research on school and classroom organization (e.g., Bidwell & Dreeben, 1992; Brophy, 1986; Good, Mulryan, & McCaslin, 1992; Slavin, 1989b). Bidwell and Dreeban (1992) noted that relationships between school organization and curriculum are virtually unexplored research territory (p. 345), while Brophy (1986) argued that inquiries on school and classroom organization are needed in mathematics education. In addition, Good et al. (1992) reported a lack of process data on what happens during small-group instruction, while Slavin (1989b) asserted that researchers should clarify how grouping methods variously affect time, student incentives, instructional quality, and subsequent student outcomes. Although Gamoran (1987) noted that organizational structures do not directly affect student outcomes, Slavin (1987) and Gutiérrez and Slavin (1992) showed that certain structures are linked to student achievement. Mason and Burns (in press; also see Mason & Burns, 1994) argue that combination classes create important curriculum and instruction tradeoffstradeoffs that are likely to lead to negative effects for teaching and learning. Observational investigations confirming or rejecting this argument, however, are nearly nonexistent (Mason & Burns, in press; Miller, 1989; Veenman, 1995).
This study explores whether combination and single-grade classes differentially affect the teaching and learning of elementary school mathematics. Based on a review of the literature on combination classes (Mason & Burns, in press), observational research on two- and three-group instruction in a single-grade setting (e.g., Good, Grouws, Mason, Slavings, & Cramer, 1990; Mason & Good, 1993), and concerns from practitioners about the tradeoffs of combination classes in U.S. schools (e.g., Appalachia Educational Laboratory and Virginia Education Association [AEL/VEA], 1990; Brown & Martin, 1989; Galluzzo et al., 1990; Mason & Burns, 1995; Mason & Doepner, in press), we hypothesized that combination classes may diminish curriculum, active teaching and learning, and peer tutoring and teacher-directed remediation. To investigate these hypotheses, we reanalyzed data from a previous naturalistic study of 33 teachers who used small groups for mathematics instruction (Good, Grouws, Mason, et al., 1990), comparing the curriculum, instruction, and organizational formats used by 6 teachers of combination classes with those used by 18 teachers of single-grade classes6 single-grade teachers who predominately used traditional whole-class teaching and 12 who predominately used two within-class ability groups.
Combination classes, which have a long history in schooling (Muse et al., 1987), are a significant form of grouping, especially in rural areas and in other countries (Miller, 1989; Muse et al., 1987; Veenman, 1995). For example, although Mason and Stimsons (1996) survey found that only 5 percent of the K6 classes in 12 states were combination classes, Laukkanen (1978) found 29 percent of the primary students in Finland placed in such classrooms, a percentage similar to that in England (Her Majestys Inspectors of Schools, 1978) and British Columbia (Craig & McLellan, 1987). Furthermore, 53 percent of the elementary school classes in the Netherlands (Commissie Evaluatie Basisonderwijs, 1994) and 85 percent of the elementary schools in Western Australia use some type of combination class (Pratt & Treacy, 1986).
Despite this widespread use and numerous comparative studies, comprehensive research on combination classes is surprisingly sparse, and conclusions about their effects are controversial. Veenman (1995) reviewed the results of 45 combination class outcome studies, computing effects sizes for 34 achievement and 13 affective comparisons. He found no achievement differences (median effects size of .00) and small affective effects favoring combination classes (median effects size of +.10), concluding that parents, teachers, and administrators need not worry about the academic progress or socio-emotional adjustment of students in multigrade . . . classes. These classes are simply no worse, and simply no better, than single-grade . . . classes (p. 367). However, Mason and Burnss (in press) review of 44 combination class studies led them to assert that (a) most outcome studies contained selection biases (i.e., better students and teachers tend to be systematically assigned to combination classes) and (b) these biases, practitioners views, and observational research point to a conclusion that combination classes have at least a small negative effect on student achievement.
Mason and Burns (in press) report 12 studies on the views that educators have about combination classes. Although most of these inquiries are limited in scope, superficially analyzed, and lacking in sound research design, several potentially instructive findings have consistently emerged. For example, a large majority of teachers and principals report a preference for single grades because of the extra planning and effort, classroom management problems, and diminished curriculum and instruction ostensibly created by combination classes (e.g., AEL/VEA, 1990; Bennett, OHare, & Lee, 1983; Pratt & Treacy, 1986). In contrast, however, some educators assert that combination classes lead to increases in social growth, peer tutoring, independent learning skills, and individualized instruction (e.g., Chace, 1961; Hoen, 1972; Mason & Burns, 1995).
Mason and Burns (in press) also report 8 observational studies that have examined combination classes. Only one of these studies, however, was conducted in the United States (Foshay, 1948), and the majority (including Foshay) are narrow in scope, presenting limited evidence. Still, two major findings appear to have emerged from these studies: First, combination class teachers, unlike their single-grade and nongraded class counterparts, tend to teach two distinctly different curriculums, maintaining grade levels and delivering separate lessons to each grade-level group. Second, this division of the combination class teachers time between or among grade-level groups appears to lead to less direct instruction, less curriculum adaptation, more independent work, more waiting for teachers, slightly lower time on-task, and an overall more complex teaching and learning environment.
If combination teachers regularly form two groups, whether by grades or by ability levels, and use two distinctly different curricula, it seems plausible that curriculum and instruction would tend to be reduced in these classes. Assuming a robust and challenging curriculum at each grade level, it makes sense that teachers would have problems delivering two curricula in the amount of time normally set aside for one curriculum. In addition, corrective feedback and individual remediation may decline in combination classes as teachers press to present two lessons in the same amount of time. It may be that two-group formats also discourage teachers from using certain materials or activities (e.g., manipulatives, calculators, problem solving, cooperative learning projects) because of time constraints in using and organizing such materials or because of a perception that such activities would tend to distract independent work or instructional presentations with a second group of students working on different objectives. Indeed, researchers have identified important negative tradeoffs in teaching two groups in single-grade classes (e.g., Brophy, 1986; Good, Grouws, Mason, et al., 1990; Leinhardt & Bickel, 1989; Mason & Good, 1993; Slavin, 1989ab; Smith & Geoffrey, 1968), and Good et al. (1992) conclude that two- and three-group teaching (at least that which is based on ability grouping) leads to less than optimal learning environmentsespecially for students who are placed in low-achievement groups.
In sum, although there is good reason to suspect that important negative tradeoffs occur in combination classes, observational research on combination classes is sparse, mixed, and generally lacking a focus. Few studies have contrasted combination and single-grade classes, and comparisons between administratively expedient and more developmental1 multiage and nongraded approaches are lacking. Moreover, comprehensive observational research on combination classes in U.S. schools is nonexistent. Research on whether combination classes affect curriculum coverage or opportunity to learn, direct instruction, active teaching and learning, and corrective feedbackeach of which has been linked to teaching effectiveness or effective mathematics teaching (e.g., Brophy & Good, 1986; Cooley & Leinhardt, 1980; Good, Grouws, & Ebmeier, 1983; Rosenshine & Stevens, 1986)would be an important contribution to the literature on school and classroom organization.
PURPOSE AND QUESTIONS OF THE STUDY
Thus, in this study we sought to compare mathematics instruction in combination classes with that in two common types of single-grade classes: (1) traditional single-grade classes in which whole-class formats were generally used and (2) single-grade classes in which within-class ability-grouped formats (two-group approaches) were generally used. The study aimed to determine whether high-inference measures of teacher-directed variables (e.g., meaningful presentation, emphasis on higher-order thinking) and independent-group variables (e.g., time on-task, student cooperation) differed among these three types of classes. Furthermore, the study sought to examine whether the content (e.g., computation, problem solving) and nature of the curriculum (e.g., paper and pencil, activity-oriented assignments) during independent seatwork varied among these three organizational structures.
The study resulted from widespread practitioner interest in whether combination classes differ from traditional single-grade classesinterest that led to a collaborative school-university investigation of combination classes (Mason & Wilson, 1991)as well as the availability of data from a recent study of small-group teaching in mathematics (Good, Grouws, Mason, et al., 1990). This latter study found that teachers used six types of classroom formatswhole-class ad hoc, two groups, three groups, four or more groups (mixed groups), heterogeneous work groups (cooperative learning), and individualized groupingand that certain formats were linked to important instructional tradeoffs. Indeed, results suggested that two- and three-group teachers used some of the worst features of whole-class teaching and none of the strengths inherent in small-group teaching, as they (1) engaged in little lesson development (those portions of the lesson in which teachers set up or clarify student inquiries or directly explain and illustrate mathematics skills or concepts in order to make mathematics meaningful), (2) promoted lengthy independent student work, and (3) assigned seatwork tasks that emphasized drill rather than problem solving. These and other findings related to the negative tradeoffs of two-group teaching (e.g., Mason & Good, 1993) suggested important implications for the teaching of mathematics in combination classesa structure in which teachers predominately use two groups. Because 6 of the 18 two-group teachers in the small-group study sample taught combination classes and 6 of the 33 teachers predominately taught mathematics using a whole-class approach, we decided to investigate whether instruction in combination classes varied from traditional single-grade classes as well as from the remaining 12 two-group teachers who used a within-class ability-grouped approach.
Although qualitative and quantitative data were collected on numerous classroom dimensions, this investigation focused on four questions:
1. Do the organizational strategies used by combination teachers for mathematics instruction differ from those of traditional single-grade and within-class ability-grouped teachers?
2. Do teaching functions (e.g., development, controlled practice, checking seatwork) used by combination class teachers for mathematics instruction vary from those used by traditional single-grade and within-class ability-grouped teachers?
3. Do combination teachers curriculum and use of materials for mathematics differ from those of traditional single-grade and within-class ability-grouped teachers?
4. In comparison to traditional single-grade and within-class ability-grouped classes and for certain teaching goals, what tradeoffs appear to occur when teaching mathematics in combination classes?
The 1990 inquiry (Good, Grouws, Mason, et al., 1990), from which the data in this study were drawn, included 206 observations of 33 teachers in 21 schools in three midwestern districts: a small suburban district (city population about 40,000), a middle-sized district (city population about 130,000), and a large urban district (city population about 500,000). Since we were interested in comparing only three organizational formats in the present study, we included only 24 of the 33 teachers (6 combination teachers, 6 single-grade teachers who used within-class ability grouping, and 12 single-grade teachers who used whole-class teaching) and their corresponding 153 observations (4 to 8 visits were conducted with each teacher, with an overall average of 6.4 visits per teacher). These 24 teachers were drawn from 19 of the original 21 schools in the three districts.
All schools in the smaller district were included in the original Good, Grouws, Mason, et al. study (1990), while schools in the urban and suburban districts were randomly selected. Each district had participated in a previous survey (Good, Grouws, & Mason, 1990) aimed at identifying teachers who frequently used small-group teaching for mathematics. Subsequently, teachers who reported using small groups were solicited by project staff or district administrators for participation in the observational study. Three of the combination teachers taught in the middle-sized district, while three taught in the urban district. Of the single-grade teachers in the present study, 10 taught in the small suburban district, 5 taught in the middle-sized district, and 3 taught in the urban district.
Observations of entire mathematics lessons were conducted over a three-to four-month period during the middle of the school year. All but two of the classes were at the third-, fourth-, fifth-, and sixth-grade levels. Students in single-grade classes were assigned heterogeneously; students in combination classes were also quite heterogeneous, since most were from small schools without the flexibility for purposeful assignment (thus, combination classes resulted from low enrollments rather than philosophical considerations). Classes ranged from 21 to 28 students, and according to teachers, observers, and administrators, generally contained a normal distribution of student abilities and all socioeconomic levels.
The instrument used to collect teaching process data was developed in five phases: First, we observed and videotaped elementary school lessons involving the small-group teaching of mathematics. Second, we wrote narratives to describe classroom activities. Third, we reviewed these narratives and the videotapes to develop definitions and a tentative coding system. Fourth, to clarify definitions, procedures, and guidelines, we applied this system to additional videotapes and classroom observations of small-group lessons. During this phase, pilot reliability tests were conducted. Fifth, the instrument was refined slightly when it was applied to additional videotapes and classroom observations during the training of observers.
The instrument was developed to address the major questions of the original study. However, because its measures were conceived to examine variables across different purposes and organizational structures, the instrument was appropriate for answering questions in the present reanalysis. Low-inference measures were based on important variables from previous research on effective whole-class teaching. High-inference measures were based on variables cited frequently in the literature as key to effective small-group teaching and learning (e.g., cooperative learning, higher-level cognitive work, group interaction).
The instrument focused on four major areas: (1) format use, (2) content and nature of work assigned, (3) time spent in twelve teaching functions, and (4) high-inference ratings of six teacher-directed and six independent-group variables. Coders began format coding by recording the number of groups that each teacher formed in the class and the number of students in each group. In addition, all teaching function times were categorized according to the format used (e.g., whole-class, two groups, three groups or more). In measuring the content and nature of the work, coders documented specific lesson objectives; sample problems; activity descriptions; and textbook, workbook, and project assignments. Further, observers coded the type of assignment (e.g., computation, problem solving, concept development), whether they worked cooperatively or independently, and whether they worked on paper-and-pencil or activity-oriented tasks. Finally, coders documented the time that teachers directed students on estimation, calculators, problem solving, and computer work.
Using a coding form divided into 30-second intervals, coders took careful notes on what teachers did in their classes (e.g., teacher tells students to take out homework, teacher provides answers to section one, teacher asks students to provide answers to section two, etc.). They then analyzed these descriptive notes, categorizing the time spent according to the coding system definitions of 12 teaching functions: directions, checking homework, review, controlled practice over review material, development, controlled practice over newly developed material, supervising seatwork, checking seatwork, testing, transition, nonmathematics activities, and other activities (those not fitting into the previous 11 categories).
Finally, observers used a five-point scale (one = low, five = high) to rate each classroom group separately on six teacher-directed and six independent-group variables. The six teacher-directed variables were meaningful presentation, accomplishment, accountability, managerial routines, emphasis on higher-order thinking, and teacher use of manipulatives. The six independent-group variables were time on-task, group interaction, student cooperation, higher-cognitive student behavior, student use of manipulatives, and group self-management. Further details on the coding instrument and definitions of variables can be found in Good, Mason, and Grouws (1987).
In preparing for classroom coding, the seven observers, all of whom were certified teachers, participated in an extensive training program. This program included a review of coding definitions, instruction on coding procedures and guidelines, and practice coding using videotapes of small-group mathematics lessons. We asked coders to suspend any preconceptions about the forms that teaching should take and to recognize that effective teaching could occur in a variety of formats. Finally, coders were checked for reliability by using videotapes and observations of mathematics lessons in elementary school classrooms. These sessions continued until agreement on actual in-class lesson coding (two or more coders) exceeded 80 percent in all coding categories (time-spans, application of definitions, high-inference measures).
Table 1 categorizes the 153 lessons taught by the 24 teachers according to classroom type and macro formats used. Macro format refers to how a teacher predominately organizes students for mathematics. Generally, for example, a teacher might organize his or her class into a whole-class, two- group, or cooperative learning format. Macro formats may differ, however, from a teachers micro formathow they temporarily organize the class within a given mathematics period for specific instructional purposes. For example, although a teacher may use a whole-class format nearly every class period (macro format), he or she might, after providing direct instruction to the whole group during a particular lesson, decide to form two groups briefly one for remedial instruction and another for reinforcement or enrichmentbefore returning to a whole-class format later in the period. As can be seen in Table 1, combination class teachers used two groups for 37 of the 38 lessons. In contrast, traditional single-grade teachers used a whole-class approach for 32 of their 44 lessons, while single-grade teachers who formed within-class ability groups used two groups during 57 of their 71 lessons.
All macro format x type of classroom structure cells with 10 or fewer observations were dropped from this study, with the exception of an initial examination of how teachers varied in their use of whole-class and small-group teaching (Table 2). This decision seemed reasonable because we were most interested in examining how teaching was influenced by the type of classroom organizational structure, and including small numbers of alternative-structure lessons with the predominant formats would have confounded the analyses, increasing the probability of teacher and interaction effects. Consequently, for all but the Table 2 analysis, we included the 32 traditional single-grade lessons in which whole-class teaching was used and the 37 combination class and 57 within-class ability-grouped lessons in which two groups were used.
The data for this study were analyzed by using percentages, means, standard deviations, analysis of variance (ANOVA) procedures, and Scheffé tests. First, to examine how the sample of 24 teachers varied in their classroom organization, means and standard deviations were computed on the time each teacher spent in whole-class and small-group micro-format teaching and nonacademic functions across all whole-class and two-group lessons. Thus, the Table 2 analysis includes the one combination class lesson in which whole-class teaching was used, the 8 whole-class lessons taught by within-class ability-grouped teachers, and the 10 two-group lessons taught by traditional single-grade teachers who typically used whole-class methods, excluding only the 8 quite atypical other lessons (see Table 1).
Second, as appropriate, percentages or means were computed on the remediation format, wait time, interaction, and nature of content and materials variables (Tables 3 and 4). As noted above, these and the following analyses included only those lessons in which traditional single-grade teachers used a whole-class format and in which combination and within-class ability-grouped teachers used two groups. Third, means and standard deviations were computed by classroom type for the time spent in each of the 12 teaching functions (Table 5) and for the ratings of the 12 high-inference variables (Table 6). For these analyses, times and ratings were pooled across teachers as well as the groups used by both types of two-group teachers. Finally, ANOVA procedures and Scheffé tests were used to determine whether significant differences existed among classroom types on the 12 teaching functions and the 12 high-inference ratings. A minimum confidence level of p < .05 was used for these ANOVA and Scheffé tests.
Results are organized into four sections. First, a description of selected combination and single-grade teachers use of whole-class and small-group micro formats is presented to illustrate the variation among selected teachers. Second, measures are presented of combination and single-grade teachers use of curriculum and materials. Third, summing across teachers and small groups, we present a comparison of the time spent in the 12 teaching functions used in the three types of classes. Fourth, observers ratings of the 12 high-inference variables are compared by type of class.
DESCRIPTION OF TEACHING FORMATS
Macro and Micro Formats
As noted above, Table 1 shows that combination and single-grade teachers varied in their use of macro-format strategies during their mathematics lessons. Combination teachers used two groups during 97 percent of the observed lessons, organizing students by grade and using different textbooks, content, and instruction with each grade-level group. In contrast, teachers of single-grade classes more frequently used alternative grouping strategies. Within-class ability-grouped teachers, organizing their students by ability according to previous performance in mathematics, used alternative macro formats (whole class, other) during 20 percent of their observed lessons. They employed the same textbook but typically different content and instruction with each ability group, although a few teachers occasionally extended or differentiated the assignment after a common presentation to all students. By definition, of course, teachers of traditional single-grade classes used primarily whole-class instruction; however, similar to within-class ability-grouped teachers, they employed alternative grouping strategies 27 percent of the time, strategies such as two groups, cooperative learning, and three or more groups.
Combination and single-grade teachers also differed in their use of micro formats, although within-group variation was substantial. Table 2 presents the total time that teachers from each structure spent in whole-class and small-group formats as well as nonacademic functions. Table 2 also shows the time spent by selected teachers from each of the three classroom organizational structures (teachers were selected arbitrarily simply to illustrate the wide variation). Overall, whole-class teaching in combination classes (8.8 of 51.6 minutes = 17%) amounted to less than half of that in within-class ability-grouped classes (18.4 of 52.8 minutes = 35%) and less than one-fourth of that in traditional single-grade classes (37.3 of 47.8 = 78%). Further, teachers use of nonacademic functions (transition, nonmathematics activities, and other) varied greatly among the three classroom types, no doubt a reflection of their organizational structures: Traditional single-grade teachers spent about 8 percent of their lessons in these three functions (4 of 47.8 minutes), while within-class ability-grouped and combination class teachers spent about 12 percent (6.3 of 52.8 minutes) and 20 percent (10.1 of 51.6 minutes) of their lessons in these functions, respectively.
Table 2 clearly illustrates that teachers within each of the three groups varied widely in their micro-format use. For example, combination teacher 10 averaged over 17 minutes in whole-class teaching, while combination teacher 11 was not observed using such a format. Similarly, traditional single-grade teacher 3 averaged over 14 minutes of small-group teaching, while single-grade teacher 4 did not use a small-group format.
In addition to their usual small-group formats (e.g., two groups), teachers often form temporary ad hoc groups for reteaching or remedial instruction, strategies that require students to work independently and to wait for assistance when they have questions. Hence, we asked observers to document the predominant type of remediation that teachers provided students (e.g., individual, small groups, mixed approaches), to code a Yes or No on the question Are there times during the period when most students in this group are waiting for the teacher for directions or clarification? and to indicate whether students were allowed to interact during their independent seatwork (perhaps to assist one another in lieu of the teacher). Although these exploratory measures are not detailed, they provide broad estimates of these variables.
Table 3 presents a comparison of the remediation approaches used by the three types of teachers as well as the presence of wait time and student interaction during independent seatwork. As shown in Table 3, observers found that the lessons of combination and within-class ability-grouped teachers were more likely to exclude remediation than the lessons of traditional single-grade teachers. However, Table 3 also shows that within-class ability-grouped lessons tended to include fewer instances of waiting for directions or teacher assistance and less interaction among students during independent seatwork than those same aspects in combination and traditional single-grade lessons.
DESCRIPTION OF CONTENT AND MATERIALS
Table 4 presents the type of content and the nature of materials that predominated students independent-group assignments as well as the amount of teacher-directed time spent on estimation, problem solving, calculators, and computers. Table 4 shows that computation assignments tended to be more frequent in combination (lower grade = 68%, upper grade = 69%) and within-class ability-grouped classes (low group = 38%, high group = 55%), while concept development (40%) and problem-solving activities (30%) tended to be more frequent in traditional single-grade classes.
Focusing on the nature of materials during seatwork, Table 4 shows that independent paper-and-pencil activities were predominant with each of the three types of classroom structures. However, activity-oriented assignments (manipulatives, measuring, hands-on projects) were used more frequently in traditional single-grade classes (45%) than in combination (lower grade = 8%, upper grade = 3%) and within-class ability-grouped classes (lower ability = 18%, higher ability = 13%).
Table 4 shows that the three groups of teachers varied greatly in their use of estimation, problem solving, calculators, and computers during instruction, though they spent little time overall on these areas. However, combination teachers spent less time on problem-solving activities (M = .7) than both types of single-grade classes (within-class ability-grouped M = 3.6, traditional single-grade M = 3.7). Further, traditional single-grade teachers spent more time on computers (M = 1.4) than combination teachers (M = 0) and on estimation (M = 1.8) than combination (M = .5) and within-class ability-grouped teachers (M = .5).
USE OF TEACHING FUNCTIONS
Table 5 shows the time spent in 12 teaching functions by the three types of classes as well as F-values derived from ANOVA comparisons of these three organizational structures. As can be seen, significant differences were found among the groups on 9 of the 12 comparisons. Scheffé tests found that while combination teachers spent more time on controlled practice over review work (M = 3.9, F = 3.05, p < .05) than traditional single-grade teachers (M = .6), within-class ability-grouped teachers spent more time on directions or overviews (M = 8.1, F = 8.63, p < .001) than combination (M = 4.6) and traditional single-grade teachers (M = 5.6). Further, combination class teachers spent more time checking seatwork (M = 8.3, F = 8.24, p < .001) than traditional single-grade teachers (M = 1.8) and in other activities (M = 5.1, F = 5.27, p < .01) than within-class ability-grouped teachers (M = 1.8). Finally, combination teachers spent more time in transition (M = 4.1, F = 6.24, p < .01) than traditional single-grade teachers (M = 1.6), and within-class ability-grouped teachers spent more time in nonmathematics activities (M = 1.5, F = 4.84, p < .01) than traditional single-grade teachers (M = 0.1).
Table 6 also shows that traditional single-grade teachers spent more time on development (M = 17.4, F = 5.69, p < .01) than combination (M = 10.0) and within-class ability-grouped teachers (M = 10.3) as well as more time on supervising seatwork (M = 15.1, F = 3.99, p < .05) than combination teachers (M = 8.2).
Not reflected in these comparisons is the fact that students in combination and within-class ability-grouped classes, because of their involvement in a two-group approach, generally received only about half of the time that teachers allocated to each teaching function (teachers delivered some functions, of course, to the whole class). For example, while it appears that students in combination and within-class ability-grouped classes received more time on directions than those in traditional single-grade classes, in most cases they actually received lessthat portion allocated to their small group (about half of the time). Taking this divided time factor into account, it can be seen that students in traditional single-grade classes received nearly three times as much time on lesson development and supervised seatwork as students in combination and within-class ability-grouped classes.
Table 6 presents the means, standard deviations, and F-values for the 12 high-inference measures of teacher-directed and independent-group variables in the three types of classrooms. With the exception of the accomplishment variable, ANOVA procedures showed significant differences among the three groups on each of the teacher-directed ratings. Traditional single-grade lessons were rated as having more meaningful presentations (M = 3.5, F = 9.53, p < .0001) than both combination (M = 2.7) and within-class ability-grouped lessons (M = 2.4). Corresponding with this finding, observers rated the lessons of traditional single-grade teachers higher in use of manipulatives (M = 2.1, F = 5.64, p < .01) than those of both combination (M = 1.4) and within-class ability-grouped teachers (M = 1.3). However, combination class lessons were rated higher on accountability strategies (M = 3.7, F = 4.22, p < .01) than traditional single-grade lessons (M = 3.0), and managerial routines were rated higher in combination classes (M = 3.6, M = 6.46, p < .01) and within-class ability-grouped classes (M = 3.3) than in traditional single-grade classes (M = 2.8). Notably, higher-level thinking was rated higher in traditional single-grade (M = 2.9, F = 7.95, p < .001) and within-class ability-grouped classes (M = 2.6) than in combination classes (M = 2.0).
Focusing on the independent-group variables, significant differences were found among groups on time on-task (F = 5.82, p < .01), higher-level cognitive behavior (F = 9.54, p < .0001), student cooperation (F = 4.79, p < .01), student use of manipulatives (F = 12.42, p < .0001), and group self-management (F = 11.73, p < .0001). Scheffé tests showed that within-class ability-grouped classes were rated higher on time on-task (M = 3.4) and group self-management (M = 3.6) than traditional single-grade classes (time on-task M = 2.7, group self-management M = 2.5). On the higher-order cognitive behavior variable, observers rated lessons in traditional single-grade (M = 2.8) and within-class ability-grouped classes (M = 2.3) as higher than lessons in combination classes (M = 1.8). Furthermore, traditional single-grade lessons were rated higher on use of manipulatives during independent-group work (M = 2.3) than both combination (M = 1.2) and within-class ability-grouped lessons (M = 1.3). Finally, cooperation was rated higher during traditional single-grade lessons (M = 2.5) than during combination (M = 1.8) and within-class ability-grouped lessons (M = 1.8).
This study compared mathematics instruction in combination classes with that in single-grade classes. Combination classes, formed expediently as a result of enrollment shortages and embedded in a graded system of schooling, are often portrayed as being fundamentally different from multiage or nongraded classes (e.g., Mason & Stimson, 1996) but also as naturally encouraging teachers to adopt the innovative strategies found in multiage/nongraded classes (e.g., integrated curriculum, peer tutoring, and individualized instruction). Thus, while we were interested in whether combination teachers tend to use more innovative, developmental approaches when faced with combination classes, we did not focus on instruction in multiage or nongraded classesclasses that are generally formed deliberately to abolish a graded system of education and to implement a more developmental philosophy of teaching and learning (Anderson & Pavan, 1993; Goodlad & Anderson, 1987). Because observational research on combination classes in U.S. schools was nearly nonexistent, we sought to generate and explore some hypotheses on how teachers approach these classes, and examine whether their approaches were similar to those of teachers in other countries.
Because our study was exploratory and limited to a sample of 24 teachers and a specific set of time-use and high-inference measures, our results may not reflect practice in some schools or districts. For example, a small percentage of teachers apparently use an integrated, whole-class approach when faced with combination classes (Mason & Burns, 1995), and some schools or districts may provide special in-service programs that significantly alter teaching and learning processes in combination classes. In addition, although these findings might lead to an inference that achievement is diminished in combination classes, other measures may have found more positive aspects of mathematics teaching and learning in combination classes; and as Burns (1996) points out, it may be that combination teachers simply work harder to negate any potentially negative effects. Furthermore, results of the present study may have been skewed by the sample selected in the original studyan inquiry focused on the small-group teaching of mathematics. That is, single-grade teachers in this study may have used more two-group teaching than a random sample of single-grade teachers (an event that would, however, exacerbate the present differences found).
Despite the limitations of this exploration, the study provides instructive findings. To begin with, several findings replicate those of other (mostly non-U.S.) studies of combination classes. In the area of classroom organization, our results correspond to naturalistic investigations in Australia (Pratt & Treacy, 1986) and the Netherlands (Veenman, Lem, & Voeten, 1987; Veenman, Voeten, & Lem, 1987), results that showed that combination teachers used two-group approaches for teaching mathematics, alternating instruction with one grade-level group while the other grade-level group worked independently on assigned seatwork. Traditional single-grade teachers, similar to Pratt and Treacys (1986) finding, typically used whole-class approaches, occasionally forming small groups for purposes of remediation, enrichment, or cooperative learning activities. Within-class ability-grouped teachers, by definition, used two groups for instruction, but they used whole-class and alternative grouping approaches more frequently (14 of 71 lessons) than combination teachers (1 of 38 lessons).
Other findings from the present study, however, differ from those of Foshay (1948), Pratt and Treacy (1986), Veenman, Lem, et al. (1987), Veenman, Voeten, et al. (1987), and Galton, Simon, and Croll (1980). Although Foshay found no organizational format differences between the two combination and four single-grade classes in his study, we found striking macro and micro format differences. Similarly, although Pratt and Treacy found no time-on-task differences between combination and single-grade classes, and Veenman, Voeten, et al. found time-on-task measures favoring single-grade classes, observers in our study rated time on task during independent work higher in combination classes than in traditional single-grade classes. Furthermore, although Pratt and Treacy and Veenman, Voeten, et al. reported no curriculum differences between combination and single-grade classes (Veenmans data, however, showed that combination teachers spent more time on computation and less time on word problems and concept development), observers in the present study rated curriculum higher in single-grade classes than that in combination classes. Observers comments indicated that the higher time-on-task rating in combination classes was partially linked to the fact that, relative to traditional single-grade teachers, combination teachers provided their students fewer activity-oriented assignments, lower-level curriculum tasks during seatwork, less cooperative work with their peers, and less manipulative use. These strategies appeared to be aimed at eliminating distractions and maintaining order and time on task with the independent group while teachers provided direct instruction or active learning opportunities to the other grade-level group.
Although some advocate combination classes as a vehicle for innovative curriculum and instruction, findings from this study suggest that combination classes created several negative tradeoffs for important processes of teaching and learning. Relative to traditional single-grade classes, most combination teachers provided less active teaching and learning, less individualized attention to students, and fewer challenging curriculum tasks. In addition, while some assert that combination classes lead students to opportunities for social growth, peer tutoring, and independent learning, these data provide little support for such claims. Somewhat surprisingly, findings from this study suggest that the two-group approaches that predominated in these combination classes differed in subtle but potentially important ways from those used in within-class ability-grouped classesa two-group approach that has been found to be effective in a single-grade context (Slavin, 1987).
ORGANIZATIONAL CONSTANTS AND VARIATION
Unsurprisingly, this study found that simple descriptions of teaching and learning in combination and single-grade classes were futile. Although combination and within-class ability-grouped teachers typically used two groups for mathematics instruction and single-grade teachers typically used whole-class teaching, the amount of time given to whole-class and small-group micro formats and how these formats were used for mathematics instruction varied greatly among class types and within each group (Table 2). In addition, individual teachers used the 12 teaching functions in diverse ways to present mathematics content, monitor student understanding, and reinforce or apply this content and understanding.
For example, a few combination teachers occasionally used a whole-class format for cooperative learning activities that spanned grade levels, and several traditional single-grade teachers used two-group approaches for remediation and enrichment or differential presentations of curriculum. Furthermore, while one combination teacher provided thorough and meaningful development of mathematics concepts (averaging over 24 minutes per lesson), three of the six combination teachers averaged less than 6 minutes per lesson on development. Similarly, while two single-grade teachers averaged over 22 minutes of development, two averaged less than 10 minutes in this function.
However, in the 24 classes studied there was limited variation in the mathematics curriculum presented to students. Skill-oriented computation completely removed from application tended to dominate both combination and single-grade classes, although single-grade classes (both traditional and within-class ability-grouped) contained more applications and concept-development activities than combination classes, subsequently receiving higher ratings from observers on higher-level thinking and cognitive behavior.
High-inference ratings and the time spent by teachers in the various teaching functions appeared to show that combination classes, relative to single-grade classes, created important trade-offs for curriculum, instruction, and student incentives. For example, combination teachers spent significantly less time on directions, supervising seatwork, and lesson development. Directions often provide students important academic orientation, whereas thorough development provides students opportunities for (1) concept formation, (2) procedural knowledge important for understanding and performing mathematics, or (3) both. Furthermore, supervising seatwork often provides opportunities for teachers to review student work, share feedback, and address student misunderstanding through roving tutorials. As noted previously, these proactive teaching function differences are actually more significant than they first appear, since combination students, being in one of two groups, typically receive about one-half of the total time in most functions.
Predictably, perhaps, the proactive time on directions, development, and supervising seatwork that was apparently diminished in combination classes appears to have resulted in teachers spending more time later in checking seatwork and controlled practice over review content, functions that might be termed more reactive. In any event, these teaching function differences were reflected in observers ratings of related high-inference variables, namely meaningful presentation, accomplishment, higher-level thinking emphasis, and teacher use of manipulatives, effects that would apparently not bode well for student achievement of basic skills (Good et al., 1983; Leinhardt & Bickel, 1989).
In addition to these instructional function differences, our results suggest that combination classes create management trade-offs when compared with traditional single-grade classes. Combination teachers spent more than twice as much time on nonacademic functions (transition, nonmathematics, other) than traditional single-grade teachers. Part of this time, of course, is created by the logistics of managing two groups, preparing different materials, and using additional textbooks. The press to manage two groups and ensure uninterrupted instructional time appears to have resulted in combination classes being rated higher by observers on accountability, managerial routines, and group self-management.
Although these higher classroom management ratings would first appear to be a positive for combination classes, the associated press to provide two presentations, to monitor two groups, and to adapt instruction to meet individual needs appears to lead to curriculum and student incentive trade-offs in these classes. Curriculum content data indicated that most combination class lessons focused on computation-oriented presentations and simple reinforcement assignments, activities apparently designed to ensure success and noninterruption rather than deeper understanding, application, challenge, or the potential for encouraging peer or teacher interaction and assistance. Although a few teachers provided concept development activities and verbal or nonroutine problem-solving assignments, activity-oriented problems or application-oriented projects were rare. Indeed, a majority of the 126 lessons were exclusively skill-, textbook-, and reinforcement-based. Although this lack of higher-level content, concept development, and orientation toward active and applied learning was also found in single-grade classes, less challenge and variation on these curriculum aspects were found in combination classes.
The press toward maintaining order may also discourage students from working cooperatively to solve problems or from assisting others when they are stuck on a problem. Although observers rated combination classes as slightly higher on group interaction than single-grade classes, much of this interaction was not oriented toward cooperative work; observers comments indicated that students in combination classes often interacted surreptitiously about their work, and observers ratings showed that combination students worked cooperatively less often than single-grade students. Thus, these results fail to support the notion that combination classes lead to increased peer tutoring or social activity, at least in mathematics. This overall lack of peer interaction and cooperation is unfortunate, especially since the two-group approach used in combination and within-class ability-grouped classes ostensibly provides a context in which teacher assistance is frequently unavailable, thereby providing opportunities for peer tutoring and assistance among students.
COMBINATION AND WITHIN-CLASS ABILITY-GROUPED CLASSES
Although most might predict significant instructional differences between combination classes in which two-group teaching is used and single-grade classes in which whole-class teaching predominates, many might argue that combination classes do not differ from single-grade classes in which teachers use within-class ability grouping (two-groups)a structure used frequently for mathematics (Mason, 1995). This argument may be called into question, however, by fundamental differences between these two formats. Combination classes typically include students grouped on the basis of age or grade level, and their teachers must confront and manage the multiple textbooks and ancillary materials from two gradestexts and materials with which they are often unfamiliar. Further, since two grades are typically more diverse than two within-class ability groups from a single grade, combination teachers are faced with providing additional enrichment and remediation to students from each grade.
In contrast, within-class ability-grouped teachers typically organize students on the basis of specific skills in particular subjects in order to tailor the curriculum and instruction to their needs. Thus, they typically face two relatively homogeneous groups, use one textbook, and allow occasionally for fluid group membership. Leinhardt and Bickel (1989; see also MacKenzie, 1983; Slavin, 1989a) explain that small-group instruction is more likely to be effective when (1) groups are formed according to specific skills within the subject, (2) group size is determined by who fits rather than expedient numbers, (3) there are different groups for different subjects, and (4) small-group and whole-group instructional approaches are interspersedcharacteristics that fit within-class ability groups but not combination classes.
These and perhaps other subtle differences between combination and within-class ability-grouped classes are certainly open to empirical investigation. However, this study suggests distinct curriculum and instruction differences between the 6 combination classes and the 12 within-class ability-grouped classes. Results show that combination teachers tended toward less flexible use of the various macro formats in their teaching, employing alternative formats only once during the 38 observed lessons (3%). In contrast, within-class ability-grouped teachers more frequently chose alternative macro formats, using whole-class, cooperative learning, and three or more groups during 14 of the 71 observations (20%). Moreover, combination teachers tended toward less flexible use of micro formats within their lessons than within-class ability-grouped teachers. For instance, within-class ability-grouped teachers averaged over 18 minutes in a whole-class format (see Table 2), but combination teachers averaged less than 9 minutes using this approach.
Although both combination and within-class ability-grouped teachers generally used two-group formats, the results of this study also suggest subtle curriculum and classroom management differences between these two types of classes. Teachers of within-class ability-grouped classes tended to use problem solving more frequently during the instructional portion of their lessons than teachers of combination classes. Furthermore, observers rated higher-level thinking in teacher-directed lessons as higher during within-class ability-grouped lessons than combination class lessons, and ratings of higher cognitive behavior during independent-group activities also favored within-class ability-grouped classes. It appears that within-class ability-grouped teachers, perhaps because of their homogeneous groups and more limited use of second texts, were able to more successfully challenge students with higher-level curriculum objectives, though meaningful presentations and use of manipulatives showed no advantages. However, these differences tend to confirm what teachers and principals reports and nonU.S. observational studies have indicated: Combination classes are more complex and more difficult teaching and learning environments for teachers than are single-grade classes, apparently forcing teachers toward curriculum and classroom management issues in subtle ways. Indeed, the results of this study show that at least some differences between combination and single-grade classes remain even when the comparisons are limited to single-grade classes that employed within-class ability-grouped teaching.
This exploratory work points to the need for a similar investigation with a larger sample. Any such investigation, however, should be expanded beyond mathematics to see, for example, how teachers address other subjects and if, as a result of the need to manage two grades, curriculum coverage is diminished in these subjects (e.g., see Mason & Burns, in press). Studies that link combination class teaching behaviors or clusters of behaviors to student motivation and achievement would also be important. Future research might also compare expert and novice combination teachers or different approaches to teaching combination classes. For example, such investigations might explore whether, as some practitioners assert, whole-class, integrated approaches are generally more effective for teaching combination classes than two groups and, if so, under what conditions.
Research that explores students perceptions of combination classes and that compares student affect and achievement in these classes would also be informative. Such research, however, should include samples of both heterogeneous and purposefully assigned students, since educators report classroom assignment as an important factor in organizing effective combination classes. For example, samples of rural and year-round schools should be included, as these schools typically include hetereogeneous groups of students and, according to teachers and principals, more difficult teaching circumstances.
Finally, subsequent research might profitably explore classrooms that are using distinctive alternative approaches to mathematics teaching (e.g., Logo environments, Kamii-type instruction where students are never told algorithms, etc.). Researchers should examine unorthodox approaches to mathematics instruction, collecting careful data to see the extent to which these approaches differentially affect students learning and attitudes.
This exploratory observational study suggests that combination classes are a more complex and challenging task for teachers than are traditional single-grade classes, apparently confirming hypotheses from Smith and Geoffreys (1968) microethnography of a split-level classroom as well as findings from more recent naturalistic research on combination classes in Europe (e.g., Veenman, Lem, et al., 1987). Because combination classes are a significant form of classroom grouping in many countries around the world (Veenman, 1995), further research on how combination classes differ from single-grade classes and on how teachers can more effectively manage and instruct these classes would clearly contribute to the literature on teaching as well as to issues of school and classroom organization. Although some educational reformers claim that teaching lacks variety and that combination classes lead to innovative curriculum and instruction, this exploration showed that (1) mathematics teaching (at least classroom organization and instruction) varied greatly both among and within particular formats and (2) combination classes led to fewer instances of peer tutoring, individualized instruction, and integrated or continuous progress curriculum.
In contrast, however, too little variation was found in the mathematics curriculum in these classes. Mathematics educators (e.g., National Council of Teachers of Mathematics [NCTM], 1989, 1991) advocate active learning (creating, discovering, gathering), real-world applications, use of technology (computers, calculators), and student activities aimed at problem solving in genuine situations, strategies aimed at developing mathematical poweran individuals abilities to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve nonroutine problems (NCTM, 1989, p. 5).
Relative to combination classes, the curriculum in single-grade classes was rated as more meaningfully presented, as more challenging cognitively, as more oriented toward conceptual understanding, and as more often employing cooperative groups for collaborative problem solving. However, the statistical means of both single-grade and combination classes were far short of NCTMs vision of mathematics curriculum and instruction. Research that develops effective models for teaching such powerful mathematics in combination as well as in single-grade classes would be valuable.
We thank Doris Wilson for initiating the combination class project and Barbara Bussman, Ron Combs, Roger Kelly, Susan Kenny, and Catherine Mulryan for their assistance in collecting data for this study. Financial support and secretarial assistance were provided by the California Educational Research Cooperative at University of California, Riverside, and the Center for Research in Social Behavior at University of Missouri-Columbia. Furthermore, Robert Burns, Rodney Ogawa, and Janet Stimson provided helpful suggestions on an earlier draft of this article. Finally, we wish to acknowledge support from the National Science Foundation (NSF Grand MDR 8550619), which made this work possible. The ideas presented here, however, are those of the authors, and no endorsement from NSF should be inferred.
Correspondence concerning this study should be addressed to DeWayne A. Mason, School of Education, University of California, Riverside, CA. 92521 (E-mail to firstname.lastname@example.org).
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