Journal American Educational Research

American Educational Research Journal http://aerj.aera.net

Improve: A Multidimensional Method For Teaching Mathematics in Heterogeneous Classrooms Zemira R. Mevarech and Bracha Kramarski Am Educ Res J 1997 34: 365 DOI: 10.3102/00028312034002365 The online version of this article can be found at: http://aer.sagepub.com/content/34/2/365

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American Educational Research Journal Summer 1997, Vol 34, No. 2, pp. 365-394

IMPROVE: A Multidimensional Method for Teaching Mathematics in Heterogeneous Classrooms Zemira R. Mevarech Bracha Kramarski Bar-Ilan University The purpose of the present research was to design an innovative instructional method for teaching mathematics in heterogeneous classrooms (with no tracking) and to investigate its effects on students'mathematics achievement. The method is based on current theories in social cognition and metacognition. It consists of three interdependent components: metacognitive activities, peer interaction, and systematic provision of feedback-corrective-enrichment. The method is called IMPROVE, the acronym of which represents all the teaching steps that constitute the method: Introducing the new concepts, Metacognitive questioning. Practicing, Reviewing and reducing difficulties. Obtaining mastery, Verification, and Enrichment. The research includes two studies, both implemented in seventh grades: One focused on in-depth analyses of students' information processing under the different learning conditions (N = 247), and one investigated the development of students' mathematical reasoning over a full academic year (N = 265). Results of both studies showed that IMPROVE students significantly outperformed the nontreatment control groups on various measures of mathematics achievement. The theoretical and practical implications of the research are discussed.

ZEMIRA R. MEVARECH is an Associate Professor in the School of Education, BarIlan University, Ramat-Gan, 52900, Israel. Her specializations are mathematics education, metacognition, and cooperative-mastery learning. BRACHA KRAMARSKI is a Research Associate at Bar-Ilan University, Ramat-Gan, 52900, Israel. Her specializations are mathematics education, metacognition, and cooperative learning.


Mevarech and Kramarski or almost half a century, the issue of grouping and tracking has remained on the agenda. For many, the essential question is not whether differentiation is socially just but rather whether it leads to higher achievement gains than learning in heterogeneous classrooms. Studies in this area have been dominated by two different traditions of research. One involves comparisons of ability-grouped to heterogeneous placements. This research has shown little or no impact of tracking and grouping on student achievement (e.g., Kulik & Kulik, 1982; Slavin, 1990). The other, rooted in the sociology tradition, involves comparisons of the progress made by students in different ability groups or tracks (Oakes, Gamoran, & Page, 1992). This research has indicated that structural features of schools and classrooms such as tracking and grouping result in significant gains for high-track students and a corresponding loss for low-track students (e.g., Gamoran, 1993; Oakes et al., 1992). In particular, these researchers have shown the dynamics of high school tracking decisions and the processes by which these organizational features affect the depressed academic achievement of low-track and lowability groups (e.g., Oakes & Guiton, 1995; Useem, 1992). Evidently, even in the mid-1990s, "students . . [were] . . unevenly distributed among tracks, with low income and minority students more likely to be in low ability classes of the non-college-bound" (Oakes & Guiton, 1995, p. 3). Structural explanations followed by meta-analysis findings have led to recommendations for the current school restructuring reform in the USA and other countries (Cohen & Lotan, 1995), calling for restructuring schools and classrooms in a way that would untrack the secondary schools and ungroup pupils at the elementary level (e.g., Massachusetts Board of Education, 1990; National Education Association, 1990). As Slavin (1990) indicated: "If the effects of ability grouping on student achievement are zero, then there is little reason to maintain the practice" (p. 492). At present, however, no study has claimed that the variance in students' prior knowledge, as found in heterogeneous classrooms, could be utilized to enhance achievement gains more than teaching in between-class ability grouping. In spite of these proposals, many schools and districts still view tracking and grouping as the only workable means to instructing secondary students with different prior knowledge. Teachers who tried to teach in heterogeneous classrooms faced enormous difficulties because in many of these classrooms the range of instructional levels is more than five grades per classroom (Fuchs, Fuchs, Hamlet, Phillips, & Bentz, 1994). Similar findings have been found also in other countries (Hativa, 1988). According to the National Education Association (1990), special support for smaller classes and intensive staff development is needed for considering alternatives to grouping and tracking. Currently, most attempts (e.g., Davidson & KroU, 1991; Slavin, 1990) to cope with diversity, particularly within the mathematics classroom, have been based on cooperative learning embedded within facilitative conditions, such as explicit training in giving and receiving help (e.g., Webb, 1989; Webb & Farivar, 1994), provision of systematic feedbackcorrective-enrichment procedures (Fuchs et al., 1994; Mevarech, 1985, 1991;

F

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IMPROVE: A Multidimensional Method for Teaching Mevarech & Susak, 1993), and special reward structures (Slavin, 1980, 1989, 1991). Meloth and Deering (1994) indicated that "cooperation is beneficial in large part because students share ideas, explain their reasoning, and provide assistance to peers as they work together" (p. 20). Yet, at the same time, studies have shown that a primary source of difficulty in working in small groups lies in students' inability to actively monitor and subsequently regulate the cognitive processes engaged in during collaborative problem solving (King, 1989, 1994). Although it is largely believed that small problem-solving groups provide natural settings for interpersonal monitoring, special training is needed for facilitating the activation of metacognitive processes within the cooperative settings. The question that always remains open is how students working in small groups can be trained to activate metacognitive processes.

Knowledge Construction Within Cooperative Settings Over the last decade, many researchers have studied cooperative settings from a cognitive information processing perspective (e.g.. Brown & Palincsar, 1989; King, 1989, 1994; Meloth & Deering, 1994). According to constructivist theories, "learning occurs not by recoding information but by interpreting it" (Resnick, 1989, p. 2). When learners encounter new information, they try to relate it to their preexisting knowledge and personal experience (e.g.. King, 1994; Mayer, 1992; Wittrock, 1989). By that, the learners construct meaningful links between unfamiliar and familiar knowledge. Sometimes, learners elaborate the new information in order to integrate it within the existing schema. In other cases, the existing schema are restructured to adapt the new information. Learning is, thus, a process of knowledge construction, not of knowledge recording or absorption. Effective learning depends, therefore, on "the intentions, self-monitoring, elaborations, and representational constructions of the individual learners . . . . This is very different from theories that grew out of earlier associationist and behaviorist psychologies" (Resnick, 1989, pp. 2-3). In the case of mathematics, during the sense-making process of constructing relationships between the new and existing knowledge, individuals often analyze the problem to identify what the problem is all about, refer to specific strategies, tactics, or principles that are already in memory, and compare the problem at hand with problems that have been solved before (e.g., Artz & Armour-Thomas, 1992; Polya, 1973; Schoenfeld, 1985, 1987). The fact that knowledge construction relates to prior knowledge makes the learning in small groups particularly beneficial because the diversity in students' prior knowledge can be utilized during the interaction so that each team-member's contribution is accumulated to provide a larger base of resources for the group's knowledge construction (Schoenfeld, 1987). Given that knowledge construction is an internal cognitive process carried out by the individual learner, a number of recent studies have suggested instructional interventions for training students to self-regulate the learning in cooperative settings. One way is by formulating and answering 367


Mevarech and Kramarski questions that focus on information processing procedures (e.g., King, 1994). Because knowledge construction occurs when individuals generate relationships between the newly encountered information and their prior knowledge (e.g., Mayer, 1992) and because one aspect of control and regulation has to do with "the decisions one makes concerning when, why, and how one should explore a problem, plan a course of action, monitor one's actions, and evaluate one's progress" (Lester, Garofalo, & Kroll, 1989, p. 1), there is reason to suppose that guiding students to formulate certain kinds of questions may elicit elaborate explanations; justifications of when, why, and how to use strategies/tactics/principles; inferences about the introduced concepts; and new perspectives on some aspects of the existing knowledge. These questions concern (a) the structure of the problem, (b) connections between the new and existing knowledge, and (c) specific strategies/tactics/principles that are appropriate for solving the new problem. Several current research projects provide indirect support for this notion. For example, Mevarech and Susak (1993) reported that students who learned to generate various kinds of questions in a cooperative-mastery learning condition developed a higher level of creative thinking than students who learned to generate questions either in cooperative settings with no feedback-correctives or in noncooperative settings with or without feedback-correctives. King (1989, 1990) reported that college and highschool students working in pairs who were trained to ask specific contentfree questions of the form: "How are . . . and . . . alike?" and "What would happen if . . . ?" demonstrated higher achievement of course content than did students who were encouraged to ask questions with no specific guidance. Both groups attained higher achievement gains than the nontreatment control group. In her studies. King (1989, 1991) distinguished between comprehension questions of the form: "Describe . . . in your own words" and connection questions such as: "How are . . . and . . . similar?" "What is the difference between . . . and . . . ?" King (1994) reported that pairs of fourth- and fifth graders who studied the material by asking and answering one another's self-generated questions designed to promote connections among ideas within the lesson as well as with prior knowledge/ experience were engaged in more complex knowledge construction than those trained in lesson-based questioning only and students in the control group. These results extended the findings of a previous study (King, 1989) that showed that fifth graders who were trained to formulate and answer various kinds of questions during programming in LOGO developed a higher level of problem solving performance than students who were simply instructed to ask and answer questions with their partners during problem solving. Also in this study, both groups obtained higher scores than the control group which was not instructed to ask questions. In mathematics, Schoenfeld (1985, 1987) suggested using metacognitive questioning to help college students learn to regulate their problem-solving performance. Schoenfeld (1987) reported that students who were trained to stop periodically during problem solving and ask themselves questions such 368


IMPROVE: A Multidimensional Method for Teaching as "What am I doing right now?" "Why am I doing it?"and "How does it help me?" were able to control and reflect on the problem-solving processes and consequently improved performance. Given these studies, we hypothesized that training students working in cooperative settings embedded within feedback-corrective enrichment to use a set of comprehension and connection questions in combination with questions regarding strategies, tactics, or principles would elicit elaborate explanations and enhance mathematical reasoning. Furthermore, consonant with the metacognitive enhancement hypothesis, we expected the effects to be stronger on mathematics reasoning than on basic skills that can be acquired through individualized practice.

The Present Study On the basis of the research reviewed above, we developed a multidimensional instructional method that aimed at enhancing mathematical reasoning. The method involves three interdependent components: (a) facilitating both strategy acquisition and metacognitive processes; (b) learning in cooperative teams of four students with different prior knowledge: one high, two middle, and one low-achieving student; and (c) provision of feedback-correctiveenrichment that focuses on lower and higher cognitive processes. The method is implemented in heterogeneous classrooms with no ability trackings, where students from different backgrounds and with different prior knowledge learn together. We call the method IMPROVE, because it furnishes the psychological foundation for our program. IMPROVE is the acronym of all the teaching steps that constitute the method: Introducing new concepts, Metacognitive questioning, Practicing, Reviewing and reducing difficulties, Obtaining mastery, Verification,and Enrichment. In particular, after the teacher introduces the new concepts to the whole class, students work in small heterogeneous groups. Students take turns in asking and answering three kinds of metacognitive questions: comprehension questions, strategic questions, and connection questions. Comprehension questions oriented the students to articulate the main ideas in the problem (e.g., "Describe . . . in your own words"), classify the problem into an appropriate category (e.g., "This is a rate problem of the form cost-perunit rate"; "This is a simplification problem with a negative multiplier"), and elaborate the new concepts (e.g., "The definition of... is ..."; "The meaning of. . . is . . ."; "The given are . . .";"The unknown is . . . "). Strategic questions refer to strategies appropriate for solving the problem. When the unit focuses on specific mathematics principles, students have to select the

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Mevarech and Kramarski principle, justify their decision, and describe the application of the principle to the given problem. When the unit focuses on algebra word problems, students are prompted to use diagrams and tables (e.g., Polya, 1973; Schoenfeld, 1985). Finally, connection questions refer to the similarities and differences between the problem at hand and the problems they have previously solved. Using connection questions, students learned to distinguish between equivalent problems sharing the same mathematical structure and the same story context, similar problems sharing the same story context but having different mathematical structures, isomorphic problems sharing the same mathematical structures but having a different story context, and unrelated problems sharing neither the mathematical structure nor the story context (Weaver & Kintsch, 1992). In addition, students learned to distinguish between different kinds of quantities, propositions, and procedures. The following excerpt shows how students responded to the connection question: What are the differences between the problem at hand (Problem 1) and the previous problem (Problem 2)? Student A:

In Problem 1, the numbers were . . . in Problem 2 the numbers are . . . .

Student B:

Oh, no, this is not important; she (the teacher) does not mean the numbers, liie important differences are not in the numbers but rather in the structure of the problem. Problem 1 is about. . . whereas Problem 2 is about....

These metacognitive questions were constructed and arranged to follow the 4-stage model of the problem-solving process: orientation and problem identification, organization, execution, and evaluation (Garofalo & Lester, 1985; Polya, 1973; Schoenfeld, 1985). The questions were deliberately designed to help students to be aware of the problem-solving process and to self-regulate their progress. According to Schoenfeld (1987), self-regulation involves (a) understanding what the problem is all about before attempting a solution, (b) planning the solution, (c) monitoring or keeping track of how well things are going during the solution, and (d) allocating resources, or deciding what to do while working on the problem. Schoenfeld (1987) further indicated that awareness and self-regulation are two important aspects of metacognition. (For more information about the metacognitive questions, including examples, see the "Procedure" section.) In addition to emphasizing metacognitive activities, IMPROVE assumes that cooperative-mastery learning (Mevarech, 1985,1991; Mevarech & Susak, 1993; Slavin & Karweit, 1984) based on peer interaction and the systematic provision of corrective/enrichment feedback enhances mathematical thinking. Several factors make cooperative-mastery learning particularly promising for enhancing mathematical thinking in heterogeneous classrooms. First, peer interaction provides ample opportunities for students to articulate their thoughts, explain their mathematical reasoning (Slavin, 1980), and use a large base of mathematical resources composed of each team member's prior knowledge (Schoenfeld, 1987; Wistedt, 1994). Second, with appropri370


IMPROVE: A Multidimensional Method for Teaching ate guidance, cooperative learning can enhance the provision of on-line strategic management of cognitive resources for students with different ability and prior knowledge (Webb, 1989). Finally, feedback-correctiveenrichment adapts the learning time to the needs of each individual student and thereby enables students to attain mastery on the tasks and deepen their mathematical thinking (Mevarech, 1985, 1991). Recently, Fuchs et al. (1994) showed that implementing classwide curriculum-based measurement (CBM) in heterogeneous classrooms helps general educators meet the challenge of student diversity. In this study, CBM teachers who received instructional recommendations designed better instructional programs and effected greater achievement for their students than CBM teachers who did not receive instructional recommendations or control teachers. Similar findings were reported by Mevarech (1985, 1991) who showed that implementing feedback-corrective-enrichment activities in either cooperative or individualized settings promoted higher mathematics achievement than learning in cooperative/individualized settings with no feedback-corrective-enrichment. This is in contrast to the traditional approach of teaching mathematics, which links higher cognitive processes more directly with individualized learning in ability tracking where each student learns according to his or her ability. The purposes of the present study were, therefore, threefold: (a) to design an instructional intervention based on metacognitive training, cooperative learning, and the provision of corrective/enrichment feedback that can be orchestrated in the ongoing mathematics curriculum over an entire academic year; (b) to assess the effects of the method on mathematics achievement; and (c) to examine who benefits from the method and in what perspectives.

Study I The primary focus of Study I is on the effects of IMPROVE on mathematical achievement of different ability groups. In particular. Study I examines (a) the extent to which the effects of IMPROVE are likely to be found on mathematics reasoning as well as on basic skills and (b) who benefits from IMPROVE and in what perspectives. The data are multilevel, with classrooms nested within treatment and crossed with ability, and thus were analyzed using hierarchical linear modeling (Bryk & Raudenbush, 1992). Method Participants. Participants were 247 seventh-grade students who studied in four junior high schools. Three classrooms (TV = 99) implemented IMPROVE, and five homeroom classes (TV = 148) served as a nontreatment control group. Experimental classrooms were randomly selected from the entire pool (10 junior high schools) of seventh-grade classes using IMPROVE, whereas the nontreatment control classes were randomly selected from a different district where students were tracked by ability and prior knowledge. Schools were similar in terms of size, mathematics level assessed prior to the beginning of the study (see below), and socioeconomic status as defined 371


Mevarech and Kramarski by the Israel Ministry of Education. Within schools, classes are normally distributed in terms of students' ability and prior knowledge. In Israel, as in the USA, the range of instructional levels within heterogeneous classrooms is more than 5 years per classroom (Hativa, 1988). Three years prior to the beginning of the study, we introduced the innovative method to all principals and mathematics department heads in one district, who then discussed it with their entire faculty. Because teachers have the autonomy to decide which teaching method to use in their classrooms, each faculty chose whether or not to implement IMPROVE. Out of the 10 schools, 6 schools (TV = 1,329) decided to implement IMPROVE, while the other 4 schools (7V= 997) decided to continue using the "traditional" method of instruction. A year later, mathematics achievement of the two groups was compared. Results showed that standardized score of IMPROVE schools increased more than those of the nontreatment control schools. On the basis of this comparison, IMPROVE was implemented in the entire district. At the time of the study, this district served a student population of about 40,000 students. It included 10 junior high schools that accepted students from 26 elementary schools. Each junior high school is an integrated school composed of students from different socioeconomic status as defined by the Israel Ministry of Education. In each school, mathematics is taught in heterogeneous classes with no groupings or ability trackings. Measures. A 36-item test was used to assess students' mathematics achievement. The test covered the following topics: rational numbers, identification of rational numbers on the number axis, operations with rational numbers, order of operations, and the basic laws of mathematics operations. The test was composed of two kinds of items: one kind (25 items) was based on the traditional evaluation procedure involving multiplechoice items regarding basic factual knowledge (e.g., which of the following numbers is the smallest?) and open-ended computation problems; the other (11 items) did not require (or invite) computations but rather was specifically designed to assess students' mathematics reasoning. These 11 items included problems that ask examinees to draw conclusions about possible outcomes and make algebraic generalizations (e.g., "if a > 0 and b < 0, is their difference a positive number? Please explain your reasoning"); evaluate mathematical expressions and decide whether or not the expressions are always true, always false, or sometimes true and sometimes false; resolve mathematical conflicts (e.g., "Ron argues that X/X (X ^ 0) is always 1; Sarah argues that the value of X/X (X ^ 0) depends on the value of X. Who is correct? Please explain your reasoning"); evaluate mathematical expressions and decide which mathematical laws are appropriate for solving them; and analyze the structure of mathematical problems. On these items, students were asked to give a final answer and to explain their reasoning. Problems of both kinds were mixed. To ensure that the test was not biased toward one group, content validity of the test was assessed by mathematics department heads of the experimental and control groups, as well as by the Israel mathematics superintendent. KR reliability coefficient = .91. 372


IMPROVE: A Multidimensional Method for Teaching Scoring: For each item, students received a score of either 1 (correct answer) or 0 (incorrect answer), and a total score ranging from 0 to 36. To gain a deeper understanding of students' mathematics reasoning, in addition to the above analysis, we analyzed separately students' mathematics explanations provided for the 11 items that require mathematical reasoning. Special scoring criteria were developed by two experts in mathematics education for each item on the qualitative component of the reasoningexplanation part, as follows: One item was assigned a score between 0 and 6, 7 items were assigned a score between 0 and 5, and 3 items were assigned a score between 0 and 4. For example, students were presented with a mathematical conflict. They had to solve the conflict and explain their reasoning. One point was assigned for an explanation based on a specific example demonstrating the correctness of the answer; 2 points were assigned for a partial answer based on a correct mathematics law that was described imprecisely; 3 points were assigned to an explanation based on an appropriate law referring to only one part of the conflict, and 4 points were assigned to an explanation based on an appropriate law referring to both parts of the conflict. These data are termed mathematical-reasoning. For the sake of clarity, all scores are presented in terms of percent correct. In addition, all students were tested at the beginning of the school year (prior to implementing IMPROVE) by a 33-item test that focused on arithmetics skills taught in elementary schools. The test covered the following content: whole numbers (10 items), fractions (10 items), decimals (5 items), percent (3 items), and word problems (5 items). KR reliability coefficient = .73. Procedure and materials. All instruction, training, practice, and testing connected with this study was designed as part of the regular mathematics curriculum for these students. As such, all the activities were carried out in the students' classrooms with their regular mathematics teachers under supervision of the investigators. All classes, experimental and control, used the same mathematics textbook (Rabinson, 1990), and all learned mathematics 5 times a week, according to the mathematics curriculum suggested by the Israel Ministry of Education. Appendix A presents the content of the course. In addition to the textbook, students under each condition used learning materials which covered problems and exercises similar to those in the textbook but which emphasized the unique components of the treatment. The learning materials of the IMPROVE group included student activities embedded within metacognitive questioning, formative tests, and corrective and enrichment activities. Except for the tests, all other materials (i.e., student activities, corrective materials, and enrichment materials) were specially designed for cooperative settings. The tests were taken individually. The learning materials of the control group included individualized practicing activities with no metacognitive questioning and quizzes with no correctiveenrichment activities. Below is a detailed description of the teaching methods. Treatment. The IMPROVE Method: IMPROVE consists of three interdependent components: metacognitive questioning, cooperative learning, 373


Mevarech and Kramarski and systematic provision of feedback-corrective-enrichment. Thus, in describing the method, we first focus on each component separately and then on how the components are combined. To facilitate the report, we demonstrate students' work by describing a team solving the following problem: Two express trains (with no stops) left the central station on parallel tracks. Train A went 60 km/h and Train B went 90 km/h. Train B left the station one hour later than Train A and reached the last station one hour earlier than Train A. What is the distance between the central and the last stations? Metacognitive questioning: A series of individual hand-held strategy prompt cards were developed for each unit. Students used these cards during the practice and subsequent lesson-discussion sessions to prompt their discussion. Each student in his or her turn tried to solve a problem and explain his or her reasoning by answering the questions printed on the cards. When all team members agreed on the answer, another student tried to solve the problem in the same way. When the student failed to solve the problem, or when no agreement was obtained, students discussed the problem by applying the metacognitive prompts printed on the cards. The cards for each unit involved three metacognitive questions: (a) comprehension question: What's in the problem; (b) connection question: What are the differences between the problem you are working on and the previous problem(s); and (c) strategic question: What is the strategy/tactic/principle appropriate for solving the problem? The comprehension questions were designed to prompt students to reflect on the problem before solving it. In addressing a comprehension question, students had to read the problem aloud, describe the concepts in their own words, and discuss what the concepts meant or into which category the problem could be classified. For example, in solving algebra word problems, students used the classification procedure suggested by Mayer (1982). In solving other kinds of problems, students named the principle, skill, or concept on which the problem was based. The following example shows how a student approached the problem by first classifying it and then referring to the essential features of the problem with little focus on the numbers: Student A:

Let's see what we have here. We first have to read the problem aloud and then say it in our own words. (The student reads the problem aloud.) Well, this is a problem about distance, time, and speed. It tells us about two trains that left the same station in the same direction with different speeds. The fast train left one hour later and reached the last station one hour earlier. We have to find the distance.

In addressing the connection questions, students have to focus on the similarities and differences between the problem they are working on and 374


IMPROVE: A Multidimensional Method for Teaching the previous problem or set of problems that they have already solved. In doing so, students gradually learn to distinguish between the surface and the deep mathematical structures of a problem, different kinds of mathematical structures, propositions, principles, and strategies. The following excerpt shows how the team approached the problem after they heard Student A: Student B:

That's the same (problem) as before. Both (problems) are distance-time-speed problems, and in both we have two trains leaving from the same place.

Student C:

Yes, but here it's different, because the trains did not leave at the same time.

Student D: That makes it more difficult because we do not know when they left, and so we cannot figure out the time. Student A:

Let's call the time x.

Student B:

Good idea. (She writes down and says:) The time of Train A and Train B is .x: (a wrong response).

Student A: x and x? Student B:

Yes, remember what the teacher said: x and x.

Student A:

No, you are wrong; the time of Train A is x, but Train B left one hour later; therefore its time is x-^1. (Again a wrong response.)

Student B:

Why x-^1?

Student A:

'cause it's one hour later (a wrong response).

Student D:

If it's one hour later, it has to be x-1,

Student A:

No, see, if Train A left the station at 7, then Train B left the station at 8; therefore, it has to be xplus 1 (a wrong response).

Student D: Yeah, but x is not the time they left but the time they traveled. So, if one train left after the other, it traveled less time; if it (Train A) reached the last station at 12, it went 5 hours, and Train B only 4 hours. [At this point, the students did not consider the fact that Train B did not only leave the central station one hour later, but it also reached the last station 1 hour earlier. Through their discussion of the problem, they revealed that information, too.] The strategic questions are designed to prompt students to think which 375


Mevarech and Kramarski strategies are appropriate for solving the given problem and for what reasons. In addressing the strategic questions, students had to describe the "what" (e.g., "What strategy/tactic/principle can be used in order to solve the problem?"), the "why" (e.g.,"Why is this strategy/tactic/principle most appropriate for solving the problem?"), and the "how" (e.g., "How can the suggested plan be carried out?"). Because different kinds of problems require different kinds of strategies, students were introduced during the year to a large repertoire of strategies from which they had to select the appropriate ones. For example, in discussing the above problem, one student suggested constructing a table, plugging the numbers (and variables) in, transforming the table into an equation, solving the equation, and "seeing what happens"; another suggested drawing a diagram to represent the time of the two trains; all used the formula. Students were told that asking and answering the metacognitive questions would help them understand and remember the material presented in mathematics classes. In the first lesson, students were taught how to use the metacognitive questions in solving mathematics problems. In addition, in each lesson, teachers presented the new materials by using the metacognitive questioning, thus modeling the use of metacognitive questioning. Furthermore, teachers encouraged students to use the metacognitive questions by using the strategy prompt cards. Each student in his/her turn asked the metacognitive questions and tried to answer them. When difficulties arose, the team discussed the problem by using the metacognitive questioning technique. Throughout all interactions, students were encouraged to articulate their mathematical reasoning and to use appropriate mathematics language in speaking, writing, and reading. When a student did not use the mathematical language correctly, other team members helped him/her to overcome the difficulty. This was seen, for example, when: (a) students read a problem aloud or articulated their mathematical thinking while leaving out a word denoting an unfamiliar concept (e.g., instead of reading, "Find the rate of change,'' Student A read, "Find the change,'' and continued to solve the problem accordingly. Student B asked her to reread the problem, and, when she did, she made the same mistake. Student B responded: "It's not the change that we have to find, it's the rate of change." She continued, ''Change is . . . and rate of change is . . ."); (b) students confused the symbol (-) denoting a negative quantity with the symbol denoting subtraction (e.g., instead of reading -6 as "minus 6" [a negative number], students said, "Six minus," and continued the exercise by subtracting a quantity from 6. Other students detected the error and corrected it); and (c) Students used the units incorrectly, as in the following excerpt: Student A:

Train A went at the speed of 60 km.

Student B:

You mean 60 km/h—^km is a measure of distance. We have speed here, so it has to be km/h.

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IMPROVE: A Multidimensional Method for Teaching [Later, Student A was able to identify an error by simply looking at the units: "What did we get here? 30? It's 30 hours . . . it does not make sense!" She solved the problem again and corrected the mistake.] Cooperative settings: Generally speaking, the cooperative technique followed the method suggested by Brown and Palincsar (1989). Students learned in heterogeneous teams constituted of 4 students: one high, two middle, and one low achieving. To construct the teams, all students were tested in the first week of the year on topics that were prerequisites for studying the first mathematics unit. On the basis of these scores, students were assigned to heterogeneous 4-student teams. During the year, team members were exchanged according to students' progress to maintain team heterogeneity. Following this exam, students started to learn in small groups as follows: Each session began with the teacher's short presentation (about 10 minutes) of the new materials to the whole class using the question-answering technique similar to that described by Marx and Walsh (1988). The teacher begins by providing answers to three metacognitive questions: "What's in the problem? What are the differences/similarities between . . . and . . . ? and What strategies/tactics/principles are appropriate for solving the problem at hand?" Then the teacher models aloud strategies/tactics/principles for completing the task and explains why they are likely to solve the task. Last, the teacher considers aloud how to check the answer and what to do if the plan does not work. While explaining the new materials, the teacher adapts the instruction to student diversity by selecting examples of different difficulty levels, providing various explanations based on different kinds of preexisting knowledge and thinking levels (e.g., intuitive vs. formal), and using the metacognitive questioning in different ways (e.g., focusing on different kinds of differences/similarities or using different strategies for solving a problem). Following the introduction, students started to work in small groups using the materials we designed. Each student, in his or her turn, read a problem aloud and solved it by using the metacognitive questions as described above. Whenever there was no consensus, the team discussed the issue until the disagreement was resolved. Talking about the problem, explaining it to one another, comparing it to what was already known, approaching it from different perspectives, balancing the perspectives against one another, and proceeding according to what seems to be the best option at the time, students actually used the diversity in their own prior knowledge to self-regulate their learning. When all team members agreed on a solution, they wrote it down on their answer sheets. Students' answers included the final solution, mathematical explanations, and a sample of metacognitive responses (e.g., "This is a problem about. . . ," "the difference between this problem and the previous problem is . . . ," "the mathematical principle appropriate for solving the problem is . . . because . . . . " ) . When none of the team members knew how to solve a problem, they asked for teacher 377


Mevarech and Kramarski assistance. At the end of the lesson, the teacher reviewed the main ideas of the lesson with the entire class. When common difficulties were observed, the teacher provided additional explanations to the whole class. When students worked in small groups, the teacher joined one team for 10-15 minutes and worked with the team as an additional team member. When the teacher's turn arrived, he or she modeled the use of the metacognitive questioning in solving the problems. The teacher read the problem aloud, used the metacognitive questions, and explained each step of the solution. Teachers listened to how students coped with the problems and provided assistance when needed. Teachers worked with each team at least once a week. Feedback-corrective-enrichment'. At the end of each unit, approximately every 10 lessons, students were administered a formative test that focused on the main ideas taught in the unit (Bloom, 1976). The formative tests included a large percentage of higher level mathematics problems as defined by Bloom's (1956) taxonomy. Students who did not attain mastery (80% correct) on the formative tests were given corrective activities to do, whereas others worked on enrichment activities related to the unit. The enrichment activities included challenging tasks that focused on mathematical reasoning rather than on lower order skills as defined by Bloom (1956). During the corrective-enrichment sessions, the teacher worked with either the slow- or the high-achieving students as she saw fit. In contrast to all other sessions where students worked in heterogeneous teams, at the corrective-enrichment session, students worked in relatively homogeneous teams. At the end of this session, students who had to correct their learning took a parallel form of the formative test to ensure that mastery had been attained. Given practical constraints, students were provided with only one opportunity to correct their learning. After that, all students continued to study the next unit. Thus, the diversity in students' prior knowledge was used by three different agencies: (a) the students when they interacted with one another in the small groups using the metacognitive questioning; (b) the teachers when they introduced the new materials, explained the topics, modeled aloud the solution processes, designed practicing activities, and provided feedback-corrective-enrichment; and (c) the learning materials that included not only a wide range of problems gradually increasing in the degree of difficulty and complexity but also corrective/enrichment activities that were designed to meet the special needs of the various groups of students in the heterogeneous classroom. The nontreatment control group. The control group was exposed to traditional methods of teaching mathematics. Thus, students in the control group learned the same topics as the IMPROVE group, but they did not receive any deliberate training in metacognitive questioning, nor were they exposed to cooperative learning and systematic feedback-corrective-enrichment that was an integral part only of the IMPROVE method. Under the control condition, the teacher introduced the new material to the whole class by using the questioning-answering technique, after which students were 378


IMPROVE: A Multidimensional Method for Teaching involved in individualized practicing activities. Students were administered quizzes and received feedback, but they were not exposed to systematic corrective-enrichment activities. Like many other mathematics teachers, also in the present study, the control teachers asked questions that can be considered as metacognitive in nature, but they did not do it deliberately and systematically as did the IMPROVE teachers. TTie teachers did not term those questions metacognitive questions, and the students were not aware of their facilitative effects, nor were they guided to use these kinds of questions in their problem-solving processes. Furthermore, the control students were not asked to write down responses to metacognitive questions, as none of their learning materials included such questions. Similarly, teachers introduced strategies also to the control group, but the teachers did not do it deliberately, and the students were not aware of the importance of the strategies. For example, in learning to solve word problems, control teachers used the 4-stage model described above, but they did not explicitly mention that this is a useful strategy for solving word problems, nor did they discuss it with their students. Issues relating to when, how, and why a strategy is useful were not deliberately discussed in the control group. During the semester, both the control and the IMPROVE students were administered two summative tests that were designed by the teachers independently of the present study. As indicated, the nontreatment control group was tracked into three groups: low-, middle-, and high-achieving students. For the purposes of tracking, all students were tested at the beginning of the school year by the school's counselors. Then, three homeroom classes were redivided for mathematics lessons into three homogeneous groups according to students' prior knowledge in mathematics, as follows: Students who scored A or B (more than 80% correct) were assigned to the higher achieving track; students who scored F (less than 65% correct) were assigned to the lower achieving track; and students who scored C or D were assigned to the middle-achieving track. The assignment of students to tracks was carried out by school counselors independently of the present study. Although all three tracking groups learned the same mathematics topics following the suggestion of the Israel Ministry of Education, the practicing in each tracking group was adapted to students' prior knowledge in mathematics. For example, all three ability groups learned to solve linear equations with one variable. The high-achieving students practiced equations with fractions and variables in the denominator; the middle group practiced equations with "simple" fractions and variables appearing only in the numerator; the low-achieving group solved equations with whole numbers. Appendix B describes the seventh grade curriculum for low- middle- and high-achieving tracks. In addition, students in each track were administered quizzes and tests designed by the teachers to assess mathematics achievement as taught in each track. Occasionally, control teachers used corrective activities, but they did not use enrichment materials. Teacher training. Prior to the beginning of the study, IMPROVE teachers were exposed to a 2-day, in-service training that focused on general issues 379


Mevarech and Kramarski in mathematics education, the theoretical background of the method, and its practical factors. The teachers went over the learning materials and learned how to use them with the students. These materials included the explicit lesson plans, the strategy prompt cards, use of examples and cognitive modeling followed by practice with corrective feedback. In addition, mathematics mentors met the teachers in the experimental schools every other week to discuss the ongoing instruction and to plan the coming activities. The 2-day in-service summer training and the teacher mentors were available only to the experimental group as part of the method.

Results and Discussion The data were analyzed in two steps. First, to examine possible aptitude X treatment interactions, students' pretest scores were classified into three ability groups (low, middle, and high), according to their pretreatment achievement in arithmetic, as follows. Students who scored A or B (more than 80% correct) were assigned to the high-achieving track (A^ = 109); students who scored C or D (scores ranged between 65% and 80% correct) were assigned to the middle-achieving track (A'^ 71); and the rest, students who scored F, were assigned to the low-achieving track (A^ = 67). Second, a 2 X 3 ANOVA with classrooms nested in treatments and crossed with ability was performed to examine the effects of the method on students' mathematics achievement. The two factors were treatment (experimental vs. control) and ability (low-, middle-, and high-achieving students). Table 1 presents the mean scores and standard deviations of students' performance in mathematics by time (pretest and posttest), treatment (experimental and control), and ability (low-, middle-, high-achieving students). The 2 X 3 ANOVA with classrooms nested in treatment and crossed with ability indicated that the treatment effect (calculated by using the classrooms nested in treatment as an error term) was significant (F(l, 6) = 14.78;/> < .01), the ability effect was significant (F(2, 12) = 154.85,/> < .001), but the ability X treatment interaction was not significant (F (2, 12) = 2.62, p > .05). According to Table 1, although prior to the beginning of the study no significant differences were found between the experimental and control groups (F< 1.00, p > .05), at the end of the first semester IMPROVE students significantly outperformed the control group. The overall mean score on the Introduction-to-Algebra test of the control group was 68.03 (SD = 22.64), while that of the experimental group was 74.72 CSD = 20.55). Table 1 further shows that within each ability group, IMPROVE students tended to outperform the control group. Post-hoc analyses indicated significant differences between the experimental and control groups for the middle and higher achieving students Q values = 3.56 and 2.86, both/> values < .05), but not for the lower-achieving students. The positive effects of IMPROVE were found not only on the overall scores of Introduction-to-Algebra but also on the mathematical reasoning scores. The 2 X 3 ANOVA with classrooms nested in treatment and crossed with ability showed significant effects for the treatment (F (1,6) = 13.00; 380


IMPROVE: A Multidimensional Method for Teaching Table 1 Mean Scores and Standard Deviations (in Terms of Percent Correct) By Time, Treatment, and Abiiity on Introduction-to-Algebra and (Mathematics Reasoning CONTROL

IMPROVE

Treatment Ability

Low

Middle

High

Low

Middle

High

Pretest M S

37.76 8.17

66.46 6.63

86.33 6.48

40.20 9.87

66.89 7.93

86.85 5.42

Introduction-to-Algebra M S

48.28 16.31

79.52 10.06

88.20 8.55

45.97 21.03

67.24 18.97

82.27 13.05

Mathematical Reasoning M S

41.37 14.41

64.00 16.38

74.74 12.98

32.60 20.68

50.98 17.79

67.47 15.34

p < .01) and ability (F (2,12) = 82.66, p < .001), but the ability X treatment interaction was not significant (F< 1.0,p > .05). The mathematical reasoning mean score of the control group was 53.15 (SD = 22.53), while that of the experimental group was 62.56 (SD = 19-96). Post-hoc analyses of the mathematical reasoning scores indicated significant differences between the experimental and control groups for each achievement level (lvalues = 2.04, 3.02, and 2.60, for the lower, middle-, and higher achieving students, respectively; allp values < .05). These findings extend the findings of Lester et al. (1989) to a wider range of mathematics topics (only word problems vs. introduction to algebra), different populations (one advanced and one regular classroom vs. heterogeneous classrooms), and different countries (USA vs. Israel). In addition, the results are in line with a previous large-scale study (Mevarech, 1994) showing the beneficial effects of IMPROVE as compared to a nontreatment control group in which mathematics was taught in heterogeneous classrooms with no ability grouping. The findings regarding mathematical reasoning need further consideration. Why did the higher, middle-, and lower achieving students benefit from IMPROVE? One reason may be related to the learning materials used by IMPROVE students that contained a large percent of challenging mathematics tasks embedded within metacognitive questioning which prompts mathematical thinking. For example, when "new" mathematical operations were introduced (e.g., powers and roots), the enrichment activities presented the mathematical operation n\ followed by interesting tasks related to this operation. Usually, seventh graders are not involved in such challenging tasks. In the meantime, while the high achievers worked on this type of 381


Mevarech and Kramarski task, the low achievers corrected their learning by using learning activities that were different from those used in the initial instruction. Another reason may be associated with the structure of the treatment that provides teachers with specific recommendations for when and how to incorporate the corrective-enrichment activities into their instructional plans. As described above, IMPROVE devotes special time for corrective/enrichment activities that is built into the curriculum table (every other week, one period was devoted to these activities). This is not to say that control teachers did not implement corrective activities. But they did not do it on a regular basis, nor did they emphasize the importance of the corrective/enrichment activities that provide each student the opportunity to reach a mastery level. These findings echo the findings of Fuchs et al. (1994) who showed that students who were exposed to CBM followed by corrective instruction scored higher on mathematics achievement tests than students who were exposed only to the feedback with no instructional recommendations, or the control (with no feedback and no instructional recommendations). Furthermore, teachers who implemented CBM with instructional recommendations tended to use peer tutoring and cooperative learning more often than teachers who implemented only CBM with no instructional recommendations or the control. Whereas IMPROVE students at all achieving levels outperformed their counterparts on the mathematical reasoning, significant differences between the experimental and control groups on the Introduction-to-Algebra were observed only for the middle- and higher achieving students. Several reasons may explain this finding. First, it is possible that in both groups the lower achieving students did not attempt the advanced knowledge computation items that require several computational steps. Second, the examination did not tap the differences (if any exist) between the experimental and control groups because it did not include enough items that were sensitive to the variations in the performance of the lower achieving students. Finally, being trained to articulate their reasoning, lower achieving students were better able to provide explanations but not to solve computational problems and basic factual knowledge items than their counterparts in the control group. The findings of Study I raise two related questions. First, the study examined students' achievement at the end of the first semester. It is not known to what extent the effects of IMPROVE can be maintained throughout a whole academic year. Second, the study focused on Introduction-toAlgebra. One may argue that performance on that topic is better for the IMPROVE students because the topic is considered to be less abstract, often relates to real contexts, and is within the experiential realm of seventh graders. By comparison, teaching algebra involves abstract concepts, such as variables and expressions, and thus may be inappropriate for heterogeneous classrooms where high-achieving students may progress much faster than low-achieving students. To gain further information on the effects of IMPROVE on achievement in algebra, we designed an additional study in which seventh graders were followed through a whole academic year. 382


IMPROVE: A Multidimensional Method for Teaching Study II Participants Participants were 265 seventh-grade students who studied in four junior high schools. Six intact classrooms (A^ = 164), randomly selected from the entire pool of classrooms (50 classrooms) using IMPROVE, served as the experimental group, whereas three classrooms (TV = 101), randomly selected from a different district in which mathematics was taught in ability tracks, were used as a nontreatment control group. Materials, treatments, and teacher training were identical to those employed in Study I, except that we designed the learning materials for the experimental group to be used during the whole academic year. (Learning materials for IMPROVE are now available for seventh, eighth, and ninth grades.) The schools were similar in terms of size, socioeconomic background, and pretest achievement in arithmetic (see below). Measures A 48-item test was used to assess students' performance in algebra. The test covered the following units: numerals and rational numbers, variables and algebraic expressions, substitutions in algebraic expressions, linear equations with one variable, converting words into symbols, and using equations to solve problems of different kinds. As in Study I, the Israel mathematics supervisor and the mathematics department heads of both the experimental and control groups reviewed the test prior to using it and determined its validity for assessing seventh-grade achievement in algebra. Items that were rated as having low validity were excluded from the tests. Scoring: Students received a score of either 1 (correct answer) or 0 (incorrect answer), and a total score ranging from 0 to 48. The scores are reported in terms of percent correct. KR reliability coefficient = .94. In addition, all students were tested at the beginning of the school ye^r by a pretest that focused on arithmetic skills taught at the elementary school. The pretest was the same as that used in Study I. Results and Discussion As in Study I, the data are multilevel, with classrooms nested within treatment and crossed with ability. Thus, the data were analyzed by a 2 (treatment: experimental vs. control) X 3 (Ability: higher, middle, and lower achieving students) ANOVA with classrooms nested within treatment and crossed with ability. To examine the treatment X ability interaction, students were assigned to three achievement levels, as in Study I, The number of students in each ability group was 76, 92, and 97 lower, middle, and higher achieving students, respectively. Table 2 presents the mean scores and standard deviations by time (pretest vs. posttest), treatment (experimental vs. control), and aptitude (low-, middle, and high-achieving students). The 2 X 3 ANOVA with classrooms nested in treatment and crossed with ability showed significant differences between the experimental and control 383


Mevarech and Kramarski groups on students' performance in algebra, in favor of the experimental group (Kl, 7) = 6.31, p < .04). The aptitude X treatment interaction was not significant, (F(2, 14) < 1.00, p > .05). As may be seen from Table 2, although at the beginning of the school year there were no significant differences between the experimental and control groups (F< 1.00, p > .05), at the end of the year, the overall algebra mean score of the IMPROVE students was 73.08 (SD = 22.99), whereas that of the control group was 63.98 (SD = 24.68). Further analysis of students' performance indicates significant differences between the experimental and control groups in favor of IMPROVE on all topics except operations with algebraic expressions (F(l,7) = 2.73, p > .05). The F values of the treatment effects (with classrooms nested in schools used as an error term) were 7.96, 5.72, and 6.28 for rational numbers and numerals, substitution in algebraic expressions, and algebraic word problems (including converting words into symbols and using equations for solving word problems), respectively; the largest F value of the ability X treatment interaction was .54, allp values > .05. It is possible that operations with algebraic expressions were less affected by IMPROVE, because this Table 2 Mean Scores and Standard Deviations (in Terms of Percent Correct) on Algebra by Treatment IMPROVE

Treatment

CONTROL

Ability

Low

Middle

High

Low

Middle

High

Pretest M S

50.71 13.29

76.20 5.59

92.19 4.33

52.32 10.40

74.52 5.65

90.39 4.35

Algebra (Overall) M S

55.90 20.95

73.94 21.09

86.58 16.60

52.65 30.08

63.72 18.88

71.77 22.30

Numerals M S

68.57 20.24

80.88 17.16

90.30 13.72

63.21 28.35

75.09 15.38

77.95 19.76

Substitution M S

55.00 21.96

71.04 25.78

83.91 20.27

49.62 34.17

62.50 25.28

70.38 27.37

Expressions M S

53.18 28.11

71.92 30.84

87.06 19.26

51.58 29.20

57.84 25.73

73.30 28.76

Word Problems M S

46.32 27.92

70.44 24.84

84.57 20.47

45.45 34.97

57.15 22.64

66.20 27.87

384


IMPROVE: A Multidimensional Method for Teaching topic requires a lot of practicing, whereas IMPROVE emphasizes articulation of mathematical reasoning, comprehension, and resolution of mathematical conflicts. Interestingly, the positive effects of IMPROVE were observed not only on topics taught just before the time of the examination, but also on those introduced in the first semester, about 6 to 8 months before the administration of the final test (F (1,7) = 6.92, p < .03). Although there is a treatment main effect and no interaction between level of achievement and treatment, post-hoc analysis indicated significant differences between the experimental and control groups for the higher achieving students and marginally significant for the middle-achieving students (lvalues = 3.44 and 2.34;p values < .05 and .052, respectively), but not for the lower achieving students. These findings are in line with the findings of Study I. It is possible that lower achieving students need additional support in order to further enhance their achievement. Interestingly, according to Table 2, the mean score of the middle-achieving students who were exposed to IMPROVE was similar to the mean score of the higher achieving students who were in the control group. The findings of Study II show the positive effects of IMPROVE over the entire academic year. The effects were observed on the overall mean scores as well as on each topic, except for operations with algebraic expressions that require a lot of practicing. These findings suggest that one may exploit the diversity in students' prior knowledge as a resource for enhancing mathematics achievement more than the teaching in between-class ability grouping. The fact that IMPROVE was implemented during an entire academic year may have practical implications for teachers who work in heterogeneous classrooms and do not like to (or cannot) change the structure of the class during the school year. This is particularly important because, throughout the whole year, the progress of one academic group did not come at the expense of the other groups.

General Discussion and Conclusions Summary of Results The present research shows that (a) mathematics can be taught in heterogeneous classrooms with no tracking; (b) the move from tracking to heterogeneous classrooms should be followed by alternative methods such as IMPROVE, which is based on metacognitive training, cooperative learning, and systematic provision of feedback-corrective-enrichment; (c) IMPROVE highly facilitates mathematics achievement; (d) the effects were particularly strong on mathematics reasoning (Study I); and (e) IMPROVE can be implemented throughout an entire school year (Study II). (At present, IMPROVE is implemented in Grades 7 to 9 in 35 schools). Interpretations and Unanswered Questions Although implementing IMPROVE in heterogeneous classrooms produced beneficial effects on students' mathematical reasoning, several questions still 385


Mevarech and Kramarski remain on both theoretical and practical levels. First, the present research investigated the combined effects of the three components that constitute IMPROVE: metacognitive questioning, cooperative learning, and the provision of feedback-corrective-enrichment. It showed that the three components interacted to produce the positive results. It would be interesting to examine the separate effects of each of the three components as well as the effects of all the possible combinations of these components. To address this issue, a different experimental design is needed. For example, cooperativemastery learning embedded within metacognitive training may be compared with cooperative-mastery learning with no metacognitive training. This issue, however, is beyond the scope of this research and merits future research. The second issue relates to the relationships between cognitive interaction and mathematical understanding. In particular, what kinds of cognitive responses are promoted in small learning groups guided by metacognitive questioning and feedback-corrective-enrichment? Students construct knowledge when involved in the IMPROVE program, but they also construct knowledge when engaged in the activities provided in the control classrooms. Yet, there are differences in the knowledge constructed under the two conditions. Our observations showed that cognitive responses of IMPROVE students often consisted of multiple perspectives and verbal explanations supported by evidence and mathematical principles. Marx and Walsh (1988) termed these kinds of cognitive responses comprehensive and opinion plans. In using comprehensive plans, students "draw inferences about information or procedures, apply previously learned procedures to new situations, select appropriate procedures for new problems, and paraphrase or transform information or procedure." In applying opinion plans, students "comprehend the nature of the problem, examine evidence, and state an opinion based in part on that evidence" (Marx & Walsh, 1988, pp. 212-213). This is in contrast to the cognitive responses provided by the control students that often consisted of only the final answer without showing the need to elaborate, seek relationships, or justify the answer. As an example, consider the following problem taken from the final exam: If a and b are inverse numbers ( | a | = - | b | ), is the following statement ab > a+b definitely true/possibly true/never true? Please explain your reasoning. The control group rarely explained their reasoning, and, when they did, they seldom used mathematical principles to support their reasoning. Instead, the ones who justified their responses gave a specific example that illustrates the correctness of their answer, as seen in the following example: "It is never true because 3 + (-3) = 0 and 3 x (-3) = -9." IMPROVE students often pursued the problem as follows: a and b are inverse numbers; therefore, their sum is 0, because 0 is the identity element with respect to addition, but not with respect to multiplication. On the other hand, if a is ^ positive number, then b is a negative number (because they are opposite numbers), and their

386


IMPROVE: A Multidimensional Method for Teaching product is a negative number (according to the multiplication principle). Since a negative number is always smaller than 0, the statement is never true." To illustrate their answers, many students provided a specific example. The third issue refers to the question of how the diversity in students' prior knowledge can be utilized when students learn in small heterogeneous groups. This question is of special interest because several features of the situation provide a solid base of information about how children learn. Our observations showed that utilizing the diversity in students' prior knowledge is related to the type of cognitive plans students use in accomplishing classrooms tasks. The more the instructional tasks were based on comprehension and opinion plans, the more relevant the diversity in students' prior knowledge was, and the more power it had to affect students' use of such cognitive plans. We found that, when tasks required automatic application of procedural plans, utilizing the diversity in students' prior knowledge was not relevant to accomplishing the task. In this case, students' interaction was minimal with almost no explicit use of metacognitive activities. Similar findings were reported by Mevarech (1991). In contrast, utilizing the diversity in students' prior knowledge was most relevant when learners drew inferences about information or procedures, applied previously learned procedures to new situations, and selected appropriate procedures for solving new problems. In such situations, each learner approaches the task from a different perspective, applies different strategies, and brings to the learning situation his or her prior knowledge and experience. Utilizing the diversity in students' prior knowledge was also relevant when learners had to state an opinion, justify what they did, and explain their reasoning. As an example, consider how a team solved the following problem: Is the sum of two consecutive numbers (numbers following each other) an even number or an odd number? Student A: It's an odd number because 2 + 3 is 5, and 5 is odd. Student B: That does not say anything. It's just an example. Student C: Yeah, but so is 4 + 5, 6 + 7, 7 + 8 . . . Student B:

So what? These are also examples. We cannot check all the numbers in the world.

Student D: Let's think, what's in the problem? Two consecutive numbers; what do we need? Their sum. Two consecutive numbers can be a and « + i; their sum is2a + 1\ therefore it's odd. Student A:

I don't get it. 387


Mevarech and Kramarski Student B:

We can call a number a, agree? (Student A nods yes.) Its consecutive is a+1. (Student A nods yes again.) Their sum isa + (a+1) which is 2a+l. Now, 2a is even because it's a product of 2; if you add 1 it becomes odd. OK?

Student A: Yes, let's write it down. This example shows how all students, regardless of their prior knowledge, participated in the interaction. Similar findings were recently reported by Cohen and Lotan (1995) who used a different method to facilitate cognitive interaction in heterogeneous classrooms. It also shows how the cognitive interaction that took place under IMPROVE promoted mathematical understanding. First, it raised the need to state mathematical ideas in a formal way and to use correct mathematical language. Second, it provided students the opportunities to realize that a problem can be solved in different ways and that in mathematics, as in other areas, there are conflicting ideas that should be discussed and resolved. Finally, it encouraged students to look at aspects that they had not noticed by themselves and to compare their own. solutions with those of others. Learning in tracks rarely exposes low achievers to similar kinds of mathematical discussions, nor does it encourage high achievers to articulate their reasoning in such elaborate ways. This is in line with Schoenfeld (1987): While the individual might generate one possibility and go off in pursuit of it, a group might generate three or four—^and precisely because they have more than one option, the group will have to decide among them. In consequence, the individual might have to formulate and defend one point of view, listen to and evaluate others, and finally take part in a group decision regarding which one(s) to pursue and for how long. These are precisely the self-regulation skills the individual needs to develop, and it is difficult to imagine a context (cooperative setting) in which they could develop more naturally. (p. 211) Although the present research did not focus on teachers, our impression based on talks with teachers, principals, and the superintendent indicates that IMPROVE teachers experienced a dramatic change in their attitudes and perceptions. Prior to implementing IMPROVE, all teachers taught mathematics in tracks using the traditional method of instruction. They had never used cooperative learning, nor did they employ a metacognitive training and/or systematic feedback-corrective-enrichment. They perceived the teaching in tracks as the best way of teaching mathematics, and they were sure that teaching mathematics in heterogeneous classrooms was impossible. When first starting to implement IMPROVE, teachers were still skeptical. They worried that the method would exert debilitating effects on their students' achievement, and they were not sure that high-achieving students would benefit from the method. At the end of the school year, most teachers realized that there are alternative ways for teaching mathematics. In fact, some teachers internalized the innovative method so deeply that, a year 388


IMPROVE: A Multidimensional Method for Teaching later, when we asked teachers to use the traditional method of instruction for research purposes, they had tremendous difficulties. The issue of changes in teachers' perceptions is still open. Future Research The research reported here raises several questions for future research. First, it focused on seventh-graders who were starting to study algebra. Future studies may investigate additional variables that have not been examined in the present study. For example, little is known of the effects of IMPROVE on students' motivation, self-esteem, and psychosocial relationships. These issues are particularly important in heterogeneous classrooms where students with different backgrounds and prior knowledge learn together. Second, training students to activate metacognitive questioning in one area (e.g., mathematics) always raises the question of the extent to which students can transfer that knowledge to solving problems in other areas. For example, to what extent do students who were exposed to IMPROVE apply similar learning processes in science classes or in reading-comprehension classes? From an information processing perspective, asking self-monitoring questions may lead students to be aware of problem-solving processes and may by that induce students to activate the 4-phase, problem-solving model suggested by Polya (1973). It is possible that once students are trained to activate domain-specific metacognitive questions in one area, they can easily learn to use similar domain-specific metacognitive questions in other areas. On a practical level, following students during a whole academic year introduces several developmental questions such as: How does the use of metacognitive questioning, cooperative learning, and corrective-enrichment activities vary as a function of time? During the year, who are likely to find more difficulties in using the metacognitive questioning? To what extent are developmental patterns of low-achieving students different from those of high-achieving students? Is it always the same students who consistently require corrective feedback? And how does the percentage of students achieving mastery on the first administration or after corrective feedback is provided vary during the year? In the present study, the teachers modeled the use of the metacognitive questioning in front of the entire class and during the work with each team. In addition, students were continually encouraged to use metacognitive questioning and answering. Although apparent difficulties have not been observed, either in our study or in previous research (e.g.. Brown & Palincsar, 1989; King, 1994), collecting data about the use of metacognitive questioning at the beginning of the study and on multiple occasions during the year would reveal important information about the stability and change in students' cognitive and metacognitive behavior. Finally, the present study did not provide systematic observations of student interactions. Undoubtedly, it would be helpful to examine students' conversation and how they work together, because this is a central component of IMPROVE. In particular, it will be interesting to examine the extent 389


Mevarech and Kramarski to which the positive effects of IMPROVE are due to emphasis on articulation and correct language that, in turn, are enhanced by metacognitive questioning, cooperative learning, and feedback-corrective-enrichment. Although some of the observations reported here showed how students helped one another to learn rather than simply telling the answer, a closer look at what happens in peer groups during the implementation of IMPROVE would help to explain (a) why and how the program works and (b) how the program can be modified so that it works more effectively in the future. Summary and Conclusions Given the current reform in school structure and demographic changes indicating that student diversity will continue to grow (Fuchs et al., 1994), educational practitioners, administrators, and teachers are deeply concerned with instructional methods appropriate for heterogeneous classes. IMPROVE, like Success for All (Slavin, 1993) and other cooperative methods (e.g., Slavin, 1990), requires a serious commitment to restructuring junior high schools. It shows that tracking is not a necessary condition for teaching mathematics. It also shows that when a school makes a commitment to implementing IMPROVE it can succeed. To ensure success, we worked with the entire school mathematics faculty, not just with a single teacher. By that, we achieved a change that could be maintained over a long time. The findings summarized in this research offer practical evidence that we can successfully teach all students higher cognitive processes and that the progress of the slow learners does not necessarily come at the expense of the fast learners, or vice versa. It also shows that instructional methods that are derived from current sociocognitive and metacognitive theories are practical and workable in modifying the learning processes that take place within heterogeneous classrooms and that the beneficial effects of the method can be maintained through an entire school year. The potential of IMPROVE to enhance other schooling outcomes and to affect other kinds of students merits future research. APPENDIX A The Seventh Grade Curriculum for Heterogeneous Classrooms The Properties of Mathematical Operations Commutative and associative properties Identity elements for addition and multiplication The distributive property Order of operations Converting words into mathematical symbols Rational Numbers Positive and negative numbers The number line and the number order Addition and subtraction Opposite numbers 390


APPENDIX A (continued) Absolute value Multiplication and division Division by zero Reciprocal numbers Cartesian system and ordered pairs Algebraic Expressions Variables Algebraic expressions and equations Substitutions and substitution sets Converting words into algebraic expressions Word problems with one variable Percents and problems with percents Sets Sets, subsets, and elements Venn diagrams Intersection and union of sets Descriptive Statistics What is descriptive statistics? Frequencies Displays of data Bar graphs, histograms, pictograms, and pie diagrams Central tendency measures

APPENDIX B The Seventh Grade Curriculum for Low-, iVIiddle-, and High-Achieving Tracl(s High

Middle

Low

The Properties of Mathematical Operations Commutative and associative properties Identity elements for edition and multiplication The distributive property Order of operations Converting words into mathematical symbols

+ + + + +

+ + + + +

+ +

Rational Numbers Positive and negative numbers The number line and the number order Addition and subtraction Opposite numbers Absolute value Multiplication and division Division by zero Reciprocal numbers Cartesian system and ordered pairs

+ + + + + + + + +

+ + + + + + + + +

+ +

+ + + +

+

+

Algebraic Expressions Variables, algebraic expressions, and equations

391


APPENDIX B {conWmed) Substitutions Substitution sets Converting words into algebraic expressions Word problems with one variable Percents Problems with percents

+ + + + + +

+ + + + + +

Sets Sets, subsets, and elements Venn diagrams Intersection and union of sets

+ + +

+ + +

-

Descriptive Statistics What is descriptive statistics? Frequencies Displays of data Bar graphs, histograms, and pie diagrams Pictograms Central tendency measures

+ + + + + +

+ + + +

—

—

+

+ + +

-

Notes Part of this article was presented at the Annual Meeting of the American Educational Research Association, New Orleans, 1994. Many thanks to Ms. Dorit Ortal, the superintendent of the district, for her great assistance in all stages of this research and to Dr. J. Reif for his helpful comments.

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