Implementation of multiple solution connecting tasks: do students' attitudes support teachers' reluctance?

Abstract

This study focuses on the pitfalls associated with incorporating multiple solution connecting tasks in classroom instruction. We found implementation of these standard-based materials problematic due to teachers' reluctance to use them in their classes. Research questions probed the teachers' explanations for their lack of enthusiasm. The tasks were implemented with students of two tenth-grade classes on the basic and medium level. Contrary to the teachers' opinions, unexpected number of the students demonstrated positive attitudes to coping with connecting tasks and were able to differentiate between different solutions. Although teachers thought that students would resist receiving and providing peer explanations, the students mostly did not verify such a resistance. Moreover, students learning basic-level mathematics preferred providing explanations to their peers, while students on the medium level preferred receiving explanations. We believe that a special intervention is needed to convince mathematics teachers to employ connecting tasks in their classrooms. Presenting teachers with the results of this study may constitute one of the parts of such an intervention.

Introduction

In this paper a task that may be attributed to different topics in the mathematics curriculum and thus may be solved in different ways is called a multiple-solution-connecting-task (shortly connecting task). The implementation of connecting tasks, which are under consideration in this paper, may serve as an effective model of mathematics teaching (Cooney, 2001; Simon, 1997), which entails student-centered learning and fosters construction of students' mathematical understanding. This study is based on the belief that mathematical thinking involves looking for connections, while making connections is essential for constructing mathematical understanding (NCTM, 2000). When students begin to see connections across different branches of mathematics they develop their mathematical integrity. We connect theory and practice through holding that solving problems in different ways is a powerful mathematical activity that promotes constructions of mathematical connections.

Example

The problem:

Find the distance between the point A (1, 6) and the straight line l : y = x-1.

This problem is borrowed from analytical geometry and usually is solved in mathematics classes in one particular way. A connecting task focused on this problem asks students to learn how to solve it in different ways. Figure 1--"map of the task"--presents a variety of the solution paths for this mathematical problem. It establishes connections between the concepts by means of four equivalent definitions. In this way different geometrical concepts (i.e., circle, area, perpendicular lines) are connected by the concept of distance. Additionally, within each approach different solution paths are possible. These paths connect different mathematical topics and concepts. For example, the minimal value of a quadratic function (in one of the solutions can be found in different ways (see Figure 1).

Theoretical Background

As literature shows, connections form an essential part of mathematical understanding (e.g., Hiebert & Carpenter, 1992; Kieren, 1990; Sfard, 1991; Sierpinska, 1990; Skemp, 1987). Skemp, for example, discussed two types of mathematical understanding, relational and instrumental. Relational understanding determines one's ability to answer such questions as "How should I approach the problem?" "Why should I do this?" Instrumental understanding facilitates answering the question, "What procedure should I use to solve the problem?" without necessarily knowing the answers to the first two questions. The two types of understanding are critical for making progress in learning mathematics. Instrumental understanding determines the speed of solving procedures and the immediacy of the results. Yet without mathematical connections between different procedures and concepts this kind of understanding may overload the memory. Instrumental understanding without relational understanding may produce no results in thinking about how familiar concepts and procedures may help in new situations. Relational understanding includes connections between different mathematical concepts, helps students to progress on the basis of previous knowledge, and forms students' expectations of the mathematical ideas to be related. Rich mathematical tasks prompt students' relational understanding, which helps students think: "How is this problem similar to what I have done before? How is it different?"

[FIGURE 1 OMITTED]

NCTM (2000) stresses the importance of teaching and learning mathematics through construction of mathematical connections of varied nature: among different mathematical concepts and procedures, among different branches of mathematics, between mathematics and other scientific fields. One of the well recognized ways for developing connectedness of one's mathematical knowledge is solving problems in different ways (Dhombres, 1993; House & Coxford, 1995; NCTM, 2000; Polya, 1963,1973,1981; Schoenfeld, 1983, 1988; Vinner, 1989). Stigler and Hiebert (1999), in their comparative analysis of mathematics lessons in the United States, Germany, and Japan, found that encouraging the idea that there could be multiple solutions to problems enhanced the lessons' quality. Schoenfeld (1983) points out that awareness of the chance of solving problems in different ways helps students not to give up when working on them. Dhombres (1993) suggests that providing two different proofs for a particular theorem opens different routes for solvers in their mathematical knowledge, each of which may be available when appropriate.

Despite the importance of the implementation of connecting tasks, teachers seldom solve problems in different ways either for themselves or in their classrooms (Leikin, 2003; Ma, 1999; Schoenfeld, 1988). NCTM (2000) standards stress that "problem selection is especially important because students are unlikely to learn to make connections unless they are working on problems or situations that have the potential for suggesting such linkages. Teachers have to take special initiatives to find such integrative problems" (p. 359). We assumed that constructing or discovering such tasks is a very difficult assignment for mathematics teachers, who usually are over-loaded with their regular responsibilities, therefore we developed a collection of such tasks (Leikin, Gurevich & Mednikov, 2002) to support and encourage their implementation in a mathematics class. The example provided earlier in this paper is borrowed from one of the mathematical activities presented in the collection.

Implementing in Practice

To design or identify connecting tasks as well as to develop classroom activities focused on connecting tasks one needs deep and connected mathematical knowledge as well as awareness of different aspects of pedagogy of mathematics. Accordingly a development team consisting of high school mathematics teacher, a professional mathematician, and a mathematics teacher educator designed a collection of learning activities focused on multiple-solutions-connecting-tasks (Leikin et al, 2002). The team worked collaboratively, each member having a particular responsibility in the design process. All three members discussed each other's contributions to the collection.

Each activity in the collection addressed the three components of the teaching triad (Jaworski, 1994): mathematical challenge, sensitivity to students, and management of learning. Overall the collection consists of six learning units, each of which includes students' activities and a teacher's guide. The collection exposes different types of mathematical connections: between mathematical objects and mathematical tools through solving a problem in different ways, or through solving different problems in similar ways. Solving problems in different ways and constructing mathematical connections constitute mathematical challenge of the tasks. Some learning units include several versions of the mathematical activity assigned to different learning levels. Note that the mathematical activities developed for students of different levels vary in the recommended learning settings and in the explanations to the solution provided in the ready-to-use materials. In this way the collection assists sensitivity to students and encourages student-centered management of learning.

Most of the activities were designed on the basis of cooperative learning methods as small-group learning methods have been shown to increase learners' communications, and consequently to raise the effectiveness of learning processes in mathematics in general and to evoke a richer problem-solving procedure in particular (Good, Mulryan & McCaslin, 1992; Goos, Galbraith & Renshaw, 2002; Leikin & Zaslavsky, 1997; Noddings, 1985; Webb, 1991; Webb & Farivar, 1999). Noddings' analysis of small groups as a setting for research on mathematical problem solving and use of "pair problem-solving" in other studies in the past (e.g. Schoenfeld, 1985) showed pair problem-solving to be an effective tool to evoke richer problem-solving procedures. As these studies suggested, learners working in pairs are far more verbal than learners working alone, and their solutions are often more elaborate and advanced. Cooperative learning procedures were also shown to improve individual performance as meaningful peer explanations were provided (Webb, 1991; Webb & Farivar, 1999; Vye, Goldman, Voss, Hmelo, & Williams, 1997). However, an optimum small group setting is one in which students consistently give each other detailed explanations about how to solve the problems and give each other opportunities to demonstrate their level of understanding (Webb, 1991; Webb & Farivar, 1999).

Consequently, we recommended that students be involved in peer explanations throughout the activities. In order to facilitate students' explanations, most of the activities for medium and basic level students included worked out examples. The decision to use worked out examples was based on multiple studies which demonstrated that coping with worked-out examples facilitates the acquisition of knowledge required for problem solving in mathematics and science (e.g., Ward & Sweller, 1990; Zhu & Simon, 1987).

Appendix 1 shows the structure of the collection, including the topics from which the problems were borrowed, types of mathematical connections expected to be learned in each unit, the recommended teaching method for each unit, and an evaluation of the difficulty level of each activity.

The Study

The Purpose of the Study

In order to disseminate teaching mathematics with connections by means of multiple-solutions-connecting-tasks our collection was presented to 30 high school mathematics teachers who took part in a seven-hour workshop. Prior to our experiment, none of these teachers had any experience in teaching multiple-solution-connecting-tasks as outlined in this paper. Thus, they took part in the workshop to learn about connecting tasks. The teachers were asked to implement the materials in their classes. Unexpectedly, only seven of these teachers volunteered to participate in the teaching experiment. Moreover, only two of these teachers agreed to collaborate fully with the researchers, namely to apply recommended cooperative learning settings, to allow classroom observations, to be interviewed, and to ask their students to complete the questionnaires.

In this way we realized that the teachers were reluctant to implement connecting tasks in their classes. To understand reasons for this reluctance, at the end of the workshop, we managed a whole group discussion in which the teachers were asked to explain their unwillingness. Analysis of the discussions with the teachers revealed several reasons the teachers used to explain their lack of enthusiasm regarding implementation of the suggested (ready-to-use) mathematical activities.

First, the teachers thought that these activities might be appropriate (if at all) only for students who learned mathematics on the high level. The teachers conjectured that students who studied mathematics on the basic and medium level would not like the activities: they would be confused by the multiple choices of the solution strategies.

Second, the teachers assumed that the recommended cooperative learning setting "will not work" because these students did not have "explaining skills". Basic-level and medium-level students would not be able to explain the solutions to their peers, and explainers and explainees alike might be "very negative" about explanations that constituted an integral part of the recommended learning setting.

Based on these teachers' negativistic interferences, our study was aimed at examining (a) students' attitudes to coping with the connecting tasks, (b) students' attitudes to giving and receiving explanations, and (c) the ways in which students characterized different ways of solution.

The Setting

The materials were tried out with three different groups of students. Group A included 22 tenth-grade students learning mathematics on the basic level. Group B included 23 tenth-grade students learning mathematics on the medium level. Group C consisted of 4 twelfth-grade students learning mathematics on the medium level. Each group worked in slightly different learning settings, and correspondingly followed a slightly different research procedure (see below). Data were analyzed and summarized for each group separately.

For Groups A and B one of the authors, who had joined the team for the research purposes, took written field notes during the activity, which took two lesson periods for each group. At the end of the activity the students of groups A and B were asked to complete written attitude questionnaires concerning their new experience with connecting tasks (See Figure 2). The mathematics teacher of Group A was interviewed. The interview was videotaped.

Students in Group C took part in two special meetings each taking up about two 45-minute lessons. The students in Group C were videotaped while working together in a Jigsaw setting. During each of the two meetings they experienced solving the problem--presented in the Example earlier--in two out of the four ways. The students twice filled out the attitude questionnaire concerning coping with connecting tasks: once at the end of each meeting.

Transcripts were made for all the videotapes. The data were analyzed and categorized by the researchers based on the transcribed videos and students' answers to the questionnaires.

Students' Attitudes to Coping with Connecting Tasks

Our observations in the classrooms as well as the results of the written questionnaires revealed diversity in students' attitudes to solving problems in different ways, depending on the student's learning level. Students who performed better in mathematics were more willing to take part in the activity. They said that they were able to enjoy this mathematical activity, and some of them wondered why they were encountering that kind of assignment for the first time. On the other hand, students who usually demonstrated poor mathematical performance were frustrated; they wanted to "go back" to the regular setting of "getting prescriptions for how to get the correct answer". We observed these differences within each one of the classes, and they were supported to some extent by comparison of the questionnaires obtained from Group A (basic level) and Group B (medium level). Table 1 summarizes main findings from the questionnaires in these two groups.

Table 1 shows that the medium-level students were more positive towards their participation in the activity than the basic-level students: 64% (14 of 22 students) vs. 87% (20 of 23 of students) having positive attitudes; 22% (5 of 22 students) vs. 9% (2 of 23 students) having negative attitudes. It was surprising that 64% of the students in the basic-level class were positive about solving problems in different ways, considering the impression we got while observing the activity. We found that the main motives for being positive about coping with the connecting tasks were similar in the two classes. We identified two main categories of these motives for the two groups of students: (a) having the option of choosing an easier solution, and (b) judging mathematics lessons as more interesting. Note that at this stage of the work on the task the students had not yet discussed mathematical connections appropriate to the task, thus they did not address this issue.

The following citations from the students' questionnaires demonstrate that the students in our experiment explained the advantages of the connecting tasks in ways similar to those we tried to present to the teachers in our workshops. The majority of the replies were as follows:

Mila:    This [activity] opened for me a new way of thinking. In other
         words, you don't need to solve all the problems in the same
         way, but you can use different ways and this is better. It's
         easier. I can choose what's easier for me.
Tom:     Each of us understands in his own way. This [coping with
         connecting tasks] gives richer opportunities and more chances
         for more students to understand better. Pupils can choose
         different ways of solutions and they are not bound to one
         specific way.
Shelly:  This makes me think that you always can find a solution. If one
         way doesn't work try another way.
Sean:    It's nice to see that there are different solutions to one
         problem. This proves that the solution is real and not
         accidental, and this shows you how clever mathematics is.

True, some pupils expressed opinions that echoed the teachers' thoughts about the activities focused on connecting tasks. Yet only a small number of pupils (see Table 1) expressed negative attitudes, like the following:

Sharon:  It confused me pretty much because these are solutions that are
         specific for this problem, and other problems will probably
         need different solutions. So this is confusing, and I prefer
         one particular solution that will fit most of the problems that
         we learn in school.

The distribution of students' references to these categories in their questionnaires differed in groups A and B. For the students in group A (basic level) the dominant reason for liking connecting tasks was the option to choose an easier solution: 86% of students with positive attitudes (12 of 14) wrote that kind of justification. Only 14% of these students (2 of 14) mentioned that "this way of learning is more interesting." Distribution of students' motives for being positive towards the connecting tasks was different in Group B (medium level). Precisely 40% (8 of 20) of students in Group B liked dealing with connecting tasks because of the option to choose the easiest solution; 55% (11 of 20) of students in this group found this activity to arouse interest and to variegate the subject. Five out of 20 (25%) students with positive attitudes in Group B mentioned an additional reason for liking connecting tasks. They wrote this kind of task enabled them to see the beauty of mathematics. Students in Group A did not refer to this category at all. Note that contrary to group A, in which the students mentioned only one reason for (dis)liking connecting tasks, some students in Group B mentioned more than one reason.

Data obtained from students in Group C show that students' attitudes to coping with the connecting tasks may change through their involvement in activities of this kind. These students completed the questionnaires twice, and even during such a short period of time we saw some changes in their answers. The first time, they reported that solving problems in different ways was confusing and they felt resistance to dealing with connecting tasks. The second time, they reported that their resistance had become weaker, if it had not disappeared altogether, and they considered the option to choose an easier solution a great advantage of the setting.

Providing and Receiving Explanations.

As noted earlier, explaining is an integral part of most of the activities included in the collection. In this study we deemed explaining to be a tool for exploring and solving problems in different ways. Through the lens of explanations we might see how the students cope with the problem. We took the students' ability to explain the solution as an indication of their understanding of it (Kieren, 1990). If students reported that they had benefited from their peers' explanations, we regarded these explanations as indicating understanding by the explainers. A negative attitude to receiving explanations could be due to the quality of the explanations or to the lack of understanding by their recipients.

Students in both groups (A and B) were asked what they preferred: giving or receiving explanations. Table 2 shows the students' answers.

We found the results shown in Table 2 surprising. One would have anticipated that most of the basic-level students would prefer receiving explanations. Instead, more than half of the students who answered the question (55%: 10 of 18 students) preferred giving explanations to their peers, and they found giving explanations helpful. Only 18% of the students (3 of 18) preferred learning by receiving explanations. Two students in Group A stated that they liked neither giving nor receiving peers' explanations: they preferred the teacher's explanations, which (in their opinion) were easier to understand. By contrast, 13 out of 20 students from Group B who answered the question (65%) preferred learning from peers' explanations. Five out of 20 students from this group (25%) preferred giving explanations. Some of the students in Groups A and B liked receiving and giving explanations, depending on the quality of the explanations.

Overall the reasons noted by the students in the questionnaires for preferring giving or receiving peer explanations also differed for groups A and B. However, in both groups the main reason for preferring giving explanations was the students' feeling that by doing so they deepened their own understanding. Six students of the basic level and three students of the medium level mentioned this reason. Some examples of what they wrote are the following:

Tami:  When explaining, I may clarify the solution, repeat the material,
       and verbalize my thinking. This helps me to understand the
       solution better.
Sean:  I prefer to learn through explaining to other people, because I
       learn myself and I also help others.
Anat:  Explaining helps. When somebody else explains to me, I often
       can't understand his thinking, or even the teacher's thinking.
       When I repeat and verbalize my thoughts and explain to others
       what I think myself, I understand it better and memorize it.

Those who preferred receiving explanations felt it was the more usual way, conducted in most of the lessons in the form of the teacher's explanations:

Keren:  I prefer learning when my friends explain to me because this is
        a familiar and common way of learning. But I also felt
        satisfaction when I managed to explain to my friend and she
        understood.

Two main reasons for the preference for giving explanations were put forward by students in Group A only. One was the possibility of checking themselves while explaining, and the other was the difficulty in understanding explanations given by peers. On the other hand, only students from Group B mentioned that they felt satisfaction when they gave explanations. Those who preferred receiving explanations in Group B found it easier and less threatening (7 out of 13 students gave this reason). Three out of 13 students in Group B preferred receiving explanations because they could listen to those who understood better. We speculate here that the differences in students' preferences, and differences in the reasons for these preferences, found in Groups A and B were due to the quality of explanations provided by the students. We suggest that the explanations of the medium-level students were better than those of basic-level students.

Groups A and B were asked to work in slightly different settings. Students of group B were required to give explanations to other students and to listen to peers' explanations. Students in Group A could explain or ask for explanations according to their wishes and need. Thus, only students in Group A were asked whether they gave or received explanations. Fourteen out of 22 students (64%) reported that they gave explanations to their peers. Two of these students doubted whether these explanations were clear or helpful. Only eight out of 22 (36%) students reported that they received explanations and only four of them found peer explanations beneficial.

We found a possible interpretation for this gap in reports on giving and receiving explanations in the interview with the teacher of Group A conducted after the lesson. The teacher said that occasionally she asked more successful students to give explanations to students who encountered obvious difficulties in their learning, which she observed. These results support the above findings about students' preferences regarding giving and receiving explanations. Students in Group A who provided explanations to other students were usually those who understood; the other students had difficulty understanding the solutions. While explaining, those who explained felt they were learning and those who received explanations did not.

However the teacher held the opinion that because of their difficulties some students lost interest in the lesson, and those who tried to help them, at her request, were unable to help; moreover those who explained became frustrated on account of the explaining activity. She said:

    They were frustrated because I told them, "If you understood,
    explain to your friend in the [small] group". While trying to
    explain they called me and said, "It's impossible. They [the
    students who were receiving explanations] don't understand".

This observation may explain students' negative attitudes to providing explanations.

An additional finding rose from the analysis of transcribed videos of Group C. Similarly to Webb & Farivar (1999) we found that students' explanations became more elaborated through their experiences in providing explanations. For example, during the first session students' explanations focused mainly on the procedures and algorithms from the worked-out examples and mainly were based on reading of the worked-out examples.

Ruth:  There is a formula for the distance between the two points ... Do
       you remember?...
Ruth:  Here it is. X minus 1 because you know that Y is X-l. So we
       substitute X-l instead of Y. Minus 6 because you need minus.

During the second learning session students' explanations were more complex, connected and included justifications for the procedures performed. These explanations included students' independent reasoning; the sequence of explanations sometimes was different from that presented in the worked out example. For example, in the episode below, Ruth did not follow exactly the solution path presented in the example. She tried to present to her peer the whole picture of the solution starting from the end, comparing different solution paths and trying to reason about why a certain path is preferable.

Ruth:  This is good. It's amusing. Our goal is to find the distance
       between A(1, 6) and the straight line Y=X-1. O.K.? We do this
       using the formula of area of the triangle. What does it mean? It
       means that the area can be found in two ways: The first is half
       AB multiplied by AE. The second is half AM by AE ...
       Now the question is what is easier AM by AE? No! It is easier to
       find the perpendiculars and if you find them you have the
       area ...
       BE multiplied by half AM equals the known area. So you have only
       AM to find ...
       So the question is how you find the perpendiculars.

Based on this finding we speculate that quality of students' explanations may develop through their explaining activity; thus, students may benefit more from the explanations when they become more experienced in providing explanations. Consequently their attitudes toward connecting tasks may become more positive both for those who provide and receive explanations.

Characteristics of Solutions

As noted earlier, the teachers at the workshop on implementation of connecting tasks in school doubted that students of the basic and medium levels would be able to differentiate between several solutions, characterize them in different ways or put forward any reason for preferring some ways of problem-solving to others. In this study we examined how students characterized different ways of problem solving. They were asked to describe each of the ways of solutions that they had learned during the activity. Table 3 presents for each characteristic the number of students who referred to it. We also examined relationship between students' attitudes to coping with connecting tasks and the problem-solving characteristics that they referred to. This relationship is described below for the cases in which the connections between the characteristic and the attitudes were transparent. Note that some students did not answer this question at all. On the other hand, several students referred to some characteristics two or three times (for example, when they found some solutions easy and others difficult).

Seven main categories for the ways in which students characterized the problems were identified. The most frequent characteristic mentioned by the students was difficulty (ease) of the solution. This characteristic appeared in 14 questionnaires from Group A and in 22 questionnaires from Group B. Most of the students wrote, "This solution is easy", while others found some of the solutions more difficult. On the videotapes with Group C we observed some conversations in which the girls were surprised by the ease of the particular solutions.

Ruth:  Nitzan, look this is really simple, this is very very simple.

Students also compared the solutions from point of view of the difficulty of solutions. Sometimes they agreed about the difficulty of the solutions (as in the following excerpt), in other cases they did not:

Agreement:
Ruth:    My solution is easier.
Nitzan:  You're right. Your solution is easier.

Disagreement:
Ruth:    Look, this way is easier.
Nitzan:  No, I do not think so. I think that the other one is easier.

The next popular characteristic, referred to mostly by students in Group B (15 of 23 students), was the length of the solutions. Only one of the students in Group A mentioned this characteristic. Thirteen out of the 15 students in Group B who found some of the solutions shorter than others were positive with respect to coping with different ways of solutions to mathematical problems. We speculate here that the option to choose shorter solutions out of several that the students learned reinforced the students' positive attitude to the activity.

Equal numbers of students from Groups A and B (7 students in each group) expressed their personal liking ("I (don't) like this way of solution") for the various ways of solving the problems. Five students in Group A and five in Group B found some of the solutions convincing. Four students in Group A indicated some solutions as unconvincing. We found that most of the students who described some of the ways of solving the problem convincing had positive attitudes to the activity. Leikin (2003) found that teachers' opinions on whether a solution was convincing significantly correlated with teachers' preferences in teaching problem-solving using this solution. We speculate here that teachers' craft knowledge includes intuitions regarding the importance of the persuasive power of mathematical solutions for students' attitudes to mathematics lessons.

Nine students, from the Group A only, characterized some of the solutions as challenging. As shown earlier, five students in Group A had negative attitudes to the activity. Three of these five students referred to the mathematical challenge of the solutions. It would seem quite natural for students who learned mathematics at the basic level to characterize mathematical challenge negatively. Yet surprisingly we found that six basic-level students who described some solutions as challenging had positive attitudes to the activity. Another unexpected observation was that medium-level students did not address this problem-solving characteristic at all. We suggest two possible explanations for this finding. On the one hand, the level of the mathematical tasks probably suited this group of students better, so they did not sense a special challenge when coping with the task. On the other hand, perhaps the students who learned mathematics at the medium level had more experiences with tasks constituting a mathematical challenge, and did not find it necessary to use this property to describe solution strategies.

An additional observation arising from the students' questionnaires concerns the explanations that students give to their peers. Seven students from Group A characterized the solutions in terms of ease or difficulty of explaining the solutions. Only one of the students in Group B addressed this characteristic. We suggest that since Group A had more students who preferred giving explanation than Group B, these students were more sensitive to this feature of the learning activity.

Discussion

This paper presents a case study on the implementation of standards-based mathematical activities encouraging construction of mathematical connections of different types. Following recommendations of the NCTM Standards (NCTM, 2000) we considered the connecting tasks described in this paper as a tool for developing students' relational understanding. The connecting tasks presented here are taken as mathematical problems that can be solved in different ways, each belonging to a different curricular topic. We found it very difficult to implement our activities in mathematics classes mainly on account of teachers' reluctance. This reluctance is consistent with findings of other studies attesting that sometimes teachers do not accept solutions students come up with that differ from those they have encountered in the mathematics lessons (e.g., Schoenfeld, 1988). Such teachers' instructional behavior is rooted in their content knowledge and in their beliefs (Cooney, 2001; Leikin, 2003; Schoenfeld, 2000; Sullivan & Mousley, 2001; Thompson, 1992). We think that teachers' prescriptive knowledge (in Kennedy's terms, 2002) has the strongest influence on their decisions. Since this kind of task is not included in final school examinations, and the students are never asked to solve problems in different ways, many teachers find the implementation of connecting tasks pointless.

Teachers were reluctant to implement our activities in their classes because they believed the students (especially at basic and medium levels) would not be willing to handle the connecting tasks. The teachers were skeptical about the peer explanations recommended in our activities, and about the students' ability to see any great difference between the different solutions. Consequently, the research questions in this case study were motivated by the reasons put forward by teachers against employing our ready-to-use activities in their classes. First, we examined students' attitudes to coping with the connecting tasks, second, students' attitudes to explaining activities, and, finally, the ways in which students described different ways of solution. This study showed that the teachers' conjectures regarding employing connecting tasks were for the most part mistaken.

The results of our study were surprising even to us. As expected, we found that students' attitudes were indeed contingent on the level of learning, namely more medium-level students had positive attitudes than basic-level students. Yet we were surprised by the number of basic-level students who reported that they liked this kind of mathematical activity. Analysis showed that the reasons for (dis)liking coping with the connecting tasks were similar for both groups, but the distribution of students' answers with respect to the various reasons differed for the two groups of participating students. We suggest that some of the negative attitudes demonstrated by students in their questionnaires stemmed from the abnormal intensity of mathematics lessons. Ordinarily, when in a mathematics lesson a teacher presents different solutions, or various students solve problems differently (as, for example, shown in Leikin & Dinur, 2003), students may ignore different solutions concerned with other things. They may say, "I have a solution, I'm not interested in anything else". But in our activity the students were required to solve problems in diverse ways so they could not ignore different solutions. They had to be attentive throughout the lesson, which they felt was more concentrated and more difficult. From our findings with Group C it seems that if such activities constitute a part of classroom culture (Cobb, 2000), and connecting tasks become routine in mathematics lessons, students' attitudes may become even more positive than in our study. More students might be less confused and able to see the advantages of mathematical activities of such a kind.

The results regarding peer explanations were also somewhat unexpected. The fact that many basic-level students preferred learning by giving explanations to learning by receiving explanations was surprising for the researchers as well as the teacher. We found that these students liked to explain solutions to their peers because they felt satisfaction by such activity, they tested themselves, and at last believed they knew what they had learned.

Contrary to the teachers' concerns, we saw that the students used different features to describe different ways of solution. The students attributed different qualities to different solution strategies; they considered some solutions easier, shorter, or more convincing. Moreover, we found no consensus among the students as to which of the solution strategies was easier and which more difficult, or which solution was more convincing or less. Naturally, students' opinions regarding the difficulty of the particular problem-solving strategy often differed from their teacher's views. As noted earlier, Leikin (2003) found that teachers chose to teach solutions that they considered convincing. This opinion of the teachers might also differ from that of the students, and thus exert a negative effect on students' attitudes.

Leikin (2003) demonstrated a variety of relationships between teachers' problem-solving preferences and the ways in which the teachers characterize the problems. In this study we did not analyze these relationships, but our observations showed that a relationship existed between the students' attitudes to coping with connecting tasks and the ways in which students described the solutions. Those ways were very similar to those identified by Leikin. Teachers referred to all the characteristics that students mentioned in their questionnaires, and two more: difficulty to teach and difficulty to understand. Naturally the students did not address difficulties associated with teaching mathematics, nor did they specify their own difficulties (e.g., in solving or in understanding).

From Table 3 it follows that students tended to characterize solutions in a "positive mode" more than in a negative one. For example, eight students in group A found some of the solutions easy (positive mode), three students characterized some of the solutions as "difficult" (negative mode), and three students found some of the solutions easy and some difficult. This tendency was evident with respect to all the characteristics. We suggest that this tendency to characterize problem-solving strategies in positive ways supports our findings on students' positive attitudes to coping with connecting tasks.

This study was based on our need to overcome teachers' reluctance to implement the connecting tasks in their classes. We are planning a more accurate study, which will analyze cognitive and affective processes taking place in students when they cope with connecting tasks. However, we feel hesitant about implementing these kinds of standard-based materials. Unless educational authorities support activities of this kind we believe it will be hard to enhance teachers' enthusiasm for implementing connecting tasks. We observed the extreme surprise of the teacher of Group A at the results of her students' attitudes questionnaires. Although she volunteered to take part in our experiment she could hardly believe that her students were so positive about their participation in the activity. This teacher's stance regarding the activity was based in her lesson observations. Note that her doubt about our research findings was consistent with the dissonance between the impression we got from the lesson observations and the data obtained from the attitude questionnaires. We propose several explanations to such a dissonance. First, we assume that, rather naturally, students who do not like what happens in the classroom are move vociferous than those who enjoy the lesson. Second, teachers may feel very badly even if only one of the students in the class expresses his/her dissatisfaction and often are more attentive to the students with negative attitudes to the lessons than to those who have positive attitudes. We consider this dissonance between what can be learned from the written questionnaires and what can be learned from classroom observations being a serious barrier to implementation of the connecting tasks in mathematics classes. We suggest that in order to convince mathematics teachers to implement the activities in their classes we need them to see students liking these kind of activities and benefiting from them. As shown in this paper, only a deep analysis of learning processes encountered in the lessons of this type together with a long-term implementation of the connecting tasks may develop teachers' enthusiasm. On the other hand, to implement the activities the teachers have to be convinced they work well. We leave this question open to discussion. We will be happy to receive readers' replies and suggestions. Acknowledgement: We would like to thank Irit Peled for her helpful comments on the earlier versions of the paper.

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Roza Leikin, Anat Levav-Waynberg, Irena Gurevich, Leonid Mednikov

University of Haifa

Appendix 1: The structure of the collection

The topic                                                    Recommended
from which the                                               cooperative
problem was     Types of connections between     Difficulty  learning
borrowed        concepts                         level       setting*

Distance        Using equivalent mathematical    Basic       Jigsaw
between a       statements for one particular    Medium      Exchange of
point and a     mathematical concept in                      knowledge
straight line   solving a particular problem.    High        Solving
                Using different mathematical                 problems in
                tools in solving one particular              groups of
                problem.                                     support
Optimization    Using different mathematical     Basic-      Jigsaw
geometric       tools in solving one particular  Medium
problems        problem                          Medium-
                Solving different mathematical   High
                problems in similar ways
Average         Using different geometrical      High        Puzzling
values          models for proving algebraic                 out a
                inequalities.                                problem
                Relationships between algebraic
                and geometric proofs.
Angles in a     Using different geometric        Medium      Puzzling
triangle        concepts in solving one                      out a
                particular problem                           problem
                                                 High
Arithmetic      Using equivalent mathematical    Basic       Exchange of
sequences       definitions for one particular               knowledge
                mathematical concept in solving              in pairs
                problems.                        Medium
                Relating algebraic and
                geometric problem-solving
                tools.
Solving         Using mathematical tools from    High        Homework
geometric       different branches of                        projects
problems        mathematics in solving
using           geometric problems.
different       Using different mathematical
mathematical    concepts in solving geometric
tools.          problems.
                Solving different mathematical
                problems in similar ways

* Cooperative learning settings always concluded with a whole-class
discussion.

1. Do you like coping with the connecting tasks? Why?
2. During the activity you sometimes explained the solutions to your.
   peers and sometimes received explanations from them. What did you
   prefer more (explaining or receiving explanations)
3. What can you say about each one of the ways for solution mat you
   learned in this activity? How would you characterize each one of
   them?

Figure 2. The shared part of the Attitude questionnaire (for Groups, A,
B, and C)

Table 1: Students' attitudes to coping with connecting tasks

                                           Group A      Group B
                                           Basic level  Medium level
Attitude  Reasons for the attitude         (N=22)       (N=23)

Positive                                   14           20
          Gives the option to choose the   12            8
          easier solution
          Enables variety and increases     2           11
          interest
          Shows the beauty of mathematics  --            5
Negative                                    5            2
          Difficult and confusing           2            1
          Unnecessary and irrelevant        3            1
No answer                                   3            1

Table 2: Students' attitudes to learning by giving and receiving
explanations.

                                 A (Basic level)  B (Medium level)
Preference for            Group  (N=22)           (N=23)

Learning by giving               10                5
explanations
Learning by receiving             3               13
explanations
Both by receiving and by          3                2
giving explanations
Learning from teacher's           2               --
explanations
No answer                         4                3

Table 3: Characterization of different ways of solution by the students.

              Characteristic        Group A  Total      Group B  Total
Category      of the solutions      N=22     no.        N=23     no.

Difficulty    Easy              11           14     21           22
when solving  Difficult             6                   9
Solution      Short              1            1     13           15
length        Long                                      7
Convincing    Convincing         5            7      5            5
power         Not convincing        4
Challenge     Demands thinking   9            9
              Does not demand       2
              thinking
Explaining    Easy to explain    6            7      1            1
              Difficult to          4
              explain
Interest      Interesting/       1            1      4            4
              beautiful/
              sophisticated/
              elegant/
              Convenient/nice
Familiarity   Familiar           1                   3            3