Educ Stud Math (2009) 72:307–324 DOI 10.1007/s10649-009-9196-y
Constructing mathematics in an interactive classroom context Paul Ngee-Kiong Lau & Parmjit Singh & Tee-Yong Hwa
Published online: 29 May 2009 # Springer Science + Business Media B.V. 2009
Abstract This paper investigates the nature of the interaction between the teacher and students as they worked on different mathematics activities in a single classroom over a 10month period. Sociocultural theories and the Vygotskian zone of proximal development provide the main framework for examining the teaching and learning processes and explaining the incorporation of a four-phase lesson plan as increasing participation of the teacher and students in the teaching and learning process. Drawing on the analyses of discourse from videotaped lessons and the interviews with the teacher and students, five different types of interactions that emphasized mathematical sense-making and justification of ideas and arguments were identified. Excerpts from transcriptions of such interactions are provided to illustrate the learning practices, either academic or non-academic, that students developed in response to these interactions. Keywords Interactions . Mathematics education . Teaching . Learning . Vygotsky
1 Introduction The mathematics skills required for youth of today and adults of tomorrow to function in the workplace are different from that for youth and adults of yesterday. Mathematics education was primarily a process of transmission of knowledge. Educators believed that a fixed knowledge base existed out there, and the aim of education was to deliver that knowledge to those who were learning. This called for the compilation of well-organized and clearly composed curriculum documents that comprised this knowledge base. Drawing on the perspectives of behaviorist models, the teacher’s drill provided practice for this knowledge base that led to mastery (Freire, 1972). However, this way of teaching and learning has changed to a conception that students actively construct their own mathematics (Kilpatrick, 1987; Von Glasersfeld, 1995). The construction process can be accomplished by students making connections, building mental schemata, and developing new mathematics based on their prior knowledge through interactions with others (Vygotsky, 1978; Goos 2004). Such a perspective assumes mathematics teaching and learning as social in nature. P. N.-K. Lau (*) : P. Singh : T.-Y. Hwa Universiti Teknologi MARA, Shah Alam, Malaysia e-mail:
[email protected]
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In the last few decades, there has been an upsurge of interest in instruction that focuses on the social aspect of teaching and learning. Educators realize that the social context of teaching and learning has the potential to enhance the construction process of mathematics meanings in students. On the one hand, teaching could be viewed as an activity in which teachers act as guides for students’ constructive processes towards, not only the taken-asshared mathematical meanings, but also the mathematical ways of knowing. On the other hand, learning could also be viewed as an active, constructive activity in which students wrestle through barriers that arise as they participate in the mathematical practices in the classrooms (Cobb, Yackel & Wood, 1992). Such instruction requires an interactive classroom context, which ensures active participation of students in their learning. Another crucial aspect to focus on is dialogue. How teachers and students or students and students talk constitutes in large measure such practices. According to Martin (1985): A good [dialogue] is neither a fight nor a contest. Circular in nature, cooperative in manner and constructive in intent, it is an interchange of ideas by those who see themselves not as adversaries but as human beings come together to talk and listen and learn from one another. (p. 10) Teachers who intend to give their students authentic problem-solving experiences in mathematics need to help them talk like expert mathematicians. Students will then be engaged constructively in mathematical discussions while solving problems by proposing, formulating, conjecturing, and justifying mathematical ideas and be evaluating the mathematical ideas of their peers (Richards, 1991). 1.1 Research questions This paper draws on the findings of a larger study carried out in a Malaysian secondary school that aimed to investigate the impact of social interactions on students’ learning. The significance of the Malaysian context is elaborated in a later section of the paper. This paper addresses two research questions of the study. First, how is a four-phase lesson plan that blends whole class and small group interactions developed to promote an interactive classroom context? The aim was to investigate whether it was possible to change a traditional classroom culture to an interactive classroom context. With this lesson plan, we found that it was possible for a teacher to reach the full range of competencies in a multiracial and large class. Second, what are the different types of teacher–student interaction that can enhance students’ learning in a classroom? While we encouraged small group interactions, in this paper, we chose to report on the different types of teacher–student interaction as we were interested in the role of the teacher in promoting an interactive classroom context. We investigated patterns of discourse that mediated meaning shaping and justification of ideas. These aims are important because the findings can provide an illustration of the ways instructional practices and learning outcomes are linked and how mathematics learning occurs in social contexts (Forman, 2003).
2 Vygotsky’s zone of proximal development It is generally accepted that the Vygotskian school of thought probably has the most profound influence on the formation of many sociocultural theories (Sfard, Forman & Kieran, 2001; Forman, 2003). Vygotsky emphasized concept formation as a major issue in the cognitive development of a child. The process of concept formation should be studied
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by referring to the means by which the operation is accomplished, including the use of tools, the mobilization of the appropriate means, and the means by which people learn to organize and direct their behavior. Based on this process, Vygotsky conceptualized the idea of the zone of proximal development (ZPD). He wrote that children who by themselves are able to perform a task at a particular cognitive level, in cooperation with adults or more capable peers, will be able to perform at a higher level, and this difference between the two levels is the child’s ZPD. He also wrote as follows: “Every function in the child’s cultural development appears twice... First, on the social and later on the psychological level; first, between people as an interpsychological category and then inside the child as an intrapsychological category” (p. 128). The process by which the social category performed externally transforms to the psychological category executed internally is called internalization. This internalization occurs within the ZPD since the social interaction may awaken mental functions lying in embryonic stage. Vygotsky’s theory has then been applied to educational research on how children learn through interaction with others (Wood, Bruner & Ross, 1976; Greenfield, 1984; Stone, 1993; Goos, 2004). Similarly, the ZPD was the theoretical framework for the study presented in this paper, where the main aspects, namely, teacher intervention and peer collaboration, were drawn in guiding data analysis. 2.1 Teacher intervention The first aspect of interest for this paper is teacher intervention. It is drawn based on the distance between the cognitive levels when a child performs a task alone and in cooperation with adults. Wood et al. (1976) were the first to study this aspect of the ZPD. They believed that the acquisition of skills by a child is an activity in which the readily relevant skills are combined to meet new, more complex task requirements. This activity is only successful through the scaffolding of a tutor. Since then, the term scaffolding was associated with teacher–student interaction where a teacher structured tasks to facilitate students’ learning that would otherwise be beyond their reach. Greenfield (1984) defined the scaffolding process in a learning situation as “... the teacher’s selective intervention provides a supportive tool for the learner, which extends his or her skills, thereby allowing the learner successfully to accomplish a task not otherwise possible” (p. 118). This first phase of Vygotskian inspired studies was criticized for the imposition of a structure on the learner. The metaphor implied by scaffolding rests on the question of who is constructing the structure. Too often, the teacher is the builder, and the learner is expected to accept a predetermined structure. Searle (1984) wrote as follows: “The children’s understanding, valuing and excitement for the personal experiences were negated as the children were led to report the experience in an appropriate form” (p. 481). Stone (1993) suggested that a learner is continually trying to interpret the adult’s intervention, whether verbal or non-verbal, in the context relevant to him. Unless the adult and the learner have a shared context at that point, the implications of the interaction will not be realized. Since then, the second phase of such studies recognizes the need to give attention to the context and the interpersonal relationship of social interactions (Forman, 2003; Goos, 2004). 2.2 Peer collaboration The second aspect draws on the notion of ZPD in terms of peer collaboration, based on the distance between the cognitive levels when a child performs a task alone and in cooperation
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with more capable peers. From an educational perspective, there is learning potential, where students require their peers’ contributions to make progress in their learning. Peer collaboration provides the right condition for the free exchange of ideas and reciprocal feedback between mutually respected equals when tackling a task (Damon, 1984). Hence, it can foster students’ self-esteem, pro-social behavior, and scholarly achievement. In the classroom, the teacher is the adult, and any student can always identify the more capable peers. The two aspects of ZPD reviewed above highlight the importance of teachers incorporating social interactions that can facilitate active participation of students in their mathematics learning. Since it is impossible for a teacher to reach the whole class using one-to-one interaction, an alternative approach that better fits classroom realities emphasizes interactions among the whole class. Many studies have attempted to identify such an approach that can engage all students actively in authentic mathematics problem solving (McClain & Cobb, 2001; Goos, 2004). Most Malaysian teachers are familiar with ZPD and sociocultural theories through attending in-service courses and seminars. However, they lack support and guidance in implementing both aspects of teacher intervention and peer collaboration in the teaching and learning process. This paper provides such a level of detail by analyzing teacher– student interactions for teacher intervention and student–student interactions for peer collaboration in the teaching and learning process by a teacher trying to create an interactive classroom context.
3 Methods This study was predominantly qualitative. The methodology was formulated through incorporating features of naturalistic inquiry and the spiral nature, interventional and group work of action research. 3.1 Malaysian background This paper draws on the findings of a larger study carried out in a Form Four class of a Malaysian secondary school. Malaysia is a multi-racial country, with Malay, Chinese, and Indian as the three main races, and many other minority natives. Thus, one issue related to mathematics classes involves multiple perspectives of a culturally, linguistically, and ability-diverse student body. Most of the classes have more than 30 students. The large number of students in a class asserts another huge pressure on teachers delivering lessons effectively to students. Most of the teachers teaching Form Four classes are graduates with teaching diplomas. The Form Four students had gone through 6 years of primary education and at least 3 years of secondary education. Topics for Form Four mathematics include Standard Form of Numbers, Quadratic Equations, Set, Straight Lines, Statistics, Probability, Circle, Trigonometry, Angles of Elevation and Depression, and Lines and Planes in Three Dimensions. Education in Malaysia is always examination-inclined. Mathematics education is still a process of transmission of knowledge, rather than a process of meaning construction. Direct instruction is considered to be the best approach to the teaching and learning of mathematics. A typical lesson consists of introducing a new concept through examples, during which the teachers present students with step-by-step instruction. Then, students are given notes and problems in the textbook with teachers monitoring them working on those problems. All students sit for the Malaysian Certificate of Examination after at least 5 years
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in secondary schools. Over the years, the performance in mathematics of students has declined. Malaysia took part in the Trends in International Mathematics and Science Study (TIMSS) for the years 1999, 2003, and 2007. The average mathematics scale scores dropped from 519 to 508 and 474. Our students fall under the Intermediate Benchmark where “students can apply basic mathematical knowledge in straightforward situations” (Gonzales et al., 2008, p. 13). Malaysian teachers are urged to cultivate an interactive classroom context for effective mathematics learning, through in-service courses and seminars organized by relevant education authorities. However, the result is daunting. The multi-racial nature and large size of the classes hinder teachers’ interaction with students. Interacting in such classrooms is a challenge and can be emotionally exhausting when a teacher tries various prompts before hitting the one that helps a student. Such interaction is costly in terms of teacher time and effort. Hence, teachers seldom give students ample time to respond to their prompts. Either they respond to their own prompts, or the interaction is usually brief due to limited time for teachers to listen to students’ discussions and sensitively respond to students’ ideas. 3.2 Participants and setting Samarahan School, located in the State of Sarawak, was invited and agreed to participate in this study. The school had a population of around 800 students. According to the school, the majority of these students came from families with average incomes. When the study was carried out, the school had a team of very experienced and dedicated teachers. The passing percentages for mathematics subjects were higher than the national passing percentages for the Lower Secondary Examination and the School Certificate Examination. Permissions to carry out the study in this school were obtained from the Educational Planning and Research Division of Malaysian Ministry of Education and Sarawak State Education Department before meeting the principal to nominate a teacher and students who were willing to be the participants. The study was conducted with a class of Form Four students aged between 16 and 17 years. All Form Four students have to take the subject mathematics, which is a prerequisite for entry into most university courses. Beside the objective of learning the different mathematical skills of the subject, other important emphases include problem solving, communication, application, and connected knowing. Hence, students will be assessed not only on their mastery of mathematics skills but also their ability to solve application problems and justify their methods. Many teachers are unwilling to participate in any research study involving the teaching and learning process. They feel threatened professionally. Furthermore, this participation will take away much of their time needed to cover the syllabus for students to sit for highstakes examinations. However, the senior assistant of the school, who was a qualified mathematics teacher having 15 years’ experience in teaching secondary school mathematics, offered to participate in the study. He achieved good mathematics results through teaching the subject using direct instruction. Being the senior assistant, he was responsible for planning professional development programs for all the teachers in the school to include the different approaches of cooperative learning, inquiry learning, contextual learning, and constructivism emphasized by the Malaysian Ministry of Education. Hence, developing teaching and learning practices through which the two aspects of teacher intervention and peer collaboration from ZPD could be enacted in the classrooms was most welcomed by him.
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The teacher agreed to start teaching as he normally would and react as he usually did to students’ responses during lessons observed. We, as researchers, assumed the role of participant–observers, so that we shared as intimately as possible in the life and activities of the classroom (Patton, 1990). We had many discussions with the teacher when conflicting interpretations of lesson episodes were negotiated. As a result, changes were made in subsequent lessons based on the two aspects of ZPD. Here, we were adopting the spiral and interventional nature of action research (Kemmis & McTaggart, 1988). Through this intervention, we were determined to provide support structures to the teacher’s confidence and self-esteem emphasizing social interactions in the teaching and learning process. 3.3 Data collection As the emphasis was on interpreting learning through a teaching approach that can cultivate an interactive classroom context by taking into consideration the two aspects of ZPD highlighted, the fieldwork for this study consisted of observing the teacher and students in the classroom and discussing the work with the teacher. One important way to strengthen the validity of such research findings is through data triangulation—the use of a variety of data sources (Patton, 1990). We collected qualitative data using three different methods, namely, audio-visual recordings, observations, and interviews. A video camera with an external sound box and microphones was used to tape the lessons in the classroom. External microphones were used to increase the clarity of the audio recording of the interactions between target students when they were involved in group work. An audio-recorder was put in the pocket of the teacher to record his utterances throughout a lesson. After we implemented a four-phase lesson plan made up of whole-class discussion, group work, reporting back, and summing up during the ninth week of the study, the recording format of each lesson was guided by this plan. It started off focusing on the teacher when the teacher and the students were engaged in the whole-class format. It then turned to the target group of students for group work. During the reporting-back phase, the focus was on the students who gave the presentations. We recorded one lesson of 80 min and two lessons of 40 min per week throughout the 10 months of the study. Excluding holidays, examinations, tests, and sport weeks, a total of 58 lessons were recorded. After each recording, the researchers viewed the tapes thoroughly, and information-rich episodes on teacher intervention and peer collaboration were identified, time-annotated, and transcribed. These information-rich episodes were those from which we could learn a great deal about issues of concern for the study (Patton, 1990). Then, the tape was replayed to the teacher, and he was asked to interpret and comment on the happenings in these episodes. Conflicting interpretations from the researchers and the teacher were negotiated, and changes were agreed upon and implemented in subsequent lessons. Such negotiations were audio-taped with portions being transcribed and inserted as field notes to the lesson concerned. For example, the teacher would not take up the suggestions from students for further discussions in a lesson as these suggestions were not intended for the lesson. However, the researchers managed to convince the teacher of the importance of taking students’ perspectives into consideration while designing instruction that could enhance their understanding of mathematics. After noticing the potential of such instruction in enhancing students’ learning, he started to value students’ suggestions and engaged them in discussing their suggestions One of the shortcomings of using audio-visual recordings is that sometimes the researchers were not able to identify any information-rich episodes from the target group. It
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was annoying to notice that groups that were not targeted were discussing actively. The researchers decided to make an ad hoc record of such cases. Such on-site observations could be biased as they had been filtered through the eyes of the researchers and were constrained for the purposes of the study. In order to avoid such deficiencies, the researchers verified the observations and their interpretations with the teacher. Such records were later inserted as field notes of the lesson. Some of the students and the teacher were also interviewed to get feedback about learning mathematics through social interactions. These interviews were audio-taped and transcribed to supplement the data collected. 3.4 Data analysis Qualitative study specifies a unit of analysis to be investigated so that the focus of data collection will be on what is happening to the participants in a setting and how these participants are affected by the setting (Patton, 1990). In our study, the analysis of data involved the teaching and learning practices of the teacher and the students in the classroom. The data were gathered and analyzed as soon as possible, and interpretations were derived. The challenge was to make sense of the data, identify patterns, and construct a framework to communicate the essence revealed. Throughout the study, three phases of preliminary analysis, episode-by-episode analysis and comparative analysis, were involved (Schoenfeld, 2000; Cobb & Whitenack, 1996; Creswell, 2003). In the preliminary analysis, the researchers organized and arranged the data into different types according to their sources. Then, patterns of interest for this study were identified. At the same time, the coding process began which involved the organization of the data by categorizing texts or segmented transcriptions into these patterns and labeling each pattern. This process resulted in three broad patterns for the study: activity, teacher–student interaction, and student–student interaction. In the episode-by-episode analysis, each text or segmented transcription under each pattern was interpreted separately. A negotiated outcome for each interpretation was reached. This interpretation process also enabled the researchers to generate a small number of themes under each pattern. For activity, we categorized data as challenging, related to prior knowledge, and promoting discussion. For teacher–student interaction, we identified type, student need, student suggestion, evaluation, and justification. The first theme, type, of teacher–student interaction addressed the second research question. For student–student interaction, thinking, communicating, and off-task talk were the three themes generated. The final comparative analysis involved a constant comparison and contrast of the selected texts or segmented transcriptions and their interpretations to develop an overview of the progress on the various themes in chronological order. One of the results was the developmental process of the four-phase lesson plan that could cultivate an interactive classroom context for the first research question.
4 Creating an interactive classroom context This part deliberates the findings for the first research question. The teacher was not the only one shaping the classroom context in the study. The students also played a very important role, from passive learners to active participants in mathematics sense-making and justification of ideas. Prior to the teacher participating in the study, he taught mathematics by direct instruction. Before the study commenced, the researchers and
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the teacher had a number of sessions discussing the possibility of applying Vygotsky’s ZPD and sociocultural theories to the teaching and learning process. The teacher commented: To be frank, actually we as teachers normally try to encourage active participation from students since it’s something that has been emphasized by our Ministry. Somehow, I think because of the time factor, we teachers end up solving the problem for our students. Besides those sessions, the support that the teacher got for the study came from the many discussions on teacher–student and student–student interactions from the lesson episodes. We negotiated our conflicting interpretations, and changes were identified and implemented in subsequent lessons. During the first 3 weeks of the study, the teacher took activities directly from the mathematics textbook or the mathematics workbooks recommended to students. Transcription 1 shows the teacher–student interaction in one such lesson. Transcription 1 Lesson on sets (25th February) T S1 T S1 T
S2 T S2 T S3 T S1 T S3 T S3 T S1 T S1
We will now study the operations for set. What are the four basic operations for whole numbers, fractions, decimals and the like? Subtract, add, divide and multiply. Can we use these operations on sets? No. We will see in the following example: ξ = {The months of a year}, P = {The months starting with the letter J}, Q = {The months with 31 days}, R = {The months with 28 or 29 days}, S = {The months with seven letters}. What are the months starting with the letter J? January and July. Any other? No. What are the months having 31 days? January, March, May, July, August, October, December. Which months have 28 or 29 days? February. What about the months with 7 letters? January and October. Now we find the intersection of two sets P∩Q = {January, July}. How many elements in this set? Two. What about P∩R? 0. What about Q∩S? January, October.
The teacher led the students through the example so as to fulfill the objectives of the lesson. By doing so, he was trying to use the ZPD by means of teacher intervention in the teaching and learning process. The responses from students were either “yes” or “no” or brief phrases such as “two” and “January, October”. When the teacher was satisfied with
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the understanding of students, he made a brief summary on intersection and invited students to contribute their examples. T
Hence, January and October are the results of Q intersect S. The intersection will result with the same elements in different sets. One of the operations that we can perform on sets is intersection. Can anybody give another example?
One of the students responded with the example of boys using neckties and spectacles. However, the teacher did not guide him to apply the operation of intersection. Instead, he started with union. S3 T S3 T S3 T
The number of boys using neckties. Boys using neckties and... Spectacles. Boys using spectacles? What is that? (No response) OK. One of the operations that we can perform on sets is intersection. The other is union.
The teacher said that the suggestion was not probed further as he would not like the lesson being side-tracked into discussions on something not intended for the lesson or students involved in off-task talk, and he had limited time to finish a tight syllabus for examination. As a result, learning happened along a well-marked path laid out by the teacher for the students. The teacher commented: It often happens... every student is talking... they could easily get side-tracked into off-task talking. Even though lots of good work can emerge but these are not necessarily what you wanted and you just have to be careful of that... especially if you’ve a huge syllabus to cover. During the discussion sessions, the researchers also suggested to the teacher that he might formulate his own mathematics activities. In the following lesson, the teacher was aiming to introduce parallel lines to the class. He prepared the following activity and gave it to the students to work in groups. Here, he was using the ZPD through peer collaboration in the teaching and learning process. T
This week, we are going to look at the relationship between straight lines. What’s the gradient of one line when compared to another? You’re given the handout. Let’s look at the first activity, solve it yourself first. Then, compare the solution with your friends. You’re asked to draw... not to sketch... Draw the following three straight lines with the same axes.
Transcription 2 shows the teacher–student interaction during the reporting-back phase of the lesson. Transcription 2 Lesson on straight lines (26th March) T S1 T
OK?... Who wants to try first? Nurul, can you draw one of the lines? You can choose to draw any of the three. Which one do you choose? y ¼ 2x þ 2 (and drawing the line). y ¼ 2x þ 2. Everybody pays attention to the front. This is important because you must be able to imagine the shape of a straight line. Why did you mark 2?
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S1 T S1 T S1 T S1 T
y-intercept. OK, y-intercept. Can you show how you get this point? 2=2/1, y=2 and x=1. 2 equals to 2/1, right? Right. Why is it y here? Find gradient. OK... What’s the definition for gradient? The difference in y divided by the difference in x. OK... You move 1 along the x-axis and then move up 2 units. Can we accept this?
In this activity, the teacher asked the student to justify the solution and he legitimized it by saying “we give her a clap”. Even though he helped the student to clarify a number of issues involved in the justification, this teacher was showing his willingness to let students assume more responsibility for their learning. T S2 T
We give her a clap... Is there any other method? I see one student putting up his hand. (Drawing the line) OK, this is the first method, we can also use. This is the second method. He uses y=0. We want to find the x-coordinate when y=0. We put y=0, we get x=−1. So the point is (−1, 0).
Again, the teacher also encouraged other solutions. This time, he legitimized the alternative solution as “the second method”. However, he did not insist that the student explain and justify the solution. Instead, the teacher explained “the second method” to the class. Hence, the teacher was still unable to place the whole burden of explaining and justifying the solutions on the students at this juncture. The teacher commented: I think that these lessons have made me more aware of what the students can do on their own and the sort of help they need from me to move on when they are stuck... I actually have to be a little more prepared for... It made me come up with some new ideas to introduce topics... I’m thinking of the way we introduced gradient the other day by bringing my students out of the class and using the tiled floor as the coordinate grid... The fact that we see students very willing to come up and share their solutions with the class... It sort of encourages you to keep working to come up with the ideas... During the first 5 weeks of the study, we observed that some lessons started with a whole-class discussion and ended with students working on problems individually. Others started with group work, followed by discussion on students’ solutions and ended with individual work. The teacher was also “walking a tightrope” in allowing students to pursue their personal mathematical constructions. On the one hand, he needed to listen and be sensitive to students’ responses so as not to make interventions that would inhibit their thinking. On the other hand, he wanted his students to develop the taken-as-shared mathematical meanings of the wider society. During a discussion session, he voiced his concern about these issues and he wanted to follow the lesson format consisting of induction set, content, and summary recommended by Malaysian Ministry of Education. However, we all agreed that both teacher–student and student–student interactions were beneficial to students’ learning and such interactions should reach the whole class. It took the researchers and the teacher another 3 weeks to deliberate a four-phase lesson plan consisting of a whole-class discussion, group work, reporting back, and summing up. During the first phase of any lesson, the teacher held a whole-class discussion facilitating students’ understanding of a problem and coming up with possible heuristics and strategies
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for a solution. Group work was the second phase that provided a chance for students to solve a problem themselves through active discussions. At the third reporting-back phase, students were given an opportunity to explain and justify their solutions to the class. The final phase was the teacher summing up the lesson by actively discussing all solutions, justifying the legitimacy of each solution, introducing new symbols and mathematical language, and extending the problem. By adopting this four-phase lesson plan, both teacher intervention and peer collaboration using the ZPD could be enacted in the classroom. It was implemented from the ninth week. In one lesson, students were asked to work on the following problem in groups: In the diagram, ABCD is a straight line. Given that CE ¼ 3EF ¼ BD=2, find the value of cos y. Transcription 3 is a part of this lesson. Transcription 3 Lesson on trigonometry (17th September) T
Read the question carefully. CE=3EF, what does this mean?
F E
y A S1 T S1 S2 S1 S2 S3 S1 S3 T S1 T S1
B
C
D
The length of CE is three times that of EF. Then, we have 3EF =BD/2. How do we solve this problem? Correct. BD/2=3EF. Now, these are 4EF, 3EF and 5EF. We use Pythagoras Theorem. How do we find cos y? This is cos. So, adjacent divided by hypotenuse. 3/5. Again, how do you find cos y? CE=3EF, 3 þ 1 ¼ 4. So, CF=4EF. 3EF ¼ BD=2 ¼ BC. The sides of the triangle are 3, 4 and 5 by using Pythagoras Theorem. So, cos y=3/5. Cindy, what is your final answer? cos y=3/5. Can you explain to the class how you get it? BC=3EF, CF ¼ 3EF þ 1EF ¼ 4EF, by applying Pythagoras Theorem to triangle BCF, BF=5EF. cos y = adjacent divided by hypotenuse = 3/5.
S1 explained the solution to S2 and S3. He reorganized his understanding and explained in a more understandable form to S3. He improved further his explanation to the class. Hence, the responses from students evolved from brief phrases or single disconnected sentences to explanations which make sense to all. Now, the teacher puts the whole burden
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on students to explain and justify their solutions, as well as to comment on the contributions of other students. He said: Now I always remember to ask students why they do that particular step; in that way they will try to recall back and they will try to think of the reason as to why they do it. Then, they will understand it better and I can assess their problem. Throughout the study, the teacher encountered numerous situations that conflicted with his previous practice, but which were very productive in the mathematical learning of the students. For instance, previously, he would write the solution of any problem for his students. Now, the students were given an opportunity first to work in groups, then present and justify their solutions to the class. As a result, they understood the concepts better and improved in communicating skills. These spurred him to value social interactions in the teaching and learning process. In other words, both the conflicts and dilemmas the teacher encountered in the classroom as he tried to enact teacher intervention and peer collaboration using the ZPD and the discussions on lesson episodes to resolve them created a context for him to learn. The teacher’s learning and subsequent changes did not occur as isolated incidents, but consisted of connected continuous constructions and transformations. For instance, the teacher changed from frequently answering his own questions to the students to giving the students a chance to answer his questions and emphasizing that students should explain and justify their answers. Another instance is that the teacher was very willing to adopt any suggestion of students and to lead a whole-class discussion on those suggestions. Two other issues were also addressed for successful implementation of the four-phase lesson plan. First, there were some students who were often engaged in off-task talk during group work. The teacher was quite frustrated with the amount of off-task talking from the students. Not only did the off-task talking make the teacher worried about the progress of those students but they also disturbed his concentration on interacting with other students. He decided to constantly remind the students of their work, ask them to present their solutions to the class, and attend to them and ask them questions related to the task when he sensed that the students were off task. By doing so, the teacher made clear to the students his expectations that they would discuss for a solution during group work and explain and justify the solution to the class during the reporting-back phase. Second, the teacher needed to find ways to encourage students to express their thinking to others. For the students, it is one thing to think through a solution privately but quite another to express their thinking to others. One of the students commented: There were times that I didn’t like sharing my solutions with the class especially when I was not sure whether my solutions were right or not. Unless students are willing to express their thoughts and methods with others and are willing to listen to the solutions of others, they will miss the opportunity to reflect on their work and reorganize their current conceptual level of understanding. As their thinking will be subjected to evaluations and criticisms, the students need to feel “safe” in expressing their thinking to others. The teacher pointed out that: We try to inquire from students what they know about a concept... before introducing to them that particular lesson. So, with that we are able to actually know the level to enter... It’s making the students quite vulnerable because often you’re not sure that they’ve got it right... This aspect placed the teacher under the obligation of approaching the students’ suggestions in a non-evaluative way and not imposing his ways of tackling the problem on
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the students. For instance, the teacher appreciated the solution put forward by the student drawing the line y ¼ 2x þ 2 using the y-intercept and gradient method. That was why he said “give her a clap”. As the students realized that their thinking and ideas were respected and accepted, they felt obliged to reconstruct their solutions and explain and justify them to others. As a result, they improved in doing so and gained confidence in themselves and their ability. One of the students commented: When I feel easy, understand the subject and know my teacher, I’m always confident and not afraid to share with my classmates. Hence, reciprocal obligations and expectations were negotiated implicitly during the teaching and learning process. The students were very optimistic towards such a process, as one of them commented: Even though my marks haven’t improved much, but I understand concepts better now. And I’m enjoying it... I understand it and I’m asking a lot of questions. Even if I don’t get the final answer right, I always know where I go wrong. Hopefully this year has given me confidence in mathematics and my marks will go up later. 5 Teacher–student interaction The analysis of data collected yielded five different types of teacher–student interactions for the second research question that could not only foster the development of students’ conceptual understanding, but also enhance their awareness of the strategies and thinking dispositions required in problem solving. The first type is offering clarification. These interactions usually occurred during the whole-class discussion and the reporting-back phases. The main aim was to get the students to understand the activity thoroughly, to generate possible strategies for a solution, and to amplify students’ explanations. In a lesson, students were working on the figure below to find ∠DAB. Transcription 4 shows the reporting-back phase. Transcription 4 Lesson on trigonometry (30th August) T S1 S2 T
What is the answer? tan ∠DAB=6/8. ∠DAB=36.9°. Why do we use tan? Why don’t we use sin or cos? That’s a good question. Why?
D
6
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S1 T S3 T
We don’t have the hypotenuse. Can’t we find it? Use Pythagoras theorem to find it. Yah. Then, we can use sin or cos. Since we can use tan for the given values, why must we waste time to find the hypotenuse?
The teacher was not satisfied with the clarification “We don’t have the hypotenuse” and he offered his clarification. In the end, the students noticed that they could use sine or cosine instead of tangent through finding the hypotenuse first, but it would take more time for them to do so. The second type is inviting students’ participation, which usually occurred during the first three phases of a lesson, aiming to further clarify the activity or doubts, to involve students actively in their learning, and to get students to justify their solutions. Transcription 5 shows the teacher–student interaction during the reporting-back phase of a lesson. Transcription 5 Lesson on circle (26th July) T S1 T S2 T S3 T S3 T S2
Is there any other method? Yes, John? (Writing) OK. This is the 1st method, we can also use. This is the 2nd method. Why do we use 1/2r2sinθ and not 1=2base height? Jane, can you draw the height of the triangle POQ? (Drawing) Now, express h in terms of θ. Sinθ=h/16, h=16sinθ. What do you get for the area by using this? It’s the same.
P 12 16
h
Q θ O T
Good. Can you use both formulas? Give him a clap.
Through this interaction, not only S3 but the other students in the class understood that 1/2r2sinθ is derived from 1=2base height. This is important if we want students to become active participants in the teaching and learning process. By doing so, students will
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have a meaningful personal connection with their teacher and peers in the classroom. Then, emotional achievement will go hand-in-hand with cognitive achievement, enabling them to acquire not only the content knowledge, but also the individual and social skills for successful engagements in any lesson. The third type is maintaining students’ focus on the activity. As mentioned earlier, the teacher was quite frustrated with the amount of off-task talking from his students. The teacher reminded the students of their work, attended to them, and asked them questions related to the task when they were off-task. Hence, such interactions mostly occurred during group work to keep students on-task and on-track to a solution. The fourth type is reinforcing key features of the activity. In any lesson, the teacher constantly monitored students’ progress on their work. He would immediately help students who misunderstood or lacked understanding in any activity. Hence, such interactions occurred throughout the lesson to emphasize important features of the activity to increase students’ success in solving it. These features included content knowledge, strategies to solve the activity, and thinking skills. Transcription 3 is an example of this type of interaction. The questions posed by the teacher probed S1 to orchestrate the key feature CE ¼ 3EF ¼1=2BD into useful information to solve the activity successfully. The last type is evaluating students’ understandings. The teacher would constantly evaluate students’ emerging understandings from their solutions. If the emerging understandings were satisfactory, the teacher would verify the students’ solutions. Otherwise, he would immediately intervene to help the students. Hence, such interactions usually occurred during the last two phases of a lesson, reporting back and summing up, to legitimize students’ solutions. Transcription 6 is such an interaction during the reportingback phase while students were working on the problem: Given AP = BQ, find the perimeter of the shaded region. One group of students had the following solution. S=rθ, q ¼ S=r ¼ 8=4 ¼ 2 rad. QB=AP=2. PQ=8 cm. AB=6(2)=12 cm. Perimeter=22 cm + 8 cm + 12 cm=24 cm.
A
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B Transcription 6 Lesson on circle (19th July) T S1
Where do you get 22? 2+2.
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Is 2 þ 2 ¼ 22 ? The answer is the same, but it’s unsuitable. Yes. It’s true for 2 only. If you don’t believe it, is 3 + 3=32? No. So, we must write 2+2.
The perimeter of the shaded region is equal to QB þ AP þ PQ þ AB. The group wrote QB þ AP ¼ 22 instead of QB þ AP ¼ 2 þ 2. The teacher gave another example to help the students realize that (QB + AP) should not be written as 22. When the teacher was asked to comment about this, he said: After so many months, I feel I’m better at dealing with students’ misconceptions. You need to put forward an example to students that can show them that their conception is not right and that’s difficult because sometimes you can’t think of one... I need to think of an example that I can give to the students. They will do it and find out for themselves that it doesn’t work. Because l say ‘No, that’s wrong’, they might accept it but they won’t understand why.
6 Discussion and conclusion One-to-one interaction with students is time consuming, and it involves a trade-off between the number of students getting this interaction and the intensity of that interaction. When a teacher does interact with individuals or small groups, does every student benefit from this interaction since they have different zones of proximal development? Many researchers are interested in creating learning environments that can foster effective classroom interactions for mathematics learning. We have presented how an interactive classroom context can be created by adopting a four-phase lesson plan, involving whole-class discussion, group work, reporting back, and summing up. Both teacher intervention and peer collaboration using the ZPD were enacted in the classroom. It was a painful process for the teacher to negotiate the obligations and expectations of such a classroom context with his students. This is consistent with the view of Schwartz (1994) who wrote as follows: It is difficult for students well into a school career in which mathematics has been an endless series of incompletely understood calculations and manipulation ceremonies to shift gears and to exercise in class their curiosity and inventiveness... to imagine mathematics classes in which mathematics is discovered rather than covered. (p. 6) The teaching and learning of mathematics became a social process of negotiation rather than imposition. The teacher’s role changed from the sole source of mathematical knowledge to facilitator in the development of the students’ mathematical constructions. Freire (1972) wrote as follows: The teacher is no longer merely the one who teaches, but one who is himself taught in dialogue with the students, who in turn while being taught also teaches. They become jointly responsible for a process in which all grow. (p. 67) The role of students also changed from passive receivers of knowledge to coconstructors, co-investigators, and co-verifiers of knowledge. Students would be actively involved in discussions, clarifying the activity, suggesting ideas, explaining, and justifying their own ideas or providing help to justify or refute others’ ideas. Hence, they assumed
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greater responsibility for their work. Polya (1973) wrote as follows: “The teacher should help, but not too much and not too little, so that the students shall have a reasonable share of the work” (p. 1). Learning occurs in an environment of social interactions leading to understanding. Hence, teachers need to create opportunities for interacting and facilitating students making connections, building mental schemata, and developing new mathematics based on their prior knowledge. We identified five different types of teacher–student interactions that could foster the development of students’ conceptual understanding and their thinking dispositions, namely, offering clarifications, inviting students’ participation, maintaining students’ focus, reinforcing key features, and evaluating students’ understanding. The excerpts that we provided support this finding. As students verbalized their thinking through these interactions, with others questioning and evaluating them, their emerging understanding grew. Students put forward questions to the teacher or during group work such as: Can we use...? Do we use...? Why do we use...? Why do we not use...? How do you...? These questions were usually posed to them by their teacher. Hence, the students internalized some of the questions of the teacher to become their own stock of tools for questioning. One of the students commented: ... before this, I didn’t know how to start answering a word problem. Now, I start looking for information contained in it and ask a lot of questions. As a result, I am able to think of different possible methods to solve it before I choose one to answer it. The teacher used social interactions as vehicles to respond to students’ questions and answers. As a result, students learned to clarify and expand one another’s ideas. Hence, the teacher cultivated a culture among students taking their questions and comments seriously and turning them into learning opportunities (Sternberg, 1994). References Cobb, P., & Whitenack, W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30, 213–228. doi:10.1007/BF00304566. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative of the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33. doi:10.2307/ 749161. Creswell, J. W. (2003). Research design. London: Sage. Damon, W. (1984). Peer education: The untapped potential. Journal of Applied Developmental Psychology, 3, 331–343. doi:10.1016/0193-3973(84)90006-6. Forman, E. A. (2003). A sociocultural approach to mathematics reform: Speaking, inscribing and doing mathematics within communities of practice. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 333–352). Reston: NCTM. Freire, P. (1972). Pedagogy of the oppressed. Harmondsworth: Penguin Books. Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35(4), 258–291. Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Brenwald, S. (2008). Highlights from TIMSS 2007. Washington, DC: National Center for Education Statistics, Institute of Education Sciences, U. S. Department of Education. Greenfield, P. M. (1984). A theory of the teacher in the learning activities of everyday life. In B. Rogoff & J. Lave (Eds.), Everyday cognition: Its development in everyday context (pp. 118–138). Cambridge: Harvard University Press. Kemmis, S., & McTaggart, R. (1988). The action research planner. Geelong, Victoria: Deakin University Press. Kilpatrick, J. (1987). What constructivism might be in mathematics education. Proceedings of the Eleventh International conference on the Psychology of Mathematics Education, Vol. 1, Montreal, 3–27.
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