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*LEARNING FROM INTERNATIONAL COMPARISONS OF TEACHING: A CASE STUDY
Susie GrovesBrian Doiggrovesac@deakin.edu.aubadoig@deakin.edu.auDeakin University, Australia
The research that is described in this paper is part of a small-scale collaborative study, Talking Across Cultures, which investigated childrens mathematical explanations during whole-class discussion in three mathematics lessons in each of Australia, Hungary and Japan, during childrens first year at school. This paper describes examples of successful strategies for encouraging and supporting childrens mathematical discussion used by one Australian and one Japanese teacher. Commonalities and differences between these strategies and barriers to their use elsewhere are discussed. As in an earlier study, also described briefly here, Australian practice was brought into sharp relief through these comparisons.
BACKGROUND
There is another more subtle reason for studying teaching across cultures. Teaching is a cultural activity. Because cultural activities vary little within society, they are often transparent and unnoticed. Comparative research is a powerful way to unveil unnoticed but ubiquitous practices. (Stigler, Galimore, & Hiebert, 2000, p. 87)
Cross-cultural comparative research provides a powerful means of achieving better understanding of ones own practice and looking for ways of extending its boundaries. Clarke (2002) describes the purpose of studying international classroom practices as not merely to mimic them, but rather to support reflection on our own practice.
This paper reports on findings from a small-scale collaborative study, Talking Across Cultures that investigated childrens mathematical explanations during the whole-class discussion phase of three mathematics lessons in each of Australia, Hungary and Japan, during childrens first year at school. The purpose of the project was to achieve better understanding of local practices and find ways of extending their boundaries, while taking into account the cultural constraints and supports identified as part of the research. As well as investigating the types of explanations given by these children, the project also sought to identify strategies the teachers used to support high quality mathematical explanations.
This study was prompted not only by larger-scale studies, such as the TIMSS Video study (see, for example, Hollingsworth, Lokan, & McCrae, 2003), but also our previous research contrasting Australian and Japanese practice, where we found widespread agreement amongst teachers, principals, and mathematics teacher educators with the notion of mathematics classrooms being places where students construct powerful mathematical ideas through participating in whole-class dialogue and argumentation together with a realisation that current Australian primary classroom practice falls far short of this goal (Groves, Doig & Splitter, 2000).
Classrooms in Australia, Hungary and Japan were chosen as the sites for this study because, while all three countries perform well on international studies of performance in mathematics, Hungary and Japan perform particularly well in areas that require higher-order thinking for example, problem solving.
Data collection included video recording of lessons, and audio recording of interviews with three expert teachers in each of Australia, Japan and Hungary. Teacher interviews were audio-taped and transcribed. Data analysis was carried out both individually by the researchers and during face-to-face meetings, including a week-long meeting for all researchers in Australia.
In this paper we focus on one lesson from each of Japan and Australia and attempt to illustrate some of the strategies used by each of the teachers to support childrens development of mathematically sophisticated explanations, and use this as a basis for reflecting on some aspects of Australian classroom practice.
THE JAPANESE LESSON
The lesson discussed here was part of a sequence of lessons on subtraction, taking place about eight months after the beginning of the childrens first year at school.
The lesson began with the teacher reminding children of the previous days lesson on 13 9, then spending about five minutes handing back workbooks while reading aloud the comments children had written at the end of that lesson. Children were told that in this lesson they would use their prior knowledge to find 14 8. They could use magnetic tiles if they wished and any strategy they chose.
After about ten minutes, the teacher identified four different solution strategies used by the children. The teacher then asked all of the children to identify the strategies they had used and place their own magnetic name cards on the blackboard under that strategy. He drew a circle on the board and told those children who could not match their solution with any of the four strategies to place their names in the circle. These children were told that, if they later decided that their strategy belonged to one of the four on the board, they could move their name cards, or perhaps they had come up with a new idea. After name cards were placed on the board, the teacher asked children to explain their strategies.
Approximately six minutes were spent discussing each of the strategies. At the end of this time, only two children still had name cards in the circle. As time was running out, the teacher said they would talk about those childrens solutions next time, but in the meantime asked all children to look at the four strategies and comment. Most of the comments were recorded on the board and labelled with the childrens names, with about a dozen names being on the board by the end of the lesson. The teacher ended the lesson by summarising what they had done. He then asked the children to spend a few minutes reflecting on the lesson in their workbooks.
We identified a number of strategies to support childrens mathematical explanations in this lesson.
Interweaving the concrete with the abstract. A major challenge for teachers is to connect conventional mathematical activity with students real experiences. A significant feature of this lesson was the way in which the teacher combined childrens verbal explanations of their solution strategies with his and the childrens manipulation of magnetic tiles and written and symbolic representations of childrens actions. So, for example, when a girl effectively stated that if b + c = a then a b = c, the teacher asked her whether she could connect this to the solution using the tiles. Similarly, solution strategies devised by the children were specifically named for later use e.g. the Subtraction, addition strategy.
Public and permanent recording of explanations. In this lesson, the teacher and children developed a shared vocabulary for the various solution strategies. The teacher made sure that explanations were recorded in such a way that they could be retrieved when needed, so that these became the protocols for supporting students induction into the mathematical discourse relating to solutions strategies for simple subtractions. The blackboard provided a public and semi-permanent way of recording the written and symbolic representations of childrens solutions as well as the results of childrens actions. At the end of the lesson, the teacher specifically asked the students to look at the four strategies, which were clearly organised on the blackboard, and comment on them. Children used their workbooks to record their working and reflections on each lesson, together with the date. Thus they were able to refer to previous work with ease for example, one child recalled that four weeks earlier they had found that 8 + 6 = 14, and so 14 8 = 6.
Giving children ownership of ideas. While there was no imperative to include all children in discussions, by the end of the lesson about a third of the children had their name cards attached to strategies or comments on the board. Moreover, the teacher had flagged that in the next lesson they would explore one childs remark that if 14 8 = 6, then we must also have 14 6 = 8. Childrens strategies were publicly acknowledged and referred to across lessons.
Promoting high level written explanations. According to Fujii (2004), Professor Takashi Nakamura (University of Yamanashi) classifies childrens written comments about lessons into four levels: affective (related to enjoyment); descriptive (this is what I did); comparative (comparing their own solutions with those of other children, perhaps friends); and those that involve generalisation or specialisation beyond the classroom context. According to Fujii, the level at which students respond to requests to reflect on lessons depends on the exact form of the teachers question. If the teacher asks What did you learn the answer will be cognitive. On the other hand, if the teacher asks How did you feel about this lesson the answer, obviously will be affective. This teacher regards written explanations as significant and uses a special Japanese word that combines comment and impression to ask students to reflect on the lesson.
THE AUSTRALIAN LESSON
This lesson took place around the middle of the childrens first year at school. The major focus of the lesson was the Firemans ladder problem, in which children were told that a fireman was standing on the middle rung of a ladder. He goes up three more rungs to get to the top. How many rungs altogether on the ladder? They could then replace three rungs with five or any other number.
The lesson began with children sitting in a circle on the floor, taking turns to throw a die and double the number thrown. After about ten minutes, the teacher began to introduce the Firemans ladder problem. During this introduction, she spent about ten minutes eliciting from children what they understood by the middle. Children then worked by themselves or together in small groups for about 20 minutes to solve the problem and its extensions. The lesson ended with a discussion, again of about 20 minutes, where children explained their solutions and engaged in considerable debate about different answers obtained.
Even in the preliminary part of the lesson, it was clear that the teacher has both an agenda regarding the way in which she expects the children to work mathematically and a range of explicit or implicit strategies that she uses to support childrens mathematical explanations.
For example in the early part of the lesson that focussed on doubling, her comments and questions included the following:
T: Talking about doubling. I was thinking that if someone doesnt remember how to double, and we were just rolling one dice and lets say we rolled it on six. How would you tell them to work out the double if they didnt remember it straight away? ...
How did you do it? What did you do in your head? ... Who can explain what she did? ...
What is doubling? What do we do when we are doubling? ...
If you had 100 and you had to double it, or you had 50, whats the idea? What do you have to do? ...
It doesnt matter whether you know the answer or not. What do you have to do? ...
Dont tell him the answer. Tell him how to do it. ...
[B1 is doubling 4 ]
T (to B2): Why did you tell him to count another four?
B2: Because ... doubling has to be the same number. Otherwise it isnt doubling.
Later, finishing explaining to the children what needs to be done for the Firemans ladder problem, she says:
T: Heres what you can do. To work that out you can use materials in our room. You can use things we usually use. You can use paper to draw it. You can use ... sticks to make the ladder or anything else in the room you think would be good to make a ladder with ....
At the end though, we are going to work out how you did it ... were going to talk about how you did it.
Some of the key features of this lesson and strategies used by this teacher to support young childrens mathematical explanations are the following.
Focussing on the conceptual. Two long segments of the lesson focussed on the meaning of doubling and the meaning of the middle. Even when children were able to find the double of a number or the middle person in a line, the teacher persisted with asking questions like What is doubling? and How can we tell it is the middle? The focus was explicitly on the concepts involved and not the calculations or the answers for example, It doesnt matter whether you know the answer or not. What do you have to do?
Making thinking public. This teacher frequently asks young children questions like What did you do in your head? She also has many strategies for making sure that children listen to and engage with other childrens thinking and explanations.
Having and flagging high expectations. The effect of the teacher making comments such as at the end though, we are going to work out how you did it ... were going to talk about how you did it is two-fold. Firstly, it focusses children on their thinking and how they will explain it before they start working on the problem. Secondly, it flags the expectation that all children will have been able to work out the solution and that they will be able to explain their thinking.
Developing norms for discussion. The teacher makes it quite clear that it is not the answer but the logic behind it that is important. In other lessons observed, she has often asked these very young children questions such as How could you prove it? or if he didnt believe you, how could you convince him? The construction of the classroom norms for public discussion is fundamental to establishing a classroom where discussion of mathematical content is possible. In Yackel and Cobbs terms, this is the development of the socio-mathematical norms (1996) needed for public mathematical development.
LEARNING FROM SUCH STUDIES
It is relatively easy to see similarities and differences between lessons and find reasons for these that relate to the school and community norms that operate in the different countries. For example, in terms of public and permanent recording of explanations and the use of the blackboard, Australian teachers are severely hampered by the fact that many do not even have a blackboard to use. This is an unexpected consequence of attempts to make teaching more child-centered. Clearly there would need to be a major shift in teachers understanding of what child-centered teaching might embrace. For example, those Australian educators who have seen lesson vignettes from Japanese classrooms easily recognize the value of a public forum, such as a blackboard, for explicating and sharing childrens ideas in the mathematics classroom.
Another example of cultural difference in the classroom can be seen when early in one of the Hungarian lessons (see Groves, Doig & Szendrei, 2006) the teacher says lets see who can be the most clever today something that would be most unlikely to happen in either an Australian or Japanese classrooms. In Japan, the focus is on perseverance, while in Australia, talking about being clever would be seen as elitist. However, Australian teachers subscribe to the notion of challenging children to try their best, so the question becomes how to reconcile these two positions.
Clearly this is not a simple matter, particularly if one is mindful of Stigler et als (1999) view that one of the problems in effecting change in teaching practice is that teachers lack a set of shared referents for the words they use to describe classroom [practice] (p. 5). The research outlined in this paper is one attempt to provide, if not the words, visual examples to provide such shared referents.
In an earlier project, Mathematics Classrooms Functioning as Communities of Inquiry: Models of Primary Practice, we examined current models of Australian mathematics practice and investigated the extent to which these support or hinder mathematics classrooms functioning as communities of inquiry. The notion of communities of inquiry underpins the Philosophy for Children movement (see, for example, Splitter & Sharp, 1995). Key features of classrooms functioning as communities of philosophical inquiry are the development of skills and dispositions associated with good thinking, reasoning and dialogue; the use of subject matter which is conceptually complex and intriguing, but accessible; and a classroom environment characterised by a sense of common purpose, mutual trust and risk-taking. The project sought to determine the extent to which the wider local education community (primary teachers and principals, as well as mathematics teacher educators and consultants) endorse the goal of mathematics classrooms functioning as communities of inquiry. Videotape from a local Japanese school was used as a counterpoint to that obtained from local Australian classrooms.
Three edited tapes of up to 10 minutes each were produced, representing the contrasting characteristic pedagogies captured on the Australian videotapes. A similar 8-minute (sub-titled) videotape was produced for the Japanese lesson. These videotapes were the stimuli for three separate four-hour focus group meetings for randomly selected teachers (n=12), principals (n=6) and mathematics teacher educators and consultants (n=10). Discussions were based on the findings from the analysis of the ten Australian and two Japanese lessons.
Australian practice was brought into sharp relief. One of the major contrasts between the Australian and the Japanese lessons shown was the highly focused and conceptually orientated nature of the problematic situation presented to the children in the Japanese lesson. This led to considerable discussion and comment in the three focus groups. Teachers and principles contrasted the multitude of tasks not problems in the Australian lessons the huge range of things explored with the genuinely problematic situation presented in the Japanese lesson and the fact that it concentrated on one area in depth. The outcomes-based Australian curriculum was believed to be a major obstacle to more conceptually focused teaching, with one mathematics educator writing that we need to wean ourselves from dependence on ticking outcomes boxes and one principal commenting during discussion that the testing program dictates the curriculum. Surprisingly, even this minimal exposure to classroom practice from another culture challenged principals, teachers, and educators understandings of what constitutes exemplary practice. (For more details about this study, see Groves, Doig and Splitter, 2000.)
CONCLUSION
The explication of the cultural differences in mathematics classroom practice in these different countries provides an example of a strategy for revealing both our own cultural pedagogical artifacts and those of other cultures. We expect the aspects revealed in these studies to provide a basis for further work, by ourselves and other researchers.
Although some cultural differences may be seen as superficial, our findings suggest that, although we may wish to import successful pedagogical practices from elsewhere, a considerable effort would be needed to actually embed them into our own current practice. We believe that the use of video vignettes of classroom practice offers the opportunity for teachers to start the discussion about their pedagogical practices with shared referents.
REFERENCES
Clarke, D.J. (2002). Developments in International Comparative Research in Mathematics Education: Problematising Cultural Explanations. In S. L. Y. Yam & S. Y. S. Lau (Eds.) ICMI Comparative Study Conference 2002: Pre-Conference Proceedings (pp. 7-16). Hong Kong: University of Hong Kong.
Fujii, T. (2004). Personal communication. 26 September 2004.
Groves, S., Doig, B., & Splitter, L. (2000). Mathematics classrooms functioning as communities of inquiry: Possibilities and constraints for changing practice. In T. Nakahara & M. Koyama (Eds), Proceedings of the Twenty-fourth Conference of the International Group for the Psychology of Mathematics Education. (Vol. III, pp.18). Hiroshima, Japan: Hiroshima University.
Groves, S., Doig, B., & Szendrei, J. (2006). Talking across cultures: An international study of young children's mathematical explanations, CIEAEM 58 Congress: Changes in society: A challenge for mathematics education, pp. 259-264, CIEAEM, Srni, Czech Republic.
ADDIN EN.REFLIST Hollingsworth, H., Lokan, J., & McCrae, B. (2003). Teaching mathematics in Australia: Results from the TIMSS 1999 video study. Camberwell: The Australian Council for Educational Research.
Splitter, L. J. & Sharp, A. M. (1995). Teaching for better thinking: The classroom community of inquiry. Melbourne: The Australian Council for Educational Research.
Stigler, J.W., Gonzalez, P., Kawanaka, T., Knoll, S., & Serrano, A. (1999). The TIMSS videotape classroom study: Methods and findings from an explanatory research project on eighth-grade mathematics instruction in Germany, Japan and the United States. (NCES 99074). Washington, DC: US Government Printing Office.
Stigler, J., Gallimore, R., & Hiebert, J. (2000). Using video surveys to compare classrooms and teaching across cultures. Educational Psychologist, 35(2), 87100.
Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458477.
An earlier version of part of this paper was presented at CIEAEM 58 (Groves, Doig, & Szendrei, 2006) under the title Talking Across Cultures: An International Study of Young Childrens Mathematical Explanations.
Talking Across Cultures was funded by the Deakin University Quality Learning Research Priority Area. The project team was Susie Groves, Brian Doig (Deakin University), Toshiakira Fujii, Yoshinori Shimizu (Tokyo Gakugei University) and Julianna Szendrei (Etvs Lornd University, Budapest.
Mathematics classrooms functioning as communities of inquiry: Models of primary practice was funded by the Australian Research Council.
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