# Insights from students’ private work in their notebooks: how do students learn from the teacher’s examples?

## Abstract

Students’ seatwork plays an important part in their learning in their lessons, and very often, students record their private work in the notebooks during seatwork. The students’ private work in their notebooks reflects students’ learning and thinking, representing explicit learning outcomes. The students’ private work in their notebooks of 14 mathematics lessons of an eighth-grade Hong Kong classroom was analyzed. The mathematical tasks used in the lessons were categorized with the Trends in International Mathematics and Science Study (TIMSS) cognitive domains framework. The implementation of the tasks was recorded in cycles of teacher’s examples (TEs) and students’ exercises (SEs). By comparing the methods employed by the students and the teacher, the students’ methods were found to be mainly imitation or partial imitation regardless of the cognitive domains of the students’ exercises. The students’ perspectives on the instructional practice expressed in the post-lesson interviews were used as a triangulation for the results. The results showed that the students appreciated the teacher’s explanation and demonstration in the teacher’s exposition. Finally, the authors argue that the high percentages of imitation of teacher’s methods not only are due to the students’ choice, but also are influenced by the Confucian heritage cultures.

## Keywords

Students’ private work Learning Cognitive domains Imitation## Notes

### Acknowledgments

The project is funded by General Research Fund, Research Grants Council, Hong Kong SAR, China.

## References

- Biggs, J. (1998). Chapter 3 Learning from the Confucian heritage: So size doesn’t matter?
*International Journal of Educational Research, 29*, 723–738.CrossRefGoogle Scholar - Bishop, A. J. (1991).
*Mathematical enculturation: A cultural perspective on mathematics education*. Doedrecht: Kluwer Academic Publisher.Google Scholar - Cai, J., & Wang, T. (2010). Conceptions of effective mathematics teaching within a cultural context: Perspectives of teachers from China and the United States.
*Journal of Mathematics Teacher Education, 13*(3), 265–287.CrossRefGoogle Scholar - Clarke, D., Keitel, C., & Shimizu, Y. (Eds.). (2006).
*Mathematics classrooms in twelve countries: The insiders’ perspective*. Rotterdam: Sense Publishers.Google Scholar - Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction.
*Educational Psychologist, 23*(2), 167–180.CrossRefGoogle Scholar - Fried, M. N. (2008). Between public and private: Where students’ mathematical selves reside. In L. Radford, G. Schubring, & F. Seeger (Eds.),
*Semiotics in mathematics education: Epistemology, history, classroom, and culture*(pp. 121–137). Rotterdam: Sense Publishers.Google Scholar - Fried, M. N., & Amit, M. (2003). Some reflections on mathematics classroom notebooks and their relationship to the public and private nature of student practices.
*Educational Studies in Mathematics, 53*(2), 91–112. doi: 10.1023/a:1025572900956 CrossRefGoogle Scholar - Gu, L.-Y., Huang, R.-J., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.),
*How Chinese learn mathematics: Perspective from insiders*(pp. 309–347). Singapore: World Scientific Publishing Company.CrossRefGoogle Scholar - Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning.
*Journal for Research in Mathematics Education, 28*, 524–549.CrossRefGoogle Scholar - Hino, K. (2006). The role of seatwork in three Japanese classrooms. In D. Clarke, C. Keitel, & Y. Shimizu (Eds.),
*Mathematics classrooms in twelve countries: The insider’s perspective*(pp. 59–74). Rotterdam: Sense Publishers.Google Scholar - Huang, R., & Leung, F. K. S. (2004). Cracking the paradox of the Chinese learners: Looking into the mathematics classrooms in Hong Kong and Shanghai. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.),
*How Chinese learn mathematics: Perspectives from insiders*(pp. 384–381). Singapore: World Scientific Publishing Company.Google Scholar - Jablonka, E. (2006). Student(s) at the front: Forms and function in six classrooms from Germany, Hong Kong and the United States. In D. Clarke, J. Emanuelsson, E. Jablonka, & I. A. C. Mok (Eds.),
*Making connections: Comparing mathematics classrooms around the world*(pp. 107–126). Rotterdam: Sense Publishers.Google Scholar - Kaur, B. (2009). Characteristic of good mathematics teaching in Singapore grade 8 classrooms: A juxtaposition of teachers’ practice and students’ perception.
*ZDM, 41*, 333–347. doi: 10.1007/s11858-009-0170-z CrossRefGoogle Scholar - Leung, F. K. S. (2001). In search of an East Asian identity in mathematics education.
*Educational Studies in Mathematics, 47*, 35–51.CrossRefGoogle Scholar - Leung, F. K. S. (2005). Some characteristics of East Asian mathematics classrooms based on data from the TIMSS 1999 video study.
*Educational Studies in Mathematics, 60*, 199–215.CrossRefGoogle Scholar - Leung, F. K. S., & Park, K. (2002). Competent students, competent teachers?
*International Journal of Educational Research, 37*, 113–129.Google Scholar - Li, S. (2006). Practice makes perfect: A key belief in China. In F. K. S. Leung, K.-D. Graf, & F. J. Lopez-Real (Eds.),
*Mathematics education in different cultural traditions: A comparative study of East Asia and the Wes*t (pp. 129–138). New York: Springer.Google Scholar - Lui, K. W., & Leung, F. K. S. (2013). Curriculum traditions in Berlin and Hong Kong: A comparative case study of the implemented mathematics curriculum.
*ZDM, 45*, 35–46. doi: 10.1007/s11858-012-0387-0 CrossRefGoogle Scholar - Mok, I. A. C. (2009). In search of an exemplary mathematics lesson in Hong Kong: An algebra lesson on factorization of polynomials.
*ZDM, 41*, 319–332. doi: 10.1007/s11858-009-01668-8 CrossRefGoogle Scholar - Mok, I. A. C., & Lopez-Real, F. (2006). A tale of two cities: A comparison of six teachers in Hong Kong and Shanghai. In D. Clarke, C. Keitel, & Y. Shimizu (Eds.),
*Mathematics classrooms in twelve countries: The insider’s perspective*(pp. 237–246). Rotterdam: Sense Publishers.Google Scholar - Mok, I. A. C., Kaur, B., Zhu, Y., & Yau, K. W. (2013). What really matters to students? A comparison between Hong Kong and Singapore mathematics lessons. In B. Kaur, G. Anthony, M. Ohtani, & D. Clarke (Eds.),
*Student voice in mathematics classrooms around the world*(pp. 189–208). Rotterdam: Sense Publishers.Google Scholar - Mullis, I. V. S., Martin, M. O., Ruddock, G. J., O’Sullivan, C. Y., & Preuschoff, C. (2009).
*TIMSS 2011 assessment frameworks*(pp. 40–46). Chestnut Hill: TIMSS & PIRLS International Study Centre, Boston College.Google Scholar - Mullis, I. V. S., Martin, M. O., Foy, P & Arora, A. (2012)
*TIMSS 2011 international results in mathematics.*(pp. 139-171). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.Google Scholar - National Council of Teachers of Mathematics. (1991).
*Professional standards for teaching mathematics*. Reston, VA: Author.Google Scholar - Nixon, R. S., & Barth, K. N. (2014). A comparison of TIMSS items using cognitive domains.
*School Science and Mathematics, 114*(2), 65–75.CrossRefGoogle Scholar - Organization for Economic Co-operation and Development (OECD). (2010).
*PISA 2009 results: What students know and can do: Student performance in reading, mathematics and science*(Vol. 1). Paris: OECD.Google Scholar - Park, K., & Leung, F. K. S. (2006). Mathematics lessons in Korea: Teaching with systematic variation. In D. Clarke, C. Keitel, & Y. Shimizu (Eds.),
*Mathematics classrooms in twelve countries: The insider’s perspective*(pp. 247–261). Rotterdam: Sense Publishers.Google Scholar - Serrano, A. M. (2012). A cross-cultural investigation into how tasks influence seatwork activities in mathematics lessons.
*Teaching and Teacher Education, 28*, 806–817.CrossRefGoogle Scholar - Stigler, J. W., Gonzales, P., Kawanaka, T., Knoll, S., & Serrano, A. (1999).
*The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States*. Washington, D. C.: U.S. Department of Education National Center for Education Statistics.Google Scholar - Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms.
*American Educational Research Journal, 33*(2), 455–488.CrossRefGoogle Scholar - Vygotsky, L. S. (1978).
*Mind in society: The development of higher psychological process*. Cambridge: Harvard University Press.Google Scholar - Watkins, D. A., & Biggs, J. B. (Eds.). (2001).
*Teaching the Chinese learner: Psychological and pedagogical perspectives*. Hong Kong: Comparative Education Research Centre, The University of Hong Kong.Google Scholar - Wong, N. Y. (2006). From “entering the way” to “exiting the way”: In search of a bridge to span “basic skills” and “process abilities”. In F. K. S. Leung, K.-D. Graf, & F. J. Lopez-Real (Eds.),
*Mathematics education in different cultural traditions: A comparative study of East Asia and the West*(pp. 111–128). New York: Springer.Google Scholar