# Understanding Linear Function: a Comparison of Selected Textbooks from England and Shanghai

## Abstract

This study describes a comparison of how worked examples in selected textbooks from England and Shanghai presented possible learning trajectories towards understanding linear function. Six selected English textbooks and one Shanghai compulsory textbook were analysed with regards to the understanding required for pure mathematics knowledge in linear function. Understanding was defined as being at five levels: Dependent Relationship, Connecting Representations, Local Properties Noticing, Object Analysis and Inventising. These levels were developed by examining the most prominent theories from the existing literature on understanding function. Findings suggested that the English textbooks constrained the structural aspect of understanding linear function due to a point-to-point view of function, while the Shanghai textbook which focussed on a variable view of function overemphasised the algebraic approach. The discussion explored the drawbacks to each approach and what teachers or textbook writers could do to balance these two approaches in order to facilitate students’ understanding towards a structural view of linear function.

### Keywords

Comparative study Linear function Textbook analysis Understanding### References

- Bao, J. (2002).
*Comparative study on composite difficulty of Chinese and British School mathematics curricula.*(Unpublished Doctoral Dissertation), East China Normal University, Shanghai, China.Google Scholar - Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Durán, R., Reed, B. S. & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra.
*American Educational Research Journal, 34*(4), 663–689.CrossRefGoogle Scholar - Cai, J. (1995). A cognitive analysis of US and Chinese students’ mathematical performance on tasks involving computation, simple problem solving, and complex problem solving [Monograph].
*Journal for Research in Mathematics Education Monographs, 7.*Google Scholar - Clarke, D. (2003). International comparative research in mathematics education. In K.Leithwood, & P. Hallinger (Eds.),
*Second international handbook of mathematics education*(pp. 143–184). Dordrecht, The Netherlands: Kluwer.Google Scholar - Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme.
*The Journal of Mathematical Behavior, 15*(2), 167–192.CrossRefGoogle Scholar - Coughlan, S. (2014, March 12). Shanghai teachers flown in for maths.
*BBC News*. Retrieved August 8, 2014 from http://www.bbc.co.uk/news/education-26533428 - DeMarois, P. & Tall, D. (1996). Facets and layers of the function concept. In L. Puig & A. Gutierrez (Eds.),
*Proceedings of the Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 297–304). Valencia, Spain: University of Valencia.Google Scholar - Department for Education (2013, December 2).
*Mathematics programmes of study: Key stages 4 national curriculum in England*. Retrieved from https://www.gov.uk/government/consultations/national-curriculum-reform-england-ks4-english-and-mathematics. - Doorman, M., Drijvers, P., Gravemeijer, K., Boon, P. & Reed, H. (2012). Tool use and the development of the function concept: From repeated calculations to functional thinking.
*International Journal of Science and Mathematics Education, 10*(6), 1243–1267.CrossRefGoogle Scholar - Dreyfus, T. & Eisenberg, T. (1982). Intuitive functional concepts: A baseline study on intuitions.
*Journal for Research in Mathematics Education, 13*(5), 360–380.CrossRefGoogle Scholar - Dubinsky, E. & McDonald, M. (2002). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, M. Artigue, U. Kirchgräber, J. Hillel, M. Niss & A. Schoenfeld (Eds.),
*The teaching and learning of mathematics at university level*(Vol. 7, pp. 275–282). Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar - Fan, L. (2013). Textbook research as scientific research: Towards a common ground on issues and methods of research on mathematics textbooks.
*ZDM, 45*(5), 765–777.CrossRefGoogle Scholar - Fan, L. & Zhu, Y. (2007). Representation of problem-solving procedures: A comparative look at China, Singapore, and US mathematics textbooks.
*Educational Studies in Mathematics, 66*(1), 61–75.CrossRefGoogle Scholar - Fan, L., Zhu, Y. & Miao, Z. (2013). Textbook research in mathematics education: development status and directions.
*ZDM, 45*(5), 633–646.CrossRefGoogle Scholar - Foxman, D. (1999).
*Mathematics textbooks across the world: Some evidence from the Third International Mathematics and Science Study (TIMSS)*. Slough, UK: National Federation for Educational Research.Google Scholar - Gagatsis, A. & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving.
*Educational Psychology, 24*(5), 645–657.CrossRefGoogle Scholar - Habre, S. & Abboud, M. (2006). Students’ conceptual understanding of a function and its derivative in an experimental calculus course.
*The Journal of Mathematical Behavior, 25*(1), 57–72.Google Scholar - Healy, L. & Hoyles, C. (1999). Visual and symbolic reasoning in mathematics: Making connections with computers?
*Mathematical Thinking and Learning, 1*(1), 59–84.CrossRefGoogle Scholar - Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York, NY: Macmillan.Google Scholar - Hitt, F. (1998). Difficulties in the articulation of different representations linked to the concept of function.
*The Journal of Mathematical Behavior, 17*(1), 123–134.CrossRefGoogle Scholar - Howse, P. (2014, February 18). Shanghai visit for minister to learn maths lessons.
*BBC News*. Retrieved August 8, 2014, from http://www.bbc.co.uk/news/education-26228234. - Howson, G. (2013). The development of mathematics textbooks: Historical reflections from a personal perspective.
*ZDM, 45*(5), 647–658.CrossRefGoogle Scholar - Jerrim, J. & Choi, Á. (2014). The mathematics skills of school children: How does England compare to the high-performing East Asian jurisdictions?
*Journal of Education Policy, 29*(3), 349–376.CrossRefGoogle Scholar - Jia, P.-Z. (2004). Six cognitive stages in the teaching for function concept.
*Journal of Mathematics Education, 13*(3), 79–81.Google Scholar - Johansson, M. (2003).
*Textbooks in mathematics education: A study of textbooks as the potentially implemented curriculum.*(Licentiate Thesis, Luleå University of Technology, Department of Mathematics).Google Scholar - Jones, K. & Fujita, T. (2013). Interpretations of national curricula: The case of geometry in textbooks from England and Japan.
*ZDM, 45*(5), 671–683.CrossRefGoogle Scholar - Kieran, C. (1997). Mathematical concepts at the secondary school level: The learning of algebra and functions. In T. Nunes & P. Bryant (Eds.),
*Learning and teaching mathematics: An international perspective*(pp. 133–158). Hove, UK: Psychology Press.Google Scholar - Kleiner, I. (2009). Evolution of the function concept: A brief survey. In M. Anderson, V. Katz & R. Wilson (Eds.),
*Who gave you the epsilon?: And other tales of mathematical history*(Vol. 20, pp. 14–26). Washington, DC : Mathematical Association of America.Google Scholar - Leinhardt, G., Zaslavsky, O. & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching.
*Review of Educational Research, 60*(1), 1–64.CrossRefGoogle Scholar - Leung, F. K. (1995). The mathematics classroom in Beijing, Hong Kong and London.
*Educational Studies in Mathematics, 29*(4), 297–325.CrossRefGoogle Scholar - Li, Y. (2000). A comparison of problems that follow selected content presentations in American and Chinese mathematics textbooks.
*Journal for Research in Mathematics Education, 31*(2), 234–241.Google Scholar - Llinares, S. (2000). Secondary school mathematics teacher’s professional knowledge: A case from the teaching of the concept of function.
*Teachers and Teaching, 6*(1), 41–62.Google Scholar - Love, E. & Pimm, D. (1996). ‘This is so’: A text on texts. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.),
*International handbook of mathematics education*(pp. 371–409). Dordrecht, The Netherlands: Kluwer.Google Scholar - Lue, Y.-T. (2013). A study on the horizontal transformations of elementary functions.
*Journal of Modern Education Review, 3*(6), 443–153.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Mullis, I. V., Martin, M. O., Gonzalez, E. J. & Chrostowski, S. J. (2004).
*TIMSS 2003 international mathematics report: findings from IEA’s trends in International Mathematics and Science Study at the fourth and eighth grades*. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.Google Scholar - Oehrtman, M., Carlson, M. & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.),
*Making the connection: research and teaching in undergraduate mathematics education*(pp. 27–42). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar - Park, K. & Leung, F. K. (2006). A comparative study of the mathematics textbooks of China, England, Japan, Korea, and the United States. In F. S. Leung, K.-D. Graf & F. Lopez-Real (Eds.),
*Mathematics education in different cultural traditions—A comparative study of East Asia and the West*(Vol. 9, pp. 227–238). Berlin, Germany: Springer.Google Scholar - Pepin, B. & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms.
*ZDM, 33*, 158–175.Google Scholar - Pirie, S. E. B. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it?
*Educational Studies in Mathematics, 26*(2–3), 165–190.Google Scholar - Ronda, E. R. (2009). Growth points in students’ developing understanding of function in equation form.
*Mathematics Education Research Journal, 21*(1), 31–53.CrossRefGoogle Scholar - Schwarz, B. & Dreyfus, T. (1995). New actions upon old objects: A new ontological perspective on functions.
*Educational Studies in Mathematics, 29*(3), 259–291.CrossRefGoogle Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*(1), 1–36.CrossRefGoogle Scholar - Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification—The case of algebra. In P. Cobb (Ed.),
*Learning mathematics—Constructivist and interactionist theories of mathematical development*(pp. 87–124). Dordrecht, The Netherlands: Springer.Google Scholar - Shanghai City Education Committee (2004).
*Shanghai City primary and secondary mathematics curriculum standard*. Shanghai, China: Shanghai Education.Google Scholar - Sierpinska, A. (1994).
*Understanding in mathematics*. London, UK: Falmer.Google Scholar - Skemp, R. R. (1971).
*The psychology of learning mathematics*. Harmondsworth, UK: Penguin.Google Scholar - Slavit, D. (1997). An alternate route to the reification of function.
*Educational Studies in Mathematics, 33*(3), 259–281.CrossRefGoogle Scholar - Son, J.-W. & Senk, S. L. (2010). How reform curricula in the USA and Korea present multiplication and division of fractions.
*Educational Studies in Mathematics, 74*(2), 117–142.CrossRefGoogle Scholar - Stein, M. K., Baxter, J. A. & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing.
*American Educational Research Journal, 27*(4), 639–663.CrossRefGoogle Scholar - Stein, M. K., Remillard, J. & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics*(pp. 319–370). Charlotte, NC: Information Age.Google Scholar - Valverde, G., Bianchi, L., Wolfe, R., Schmidt, W. & Houang, R. (2002).
*According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks*. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar - Whitburn, J. (1995). The teaching of mathematics in Japan: An English perspective.
*Oxford Review of Education, 21*(3), 347–360.CrossRefGoogle Scholar - Zachariades, T., Christou, C., & Papageorgiou, E. (2002).
*The difficulties and reasoning of undergraduate mathematics students in the identification of functions*. Paper presented at the 10th ICME Conference, Crete, Greece.Google Scholar - Zeng, G.-G. (2002). Investigation on students’ cognitive development of function concept.
*Journal of Mathematics Education, 11*(2), 99–102.Google Scholar - Zhang, D., & Yu, B. (2013).
*Mathematics education of Chinese way*. Shanghai, China: Shanghai Educational Publishing House. [In Chinese]. Google Scholar - Zhu, Y. & Fan, L. (2006). Focus on the representation of problem types in intended curriculum: A comparison of selected mathematics textbooks from mainland China and the United States.
*International Journal of Science and Mathematics Education, 4*(4), 609–626.CrossRefGoogle Scholar