# How reform curricula in the USA and Korea present multiplication and division of fractions

- 859 Downloads
- 19 Citations

## Abstract

In order to give insights into cross-national differences in schooling, this study analyzed the development of multiplication and division of fractions in two curricula: *Everyday Mathematics* (EM) from the USA and the 7th Korean mathematics curriculum (KM). Analyses of both the content and problems in the textbooks indicate that multiplication of fractions is developed in KM one semester earlier than in EM. However, the number of lessons devoted to the topic is similar in the two curricula. In contrast, division of fractions is developed at about the same time in both curricula, but due to different beliefs about the importance of the topic, KM contains five times as many lessons and about eight times as many problems about division of fractions as EM. Both curricula provide opportunities to develop conceptual understanding and procedural fluency. However, in EM, conceptual understanding is developed first followed by procedural fluency, whereas in KM, they are developed simultaneously. The majority of fraction multiplication and division problems in both curricula requires only procedural knowledge. However, multistep computational problems are more common in KM than in EM, and the response types are also more varied in KM.

### Keywords

Textbook analysis Multiplication of fractions Division of fractions Reform curriculum### References

- Aksu, M. (1997). Student performance in dealing with fractions.
*The Journal of Educational Research, 90*(6), 375–380.Google Scholar - Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratioand proportion. In D. Grouws (Ed.),
*Handbook of research onmathematics teaching and learning*(pp. 296–333). New York: Macmillan.Google Scholar - Burns, M. (2000).
*About teaching mathematics: A K-8 resource, 2nd ed.*. Sausalito: Math Solutions Publication.Google Scholar - Carroll, W. M. (1998). Middle school students’ reasoning about geometric situations.
*Mathematics Teaching in the Middle School, 3*, 398–403.Google Scholar - Carroll, W. M., & Isaacs, A. (2003). Achievement of students using the University of Chicago School Mathematics Project’s Everyday Mathematics. In S. L. Senk & D. R. Thompson (Eds.),
*Standards based school mathematics curricula: What are they? What do students learn?*(pp. 79–108). Mahwah: Erlbaum.Google Scholar - Collopy, R. (2003). Curriculum materials as a professional development tool: How mathematics textbook affected two teachers’ learning.
*Elementary School Journal, 103*, 287–311.CrossRefGoogle Scholar - Cramer, K., & Bezuk, N. (1991). Multiplication of fractions: Teaching for understanding.
*Arithmetic Teacher, 39*(3), 34–37.Google Scholar - Dossey, J., Halvorsen, K., & McCrone, S. (2008).
*Mathematics education in the United States 2008, A capsule summary fact book*. Reston: National Council of Teachers of Mathematics.Google Scholar - Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving vertical problems in multiplication and division.
*Journal for Research in Mathematics Education, 16*, 3–17.CrossRefGoogle Scholar - Flanders, J. R. (1987). How much of the content in mathematics textbooks is new?
*Arithmetic Teacher, 35*, 18–23.Google Scholar - Fuson, K. C., Stigler, J. W., & Bartsch, K. (1988). Grade placement of addition and subtraction topics in Japan, mainland China, the Soviet Union, Taiwan, and the United States.
*Journal for Research in Mathematics Education, 19*(5), 449–456.CrossRefGoogle Scholar - Harding, D. C. (1995).
*A comparative study of German and American school mathematics textbooks in their approach to problem solving*. Doctoral dissertation, University of Chicago (UMI: AAT T-30763).Google Scholar - Husen, T. (Ed.). (1967).
*International study of achievement in mathematics: A comparison of twelve countries, Vols. I and II*. New York: Wiley.Google Scholar - Kaufman, D. (1997). Collaborative approaches in preparing teachers for content-based and language enhanced settings. In M. A. Snow & D. M. Brinton (Eds.),
*The content-based classroom: Perspectives on integrating language and content*(pp. 175–186). London: Longman.Google Scholar - Kennedy, L. M., Steve, T., et al. (1997).
*Guiding children’s learning of mathematics*. Belmont: Wadsworth/Thomson Learning.Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001).
*Adding it up: Helping children learn mathematics*. Washington: National Academy Press.Google Scholar - Kouba, V., & Franklin, K. (1993). Multiplication and division: Sense making and meaning. In R. J. Jensen (Ed.),
*Research ideas for the classroom: Early childhood mathematics*(pp. 103–126). New York: Macmillan.Google Scholar - Li, Y. (2002). 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.CrossRefGoogle 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 - Mack, N. K. (1998). Building a foundation for understanding the multiplication of fractions.
*Teaching Children Mathematics, 5*(1), 34–38.Google Scholar - Mayer, R. E., Sims, V., & Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States.
*American Educational Research Journal, 32*(2), 443–460.Google Scholar - McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. J., et al. (1987).
*The underachieving curriculum: Assessing U.S. school mathematics from an international perspective*. Champaign: Stipes.Google Scholar - Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: An empirical approach.
*Educational Studies in Mathematics, 56*, 255–286.CrossRefGoogle Scholar - Mullis, I. V. S., Martin, M. O., Foy, P., Olson, J. F., Preuschoff, C., Erberber, E., et al. (2008).
*TIMSS 2007 International mathematics report: Findings from IEA’s Trends in International Mathematics and Science Study at the fourth and eighth grades*. Chestnut Hill: TIMSS & PIRLS International Study Center, Boston College.Google Scholar - National Council of Teachers of Mathematics. (2006).
*Curriculum focal points for kindergarten through grade 8 mathematics: A quest for coherence*. Reston, VA: NCTM.Google Scholar - NCTM. (1989).
*Curriculum and evaluation standards for school mathematics*. Reston: National Council of Teachers of Mathematics.Google Scholar - NCTM. (2000).
*Principles and standards for school mathematics*. Reston: National Council of Teachers of Mathematics.Google Scholar - Nicely, R. F. (1985). Higher-order thinking skills in mathematics textbooks.
*Educational Leadership, 42*(7), 26–30.Google Scholar - Nicely, R. F., Fiber, H. R., & Bobango, J. C. (1986). The cognitive content of elementary school mathematics textbooks.
*Arithmetic Teacher, 34*, 60–61.Google Scholar - Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis.
*Mathematical Thinking and Learning, 1*(4), 279314.CrossRefGoogle Scholar - OECD. (2003).
*Learning for tomorrow’s world—First results from PISA.*Organization for Economic Co-operation and Development.Google Scholar - OECD. (2006).
*Education at a glance 2006: Briefing note for the United States*. Paris: Organisation for Economic Co-Operation and Development.Google Scholar - Reys, B. J., Reys, R. E., & Chavez, O. (2004). Why mathematics textbooks matter.
*Educational Leadership, 61*(5), 61–66.Google Scholar - Romberg, T. (1983). A common curriculum for mathematics. In G. D. Fenstermacher & J. Goodlad (Eds.),
*Individual differences and the common curriculum*(pp. 121–159). Chicago: National Society for the Study of Education.Google Scholar - Schmidt, W. H., McKnight, C., Cogan, L. S., Jakwerth, P. M., & Houang, R. T. (1999).
*Facing the consequences: Using TIMSS for a closer look at U. S. mathematics and science education*. Boston: Kluwer.Google Scholar - Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., Cogan, L. S., et al. (2001).
*Why schools matter: A cross-national comparison of curriculum and learning*. San Francisco: Jossey-Bass.Google Scholar - Senk, S. L., Beckmann, C. E., & Thompson, D. R. (1997). Assessment and grading in high school mathematics classrooms.
*Journal for Research in Mathematics Education, 28*(2), 187–215.CrossRefGoogle Scholar - Senk, S. L., & Thompson, D. R. (Eds.). (2003).
*Standards-based school mathematics curricula: What are they? What do students learn?*Mahwah: Erlbaum.Google Scholar - Silver, E. A., & Kenney, P. A. (Eds.) (2002)
*Results from the seventh assessment of educational progress.*Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge.
*Journal of Mathematical Behavior, 20*, 267–307.CrossRefGoogle Scholar - Stigler, J. W., Fuson, K. C., Ham, M., & Kim, M. S. (1986). An analysis of addition and subtraction word problems in American and Soviet elementary mathematics textbooks.
*Cognition and Instruction, 3*, 153–171.CrossRefGoogle Scholar - Stigler, J. W., & Hiebert, J. (1999).
*The teaching gap—Best ideas from the world’s teachers for improving education in the classroom*. New York: The Free Press.Google Scholar - Tabachneck, H. J. M., Koedinger, K. R., & Nathan, M. J. (1995). A cognitive analysis of the task demands of early algebra. In J. D. Moore & J. F. Lehman (Eds.),
*Proceedings of the Seventeenth Annual Conference of the Cognitive Science Society*(pp. 397–402). Hillsdale: Erlbaum.Google Scholar - Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to principles and standards in school mathematics*(pp. 95–113). Reston: National Council of Teachers of Mathematics.Google Scholar - Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning.
*Journal for Research in Mathematics Education, 30*(4), 390–416.CrossRefGoogle Scholar - Tzur, R. (2004). Teacher and students’ joint production of a reversible fraction conception.
*Journal of Mathematical Behavior, 23*, 92–114.CrossRefGoogle Scholar - Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002).
*According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks*. Dordrecht: Kluwer.Google Scholar - Van de Walle, J. (2001).
*Elementary and middle school mathematics: Teaching developmentally, 4th ed.*. New York: Addison Wesley Longman.Google Scholar - Watanabe, T. (2003). Teaching multiplication: An analysis of elementary school mathematics teachers’ manual from Japan and the United States.
*Elementary School Journal, 104*, 111–125.CrossRefGoogle Scholar - What Works Clearinghouse. (2007).
*Intervention report, Everyday Mathematics.*Retrieved April 30, 2007 from http://ies.ed.gov/ncee/wwc/pdf/WWC_Everyday_Math_043007.pdf. - Wong, N. Y., Lam, C. C., & Chan, C. C. (2002). The current state of the “lived space” of mathematics learning.
*Hiroshima Journal of Mathematics Education, 10*, 27–52.Google Scholar - Wu, H. S. (April, 2005). Key mathematical ideas in grades 5–8. Paper presented at the Annual Meeting of the NCTM, Anaheim, CA. Retrieved September 12, 2005 from http://math.berkeley.edu/∼wu/NCTM2005a.pdf.
- Yan, Z. &, Fan, L. (2004, July).
*An analysis of representation of problem types in China and US Mathematics Textbook*. Paper presented at ICME-10, Copenhagen, Denmark.Google Scholar