Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

The development of studients' algebraic thinking in earlier grades:

A cross-cultural comparative perspective

  • Analyses
  • Published:
Zentralblatt für Didaktik der Mathematik Aims and scope Submit manuscript

Abstract

We analyzed how algebraic concepts and representations are introduced and developed in the Chinese, South Korean, and Singaporean elementary curricula, and in selected Russian and U.S. elementary curricula. In all five curricula, the main goal for learning algebraic concepts is to deepen students' understanding of quantitative relationships, but the emphases and approaches to helping students deepen their understanding of quantitative relationships are very different. Based on the analyses of the five curricula, we discuss four issues related to the development of algebraic thinking in earlier grades: (1) To what extent do curricula expect students in early grades to think algebraically? (2) What level of formalism should we expect of students in the early grades? (3) How can we help students make a smooth transition from arithmetic to algebraic thinking? and (4) Are authentic applications necessary for students in early grades?

Kurzreferat

In dem Beitrag wird analysiert, wie algebraische Begriffe und Repräsentationen in Grundschulcurricula aus China, Russland, Südkorea, Singapur und USA eingeführt und entwickelt werden. In allen fünf Curricula besteht das Hauptziel des Lernens von algebraischen Begriffen darin, das Verständnis von Lernenden hinsichtlich quantitativer Beziehungen zu vertiefen. Jedoch sind die Schwerpunkte und die Ansätze der fünf Curricula zur Vertiefung des Verständnisses der Schülerinnen und Schüler von quantitativen Beziehungen sehr unterschiedlich. Auf der Basis der Analysen dieser fünf genannten Curricula diskutieren wir vier Themen in Zusammenhang mit der Entwicklung von algebraischem Denken in unteren Jahrgangsstufen: (1) Inwieweit erwarten Curricula von Lernenden in unteren Klassenstufen, algebraisch zu denken? (2) Welchen Grad an Formalismus dürfen wir bei Lernenden der unteren Klassenstufen erwarten? (3) Wie können wir Lernenden zu einem angemessenen. Übergang von arithmetischem zu algebraischem Denken verhelfen? und (4) Sind authentische Anwendungen für Lernende der unteren Jahrgangsstufen erforderlich?

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • AAAS (2000): Algebra for all: Not with today's texbooks. See www.project2061.org/newsinfo/press/r1000426.htm

  • Cai, J. (2004a): Introduction to the special issue on developing algebraic thinking in the earlier grades from an international perspective.—In: The Mathematics Educator (Singapore), 8(No.1), p. 1–5

  • Cai, J. (2004b): Developing algebraic thinking in the earlier grades: A case study of the Chinese mathematics curriculum. —In: The Mathematics Educator (Singapore), 8(No. 1), p. 107–130

  • Carpenter, T. P.; Franke, M. L.; Levi, L. (2003): Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann

    Google Scholar 

  • Curriculum Planning & Development Division (2000): Mathematics syllabus primary.—Singapore: Ministry of Education

    Google Scholar 

  • Davydov, V. V.; Gorbov, S. F.; Mikulina, G. G.; Saveleva, O. V. (1999): Mathematics: class 1. J. Schmittan (Ed.)— Binghamton, NY: State University of New York

    Google Scholar 

  • Division of Elementary Mathematics (1999): Mathematics: Elementary school textbook (number 1).—Beijing, China: People's Education Press

    Google Scholar 

  • Driscoll, M. (1999): Fostering algebraic thinking: A guide for teachers grades 6–10.—Portsmouth, NH: Heinemann

    Google Scholar 

  • Freeman, D. J.; Porter, A. C. (1989): Do textbooks dictate the content of mathematics instruction in elementary school?—In: American Educational Research Journal, Vol. 26, p.403–421

  • Kaput, J. (1999): Teaching and learning a new algebra. In: E. Fennema; T. Romberg (Eds), Mathematics classrooms that promote understanding, Mahwah, NJ: Erlbaum, p. 133–155

    Google Scholar 

  • Kieran, C. (2004): Algebraic thinking in the early grades: What is it?—In: The Mathematics Educator (Singapore) 8(No. 1), p. 139–151

  • Lew, H. C. (2004): Developing algebraic thinking in the earlier grades: A case study of the South Korean elementary school mathematics curriculum.—In: The Mathematics Educator (Singapore) 8(No. 1), p. 88–106

  • Mathematical Sciences Education Board (1998): The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium.—Washington D.C.: National Research Council

    Google Scholar 

  • Moyer, J. C.; Huinker, D.; Cai, J. (2004): Developing algebraic thinking in the earlier grades: A case study of the U.S. Investigation series.—In: The Mathematics Educator (Singapore) 8(No. 1), p. 6–38

  • National Council of Teachers of Mathematics. (1989): Curriculum and evaluation standards for school mathematics.—Reston, VA: Author

    Google Scholar 

  • National Council of Teacher of Mathematics (2000): Principles and standards for school mathematics.—Reston, VA: Author

    Google Scholar 

  • Ng, S. F. (2004) Developing algebraic thinking: A case study of the Singaporean primary school curriculum.—In: The Mathematics Educator (Singapore) 8(No. 1), p. 39–59

  • Post, T. R.; Behr, M. J.; Lesh, R. (1988): Proportionality and the development of prealgebra understandings.—In: A. Coxford; A. Shulte (Eds.), The ideas of algebra, K-12. Reston, VA: NCTM (1988 Yearbook), p. 78–90.

    Google Scholar 

  • Russell, S. J.; Economopoulos, K.; Murray, M.; Mokros, J.; Goodrow, A. (1998): Implementing theInvestigations in Number, Data, and Space curriculum: Grades K, 1 and 2.— Menlo Park: CA: Dale Seymour Publications

    Google Scholar 

  • Schifter, D. (1999): Reasoning about operations: Early algebraic thinking in grades K-6.—In: L. V. Stiff, F. R. Curcio (Eds.), Developing mathematical reasoning in grades K-12. Reston, VA: NCTM (1999 Yearbook), p. 62–81

    Google Scholar 

  • Schmidt, W. H.; McKnight, C. C.; & Raizen, S. A. (1996): Characterizing pedagogical flow: An investigation of mathematics and science teaching in six countries.—The Netherlands: Kluwer Academic Publishers

    Google Scholar 

  • Schmidt, W. H.; McKnight, C. C.; Houang, R. T.; Wang, H.; Wiley, D. E.; Cogan, L. S.; Wolfe, R. G. (2002): Why schools matter: A cross-national comparison of curriculum and learning.—San Francisco, CA: Jossey-Bass

    Google Scholar 

  • Schmittau, J.; Morris, A. (2004): The development of algebra in the elementary mathematics curriculum of V. V. Davydov.— In: The Mathematics Educator (Singapore) 8(No. 1), p. 60–87

  • Silver, E. A.; Kenney, P. A. (2001): Results from the sixth mathematics assessment of the National Assessment of Educational Progress.—Reston, VA: NCTM

    Google Scholar 

  • Stacey, K.; Chick H.; Kendal M. (2004): The Future of the Teaching and Learning of Algebra: The 12th, ICMI Study.— The Netherlands: Kluwer Academic Publishers

    Book  Google Scholar 

  • Usiskin, Z. (1995): Why is algebra important to learn?—In: B. Moses (Ed.), Algebraic thinking in grades K-12: Readings from NCTM's school-based journals and other publications. Reston, VA: NCTM, p. 16–21.

    Google Scholar 

  • van Dooren, W.; Verschaffel, L.; Onghena, P. (2002): The impact of preservice teachers' content knowledge on their evaluation of students' strategies for solving arithmetic and algebra word problems.—In: Journal for Research in Mathematics Education, Vol. 33, p. 319–351

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, 523 Ewing Hall, 19716, Newark, De, USA

    Jinfa Cai

  2. Dept. of Math Education, Korea National University of Education, Cheongwon, 363-791, Chungbuk, Korea

    Hee Chan Lew

  3. School of Education, University of Delaware, 19716, Newark, De, USA

    Anne Morris (Associate)

  4. Dept. of Mathematics, Statistics and Computer Science, Marquette University, P.O. Box 1881, 53201-1881, Milwaukee, WI

    John C. Moyer

  5. Mathematics and Mathematics Education Academic Group, National Institute of Education, 1 Nanyang Walk, 637 616, Singapore

    Swee Fong Ng (Associate)

  6. School of Education and Human Development, SUNY At Binghamton, P. O. Box 6000, Binghamton, NY, 13902-6000

    Jean Schmittau (Associate Prof., Dr.)

Authors
  1. Jinfa Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hee Chan Lew
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anne Morris
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. John C. Moyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Swee Fong Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jean Schmittau
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cai, J., Lew, H.C., Morris, A. et al. The development of studients' algebraic thinking in earlier grades:. Zentralblatt für Didaktik der Mathematik 37, 5–15 (2005). https://doi.org/10.1007/BF02655892

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02655892

ZDM-Classification

Search

Navigation