Skip to main content
SpringerLink
Log in

The Fifth Lesson: Students’ Responses to a Patterning Task Across the Four Countries

  • Chapter
  • First Online:
Encountering Algebra

  • 614 Accesses

Abstract

The chapter reports a comparative analysis of problem-solving activities in classrooms in the four countries. The problem students work with is a patterning task, taken from a major international comparative study. The idea of analyzing this group work across countries is to see what traces of algebra learning may be discerned in the work of the students. The results show the complexities of early algebra learning by presenting in detail the obvious variations in how students tackle the problem in more or less algebraic ways. In the small-group work in classrooms, students pool their ideas allowing for the discourse to build on itself in a cumulative manner.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Sfard coined the term commognition to signal the close relationship between communication and cognition in her socio-cultural framework for the learning of mathematics.

  2. 2.

    Radford (2002) use the wording objectification in a more general sense than Sfard (2008). The process of knowledge objectification that Radford delineates is an ongoing process of becoming aware of cultural ways of reasoning. However, Sfard uses objectification in order to describe the movement from talking about mathematical objects as processes to explaining them as objects.

  3. 3.

    See Chap. 5, this volume, classroom B for more information.

  4. 4.

    See Chap. 7 in this volume, Miss A’s classroom for more information.

  5. 5.

    These students normally speak Swedish, so the inclusion of Finnish words is a break in the ordinary discourse.

  6. 6.

    Here the term “algebraic equation” is defined as an equation involving at least one alpha-numerical letter (variable).

References

  • Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 115–136). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Berg, C. V. (2009). Developing algebraic thinking in a community of inquiry: Collaboration between three teachers and a didactician. Diss., The University of Agder, Kristiansand, Norway.

    Google Scholar 

  • Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM: The International Journal on Mathematics Education, 40(3), 349–361.

    Google Scholar 

  • Carraher, D., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.), Handbook of research in mathematics education (pp. 669–705). Greenwich, CT: Information Age Publishing.

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D., & Schwartz, J. (2008). Early algebra is not the same as algebra early. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 235–272). New York, NY: Routledge.

    Google Scholar 

  • Caspi, S., & Sfard, A. (2012). Spontaneous meta-arithmetic as a first step toward school algebra. International Journal of Educational Research, 51–52, 45–65.

    Article  Google Scholar 

  • de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: Theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics, 80, 133–152.

    Article  Google Scholar 

  • Filloy, E., & Rojano, T. (1984). From an arithmetical to an algebraic thought. In J. Moser (Ed.), Proceedings of PME-NA VI (pp. 51–56). Madison, WI: PME.

    Google Scholar 

  • Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.

    Google Scholar 

  • Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.

    Google Scholar 

  • Hitt, F., & González-Martín, A. S. (2016). Generalizations, covariation, functions, and calculus. In A. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education: The journey continues (pp. 3–38). Rotterdam: Sense.

    Chapter  Google Scholar 

  • Kaput, J. J. (2000). Teaching and learning a new algebra with understanding. Dartmouth, MA: National Center for Improving Student Learning and Achievement in Mathematics and Science.

    Google Scholar 

  • Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York, NY: Macmillan.

    Google Scholar 

  • Kieran, C. (2007b). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester (Ed.), Second Handbook of research on mathematics teaching (pp. 707–762). Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Kieran, K. (Ed.). (2018). Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice. New York, NY: Springer.

    Google Scholar 

  • Mason, J. (1996). Expressing generality and the roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (pp. 65–86). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom. The case of function. International Journal of Educational Research, 51, 10–27.

    Article  Google Scholar 

  • Nardi, E., Ryve, A., Stadler, E., & Viirman, O. (2014). Commognitive analysis of the learning and teaching of mathematics at university level: The case of discursive shifts in the study of calculus. Research in Mathematics Education, 16(2), 182–198.

    Article  Google Scholar 

  • Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237–268.

    Article  Google Scholar 

  • Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23.

    Google Scholar 

  • Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70, 111–126.

    Article  Google Scholar 

  • Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. Research in Mathematics Education, 12(1), 1–19.

    Article  Google Scholar 

  • Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM: The International Journal of Mathematics Education, 46(3), 349–361.

    Article  Google Scholar 

  • Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 1–23). New York: Springer.

    Google Scholar 

  • Reinhardtsen, J. (2012). The introduction of algebra: Comparative studies of textbooks in Finland, Norway, Sweden and USA (Master’s thesis at the University of Agder). Kristiansand: University of Agder.

    Google Scholar 

  • Reinhardtsen, J., Carlsen, M., & Säljö, R. (2015). Capturing learning in classroom interaction in mathematics: Methodological considerations. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 1474–1481). Prague: Charles University in Prague.

    Google Scholar 

  • Remillard, K. S. (2014). Identifying discursive entry points in paired-novice discourse as a first step in penetrating the paradox of learning mathematical proof. The Journal of Mathematical Behavior, 34, 99–113.

    Article  Google 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, 1–36.

    Article  Google Scholar 

  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York, NY: Cambridge University Press.

    Book  Google Scholar 

  • Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191–228.

    Article  Google Scholar 

  • Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147–164.

    Article  Google Scholar 

  • Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 386–420). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • TIMSS & PIRLS. (2009). TIMSS 2007 user guide for the international database: Released items, mathematics eighth grade. TIMSS & PIRLS International Study Center: Boston College. Retrieved from http://timss.bc.edu/timss2007/items.html

  • Vergnaud, G. (1982). Cognitive and developmental psychology and research in mathematics education: Some theoretical and methodological issues. For the Learning of Mathematics, 3(2), 31–41.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Agder, Kristiansand, Norway

    Jorunn Reinhardtsen

  2. Department of Psychology, University of California, Los Angeles, CA, USA

    Karen B. Givvin

Authors
  1. Jorunn Reinhardtsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karen B. Givvin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jorunn Reinhardtsen .

Editor information

Editors and Affiliations

  1. Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden

    Cecilia Kilhamn

  2. Department of Education, Communication and Learning, University of Gothenburg, Gothenburg, Sweden

    Roger Säljö

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Reinhardtsen, J., Givvin, K.B. (2019). The Fifth Lesson: Students’ Responses to a Patterning Task Across the Four Countries. In: Kilhamn, C., Säljö, R. (eds) Encountering Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-17577-1_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-17577-1_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-17576-4

  • Online ISBN: 978-3-030-17577-1

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation