Skip to main content
SpringerLink
Log in

The Cognitive Demand of a Gifted Student’s Answers to Geometric Pattern Problems

Analysis of Key Moments in a Pre-algebra Teaching Sequence

  • Chapter
  • First Online:
Mathematical Creativity and Mathematical Giftedness

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Mathematically gifted students require specific teaching methodologies to foster their interest in mathematics and their engagement in solving problems. Geometric pattern problems are an interesting context in which to introduce algebra to those students. We present the case of a nine-year-old student engaged in a teaching unit based on geometric pattern problems that was aimed at helping him start learning algebra, equations, and algebra word problems. To analyze and assess the cognitive effort the student made to solve the problems, we used a particularization to this context of the cognitive demand model. We analyzed answers typical of the different kinds of problems posed throughout the teaching unit, showing the student’s learning trajectory and related characteristics of mathematical giftedness.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

References

  • Amit, M., & Neria, D. (2008). “Rising to the challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 111–129.

    Article  Google Scholar 

  • Benedicto, C., Gutiérrez, A., & Jaime, A. (2017). When the theoretical model does not fit our data: A process of adaptation of the Cognitive Demand model. To appear in the Proceedings of the 10th Congress of European Research in Mathematics Education (CERME10). Dublin, Ireland: ERME. Available in http://www.uv.es/angel.gutierrez/archivos1/textospdf/BenyOtros17.pdf.

  • Benedicto, C., Jaime, A., & Gutiérrez, A. (2015). Análisis de la demanda cognitiva de problemas de patrones geométricos. In C. Fernández, M. Molina, & N. Planas (Eds.), Investigación en Educación Matemática XIX (pp. 153–162). Alicante, Spain: SEIEM. Available in http://www.seiem.es/docs/actas/19/ActasXIXSEIEM.pdf.

  • Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers’ classrooms. Journal for Research in Mathematics Education, 40(2), 119–156.

    Google Scholar 

  • Budak, I. (2012). Mathematical profiles and problem solving abilities of mathematically promising students. Educational Research and Reviews, 7(16), 344–350.

    Google Scholar 

  • Cai, J., & Knuth, E. J. (Eds.). (2011). Early algebraization. Heidelberg, Germany: Springer.

    Google Scholar 

  • Diezmann, C. M., & Watters, J. J. (2002). Summing up the education of mathematically gifted students. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.), Proceedings of the 25th Annual Conference of the MERGA (pp. 219–226). Sidney, Australia: MERGA.

    Google Scholar 

  • Filloy, E., Puig, L., & Rojano, T. (2008). Educational algebra. New York: Springer.

    Book  Google Scholar 

  • Freiman, V. (2006). Problems to discover and to boost mathematical talent in early grades: A challenging situations approach. The Montana Mathematics Enthusiast, 3(1), 51–75.

    Google Scholar 

  • Fritzlar, T., & Karpinski-Siebold, N. (2012). Continuing patterns as a component of algebraic thinking—An interview study with primary school students. In Pre-proceedings of the 12th International Congress on Mathematical Education (pp. 2022–2031). Seoul, South Korea: ICMI.

    Google Scholar 

  • García-Reche, A., Callejo, M. L., & Fernández, C. (2015). La aprehensión cognitiva en problemas de generalización de patrones lineales. In C. Fernández, M. Molina, & N. Planas (Eds.), Investigación en Educación Matemática XIX (pp. 279–288). Alicante, Spain: SEIEM. Available in http://www.seiem.es/docs/actas/19/ActasXIXSEIEM.pdf.

  • Gavin, M. K., Casa, T. M., Adelson, J. L., Carroll, S. R., & Sheffield, L. J. (2009). The impact of advanced curriculum on the achievement of mathematically promising elementary students. Gifted Child Quarterly, 53(5), 188–202.

    Article  Google Scholar 

  • Greenes, C. (1981). Identifying the gifted student in mathematics. Arithmetic Teacher, 28, 14–18.

    Google Scholar 

  • Kennard, R. (2001). Teaching mathematically able children. London, UK: David Fulton.

    Google Scholar 

  • Krutetskii, V. A. (1976). The psychology of mathematical abilities in school-children. Chicago: The University of Chicago Press.

    Google Scholar 

  • Miller, R. C. (1990). Discovering mathematical talent. Washington, DC: ERIC. Available in http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED321487.

  • National Library of Virtual Manipulatives (NLVM). (2016). Algebra balance scales. Logan: Utah State University. Applet available in http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2.html.

  • Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers’ mathematical patterning. Journal for Research in Mathematics Education, 42(3), 237–268.

    Article  Google Scholar 

  • Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237–268.

    Article  Google Scholar 

  • Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th PME-NA Conference (Vol. 1, pp. 2–21). Merida, Mexico: PME-NA.

    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. (2011). Embodiment, perception and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.), Proceedings of the 35th PME Conference (Vol. 4, pp. 17–24). Ankara, Turkey: PME.

    Google Scholar 

  • Rivera, F. D. (2010). Second grade students’ preinstructional competence in patterning activity. In M. F. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th PME Conference (Vol. 4, pp. 81–88). Belo Horizonte, Brazil: PME.

    Google Scholar 

  • Rivera, F. D. (2013). Teaching and learning patterns in school mathematics. New York: Springer.

    Book  Google Scholar 

  • Rivera, F. D., & Becker, J. R. (2005). Figural and numerical modes of generalizing in algebra. Mathematics Teaching in the Middle School, 11(4), 198–203.

    Google Scholar 

  • Silver, E. A., & Mesa, V. (2011). Coordinating characterizations of high quality mathematics teaching: Probing the intersection. In Y. Li & G. Kaiser (Eds.), Expertise in mathematics instruction (pp. 63–84). New York: Springer.

    Chapter  Google Scholar 

  • Smith, M. S., Hillen, A. F., & Catania, C. L. (2007). Using pattern tasks to develop mathematical understanding and set classroom norms. Mathematics Teaching in the Middle School, 13(1), 38–44.

    Google Scholar 

  • Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350.

    Google Scholar 

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

    Article  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.

    Article  Google Scholar 

  • Warren, E. (2005). Young children’s ability to generalize the pattern rule for growing patterns. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th PME Conference (Vol. 4, pp. 305–312). Melbourne, Australia: PME.

    Google Scholar 

  • Warren, E., Trigueros, M., & Ursini, S. (2016). Research on the learning and teaching of algebra. In A. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 73–108). Rotterdam, The Netherlands: Sense.

    Chapter  Google Scholar 

  • Wijaya, A., Van den Heuvel-Panhuizen, M., & Doorman, M. (2015). Opportunity-to-learn context-based tasks provided by mathematics textbooks. Educational Studies in Mathematics, 89(1), 41–65.

    Article  Google Scholar 

Download references

Acknowledgements

The results presented in this chapter are partially funded by the research projects EDU2012-37259 (MINECO) and EDU2015-69731-R (MINECO/ERDF), funded by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund, and GVPROMETEO2016-143, funded by the Valencian Government.

Author information

Authors and Affiliations

  1. Departamento de Didáctica de la Matemática, Universidad de Valencia, Av. de los Naranjos 4, 46022, Valencia, Spain

    Angel Gutierrez, Clara Benedicto, Adela Jaime & Eva Arbona

Authors
  1. Angel Gutierrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clara Benedicto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Adela Jaime
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eva Arbona
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Angel Gutierrez .

Editor information

Editors and Affiliations

  1. Faculty of Letters and Science, Oil & Gas University of Ploiești, Ploiești, Romania

    Florence Mihaela Singer

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gutierrez, A., Benedicto, C., Jaime, A., Arbona, E. (2018). The Cognitive Demand of a Gifted Student’s Answers to Geometric Pattern Problems. In: Singer, F. (eds) Mathematical Creativity and Mathematical Giftedness. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-73156-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-73156-8_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73155-1

  • Online ISBN: 978-3-319-73156-8

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation