Skip to main content
SpringerLink
Log in

Algebra in the Middle School: Developing Functional Relationships Through Quantitative Reasoning

  • Chapter
Early Algebraization

Part of the book series: Advances in Mathematics Education ((AME))

Abstract

Understanding function is a critical aspect of algebraic reasoning, and building functional relationships is an activity encouraged in the younger grades to foster students’ relational thinking. One way to foster functional thinking is to leverage the power of students’ capabilities to reason with quantities and their relationships. This paper explicates the ways in which reasoning directly with quantities can support middle-school students’ understanding of linear and quadratic functions. It explores how building quantitative relationships can support an initial function understanding from a covariation perspective, and later serve as a foundation to build a more flexible view of function that includes the correspondence perspective.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Bednarz, N., Kieran, C., & Lee, L. (1996). Approaches to Algebra: Perspectives for Research and Teaching. Boston, MA: Kluwer Academic Publishers.

    Google Scholar 

  • Buck, J. C. (1995). Fostering connections between classes of polynomial functions. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics education (17th, Columbus, OH, October, 1995).

    Google Scholar 

  • Bussi, M. G. B., & Mariotti, M. A. (1999). Semiotic mediation: From history to the mathematics classroom. For the Learning of Mathematics, 19(2), 27–35.

    Google Scholar 

  • Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. Research in Collegiate Mathematics Education III, Conference Board of the Mathematical Sciences, Issues in Mathematics Education, 7, 114–163.

    Google Scholar 

  • Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Mathematical Association of America Research Sampler.

    Google Scholar 

  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33, 352–378.

    Article  Google Scholar 

  • Carlson, M. P., Smith, N., & Persson, J. (2003). Developing and connecting calculus students’ notions of rate of change and accumulation: The fundamental theorem of calculus. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the Joint Meeting of PME and PMENA (Vol. 2, pp. 165–172). Honolulu, HI: CRDG, College of Education, University of Hawai’i.

    Google Scholar 

  • Carraher, D. W., & Schliemann, A. D. (2002, April). Designing and implementing early algebra activities: From finding unknowns to representing variables. Paper presented at the annual meeting of the National Council of Teachers of Mathematics, Las Vegas, NV.

    Google Scholar 

  • Chazan, D. (2000). Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom. New York, NY: Teachers College Press.

    Google Scholar 

  • Chazan, D. (2006). “What if not?” and teachers’ mathematics. In F. Rosamund & L. Copes (Eds.), Educational Transformations: Changing Our Lives Through Mathematics; A Tribute to Stephen Ira Brown (pp. 3–20). Bloomington, Indiana: AuthorHouse.

    Google Scholar 

  • Confrey, J., & Smith, E. (1992). Function probe: Multi-representational software for Learning about functions. New York State Association for Computers and Technology in Education 6 (pp. 60–64).

    Google Scholar 

  • Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. In P. Cobb (Ed.), Learning Mathematics (pp. 31–60). Dordrech, Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential function. Journal for Research in Mathematics Education, 26(1), 66–86.

    Article  Google Scholar 

  • Cooney, T., & Wilson, M. (1996). Teachers’ thinking about functions: Historical and research perspectives. In T. A. Romberg & E. Fennema (Eds.), Integrating Research on the Graphical Representation of Functions (pp. 131–158). Hillside, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Dreyfus, T., & Halevi, T. (1991). QuadFun: A case study of pupil computer interaction. Journal of Computers in Mathematics and Science Teaching, 10(2), 43–48.

    Google Scholar 

  • Ellis, A. B. (2007). The influence of reasoning with emergent quantities on students’ generalizations. Cognition and Instruction, 25(4), 439–478.

    Article  Google Scholar 

  • Ellis, A. B. (2009, April). A quantitative understanding of quadratic growth: What is DiRoG and why does it matter? Paper presented at the Annual Meeting of the American Educational Research Association, Washington, DC.

    Google Scholar 

  • Ellis, A. B., & Grinstead, P. (2008). Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. Journal of Mathematical Behavior, 27(4), 277–296.

    Article  Google Scholar 

  • English, L., & Warren, E. (1995). General reasoning processes and elementary algebraic understanding: Implications for instruction. Focus on Learning Problems in Mathematics, 17(4), 1–19.

    Google Scholar 

  • Farenga, S. J., & Ness, D. (2005). Algebraic thinking part II: The use of functions in scientific inquiry. Science Scope, 29(1), 62–64.

    Google Scholar 

  • Kaput, J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Research on Mathematics Teaching and Learning (pp. 515–556). New York, NY: Macmillan Publishing.

    Google Scholar 

  • Kaput, J. (1995). Long term algebra reform: Democratizing access to big ideas. In C. Lacampagne, W. Blair, & J. Kaput (Eds.), The Algebra Initiative Colloquium (pp. 33–52). Washington, DC: U.S. Department of Education.

    Google Scholar 

  • Kieran, C. (1996). The changing face of school algebra. In 8th International Congress on Mathematical Education, Selected Lectures (pp. 271–286). S.A.E.M. THALES.

    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.

    Google Scholar 

  • Lobato, J., Ellis, A. B., & Muñoz, R. (2003). How “focusing phenomena” in the instructional environment afford students’ generalizations. Mathematical Thinking and Learning, 5(3), 1–36.

    Article  Google Scholar 

  • MacGregor, M., & Stacey, K. (1993). Seeing a pattern and writing a rule. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F. L. Lin (Eds.), Proceedings of the 17th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 181–188). Tsukuba, Japan: PME Program Committee.

    Google Scholar 

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

    Google Scholar 

  • Mokros, J., Economopoulos, K., & Russel, S. J. (1995). Beyond Arithmetic: Changing Mathematics in the Elementary Classroom. Palo Alto, CA: Dale Seymour.

    Google Scholar 

  • Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel & E. Dubinsky (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 175–193).

    Google Scholar 

  • Monk, G. S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In E. Dubinsky, A. H. Shoenfield, & J. J. Kaput (Eds.), Research in Collegiate Mathematics Education I (Vol. 4, pp. 139–168). Providence, RI: American Mathematical Society.

    Google Scholar 

  • National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

    Google Scholar 

  • Nemirovsky, R., Tierney, C., & Ogonowski, M. (1993). Children, additive change, and calculus (Working Paper 2-93). Cambridge: TERC.

    Google Scholar 

  • Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Mathematics in the Streets and in Schools. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Orton, A., & Orton, J. (1994). Students’ perception and use of pattern and generalization. In J. P. da Ponto & J. F. Matos (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education (Vol. III, pp. 407–414). Lisbon, Portugal: PME Program Committee.

    Google Scholar 

  • Rasmussen, C. L. (2000). New directions in differential equations: A framework for interpreting students’ understandings and difficulties. Journal of Mathematical Behavior, 20, 55–87.

    Article  Google Scholar 

  • Romberg, T. A., Carpenter, T. P., & Fennema, E. (1993). Toward a common research perspective. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating Research on the Graphical Representation of Functions (pp. 1–9). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Roschelle, J., Kaput, J., & Stroup, W. (2000). SimCalc: Accelerating students’ engagement with the mathematics of change. In M. J. Jacobson & R. B. Kozma (Eds.), Innovations in Science and Mathematics Education: Advanced Designs for Technologies of Learning (pp. 47–75). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Schliemann, A. D., Araujo, C., CassundĂ©, M. A., Macedo, S., & NicĂ©as, L. (1998). Multiplicative commutativity in school children and street sellers. Journal for Research in Mathematics Education, 29(4), 422–435.

    Article  Google Scholar 

  • Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2001). When tables become function tables. In Proceedings of the XXV Conference of the International Group for the Psychology of Mathematics Education (pp. 145–152). Utrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Schliemann, A. D., Carraher, D. W., Brizuela, B. M., Earnest, D., Goodrow, A., Lara-Roth, S., & Peled, I. (2003). Algebra in the elementary school. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PME-NA (Vol. 4, pp. 127–134). Honolulu, HI: CRDG, College of Education, University of Hawai’i.

    Google Scholar 

  • Schliemann, A. D., Carraher, D. W., & Brizuela, B. (2007). Bringing Out the Algebraic Character of Arithmetic: From Children’s Ideas to Classroom Practice. Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30(4), 362–390.

    Article  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 

  • Slavit, D. (1997). An alternative route to the reification of functions. Educational Studies in Mathematics, 33, 259–281.

    Article  Google Scholar 

  • Smith, E. (2003). Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 curriculum. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 136–150).

    Google Scholar 

  • Smith, E., & Confrey, J. (1994). Multiplicative structures and the development of logarithms: What was lost by the invention of function. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 333–364). Albany, NY: State University of New York Press.

    Google Scholar 

  • Smith, J., & Thompson, P. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 133–160). New York: Erlbaum.

    Google Scholar 

  • Stacey, K., & MacGregor, M. (1997). Building foundations for algebra. Mathematics Teaching in the Middle School, 2(4), 253–260.

    Google Scholar 

  • Steffe, L., & Izsak, A. (2002). Pre-service middle-school teachers’ construction of linear equation concepts through quantitative reasoning. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Noony (Eds.), Proceedings of the Twenty-Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1163–1172). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

    Google Scholar 

  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181–234). Albany, NY: SUNY Press.

    Google Scholar 

  • Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.

    Article  Google Scholar 

  • Yerushalmy, M. (2000). Problem solving strategies and mathematical resources: A longitudinal view on problem solving in a function based approach to algebra. Educational Studies in Mathematics, 43(2), 125–147.

    Article  Google Scholar 

  • Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), CBMS Issues in Mathematics: Research in Collegiate Mathematics Education IV (Vol. 8, pp. 103–127). Providence: American Mathematical Society.

    Google Scholar 

  • Zaslavsky, O. (1997). Conceptual obstacles in the Learning of quadratic functions. Focus on Learning Problems in Mathematics, 19(1), 20–44.

    Google Scholar 

  • Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of function translation: Obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22(4), 435–448.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Education, University of Wisconsin-Madison, Madison, USA

    Amy B. Ellis

Authors
  1. Amy B. Ellis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amy B. Ellis .

Editor information

Editors and Affiliations

  1. Dept. Mathematical Sciences, University of Delaware, Ewing Hall 501, Newark, 19716, Delaware, USA

    Jinfa Cai

  2. University of Wisconsin-Madison, Teacher Education Building, 225 N. Mills Street, Madison, 53706, Wisconsin, USA

    Eric Knuth

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ellis, A.B. (2011). Algebra in the Middle School: Developing Functional Relationships Through Quantitative Reasoning. In: Cai, J., Knuth, E. (eds) Early Algebraization. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_13

Download citation

Publish with us

Policies and ethics

Search

Navigation