Skip to main content

Examining the Role of Prior Experience in the Learning of Algebra

  • Chapter
  • First Online:
And the Rest is Just Algebra

Abstract

Widespread emphasis on developing students’ algorithmic competency and symbol manipulation has resulted in students failing to think analytically and critically. If students are not encouraged to think flexibly about arithmetic and algebra in school, then this needs to be addressed by developmental courses and tasks designed to change the procedural orientation and superficial, fragmented knowledge of too many of our undergraduate students. Those who teach mathematics at the postsecondary level often dismiss the increasing number of students enrolled in precollege mathematics courses as “not my problem,” not realizing that “just algebra” is the downfall for many college students. Learning “just algebra” is a much more complex task than it appears. In this chapter, prior knowledge will be shown to have become problematic for many students, and we provide evidence of the need to improve the effectiveness of our own teaching and that of our future teachers in ways that help students develop deeper understanding of mathematics and promote mathematical thinking.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
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

Similar content being viewed by others

Notes

  1. 1.

    Portions of the collected data have been previously reported (McGowen, 1998; McGowen & Tall, 2013), but the accumulated data of 516 students have not been reported previously.

References

  • American Mathematical Association of Two-Year Colleges. (2004). Beyond crossroads: Implementing mathematics standards in the first two years of college. Memphis, TN: American Mathematical Association of Two Year Colleges.

    Google Scholar 

  • Attewell, P., Lavin, D., Domina, T., & Levey, T. (2006). New evidence on college remediation. Journal of Higher Education, 77(5), 886–924.

    Article  Google Scholar 

  • Ausubel, D. P., Novak, J. D., & Hanesian, H. (1968). Educational psychology: A cognitive view.

    Google Scholar 

  • Bailey, T. (2009). Challenge and opportunity: Rethinking the role and function of developmental education in community college. New Directions for Community Colleges, 145, 11–30.

    Article  Google Scholar 

  • Barker, W., Bressoud, D., Epp, S., Ganter, S., Haver, B., & Pollatsek, H. (2004). Undergraduate programs and courses in the mathematical sciences: CUPM curriculum guide, 2004. Washington, DC: Mathematical Association of America.

    Google Scholar 

  • Bautsch, B. (2013). Hot topics in higher education: Reforming remedial education. In National Conference of State Legislatures. http://www.ncsl.org/documents/educ/REMEDIALEDUCATION_2013.pdf.

  • Beaton, A. E. (1996). Mathematics achievement in the middle school years. IEA’s third international mathematics and science study (TIMSS). Chestnut Hill, MA: Boston College.

    Google Scholar 

  • Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education, 5(1), 7–74.

    Article  Google Scholar 

  • Blair, R. M., Kirkman, E. E., Maxwell, J. W., & American Mathematical Society. (2013). Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2010 CBMS survey (pp. 131–136, Table TYE4).

    Google Scholar 

  • Breneman, D. W., & Haarlow, W. N. (1998). Remediation in higher education. A symposium featuring “Remedial education: Costs and consequences”. Fordham Report, 2(9), 9.

    Google Scholar 

  • Bright, G. W., & Joyner, J. M. (2003). Dynamic classroom assessment: Linking mathematical understanding to instruction. Vernon Hills, IL: ETA/Cuisenaire.

    Google Scholar 

  • Brothen, T., & Wambach, C. A. (2004). Refocusing developmental education. Journal of Developmental Education, 2, 16–18, 20, 22, 33.

    Google Scholar 

  • Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), CBMS issues in mathematics education: Research in collegiate mathematics education III (Vol. 7, pp. 114–162).

    Google Scholar 

  • Cohen, D. (1995). Crossroads in mathematics: Standards for introductory college mathematics before calculus. Memphis, TN: AMATYC.

    Google Scholar 

  • Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

    Google Scholar 

  • Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Part I and Part II. Washington, DC: The Mathematical Association of America in cooperation with American Mathematical Society.

    Book  Google Scholar 

  • Crane, S. (1972). The wayfarer. In The complete poems of Stephen crane. Ithaca, NY: Cornell University Press.

    Google Scholar 

  • Crane, S. (1972). The complete poems of Stephen Crane (Vol. 130). Cornell University Press.

    Google Scholar 

  • Davis, G. E., & McGowen, M. A. (2002). Function machines & flexible algebraic thought. In Proceedings of the 26th international group for the psychology of mathematics education (Vol. 2, pp. 273–280). Norwick, UK: University of East Anglia.

    Google Scholar 

  • Davis, G. E., & McGowen, M. A. (2007). Formative feedback and the mindful teaching of mathematics. Australian Senior Mathematics Journal, 21(1), 19.

    Google Scholar 

  • De Lima, R. N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18.

    Article  Google Scholar 

  • DeMarois, P. (1998). Facets and layers of function for college students in beginning algebra (Doctoral dissertation, University of Warwick).

    Google Scholar 

  • DeMarois, P., & McGowen, M. (1996b). Understanding of function notation by college students in a reform developmental algebra curriculum. In Proceedings of the 18th annual meeting of the North American chapter of the international group for the psychology of mathematics education, Panama City, Florida (Vol. 1, pp. 183–186).

    Google Scholar 

  • DeMarois, P., McGowen, M., & Whitkanack, D. (1996). Applying algebraic thinking to data (Preliminaryth ed.). Glenview, IL: Harper Collins Publishers.

    Google Scholar 

  • Ganter, S., & Barker, W. (2004). The curriculum foundations project: Voices of the partner disciplines. AMC, 10, 12.

    Google Scholar 

  • Krutetskii, V. A. (1969). Mathematical aptitudes. In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics (Vol. II, pp. 113–128). Chicago, IL: University of Chicago Press.

    Google Scholar 

  • McGowen, M. A. (1998). Cognitive units, concept images, and cognitive collages: An examination of the processes of knowledge construction. (Doctoral dissertation, University of Warwick). ERIC: ED466377. http://www.dissertation.com/book.php?method=ISBN&book=1612337732.

  • McGowen, M. A. (2006). Developmental algebra. In N. Baxter-Hastings (Ed.), MAA Notes 69: A fresh start for collegiate mathematics (pp. 369–375).

    Google Scholar 

  • McGowen, M., DeMarois, P., & Tall, D. (2000). Using the function machine as a cognitive root for building a rich concept image of the function concept. In Proceedings of the 22nd annual meeting of the North American chapter of the international group for the psychology of mathematics, Tucson, AZ (pp. 247–254).

    Google Scholar 

  • McGowen, M. A., & Tall, D. O. (2010). Metaphor or met-before? The effects of previous experience on practice and theory of learning mathematics. The Journal of Mathematical Behavior, 29(3), 169–179.

    Article  Google Scholar 

  • McGowen, M. A., & Tall, D. O. (2013). Flexible thinking and met-befores: Impact on learning mathematics. The Journal of Mathematical Behavior, 32(3), 527–537.

    Article  Google Scholar 

  • National Center on Education and the Economy (2013). What does it mean to be college and work ready? The Mathematics Required of First-Year Cmmunity College Students. Washington, DC. Available online at: http://www.ncee.org/college-and-work-ready.

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

    Google Scholar 

  • National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

    Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

    Google Scholar 

  • National Research Council. (2000). Mathematics education in the middle grades: Teaching to meet the needs of middle grades learners and to maintain high expectations. In Proceedings of national convocation and action conferences/center for science, mathematics, and engineering education. Washington, DC: National Academy Press.

    Google Scholar 

  • National Research Council. (2001). Educating teachers of science, mathematics, and technology: New practices for the new millennium. Washington, DC: National Academy Press.

    Google Scholar 

  • Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics (pp. 27–42). Washington, DC: Mathematical Association of America.

    Chapter  Google Scholar 

  • Recorde, R. (1543). The grounde of artes. London: Reynold Wolff. http://www.alibris.com/The-Grounde-of-Artes-Robert-Recorde/book/14453565.

  • 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), 1–36.

    Article  Google Scholar 

  • Skemp, R. (1987). The psychology of learning mathematics expanded (Americanth ed.). Hillsdale, NJ: Lawrence Erlbaum & Associates, Publishers.

    Google Scholar 

  • Smith, J. P., III. (1996). Efficacy and teaching mathematics by telling: A challenge for reform. Journal for Research in Mathematics Education, 27, 387–402.

    Article  Google Scholar 

  • Stigler, J. W., Givvin, K. B., & Thompson, B. J. (2010). What community college developmental mathematics students understand about mathematics. MathAMATYC Educator, 1(3), 4–16.

    Google Scholar 

  • Stump, S. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 2(11), 124–144.

    Article  Google Scholar 

  • Tall, D. O. (2004). The three worlds of mathematics. For the Learning of Mathematics, 23(3), 29–33.

    Google Scholar 

  • Tall, D., McGowen, M., & DeMarois, P. (2000). The function machine as a cognitive root for the function concept. In Proceedings of 22nd annual meeting of the North American chapter of the international group for the psychology of mathematics education. Tucson, AZ (pp. 255–261).

    Google Scholar 

  • Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education. I. CBMS issues in mathematics education (Vol. 4, pp. 21–44).

    Google Scholar 

  • Thompson, P. (1996). If you say it, will they hear? In A slide presentation given at the American Mathematical Association of two-year colleges annual conference, Long Beach, CA.

    Google Scholar 

  • Tucker, A. (1995). Models that work: Case studies in effective undergraduate mathematics programs. MAA Notes No. 38.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mercedes McGowen .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

McGowen, M. (2017). Examining the Role of Prior Experience in the Learning of Algebra. In: Stewart, S. (eds) And the Rest is Just Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-45053-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45053-7_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45052-0

  • Online ISBN: 978-3-319-45053-7

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics