# Using Variables in School Mathematics: Do School Mathematics Curricula Provide Support for Teachers?

- 310 Downloads
- 1 Citations

## Abstract

This study employed content analysis to examine 3 popular middle-grades mathematics curricula in the USA on the support they provide for teachers to implement concepts associated with variables in school mathematics. The results indicate that each of the 3 curricula provides some type of support for teachers, but in a varied amount and quality. More specifically, whereas the *University of Chicago School Mathematics Project* (*UCSMP*) curriculum provides support for teachers on several aspects of using variables in school mathematics, the support found in the *Connected Mathematics 2* and the *Math Connects* curricula focused mainly on one conception of variables—namely, the use of variables as *quantity that varies* in the Connected Mathematics 2 curriculum and the use of variables as *specific unknowns* in the Math Connects curriculum. Overall, the UCSMP curriculum provides the most support for teachers, followed by the Connected Mathematics 2 curriculum, with the Math Connects curriculum recording the least support for teachers to enact variable concepts. It is worth pointing out that, although the 3 curricula collectively provide guidance for teachers to implement variable ideas within meaningful real-world contexts, the supports identified in the respective curriculum were not sufficient in addressing all of the areas that are essential for teaching the many concepts associated with variables in school mathematics effectively. Recommendations for curriculum developers and for international researchers with interest in the roles of variables in school mathematics are provided.

## Keywords

Content analysis Curriculum School mathematics Teachers Variables## References

- Asquith, S., Stephens, A., Knuth, J. & Alibali, W. (2007). Middle-school math teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable.
*Mathematical Thinking and Learning, 9*, 247–270.CrossRefGoogle Scholar - Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division.
*Journal for Research in Mathematics Education, 21*(2), 132–144.CrossRefGoogle Scholar - Ball, D. & Cohen, D. (1996). Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform?
*Educational Researcher, 25*, 6–8.Google Scholar - Ball, D. L., Hill, H. C. & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?
*American Educator*,*29*(1), 14–17, 20–22, 43–46.Google Scholar - Blömeke, S. & Klein, P. (2013). When is a school environment perceived as supportive by beginning math teachers? Effects of leadership, trust, autonomy and appraisal on teaching quality.
*International Journal of Science and Mathematics Education, 11*(4), 1029–1048.CrossRefGoogle Scholar - Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford & A. P. Shulte (Eds.),
*The ideas of algebra, K-12*(pp. 20–32). Reston, VA: NCTM.Google Scholar - Boyer, C. B. (1991).
*A history of mathematics*. New York, NY: Wiley.Google Scholar - Boz, N. (2002). Prospective teachers’ subject matter and pedagogical content knowledge of variables.
*Proceedings of British Society for Research into Learning Mathematics*,*22*(3). Retrieved from http://www.bsrlm.org.uk/informalproceedings.html. - Boz, N. (2007). Interactions between knowledge of variables and knowledge about teaching variables.
*Sosyal Bilimler Arastırmaları Dergisi, 1*, 1–18.Google Scholar - Carter, T. A. & Capraro, R. M. (2005). Stochastic misconceptions of pre-service teachers.
*Academic Exchange Quarterly, 9*, 105–111.Google Scholar - Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception.
*Journal for Research in Mathematics Education, 13*, 16–30.Google Scholar - Clement, J., Lochhead, J. & Monk, G. (1981). Translation difficulties in learning mathematics.
*The American Mathematical Monthly, 8*, 286–290.CrossRefGoogle Scholar - Cohen, J. A. (1960). A coefficient of agreement for nominal scales.
*Educational and Psychological Measurement, 20*, 37–46.CrossRefGoogle Scholar - Collis, K. F. (1975).
*A study of concrete and formal operations in school mathematics: A Piagetian viewpoint*. River Road Mystic, CT: Lawrence Verry.Google Scholar - Common Core State Standards Initiative (2010).
*Common core state standards for mathematics*. Washington, DC: National Governors Association Center for Best Practices.Google Scholar - Cramer, K., Post, T. & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.),
*Research ideas for the classroom*(pp. 159–178). NewYork, NY: Macmillan.Google Scholar - Dogbey, J. & Kersaint, G. (2012). Treatment of variables in popular middle-grades mathematics textbooks in the USA: Trends from 1957 through 2009.
*International Journal for Mathematics Teaching and Learning, 2*(1), 1–30.Google Scholar - Eisenberg, T. (1991). Functions and associated learning difficulties. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 140–152). Dordrecht, The Netherlands: Kluwer.Google Scholar - Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept.
*Journal for Research in Mathematics Education, 24*(2), 94–116.CrossRefGoogle Scholar - Gray, S. S., Loud, B. J. & Sokolowski, C. P. (2005). Undergraduates’ errors in using and interpreting algebraic variables: A comparative study. In G. M. Lloyd, M. R. Wilson, J. L. Wilkins & S. L. Behm (Eds.),
*Proceedings of the 27th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education [CD-ROM]*. Eugene, OR: All Academic.Google Scholar - Grouws, D. A., Smith, M. S. & Sztajn, P. (2004). The preparation and teaching of U.S. mathematics teachers: Grades 4 and 8. In P. Kloosterman & F. Lester (Eds.),
*The 1990 through 2000 mathematics assessment of the national assessment of educational progress: Results and interpretations*(pp. 221–269). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Haller, S. K. (1997).
*Adopting probability curricular: The content and pedagogical content knowledge of middle grades teachers*. (Unpublished doctoral dissertation). MA: University of Minnesota.Google Scholar - Heinz, K. (2000).
*Conceptions of ration in a class of pre-service and practicing teachers*. (Unpublished Doctoral Dissertation). Pennsylvania, PA: Pennsylvania State University.Google Scholar - Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S. Wagner & C. Kieran (Eds.),
*Research issues in the learning and teaching of algebra*(pp. 60–86). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Huntley, M. A. (2008). A framework for analyzing differences across mathematics curricula.
*Journal of Mathematics Education Leadership, 10*(2), 10–17.Google Scholar - Jones, D. L. & Tarr, J. E. (2007). An examination of the levels of cognitive demand required by probability tasks in middle grades mathematics textbooks.
*Statistics Education Research Journal, 6*(2), 4–27.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 - Krippendorff, K. (2004).
*Content analysis: An introduction to its methodology*(2nd ed.). Thousand Oaks, CA: Sage.Google Scholar - Küchemann, D. (1978). Children’s understanding of numerical variables.
*Mathematics in School, 7*(4), 23–26.Google Scholar - Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.),
*Children’s understanding of mathematics: 11–16*(pp. 102–119). London, UK: John Murray.Google Scholar - Leitzel, J. R. (1989). Critical considerations for the future of algebra instruction. In Wagner & C. Kieran (Eds.),
*Research issues in the learning and teaching of algebra*(pp. 25–32). Hillsdale, NJ: Erlbaum.Google Scholar - Lockhead, J. (1980). Faculty interpretations of simple algebraic statements: The professor’s side of the equation.
*Journal of Mathematical Behaviour, 3*, 29–37.Google Scholar - Lockhead, J. & Mestre, J. (1988). From words to algebra: Mending misconceptions. In A. Coxford & A. Shulte (Eds.),
*The ideas of algebra*,*K-12 (NCTM Yearbook)*(pp. 127–136). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the USA*. Mahwah, NJ: Erlbaum.Google Scholar - MacGregor, M. & Stacey, K. (1993). Cognitive models underlying students’ formulation of simple linear equations.
*Journal for Research in Mathematics Education, 24*, 217–232.Google Scholar - MacGregor, M. & Stacey, K. (1997). Students’ understanding of algebraic notation.
*Educational Studies in Mathematics, 33*, 1–19.CrossRefGoogle Scholar - McKnight, C., Magad, A., Murphy, T. J. & McKnight, M. (2000).
*Mathematics education research: A guide for the research mathematician*. Providence, RI: American Mathematical Society.Google Scholar - Mestre, J. & Gerace, W. (1986). A study of the algebra acquisition of Hispanic and Anglo ninth graders: Research findings relevant to teacher training and classroom practice.
*The Journal for the National Association for Bilingual Education, 10*(2), 137–167.Google Scholar - Meyer, M. R. & Langrall, C. W. (Eds.). (2008).
*A decade of middle school mathematics curriculum implementation: Lessons learned from the Show-Me Project*. Greenwich, CT: Information Age.Google Scholar - Miles, M. B. & Huberman, A. M. (1994).
*Qualitative data analysis: An expanded sourcebook*. Thousand Oaks, CA: Sage.Google Scholar - Misailidou, C. & Williams, J. (2002).
*Ratio: Raising teachers’ awareness of children’s thinking*. Greece: Paper presented at Second International Conference on the Teaching of Mathematics.Google Scholar - Mohr, D. J. (2008). Pre-service elementary teachers make connections between geometry and algebra through the use of technology.
*Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal, 3*. Retrieved from http://www.k-12prep.math.ttu.edu/journal/journal.shtml. - National Council of Teachers of Mathematics. (1989).
*Curriculum and evaluation standards for school mathematics*. Reston, VA: Author.Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston, VA: Author.Google Scholar - National Council of Teachers of Mathematics (2006).
*Curriculum focal points*. Retrieved from http://www.nctmmedia.org/cfp/full_document.pdf. Accessed 27 Jan 2007. - National Research Council (2004).
*On evaluating curricular effectiveness: Judging the quality of K-12 mathematics evaluations*. Washington, DC: National Academies Press.Google Scholar - Philipp, R. (1992). The many uses of algebraic variables.
*The Mathematics Teacher, 85*(7), 557–561.Google Scholar - Pittman, M., Koellner, K. & Brendefur, J. (2007, October)
*Analyzing teacher content knowledge of probability: The maze problem*. Paper presented at the annual meeting of the PME, the North American Chapter, the University of Nevada, Reno, NA.Google Scholar - Riley, K. R. (2010). Teachers’ understanding of proportional reasoning. In P. Brosnan, D. B. Erchick & L. Flevares (Eds.),
*Proceedings of the 32nd annual meeting of the PME, the North American Chapter*(pp. 1055–1061). Columbus, OH: The Ohio State University.Google Scholar - Robitaille, D. F. & Travers, K. J. (1992). International studies of achievement in mathematics. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 687–723). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Schoenfeld & Arcavi (1988). On the meaning of variable.
*Mathematics Teacher, 81*, 420–427.Google Scholar - Senk, S. L. & Thompson, D. R. (2003).
*Standards-based school mathematics curricula: What are they? What do students learn?*Mahwah, NJ: Erlbaum.Google Scholar - Simon, M. & Blume, G. (1994). Building and understanding multiplicative relationships.
*Journal for Research in Mathematics Education, 25*(5), 472–494.CrossRefGoogle Scholar - Stone, P. J., Dunphy, D. C., Smith, M. S. & Ogilvie, D. M. (1966).
*The general inquirer: A computer approach to content analysis*. Cambridge, MA: MIT Press.Google Scholar - Ursini, S. & Trigueros, M. (1997). Understanding of different uses of variable: A study with starting college students.
*Proceedings of the XXI PME International Conference, 4*, 254–261.Google Scholar - Usiskin, Z. (1988). Conceptions of school algebra and uses of variable. In A. F. Coxford & A. P. Shulte (Eds.),
*The ideas of algebra, K-12*(pp. 8–19). Reston, VA: NCTM.Google Scholar - Usiskin, Z. (1999). Conceptions of school algebra and uses of variables. In B. Moses (Ed.),
*Algebraic thinking, Grades K-12: Readings from the NCTM’s school-based journals and other publications*(pp. 316–320). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Viktora, S. S., Cheung, E., Highstone, V. & Capuzzi, C. (2008).
*UCSMP: Transition mathematics*(3rd ed., pp. 68–69). New York, NY: Wright Group/McGraw Hill.Google Scholar - Wagner, S. (1981). An analytical framework for mathematical variables.
*Proceedings of the Fifth PME Conference*(pp. 165–170). France, Grenoble.Google Scholar - Wagner, S. (1999). What are these things called variables. In B. Moses (Ed.),
*Algebraic thinking, Grades K-12: Readings from the NCTM’s school-based journals and other publications*(pp. 316–320). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - White, P. & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus.
*Journal for Research in Mathematics Education, 27*, 79–95.Google Scholar