# Promoting middle school students’ proportional reasoning skills through an ongoing professional development programme for teachers

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## Abstract

Proportional reasoning, the ability to use ratios in situations involving comparison of quantities, is essential for mathematical competence, especially in the middle school years, and is an important determinant of success beyond school. Research shows students find proportional reasoning and its foundational concepts difficult. Proportional reasoning does not always develop naturally, however some research suggests that with targeted teaching, its development can be promoted. This paper reports on a large Australian study involving over 130 teachers and their students. A major goal of the study was to investigate the efficacy of ongoing teacher professional development for promoting middle years students’ proportional reasoning. A series of professional development workshops was designed to enhance the teachers’ understanding of proportional reasoning and to extend their repertoire of teaching strategies to promote their students’ proportional reasoning skills. The workshop design was informed by research literature on proportional reasoning teaching and learning as well as the results of a diagnostic instrument administered to over 2500 middle years students prior to the professional development. Between workshops, the teachers implemented a variety of targeted teaching activities. This paper reports on pre- and post- instrument student data collected at the beginning and end of the first year of the project (i.e., after completion of half of the workshops). The findings suggest that targeted professional development and explicit teaching can make a difference to students’ proportional reasoning.

### Keywords

Proportional reasoning Middle school mathematics Numeracy development Teacher professional development## Notes

### Acknowledgments

This research was funded by Australian Research Council, in partnership with Department of Education and Child Development SA and individual Education Queensland schools.

### References

- Ahl, V. A., Moore, C. F., & Dixon, J. A. (1992). Development of intuitive and numerical proportional reasoning.
*Cognitive Development, 7*, 81–108.CrossRefGoogle Scholar - Akatugba, A. H., & Wallace, J. (2009). An integrative perspective on students’ proportional reasoning in high school physics in a West African context.
*International Journal of Science Education, 31*(11), 1473–1493.CrossRefGoogle Scholar - Allen, J. P., Pianta, R. C., Gregory, A., Mikami, A. Y., & Lun, J. (2011). An interaction-based approach to enhancing secondary school instruction and student achievement.
*Science, 333*(6045), 1034–1037.CrossRefGoogle Scholar - Barab, S., & Squire, K. (2004). Design-based research: Putting a stake in the ground.
*Journal of the Learning Sciences, 13*(1), 1–14.CrossRefGoogle Scholar - Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 296–333). New York: Macmillan.Google Scholar - Ben-Chaim, D., Keret, Y., & Ilany, B.-S. (2007). Designing and implementing authentic investigative proportional reasoning tasks: The impact on pre-service mathematics teachers’ content and pedagogical knowledge and attitudes.
*Journal of Mathematics Teacher Education, 10*, 333–340.CrossRefGoogle Scholar - Boyer, T. W., & Levine, S. C. (2012). Child proportional scaling: Is 1/3 = 2/6 = 3/9 = 4/12?
*Journal of Experimental Child Psychology, 111*, 516–533.CrossRefGoogle Scholar - Boyer, T. W., Levine, S. C., & Huttenlocher, J. (2008). Development of proportional reasoning: Where young children go wrong.
*Developmental Psychology, 44*(5), 1478–1490.CrossRefGoogle Scholar - Bright, G. W., Joyner, J. M., & Wallis, C. (2003). Assessing proportional thinking.
*Mathematics Teaching in the Middle School, 9*(3), 166–172.Google Scholar - Chandrasegaran, A. L., Treagust, D. F., & Mocerino, M. (2008). An evaluation of a teaching intervention to promote students’ ability to use multiple levels of representation when describing and explaining chemical reactions.
*Research in Science Education, 38*, 237–248.CrossRefGoogle Scholar - Clarke, D., & Hollingsworth, H. (2002). Elaborating a model of teacher professional growth.
*Teaching and Teacher Education, 18*, 947–967.CrossRefGoogle Scholar - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.CrossRefGoogle Scholar - Coenders, F. (2010).
*Teachers’ professional growth during the development and class enactment of context-based chemistry student learning material*. Enschede, The Netherlands: University of Twente. Retrieved from http://doc.utwente.nl/71370/1/thesis_F_Coenders.pdf - Cortina, J. L., Visnovska, J., & Zuniga, C. (2014). Unit fractions in the context of proportionality: Supporting students’ reasoning about the inverse order relationship.
*Mathematics Education Research Journal, 26*, 79–99.CrossRefGoogle Scholar - Cramer, K. M., & Post, T. (1993). Connecting research to teaching proportional reasoning.
*Mathematics Teacher, 86*(5), 404–407.Google Scholar - English, L. D., & Halford, G. S. (1995).
*Mathematics education: Models and processes*. Mahwah: Lawrence Erlbaum Associates.Google Scholar - Fielding-Wells, J., Dole, S., & Makar, K. (2014). Inquiry pedagogy to promote emerging proportional reasoning in primary students.
*Mathematics Education Research Journal, 26*, 47–77.CrossRefGoogle Scholar - Goldsmith, L. T., Doerr, H. M., & Lewis, C. C. (2014). Mathematics teachers’ learning: A conceptual framework and synthesis of research.
*Journal of Mathematics Teacher Education, 17*, 5–36.CrossRefGoogle Scholar - Guskey, T. R. (1986). Staff development and the process of teacher change.
*Educational Researcher, 15*(5), 5–12.CrossRefGoogle Scholar - Guskey, T. R., & Yoon, K. S. (2009). What works in professional development.
*Phi Delta Kappan, 90*(7), 495–500.CrossRefGoogle Scholar - Halsam, F., & Treagust, D. F. (1987). Diagnosing secondary students’ misconceptions of photosynthesis and respiration in plants using a two-tier multiple choice instrument.
*Journal of Biological Education, 21*, 203–211.CrossRefGoogle Scholar - Hawley, W. D., & Valli, L. (1999). The essentials of effective professional development: A new consensus. In L. Darling-Hammond & G. Sykes (Eds.),
*Teaching as the learning profession: Handbook of policy and practice*(pp. 127–150). San Francisco: Jossey-Bass Publishers.Google Scholar - Hiebert, J. S., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 371–404). Reston: National Council of Teachers of Mathematics.Google Scholar - Hilton, A., Hilton, G., Dole, S., & Goos, M. (2013). Development and application of a two-tier diagnostic instrument to assess middle years students’ proportional reasoning.
*Mathematics Education Research Journal, 25*, 523–545.CrossRefGoogle Scholar - Howe, C., Nunes, T., & Bryant, P. (2011). Rational number and proportional reasoning: Using intensive quantities to promote achievement in mathematics and science.
*International Journal of Science and Mathematics Education, 9*, 391–417.CrossRefGoogle Scholar - Ingvarson, L., Meiers, M., & Beavis, A. (2005). Factors affecting the impact of professional development programs on teachers’ knowledge, practice, student outcomes & efficacy.
*Education Policy Analysis Archives, 13*(10). Retrieved from http://epaa.asu.edu/epaa/v13n10/ - Jones, G., Taylor, A., & Broadwell, B. (2009). Estimating Linear Size and Scale: Body rulers.
*International Journal of Science Education, 31*(11), 1495–1509.CrossRefGoogle Scholar - Justi, R., & Van Driel, J. (2006). The use of the Interconnected Model of Teacher Professional Growth for understanding the development of science teachers’ knowledge on models and modelling.
*Teaching and Teacher Education, 22*, 437–450.CrossRefGoogle Scholar - Kastberg, S. E., D’Ambrosio, B., & Lynch-Davis, K. (2012). Understanding proportional reasoning for teaching.
*Australian Mathematics Teacher, 68*(3), 32–40.Google Scholar - Lamon, S. J. (1993). Ratio and proportion: Connecting content and children’s thinking.
*Journal for Research in Mathematics Education, 24*(1), 41–61.CrossRefGoogle Scholar - Lamon, S. J. (2005).
*Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers*. Mahwah: Lawrence Erlbaum.Google Scholar - Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 629–668). Charlotte: Information Age Publishing.Google Scholar - Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning.
*Mathematics Teaching in the Middle School, 6*(4), 254–261.Google Scholar - Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 93–118). Reston: Lawrence Erlbaum & National Council of Teachers of Mathematics.Google Scholar - Lo, J., & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader.
*Journal for Research in Mathematics Education, 28*(2), 216–236.CrossRefGoogle Scholar - Lobato, J., Orrill, C. H., Druken, B., & Jacobson, E. (2011).
*Middle school teachers’ knowledge of proportional reasoning for teaching*. Paper presented in the Symposium Extending, expanding, and applying the construct of mathematical knowledge for teaching, at the Annual meeting of the American Educational Research Association, New Orleans, USA. Google Scholar - Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010).
*Designing professional development for teachers of science and mathematics*(3rd ed.). Thousand Oaks: Corwin.Google Scholar - Luke, A., & McArdle, R. (2009). A model for research-based state professional development policy.
*Asia-Pacific Journal of Teacher Education, 37*(3), 231–251.CrossRefGoogle Scholar - McKenney, S., & Reeves, T. C. (2012).
*Conducting educational design research*. Abingdon: Routledge.Google Scholar - Mamlok-Naaman, R., & Eilks, I. (2011). Different types of action research to promote chemistry teachers’ professional development - A joined theoretical reflection on two cases from Israel and Germany.
*International Journal of Science and Mathematics Education, 2011*, 1–30.Google Scholar - Martin, A. M., & Hand, B. (2009). Factors affecting the implementation of argument in the elementary science classroom: A longitudinal case study.
*Research in Science Education, 39*(17–38).Google Scholar - Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning.
*Journal of Mathematical Behavior, 22*, 335–368.CrossRefGoogle Scholar - Nabors, W. (2003). From fractions to proportional reasoning: A cognitive schemes of operation approach.
*Journal of Mathematical Behavior, 22*, 133–179.CrossRefGoogle Scholar - National Council of Teachers of Mathematics. (1989).
*Curriculum and evaluation, standards for school mathematics*. Reston: Author.Google Scholar - Osborne, J., Simon, S., Christodoulou, A., Howell-Richardson, C., & Richardson, K. (2013). Learning to argue: A study of four schools and their attempt to develop the use of argumentation as a common instructional practice and its impact on students.
*Journal of Research in Science Teaching, 50*(3), 315–347.CrossRefGoogle Scholar - Özmen, H. (2008). The influence of computer-assisted instruction on students’ conceptual understanding of chemical bonding and attitude toward chemistry: A case study for Turkey.
*Computers and Education, 51*(1), 423–438.CrossRefGoogle Scholar - Reeves, T. C., McKenney, S., & Herrington, J. (2011). Publishing and perishing: The critical importance of educational design research.
*Australasian Journal of Educational Technology, 27*(1), 55–65.CrossRefGoogle Scholar - Siebert, D. (2002). Connecting informal thinking and algorithms: The case of division of fractions. In B. Litwiller & G. Bright (Eds.),
*Making sense of fractions, ratios, and proportions: 2002 Yearbook*(pp. 247–256). Reston: National Council of Teachers of Mathematics.Google Scholar - Siemon, D., Izard, J., Breed, M., & Virgona, J. (2006). The derivation of a learning assessment framework for multiplicative thinking. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.),
*Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education*(pp. 5-113 – 5-120). Prague: Charles University.Google Scholar - Sowder, J. (2007). The mathematical education and development of teachers. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 157–223). Reston: National Council of Mathematics Teachers.Google Scholar - Sowder, J., Armstrong, B., Lamon, S. J., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers to teach multiplicative structures in the middle grades.
*Journal of Mathematics Teacher Education, 1*, 127–155.CrossRefGoogle Scholar - Staples, M. E., & Truxaw, M. P. (2012). An initial framework for the language of higher-order thinking mathematics practices.
*Mathematics Education Research Journal, 24*(3), 257–281.CrossRefGoogle Scholar - Taber, S. B. (2002). Go ask Alice about multiplication of factions. In B. Litwiller & G. Bright (Eds.),
*Making sense of fractions, ratios, and proportions: 2002 Yearbook*(pp. 61–71). Reston: National Council of Teachers of Mathematics.Google Scholar - Tan, K.-C. D., & Treagust, D. F. (1999). Evaluating students’ understanding of chemical bonding.
*School Science Review, 81*(294), 75–83.Google Scholar - Thompson, A. G., & Thompson, P. W. (1996). Talking about rates conceptually, Part II: Mathematical knowledge for teaching.
*Journal for Research in Mathematics Education, 27*, 2–24.CrossRefGoogle Scholar - Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature.
*Educational Studies in Mathematics, 16*, 181–204.CrossRefGoogle Scholar - Tsui, C.-Y., & Treagust, D. F. (2009). Evaluating secondary students’ scientific reasoning in genetics using a two-tier diagnostic instrument.
*International Journal of Science Education, 32*(8), 1073–1098.CrossRefGoogle Scholar - Tüysüz, C. (2009). Development of a two-tier diagnostic instrument to assess students’ understanding in chemistry.
*Scientific Research and Essay, 4*(6), 626–631.Google Scholar - Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010).
*Elementary and middle school mathematics: Teaching developmentally*. Boston: Allyn & Bacon.Google Scholar - Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralisation.
*Cognition and Instruction, 23*(1), 57–86.CrossRefGoogle Scholar - Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication … back: The development of students’ additive and multiplicative reasoning skills.
*Cognition and Instruction, 28*(3), 360–381.CrossRefGoogle Scholar - Watson, J., & Beswick, K. (2011). School pupil change associated with a continuing professional development programme for teachers.
*Journal of Education for Teaching, 37*(1), 63–75.CrossRefGoogle Scholar - Witterholt, M., Goedhart, M., Suhre, C., & van Streun, A. (2012). The interconnected model of professional growth as a means to assess the development of a mathematics teacher.
*Teaching and Teacher Education, 28*, 661–674.CrossRefGoogle Scholar