Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property

Alexander Meyer

doi:10.12973/eurasia.2016.1307a

Full Text (PDF)

Alexander Meyer ¹ ^*

More Detail

¹ TU Dortmund University^* Corresponding Author

Abstract

Background:
Flexibility in transforming algebraic expressions is recognized as fundamental for a rich procedural knowledge. Here, flexibility in-depth is proposed as the ability to apply one strategy to a wide range of unfamiliar expressions.

Material and methods:
In this study, design experiments with four groups of students were conducted to support students’ flexibility in-depth in regard to the application of the distributive law with the help of worked examples. The data from these experiments was analyzed qualitatively with the method of content analysis.

Results:
Students’ flexibility in-depth translates into their abilities to reconstruct the distributive property within an expression via its perceived relevant structural features. The students’ use individual cornerstones for this reconstruction, for example worked examples or certain focal points in an expression like the multiplication sign.

Conclusions:
Transforming expressions and especially applying formulas to algebraic expressions is thus an interpretative and reconstructive process.

Keywords

algebra
procedural knowledge
distributive property
flexibility
algebraic symbols

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2016, Volume 12, Issue 10, 2719-2732

https://doi.org/10.12973/eurasia.2016.1307a

Publication date: 25 Jul 2016

Article Views: 1802

Article Downloads: 1294

Open Access References How to cite this article

References

Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the Learning of Mathematics, 25(2), 42-47.
Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115- 131.
Beishuizen, M. (2001). Different approaches to mastering mental calculation strategies. In J. Anghileri (Ed.), Principles and practices in arithmetic teaching (pp. 119-130). Buckingham: Open University Press.
Clement, John. "Analysis of Clinical Interviews: Foundations and Model Viability." In R. Lesh & A. Kelly (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 547-589). Hillsdale: Lawrence Erlbaum, 2000.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.
Guo, J., & Pang, M. F. (2011). Learning a mathematical concept from comparing examples: The importance of variation and prior knowledge. European Journal of Psychology of Education, 26(4), 495-525.
Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41(5), 535-540.
Hoch, M., & Dreyfus. (2004). Structure sense in high school algebra: The effect of brackets. In M.J. Hoines & A.B. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 49-56). Bergen, NO: PME.
Hoch, M., & Dreyfus, T. (2005). Students’ difficulties with applying a familiar formula in an unfamilar context. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 145- 152). Melbourne. PME
Kilpatrick, J., Swafford, J. & Findell, B. (Eds., 2001). Adding it up: Helping children learn mathematics. Washington D.C.: National Academy Press.
Kieran, C. (2011). Overall commentary on early algebraization: Perspectives for research and teaching. In J. Cai & E. Knuth (Eds.), Early algebraization. A global dialogue from multiple perspectives (pp. 579-593). Berlin, Heidelberg: Springer.
Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In K. Leatham (Ed.), Vital directions for mathematics education research (pp. 153-171). New York: Springer.
Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 274.
Lagrange, J. -B. (2003). Learning techniques and concepts using CAS: A practical and theoretical reflection. In J. Fey; A. Cuoco; C. Kieran; L. McMullin; R. M. Zbiek (Eds), Computer algebra systems in secondary school mathematics education (pp. 269-283). Reston, VA: NCTM.
Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173-196.
Mason, J. (2004). Doing≠ construing and doing+ discussing≠ learning: The importance of the structure of attention. ICME 10 Regular Lecture. Retrieved from http://math.unipa.it/~grim/YESS-5/ICME%2010%20Lecture%20Expanded.pdf
Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 365-380). Dordrecht: Springer.
Newton, K. J., Star, J. R., & Lynch, K. (2010). Understanding the Development of Flexibility in Struggling Algebra Students. Mathematical Thinking and Learning 12(4), 282-305.
Proulx, J. (2013). Mental mathematics, emergence of strategies, and the enactivist theory of cognition. Educational Studies in Mathematics, 84(3), 309-328.
Radford, L. (2006). Elements of a cultural theory of objectification. Revista Latinoamericana De Investigación En Matemática Educativa, 9, 103-129.
Rittle-Johnson, B., & Schneider, M. (2014). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition. Oxford: Oxford University Press.
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561-574.
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836-852.
Rojano, T., Filloy, E., & Puig, L. (2014). Intertextuality and sense production in the learning of algebraic methods. Educational Studies in Mathematics, 87(3), 389-407.
Rüede, C. (2012). The structuring of an algebraic expression as the production of relations. Journal für Mathematik-Didaktik, 33(1), 113-141.
Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132-135.
Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565-579.
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102(4), 408-26.
Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50(1), 29- 47.
Torbeyns, J., Smedt, B., Ghesquière, P., & Verschaffel, L. (2009). Jump or compensate? Strategy flexibility in the number domain up to 100. ZDM Mathematics Education, 41(5), 581-590.
Van den Akker, J., Gravemeijer, K., McKenney, S. & Nieveen, N. (Eds., 2006): Educational Design Research. London: Routledge.
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335-359.
Watson, A. & Mason, J. (2006) Seeing An Exercise As a Single Mathematical Object: Using Variation to Structure Sense-making. Mathematical thinking and learning 8(2), 91-111.

How to cite this article

APA

Meyer, A. (2016). Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property. Eurasia Journal of Mathematics, Science and Technology Education, 12(10), 2719-2732. https://doi.org/10.12973/eurasia.2016.1307a

Vancouver

Meyer A. Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property. EURASIA J Math Sci Tech Ed. 2016;12(10):2719-32. https://doi.org/10.12973/eurasia.2016.1307a

AMA

Meyer A. Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property. EURASIA J Math Sci Tech Ed. 2016;12(10), 2719-2732. https://doi.org/10.12973/eurasia.2016.1307a

Chicago

Meyer, Alexander. "Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property". Eurasia Journal of Mathematics, Science and Technology Education 2016 12 no. 10 (2016): 2719-2732. https://doi.org/10.12973/eurasia.2016.1307a

Harvard

Meyer, A. (2016). Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property. Eurasia Journal of Mathematics, Science and Technology Education, 12(10), pp. 2719-2732. https://doi.org/10.12973/eurasia.2016.1307a

MLA

Meyer, Alexander "Flexibly Applying the Distributive Law – Students’ Individual Ways of Perceiving the Distributive Property". Eurasia Journal of Mathematics, Science and Technology Education, vol. 12, no. 10, 2016, pp. 2719-2732. https://doi.org/10.12973/eurasia.2016.1307a