COUNTEREXAMPLES: CHALLENGES FACED BY ELEMENTARY STUDENTS WHEN TESTING A CONJECTURE ABOUT THE RELATIONSHIP BETWEEN PERIMETER AND AREA

Wanty Widjaja, Colleen Vale

Abstract


One pedagogical approach to challenge a persistent misconception is to get students to test a conjecture whereby they are confronted with the misconception. A common misconception about a ‘direct linear relationship’ between area and perimeter is well-documented. In this study, Year 4-6 students were presented with a conjecture that a rectangle with a larger perimeter will always have a larger area. Eighty-two (82) students’ written responses from three elementary schools in Victoria, Australia were analyzed. The findings revealed that Year 4-6 students could find multiple examples to support the conjecture but they struggled to find counterexamples to refute the conjecture. The findings underscored the importance of developing elementary school students’ capacity to construct counterexamples and recognize that it is sufficient to offer one counterexample in refuting a conjecture about all cases. Implications for ­teaching practice to support investigating and testing a conjecture are discussed.

Keywords


Counterexamples; Conjectures; Perimeter; Area; Elementary Students; Justifying

Full Text:

PDF

References


Australian Academy of Science (AAS) (2020). reSolve: Maths by Inquiry Special Topic 5 Assessing Reasoning. Retrived from https://www.resolve.edu.au/

Australian Curriculum and Assessment Authority (ACARA) (nd). The Australian Curriculum: Mathematics. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics

Buchbinder, O., & Zaslavsky, O. (2019). Strengths and inconsistencies in students’ understanding of the roles of examples in proving. Journal of Mathematical Behavior, 53, 129-147. http://doi.org/10.1016/j.jmathb.2018.06.010

Campbell, T. G., King, S. & Zelkowski, J. (2020). Comparing middle grade students’ oral and written arguments. Research in Mathematics Education. http://doi.org/10.1080/14794802.2020.1722960

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth: Heinemann.

Cavanagh, M. (2007). Year 7 students' understanding of area measurement. In K. Milton, H. Reeves, & T. Spencer (Eds.), Proceedings of the 21st biennial conference of the Australian Association of Mathematics Teachers Inc. (pp. 136-143). Adelaide: Australian Association of Mathematics Teachers.

Chen, C.-L., & Herbst, P. (2013). The interplay among gestures, discourse, and diagrams in students’ geometrical reasoning. Educational Studies in Mathematics, 83(2), 285–307. http://doi.org/10.1007/s10649-015-9652-9

Cohen, D. & Ball, D. (2001). Making change: instruction and its improvement. Phi Delta Kappan, 73-77.

Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks, CA: Sage.

Davis, P. J. & Hersch, R. (1981). The mathematical experience. London: Penguin Books.

De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and area of similar plane figures, Educational Studies in Mathematics, 35, 65–83. http://doi.org/10.1023/A:1003151011999

Ellis, A. B. (2007). A taxonomy for categorizing generalizations: Generalizing Actions and Reflection Generalizations. Journal of the Learning Sciences, 16(2), 221-261. http://doi.org/10.1080/10508400701193705

Ellis, A. B., Ozgur, Z., Vinsonhaler, R., Dogan, M. F., Carolan, T., Lockwood, E., Lynch, A., Sabouri, P., Knuth, E. & Zaslavsky, O. (2019). Student thinking with examples: the criteria-affordances-purposes strategies framework. Journal of Mathematical Behavior, 53, 263-283. http://doi.org/10.1016/j.jmathb.2017.06.003

Fernández, C. H., De Bock, D., Verschaffel, L., & Van Dooren, W. (2014). Do students confuse dimensionality and “directionality”? Journal of Mathematical Behavior, 36, 166-176. http://doi.org/10.1016/j.jmathb.2014.07.001

Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183-194. http://doi.org/10.1007/s10649-008-9143-3

Grant, T. J. & Kline, K. (2003). Developing building blocks of measurement with young children. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement: 2003 Yearbook (pp. 46–56). Reston, VA: National Council of Teachers of Mathematics.

Jeannotte, D. & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16. http://doi.org/10.1007/s10649-017-9761-8

Kamii, C. & Clark, F. (1997). Measurement of length: The need for a better approach to teaching. School Science and Mathematics, 97(3), 116–121. http://doi.org/10.1111/J.1949-8594.1997.TB17354.X

Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. The Journal of Mathematical Behavior, 53, 256-262. http://doi.org/10.1016/j.jmathb.2017.06.002

Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. Journal of Mathematical Behavior, 29(1), 1-10. http://doi.org/10.1016/j.jmathb.2010.01.003

Komatsu, K. (2016). A framework for proofs and refutations in school mathematics: increasing content by deductive guessing. Educational Studies in Mathematics, 92(2), 147-162. http://doi.org/10.1007/s10649-015-9677-0

Lakatos, I. (1976). Proofs and refutations: the logic of mathematical discovery. Cambridge: Cambridge University Press.

Limon, M. (2001). On the cognitive conflict as an instructional strategy for conceptual change: a critical appraisal. Learning and Instruction, 11(4-5), 357-380. https://doi.org/10.1016/S0959-4752(00)00037-2

Lin, P.-J., & Tsai, W.-H. (2016). Enhancing students’ mathematical conjecturing and justification in third grade classrooms: The sum of even/odd numbers. Journal of Mathematics Education, 9(1), 1–15.

Livy, S., Muir, T., & Maher, N. (2012). How do they measure up? primary pre-service teachers’ mathematical knowledge of area and perimeter. Mathematics Teacher Education and Development, 14(2), 91-112.

Lobato, J., Hohensee, C., & Rhodehamel, B. (2013). Students’ mathematical noticing. Journal for Research in Mathematics Education, 44(5), 809–850. https://doi.org/10.5951/jresematheduc.44.5.0809

Loong, E., Vale, C., Widjaja, W., Herbert, E.S., Bragg, L. & Davidson, A. (2018). Developing a rubric for assessing mathematical reasoning: A design-based research study in primary classrooms. In J. Hunter, L. Darragh & P. Perger (eds.) Making Waves, Opening Spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia, p. 503-510). Auckland, NZ: MERGA.

Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51. https://doi.org/10.2307/749097

Mason, J. (1982). Thinking mathematically. London: Addison-Wesley.

Mason, J. (2019). Relationships between proof and examples: comments arising from the papers in this issue. Journal of Mathematical Behavior, 53, 339-347. https://doi.org/10.1016/j.jmathb.2017.07.005

Markovits, H., Brisson, J., de Chantal, P-L. & St-Onge, C. M. (2016). Elementary school children know a logical argument when they see one. Journal of Cognitive Psychology, 28(7), 877-883, http://doi.org/10.1080/20445911.2016.1189918

Moyer, S. P. (2001). Using representations to explore perimeter and area. Teaching Children Mathematics, 8(1), 52-59.

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

NRICH. (2018). Magic Vs. Retrieved from https://nrich.maths.org/6274.

Pedemonte, B., & Buchbinder, O. (2011). Examining the role of examples in proving processes through a cognitive lens: The case of triangular numbers. ZDM Mathematics Education, 43(2):257-267. https://doi 10.1007/s11858-011-0311-z

Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also) explain. Focus on Learning Problems in Mathematics, 19(3), 49-61.

Schifter, D. (2009). Representation-based proof in the elementary grades. In D. A. Stylianou, M. Blanton, & E. Kieran (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 71-86): Taylor & Francis.

Small, M. (2011). One, two, infinity. Retrieved from http://www.onetwoinfinity.ca/

Soto-Johnson, H., & Fuller, E. (2012). Assessing proofs via oral interviews. Investigations in Mathematics Learning. 4(3), 1-14. https://doi.org/10.1080/24727466.2012.11790313

Sowder, L. & Harel, G. (1998). Types of students’ justifications. The Mathematics Teacher. 91(8), 670-675. https://www.jstor.org/stable/27970745

Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. Journal of Mathematical Behavior, 31, 447-462. http://dx.doi.org/10.1016/j.jmathb.2012.07.001

Staples, M. E. (2014). Supporting student justification in middle school mathematics classrooms: teachers' work to create a context for justification. CERME Publications. 4.

Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1-20. https://doi.org/10.1007/s10649-006-9038-0

Stylianides, A. J., & Al-Murani, T. (2010). Can a proof and a counterexample coexist? Students' conceptions about the relationship between proof and refutation. Research in Mathematics Education, 12(1), 21-36. https://doi.org/10.1080/14794800903569774

Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving Journal of Mathematics Teacher Education, 11(4), 307-332. https://doi/10.1007/s10857-008-9077-9

Tan Sisman, G., & Aksu, M. (2016). A study on sixth grade students’ misconceptions and errors in spatial measurement: length, area, and volume. International Journal of Science and Mathematics Education, 14(7), 1293-1319. https://doi.org/10.1007/s10763-015-9642-5

Tirosh, D., & Graeber, A. O. (1990). Evoking cognitive conflict to explore preservice teachers' thinking about division. Journal for Research in Mathematics Education, 21(2), 98-108. https://doi.org/10.2307/749137

Tirosh, D., & Stavy, S. (1999). Intuitive rules: A way to explain and predict students reasoning. Educational Studies in Mathematics, 38(1/3), 51–66. http://dx.doi. org/10.1023/A:1003436313032

Vale, C., Widjaja, W., Herbert, E. S., Loong, E. & Bragg, L.A. (2017). Mapping variation in children’s mathematical reasoning: The case of ‘What else belongs?”International Journal of Science and Mathematics Education, 20(4), 357-383. http://doi.org/10.1007/s10857-015-9341-8

Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004). Remedying secondary school students' illusion of linearity: A teaching experiment aiming at conceptual change. Learning and Instruction, 14(5), 485-501. https://doi.org/10.1016/j.learninstruc.2004.06.019

Van Dooren, W., De Bock, D. Janssens, D. & Verschafell, L. (2008). The linear imperative: an inventory and conceptual analysis of students’ overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311-342. https://doi.org/10.2307/30034972

Watson, A. & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.

Watson, J. M. (2007). The role of cognitive conflict in developing students' understanding of average. Educational Studies in Mathematics, 65(1), 21-47. https://doi.org/10.1007/s10649-006-9043-3

Widjaja, W., Vale, C., Herbert, S., Y-K. Loong, E. & Bragg, L. (2021). Linking comparing and contrasting, generalising and justifying: A case study of primary students’ levels of justifying. Mathematics Education Research Journal, 33(3), 589-612. https://doi: 10.1007/s13394-020-00323-0

Yeo, J. K. (2008). Teaching area and perimeter: Mathematics-Pedagogical-Content-Knowledge-in-Action. In M. Goos, R. Brown, & K. Makar (Eds.) Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia. (621-627). Brisbane: MERGA.

Yopp, D. A. (2013). Counterexamples as starting points for reasoning and sense making. The Mathematics Teacher, 106(9), 674-679. https://doi.org/10.5951/mathteacher.106.9.0674

Zaslavsky, O., Nickerson, S. D., Stylianides A. J., Kidron I., & Winicki-Landman, G. (2011) The Need for Proof and Proving: Mathematical and Pedagogical Perspectives. In Hanna G., de Villiers M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_9

Zaslavsky, O., & Ron, G. (1998). Students’ understanding of the role of counter examples. In A. Olivier & K. Newstead (Eds.), The 22nd conference of the international group for the psychology of mathematics education, Vol. 4, 225-232. Stellenboch, South Africa: PME.

Zazkis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary. Educational Studies in Mathematics, 68(3), 195-208. https://10.1007/s10649-007-9110-4




DOI: https://doi.org/10.22342/jme.12.3.14526.487-506

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Journal on Mathematics Education
Doctoral Program on Mathematics Education
Faculty of Teacher Training and Education, Universitas Sriwijaya
Kampus FKIP Bukit Besar
Jl. Srijaya Negara, Bukit Besar
Palembang - 30139
email: jme@unsri.ac.id
p-ISSN: 2087-8885 | e-ISSN: 2407-0610

Creative Commons License
Journal on Mathematics Education (JME) is licensed under a Creative Commons Attribution 4.0 International License.


View My Stats