Abstract

A critical part of supporting the development of students’ algorithmic thinking is understanding the challenges that emerge when students engage with algorithmatizing tasks—tasks that require the creation of an algorithm. Knowledge of these challenges can serve as a basis upon which educators can build effective strategies for enhancing students’ algorithmic thinking skills. This paper presents three illustrative cases of emergent challenges evident as students grapple with the process of creating an algorithm. The first challenge highlights discrepancies between the method with which students solve a problem and the algorithm they create, and claim would, when implemented, solve the same problem. The second challenge pertains to the persistence of students’ normatively incorrect algorithms, despite going through multiple iterations of testing and revising. Finally, the third challenge concerns issues around the use of test problems for supporting students in their creation of generalized algorithms. These three challenges are discussed using student data (illustrative cases) from three different mathematical algorithmatizing tasks. Suggestions are put forth for addressing some of these challenges, with a particular emphasis on practical pedagogical suggestions for cultivating students’ mathematical thinking in the context of algorithmatizing tasks. 

Keywords

References

• Ashlock, R. (2001). Error patterns in computations: Using error patterns to improve instruction. Prentice Hall.
• Booth, J. L., Barbieri, C., Eyer, F., & Paré-Blagoev, E. J. (2014). Persistent and pernicious errors in algebraic problem solving. The Journal of Problem Solving, 7(1), 3. https://doi.org/10.7771/1932-6246.1161
• Braun, V. & Clarke, V. (2012) Thematic analysis. In H. Cooper, P. M. Camic, D. L. Long, A. T. Panter, D. Rindskopf, & K. J. Sher (Eds.), APA handbook of research methods in psychology, Vol. 2: Research designs: Quantitative, qualitative, neuropsychological, and biological (pp. 57–71). American Psychological Association.
• Brown, G., & Quinn, R. J. (2006). Algebra students' difficulty with fractions: An error analysis. Australian Mathematics Teacher, 62(4), 28-40.
• Campbell, P. F., Rowan, T. E., & Suarez, A. R. (1998). What criteria for student-invented algorithms? In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics, 1998 yearbook (pp. 49–55). NCTM.
• Carroll, W. M. (2000). Invented computational procedures of students in a standards-based curriculum. The Journal of Mathematical Behavior, 18(2), 111-121. https://doi.org/10.1016/S0732-3123(99)00024-3
• Clement, J. (2000). Analysis of clinical interviews: Foundation and model viability. In A. E. Kelly, & R. Lesh (Eds.). Handbook of research design in mathematics and science education (pp. 547–589). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410602725
• Czocher, J. A. (2018). How does validating activity contribute to the modeling process? Educational Studies in Mathematics, 99(2), 137-159. https://doi.org/10.1007/s10649-018-9833-4
• Czocher, J., Stillman, G., & Brown, J. (2018). Verification and validation: what do we mean? In Hunter, J., Perger, P., & Darragh, L. (Eds.), Proceedings of the 41st annual conference of the mathematics education research group of Australasia (pp. 250-257). MERGA.
• Clarke, D. M. (2005). Written algorithms in the primary years: Undoing the good work. In M. Coupland, J. Anderson, & T. Spencer (Eds.), Making Mathematics Vital (pp. 93-98). AAMT.
• Darminto, B. (2020). The Influence of Mathematical Logic Ability Toward Computer Programming to Solve Mathematical Problems. In Proceedings of the 2nd international conference on education, ICE 2019. Universitas Muhammadiyah Purworejo, Indonesia. http://doi.org/10.4108/eai.28-9-2019.2290992
• De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. https://doi.org/10.1023/A:1021205413749
• Fischbein, E. (1987). Intuition in science and mathematics. D. Reidel. https://doi.org/10.1007/0-306-47237-6
• Futschek, G., & Moschitz, J. (2010). Developing algorithmic thinking by inventing and playing algorithms. Proceedings of the 2010 constructionist approaches to creative learning, thinking and education: Lessons for the 21st century (pp. 1-10). Comenius University, Slovakia.
• Groth, R. (2007). Understanding teachers’ resistance to the curricular inclusion of alternative algorithms. Mathematics Education Research Journal, 19(1), 3-28. https://doi.org/10.1007/BF03217447
• Knuth, D. E. (1974). Computer science and its relation to mathematics. The American Mathematical Monthly, 81(4), 323-343. https://doi.org/10.1080/00029890.1974.11993556
• Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2012). An exploratory framework for handling the complexity of mathematical problem posing in small groups. The Journal of Mathematical Behavior, 31(1), 149-161. https://doi.org/10.1016/j.jmathb.2011.11.002
• Lakatos, I. (1976). Proofs and refutations. Cambridge University Press. https://doi.org/10.1017/CBO9781139171472
• Lew, K., & Zazkis, D. (2019). Undergraduate mathematics students’ at-home exploration of a prove-or-disprove task. The Journal of Mathematical Behavior, 54, 100674. https://doi.org/10.1016/j.jmathb.2018.09.003
• Madell, R. (1985). Children’ s natural processes. Arithmetic Teacher, 32, 20-22. https://doi.org/10.5951/AT.32.7.0020
• Maher, C. A., & Sigley, R. (2020). Task-based interviews in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 821-824). Springer. https://doi.org/10.1007/978-3-030-15789-0_147
• Marrongelle, K. (2007). The function of graphs and gestures in algorithmatization. The Journal of Mathematical Behavior, 26(3), 211–229. https://doi.org/10.1016/j.jmathb.2007.09.005
• Mingus, T. T., & Grassl, R. M. (1998). Algorithmic and recursive thinking: Current beliefs and their implications for the future. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics, 1998 yearbook (pp. 32-43). NCTM.
• Moala, J. G. (2014). On being stuck: a preliminary investigation into the essence of mathematical problem solving. Master’s thesis. The University of Auckland.
• Moala, J. G. (2019). Creating algorithms by accounting for features of the solution: the case of pursuing maximum happiness. Mathematics Education Research Journal, 33(2), 263-284. https://doi.org/10.1007/s13394-019-00288-9
• Moala, J.G., Yoon, C., & Kontorovich, I. (2019). Localized considerations and patching: Accounting for persistent attributes of an algorithm on a contextualized graph theory task. The Journal of Mathematical Behavior, 55, 100704. https://doi.org/10.1016/j.jmathb.2019.04.003
• Morrow, L. J., & Kenney, M. J. (1998). The teaching and learning of algorithms in school mathematics. 1998 yearbook. NCTM.
• Peressini, D., & Knuth, E. (1998). The importance of algorithms in performance-based assessments. In L. J. Morrow & M. J. Kenney (Eds), The teaching and learning of algorithms in school mathematics (pp. 56–68). NCTM.
• Pinto, A., & Cooper, J. (2023). “This cannot be”—refutation feedback and its potential affordances for proof comprehension. Educational Studies in Mathematics, 113, 287-306. https://doi.org/10.1007/s10649-022-10190-0
• Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7(1), 51–73. https://doi.org/10.1207/s15327833mtl0701_4
• Ross, K. A. (1998). Doing and proving: The place of algorithms and proofs in school mathematics. The American Mathematical Monthly, 105(3), 252-255. https://doi.org/10.1080/00029890.1998.12004875
• Selden, A., & Selden, J. (1998). The role of examples in learning mathematics. The Mathematical Association of America Online. https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning/examples-in-learning-mathematics
• Sierpinska, A. (1994). Understanding in mathematics. Falmer Press. https://doi.org/10.4324/9780203454183
• Sinclair, N., Watson, A., Zazkis, R., & Mason, J. (2011). The structuring of personal example spaces. The Journal of Mathematical Behavior, 30(4), 291-303. https://doi.org/10.1016/j.jmathb.2011.04.001
• Son, J. (2016). Moving beyond a traditional algorithm in whole number subtraction: Preservice teachers’ responses to a student’s invented strategy. Educational Studies in Mathematics, 93(1), 105-129. https://doi.org/10.1007/s10649-016-9693-8
• Stephens, M., & Kadijevich, D. M. (2019). Computational/algorithmic thinking. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 483–451). Springer. https://doi.org/10.1007/978-3-319-77487-9_100044-1
• Tupouniua, J. G. (2019). Exploring mechanisms by which student-invented algorithms in mathematics emerge [Doctoral Thesis]. The University of Auckland.
• Tupouniua, J. G. (2020a). Explicating how students revise their algorithms in response to counterexamples: building on small nuanced gains. International Journal of Mathematical Education in Science and Technology, 53(7), 1711-1732. https://doi.org/10.1080/0020739X.2020.1837402
• Tupouniua, J. G. (2020b). Finding correct solution(s) ⇏ creating correct algorithm(s): Shedding more light on how and why. The Mathematician Educator, 1(2), 102-121.
• Tupouniua, J. G. (2023). Differentiating between counterexamples for supporting students’ algorithmic thinking. Asian Journal for Mathematics Education, 1(4), 475–493. https://doi.org/10.1177/27527263221139869
• Van Dooren, W., De Bock, D., Depaepe, F., Janssens, D., & Verschaffel, L. (2003). The illusion of linearity: Expanding the evidence towards probabilistic reasoning. Educational Studies in Mathematics, 53(2), 113-138. https://doi.org/10.1023/A:1025516816886
• Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33-35. https://doi.org/10.1145/1118178.1118215
• Wu, W. R., & Yang, K. L. (2022). The relationships between computational and mathematical thinking: A review study on tasks. Cogent Education, 9(1), 2098929. https://doi.org/10.1080/2331186X.2022.2098929
• Zazkis, R., & Chernoff, E. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195–208. https://doi.org/10.1007/s10649-007-9110-4

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