Incorporating history of mathematics in open-ended problem solving: An empirical study

Ioannis Rizos; Nikolaos Gkrekas

doi:10.29333/ejmste/13025

Full Text (PDF)

Ioannis Rizos ¹ ^* , Nikolaos Gkrekas ¹

More Detail

¹ Department of Mathematics, University of Thessaly, Lamia, GREECE^* Corresponding Author

Abstract

In this paper, we present a research project we conducted with 27 undergraduate students in a history of mathematics course in Greece during the academic year 2022-2023. In our study, we presented participants with an open-ended problem with historical background and evaluated their reactions and solving strategies. To reach findings we collected data via worksheets, questionnaires and interviews. We intended to focus on students’ techniques for open-ended issues while also investigating whether and how History of Mathematics may be included into its instruction. The results showed that such type of problems is able to catch the participants’ attention and support them in experimentation and development of multiple problem solving strategies. The students acquired a positive attitude towards the entire process, and they would like to repeat it in other university courses, too. This study might pave the way for a new curriculum that includes historically inspired open-ended assignments in school and university practice.

Keywords

open-ended problems
problem solving
hundred fowls problem
history of mathematics
mathematics education

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, Volume 19, Issue 3, March 2023, Article No: em2242

https://doi.org/10.29333/ejmste/13025

Publication date: 03 Mar 2023

Article Views: 2008

Article Downloads: 1213

Open Access References How to cite this article

References

Arcavi, A., Bruckheimer, M., & Ben-Zwi, R. (1987). History of mathematics for teachers. For the Learning of Mathematics, 7(2), 18-23.
Arthur, Y. D., Appiah, S. K., Amo-Asante, K., & Asare, B. (2022). Modeling student’s interest in mathematics: Role of history of mathematics, peer-assisted learning, and student’s perception. EURASIA Journal of Mathematics, Science and Technology Education, 18(10), em2168. https://doi.org/10.29333/ejmste/12458
Attard, A., Di Iorio, E., Geven, K., & Sánta, R. (2010). Student-centered learning: Toolkit for students, staff and higher education institutions. The European Students’ Union. https://www.esu-online.org/wp-content/uploads/2017/10/SCL_toolkit_ESU_EI.compressed.pdf
Awofala, A. O. A., & Ajao, E. A. (2021). Trends in research in problem solving in mathematics. International Journal of Social Sciences & Educational Studies, 8(3), 183-199. https://doi.org/10.23918/ijsses.v8i3p183
Baba, T., & Shimada, I. (2019). Socially open-ended problems for enriching student learning with mathematical models and social values. In P. Clarkson, W. T. Seah, & J. Pang (Eds), Values and valuing in mathematics education: Scanning and scoping the territory (pp. 171-183). Springer. https://doi.org/10.1007/978-3-030-16892-6_12
Bahar, A., & Maker, C. J. (2015). Cognitive backgrounds of problem solving: A comparison of open-ended vs. closed mathematics problems. EURASIA Journal of Mathematics, Science and Technology Education, 11(6), 1531-1546. https://doi.org/10.12973/eurasia.2015.1410a
Beghetto, R. A. (2017). Inviting uncertainty into the classroom. Educational Leadership, 75(2), 20-25.
Bicer, A., Marquez, A., Colindres, K. V. M., Schanke, A. A., Castellon, L. B., Audette, L. M., Perihan, C., & Lee, Y. (2021). Investigating creativity-directed tasks in middle school mathematics curricula. Thinking Skills and Creativity, 40, 100823. https://doi.org/10.1016/j.tsc.2021.100823
Bobis, J., Russo, J., Downton, A., Feng, M., Livy, S., McCormick, M., & Sullivan, P. (2021). Instructional moves that increase chances of engaging all students in learning mathematics. Mathematics, 9(6), 582. https://doi.org/10.3390/math9060582
Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83, 37-55. https://doi.org/10.1007/s10649-012-9441-7
Brandenburg, R., & Nevenzeel, K. (2007). The nine chapters on the history of Chinese mathematics. https://www.astro.rug.nl/~nevenzeel/Study/PGvdW_t=9C_HCM_a=RB,KN.pdf
Brousseau, G., & Warfield, V. (2014). Didactic situations in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-94-007-4978-8_47
Buckley, S., & Sullivan, P. (2021). Reframing anxiety and uncertainty in the mathematics classroom. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-021-00393-8
Calabrese, J. E., Capraro, M. M., & Thompson, C. G. (2022). The relationship between problem posing and problem solving: A systematic review. International Education Studies, 15(4), 1-8. https://doi.org/10.5539/ies.v15n4p1
Campbell, D. M., & Higgins, J. C. (2019). Mathematics: People, problems, results. Chapman and Hall/CRC. https://doi.org/10.1201/9781351074315
Chorlay, R., Clark, K. M. & Tzanakis, C. (2022). History of mathematics in mathematics education: Recent developments in the field. ZDM Mathematics Education, 54, 1407-1420. https://doi.org/10.1007/s11858-022-01442-7
Cifarelli, V. V., & Cai, J. (2005). The evolution of mathematical explorations in open-ended problem-solving situations. The Journal of Mathematical Behavior, 24, 302-324. https://doi.org/10.1016/j.jmathb.2005.09.007
Clark, K. M. (2012). History of mathematics: illuminating understanding of school mathematics concepts for prospective mathematics teachers. Educational Studies in Mathematics, 81, 67-84. https://doi.org/10.1007/s10649-011-9361-y
Cuomo, S. (2007). Pappus of Alexandria and the mathematics of late antiquity. Cambridge University Press.
Deep, S., Mohd-Salleh, B., & Othman, H. (2019). Study on problem-based learning towards improving soft skills of students in effective communication class. International Journal of Innovation and Learning, 25(1), 17-34. https://doi.org/10.1504/IJIL.2019.096512
Denzin, N. K. (2009). The research act: A theoretical introduction to sociological methods. Transaction Publishers Rutgers.
Devlin, K. (2011). The man of numbers. Fibonacci’s arithmetic revolution. Walker Books.
Di Martino, P., Gregorio, F., & Iannone, P. (2022). The transition from school to university mathematics in different contexts: affective and sociocultural issues in students’ crisis. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-022-10179-9
Dostál, J. (2015). Theory of problem solving. Procedia-Social and Behavioral Sciences, 174, 2798-2805. https://doi.org/10.1016/j.sbspro.2015.01.970
Euler, L. (1828). Elements of algebra. Longman, Rees, Orme and Co.
Fatah, A., Suryadi, D., Sabandar, J., & Turmudi. T. (2016). Open-ended approach: An effort in cultivating students’ mathematical creative thinking ability and self-esteem in mathematic. Journal on Mathematics Education, 7(1), 11-20. https://doi.org/10.22342/jme.7.1.2813.9-18
Furinghetti, F. (2007). Teacher education through the history of mathematics. Educational Studies in Mathematics, 66, 131-143. https://doi.org/10.1007/s10649-006-9070-0
Furinghetti, F. (2020). Rethinking history and epistemology in mathematics education. International Journal of Mathematical Education in Science and Technology, 51(6), 967-994. https://doi.org/10.1080/0020739X.2019.1565454
Goos, M., Galbraith, P., & Renshaw, P. (1999). Establishing a community of practice in a secondary mathematics classroom. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 36-61). Falmer Press.
Grabiner, J. (1995). Descartes and problem-solving. Mathematics Magazine, 68(2), 83-97. https://doi.org/10.2307/2691183
Gupta, R. C. (2008). Mahavira. In H. Selin (Ed.), Encyclopedia of the history of science, technology and medicine in non-Western cultures. Springer https://doi.org/10.1007/978-1-4020-4425-0_9267
Hadley, J., & Singmaster, D. (1992). Problems to sharpen the young. The Mathematical Gazette, 76(475), 102-126. https://doi.org/10.2307/3620384
Hafidzah, N. A., Aziz, Z. & Irvan, I. (2021). The effect of open ended approach on problem solving ability and learning independence in students’ mathematics lessons. Indonesian Journal of Education and Mathematical Science, 1(1), 11-18. https://doi.org/10.30596/ijems.v2i1.6176
Hogarth, R. M., & Kunreuther, H. (1995). Decision making under ignorance: Arguing with yourself. Journal of Risk and Uncertainty, 10, 15-36. https://doi.org/10.1007/BF01211526
Ibrahim, I., & Widodo, S. A. (2020). Advocacy approach with open-ended problems to mathematical creative thinking ability. Infinity, 9(1), 93-102. https://doi.org/10.22460/infinity.v9i1.p93-102
Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., da Silva, C. M. S., & Weeks, C. (2000). The use of original sources in the mathematics classroom. In J. Fauvel, & J. van Maanen (Eds), History in mathematics education: The ICMI study (pp. 291-328). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47220-1_9
Jahnke, H. N., Jankvist, U. T., & Kjeldsen, T. H. (2022). Three past mathematicians’ views on history in mathematics teaching and learning: Poincaré, Klein, and Freudenthal. ZDM Mathematics Education, 54, 1421-1433. https://doi.org/10.1007/s11858-022-01376-0
Jamieson, S. (2004). Likert scales: How to (ab)use them. Medical Education, 38(12), 1217-1218. https://doi.org/10.1111/j.1365-2929.2004.02012.x
Kangshen, S., Crossley, J. N., & Lun, A. W. C. (1999). The nine chapters on the mathematical art: Companion and commentary. Oxford University Press.
Katz, V. J. (2009). A history of mathematics. An introduction. Pearson Education, Inc.
Kelle, U., Kühberger, C., & Bernhard, R. (2019). How to use mixed-methods and triangulation designs: An introduction to history education research. History Education Research Journal, 16(1), 5-23. https://doi.org/10.18546/HERJ.16.1.02
Keren, G., & Gerritsen, L. E. M. (1999). On the robustness and possible accounts of ambiguity aversion. Acta Psychological, 103(12), 149-172. https://doi.org/10.1016/S0001-6918(99)00034-7
Kosyvas, G. (2016). Levels of arithmetic reasoning in solving an open-ended problem. International Journal of Mathematical Education in Science and Technology, 47(3), 356-372. https://doi.org/10.1080/0020739X.2015.1072880
Leavy, A., & Hourigan, M. (2020). Posing mathematically worthwhile problems: Developing the problem-posing skills of prospective teachers. Journal of Mathematics Teacher Education, 23, 341-361. https://doi.org/10.1007/s10857-018-09425-w
Libbrecht, U. (1973). Chinese mathematics in the thirteenth century. MIT Press.
Liljedahl, P., Santos-Trigo, M., Malaspina, U., & Bruder, R. (2016). Problems solving in mathematics education. Springer. https://doi.org/10.1007/978-3-319-40730-2
Mamona-Downs, J., & Downs, M. (2013). Problem solving and its elements in forming proof. The Mathematics Enthusiast, 10(1), 137-162. https://doi.org/10.54870/1551-3440.1263
Manz, E. (2018). Designing for and analyzing productive uncertainty in science investigations. In J. Kay, & R. Luckin (Eds.), Rethinking learning in the digital age: Making the learning sciences count. International Society of the Learning Sciences.
Meadows, M., & Caniglia, J. (2021). That was then … This is now: Utilizing the history of mathematics and dynamic geometry software. International Journal of Education in Mathematics, Science, and Technology, 9(2), 198-212. https://doi.org/10.46328/ijemst.1106
Mousoulides, N., & Sriraman, B. (2020). Heuristics in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-030-15789-0_172
Moustapha-Correa, B., Bernades, A., Giraldo, V., Biza, I., & Nardi, E. (2021). Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks. The Journal of Mathematical Behavior, 61, 100840. https://doi.org/10.1016/j.jmathb.2021.100840
Murwaningsih, T. & Fauziah, M. (2022). The development of open-ended questions test to measure divergent thinking skills: The development of open-ended questions test. In Proceedings of the 5^th International Conference on Learning and Quality Education (pp. 1-7). Association for Computing Machinery. https://doi.org/10.1145/3516875.3516876
NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
Palincsar, A. S., & Brown, A. L. (1984). Reciprocal teaching of comprehension-fostering and comprehension-monitoring activities. Cognition and Instruction, 1(2), 117-175. https://doi.org/10.1207/s1532690xci0102_1
Papadopoulos, I. (2020). Using tasks to bring challenge in mathematics classroom. Journal of Pedagogical Research, 4(3), 375-386. https://doi.org/10.33902/JPR.2020063021
Pappas, T. (1989). The joy of mathematics. Wide World Publications/Tetra.
Pehkonen, E. (2017). Open tasks in mathematics: Experiences with one problem field. Magistra Iadertina, 12(2), 9-19. https://doi.org/10.15291/magistra.1487
Pehkonen, E., Näveri, L., & Laine, A. (2013). On teaching problem solving in school mathematics. CEPS Journal, 3(4), 9-23. https://doi.org/10.26529/cepsj.220
Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press. https://doi.org/10.1515/9781400828678
Polya, G. (1954). Mathematics and plausible reasoning. Princeton University Press. https://doi.org/10.1515/9780691218304
Polya, G. (1962). Mathematical discovery. John Wiley.
Poulos, A., & Mamona-Downs, J. (2018). Gifted students approaches when solving challenging mathematical problems. In F. Singer (Ed.) Mathematical creativity and mathematical giftedness. Springer. https://doi.org/10.1007/978-3-319-73156-8_12
Puspitasari, L., In’am, A., & Syaifuddin, M. (2019). Analysis of students’ creative thinking in solving arithmetic problems. International Electronic Journal of Mathematics Education, 14(1), 49-60. https://doi.org/10.12973/iejme/3962
Rahayuningsih, S., Sirajuddin, S., & Ikram, M. (2021). Using open-ended problem-solving tests to identify students’ mathematical creative thinking ability. Participatory Educational Research, 8(3), 285-299. https://doi.org/10.17275/per.21.66.8.3
Randles, C., Overton, T., Galloway, R., & Wallace, M. (2018). How do approaches to solving open-ended problems vary within the science disciplines? International Journal of Science Education, 40(11), 1367-1390. https://doi.org/10.1080/09500693.2018.1503432
Rizos, I., & Gkrekas, N. (2022a). Teaching and learning sciences within the COVID-19 pandemic era in a Greek university department. U.Porto Journal of Engineering, 8(1), 73-83. https://doi.org/10.24840/2183-6493_008.001_0008
Rizos, I., & Gkrekas, N. (2022b). The historical background of a famous indeterminate problem and some teaching perspectives. Journal of Mathematics and Statistics Studies, 3(1), 1-8. https://doi.org/10.32996/jmss.2022.3.1.1
Rohmah, M., & Sutiarso, S. (2018). Analysis problem solving in mathematical using theory Newman. EURASIA Journal of Mathematics, Science and Technology Education, 14(2), 671-681. https://doi.org/10.12973/ejmste/80630
Rowland, T. (1995). Hedges in mathematics talk: Linguistic pointers to uncertainty. Educational Studies in Mathematics, 29(4), 327-353. https://doi.org/10.1007/BF01273910
Rus, T. (2016). Milestones for computing future. Journal of Software Engineering and Applications, 9, 52-56. https://doi.org/10.4236/jsea.2016.92003
Schoenfeld, A. H (1992). Learning to think mathematically: Problem solving, metacognition and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). Macmillan.
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
Schoenfeld, A. H. (1987). Polya, problem solving, and education. Mathematics Magazine, 60(5), 283-291. https://doi.org/10.2307/2690409
Scoones, I. (2019). What is uncertainty and why does it matter? STEPS Center. https://steps-centre.org/publication/what-is-uncertainty-and-why-does-it-matter/
Sesiano, J. (2008). Abu Kamil. In H. Selin (Ed.), Encyclopedia of the history of science, technology and medicine in non-Western cultures. Springer. https://doi.org/10.1007/978-1-4020-4425-0_9198
Sesiano, J. (2009). An introduction to the history of algebra: Solving equations from Mesopotamian times to the Renaissance. American Mathematical Society.
Skovsmose, O. (2020). Critical mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-030-15789-0_34
Slavich, G. M., & Zimbardo, P. G. (2012). Transformational teaching: Theoretical underpinnings, basic principles, and core methods. Educational Psychology Review, 24, 569-608. https://doi.org/10.1007/s10648-012-9199-6
Suastika, I. K. (2021). Mathematics learning syntax model using open-ended problem solving to develop students’ creativity. Pancaran Pendidikan FKIP Universitas Jember [Education Radiation FKIP University of Jember], 6(4), 105-112. https://doi.org/10.25037/pancaran.v6i4.118
Tanjung, D. F., Syahputra, E. & Irvan, I. (2020). Problem based learning, discovery learning and open ended models: An experiment on mathematical problem solving abilities. Jurnal Teori dan Aplikasi Matematika [Journal of Mathematical Theory and Applications], 4(1), 9-16. https://doi.org/10.31764/jtam.v4i1.1736
Tarchi, C., & Pinto, G. (2016). Reciprocal teaching: Analyzing interactive dynamics in the co-construction of a text’s meaning. The Journal of Educational Research, 109(5), 518-530. https://doi.org/10.1080/00220671.2014.992583
Todorovski, B., Nordal, E., & Isoski, T. (2015). Overview on student-centered learning in higher education in Europe: Research study. The European Students’ Union. https://www.esu-online.org/wp-content/uploads/2016/07/Overview-on-Student-Centred-Learning-in-Higher-Education-in-Europe.pdf
Ulinnuha, R., Budi Waluya, S., & Rochmad, R. (2021). Creative thinking ability with open-ended problems based on self-efficacy in gnomic blended learning. Unnes Journal of Mathematics Education Research, 10(A), 20-25.
van Garderen, D. (2004). Focus on inclusion reciprocal teaching as a comprehension strategy for understanding mathematical word problems. Reading & Writing Quarterly, 20(2), 225-229. https://doi.org/10.1080/10573560490272702
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.
Waxer, M., & Morton, J. B. (2012). Cognitive conflict and learning. In N. M. Seel (Ed.), Encyclopedia of the sciences of learning. Springer. https://doi.org/10.1007/978-1-4419-1428-6_280
Weldeana, H. N., & Abraham, S. T. (2014). The effect of an historical perspective on prospective teachers’ beliefs in learning mathematics. Journal of Mathematics Teacher Education, 17, 303-330. https://doi.org/10.1007/s10857-013-9266-z
Wulandari, A. (2021). The effectiveness of the application of learning models creative problem solving and problem based instruction: The impact on students mathematics learning interest. In Proceedings of the 2^nd International Conference on Technology and Educational Science (pp. 150-155). Atlantis Press. https://doi.org/10.2991/assehr.k.210407.229
Yazgan-Sag, G., & Emre-Akdogan, E. (2016). Creativity from two perspectives: Prospective mathematics teachers and mathematician. Australian Journal of Teacher Education, 41(12), 25-40. https://doi.org/10.14221/ajte.2016v41n12.3
Yong, L. L. (1977). A critical study of the Yang Hui Suan Fa. Singapore University Press. https://doi.org/10.1007/BF00374594
Yong, L. L. (1997). Zhang Qiujian Suanjing (the mathematical classic of Zhang Qiujian): An overview. Archive for History of Exact Science, 50(3-4), 201-240.
Yuniarti, Y., Kusumah, Y. S., Suryadi, D., & Kartasasmita, B. G. (2017). The effectiveness of open-ended problems based analytic-synthetic learning on the mathematical creative thinking ability of pre-service elementary school teachers. International Electronic Journal of Mathematics Education, 12(3), 655-666. https://doi.org/10.29333/iejme/640
Zaslavsky, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics. Educational Studies in Mathematics, 60, 297-321. https://doi.org/10.1007/s10649-005-0606-5
Zengin, Y. (2018). Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course. International Journal of Mathematical Education in Science and Technology, 49(7), 1083-1098. https://doi.org/10.1080/0020739X.2018.1431850

How to cite this article

APA

Rizos, I., & Gkrekas, N. (2023). Incorporating history of mathematics in open-ended problem solving: An empirical study. Eurasia Journal of Mathematics, Science and Technology Education, 19(3), em2242. https://doi.org/10.29333/ejmste/13025

Vancouver

Rizos I, Gkrekas N. Incorporating history of mathematics in open-ended problem solving: An empirical study. EURASIA J Math Sci Tech Ed. 2023;19(3):em2242. https://doi.org/10.29333/ejmste/13025

AMA

Rizos I, Gkrekas N. Incorporating history of mathematics in open-ended problem solving: An empirical study. EURASIA J Math Sci Tech Ed. 2023;19(3), em2242. https://doi.org/10.29333/ejmste/13025

Chicago

Rizos, Ioannis, and Nikolaos Gkrekas. "Incorporating history of mathematics in open-ended problem solving: An empirical study". Eurasia Journal of Mathematics, Science and Technology Education 2023 19 no. 3 (2023): em2242. https://doi.org/10.29333/ejmste/13025

Harvard

Rizos, I., and Gkrekas, N. (2023). Incorporating history of mathematics in open-ended problem solving: An empirical study. Eurasia Journal of Mathematics, Science and Technology Education, 19(3), em2242. https://doi.org/10.29333/ejmste/13025

MLA

Rizos, Ioannis et al. "Incorporating history of mathematics in open-ended problem solving: An empirical study". Eurasia Journal of Mathematics, Science and Technology Education, vol. 19, no. 3, 2023, em2242. https://doi.org/10.29333/ejmste/13025