Metaphorical Perceptions of Vocational High School Students on Mathematical Problem Solving

Öz

Since problem solving has an important place in university level mathematics courses, determining university students' perceptions about problem solving can contribute to the field. In this regard, the aim of this study is to determine the mathematical problem solving perceptions of vocational high school students through metaphors. The case survey method was preferred for this study. Participants were 234 students studying in various programs of a vocational high school of a public university in Turkey in the fall semester of the 2019-2020 academic year. “Mathematical Problem Solving Metaphor Survey” were used as data collection tool. This survey included items that asked them to produce a metaphor for mathematical problem solving and its justification. Their written responses were analyzed using content analysis. Analysis of data indicated that participants produced 105 valid metaphors for mathematical problem solving. The most commonly used metaphors were “life”, “labyrinth”, “playing chess”, and “jigsaw puzzles”. Also, the metaphors produced by the participants were categorized under ten different conceptual categories. Metaphors were mostly gathered in the categories of “complicated” and “strategically progressive”. These categories have a statistically significant relationship with the programs in which the participants study at. Based on the categories, students’ perceptions of mathematical problem solving were discussed. Also, the study discusses instructional implications regarding how to change negative perceptions of students.

Anahtar Kelimeler

• Mathematical problem solving
• metaphor
• perception
• vocational high school students

Meslek Yüksekokulu Öğrencilerinin Matematiksel Problem Çözmeye İlişkin Metaforik Algıları

Öz

Problem çözme üniversite düzeyindeki matematik derslerinde önemli bir yere sahip olduğundan üniversite öğrencilerinin problem çözme ile ilgili algılarını belirlemek alana katkı sağlayabilir. Bu doğrultuda, bu çalışmanın amacı meslek yüksekokulu öğrencilerinin matematiksel problem çözme algılarını metaforlar aracılığıyla belirlemektir. Bunun için bu çalışmada örnek olay tarama modeli tercih edilmiştir. Araştırmanın katılımcıları 2019-2020 eğitim-öğretim yılı güz döneminde Türkiye’deki bir devlet üniversitesinin meslek yüksekokulunun çeşitli programlarında öğrenim gören 234 öğrencidir. “Matematiksel Problem Çözme Metafor Anketi” veri toplama aracı olarak kullanılmıştır. Bu ankette katılımcılardan matematiksel problem çözme ile ilgili bir metafor üretmeleri ve bunu gerekçelendirmeleri istenmiştir. Katılımcıların yazılı yanıtları içerik analizi ile analiz edilmiştir. Verilerin analizi katılımcıların matematiksel problem çözme ile ilgili 105 geçerli metafor ürettiğini göstermiştir. Katılımcıların en sık “hayat”, “labirent”, “satranç oynamak” ve “yapboz yapmak” metaforlarını ürettikleri saptanmıştır. Ayrıca katılımcıların ürettikleri metaforlar on farklı kavramsal kategori altında kategorize edilmiştir. Üretilen metaforlar en sık “komplike” ve “stratejik aşamalı” kategorilerinde toplanmıştır. Bu kategorilerin katılımcıların öğrenim gördükleri programlarla istatistiksel olarak anlamlı bir ilişkiye sahip olduğu saptanmıştır. Kategorilere dayalı olarak, öğrencilerin matematiksel problem çözme algıları tartışılmıştır. Ayrıca öğrencilerin olumsuz algılarının nasıl değiştirilebileceğine ilişkin öğretimsel çıkarımlar tartışılmıştır.

Anahtar Kelimeler

• Matematiksel problem çözme
• metafor
• algı
• meslek yüksekokulu öğrencileri

Kaynakça

1. Agresti, A. (1990). Categorical data analysis. New York: John Wiley & Sons. Retrieved March 20, 2020 from https://books.google.com.tr/books/about/Categorical_Data_Analysis.html?id=MCnvAAAAMAAJ&redir_esc=y
2. Botha, E. (2009). Why metaphor matters in education. South African Journal of Education, 29(4), 431-444. https://dx.doi.org/10.15700/saje.v29n4a287
3. Buerk, D. (1988). Mathematical metaphors from advanced placement students. Humanistic Mathematics Network Journal, 1(3), Article 10. https://dx.doi.org/10.5642/hmnj.198801.03.10
4. Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs. The Journal of Mathematical Behavior, 23(2), 135-167. https://doi.org/10.1016/j.jmathb.2004.03.004
5. Callejo, M. L., & Vila, A. (2009). Approach to mathematical problem solving and students’ belief systems: two case studies. Educational Studies in Mathematics, 72(1), 111-126. https://doi.org/10.1007/s10649-009-9195-z
6. Chapman, O. (1997). Metaphors in the teaching of mathematical problem solving. Educational Studies in Mathematics, 32(3), 201-228. https://doi.org/10.1023/A:1002991718392
7. Chiu, M. M. (2001). Using metaphors to understand and solve arithmetic problems: Novices and experts working with negative numbers. Mathematical Thinking and Learning, 3(2-3), 93-124. https://doi.org/10.1080/10986065.2001.9679970
8. Cochran, W. G. (1954). Some methods for strengthening the common χ2 tests. Biometrics, 10(4), 417-451. https://dx.doi.org/10.2307/3001616

9. Creswell, J. W. (2014). Research design: qualitative, quantitative and mixed methods approaches (4th ed.). Thousand Oaks, CA: Sage. Retrieved March 10, 2020 from https://study.sagepub.com/creswellrd4e
10. Department for Education. (2014). Statutory guidance national curriculum in England: Mathematics programmes of study. Retrieved December 10, 2019 from https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study
11. Diaz-Rojas, D., & Soto-Andrade, J. (2017, February). Enactive metaphors in mathematical problem solving. In T. Dooley & G. Gueudet (Eds.), Proceedings of CERME10 (pp. 3904-3911). DCU Institute of Education and ERME, Dublin, Ireland. Retrieved February 11, 2020 from https://hal.archives-ouvertes.fr/hal-01950551
12. Ergen, Y. (2020). ‘Does mathematics fool us?’ A study on fourth grade students’ non-routine maths problem solving skills. Issues in Educational Research, 30(3), 845-865. Retrieved October 11, 2021 from http://www.iier.org.au/iier30/ergen.pdf
13. Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh, Scotland: Oliver and Boyd. Retrieved March 20, 2020 from https://books.google.com.tr/books/about/Statistical_Methods_for_Research_Workers.html?id=I0NBAAAAIAAJ&redir_esc=y
14. Font, V., Bolite, J., & Acevedo, J. (2010). Metaphors in mathematics classrooms: analyzing the dynamic process of teaching and learning of graph functions. Educational Studies in Mathematics, 75(2), 131-152. https://doi.org/10.1007/s10649-010-9247-4
15. Freeman, G. H., & Halton, J. H. (1951). Note on exact treatment of contingency, goodness of fit and other problems of significance. Biometrika, 38(1-2), 141-149. https://dx.doi.org/10.2307/2332323
16. Guba, E. G., & Lincoln, Y. S. (1982). Epistemological and methodological bases of naturalistic inquiry. Educational Communication and Technology Journal, 30(4), 233-252. https://doi.org/10.1007/BF02765185
17. Guest, G., MacQueen, K. M., & Namey, E. E. (2012). Applied thematic analysis. Thousand Oaks, CA: Sage. https://dx.doi.org/10.4135/9781483384436
18. Hannula, M. S. (2002). Attitude towards mathematics: emotions, expectations and values. Educational Studies in Mathematic, 49(1), 25-46. https://dx.doi.org/10.1023/A:1016048823497
19. Hannula, M. S. (2015). Emotions in problem solving. In S. J. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 269-288). New York: Springer. https://doi.org/10.1007/978-3-319-17187-6_16
20. Hendriana, H., & Rohaeti, E. E. (2017). The importance of metaphorical thinking in the teaching of mathematics. Current Science, 113(11), 2160-2164. https://doi.org/10.18520/cs/v113/i11/2160-2164
21. Hodgen, J., Wake, G., & Dalby, D. (2017). Mathematics in the successful technical education of 16-19 year olds. London: Gatsby. Retrieved January 17, 2020 from https://www.gatsby.org.uk/uploads/education/reports/pdf/maths-in-international-systems.pdf
22. Innis, R. E. (1982). From perception to metaphor. In K. Bühler (Ed.), Semiotic foundations of language theory (pp. 43-53). Boston, MA: Springer. https://doi.org/10.1007/978-1-4757-0923-0_4
23. Julien, H. (2008). Content analysis. In L. M. Given (Ed.), The Sage encyclopedia of qualitative research methods (Vol 1, pp. 120-122). Los Angeles, California: Sage Publications, Inc. https://dx.doi.org/10.4135/9781412963909
24. Lachner, A., & Nückles, M. (2016). Tell me why! Content knowledge predicts process-orientation of math researchers’ and math teachers’ explanations. Instructional Science, 44(3), 221-242. https://doi.org/10.1007/s11251-015-9365-6
25. Lakoff, G. (1993). The contemporary theory of metaphor. In A. Ortony (Ed.), Metaphor and thought (2nd ed., pp. 202-251). Cambridge: Cambridge University Press. Retrieved January 11, 2020 from https://escholarship.org/content/qt4nv3j5j9/qt4nv3j5j9.pdf
26. Lakoff, G., & Johnson, M. (1980a). Metaphors we live by. Chicago: The University of Chicago Press. Retrieved January 11, 2020 from https://books.google.com.tr/books/about/Metaphors_We_Live_By.html?id=r6nOYYtxzUoC&redir_esc=y
27. Lakoff, G., & Johnson, M. (1980b). Conceptual metaphor in everyday language. The Journal of Philosophy, 77(8), 453-486. https://dx.doi.org/10.2307/2025464
28. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Retrieved January 11, 2020 from https://pages.ucsd.edu/~rnunez/COGS252_Readings/Preface_Intro.PDF
29. Larsson, R. (1993). Case survey methodology: Quantitative analysis of patterns across case studies. Academy of management Journal, 36(6), 1515-1546. https://doi.org/10.5465/256820
30. Latterell, C. M., & Wilson, J. L. (2016). Math is like a lion hunting a sleeping gazelle: Preservice elementary teachers’ metaphors of mathematics. European Journal of Science and Mathematics Education, 4(3), 283-292. Retrieved January 25, 2020 from https://eric.ed.gov/?id=EJ1107832
31. Latterell, C. M., & Wilson, J. L. (2017). Metaphors and mathematical identity: Math is like a tornado in Kansas. Journal of Humanistic Mathematics, 7(1), 46-61. https://dx.doi.org/ 10.5642/jhummath.201701.05
32. Lester, F. K., & Cai, J. (2016). Can mathematical problem solving be taught? Preliminary answers from 30 years of research. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems: advances and new perspectives (pp. 117-135). Buenos Aires: Springer Publishing. https://doi.org/10.1007/978-3-319-28023-3_8
33. Mailisman, N., Ikhsan, M., & Hajidin. (2020). Mathematics problem-solving skills of vocational high school students related to the 21st-century education. Journal of Physics: Conference Series, 1460(1):012014. https://doi.org/10.1088/1742-6596/1460/1/012014
34. Markovits, Z., & Forgasz, H. (2017). “Mathematics is like a lion”: Elementary students’ beliefs about mathematics. Educational Studies in Mathematics, 96(1), 49-64. https://doi.org/10.1007/s10649-017-9759-2
35. Martin, M. O., von Davier, M., & Mullis, I. V. S. (2020). Methods and procedures: TIMSS 2019 technical report. T. P. I. S. Center. Lynch School of Education, Boston College. Retrieved January 10, 2020 from https://timssandpirls.bc.edu/timss2019/methods/pdf/TIMSS-2019-MP-Technical-Report.pdf
36. Mason, L. (2003). High school students’ beliefs about maths, mathematical problem solving and their achievement in maths: a cross sectional study. Educational Psychology, 23(1), 73-85. https://doi.org/10.1080/01443410303216
37. Mathieu, A. (2009). Metaphors in mathematics: introduction and the case of algebraic geometry. http://dx.doi.org/10.2139/ssrn.1478871
38. McLeod, D. B. (1988). Affective issues in mathematical problem solving: some theoretical considerations. Journal for Research in Mathematics Education, 19(2), 134-141. https://doi.org/10.5951/jresematheduc.19.2.0134
39. Measuring, Selection and Placement Center [MSPC]. (2019). Yükseköğretim Kurumları sınavı (YKS) klavuzu [Council of Higher Education (CoHE) exam guide]. Retrieved January 18, 2020 https://www.osym.gov.tr/TR,15615/2019.html
40. Metha, C. R., & Patel, N. R. (1989). IBM SPSS exact tests. Cambridge, MA: IBM Corporation. Retrieved March 20, 2020 from https://www.ibm.com/docs/en/SSLVMB_27.0.0/pdf/en/IBM_SPSS_Exact_Tests.pdf
41. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: an expanded sourcebook (2nd ed.). Thousand Oaks, CA, US: Sage Publications, Inc. Retrieved July 19, 2018 from https://books.google.com.tr/books?id=U4lU_-wJ5QEC&printsec=frontcover&hl=tr#v=onepage&q&f=false
42. Ministry of Education [MoE]. (2012). Mathematics syllabus: primary one to six. Retrieved December 11, 2019 from https://www.moe.gov.sg/education/syllabuses/sciences/
43. Ministry of National Education [MoNE]. (2018a). Matematik dersi öğretim programı (İlkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar [Mathematics curriculum (Primary and secondary schools 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, and 8th grades)]. Ankara: MoNE Board of Education and Discipline. Retrieved January 27, 2019 http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=329
44. Ministry of National Education [MoNE]. (2018b). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı [High school mathematics curriculum (9th, 10th, 11th, and 12th grades)]. Ankara: MoNE Board of Education and Discipline. Retrieved January 27, 2019 http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=343
45. Ministry of National Education [MoNE]. (2019). Millî Eğitim Bakanlığı ortaöğretime geçiş yönergesi [Ministry of National Education the transition to secondary education directive]. Retrieved November 09, 2019 http://www.meb.gov.tr/meb_iys_dosyalar/2019_04/03134342_Yonerge2019.pdf
46. Mohd, N., & Mahmood, T. F. P. T. (2011). The effects of attitude towards problem solving in mathematics achievements. Australian Journal of Basic and Applied Sciences, 5(12), 1857-1862. Retrieved February 26, 2020 from http://www.ir.unikl.edu.my/jspui/handle/123456789/2129
47. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, Va: National Council of Teachers of Mathematics. Retrieved January 27, 2019 https://www.nctm.org/standards-and-positions/principles-and-standards/
48. Or., M. B., & Bal, A. P. (2021). Ortaokul öğrenci̇leri̇ni̇n problem çözme algıları ile matematik başarıları arasındaki i̇li̇şki̇ni̇n incelenmesi [Investigation of the relationship between the problem-solving perception levels and mathematics achievement of secondary school students]. Journal of Çukurova University Institute of Social Sciences, 30(2), 154-165. https://doi.org/10.35379/cusosbil.906891
49. Organization of Economic Co-operation and Development [OECD]. (2016). PISA 2015 assessment and analytical framework: science, reading, mathematic and financial literacy. Paris: PISA, OECD Publishing. https://dx.doi.org/10.1787/9789264255425-en
50. Pajares, F., & Miller, M. D. (1994). Role of self-efficacy and self-concept beliefs in mathematical problem solving: a path analysis. Journal of Educational Psychology, 86(2), 193-203. https://dx.doi.org/10.1037//0022-0663.86.2.193
51. Patton, M. Q. (2001). Qualitative research & evaluation methods. (4th ed.). Saint Paul, MN: Sage Publications. Retrieved March 10, 2020 from https://us.sagepub.com/en-us/nam/qualitative-research-evaluation-methods/book232962
52. Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.
53. Purwasi, L. A., & Fitriyana, N. (2021, January). The enhancement of mathematical reasoning and problem solving ability through metaphorical thinking approach. Journal of Physics Conference Series, (Vol. 1731, No. 1, p. 012040). IOP Publishing. https://dx.doi.org/10.1088/1742-6596/1731/1/012040
54. Reyes, L. H. (1984). Affective variables and mathematics education. The Elementary School Journal, 84(5), 558-581. Retrieved February 27, 2020 from https://www.jstor.org/stable/1001237
55. Sağlam, Y., & Dost, S. (2014). Preservice science and mathematics teachers’ beliefs about mathematical problem solving. Procedia-Social and Behavioral Sciences, 116, 303-306. https://doi.org/10.1016/j.sbspro.2014.01.212
56. Saka, E., & Durmuş, M. (2021). Sınıf öğretmeni adaylarının matematiksel problem çözmeye yönelik inançlarının ve metaforik algılarının incelenmesi [Investigation of prospective primary school teachers’ beliefs and metaforic perceptions related to mathematical problem solving]. Ekev Academic Review, 85, 129-150. https://doi.org/10.17753/Ekev1794
57. Schinck, A. G., Neale, H. W., Pugalee, D. K., & Cifarelli, V. V. (2008). Using metaphors to unpack student beliefs about mathematics. School Science and Mathematics, 108(7), 326-333. https://dx.doi.org/10.1111/j.1949-8594.2008.tb17845.x
58. Schoenfeld, A. H., & Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8(5), 484-494. https://doi.org/10.1037/0278-7393.8.5.484
59. Schommer-Aikins, M., Duell, O. K., & Hutter, R. (2005). Epistemological beliefs, mathematical problem-solving beliefs, and academic performance of middle school students. The Elementary School Journal, 105(3), 289-304. https://dx.doi.org/10.1086/428745
60. Sezgin-Memnun, D. (2015). Ortaokul öğrencilerinin matematik problemine ilişkin sahip oldukları metaforlar ve bu metaforların sınıf düzeylerine göre değişimi [Secondary school students’ metaphors about mathematical problem and change of metaphors according to grade levels]. Necatibey Faculty of Education, Electronic Journal of Science and Mathematics Education, 9(1), 351-374. https://dx.doi.org/10.17522/nefefmed.30643
61. Son, J. W., & Lee, M. Y. (2021). Exploring the relationship between preservice teachers’ conceptions of problem solving and their problem-solving performances. International Journal of Science and Mathematics Education, 19(1), 129-150. https://doi.org/10.1007/s10763-019-10045-w
62. Sturm, N., & Bohndick, C. (2021). The influence of attitudes and beliefs on the problem-solving performance. Frontiers in Education, 6, 525923. https://doi.org/10.3389/feduc.2021.525923
63. Thibodi, O. (2017). Metaphors for learning mathematics: An interpretation based on learners’ responses to an exploratory questionnaire on mathematics and learning. International Journal of Secondary Education, 5(6), 70-74. http://dx.doi.org/10.11648/j.ijsedu.20170506.11
64. Tsao, Y. (2004). A comparison of American and Taiwanese students: their math perception. Journal of Instructional Psychology, 31(3), 206-213. Retrieved July 17, 2020 from https://eric.ed.gov/?id=EJ774094
65. Turhan-Türkkan, B., & Yeşilpınar-Uyar, M. (2016). The metaphors of secondary school students towards the concept of “mathematical problem”. Çukurova University Faculty of Education Journal, 45(1), 99-130. Retrieved July 27, 2019 from https://dergipark.org.tr/tr/pub/cuefd/issue/27005/283888
66. Uygun, T., Gökkurt, B., & Usta, N. (2016). Üniversite öğrencilerinin matematik problemine ilişkin algılarının metafor yoluyla analiz edilmesi [Analysis of the perceptions of the university students about mathematics problem through metaphor]. Bartin University Journal of Faculty of Education, 5(2), 536-556. https://dx.doi.org/10.14686/buefad.v5i2.5000187677
67. van Merriënboer, J. J. G. (2013). Perspectives on problem solving and instruction. Computers & Education, 64(1), 153-160. https://dx.doi.org/10.1016/j.compedu.2012.11.025
68. Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.), Research ideas for the classroom: high school mathematics (pp. 57- 78). New York, NY: MacMillan. Retrieved May 7, 2020 from https://eric.ed.gov/?q=Charles+AND+W.L&id=ED404144
69. Wolodko, B. L., Willson, K. J., & Johnson, R. E. (2003). Preservice teachers’ perceptions of mathematics: Metaphors as a vehicle for exploring. Teaching Children Mathematics, 10(4), 224-229. Retrieved January 17, 2020 from https://link.gale.com/apps/doc/A111453661/AONE?u=anon~b7dea5df&sid=googleScholar&xid=e60ed9e5
70. Yee, S. P. (2017). Students’ and teachers’ conceptual metaphors for mathematical problem solving. School Science and Mathematics, 117(3-4), 146-157. https://dx.doi.org/10.1111/ssm.12217