Research Article

How can we help a student with Asperger syndrome to avoid the illusion of linearity?

Ioannis Rizos 1 * , Evaggelos Foykas 2
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1 Department of Mathematics, University of Thessaly, Lamia, GREECE2 Greek Secondary Education, Lamia, GREECE* Corresponding Author
Contemporary Mathematics and Science Education, 4(2), 2023, ep23021, https://doi.org/10.30935/conmaths/13404
Published Online: 18 June 2023, Published: 01 July 2023
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ABSTRACT

The choice of appropriate educational materials, visual or physical, for teaching mathematics to children on the autism spectrum, is an important topic in the context of inclusive education. In this article we first focus on some theoretical issues concerning the manipulative material, on the teaching of proportional quantities in the Greek high school and the difficulties it presents, as well as on the Asperger syndrome. We present the design, the implementation and the basic results of a teaching intervention conducted in a 14-year-old student with Asperger syndrome studying in the 8th grade of a high school in Greece on the use of manipulative materials for solving authentic real-world and word mathematical problems. The research showed that the student understood the differences between the real analogy and the illusion of linearity inherent in the problems he was given to solve using manipulatives. The paper concludes with suggestions, perspectives and limitations for teaching proportion problems and the tackle of the illusion of proportionality using manipulatives.

CITATION (APA)

Rizos, I., & Foykas, E. (2023). How can we help a student with Asperger syndrome to avoid the illusion of linearity?. Contemporary Mathematics and Science Education, 4(2), ep23021. https://doi.org/10.30935/conmaths/13404

REFERENCES

  1. Abassian, A., Safi, F., Bush, S., & Bostic, J. (2020). Five different perspectives on mathematical modeling in mathematics education. Investigations in Mathematics Learning, 12(1), 53-65. https://doi.org/10.1080/19477503.2019.1595360
  2. Abreu, R. L., Townsend, D., Mitchell, Y., Ward, J., Audette, L., & Gonzalez, K. A. (2022). LGBTQ qualitative and mixed methods research in counseling psychology: A content analysis. The Counseling Psychologist, 50(5), 708-737. https://doi.org/10.1177/00110000221092481
  3. Bakar, W. N. W., & Fauzi, F. Z. Y. (2019). Using origami to enhance visual and mathematical thinking skills among the autistics. International Journal of Education, Psychology and Counseling, 4(32), 72-78. https://doi.org/10.35631/IJEPC.432008
  4. Billingsley, B., & Bettini, E. (2019). Special education teacher attrition and retention: A review of the literature. Review of Educational Research, 89(5), 697-744. https://doi.org/10.3102/0034654319862495
  5. Bujak, K., R., Radu, I., Catrambone, R., MacIntyre, B., Zheng, R., & Golubski, G. (2013). A psychological perspective on augmented reality in the mathematics classroom. Computers & Education, 68, 536-544. https://doi.org/10.1016/j.compedu.2013.02.017
  6. Bury, S. M., Hedley, D., Uljarević, M., & Gal, E. (2020). The autism advantage at work: A critical and systematic review of current evidence. Research in Developmental Disabilities, 105, 103750. https://doi.org/10.1016/j.ridd.2020.103750
  7. Carr, M., & Seah, W. T. (2019). Mathematics education for students with autism spectrum disorder: Where are we now? Annales Universitatis Paedagogicae Cracoviensis: Studia and Didacticam Mathematicae Pertinentia, 10, 17-39. https://doi.org/10.24917/20809751.10.2
  8. Chiang, H.-M., & Lin, Y.-H. (2007). Mathematical ability of students with Asperger syndrome and high-functioning autism: A review of literature. Autism, 11(6), 547-556. https://doi.org/10.1177/1362361307083259
  9. Chico, A., Gómez, I., & Climent, N. (2022). Problem-solving by students with Asperger’s syndrome. In Proceedings of the 12th Congress of the European Society for Research in Mathematics Education.
  10. Christodoulou, T. (2022). Why are the students unable to distinguish the proportional from the non-proportional mathematical situations? A review of the relative research on the illusion of proportionality. Mediterranean Journal for Research in Mathematics Education, 19, 37-54.
  11. Coskun, M., Hajdini, A., Alnak, A., & Karayagmurlu, A. (2020). Internet use habits, parental control and psychiatric comorbidity in young subjects with Asperger syndrome. Journal of Autism and Developmental Disorders, 50(1), 171-179. https://doi.org/10.1007/s10803-019-04243-2
  12. Cox, S. K., & Root, J. R. (2020). Modified schema-based instruction to develop flexible mathematics problem-solving strategies for students with autism spectrum disorder. Remedial and Special Education, 41(3), 139-151. https://doi.org/10.1177/0741932518792660
  13. Daher, W. M. (2014). Manipulatives and problem situations as escalators for students’ geometric understanding: A semiotic analysis. International Journal of Mathematical Education in Science and Technology, 45(3), 417-427. https://doi.org/10.1080/0020739X.2013.837527
  14. 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, 311-334. https://doi.org/10.1023/A:1021205413749
  15. 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. https://doi.org/10.1023/A:1003151011999
  16. De Giambattista, C., Ventura, P., Trerotoli, P., Margari, M., Palumbi, R., & Margari, L. (2019). Subtyping the autism spectrum disorder: Comparison of children with high functioning autism and Asperger syndrome. Journal of Autism and Developmental Disorders, 49, 138-150. https://doi.org/10.1007/s10803-018-3689-4
  17. Duma, L. (2021). The groundless use of linearity in daily thinking and decision-making. Periodica-Polytechnica Social and Management Sciences, 29(2), 125-135. https://doi.org/10.3311/PPso.14900
  18. Fauziyah, N., Budayasa, I. K., & Juniati, D. (2022). Cognition processes of ASD students: Recommendations for mathematics teaching and learning process. International Journal of Instruction, 15(3), 805-830. https://doi.org/10.29333/iji.2022.15344a
  19. Gasser, B. A., Kurz, J., Dick, B., & Mohaupt, M. G. (2019). Steroid metabolites support evidence of autism as a spectrum. Behavioral Science (Basel), 9, 52. https://doi.org/10.3390/bs9050052
  20. Goldin, G. A. (2020). Mathematical representations. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-030-15789-0_103
  21. Gordon, N., Hohwy, J., Davidson, M. J., van Boxtel, J., & Tsuchiya, N. (2019). From intermodulation components to visual perception and cognition–A review. Neuroimage, 199, 480-494. https://doi.org/10.1016/j.neuroimage.2019.06.008
  22. Gunardi, O. J., Prasetio, O., & Kamadjaja, D. B. (2019). Oral surgery management in Asperger syndrome: A case report. Acta Medica Philippina, 53(6). https://doi.org/10.47895/amp.v53i6.685
  23. Haji Ismail, N. F., Shahrill, M., & Asamoah, D. (2023). Learning through virtual manipulatives: Investigating the impact of Gizmos-based lessons on students’ performance in integers. Contemporary Mathematics and Science Education, 4(1), ep23009. https://doi.org/10.30935/conmaths/12857
  24. Hurst, C., & Linsell, C. (2020). Manipulatives and multiplicative thinking. European Journal of STEM Education, 5(1), 04. https://doi.org/10.20897/ejsteme/8508
  25. Jitendra, A. K., Harwell, M. R., Im, S.-H., Karl, S. R., & Slater, S. C. (2019). Improving student learning of ratio, proportion, and percent: A replication study of schema-based instruction. Journal of Educational Psychology, 111(6), 1045-1062. https://doi.org/10.1037/edu0000335
  26. Knott, E., Rao, A. H., Summers, K., & Teeger, C. (2022). Interviews in the social sciences. Nature Reviews Methods Primers, 2, 73. https://doi.org/10.1038/s43586-022-00150-6
  27. Kohen, Z., & Orenstein, D. (2021). Mathematical modeling of tech-related real-world problems for secondary school-level mathematics. Educational Studies in Mathematics, 107, 71-91. https://doi.org/10.1007/s10649-020-10020-1
  28. Krawitz, J., & Schukajlow, S. (2020). When can making a drawing hinder problem solving? Effect of the drawing strategy on linear overgeneralizations and problem solving. Frontiers in Psychology, 11, 506. https://doi.org/10.3389/fpsyg.2020.00506
  29. Lannin, J., Barker, D., & Townsend, B. (2006). Algebraic generalisation strategies: Factors influencing student strategy selection. Mathematics Education Research Journal, 18(3), 3-28. https://doi.org/10.1007/BF03217440
  30. Larkin, K. (2016). Mathematics education and manipulatives: Which, when, how? Australian Primary Mathematics Classroom, 21(1), 12-17.
  31. Laski, E. V., Jordan, J. R., Daoust, C., & Murray, A. K. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. SAGE Open, 5(2), 1-8. https://doi.org/10.1177/2158244015589588
  32. Lindner, K. T., & Schwab, S. (2020). Differentiation and individualisation in inclusive education: A systematic review and narrative synthesis. International Journal of Inclusive Education. https://doi.org/10.1080/13603116.2020.1813450
  33. Mattila, M. L., Kielinen, M., Jussila, K., Linna, S. L., Bloigu, R., Ebeling, H., & Moilanen, I. (2007). An epidemiological and diagnostic study of Asperger syndrome according to four sets of diagnostic criteria. Journal of the American Academy of Child and Adolescent Psychiatry, 46(5), 636-646. https://doi.org/10.1097/chi.0b013e318033ff42
  34. McConkey, R. (2020). The rise in the numbers of pupils identified by schools with autism spectrum disorder (ASD): A comparison of the four countries in the United Kingdom. Support for Learning, 35, 132-134. https://doi.org/10.1111/1467-9604.12296
  35. Meke, K. D. P., Jailani, J., Wutsqa, D. U., & Alfi, F. D. (2019). Problem based learning using manipulative materials to improve student interest of mathematics learning. Journal of Physics: Conference Series, 1157, 032099. https://doi.org/10.1088/1742-6596/1157/3/032099
  36. Millard, C., & Callard, F. (2020). Thinking in, with, across, and beyond cases with John Forrester. History of Human Sciences, 33(3-4), 3-14. https://doi.org/10.1177/0952695120965403
  37. Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75-92. https://doi.org/10.1080/01443410601061462
  38. Modestou, M., & Gagatsis, A. (2008). Proportional reasoning in elementary and secondary education: Moving beyond the percentages. In A. Gagatsis (Ed.), Research in mathematics education (pp. 147-162). University of Cyprus.
  39. Modestou, M., Elia, I., Gagatsis, A., & Spanoudis, G. (2008). Behind the scenes of pseudo-proportionality. International Journal of Mathematical Education in Science and Technology, 39(3), 313-324. https://doi.org/10.1080/00207390701691541
  40. Nilholm, C. (2021). Research about inclusive education in 2020–How can we improve our theories in order to change practice? European Journal of Special Needs Education, 36(3), 358-370. https://doi.org/10.1080/08856257.2020.1754547
  41. O’Meara, N., Johnson, P., & Leavy, A. (2020). A comparative study investigating the use of manipulatives at the transition from primary to post-primary education. International Journal of Mathematical Education in Science and Technology, 51(6), 835-857. https://doi.org/10.1080/0020739X.2019.1634842
  42. Peklari, E. (2019). Mathematical skills in autism spectrum disorder. Asian Journal of Applied and Technology, 3(1), 111-123.
  43. Peltier, C., Morin, K. L., Bouck, E. C., Lingo, M. E., Pulos, J. M., Scheffler, F. A., Suk, A., Mathews, L. A., Sinclair, T. E., & Deardorff, M. E. (2020). A meta-analysis of single-case research using mathematics manipulatives with students at risk or identified with a disability. The Journal of Special Education, 54(1), 3-15. https://doi.org/10.1177/0022466919844516
  44. Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press. https://doi.org/10.1515/9781400828678
  45. Putarek, V., Vlahović-Štetić, V. (2019). Metacognitive feelings, conflict detection and illusion of linearity. Psihologijske Teme [Psychological Topics], 28(1), 171-192. https://doi.org/10.31820/pt.28.1.9
  46. Ramberg, J., & Watkins, A. (2020). Exploring inclusive education across Europe: Some insights from the European agency statistics on inclusive education. FIRE: Forum for International Research in Education, 6(1), 85-101. https://doi.org/10.32865/fire202061172
  47. Rizos, I., & Adam, M. (2022). Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), em0686. https://doi.org/10.29333/iejme/11977
  48. Rizos, I., & Foykas, E. (2023). Utilization of “Byrne’s Euclid” in the Teaching of Geometry to Students with Special Learning Difficulties: A Qualitative Research. European Journal of Education and Pedagogy, 4(2), 139-148. https://doi.org/10.24018/ejedu.2023.4.2.623
  49. Rizos, I., Kolokotronis, G., & Papanikolaou, A. M. (2023). Investigating the effectiveness of Flipped Classroom model in a Mathematics Education course in Greece. Journal of Mathematics and Science Teacher, 3(1), em021. https://doi.org/10.29333/mathsciteacher/12608
  50. Rourke, B., & McGloin, R. (2019). A different take on the Big Bang theory: Examining the influence of Asperger traits on the perception and attributional confidence of a fictional TV character portraying traits of Asperger syndrome. Atlantic Journal of Communication, 27(2), 127-138. https://doi.org/10.1080/15456870.2019.1574797
  51. Shin, M., Ok, M. W., Kang, E. Y., & Bryant, D. P. (2019). Korean elementary school teachers’ implementation of mathematics instruction for students struggling to learn mathematics in inclusive settings. Journal of Research in Special Educational Needs, 19(2), 145-157. https://doi.org/10.1111/1471-3802.12437
  52. Sulistyaningsih, D., Mawarsari, V. D., Hidayah, I., & Dwijanto (2017). Manipulatives implementation for supporting learning of mathematics for prospective teachers. Journal of Physics: Conference Series, 824, 012047. https://doi.org/10.1088/1742-6596/824/1/012047
  53. Swan, P., & Marshall, L. (2010). Revisiting mathematics manipulative materials. Australian Primary Mathematics Classroom, 15(2), 13-19.
  54. Tabatabaei, S. H., Shahrokhi, H., Gholipour, K., Iezadi, S., Rezapour, R., Naghibi, D., & Azami-Aghdash, S. (2022). The characteristics and results of parent training interventions in children with Autism spectrum disorder: A systematic review. Iran Journal Public Health, 51(3), 518-530. https://doi.org/10.18502/ijph.v51i3.8927
  55. Testoni, I., Pesci, S., De Vincenzo, C., Dal Corso, L., & Zamperini, A. (2019). Work and spirituality among people with Asperger syndrome: An exploratory study. Journal of Disability & Religion, 23, 178-196. https://doi.org/10.1080/23312521.2019.1580174
  56. Uddin, L. Q. (2022). Exceptional abilities in autism: Theories and open questions. Current Directions in Psychological Science, 31(6), 509-517. https://doi.org/10.1177/09637214221113760
  57. UNICEF. (2017). Inclusive education: Understanding article 24 of the rights of persons with disabilities. https://www.unicef.org/eca/sites/unicef.org.eca/files/IE_summary_accessible_220917_0.pdf
  58. Valori, G., Giacomone, B., Albanese, V., & Adamuz-Povedano, N. (2022). Approaching Euclidean proofs through explorations with manipulative and digital artifacts. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2022.2055503
  59. 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, 485-501. https://doi.org/10.1016/j.learninstruc.2004.06.019
  60. Vanluydt, E., Verschaffel, L., & Van Dooren, W. (2022). The role of relational preference in word-problem solving in 6-to 7-year-olds. Educational Studies in Mathematics, 110, 393-411. https://doi.org/10.1007/s10649-021-10139-9
  61. Verschaffel, L., Schukajlow, S., Star, J., & Van Dooren, W. V. (2020). Word problems in mathematics education: A survey. ZDM Mathematics Education, 52, 1-16. https://doi.org/10.1007/s11858-020-01130-4
  62. Vicedo, M., & Ilerbaig, J. (2021). Autism in Baltimore, 1938-1943. Journal of Autism and Developmental Disorders, 51, 1157-1172. https://doi.org/10.1007/s10803-020-04602-4
  63. Vos, P. (2018). How real people really need mathematics in the real world–Authenticity in mathematics education. Education Sciences, 8(4), 195. https://doi.org/10.3390/educsci8040195
  64. Wester, J. S. (2020). Students’ possibilities to learn from group discussions integrated in whole-class teaching in mathematics. Scandinavian Journal of Educational Research, 65(6), 1020-1036. https://doi.org/10.1080/00313831.2020.1788148
  65. Wright, B., Spikins, P., & Pearson, H. (2020). Should autism spectrum conditions be characterised in a more positive way in our modern world? Medicina, 56(5), 233. https://doi.org/10.3390/medicina56050233
  66. Zippert, E. L., Eason, S. H., Marshall, S., & Ramani, G. B. (2019). Preschool children’s math exploration during play with peers. Journal of Applied Developmental Psychology, 65, 101072. https://doi.org/10.1016/j.appdev.2019.101072