Developing a Number Sense-Based Instructional Design to Eliminate Student Errors Based on Mathematical Misconceptions

Year 2022, Volume: 6 Issue: 2, 493 - 522, 31.12.2022

Pelin Üredi

Abstract

The purpose of this research was to develop and evaluate a number sense-based curriculum aiming to eliminate student errors due to mathematical misconceptions. A mixed method research model as one of the qualitative research approaches was adopted in this study aiming to develop a number sense-based instructional design to eliminate student errors arisen from mathematical misconceptions in secondary school students. In preparation stage of the instructional design, 27 seventh grade students studying at a state school in Mersin province were determined as the study group. At the stage of developing and evaluating the instructional design, 24 fifth grade students studying at a state school in Mersin province were determined as the study group. In the process of developing the instructional design, "Needs Analysis Student Interview Form," "Instructional Design Teachers’ Board Interview Form," and "Expert Interview Form" were used. When all these results were evaluated together, it would be correct to say that number sense was an important prerequisite for the interpretation and use of mathematical knowledge, and it would be efficient on eliminating errors arisen from the mathematical misconceptions when it was integrated into an effective instructional design.It was possible to suggest that further researches should be conducted for determining the misconceptions of the students related to different mathematics sub-topics.

Keywords

Number sense, mathematical misconceptions, instructional design, mixed method research, secondary school students.

References

• Akay, Y. 2017. İlkokul 4.sınıf matematik dersinde oluşturulan öğretim tasarımına dayalı uygulamaların etkililiği. [The Efficiency of Implementations Based on Instructional Design Created in Primary School 4th Grade Mathematics Lesson] Doctoral dissertation. Balıkesir University, Balıkesir. https://hdl.handle.net/20.500.12462/3720
• Akkaya, S. (2018). İlkokul dördüncü sınıf matematik dersinde geometri alt öğrenme alanlarına ilişkin kavram yanılgılarının giderilmesinde oyun temelli öğretimin etkisi. [The Effect of Game Based Activities in Improving the Concept of Geometry Sub-learning Areas of Primary School Fourth Class Mathematics Course] Doctoral dissertation. İnönü University, Malatya. Retrieved from http://hdl.handle.net/11616/11379
• Arnas, Y.A. (2005). Okul öncesi dönemde matematik eğitimi. [Pre-school Mathematics Education]. Nobel Publishing.
• Altıparmak, K. & Özüdoğru, M. (2015). Hata ve kavram yanılgısı: Kesir ve parça-bütün ilişkisi. [Error and misconception: Relation of fraction and part-whole] International Journal of Human Sciences, 12(2), 1465-1483. https://doi.org/10.14687/ijhs.v12i2.3404
• Ang, L. H. & Shahrill, M. (2014). Identifying students’ specific misconceptions in learning probability. International Journal of Probability and Statistics, 3(2), 23-29. http://article.sapub.org/10.5923.j.ijps.20140302.01.html
• Apino, E. & Retnawati, H. (2017). Developing instructional design to improve mathematical higher order thinking skills of students. In Journal of Physics: Conference Series, 812(1), 1-7. https://iopscience.iop.org/article/10.1088/1742-6596/812/1/012100/pdf
• Avcu, Y. E. & Er, K. O. (2020). Developing an ınstructional design for the field of ıct and software for gifted and talented students. International Journal of Educational Methodology, 6(1), 161-183. https://pdf.ijem.com/IJEM_6_1_161.pdf
• Baki, A. & Bell, A. (1997). Ortaöğretim matematik öğretimi. [Secondary School Mathematics Teaching]. YÖK Publishing.
• Başkurt, H. (2011). İlköğretim 6, 7 ve 8. sınıf öğrencilerinin nokta, doğru ve düzlem kavramlarını algılama düzeyleri ve kavram yanılgıları. [Primary School the 6th, 7th and 8th Grade Students’ Perceptions and Misconceptions on Point, Line and Plane Concept] Unpublished Master Thesis. Erzincan University, Erzincan. https://dergipark.org.tr/tr/download/article-file/187979
• Bennison, A., Goos, M. & Geiger, V. (2020). Utilising a research-informed instructional design approach to develop an online resource to support teacher professional learning on embedding numeracy across the curriculum. ZDM, 1-15. https://doi.org/10.1007/s11858-020-01140-2
• Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339. https://doi.org/10.1177/00222194050380040901
• Bilgin, İ. & Geban, Ö. (2001). Benzeşim (analoji) yöntemi kullanarak lise 2. sınıf öğrencilerinin kimyasal denge konusundaki kavram yanılgılarının giderilmesi. [The use of anology to remove 10th grade students’ misconception related to chemical equilibrium concepts] Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 20(20), 26-32. http://efdergi.hacettepe.edu.tr/yonetim/icerik/makaleler/1013-published.pdf
• Brinkmann, A. (2003). Graphical knowledge display- mind mapping and concept mapping as efficient tools in mathematics education. Mathematics Education Review, 16(4), 35-48. https://www.researchgate.net/publication/228703184_Graphical_Knowledge_Display-Mind_Mapping_and_Concept_Mapping_as_Efficient_Tools_in_Mathematics_Education
• Büyüköztürk, Ş. (2006). Sosyal Bilimler Için Veri Analizi El Kitabı [Manual of Data Analysis for Social Science] (6th ed.). Pegem A Publishing.
• Çetin, İ. (2009). 7. ve 9. Sınıf öğrencilerinin oran ve orantı konusundaki kavram yanılgıları. [Seventh and Nineth Grade Students Misconceptions About Ratio and Proportion] Master Thesis. Selçuk University, Konya. http://hdl.handle.net/123456789/37
• Dağlı, H. (2010). İlköğretim beşinci sınıf öğrencilerinin çevre, alan ve hacim konularına ilişkin kavram yanılgıları. [Misconceptions of Elementary School Students in 5th Grades About the Perimeter, Area and Volume Concepts] Unpublished Master Thesis. Kocatepe University, Afyon. http://hdl.handle.net/11630/3124
• Dehaene, S. (2001). Précis of the number sense. Mind & language, 16(1), 16-36. https://doi.org/10.1111/1468-0017.00154
• Doğan, F.S. (2013). Geometri dersi uzay konusunda 12.sınıf öğrencilerinin hata ve kavram yanılgılarının belirlenmesi. [Identification of Misconceptions of 12th Grade Students in Space] Unpublished Master Thesis. Balıkesir University, Balıkesir. https://hdl.handle.net/20.500.12462/2486
• Doyuran, G. (2014). Ortaokul öğrencilerinin temel geometri konularında sahip oldukları kavram yanılgıları. [Secondary School Students’ Misconceptions on Based Terms in Geomerty] Unpublished Master Thesis. Dokuz Eylül University, İzmir Durkin, K. & Johnson, B.R. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37(3), 21-29. https://asset-pdf.scinapse.io/prod/2020357307/2020357307.pdf
• Erbay, H. N. (2016). 6. Sınıf öğrencilerinin açılar konusundaki kavram bilgilerinin incelenmesi. [An investigation of the 6th grade students’ concept knowledge about angles] Akademik Sosyal Araştırmalar Dergisi, 4(36), 704-718.
• Eryılmaz, A. & Sürmeli, E. (2002, September 5-7). Üç-aşamalı sorularla öğrencilerin ısı ve sıcaklık konularındaki kavram yanılgılarının ölçülmesi [Temperature and temperature of students with three-stage questions measurement of misconceptions on the subjects][Conference presentation]. V. Ulusal Fen Bilimleri ve Matematik Eğitim Kongresi, Ankara, Turkey. https://users.metu.edu.tr/eryilmaz/TamUcBaglant.pdf
• Fer, S. (2009). Öğretim Tasarımı [Instructional Design]. Anı Publishing.
• Filiz, A. & Moralı, S. (2020). Öğrencilerin sayı duyusu stratejilerinin sınıf düzeyi ve sayı duyusu bileşenlerine göre analizi. [An analysis of students’ number sense strategies in terms of their grades and number sense components]. Manisa Celal Bayar Üniversitesi Eğitim Fakültesi Dergisi, 8(2), 39-62. https://dergipark.org.tr/en/download/article-file/1156935
• Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain source. Journal for Research in Mathematics Education, 22(3), 170–218. https://www.jstor.org/stable/749074 Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 6–11. https://www.jstor.org/stable/i40053564
• Jackson, S. L. (2008). Research methods a modular approach. Thomson Wadsworth.
• Jankvist, U. T. & Niss, M. (2018). Counteracting destructive student misconceptions of mathematics. Education Sciences, 8(2), 53-70. https://doi.org/10.3390/educsci8020053
• Jin, H. & Wong, K. Y. (2010). Training on concept mapping skills in geometry. Journal of Mathematics Education, 3(1), 104-119. Kabapinar, F. (2005). Effectiveness of Teaching via Concept Cartoons from the Point of View of Constructivist Approach. Educational Sciences: Theory & Practice, 5(1). https://www.researchgate.net/publication/265411400_Effectiveness_of_Teaching_via_Concept_Cartoons_from_the_Point_of_View_of_Constructivist_Approach
• Kaminski, E. (2002). Promoting mathematical understanding: Number sense in action. Mathematics Education Research Journal, 14(2), 133–149. https://link.springer.com/article/10.1007/BF03217358 Khodabandelou, R. & Abu Samah, S. A. (2012). Instructional Design Models For Online İnstruction: From the Perspective of Iranian Higher Education. ProcediaSocial and Behavioral Sciences, 67, 545-552. https://www.researchgate.net/publication/257718374_Instructional_Design_Models_for_Online_Instruction_From_the_Perspective_of_Iranian_Higher_Education Kiriş, B. (2008). İlköğretim altıncı sınıf öğrencilerinin 'nokta, doğru, doğru parçası, ışın ve düzlem' konularında sahip oldukları kavram yanılgıları ve bu yanılgı nedenlerinin belirlenmesi. [Determining elementary sixth grades’ misconceptions in points, lines, line segments and planets and the reason underlying these misconceptions]. Unpublished Master Thesis, Adnan Menderes University, Aydın. https://tez.yok.gov.tr/UlusalTezMerkezi/tezDetay.jsp?id=D3o6H8dGFjo7MAmLr8laWg&no=TF7kmkfCKhJI9iIPw85Mlw
• McIntosh, A., Reys, B. J. & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2–9. https://www.jstor.org/stable/40248053 Ministry of National Education. (2018). Milli Eğitim Bakanlığı veliler için öğretim programlarında çocuğunuzu ne bekliyor? [What does the Ministry of National Education expert your child in the curriculum for parents?]. Board of Education and Discipline Publishing. http://mufredat.meb.gov.tr/ Miles, M. & Huberman, A.M. (1994). Qualitative data analysis. Thousand Oaks, CA.
• Mohamed, M. & Johnny, J. (2010). Investigating Number Sense Among Students. Procedia-Social and Behavioral Sciences, 8, 317-324. https://doi.org/10.1016/j.sbspro.2010.12.044
• Morrison, G. R., Ross, S. M. & Kemp, J. E. (2012). Etkili öğretim tasarımı [Effective Instructional Design]. (Ed. İlhan Varank). Bahçeşehir Publishing.
• National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and evaluation standards for school mathematics. VA: NCTM.
• Ojose, B. (2015). Students' misconceptions in mathematics: Analysis of remedies and what research says. Ohio Council of Teachers of Mathematics, 72, 30-34. https://kb.osu.edu/bitstream/handle/1811/78927/OJSM_72_Fall2015_30.pdf?sequence=1
• Osborne, R. J. & Gilbert, J. (1980). A method for the investigation of concept understanding in science. European Journal of Science Education, 2(3), 311-321. https://doi.org/10.1080/0140528800020311
• Özdemir, E. & Uyangör Mert, S. (2011). Matematik eğitimi için bir öğretim tasarımı modeli. [The instructional design model for mathematics education] e-Journal of New World Sciences Academy, 6(2), 1786-1796. https://dergipark.org.tr/tr/pub/nwsaedu/issue/19820/212100
• Özkan, M. (2015). 7. sınıf öğrencilerinin çokgenlerde ve özel dörtgenlerde yaptıkları kavram yanılgılarının incelenmesi. [Analysing the Misconceptions That Students Have About Polygon and Special Quadrilaterals]. Unpublished Master Thesis, Çukurova University, Adana. https://tez.yok.gov.tr/UlusalTezMerkezi/tezDetay.jsp?id=bqVEfW_wAAF91C-QiqbDeQ&no=WFX_o-i77ONTxcMGcIYs-Q
• Paul, M. & Hlanganipai, N. (2014). The nature of misconceptions and cognitive obstacles faced by secondary school mathematics students in understanding probability: A case study of selected Polokwane secondary schools. Mediterranean Journal of Social Sciences, 5(8), 446-455. https://www.researchgate.net/publication/288222635_The_Nature_of_Misconceptions_and_Cognitive_Obstacles_Faced_by_Secondary_School_Mathematics_Students_in_Understanding_Probability_A_Case_Study_of_Selected_Polokwane_Secondary_Schools
• Pesen, C. (2008). Kesirlerin sayı doğrusu üzerindeki gösteriminde öğrencilerin öğrenme güçlükleri ve kavram yanılgıları. [Students’ Learning Difficulties and Misconceptions in Pointing the Fractions on the Number Line] İnönü Üniversitesi Eğitim Fakültesi Dergisi, 9(15), 157–168. https://dergipark.org.tr/tr/pub/inuefd/issue/8708/108722
• Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8-27. https://doi.org/10.2307/749095
• Reys, R. E. & Yang, D. C. (1998). Relationship between computational performance and number sense among sixth- and eighth- grade students in Taiwan. Journal for Research in Mathematics Education, 29(2), 225–237. https://doi.org/10.2307/749900
• Sarwadi, H. R. H. & Shahrill, M. (2014). Understanding students’ mathematical errors and misconceptions: The case of year 11 repeating students. Mathematics Education Trends and Research, 1, 1-10. https://www.academia.edu/31477961/Understanding_Students_Mathematical_Errors_and_Misconceptions_The_Case_of_Year_11_Repeating_Students Singh, P. (2009). An assessment of number sense among secondary school students. International Journal For Mathematics Teaching And Learning, 155, 1-29. https://eric.ed.gov/?id=EJ904871 Smith III, J. P., DiSessa, A. A. & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115-163. https://doi.org/10.1207/s15327809jls0302_1
• Sowder, J. T. (1992). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics, 371–389, Macmillan Publishing Co, Inc. https://psycnet.apa.org/record/1992-97586-016
• Sowder, J. & Schappelle, B. (1994). Number sense-making. Arithmetic Teacher, 342–345. https://eric.ed.gov/?id=EJ482274
• Soylu, Y. & Soylu, C. (2005). Learning difficulties of 5fh class in primary education at fraction: Ordering, adding, subtraction, multiplication in fraction and problems related to fraction. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 7(2), 101-117. https://www.semanticscholar.org/paper/Learning-Difficulties-Of-5fh-Class-In-Primary-At-In-Soylu-Soylu/ab35ea9fc396f72f30ddf44bf151b3efed7c022b
• Stefanich, G.P. & Rokusek, T. (1992). An analysis of computational errors in the use of division algorithms by fourth-grade students. School Science and Mathematics, 92(4), 201- 205. https://doi.org/10.1111/j.1949-8594.1992.tb12173.x
• Şen, H. Ş. & Erişen, Y. (2002). Öğretmen yetiştiren kurumlarda öğretim elemanlarının etkili öğretmenlik özellikleri. [Effective Teaching Specialities of Teacher Trainers Working at Teacher Training Institutions]. G.Ü. Gazi Eğitim Fakültesi Dergisi, 22(1), 99-116. https://dergipark.org.tr/tr/download/article-file/77439 Umay, A. (2003). Okul öncesi öğretmen adaylarının matematik öğretmeye ne kadar hazır olduklarına ilişkin bazı ipuçları. [Some Clues on How Much Preschool Teacher Candidates Ready to Teach Mathematics]. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 25, 194-203. https://search.trdizin.gov.tr/yayin/detay/3090/
• Ülgen, G. (2004). Kavram geliştirme, kuramlar ve uygulamalar [Concept Development, Theories and Applications]. (4. Ed.). Nobel Publishing.
• Yang, D. C. (1995). Number sense performance and strategies possessed by sixth and eighth grade students in Taiwan. Unpublished PhD Thesis, University of Missouri-Columbia. https://www.worldcat.org/title/number-sense-performance-and-strategies-possessed-by-sixth-and-eighth-grade-students-in-taiwan/oclc/690115356
• Yang, D. C. & Li, M. F. (2008). An investigation of 3rd grade Taiwanese students’performance in number sense. Educational Studies, 34(5), 443-455. https://www.tandfonline.com/doi/abs/10.1080/03055690802288494 Yıldırım, A. & Şimşek, H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri [Qualitative Research Methods in the Social Sciences]. (9th Ed.). Seçkin Publishing.
• Yılmaz, S. (2011). 7. sınıf öğrencilerinin 'doğrular ve açılar' konusundaki hata ve kavram yanılgılarının Van Hiele geometri anlama düzeyleri açısından analizi. [The Distribution of Misconceptions of that 7th Grade Students on the Subject “Line and Angle”According to Van Hiele Geometric Thinking Level]. Unpublished Master Thesis, Kastamonu University, Kastamonu. https://acikbilim.yok.gov.tr/handle/20.500.12812/100767 Zembat, I. (2010). Kavram Yanılgısı Nedir? [What is Misconception?]. In: Özmantar, M. F., Bingölbali, E. and Akkoç, H. (eds.) Matematiksel kavram yanılgıları ve çözüm önerileri. Pegem Publishing.