#### Prospective elementary and secondary school mathematics teachers’ statistical reasoning

Rabia KARATOPRAK , Gülseren KARAGÖZ AKAR, Bengü BÖRKAN

This study aims to identify errors made by primary school students when modelling word problems and to eliminate those errors through scaffolding. A 10-question problem-solving achievement test was used in the research. The qualitative and quantitative designs were utilized together. The study group of the quantitative design comprises 248 elementary 4th grade students attending nine classes at three state schools in the city centre of Kütahya, chosen with the cluster sampling method. Frequency analysis and discriminant analysis were performed to analyse the quantitative data. The qualitative data were collected through clinical interviewing. The study group with whom the clinical interviews were performed comprises 30 primary school students in the class closest to the average problem-solving achievement among the nine classes. As a result, it was observed that most of the errors made by the students were caused by the use of the number operator model, which was followed by incorrect relations, number consideration, missing critical information, an inability to determine structure and relation and incorrect diagrams. The discriminant analysis shows that the biggest contribution to discriminating between students with high and low levels of modelling achievement is made by errors originating from using the number operator model, and this type of error is followed by incorrect relations, an inability to determine structure and relation and number consideration models respectively. It was concluded that errors originating from missing critical information are mostly made by successful students and the ratio of errors originating from incorrect diagrams does not affect the distinction between successful and unsuccessful students. The research also found that the modelling cycle of students does not benefit from the interpretation and validation stages. Finally, it was seen that more than half of errors made during modelling can be corrected through scaffolding.

Word problems, modelling cycle, Error analysis and scaffolding.

Paper Details | |
---|---|

Topic | Mathematics Education |

Pages | 553 - 580 |

Issue | IEJEE, Volume 9, Issue 3 |

Date of acceptance | 13 March 2017 |

Read (times) | 83 |

Downloaded (times) | 54 |

Akay, H., Soybaş, D., & Argün, Z. (2006). Problem kurma deneyimleri ve matematik öğretiminde açık-uçlu soruların kullanımı [Problem posing experiences and using open ended questions in math education] . *Kastamonu Eğitim Dergisi, 14*(1), 129 -146.

Allen, J. (1985). Inferential comprehension: The effects of text source, decoding ability, and mode. *Reading Research Quarterly,* *20*, 603–615.

Altun, M. (2007). *Orta öğretimde Matematik Öğretimi*. [Mathemetics education in secondary school.] Bursa: Aktüel Alfa Bas. Yay.

Andersson, U. (2010). Skill Development in Different Components of Arithmetic and Basic Cognitive Functions: Findings From a 3-Year Longitudinal Study of Children with Different Types of Learning Difficulties. *Journal of Educational Psychology*, *102*(1), 115–134

Baki, A & Kartal, T (2004). Kavramsal Ve İşlemsel Bilgi Bağlamında Lise Öğrencilerinin Cebir Bilgilerinin Değerlendirilmesi [Characterize the nature of high school students’ algebra knowledge in terms of procedural and conceptual knowledge]. *Türk Eğitim Bilimleri Dergisi, 2*(1), 27-46.

Barbey, A. K. & Barsalou, L. W. (2009). Reasoning and problem solving: Models. In L. Squire (Ed.), *Encyclopedia of Neuroscience*, (pp. 35-43). Oxford: Academic Press.

Baroody, A. J., Feil, Y. & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. *Journal for Research in Mathematics Education, 38*, 115-131.

Blum, W. (2015). Quality Teaching of Mathematical Modelling: What Do We Know, What Can We Do? In S.J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education, DOI 10.1007/978-3-319-12688-3_9

[14] Blum, W., & Ferri, R. B. (2009). Mathematical modelling: Can it be taught and learnt?. *Journal of Mathematical Modelling and Application, 1*(1), 45-58.

Blum, W. & Leiß, D. (2006). “Filling up” – The problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In M. Bosch (Ed.), *CERME-4 –Proceedings of the Fourth Conference of the European Society for Research in Mathematics Education.*Guixol.

Blum, W. & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects – state, trends and issues in mathematics instruction. *Educational Studies in Mathematics, *22(1), 37-68.

Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. *ZDM, *38 (2), 86-95

Borromeo Ferri, R. (2007). Modelling problems from a cognitive perspective. In Haines, C. *et al.* (Eds.), *Mathematical Modelling: Education, Engineering and Economics* (pp. 260-270). Chichester: Horwood.

Borromeo Ferri, R. & Blum, W. (2009). Insight into Teachers’ Unconscious Behaviour in Modelling Contexts. In Lesh, R. *et al.* (Eds.), *Modeling Students’ Mathematical Modeling Competencies.* (pp. 423-432). New York: Springer.

Borromeo Ferri, R. & Blum, W. (2010). Mathematical Modelling in Teacher Education –Experiences from a Modelling Seminar. In Durand-Guerrier, V., Soury-Lavergne, S. &Arzarello, F. (Eds.), *CERME-6 – Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education. *(pp. 2046-2055). INRP, Lyon 2010.

Byrman, A. (2006). Mixed methods: A four-volume set. Thousands Oaks, CA: Sage.

Büyüköztürk , S. (2006). Sosyal bilimler için veri analizi el kitabı [Data Analyzing Handbook for Social Science] (5th Ed.). Ankara: Pegem Yayıncılık.

Büyüköztürk, Ş. & Çokluk-Bökeoğlu, Ö. (2008). Discriminant analysis: Concept and application. *Eurasian Journal of Educational Research,* *33*, 73-92.

Byrnes, J. P. &Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. *Developmental Psychology, 27*, 777–786

Çelik, A. & Özdemir, E. Y. (2011). İlköğretim öğrencilerinin orantısal akıl yürütme becerileri ile oran-orantı problem kurma becerileri arasındaki ilişki. *Pamukkale Üniversitesi Eğitim Fakültesi Dergisi,* *30*(1), 1-11.

Çetin, H. &Ertekin, E. (2011). The relationship between eighth grade primary school students’ proportional reasoning skills and success in solving equations. *International Journal of Instruction,* *4*(1), 47-62.

Charles, R. I. & Lester, F. K. (1984). An evaluation of a process-oriented instructional program in mathematical problem solving in grades 5 and 7. *Journal for Research in Mathematics Education, 15*, 15-34.

Chikalanga, I. (1992). A Suggested Taxonomy of Inferences for Reading Teachers. *Reading in a Foreign Language,* *8*(2), 697-709

Clarkson, P. C. (1991). Language comprehension errors: A further investigation. *Mathematics Education Research Journal, 3*(2), 24-33.

Clements, M. A. &Ellerton, N. (1996). *The Newman Procedure for Analysing Errors on Written Mathematical Tasks*. (Retrieved 17 August 2008 from http://users.tpg.com.au /arnold10/PAGES/newman.htm)

Cummins, D. D., Kintsch, W., Reusser, K. & Weimer, R. (1988). The role of understanding in solving word problems. *Cognitive Psychology, 20*, 405-438.

Creswell, J. W. (2014). Research design: Qualitative, quantitative, and mixed methods approaches. (4th Edition). Thousand Oaks: Sage.

Eraslan, A. & Kant, S. (2015). Modeling processes of 4th-year middle-school students and the difficulties encountered. *Educational Sciences: Theory & Practice,* *15*(3), 809-824.

Foong, P. Y. (2002). The role of problems to enhance pedagogical practices in the Singapore Mathematics Classroom. *The Mathematics Educator*, *6*(2), 15-31.

Frederiksen, N., Glaser, R., Lesgold, A. & Shafto, M. (1990). *Diagnostic monitoring of skill and knowledge acquisition. *Mahwah, NJ: Lawrence Erlbaum.

Griffin, C. C. &Jitendra, A. K. (2008). Word problem solving instruction in inclusive third grade mathematics classrooms. *Journal of Educational Research,* *102*, 187-202.

Grimm, K. J. (2008). Longitudinal associations between reading and mathematics. *Developmental Neuropsychology*, *33*(3), 410-426.

Hegarty, M., Mayer, R. E. & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. *Journal of Educational Psychology, 87*, 18–32.

Hite, S. (2009). *Improving Problem Solving by Improving Reading Skills. Math in the Middle Institute Partnership Summative Projects for MA Degree*. University of Nebraska, Lincoln.

Hong, E. (1993*). *Mental models in word problem solving: An analysis of Korean elementary students.* Paper presented at the Annual Meeting of the American Educational Research Association*, Atlanta, GA*.*

Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice. *Journal of Mathematical Behavior*, *16*(2), 145-165.

Jitendra, A. K. & Hoff, K. (1996). The effects of schema-based instruction on the mathematical word- problem solving performance of students with learning disabilities. *Journal of Learning Disabilities*, *29*, 422–432.

Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. *Child Development*, *74*(3), 834–850

Karasar, N. (2002). *Bilimsel Araştırma Yöntemleri *[Scientific Research Methods] (2nd Ed.) Ankara: Nobel Yayınevi.

Karataş, İ. & Güven, B. (2004). 8. Sınıf öğrencilerinin problem çözme becerilerinin belirlenmesi: Bir özel durum çalışması [Defining 8 grades problem solving ability: A case study]. *Milli Eğitim Dergisi*, *163*, 132-143.

Kaur, B. (1997). Difficulties with problem solving in mathematics. The Mathematics Educator, *2*(1), 93-112.

Kaur, B. & Yeap, B. H. (2009). Mathematical problem solving in Singapore schools. In B. Kaur, B. H. Yeap, & M. Kapur (Eds.), Mathematical problem solving: Yearbook 2009 (pp. 3-13). Singapore: Association of Mathematics Education and World Scientific

Kieren, T. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. Romberg (Ed.), *Rational numbers: An integration of research *(pp. 49–84). Hillsdale, NY: Erlbaum.

Kintsch, W. (1988). The role of knowledge in discourse comprehension: A construction integration model. *Psychological Review*, *95*, 163-182.

Kintsch, W. & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. *Psychological Review*, *92*(1), 109-129.

Kispal, A. (2008). Effective Teaching of Inference Skills for Reading. Literature Review. Research Report, DCSF-RR031, ED501868.

Kroll, D. L. & Miller, T. (1993). Insights from research on mathematical problem solving in the middle grades. In D. T. Owens (Ed.), *Research ideas for the classroom: Middle grades mathematics *(pp. 58-77). NY: Macmillan.

Leighton, J. P. & Sternberg, R. J. (2003). Reasoning and problem solving. In A. F. Healy & R. W. Proctor (Eds.), *Experimental Psychology* (pp. 623-648). Volume 4 in I. B. Weiner (Editor-in-Chief) *Handbook of psychology*. New York: Wiley.

Lesh, R. A. & Doerr, H. M. (2003). *Beyond constructivism: A models and modelling perspective on teaching, learning, and problem solving in mathematics education.* Mahwah: Lawrence Erlbaum.

Liang, X. (2010). Assessmentuse, self-efficacy and mathematics achievement: Comparative analysis of PISA 2003. Data of Finland, Canada and the USA. *Evaluation & Research in Education, 23*(3), 213-229.

Maaß, K. (2007). Modelling tasks for low achieving students. First results of an empirical study. In D. PittaPantazi& G. Philippou (Eds.), *CERME 5 – Proceedings of the Fi**ﬅ**h Congress of the European Society for Research in Mathematics Education* (pp. 2120–2129). Larnaca: University of Cyprus.

Maass, K. (2010). Classification scheme for modelling tasks. *Journal fürMathematik-Didaktik*, *31*(2), 285-311.

Marshall, (1995). *Schemas in problem solving*. NY: Cambridge University Press.

MEB (2005). *İlköğretim 1–5. Sınıflar Matematik Dersi Öğretim Programı.* Ankara: Talim ve Terbiye Kurulu Başkanlığı.

Moreno**, **R. & Mayer**, **R. E. (1999). Multimedia**-**supportedmetaphors for meaning making in mathematics*. Cognition and Instruction*, *17*, 215-248.

Newman, M. A. (1977). An analysis of sixth-grade pupils' errors on written mathematical tasks. *Research in Mathematical Education in Australia*, *1*, 239-258.

Niss, M. (2003). Mathematical Competencies and the Learning of Mathematics: The Danish KOM Project. In A. Gagatsis & S. Papastavridis (Eds.), *3rd Mediterranean Conference on Mathematical Education. *(pp. 115-124). Athens: The Hellenic Mathematical Society.

OECD. (2016). PISA 2015 results: What students know and can do. Student performance in reading, mathematics, and science (Vol. I). Paris: OECD.

Özsoy, G. & Ataman, A. (2009). The effect of metacognitive strategy training on problem solving achievement. *I**nternational Electronic Journal of Elementary Education, 1*(2), 67–82.

Özsoy, G., Kuruyer, H. G. & Çakıroğlu, A, (2015). Evaluation of Students’ Mathematical Problem Solving Skills in Relation to Their Reading Levels. *International Electronic Journal of Elementary Education, 8*(1), 113-132.

Panasuk, R. & Beyranevand, M. (2010). Algebra students' ability to recognize multiple representations and achievement. *International Journal for Mathematics Teaching and Learning.*

Pape, S. J. (2004). Middle school children's problem-solving behavior: A cognitive analysis from a reading comprehension perspective. *Journal for Research in Mathematics Education*, *35*, 187-219.

Passolunghi, M. C. & Pazzaglia, F. (2005). A comparison of updating processes in children good or poor in arithmetic word problem-solving. *Learning and Individual Differences, 15*, 257–269.

Passolunghi, M. C., & Pazzaglia, F. (2004). Individual differences in memory updating in relation to arithmetic problem solving*.Learning and Individual Differences*, *14*, 219–230.

Prakitipong, N. & Nakamura, S. (2006). Analysis of mathematics performance of Grade 5 students in Thailand using Newman procedure. *Journal of International Cooperation in Education, **9*(1), 111-122.

Pressley, M. (2000). What should comprehension Instruction be the Instruction of? In Kamil, M. L., Mosenthal, P. B., Pearson, P. D. & Ban, R. (Eds.), *Handbook of Reading Research* (pp. 545-562). Mahwah, NJ: LEA.

Reusser, K. (1985). *From situation to equation. On formulation, understanding and solving "situation problems"*. Institute of Cognitive Science, University of Colorado at Boulder. Technical Report l43.

Schapp, S., Vos, P. &Goedhart, M. (2011). Students overcoming blockages while building a mathematical model: Exploring a framework. In G. Kaiser, W. Blum, R. B. Ferri& G. Stillman (Eds.), *Trends in teaching and learning of mathematical modelling: Te 14. ICMTA Study* (pp. 137–146). New York, NY: Springer.

Schleicher, A. (2007). Can competencies assessed by PISA be considered the fundamental school knowledge 15-year-olds shouldpossess? *Educational Change, 8*(4), 349-357.

Schoenfeld, A. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. Voss, D. Perkins & J. Segal (Eds.), *Informal reasoning and education *(pp. 311-343).

Schwarzkopf, R. (2007). Elementary modelling in mathematics lessons: The interplay between “real-world” knowledge and “mathematical structures”. In W. Blum, P. L. Galbraith, H. W. Hillsdale, NJ: Lawrence Erlbaum Associates. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 209-216). New York: Springer.

Şekercioğlu G., Bayat N., & Bakır S. (2014). Fen maddelerini anlama testinin psikometrik niteliklerinin belirlenmesi [Psychometric Properties of Science Items Comprehension Test]. *Eğitim ve Bilim*, *39*(176), 447-455.

Singhatat, N. (1991). Analysis of mathematics errors of Iower secondary pupils in solving word problems. Penang: Seameo-Recsam

Singh. P, Rahman, A. A & Hoon, T. C. (2010). The Newman Procedure for Analyzing Primary Four Pupils Errors on Written Mathematical Tasks: A Malaysian Perspective. *Procedia Social and Behavioral Sciences,* *8*, 264–271

Smith, G. G., Gerretson, H., Olkun, S. & Joutsenlahti, J. (2010). Effect of causal stories in solving mathematical story problems. *Hacettepe Üniversitesi Eğitim Fakültesi Dergisi*., *39*, 284-295.

Soylu, Y. & Soylu, C. (2006). Matematik derslerinde başarıya giden yolda problem çözmenin rolü [The role of problem solving in mathematics lessons for success]. *İnönü Eğitim Fakültesi Dergisi*, *7*(11): 97–111.

Staub, F. C. & Reusser, K. (1995). The role of presentational structures in understanding and solving mathematical word problems. In C. A. Weaver, S. Mannes & C. R. Fletcher (Eds.), *Discourse comprehension. Essays in honor of Walter Kintsch* (pp. 285-305). Hillsdale, NJ: Lawrence Erlbaum.

Steele, D. F & Johanning, D. I. (2004). A schematic–theoretic view of problem solving and development of algebraic thinking.* Educational Studies in Mathematics*, *57*, 65–90.

Suk, H. J. (1997). The effects of language ability, text presentation style and the nature of questions inference, Doktoraldesertation, Chong-buk University Kore.

Tekin, H. (1997). *Eğitimde ölçme ve değerlendirme [Measurement and Assessment in Education].* Ankara: Mars Matbaası.

Teong, K. S. (2002). *Metacognitive Intervention Strategy and Word Problem Solving in a Cognitive-Apprenticeship-Computer-Based Environment.* National Institute of Education, Nanyang Technological University.

Tertemiz, N. (1994). *İlkokulda Aritmetik Problemlerini Çözmede Etkili Görülen Bazı Faktörler [factors affecting problem solving ability in elementary scholls]*, Hacettepe Üniversitesi Sosyal Bilimler Enstitüsü, Yayımlanmamış DoktoraTezi. Ankara.

Tashakkori, A., & Teddlie, C. (Eds.). (2010). SAGE handbook of mixed methods in social and behavioral research. (2nd Ed.). Thousand Oaks, CA: Sage.

Ulu, M., Tertemiz, N., & Peker, M. (2016a). İlköğretim 5. sınıf öğrencilerinin problem çözme sürecinde yaptıkları hata türlerinin belirlenmesi. *Kuramsal Eğitimbilim Dergisi*, *9*(4), 571-605

Ulu, M., Tertemiz, N., Peker, M. (2016b). The Effect of Comprehension and Problem Solving Strategy Training on 5th Graders’ Non-Routine Problem Solving Success. Afyon Kocatepe Üniversitesi Sosyal Bilimler Dergisi, *18*(2), 303-340.

Ulu, M. (2016). A Structural Equation Model to Explain the Effect of Fluent Reading, Literal Comprehension and Inferential Comprehension Levels of Elementary School 4th Grade Students on Success in Problem Solving. *Egitim ve Bilim,* *41*(186), 93-117.

Umay, A. (2003). Matematiksel muhakeme yeteneği [Mathematical reasoning skills]. *Hacettepe Üniversitesi Eğitim Fakültesi Dergisi*, *24*(3), 234-243.

Vacca, J. A. L., Vacca, R. T., Gove, M. K., Burkey, L. C., Lenhart, L. A. &Mckeon, C. A. (2006). *Reading and learning to read. *Boston: Allyn and Bacon.

Vaismoradi, M., Turunen, H., & Bondas, T. (2013). Content analysis and thematic analysis: Implications for conducting a qualitative descriptive study. *Nursing and Health Sciences, 15*(3), 398–405. doi: 10.1111/nhs.12048

Van Dijk, T. A. & Kintsch, W. (1983). *Strategies of discourse comprehension.* New York: Academic Press.

Verschaffel, L., De Corte, E. &Vierstraete, H. (1999). Upper elementary school pupils’ difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers. *Journal for Research in Mathematics Education,* *30*(3), 265-285.

Verschaffel, L., Greer, B. & De Corte, E. (2000). *Making sense of word problems. *Lisse, The Netherlands: Swets and Zietlinger.

Viennot, D. C. & Moreau, S. (2007). Arithmetic problems at school: When there is an apparent contradiction between the situation model and the problem model. *British Journal of Educational Psychology*, *77*, 69–80.

Vygotsky, L. S. (1978). *Mind in society: the development of higher psychological processes*. Cambridge: Harvard University Press.

White, A. L. (2005). Active mathematics in classrooms: Finding out why children make mistakes – and then doing something to help them. *Square One*, *15*(4), 127-135.

Wijaya, A., Van den Heuvel-Panhuizen, M., Doorman, M. &Robitzsch, A. (2014). Difficulties in solving context-based PISA mathematics tasks: An analysis of students’ errors. *The Mathematics Enthusiast*, *11*(3), 555–584.

Wischgoll, A., Pauli, C. &Reusser, K. (2015). Scaffolding--How Can Contingency Lead to Successful Learning When Dealing with Errors? *The International Journal on Mathematics Education*, *47*(7), 1147-1159.

Wood, D., Bruner, J. S. & Ross, G. (1976). The role of tutoring in problem solving. *Journal of Child Psychology and Psychiatry*, *17*(2), 89–100.

Xin, Z., Lin, C., Zhang, L. & Yan, R. (2007).The Performance of Chinese Primary School Students on Realistic Arithmetic Word Problems.* Educational Psychology in Practice, 23*(2), 145–159.

Yazgan, Y. & Bintaş, J. (2005). İlköğretim dördüncü ve beşinci sınıf öğrencilerinin problem çözme stratejilerini kullanabilme düzeyleri: Bir öğretim deneyi [Fourth And Fifth Grade Students’ Levels Of Using Problem Solving Strategies: A Teaching Experiment]. *Hacettepe Üniversitesi Eğitim Fakültesi Dergisi*, *28*, 210–218.

Yore, L. D., Anderson, J. O., & Hung Chiu, M. (2010). Moving PISA results into the policy arena: Perspectives on knowledge transfer for future considerations and preparations. *International Journal of Science and Mathematics Education, 8*(3), 593-603.

Yurt, E. & Sünbül, A. M. (2014). A Structural Equation Model Explaining the Mathematics Achievements of the 8th Grade Students, *Educational Sciences: Theory & Practice*, *14*(4), 1629-1653.

Zwiers, J. (2004). Making inferences and predictions. In Money, M. M. (Ed.), *Building Reading Comprehension Habits in Grades 6-12: a Toolkit of Classroom Activities* (2nd Ed.). Chapter 5, (pp. 99-122). Newark DE: International Reading Assoc.

Rabia KARATOPRAK , Gülseren KARAGÖZ AKAR, Bengü BÖRKAN

Read (times)1184
Read

Read (times)488
Read

Yoppy Wahyu PURNOMO , Didi SURYADI , Sutawanir DARWIS

Read (times)439
Read

Deniz KAYA

Read (times)100
Read

Feride ÖZYILDIRIM-GÜMÜŞ , Yeter ŞAHİNER

Read (times)66
Read