Year 2021, Volume 8 , Issue 2, Pages 357 - 375 2021-06-10

Developing a Two-Tier Proportional Reasoning Skill Test: Validity and Reliability Studies

Kübra AÇIKGÜL [1]

The main aim of this study is to develop a useful, valid, and reliable two-tier proportional reasoning skill test for middle school 7th and 8th-grade students. The research was carried out using the sequential explanatory mixed method. The study group of this research comprised of 391 (n7th-grade= 223, n8th-grade= 168) students. With validity and reliability studies, the content, face, construct, discriminant validity, and reliability coefficient of the test were examined. As a result, the two-tier proportional reasoning skill test with 12 items under 3 factors (qualitative prediction and comparison, missing value, numerical comparison) valid and reliable for adequate values specified in the literature.
Proportional reasoning, Two-tier test, Middle school
  • Afnia, P.N., & Istiyono, E. (2020, February). The development of two-tier multiple choice instruments to measure higher order thinking skills bloomian. In 3rd International Conference on Learning Innovation and Quality Education (ICLIQE 2019) (pp. 1038-1045). Atlantis Press.
  • Akkus, O., & Duatepe-Paksu, A. (2006). Construction of a proportional reasoning test and its rubrics. Eurasian Journal of Educational Research, 25, 1-10.
  • Alfieri, L., Higashi, R., Shoop, R., & Schunn, C. D. (2015). Case studies of a robot-based game to shape interests and hone proportional reasoning skills. International Journal of STEM Education, 2(4), 1-13.
  • Allain, A. (2000). Development of an instrument to measure proportional reasoning among fast-track middle school students. [Master’s thesis]. University of North Carolina State.
  • Arıcan, M. (2019). A diagnostic assessment to middle school students’ proportional reasoning. Turkish Journal of Education, 8(4), 237-257.
  • Ayan, R., & Isiksal-Bostan, M. (2019). Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement. International Journal of Mathematical Education in Science and Technology, 50(1), 65-81.
  • Behr, M., Lesh, R., & Post, T. (1988). Proportional reasoning, In M. Behr and J. Hiebert (Eds.), Number concepts and operations in the middle grades. Lawrence Erlbaum Associates.
  • Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Eds.), Handbook on research of teaching and learning (pp. 296-333). McMillan.
  • Brown, T. A. (2006). Confirmatory factor analysis for applied research. In David A. Kenny (Eds.), Methodology in the Social Sciences. The Guilford Press.
  • Bright, G. W., Joyner, J. M., & Wallis, C. (2003). Assessing proportional thinking. Mathematics Teaching in the Middle School, 9(3), 166-172.
  • Burton, S. J., Sudweeks, R. E., Merrill, P. F., & Wood, B. (1991). How to prepare better multiple-choice test items: Guidelines for university faculty. Brigham Young University Testing Services and The Department of Instructional Science.
  • Büyüköztürk, S. (2010). Sosyal bilimler için veri analizi el kitabı [Data analysis handbook for social sciences]. Pegem Akademi.
  • Cameron, A. (2004). Kurtosis. In M. Lewis-Beck, A. Bryman and T. Liao (Eds.). Encyclopedia of social science research methods. (pp. 544-545). SAGE Publications, Inc.
  • Common Core State Standards Initiative (US). Common core state standards for mathematics.
  • Cramer, K., & Post, T. (1993). Proportional reasoning. The Mathematics Teacher, 86(5), 404-407.
  • Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: research implications. In D. Owens (Eds.), Research ideas for the classroom (pp. 159-178). Macmillan Publishing Company.
  • Crocker, L., & Algina, J. (2008). Introduction to classical and modern test theory. Cengage Learning.
  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Erlbaum.
  • Cohen, L., Manion, L., & Morrison, K. (2013). Research methods in education. Routledge.
  • Creswell, J .W., & Plano Clark, V. L. (2011). Designing and conducting mixed methods research. Sage Publications.
  • Davis, L. L. (1992). Instrument review: getting the most from a panel of experts. Applied Nursing Research, 5, 194-197.
  • Dinç-Artut, P., & Pelen, M. S. (2015). 6th grade students’ solution strategies on proportional reasoning problems. Procedia-Social and Behavioral Sciences, 197, 113-119.
  • Duatepe, A., Akkus-Cıkla, O., & Kayhan, M. (2005). An investigation on students’ solution strategies for different proportional reasoning items. Hacettepe Journal of Education Faculty, 28, 73-81.
  • Ebel, R. L. (1965). Measuring educational achievement. Prentice Hall.
  • Ebel, R. L., & Frisbie, D. A. (1991). Essentials of educational measurement. Prentice Hall.
  • Field, A. (2009). Discovering statistics using SPSS. Sage Publication.
  • Fornell, C., & Larcker, D. F. (1981). Structural equation models with unobservable variables and measurement error: Algebra and statistics. Journal of Marketing Research, 18(3), 328–388.
  • Gable, R. K. (1986). Instrument development in the affective domain. Kluwer-Nijhoff Publishing.
  • Hair, J. F., Jr., Black, W. C., Babin, B. J., Anderson, R. E., & Tatham, R. L. (2014). Multivariate data analysis. Pearson New International Edition.
  • Haja, S., & Clarke, D. (2011). Middle school students’ responses to two-tier tasks. Mathematics Education Research Journal, 23(1), 67-76.
  • Haslam, F., & Treagust, D. F. (1987). Diagnosing secondary students' misconceptions of photosynthesis and respiration in plants using a two-tier multiple choice instrument. Journal of biological education, 21(3), 203-211.
  • Hatcher, L. (1994). A step-by-step approach to using the SAS® system for Factor Analysis and Structural Equation Modeling. SAS Institutte, Inc.
  • Hilton, A., Hilton, G., Dole, S., & Goos, M. (2013). Development and application of a two-tier diagnostic instrument to assess middle-years students’ proportional reasoning. Mathematics Education Research Journal, 25(4), 523-545.
  • Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6(1), 1-55.
  • Hyman, M. R., & Sierra, J. J. (2016). Open-versus close-ended survey problems. Business Outlook, 14(2), 1-5.
  • Kline, R. B. (2011). Principles and practice of structural equation modeling. Guilford Press.
  • Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–668). Information Age Publishing.
  • Lawton, C. A. (1993). Contextual factors affecting errors in proportional reasoning. Journal for Research in Mathematics Education, 24(5), 460-466.
  • Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93-118). Lawrence Erlbaum & National Council of Teachers of Mathematics.
  • Lim, K. (2009). Burning the candle at just one end: Using nonproportional examples helps students determine when proportional strategies apply. Mathematics Teaching in the Middle School, 14(8), 492–500.
  • Mersin, N. (2018). An evaluation of proportional reasoning of middle school 5th, 6th and 7th grade students according to a two-tier diagnostic test. Cumhuriyet InternationalJournal of Education, 7(4), 319–348.
  • Ministry of National Education [MoNE], (2018). Matematik dersi öğretim programı. (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar) [Mathematics curriculum. (Primary and Middle Schools 1, 2, 3, 4, 5, 6, 7 and 8th Grades) [Middle school mathematics curricula for grades 5, 6, 7, and 8)].
  • Mueller, R. O., & Hancock, G. R. (2001). Factor analysis and latent structure: Confirmatory factor analysis. In N. J. Smelser & P. B. Baltes (Eds.), International Encyclopedia of the Social and Behavioral Sciences (pp. 5239-5244). Pergamon.
  • National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics. National Council of Teachers of Mathematics.
  • Ozuru, Y., Briner, S., Kurby, C. A., & McNamara, D. S. (2013). Comparing comprehension measured by multiple-choice and open-ended problems. Canadian Journal of Experimental Psychology, 67(3), 215-227.
  • Özgün-Koca, S. A., & Altay, M. K. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48.
  • Pelen, M. S., & Dinç-Artut, P. (2015). 7th grade students’problem solving success rates on proportional reasoning problems. The Eurasia Proceedings of Educational and Social Sciences, 2, 96-100.
  • Peterson, R. F., Treagust, D. F., & Garnett, P. (1986). Identification of secondary students' misconceptions of covalent bonding and structure concepts using a diagnostic test instrument. Research in Science Education, 16, 40-48.
  • Post, T. R., Behr, M. J., & Lesh, R. (1988). Proportionality and the development of pre-algebra understandings. In A. Coxford & A. Shulte (Eds.), The ideas of algebra, K-12 (pp. 78-90). National Council of Teachers of Mathematics.
  • Reja, U., Manfreda, K. L., Hlebec, V., & Vehovar, V. (2003). Open-ended vs. close-ended questions in web questionnaires. In A. Ferligoj & A. Mrvar (Eds.), Developments in applied statistics (pp. 159–177). FDV.
  • Rudner, L. M., & Shafer, W. D. (2002). What teachers need to know about assessment. National Education Association.
  • Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43, 271-292.
  • Soyak, O., & Isiksal, M. (2017, February). Middle school students’ difficulties in proportional reasoning. Paper Presented at the CERME 10, Dublin, Ireland.
  • Suhr, D. (2006). Exploratory or confirmatory factor analysis. SAS Users Group International Conference (pp. 1- 17). SAS Institute, Inc.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics. Pearson.
  • Tamir, P. (1989). Some issues related to the use of justifications to multiple-choice answers. Journal of Biological Education, 23(4), 285-292. 10.1080/00219266.1989.9655083
  • Tamir, P. (1990). Justifying the selection of answers in multiple choice items. International Journal of Science Education, 12(5), 563-573.
  • Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational studies in mathematics, 16(2), 181-204.
  • Tsui, C. Y., & Treagust, D. (2010). Evaluating secondary students’ scientific reasoning in genetics using a two‐tier diagnostic instrument. International Journal of Science Education, 32(8), 1073-1098.
  • Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: teaching developmentally. Pearson Education, Inc.
  • Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for over generalization. Cognition and instruction, 23(1), 57-86.
  • Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28, 360–381.
  • Weinberg, S. L. (2002). Proportional reasoning: One problem, many solutions! In B. Litwiller (Eds.), Making sense of fractions, ratios, and proportions: 2002 year book (pp. 138-144). National Council of Teachers of Mathematics.
  • Wells, C. S., & Wollack, J. A. (2003). An instructor’s guide to understanding test reliability. Testing & evaluation services. University of Wisconsin.
Primary Language en
Subjects Education, Scientific Disciplines
Published Date June
Journal Section Articles

Orcid: 0000-0003-2656-8916
Author: Kübra AÇIKGÜL
Country: Turkey


Publication Date : June 10, 2021

APA Açıkgül, K . (2021). Developing a Two-Tier Proportional Reasoning Skill Test: Validity and Reliability Studies . International Journal of Assessment Tools in Education , 8 (2) , 357-375 . DOI: 10.21449/ijate.909316