Uncovering student errors in measures of dispersion: An APOS theory analysis in high school statistics education

Chiew Leng Ng 1, Cheng Meng Chew 1 2 *
More Detail
1 Universiti Sains Malaysia, MALAYSIA
2 Wawasan Open University, MALAYSIA
* Corresponding Author
EUR J SCI MATH ED, Volume 11, Issue 4, pp. 599-614. https://doi.org/10.30935/scimath/13260
Published Online: 10 May 2023, Published: 01 October 2023
OPEN ACCESS   1085 Views   740 Downloads
Download Full Text (PDF)


Despite statistics learning becoming more important during this information explosion era, many students still deem the subject complex and challenging. Measures of dispersion, a critical component of statistical knowledge that students often struggle with, have received little attention in research on statistics education. The goal of this study was to uncover students' errors in solving problems involving measures of dispersion by examining students’ response in the diagnostic test through the lens of APOS theory. The participants consisted of 85 grade 11 high school students and were then divided into three groups according to their performance to better understand the difficulties and errors made by students from different cognitive levels. The findings revealed that majority of low achievers operate at the action level, as indicated by the numerous conceptual errors discovered during the test. These students have limited conceptual understanding on the topic which required proper remedial from the educators. The study's results are discussed, as well as potential implications for education.


Ng, C. L., & Chew, C. M. (2023). Uncovering student errors in measures of dispersion: An APOS theory analysis in high school statistics education. European Journal of Science and Mathematics Education, 11(4), 599-614. https://doi.org/10.30935/scimath/13260


  • Almanasreh, E., Moles, R., & Chen, T. F. (2019). Evaluation of methods used for estimating content validity. Research in Social and Administrative Pharmacy, 15(2), 214-221. https://doi.org/10.1016/j.sapharm.2018.03.066
  • Altman, D. G., & Bland, J. M. (2005). Standard deviations and standard errors. Bmj, 331(7521), 903. https://doi.org/10.1136/bmj.331.7521.903
  • Andriani, S. P., & Nurhasanah, F. (2021). Procedural error of XIIth grade high school students in solving algebra problems based on Elbrink’s theory. Journal of Physics: Conference Series, 1796(1), 012048). https://doi.org/10.1088/1742-6596/1796/1/012048
  • Arnon, I., Cottrill, J., Dubinsky, E. D., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer. https://doi.org/10.1007/978-1-4614-7966-6
  • Bansilal, S. (2014). Using an apos framework to understand teachers’ responses to questions on the normal distribution. Statistics Education Research Journal, 13(2), 42-57. https://doi.org/10.52041/serj.v13i2.279
  • Boaler, J., & Levitt, S. D. (2019, October 23). Opinion: Modern high school math should be about data science—not Algebra 2. The Los Angeles Times. https://www.latimes.com/opinion/story/2019-10-23/math-high-school-algebra-data-statistics
  • Busi, R., & Jacobbe, T. (2014). Examining student work in the preparation of preservice elementary school teachers. The Mathematics Educator, 23(2), 23-39.
  • Canobi, K. H. (2009). Concept–procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102(2), 131-149. https://doi.org/10.1016/j.jecp.2008.07.008
  • Chamundeswari, S. (2014). Conceptual errors encountered in mathematical operations in algebra among students at the secondary level. International Journal of Innovative Science, Engineering & Technology, 1(8), 24-38.
  • Chan, S. W., Ismail, Z., & Sumintono, B. (2014). A Rasch model analysis on secondary students’ statistical reasoning ability in descriptive statistics. Procedia-Social and Behavioral Sciences, 129, 133-139. https://doi.org/10.1016/j.sbspro.2014.03.658
  • Chance, B., Garfield, J., & delMas, R. (2000). Developing simulation activities to improve students' statistical reasoning [Paper presentation]. Proceedings of the International Conference on Technology in Mathematics Education (PICTME), Auckland, NZ. Dec. 11-14, 2000. https://files.eric.ed.gov/fulltext/ED474052.pdf
  • Clark, J., Kraut, G., Mathews, D., & Wimbish, J. (2007). The fundamental theorem of statistics: Classifying student understanding of basic statistical concepts [Unpublished manuscript]. http://www1.hollins.edu/faculty/clarkkjm/stat2c.pdf
  • Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13-20. https://doi.org/10.3102/0013189X023007013
  • Confrey, J., & Kazak, S. (2006). A thirty-year reflection on constructivism in mathematics education in PME. In Á. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 305-345). Brill. https://doi.org/10.1163/9789087901127_012
  • Cooper, L. L., & Shore, F. S. (2008). Students' misconceptions in interpreting center and variability of data represented via histograms and stem-and-leaf plots. Journal of Statistics Education, 16(2). https://doi.org/10.1080/10691898.2008.11889559
  • Delmas, R., & Liu, Y. (2005). Exploring students’ conceptions of the standard deviation. Statistics Education Research Journal, 4(1), 55-82. https://doi.org/10.52041/serj.v4i1.525
  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-123). Springer, Dordrecht. https://doi.org/10.1007/0-306-47203-1_7
  • Dubinsky, E., & McDonald, M. A. (2002). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, M. Artigue, U. Kirchgräber, J. Hillel, M. Niss, & A. Schoenfeld (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275-282). Springer, Dordrecht. https://doi.org/10.1007/0-306-47231-7_25
  • Engel, J. (2017). Statistical literacy for active citizenship: A call for data science education. Statistics Education Research Journal, 16(1), 44-49. https://doi.org/10.52041/serj.v16i1.213
  • Garfield, J. B., Ben-Zvi, D., Chance, B., Medina, E., Roseth, C., & Zieffler, A. (2008). Developing students' statistical reasoning: Connecting research and teaching practice. Springer.
  • Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8(2), 53-96. https://doi.org/10.1111/j.1539-6053.2008.00033.x
  • Godden, H., Mbekwa, M., & Julie, C. (2013). An analysis of errors and misconceptions in the 2010 grade 12 mathematics examination: A focus on quadratic equations and inequalities. In Z. Davis & S. Jaffer (Eds.), Proceedings of the 19th Annual Congress of the Association for Mathematics Education of South Africa (pp. 70-79). Association for Mathematics Education of South Africa.
  • Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1), 37-63. https://doi.org/10.1207/s15327833mtl0801_3
  • Herholdt, R., & Sapire, I. (2014). An error analysis in the early grades mathematics-A learning opportunity? South African Journal of Childhood Education, 4(1), 43-60.
  • Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 65-97). Macmillan Publishing Co, Inc.
  • Horn, I., Kane, B., & Wilson, B. (2015). Making sense of student performance data: Data use logics and mathematics teachers’ learning opportunities. American Educational Research Journal, 52(2), 208–242. https://doi.org/10.3102/0002831215573773
  • Hurrell, D. (2021). Conceptual knowledge or procedural knowledge or conceptual knowledge and procedural knowledge: Why the conjunction is important to teachers. Australian Journal of Teacher Education, 46(2), 57-71. https://doi.org/10.14221/ajte.2021v46n2.4
  • Ingram, J., Pitt, A., & Baldry, F. (2015). Handling errors as they arise in whole-class interactions. Research in Mathematics Education, 17(3), 183-197. https://doi.org/10.1080/14794802.2015.1098562
  • Kula, F., & Koçer, R. G. (2020). Why is it difficult to understand statistical inference? Reflections on the opposing directions of construction and application of inference framework. Teaching Mathematics and its Applications: An International Journal of the IMA, 39(4), 248-265. https://doi.org/10.1093/teamat/hrz014
  • Lee, H., Mojica, G., Thrasher, E., & Baumgartner, P. (2022). Investigating data like a data scientist: Key practices and processes. Statistics Education Research Journal, 21(2), 3-3. https://doi.org/10.52041/serj.v21i2.41
  • Lee, H. S., & Harrison, T. (2021). Trends in teaching Advanced Placement Statistics: Results from a national survey. Journal of Statistics and Data Science Education, 29(3), 317-327. https://doi.org/10.1080/26939169.2021.1965509
  • Lee, O., & Campbell, T. (2020). What science and STEM teachers can learn from COVID-19: Harnessing data science and computer science through the convergence of multiple STEM subjects. Journal of Science Teacher Education, 31(8), 932-944. https://doi.org/10.1080/1046560X.2020.1814980
  • Lovett, J. N., & Lee, H. S. (2018). Preservice secondary mathematics teachers’ statistical knowledge: A snapshot of strengths and weaknesses. Journal of Statistics Education, 26(3), 214-222. https://doi.org/10.1080/10691898.2018.1496806
  • Lynn, M. R. (1986). Determination and quantification of content validity. Nursing Research, 35(6), 382-386.
  • Mathews, D., & Clark, J. (2003). Successful students’ conceptions of mean, standard deviation, and the Central Limit Theorem. https://www.researchgate.net/publication/253438034_Successful_Students'_Conceptions_of_Mean_Standard_Deviation_and_The_Central_Limit_Theorem
  • Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. SAGE.
  • Miller, S. P., & Hudson, P. J. (2007). Using evidence‐based practices to build mathematics competence related to conceptual, procedural, and declarative knowledge. Learning Disabilities Research & Practice, 22(1), 47-57. https://doi.org/10.1111/j.1540-5826.2007.00230.x
  • Moore, D. S. (1997). New pedagogy and new content: The case of statistics. International Statistical Review, 65(2), 123-137. https://doi.org/10.1111/j.1751-5823.1997.tb00390.x
  • Nesher, P. (1987). Towards an instructional theory: The role of student’s misconceptions. For the Learning of Mathematics - An International Journal of Mathematics Education, 7(3), 33-40.
  • Pangrazio, L., & Selwyn, N. (2021). Towards a school-based ‘critical data education’. Pedagogy, Culture & Society, 29(3), 431-448. https://doi.org/10.1080/14681366.2020.1747527
  • Piaget, J. (1968). Development and learning. Journal of Research in Science Teaching, 40, S8-S18. https://doi.org/10.1016/B978-0-08-097086-8.92013-0
  • Polit, D. F., & Beck, C. T. (2006). The content validity index: Are you sure you know what's being reported? Critique and recommendations. Research in Nursing & Health, 29(5), 489-497. https://doi.org/10.1002/nur.20147
  • Raghubar, K., Cirino, P., Barnes, M., Ewing-Cobbs, L., Fletcher, J., & Fuchs, L. (2009). Errors in multi-digit arithmetic and behavioral inattention in children with math difficulties. Journal of Learning Disabilities, 42(4), 356-371. https://doi.org/10.1177/0022219409335211
  • Riccomini, P. J. (2005). Identification and remediation of systematic error patterns in subtraction. Learning Disability Quarterly, 28(3), 233-242. https://doi.org/10.2307/1593661
  • Rittle‐Johnson, B., Fyfe, E. R., & Loehr, A. M. (2016). Improving conceptual and procedural knowledge: The impact of instructional content within a mathematics lesson. British Journal of Educational Psychology, 86(4), 576-591. https://doi.org/10.1111/bjep.12124
  • Russell, M., O’dwyer, L. M., & Miranda, H. (2009). Diagnosing students’ misconceptions in algebra: Results from an experimental pilot study. Behavior Research Methods, 41(2), 414-424. https://doi.org/10.3758/BRM.41.2.414
  • Sáenz, C. (2009). The role of contextual, conceptual and procedural knowledge in activating mathematical competencies (PISA). Educational Studies in Mathematics, 71(2), 123-143. https://doi.org/10.1007/s10649-008-9167-8
  • Saldahna, L. A., & Thompson, P. (2002). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51(3), 257-279. https://doi.org/10.1023/A:1023692604014
  • Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS Theory. The Journal of Mathematical Behavior, 39, 100-120. https://doi.org/10.1016/j.jmathb.2015.06.005
  • Siegler, R. S. (2007). Cognitive variability. Developmental Science, 10(1), 104-109. https://doi.org/10.1111/j.1467-7687.2007.00571.x
  • Siyepu, S. W. (2013). An exploration of students’ errors in derivatives in a university of technology. The Journal of Mathematical Behavior, 32(3), 577-592. https://doi.org/10.1016/j.jmathb.2013.05.001
  • Sfard, A. (1991). On the dual nature of mathematical conceptions. Educational Studies in Mathematics, 22, 1-36. https://doi.org/10.1007/BF00302715
  • Skemp, R. (1987). The psychology of learning mathematics (Expanded American ed.). Lawrence Erlbaum Associates.
  • Swan, M. (2001). Dealing with misconceptions in mathematics. In P. Gates (Ed.), Issues in Mathematics Teaching (pp. 147-165). Routledge.
  • Tishkovskaya, S., & Lancaster, G. A. (2012). Statistical education in the 21st century: A review of challenges, teaching innovations and strategies for reform. Journal of Statistics Education, 20(2). https://doi.org/10.1080/10691898.2012.11889641
  • Trigueros, M., & Okta§, A. (2019). Task design in APOS Theory. Avances de Investigacin en Educacin Matemίtica, 15, 43-55.
  • von Glasersfeld, E. (1989). Cognition, construction of knowledge, and teaching. Synthese, 80(1), 121-140. https://doi.org/10.1007/BF00869951
  • Zaidan, A., Ismail, Z., Yusof, Y. M., & Kashefi, H. (2012). Misconceptions in descriptive statistics among postgraduates in social sciences. Procedia-Social and Behavioral Sciences, 46, 3535-3540. https://doi.org/10.1016/j.sbspro.2012.06.100