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Evaluating the Understanding of the University Students (Basic Sciences and Engineering) about the Numerical Representation of the Average Rate of Change

Authors: Saeid Haghjoo, Ebrahim Reyhani, Fahimeh Kolahdouz


The present study aimed to evaluate the understanding of the students in Tehran universities (Iran) about the numerical representation of the average rate of change based on the Structure of Observed Learning Outcomes (SOLO). In the present descriptive-survey research, the statistical population included undergraduate students (basic sciences and engineering) in the universities of Tehran. The samples were 604 students selected by random multi-stage clustering. The measurement tool was a task whose face and content validity was confirmed by math and mathematics education professors. Using Cronbach's Alpha criterion, the reliability coefficient of the task was obtained 0.95, which verified its reliability. The collected data were analyzed by descriptive statistics and inferential statistics (chi-squared and independent t-tests) under SPSS-24 software. According to the SOLO model in the prestructural, unistructural, and multistructural levels, basic science students had a higher percentage of understanding than that of engineering students, although the outcome was inverse at the relational level. However, there was no significant difference in the average understanding of both groups. The results indicated that students failed to have a proper understanding of the numerical representation of the average rate of change, in addition to missconceptions when using physics formulas in solving the problem. In addition, multiple solutions were derived along with their dominant methods during the qualitative analysis. The current research proposed to focus on the context problems with approximate calculations and numerical representation, using software and connection common relations between math and physics in the teaching process of teachers and professors.

Keywords: derivative, average rate of change, context problems, numerical representation, SOLO taxonomy

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[1] Ahmadi, A., Khalili Boroujani, R., Khoshbin-e Khoshnazar, M.R., Sharifzadeh Ekbatani, M.R., Sajjadi, S.H., Atiqi, M., Mardukhi, S., & Niknam, A. (2018). Physics 3, 4th year, Secondary School, Mathematical Sciences, Organization for Educational Research and Planning, Ministry of Education, Tehran, Printing and Publishing Company of Iran Textbooks.
[2] Amiri, H., Bijanzadeh, M.H., Bahrami Samani, E., Heidari Ghezeljeh, R., Davarzani, M., Reyhani, E., Seyed Salehi, M.R., & Ghorbani Arani, M. (2016). Mathematics 1, 10th grade, Secondary School, Organization for Educational Research and Planning, Ministry of Education, Tehran, Printing and Publishing Company of Iran Textbooks.
[3] Amiri, H.R., Iranmanesh, A., Hamzeh Beigi, T., Davoudi, Kh., Rostami, M.H., Reyhani, E., Seyed Salehi, M.R., Sharghi, H., & Sadr, M. (2018). Mathematics, 9th grade, High School, Organization for Educational Research and Planning, Ministry of Education, Tehran, Printing and Publishing Company of Iran Textbooks.
[4] Amiri, M.R., Izadi, M., Zamani, I., Bahrami Samani, E., Parang, H., Min Bashian, H., & Nirou, M. (2017). The Analysis of Policies, Approved documents, Researches, and Reliable Resources with the Field of Mathematical Learning, Department of Research, Development, and Mathematics Education, Organization for Educational Research and Planning.
[5] Apostol, T. M. (2007). Calculus, Volume I, One-variable Calculus, with an Introduction to Linear Algebra (Vol. 1). John Wiley & Sons.
[6] Asadi, M.B., Ranjbari, A., Reyhani, E., Taheri Tanjani, M.T., Ghorbani Arani, M., & Min Bashian, H. (2017). Calculus 1 11th grade, Secondary School, Organization for Educational Research and Planning, Ministry of Education. Tehran: Printing and Publishing Company of Iran Textbooks.
[7] Biggs, J. & Collis, K. (1982). Evaluating the quality of learning: the SOLO taxonomy. New York: Academic Press.
[8] Biggs, J. & Collis, K. (1991). Multimodal learning and the quality of intelligent behavior. In H. Rowe (Ed.), Intelligence, Reconceptualization and Measurement (pp. 57-76). New Jersey: Laurence Erlbaum Assoc.
[9] Biggs, J. (2003). Aligning teaching for constructing learning. Higher Education Academy, 1-4.
[10] Biggs, J. (2014). Constructive alignment in university teaching. HERDSA Review of higher education, 1(1), 5-22.
[11] Biggs, J., & Tang, C. (2007). Setting the stage for effective teaching. Teaching for quality learning at university, 31-59.
[12] Biggs, J.B. & Tang, C. (2011). Teaching for Quality Learning at University. (4th Ed.).Maidenhead: McGraw Hill Education & Open University Press.
[13] Biggs, J.B. (2013). Changing Universities: A memoir about academe in different places and times. Melbourne: Strictly Literary.
[14] Caniglia, J. C., & Meadows, M. (2018). An Application of the Solo Taxonomy to Classify Strategies Used by Pre-Service Teachers to Solve" One Question Problems". Australian Journal of Teacher Education, 43(9), 75-89.
[15] Case, R. (1992). The Mind's Staircase: Exploring the conceptual underpinnings of children's thought and knowledge. New Jersey: Laurence Erlbaum Assoc.
[16] Çetin, N. (2009). The ability of students to comprehend the function-derivative relationship with regard to problems from their real life. Primus, 19(3), 232-244.
[17] Chick, H. (1998). Cognition in the formal modes: Research mathematics and the SOLO taxonomy. Mathematics Education Research Journal, 10(2), 4-26.
[18] Clement, J. (1985, July). Misconceptions in graphing. In Proceedings of the ninth international conference for the psychology of mathematics education (Vol. 1, pp. 369-375). Utrecht, The Netherlands: Utrecht University.
[19] Collis, K., Jones, B., Sprod, T., Watson, J., & Fraser, S. (1998). Mapping Development in Student’s Understanding of Vision using a Cognitive Structural Model. International Journal of Science Education, 20(1), 44-66.
[20] Davarzani, M., Reyhani, E., Seyed Salehi, M.R. Taheri Tanjani, M.T., Ghorbani, M., & Min Bashian, H. (2018). Calculus 2, 4th year, Secondary school, Mathematical Sciences, Organization for Educational Research and Planning, Ministry of Education, Tehran, Printing and Publishing Company of Iran Textbooks.
[21] De Gamboa G, Figueiras L. Conexiones en el conocimiento matemático del profesor: propuesta de un modelo de análisis
[Connections in the mathematical knowledge of the teacher: a proposed analysis model]. In: González MT, Codes M, Arnau D, Ortega T, editor. Investigación en Educación Matemática XVIII. Salamanca: SEIEM; 2014. p. 337–344.
[22] Dolores-Flores, C., Rivera-López, M. I., & García-García, J. (2019). Exploring mathematical connections of pre-university students through tasks involving rates of change. International Journal of Mathematical Education in Science and Technology, 50(3), 369-389.
[23] Eli J, Mohr-Schroeder M, Lee C. Mathematical connections and their relationship to mathematics knowledge for teaching geometry. Sch Sci Math. 2013;113(3):120–134.
[24] Fischer, K.W. & Knight, C.C. (1990). Cognitive development in real children: Levels and variations. In B. Presseisen (Ed.), Learning and thinking styles: Classroom interaction. Washington: National Education Association.
[25] Fonger, N. L. (2019). Meaningfulness in representational fluency: An analytic lens for students’ creations, interpretations, and connections. The Journal of Mathematical Behavior.
[26] Grabiner, J. V. (1983). The changing concept of change: the derivative from Fermat to Weierstrass. Mathematics Magazine, 56(4), 195-206.
[27] Gundlach, M. R., & Jones, S. R. (2015). Students' understanding of concavity and inflection points in real-world contexts: Graphical, symbolic, verbal, and physical representations. In T. Fukawa-Connelly, N. E.
[28] Hähkiöniemi, M. (2006). Associative and reflective connections between the limit of the difference quotient and limiting process. The Journal of Mathematical Behavior, 25(2), 170–184.
[29] Hauger, G. S. (2000). Instantaneous rate of change: a numerical approach. International Journal of Mathematical Education in Science and Technology, 31(6), 891-897.
[30] Karkdijk, J., van der Schee, J. A., & Admiraal, W. F. (2019). Students' geographical relational thinking when solving mysteries. International Research in Geographical and Environmental Education, 28(1), 5-21.
[31] Kendal, M., & Stacey, K. (2003). Tracing learning of three representations with the differentiation competency framework. Mathematics Education Research Journal, 15(1), 22–41.
[32] Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of educational research, 60(1), 1-64.
[33] Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235–250.
[34] Panizzon, D. (2003). Using a cognitive structural model to provide new insights into students’ understandings of diffusion. International Journal of Science Education, 25(12), 1427-1450.
[35] Panizzon, D. L., Wang, C., & Pegg, J. (2010). Exploring students' scientific understandings in different educational contexts using a cognitive structural model. In International Research in Teacher Education: Current Perspectives (pp. 107-122). ISTE Ltd.
[36] Panizzon, D., Callingham, R. A., Wright, T., & Pegg, J. (2007). Shifting sands: Using SOLO to promote assessment for learning with secondary mathematics and science teachers. In AARE (p. EJ).
[37] Park, J. (2013). Is the derivative a function? If so, how do students talk about it? International Journal of Mathematical Education in Science and Technology, 44(5), 624–640.
[38] Pegg, J. (2003). Assessment in mathematics: A developmental approach. In M. Royer (Ed.), Mathematical Cognition (pp. 227-259). Greenwich, Connecticut: Information Age Publishing.
[39] Pegg, J., & Tall, D. (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. ZDM, 37(6), 468-475.
[40] Piaget, J. (1954). The Construction of reality in the child. New York: Basic Books.
[41] Pino-Fan, L. R., Godino, J. D., & Font, V. (2018). Assessing key epistemic features of didactic-mathematical knowledge of prospective teachers: the case of the derivative. Journal of Mathematics Teacher Education, 21(1), 63-94.
[42] Pino-Fan, L., Godino, J. D., Font, V., & Castro, W. F. (2012, July). Key epistemic features of mathematical knowledge for teaching the derivative. In Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 297-304).
[43] Roorda, G., Vos, P., & Goedhart, M. (2007). The concept of derivative in modeling and applications. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling (Proceedings of ICTMA 12): Education, engineering, and economics (pp. 288–293). Chichester: Horwood Publishing.
[44] Roorda, G., Vos, P., Drijvers, P., & Goedhart, M. (2016). Solving the rate of change tasks with a graphing calculator: a case study on Instrumental Genesis. Digital Experiences in Mathematics Education, 2(3), 228-252.
[45] Roundy, D., Dray, T., Manogue, C. A., Wagner, J., & Weber, E. (2015). An extended theoretical framework for the concept of the derivative. In T. Fukawa-Connelly, N. E. Infante, K. Keene, & M. Zandieh (Eds.).
[46] Smith, S. R. (2015). Incomplete understanding of concepts: The case of the derivative. Mind, 124(496), 1163-1199.
[47] Stålne, K., Kjellström, S., & Utriainen, J. (2016). Assessing complexity in learning outcomes–a comparison between the SOLO taxonomy and the model of hierarchical complexity. Assessment & Evaluation in Higher Education, 41(7), 1033-1048.
[48] Stewart, J. (2016). Single variable calculus: Early Transcendentals. Cengage Learning.
[49] Stump, S. L. (2001). High school precalculus students' understanding of slope as a measure. School Science and Mathematics, 101(2), 81-89.
[50] Walker, J., Halliday, D., Resnick, R., & Trees, B. R. (2018). Fundamentals of physics. Wiley.
[51] Wang, X.Y., Su, Y., Cheung, S., Wong, E. and Kwong, T. (2013). An exploration of Biggs’ constructive alignment in course design and its impact on students’ learning approaches. Assessment and Evaluation in Higher Education. 38 (4), 477-91.
[52] Watson, J., Collis, K., Callingham, R., & Moritz, J. (1995). A Model for Assessing Higher Order Thinking in Statistics. Educational Research and Evaluation, 1 (3), 247-275.
[53] Wells, C. (2015). The structure of observed learning outcomes (SOLO) taxonomy model: How effective is it?.
[54] Yassine, A., Chenouni, D., Berrada, M., & Tahiri, A. (2017). A Serious Game for Learning C Programming Language Concepts Using Solo Taxonomy. International Journal of Emerging Technologies in Learning, 12(3).
[55] Zandieh, M. (1997). The evolution of student understanding of the concept of the derivative. Unpublished Ph.D. dissertation. Corvallis: Oregon State University.
[56] Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of the derivative.
[57] Zaslavsky, O., Sela, H., & Leron, U. (2002). Being sloppy about the slope: The effect of changing the scale. Educational Studies in Mathematics, 49(1), 119-140.