Journal of Research in Science, Mathematics and Technology Education

A Unique Experience Learning Calculus: Integrating Variation Theory with Problem-Based Learning

Journal of Research in Science, Mathematics and Technology Education, Online-First Articles, pp. 1-20
OPEN ACCESS VIEWS: 90 DOWNLOADS: 38 Publication date: 15 Jun 2023
The paper proposes a pedagogical approach to teaching and learning calculus differentiation formulas that synthesizes the principles of variation theory (VT) and bianshi in a problem-based learning (PBL) format. Unlike traditional approaches that view formulas procedurally, the paper adapts Steinbring’s (1989) distinction between “concept” and “symbol,” abstracting differentiation calculus formulas as “concept” (i.e., the meaning of the formula) and “symbol” (i.e., procedural knowledge about how to apply the formula). The paper then aligns this distinction with VT and bianshi pedagogies. While VT emphasizes more static elements of conceptual knowledge (e.g., highlighting the contrast between conceptual and non-conceptual features of the object of learning), bianshi broadens the concept of variation, offering more dynamic principles of variation through procedural variation (e.g., via the process of problem solving) (Gu et al., 2004). Combining VT and bianshi into a single pedagogical application yields an eight-step approach to teaching and learning calculus differentiation formulas.
bianshi, calculus, problem-based learning, variation theory
Pogorelova, L. (2023). A Unique Experience Learning Calculus: Integrating Variation Theory with Problem-Based Learning. Journal of Research in Science, Mathematics and Technology Education.
  1. Albanese, M. A., & Mitchell, S. (1993). Problem‐based learning: A review of literature on its outcomes and implementation issues. Academic Medicine, 68, 52–81.
  2. Barrows, H. S. (1996). Problem‐based learning in medicine and beyond: A brief overview. In L. Wilkerson & W. H. Gijselaers (Eds.), New directions in teaching and learning: Issue 68. Bringing problem‐based learning to higher education: Theory and practice (pp. 3–12). Jossey‐Bass.
  3. Bressoud, D. (2021, August 9). Decades later, problematic role of calculus as gatekeeper to opportunity persists. The University of Texas at Austin, Charles A. Dana Center.
  4. Bressoud, D. (2020). The strange role of calculus in the United States. ZDM–Mathematics Education, 53, 521-533.
  5. Canobi, K. H. (2009). Concept–procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102, 131–149.
  6. Chappell, K. K., & Killpatrick, K. (2003). Effects of concept-based instruction on students’ conceptual understanding and procedural knowledge of calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 13(1), 17-37.
  7. Eichler, A., & Erens, R. (2014). Teachers’ beliefs toward teaching calculus. ZDM Mathematics Education, 46, 647-659.
  8. Elshafei, D. L. (1998). A comparison of problem-based and traditional learning in algebra II. (OCLC No.: 44021973) [Doctoral dissertation, Indiana University].
  9. Ferrini-Mundy, J. & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. In J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning: Preliminary analysis and results (pp. 29-45). Washington, D.C.: Mathematical Association of America.
  10. Fothergill, L. (2011). Aspects of calculus for preservice teachers. The Mathematics Educator, 21(1), 23–31.
  11. Furner, J. & Duffy, M. L. (2002). Equity for students in the new millennium: Disabling math anxiety. Intervention in School and Clinic, 38(2), 67–74.
  12. Gallagher, S. A. (1997). Problem-based learning: Where did it come from, what does it do, and where is it going? Journal for the Education of the Gifted, 20(4), 332-362.
  13. Gray, S. S., Loud, B. J., & Sokolowski, C. P. (2009). Calculus students’ use and interpretation of variables: Algebraic vs. Arithmetic Thinking. Canadian Journal of Science, Mathematics and Technology Education, 9(2), 59-72.
  14. Gu, L., et al. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (pp. 309–348). Singapore: World Scientific.
  15. Habre, S., & Abboud, M. (2006). Students’ conceptual understanding of a function and its derivative in an experimental calculus course. The Journal of Mathematical Behavior, 25(1), 57- 72.
  16. Hung, W. (2019). Problem design in PBL. In W. Hung, M. Moallem, & N. Dabbagh (Eds.), The Wiley handbook of problem-based learning (pp. 249-272). Wiley.
  17. Hung, W., Mehl, K., & Holen, J. B. (2013). The relationships between problem design and learning process in problem‐based learning environments: Two cases. The Asia‐Pacific Education Researcher, 22(4), 635–645.
  18. Hung, W., Jonassen, D. H., & Liu, R. (2008). Problem-based learning. In J. M. Spector, J. G. van Merrienboer, M. D. Merrill, & M. Driscoll (Eds.), Handbook of research on educational communications and technology (3rd ed., pp. 485-506). Routledge.
  19. Jimo Icons. (n.d.). Process icon.
  20. Jonassen, D. H. (1997). Instructional design models for well‐structured and III-structured problem‐solving learning outcomes. Educational Technology Research and Development, 45, 65–94.
  21. Jukić, L., & Dahl, B. (2010). University students’ retention of derivative concepts 14 months after the course: Influence of ‘met-befores’ and ‘met-afters.’ International Journal of Mathematical Education in Science and Technology, 43(6), 749-764.
  22. Kattayat, S., & Josey, S. (2019). Improving students’ conceptual understanding of calculus-based physics using a problem-based learning approach on an e-learning platform applied to engineering education. 2019 Advances in Science and Engineering Technology International Conferences (ASET) (pp. 1-6). Institute of Electrical and Electronics Engineers (IEEE).
  23. Kullberg, A., Kempe, U. R., & Marton, F. (2017). What is made possible to learn when using the variation theory of learning in teaching mathematics. ZDM Mathematics Education, 49, 559-569.
  24. Kullberg, A., Martensson, P., & Runesson, U. (2016). What is to be learned? Teachers’ collective inquiry into the object of learning. Scandinavian Journal of Educational Research, 60(3), 309–322.
  25. Luneta, K, & Makonye, P. J. (2010). Learner errors and misconceptions in elementary analysis: a case study of a grade 12 class in South Africa. Acta Didactica Napocensia, 3(3), 35–46.
  26. Maciejewski, W., & Star, J. R. (2016). Developing flexible procedural knowledge in undergraduate calculus. Research in Mathematics Education, 18(3), 299-316.
  27. Maharaj, A. (2013). An APOS analysis of natural science students’ understanding of Derivatives. South African Journal of Education, 33(1), 146-164.
  28. Marton, F., & Pang, M. F. (2013). Meanings are acquired from experiencing differences against a background of sameness, rather than from experiencing sameness against a background of difference: Putting a conjecture to the test by embedding it in a pedagogical tool. Frontline Learning Research, 1(1), 24–41.
  29. Marton, F., Runesson, U., & Tsui, A. B. (2004). The space of learning. In F. Marton and A. B. Tsui (Eds.), Classroom discourse and the space of learning (pp. 17-82). Routledge.
  30. Marton, F., Tsui, A. B., Chik, P. P. M., Ko, P. Y., Lo, M. L., Mok, I. A. C., Ng, D. F. P., Pang, M. F., Pong, W. Y., & Runesson, U. (2004). Classroom discourse and the space of learning. Lawrence Erlbaum.
  31. Mokhtar, M. Z., Tarmizi, M. A. A., Tarmizi, R. A., & Ayub, A. F. M. (2010). Problem-based learning in calculus course: Perception, engagement and performance. WSEAS proceedings on latest trends on engineering education (pp. 21-25).
  32. Moust, J., Bouhuijs, P., & Schmidt, H. (2021). Introduction to problem-based learning: A guide for students (4th ed.). Routledge.
  33. Moust, J., Berkel, H., & Schmidt, H. (2005). Signs of erosion: Reflections on three decades of problem‐based learning at Maastricht University. Higher Education, 50(4), 665–683.
  34. Murphy, L. (2006). Reviewing reformed calculus.
  35. Niss, M. A. (2003). Quantitative literacy and mathematical competencies. In B. L. Madison & L. A. Steen (Eds.), Quantitative literacy: Why numeracy matters for schools and colleges (pp. 215-220). Princeton: National Council on Education and the Disciplines.
  36. Olivier, A. (1989). Handling pupils’ misconceptions.
  37. O’Neill, P. A. (2000). The role of basic sciences in a problem‐based learning clinical curriculum. Medical Education, 34, 608–613.
  38. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 15, 235-250.
  39. Othman, Z. S. B., Bin Khalid, A. K., & Mahat, A. B. (2018). Students’ common mistakes in basic differentiation topics. Proceeding of the 25th National Symposium on Mathematical Sciences. AIP Conference Proceedings 1974, 050009.
  40. Pang, M. F., et al. (2017). ‘Bianshi’ and the variation theory of learning: Illustrating two frameworks of variation and invariance in the teaching of mathematics. In R. Huang & Y. Li (Eds.), Teaching and learning mathematics through variation: Confucian heritage meets Western theories (pp. 43-67). Sense Publishers.
  41. Pyzdrowski, L. J., Sun, Y., Curtis, R., Miller, D., Winn, G., Hensel, R. A. M. (2013). Readiness and attitudes as indicators for success in college calculus. International Journal of Science and Mathematics Education, 11, 529-554.
  42. Qi, C., Wang, R., Mok, I. A. C., & Huang, D. (2017). Teaching the formula of perfect square through bianshi teaching. In R. Huang & Y. Li (Eds.), Teaching and learning mathematics through variation: Confucian heritage meets Western theories (pp. 127-150). Sense Publishers.
  43. Quezada, V. D. (2020). Difficulties and performance in mathematics competences: Solving problems with derivatives. International Journal of Engineering Pedagogy, 10(4), 35-52.
  44. Rézio, S., Andrade, M. P., & Teodoro, M. F. (2022). Problem-based learning and applied mathematics. Mathematics, 10(16), 2862.
  45. Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review.
  46. Romito, L. M., & Eckert, G. J. (2011). Relationship of biomedical science content acquisition performance to students’ level of PBL group interaction: Are students learning during PBL group? Journal of Dental Education, 75(5), 653–664.
  47. Ruslimin, A., Masriyah, & Manuharawati. (2019). Difficulties of undergraduate students to understand 2nd calculus. International Journal of Trends in Mathematics Research, 2(1), 26-30.
  48. Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525-1538.
  49. Simangunsong, S., & Parsaoran, D. (2021). The influence of problem based learning strategy with basic calculus approach. Jurnal Riset Fisika Educasi Dan Sains [Research Journal of Educational Physics and Science], 8(1), 69-80.
  50. Siti Fatimah, Y. (2019). Analysis of difficulty learning calculus subject for mathematical education students. International Journal of Scientific and Technology Research 8(3), 80-84.
  51. Siyepu, S. W. (2015). Analysis of errors in derivatives of trigonometric functions. International Journal of STEM Education, 2(16).
  52. 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.
  53. Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: a constructivist analysis of knowledge in transition. The Journal of the Learning Science, 3(2), 115-163.
  54. Socas, M. M. (1997). Math education in medium school: Difficulties, obstacles and errors in the learning of mathematics in secondary education. In L. Rico (Ed.), La educación matemática en la enseñanza secundaria [Mathematical education in secondary teaching] (pp. 125-152). Editorial Horsori.
  55. Sofronas, K. S., DeFranco, T. C., Vinsonhaler, C., Gorgievski, N., Schroeder, L., & Hamelin, C. (2011). What does it mean for a student to understand the first-year calculus? Perspectives of 24 experts. The Journal of Mathematical Behavior, 30(2), 131-148.
  56. Steinbring, H. (1989). Routine and meaning in the mathematics classroom. For the Learning of Mathematics, 9(1), 24-33.
  57. Stewart, J., Clegg, D. K., & Watson, S. (2021). Single variable calculus (9th ed.). Cengage.
  58. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.
  59. Ting, J. J., Tarmizi, R. A., Abu Bakar, A., & Aralas, D. (2018). Effects of variation theory approach in teaching and learning of algebra on urban and rural students’ algebraic achievement and motivation. International Journal of Mathematical Education in Science and Technology, 49(7), 986-1002.
  60. Toh, T. L. (2009). On in-service mathematics teachers’ content knowledge of calculus and related concepts. The Mathematics Educator, 12(1), 69–86.'%20Calculus%20Knowledge.pdf
  61. Watson, A. (2017). Pedagogy of variations: Synthesis of various notions of variation pedagogy. In R. Huang & Y. Li (Eds.), Teaching and learning mathematics through variation: Confucian heritage meets Western theories (pp. 43-67). Sense Publishers.
  62. Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209-234.
  63. Wewe, M. (2020). The profile of students’ learning difficulties in concepts mastery in calculus course. Decimal: Journal of Mathematics, 3(2), 161-168.
  64. Wijnia, L., Loyens, S. M. M., & Rikers, M. J. P. (2019). The problem-based learning process: An overview of different models. In M. Moallem, W. Hung, & N. Dabbagh, (Eds.), The Wiley handbook of problem-based learning (pp. 273-295). Wiley.
  65. Wu, L., & Liu, B. (2006). The three point of variation pedagogy. Mathematics Bulletin, 4, 18-19.
  66. Yuan, Q. (2006). The phycology analysis of variation pedagogy. Mathematics Communication, 3, 4-5.
Creative Commons License