Innovative approaches to teaching mathematics in higher education: a review and critique
DOI:
https://doi.org/10.7146/nomad.v17i2.148420Abstract
This paper provides a snapshot of emerging trends in mathematics teaching in higher education for STEM subjects (Science, Technology, Engineering and Mathematics). Overwhelmingly, papers identify a focus on conceptual understandings of mathematics in comparison to understanding that is instrumental or procedural. Calls for reform of mathematics teaching have been the basis for a range of studies; responses to these calls have embraced innovative methods for implementing changes in learning and teaching of mathematics, sometimes rooted in constructivist ideology. Observed trends have been categorised in six groups. In many studies, technology is being used as an enabler of reforms. Constraints to implementing new approaches in mathematics teaching are indicated. Discussion of contemporary research questions that could be asked as a result of the shift towards teaching mathematics in innovative ways is provided and is followed by a critique of the underlying theoretical positions, essentially that of constructivism.
References
Abaté, C. J. & Cantone, K. (2005). An evolutionary approach to mathematics education: enhancing learning through contextual modification. Primus, 15 (2), 157-176. https://doi.org/10.1080/10511970508984115
Abdulwahed, M. & Nagy, Z.K. (2011). The TriLb, a novel ICT based model of laboratory education. Computers & Education, 56 (1), 262-274. https://doi.org/10.1016/j.compedu.2010.07.023
Abramovich, S. & Grinshpan, A. (2008). Teaching mathematics to non- mathematics majors through applications. Primus, 18 (5), 411-428. https://doi.org/10.1080/10511970601182772
Alsardary, S. & Blumberg, P. (2009). Interactive, learner-centered methods of teaching mathematics. Primus, 19 (4), 401-416. https://doi.org/10.1080/10511970701678596
Alsina, C. (2001). Why the professor must be a stimulating teacher: towards a new paradigm of teaching mathematics at university level. In D. A. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 3-12). Dordrecht: Kluwer. https://doi.org/10.1007/0-306-47231-7_1
Anku, S. (1996). Fostering students' disposition towards mathematics: a case from Canadian university. Education, 116 (4), 536-542.
Atkinson, R. C. & Shiffrin, R.M. (1968). Human memory: a proposed system and its component processes. The Psychology of Learning and Motivation (Vol. 2). New York: Academic Press. https://doi.org/10.1016/S0079-7421(08)60422-3
Aydin, N. (2009). Enhancing undergraduate mathematics curriculum via coding theory and cryptography. Primus, 19 (3), 296-309. https://doi.org/10.1080/10511970701590627
Bandalos, D. L., Yates, K. & Thorndike-Christ, T. (1995). Effects of math self- concept, perceived self-efficacy, and attributions for failure and success on test anxiety. Journal of Educational Psychology, 87 (4), 611-623. https://doi.org/10.1037/0022-0663.87.4.611
Barrows, H. S. & Tamblyn, R. M. (1980). Problem-based learning: an approach to medical education. New York: Springer.
Berg, C. V. (2009). Developing algebraic thinking in a community of inquiry (Unpublished PhD thesis). University of Agder, Norway.
Brito, I., Figueiredo, J., Flores, M., Jesus, A., Machado, G. et al. (2009). Using e-learning to self regulate the learning process of mathematics for engineering students. In A. Bulucea, V. Mdladenov, E. Pop, M. Leba & M. Mastorakis (Eds.), Recent advances in applied mathematics. Proceedings of the 14th international conference on applied mathematics (MATH -09), Puerto de la Cruz, Spain (pp. 165-169). WSEAS Press. Retrieved January 9, 2013 from http://www.wseas.us/e-library/conferences/2009/tenerife/MATH/MATH-25.pdf
Brown, S., Collins, A. & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Research, 18, 32-42. https://doi.org/10.2307/1176008
Brooks, J. G. & Brooks, M. G. (1993). In search of understanding: the case for constructivist classrooms. Alexandria: Association for Supervision and Curriculum Development.
Burks, R. (2011). Survivor math: using pop culture to enhance learning mathematics. Primus, 21 (1), 62-72. https://doi.org/10.1080/10511970902870364
Burns, M. (1998). Math: facing an american phobia. Sausalito: Math Solutions Publications.
Burton, L. (2002). Methodology and methods in mathematics education research: Where is "The Why"? In S. Goodchild & L. English (Eds.), Researching mathematics classrooms: a critical examination of methodology (pp. 1-10), London: Praeger. https://doi.org/10.5040/9798216007920.0006
Campbell, N. K. & Hackett, G. (1986). The effects of mathematics task performance on math self-efficacy and task interest. Journal of Vocational Behavior, 28 (2), 149-162. https://doi.org/10.1016/0001-8791(86)90048-5
Carter, J. (2009). Lines of communication: using a WIKI in a mathematics course. Primus, 19 (1), 1-17. https://doi.org/10.1080/10511970802475124
Chang, J. M. (2011). A practical approach to inquiry-based learning in linear algebra. International Journal of Mathematical Education in Science and Technology, 42 (2), 245-259. https://doi.org/10.1080/0020739X.2010.519795
Collins, A. (1986). Different goals of inquiry teaching. Camebridge: BBN Laboratories Incorporated. Retrieved January 9, 2013 from http://thorndike.tc.columbia.edu/~david/MTSU4083/Readings/Inquiry-based%20ID/Collins-DifferentGoalsofInquiryTeaching.pdf
Confrey, J. (2000). Leveraging constructivism to apply to systemic reform. Nordic Studies in Mathematics Education, 8 (3), 7-30.
Craik, F. I. M. & Tulving, E. (1975). Depth of processing and the retention of words in episodic memory. Journal of Experimental Psychology: General, 104, 268-294. https://doi.org/10.1037/0096-3445.104.3.268
Daniels, H. (2001). Vygotsky and pedagogy. London: Routledge. https://doi.org/10.4324/9780203469576
DeLong, M. & Winter, D. (1998). Addressing difficulties with student-centered instruction. Primus, 8 (4), 340-364. https://doi.org/10.1080/10511979808965909
Dewey, J. (1998). Experience and education: the 60th anniversary edition. Indianapolis: Kappa Delta Pi. (Original work published in 1938.)
Donaldson, M. (1978). Children's minds. London: Fontana.
Doolittle, P. E. (1999). Constructivism and online education. In Proceedings of the International Conference on Teaching Online in Higher Education. Retrieved January 9, 2013 from http://en.scientificcommons.org/42452964.
Driver, R. (1995). Constructivist approaches to science teaching. In L. Steffe & J. Gale (Eds.), Constructivism in education (pp. 385-400). Hillsdale: Lawrence Erlbaum Associates.
Edwards, D. & Mercer, N. (1987). Common knowledge. London: Falmer Press. Ernest, P. (1991). The philosophy of mathematics education. London: Falmer Press.
Felder, R. M. & Silverman, L. K. (1988). Learning and teaching styles in engineering education. Engineering Education, 78 (7), 674-681.
Fielding-Wells, J. & Makar, K. (2008). Student (dis)engagement in mathematics. In AARE 2008 International education conference Brisbane: Changing climates: education for sustainable futures (pp. 1-10). AARE. Retrieved January 9, 2013 from http://aare.edu.au/08pap/mak08723.pdf
Gallegos, I. & Flores, A. (2010). Using student-made games to learn mathematics. Primus, 20 (5), 405-417. https://doi.org/10.1080/10511970802353644
Gardner, H. (1983). Frames of mind: the theory of multiple intelligences. New York: Basic Books.
Glasersfeld, E. von (1987a). Constructivism in education. In T. Husen & T. N. Postlethwaite (Eds.), International encyclopedia of education (Supplement Vol. 1, pp. 162-163). Oxford: Pergamon.
Glasersfeld, E. von (1987b). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 3-18). Hillsdale: Lawrence Erlbaum.
Hadgraft, R. G. (1998). PBL - a vital step towards a new work environment. Journal of Engineering Education, 14 (1), 14-23. https://doi.org/10.1080/17486839808412601
Hagerty, G. W. & Smith, S. (2005). Using web based interactive software to enhance college algebra. Mathematics and Computer Education, 39 (3), 183-194.
Hagerty G., Smith S. & Goodwin D. (2010). Redesigning college algebra: combining educational theory and web-based learning to improve student attitudes and performance. Primus, 20 (5), 418-437. https://doi.org/10.1080/10511970802354527
Harvey L., Drew, S. & Smith, M. (2006). The first year experience: a review of literature for the higher education academy. York: The Higher Education Academy.
Hekimoglu, S. & Kittrell, E. (2010). Challenging students' beliefs about mathematics: the use of documentary to alter perceptions of efficacy. Primus, 20 (4), 299-331. https://doi.org/10.1080/10511970802293956
Henderson, S. & Broadbridge, P. (2007). Mathematics for 21st century engineering students. In Proceedings of the 2007 AaeE Conference, Melbourne (pp. 1-8). Retrieved January 9, 2013 from http://www.aaee.com.au/conferences/papers/2007/inv_Hend.pdf
Hergenhahn, B. & Olson, M. H. (2004). An introduction to theories of learning (7th ed.). Upper Saddle River: Prentice Hall.
Hilton, P. J. (1981). Avoiding math avoidance. In L. A. Steen (Ed.), Mathematics tomorrow (pp. 73-82). New York: Springer-Verlag. https://doi.org/10.1007/978-1-4613-8127-3_8
Hillel, J. (2001). Trends in curriculum: a working group report. In D. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 59-70). Dordrecht: Kluwer. https://doi.org/10.1007/0-306-47231-7_7
Hodge, J. K. (2006). The top ten things I have learned about discovery-based teaching. Primus, 16 (2), 154-161. https://doi.org/10.1080/10511970608984143
Howson, A. G. & Kahane, J. P. (1990). The popularisation of mathematics. Cambridge University Press. https://doi.org/10.1017/CBO9781139013512
Jaworski, B. (1994). Investigating mathematics teaching: a constructivist enquiry. London: Falmer Press.
Jaworski, B. (2010). Challenge and support in undergraduate mathematics for engineers in a GeoGebra medium. MSOR Connections, 10 (1), 10-14. https://doi.org/10.11120/msor.2010.10010010
Jaworski, B., Fuglestad, A. B., Bjuland, R., Breiteig, T., Goodchild, S. & Grevholm, B. (2007). Learning communities in mathematics. Bergen: Caspar Forlag.
Jaworski, B. & Matthews J. (2011). Developing teaching of mathematics to first year engineering students. Teaching Mathematics and its Applications, 30 (4), 178-185. https://doi.org/10.1093/teamat/hrr020
Jaworski, B., Robinson, C., Matthews, J. & Croft, A. C. (2012). An activity theory analysis of teaching goals versus student epistemological positions. International Journal of Technology in Mathematics Education, 19 (4), 147-152.
Johnson, A., Kimball, R., Melendez, B., Myers, L., Rhea, K. & Travis, B. (2009). Breaking with tradition: preparing faculty to teach in a student-centered or problem-solving environment. Primus, 19 (2), 146-160. https://doi.org/10.1080/10511970802409164
Jones, M. (2008). Pedagogical literature: What can be learned and where to begin? Primus, 18 (3), 291-298. https://doi.org/10.1080/10511970601182749
King, S. O. & Robinson, C. L. (2009). Pretty lights and maths! Increasing student engagement and enhancing learning through the use of electronic voting systems. Computers & Education, 53 (1), 189-199. https://doi.org/10.1016/j.compedu.2009.01.012
Kolb, D. A. (1984). Experiential learning: experience as the source of learning and development. Englewood Cliffs: Prentice-Hall.
Kulhavy, R. W. & Wager, W. (1993). Feedback in programmed instruction: historical context and implications for practice. In J. V. Dempsey & G. C. Sales (Eds.), Interactive instruction and feedback (pp. 3-20). Englewood Cliffs: Educational Technology.
Lazakidou, G. & Retalis, S. (2010). Using computer supported collaborative learning strategies for helping students acquire self-regulated problem- solving skills in mathematics. Computers & Education, 54 (1), 3-13. https://doi.org/10.1016/j.compedu.2009.02.020
Lent, R. W., Lopez, F. G., Brown, S. D. & Gore, P. A. (1996). Latent structure of the sources of mathematics self-efficacy. Journal of Vocational Behavior. 49 (3), 292-308. https://doi.org/10.1006/jvbe.1996.0045
Lerman, S. (1996). Intersubjectivity in mathematics learning: a challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27 (2), 133-150. https://doi.org/10.5951/jresematheduc.27.2.0133
Lindfors, J. W. (1999). Children's inquiry. Using language to make sense of the world. New York: Teachers College Press.
Love, E. (1988). Evaluating mathematical activity. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 249-262). London: Hodder and Stoughton.
Lucas, A. (2009). Using peer instruction and I-Clickers to enhance student participation in calculus. Primus, 19 (3), 219-231. https://doi.org/10.1080/10511970701643970
Martin, J. (1993). Episodic memory: a neglected phenomenon in the psychology of education. Educational Psychologist, 28 (2), 169-184. https://doi.org/10.1207/s15326985ep2802_5
Matthews, K., Adams, P. & Goos, M. (2009). Putting it into perspective: mathematics in the undergraduate science curriculum. International Journal of Mathematical Education in Science and Technology, 40 (7), 891-902. https://doi.org/10.1080/00207390903199244
Maull, W. & Berry, J. (2000). A questionnaire to elicit the mathematical concept images of engineering students. International Journal for Mathematics Education in Science and Technology, 31 (6), 899-917. https://doi.org/10.1080/00207390050203388
Mayer, R. H. (1999). Designing instruction for constructivist learning. In C. M. Reigeluth (Ed.), Instructional-design theories and models: a new paradigm of instructional theory (Vol. II.) (pp. 141-159). Mahwah: Lawrence Erlbaum Associates.
McCarthy, B. (1986). The 4MAT system: teaching to learning styles with right-left mode techniques. Barrington: EXCEL Inc.
Mokhtar, M. Z., Tarmizi, R. A., Ayub, M. & Tarmizi, M. A. A. (2010). Enhancing calculus learning engineering students through problem-based learning. WSEAS Transactions on Advances in Engineering Education, 7 (8), 255-264.
Mtenga P. V. & Spainhour, L. K. (2000). Applications of mathematical software packages in structural engineering education and practice. Journal of Computing in Civil Engineering, 14 (4), 273-278. https://doi.org/10.1061/(ASCE)0887-3801(2000)14:4(273)
Niu, V. B. & Shing, W. L. (2010). Implementing problem based learning in mathematical studies using graphing calculator and real time data streamer. In Fifteenth Asian Technology Conference in Mathematics ACTM 2010, University of Malaya Kuala Lumpur, Malaysia 17-21 December. Retrieved January 9, 2013 from http://atcm.mathandtech.org/EP2010/regular/3052010_18457.pdf
Orton, T. & Roper, T. (2000). Science and mathematics: a relationship in need of counselling? Studies in Science Education, 35 (1), 123-153. https://doi.org/10.1080/03057260008560157
Pajares, F. & Miller, M. D. (1997). Mathematics self-efficacy and mathematical problem solving: implications of using different forms of assessment. Journal of Experimental Education, 65 (3), 213-229. https://doi.org/10.1080/00220973.1997.9943455
Palincsar, A. S. (1998). Social constructivist perspectives on teaching and learning. Annual Review of Psychology, 49, 345-375. https://doi.org/10.1146/annurev.psych.49.1.345
Pavlov, I. P. (1927). Conditioned reflexes. London: Clarendon Press.
Pennell, S., Avitabile, P. & White, J. (2009). An engineering-oriented approach to the introductory differential equations course. Primus, 19 (1), 88-99. https://doi.org/10.1080/10511970701474111
Peterson, E. (2009). Using a Wiki to enhance cooperative learning in a real analysis course. Primus, 19 (1), 18-28. https://doi.org/10.1080/10511970802475132
Piaget, J. (1950). The psychology of intelligence. London: Routledge.
Plowden Report. (1967). Children and their primary schools. London: Central Advisory Council for Education.
Ponte, J. P. da (2001). Investigating mathematics and learning to teach mathematics. In F.-L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 53-72). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-010-0828-0_3
Potocka, K. (2010). An entirely-online developmental mathematics course: creation and outcomes. Primus, 20 (6), 498-516. https://doi.org/10.1080/10511970802398151
Reid, T. & King, S. (2009). Pendulum motion and differential equations. Primus, 19 (2), 205-217. https://doi.org/10.1080/10511970701693942
Richardson, V. (2003). Constructivist pedagogy. The Teachers College Record, 105 (9), 1623-1640. https://doi.org/10.1177/016146810310500906
Roddick, C. D. (2001). Differences in learning outcomes: calculus & Mathematica vs. traditional calculus. Primus, 11 (2), 161-184. https://doi.org/10.1080/10511970108965986
Savery, J. R. & Duffy, T. M. (1995). Problem-based learning: an instructional model and its constructivist framework. Educational Technology, 35 (5), 31-38.
Schoenfeld, A. H. (1989). Exploration of students' mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20 (4), 338-355. https://doi.org/10.2307/749440
Skinner, B. F. (1953). Science and human behavior. New York: Macmillan.
Smith, G. H. & Wood, L. N. (2000). Assessment of learning in university mathematics. International Journal of Mathematical Education in Science and Technology, 31 (1), 125-132. https://doi.org/10.1080/002073900287444
Squire, L. R., Knowlton, B. & Musen, G. (1993). The structure and organization of memory. Annual Review of Psychology, 44, 453-495. https://doi.org/10.1146/annurev.ps.44.020193.002321
Steffe, L. P. (1983). The teaching experiement methodology in a constructivist research program. In M. Zweng, T. Green, J. Kilpatrick, H. Pollack, & M. Suydam (Eds.), Proceedings of the 4th international congress of mathematical education (pp. 469-471). Boston: BirkhÑuser.
Steffe, L. P. (1990). On the knowledge of mathematics teachers. In R. B. Davis, C. A. Maher & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education, Monograph number 4 (pp. 167-186). Reston: National Council of Teachers of Mathematics. https://doi.org/10.2307/749919
Steffe, L. P. & Gale, J. E. (Eds.). (1995). Constructivism in education. Hillsdale: Lawrence Erlbaum Associates Inc.
Steffe, L. P. & Thompson, P. W. (2000). Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education, 31 (2), 191-209. https://doi.org/10.2307/749751
Taylor, P. & Campbell-Williams, M. (1993). Discourse towards balanced rationality in the high school mathematics classroom: ideas from Habermas' critical theory. In P. C. S. Taylor & A. J. Malone (Eds.), Constructivist interpretations of teaching and learning mathematics (pp. 135-148). Perth: Curtin University of Technology.
Thorndike, E. L. (1913). Educational psychology: the psychology of learning. New York: Teachers' College Press. https://doi.org/10.1037/13051-000
Turner, P. (2008). A predictor-corrector process with refinement for first-year calculus transition support. Primus, 18 (4), 370-393. https://doi.org/10.1080/10511970601131639
TurningPoint. (2011). TurningPoint - interactive responses systems (School, Universities, Business). Available February 4, 2011 from http://www.turningtechnologies.co.uk
Tynjälä, P. (1999). Towards expert knowledge? A comparison between a constructivist and a traditional learning environment in the university. International Journal of Educational Research, 31 (5), 357-442. https://doi.org/10.1016/S0883-0355(99)00012-9
Vygotsky, L. S. (1978). Mind in society: the development of higher psychological processes. Cambridge: Harvard University Press.
Ward, B., Campbell, S., Goodloe, M., Miller, A. J., Kleja, et al. (2010). Assessing a mathematical inquiry course: Do students gain an appreciation for mathematics? Primus, 20 (3), 183-203. https://doi.org/10.1080/10511970801992921
Waldvogel, J. (2006). Teaching mathematics to engineering students at ETH: coping with the diversity of engineering studies. IDEA League Workshop on Mathematics in Engineering. Imperial College, London.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
