Sdrawkcab scitamehtam: The case for understanding mathematics backwards

Abstract

Despite the widespread acknowledgement of the need for graduates with quantitative problem-solving skills, many students enter university having relied heavily on pattern recognition techniques for high school mathematics. While these can often lead students to obtaining correct solutions for problems similar to those which they have practised, they do not lead to a deeper understanding of the material and, critically, may not develop more widely-applicable skills. Even when correct solutions are obtained, students can sometimes not understand or explain why their solution is indeed correct. Here, I present an argument in favour of avoiding predictability in question structures and, in particular, asking questions ``backwards'' to how they might traditionally be asked. Some mathematical topics readily lend themselves to such approaches --- the Fundamental Theorem of Calculus tells us that an integral problem is inherently linked to an antiderivative problem --- whereas other require much more care and subtlety to avoid routine or predictable assessments. I will discuss some preliminary results from a first year probability subject at the University of Technology Sydney (UTS) which suggest that student understanding of material can be increased when they feel that prioritising pattern recognition over problem solving is unlikely to be rewarded with high marks. References

• Khan academy. expected value. URL: https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/random-variables-discrete/e/expected_value.
• Mark Asiala, Anne Brown, David J. Devries, Ed Dubinsky, David Mathews, and Karen Thomas. A framework for research and curriculum development in undergraduate mathematics education. Maa Notes, pages 37–54, 1997.
• I. Chubb. Science, technology, engineering and mathematics in the national interest: a strategic approach. Australian Government, Canberra, 7 2013. URL: http://www.chiefscientist.gov.au/2013/07/science-technology-engineering-and-mathematics-in-the-national-interest-a-strategic-approach/.
• Ted Dunning. Accurate methods for the statistics of surprise and coincidence. Computational Linguistics, 19(1):61–74, 1993.
• Justin Kruger and David Dunning. Unskilled and unaware of it: How difficulties in recognizing one's own incompetence lead to inflated self-assessments. Journal of Personality and Social Psychology, 77(6):1121–1134, 1999. doi:10.1037/0022-3514.77.6.1121.
• Fengshan Liu. 14 - teaching inverse problems in undergraduate level mathematics, modelling and applied mathematics courses. In Qi-Xiao Ye, Werner Blum, Ken Houston, and Qi-Yuan Jiang, editors, Mathematical Modelling in Education and Culture, pages 165–172. Woodhead Publishing, 2003. doi:10.1533/9780857099556.4.165.
• David S. Moore, George P. McCabe, and Bruce A. Craig. Introduction to the Practice of Statistics. W. H. Freeman and Company, New York, 6th edition, 2009.
• Terezinha Nunes, Peter Bryant, Rossana Barros, and Kathy Sylva. The relative importance of two different mathematical abilities to mathematical achievement. British Journal of Educational Psychology, 82(1):136–156, 2012. doi:10.1111/j.2044-8279.2011.02033.x.
• L. J. Rylands and C. Coady. Performance of students with weak mathematics in first-year mathematics and science. International Journal of Mathematical Education in Science and Technology, 40(6):741–753, 2009. doi:10.1080/00207390902914130.
• R. R. Sokal and F. J. Rohlf. Biometry: The Principles and Practice of Statistics in Biological Research. W. H. Freeman and Company, New York, 2nd edition, 1981.
• Arathi Sriprakash, Helen Proctor, and Betty Hu. Visible pedagogic work: parenting, private tutoring and educational advantage in australia. Discourse: Studies in the Cultural Politics of Education, 37(3):426–441, 2016. doi:10.1080/01596306.2015.1061976.
• Jon R. Star. On the relationship between knowing and doing in procedural learning. In B. Fishman and S. O'Connor-Divelbiss, editors, Proceedings of fourth international conference of the Learning Sciences, pages 80–86. Erlbaum, 2000.
• S. Woodcock and S. Bush. Slipping between the cracks? maximising the effectiveness of prerequisite paths in uts mathematics degrees. In Mark Nelson, Tara Hamilton, Michael Jennings, and Judith Bunder, editors, Proceedings of the 11th Biennial Engineering Mathematics and Applications Conference, EMAC-2013, volume 55 of ANZIAM J., pages C297–C314, 8 2014. URL: http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7943.
• Stephen Woodcock. Development of enquiry-oriented learning in the mathematical sciences. In Mark Nelson, Dann Mallet, Brandon Pincombe, and Judith Bunder, editors, Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C1–C13, 5 2016. URL: http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/10439.
• F. P. Yee. Open-ended problems for higher-order thinking in mathematics. Teaching and Learning, 20(2):49–57, 2000. URL: https://repository.nie.edu.sg/handle/10497/365.