How best to teach trigonometry so that pupils acquire both procedural fluency and conceptual understanding (even in the basic cases of right-angled triangles) is not obvious; but, ironically, it is probably one of the least extensively researched topics – and the results that have been gathered sometimes appear, at least on the surface, contradictory. The identified complexity of the topic reflects that it “may be seen as the confluence of a number of streams of mathematical difficulty" (Pritchard and Simpson, 1999, p1247): pupils need to have a confident grasp of angle, ratio, similarity and algebraic manipulation (and, later, functions), as well as being able to shift their perspective between trigonometric imagery, scale factors and algebraic symbolism. No wonder there is the temptation to resort to vacuous mnemonics such as SOHCAHTOA and procedurally dogmatic algebraic manipulation.
Conceptual understanding of function is essential for successful reasoning about sine and cosine at the higher GCSE tier and for solving equations and differentiating trigonometric functions at A level. But because trigonometry comprises multiple concepts (as described in Key Ideas in Teaching Mathematics - Similarity, ratio and trigonometry in KS3 in Issue 115), including scale factor, ratio and function, how should one introduce the subject?
Craig Pournara (2001, University of the Witwatersrand) compared pupils’ conceptions of trigonometric ratio and function. Where pupils used calculators to produce a ratio in decimal form they were unable to relate it back to the sides of a triangle. But one pupil who treated the ratio as a scale factor was able to work with both trigonometric representations, as the output from a function and as a ratio between the sides of a triangle. Pournara’s unpublished Master’s dissertation is a significant contribution to the research, both in terms of its theoretical approach and its practical investigations; he can be contacted at the university.
Keith Weber (Weber, 2005) argues that trigonometry should be introduced via a function approach and that pupils need to understand a trigonometric function as a process rather than solely as the result of applying a set of rules. He proposes that trigonometric functions can be problematic for at least two reasons: not only are they early examples of functions which cannot be explicitly calculated, but they also map between different (to the pupils) types of mathematical objects: from angles to real numbers. Weber carried out research introducing the functions through use of the unit circle, constructing a variety of right-angled triangles on a Cartesian plane, measuring the lengths of the sides and calculating ratios. The concrete geometric approach, combined with time for reflection, was seen as essential to the conceptualisation of trigonometric function as a process. The lesson activities and follow up work are detailed here. Appreciation of the process enabled pupils to reason about the functions: for example, when asked to approximate cos 340˚ and explain for what values of x the function sin x is decreasing, and to justify their reasoning, a much higher proportion of pupils demonstrated the facility for successful reasoning.
However, the results from this research may be perceived as contradicting those of Kendal and Stacey (Kendal and Stacey, 1996). They considered the solving of right-angled triangle problems and compared the ratio approach (SOHCAHTOA) with the unit circle approach. They found that the right-angled triangle was method more effective for solving these types of problems, but they still recommended introducing the concepts of trigonometry via the unit circle.
What role can dynamic geometry tools play in getting a feel for a trigonometric function as a process? Steer et al argue for the benefits of a dynamic tool over a protractor and ruler (Steer, de Vila, Eaton part 1), combining the ratio and unit circle approach. Steer et al are closer in approach to Weber in that “the investment here is in introducing students to a complete picture of trigonometry - ratio and function”. Detailed descriptions of lessons and activities are provided in Steer, de Vila, Eaton, 2009 part 2.
It is important to recognise the different aims of these research articles: Weber was looking to address the issue of pupils’ conceptual understanding of trigonometric functions, but Kendal and Stacey were comparing ways of learning to solve right-angled triangle problems. As Pournara concludes, “Based on the interview data, it seems that learners’ conceptions of trigonometric ratio are closely tied to the methods they use”. Different aims need different strategies; our responsibility is to read the research and draw on its findings to match our context to our chosen approach as closely as we can. Let us know what you decide to do – perhaps you could contribute a picture to a future Eyes Down article.
Pritchard, L. and Simpson, A. (1999) The role of pictorial images in trigonometry problems, Zaslaysky, O, ed., Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (23rd, Haifa, Israel, July 25-30, 1999) Vol 4, p. 81.
Weber, K. (2005) Students’ Understanding of Trigonometric Functions, Mathematics Education Research Journal 2005, Vol. 17, No. 3, 91–112.
Weber, K. (2008) Teaching Trigonometric Functions: Lessons Learned from Research, Mathematics Teacher, Vol. 102, No. 2, 144 – 150.
Kendal, M. and Stacey, K. (1996) Trigonometry: Comparing Ratio and Unit Circle Methods, Technology in Mathematics Education: Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia, 322 – 329.
Steer, J., de Vila, M. and Eaton, J. (2009) Trigonometry with Year 8: Part 1, Mathematics Teaching, 214, 42 – 44.
Steer, J., de Vila, M. and Eaton, J. (2009) Trigonometry with Year 8: Part 2, Mathematics Teaching, 214i.
Pournara, C. (2001) An investigation into learners’ thinking in trigonometry at Grade 10 level, a research report submitted as part of a degree in Master of Science, School of Science Education in the Faculty of Science, University of the Witwatersrand