AN INVESTIGATION OF TRIGONOMETRIC REPRESENTATIONS AS A SOURCE OF STUDENT DIFFICULTIES OCMA 30 th Annual Conference May 28, 2010 Patricia (Trish) Byers

 Positioning the study  defining representations  The research questions  Phases of the study  developing the networks  Research results 5/28/20102Transition to College Mathematics

 “A representation is a configuration of signs, characters, icons, or objects that can somehow stand for, or “represent” something else...” (Goldin, 2003, p. 276).  xy a + b = b + a 5/28/20103Transition to College Mathematics

 What everyday objects could be represented using a sphere? 5/28/20104Transition to College Mathematics

 opposite adjacent hypotenuse  P(x, y) r P(cos , sin  ) tsin(t) 00  /2 1  0 3  /2 /Animations/SineCosineAnim.html 5/28/20105Transition to College Mathematics

1. What is the relationship between secondary and college technical mathematics treatment of trigonometry? In particular, how do the representations in secondary school college pathway courses compare to the representations used in a first semester college technology mathematics course? 2. How can this research inform the teaching and learning of trigonometry in the secondary and college educational panels? 5/28/20106Transition to College Mathematics

 Electrical engineering technology program  College Mathematics Project results from Fall 2004  MTCU Program standards –  General education  Generic  Vocational 5/28/2010Transition to College Mathematics7

MFM1P Gr. 9 Applied Foundations of Mathematics MPM1D Gr. 9 Academic Principles of Mathematics MPM2D Gr. 10 Academic Principles of Mathematics MCF3M Gr. 11 University/ College Functions MFM2P Gr. 10 Applied Foundations of Mathematics MBF3C Gr. 11 College Mathematics of Personal Finance MAP4C Gr. 12 College College and Apprenticeship Mathematics MCT4C Gr. 12 College Mathematics for College Technology Mathematics MCR3U Gr. 11 University Functions and Relations Figure 5. Secondary school mathematics pathways (adapted from Ministry of Education, 2000a). 5/28/20108Transition to College Mathematics

MFM1P Gr. 9 Applied Foundations of Mathematics MPM1D Gr. 9 Academic Principles of Mathematics MPM2D Gr. 10 Academic Principles of Mathematics MCF3M Gr. 11 University/ College Functions MFM2P Gr. 10 Applied Foundations of Mathematics MBF3C Gr. 11 College Mathematics of Personal Finance MAP4C Gr. 12 College College and Apprenticeship Mathematics MCT4C Gr. 12 College Mathematics for College Technology Mathematics MCR3U Gr. 11 University Functions and Relations Figure 6. Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000a) highlighting Pathway 1. 5/28/20109Transition to College Mathematics

MFM1P Gr. 9 Applied Foundations of Mathematics MPM1D Gr. 9 Academic Principles of Mathematics MPM2D Gr. 10 Academic Principles of Mathematics MCF3M Gr. 11 University/ College Functions MFM2P Gr. 10 Applied Foundations of Mathematics MBF3C Gr. 11 College Mathematics of Personal Finance MAP4C Gr. 12 College College and Apprenticeship Mathematics MCT4C Gr. 12 College Mathematics for College Technology Mathematics MCR3U Gr. 11 University Functions and Relations Figure 7. Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000a) highlighting Pathway 2. 5/28/201010Transition to College Mathematics

MFM1P Gr. 9 Applied Foundations of Mathematics MPM1D Gr. 9 Academic Principles of Mathematics MPM2D Gr. 10 Academic Principles of Mathematics MCF3M Gr. 11 University/ College Functions MFM2P Gr. 10 Applied Foundations of Mathematics MBF3C Gr. 11 College Mathematics of Personal Finance MAP4C Gr. 12 College College and Apprenticeship Mathematics MCT4C Gr. 12 College Mathematics for College Technology Mathematics MCR3U Gr. 11 University Functions and Relations Figure 8. Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000a) highlighting Pathway 3. 5/28/201011Transition to College Mathematics

 Phase 1:  Analysis of expert teacher interviews  Phase 2:  Analysis of postsecondary resource textbooks  Phase 3:  Analysis of secondary and college textbooks 5/28/201012Transition to College Mathematics

Unit circle Trigonometric ratios Tables Sinusoidal waveform Right triangle Figure 18. Conceptualization of the secondary school teacher’s network of trigonometric representations. Begins with the right triangle Linear, methodical, sequential steps suggesting a careful learning trajectory 5/28/201013Transition to College Mathematics

x = cos  y = sin  (trigonometric circular functions) Vector Trigonometric ratios Sinusoidal waveform Unit circle Figure 22. Conceptualization of the college teacher’s network of trigonometric representations. Begins with the unit circle; no right triangle Non-linear, multiple connections suggesting an integrated approach 5/28/201014Transition to College Mathematics

Vector on the xy -plane Domain Sinusoidal Waveform Domain Trigonometric Functions Domain (0  2  ) Trigonometric Ratios Domain (0  /2) Unit Circle Domain Figure 50. A general representation network of trigonometry. Right Triangle Domain labels - A, B, C, a, b, c labels - opp, adj, hyp, θ labels - opp, adj, hyp,  A Trigonometric functions - x, y, r,  Primary ratios -inverse ratios Secondary ratios -inverse ratios Secondary functions -inverse functions Primary functions - x, y, r,  - inverse functions amplitude/periodicity/ phase shift displacement Circle Trigonometric (circular) functions – x, y,  (0  2  ) -inverse functions -primary and secondary functions Trigonometric functions- t,  Tables Reference angles Right triangle on the xy -plane 5/28/201015Transition to College Mathematics Principles of Euclidean geometry Points/Line segments/Angles/ Triangles xy -plane Slope Transformations Functions/Relationships Accuracy & precision Proportionality Graphing

Vector on the xy -plane Domain Sinusoidal Waveform Domain Trigonometric Functions Domain (0  2  ) Trigonometric Ratios Domain (0  /2) Unit Circle Domain Building the representation network of trigonometry for pathway analysis. Right Triangle Domain labels - A, B, C, a, b, c labels - opp, adj, hyp, θ labels - opp, adj, hyp,  A Trigonometric functions - x, y, r,  Primary ratios -inverse ratios Secondary ratios -inverse ratios Secondary functions -inverse functions Primary functions - x, y, r,  - inverse functions amplitude/periodicity/ phase shift displacement Circle Trigonometric (circular) functions – x, y,  (0  2  ) -inverse functions -primary and secondary functions Trigonometric functions- t,  Tables Reference angles Right triangle on the xy -plane 5/28/201016Transition to College Mathematics

Vector on the xy -plane Domain Sinusoidal Waveform Domain Trigonometric Functions Domain (0  2  ) Trigonometric Ratios Domain (0  /2) Unit Circle Domain Building the representation network of trigonometry for pathway analysis. Right Triangle Domain labels - A, B, C, a, b, c labels - opp, adj, hyp, θ labels - opp, adj, hyp,  A Trigonometric functions - x, y, r,  Primary ratios -inverse ratios Secondary ratios -inverse ratios Secondary functions -inverse functions Primary functions - x, y, r,  - inverse functions amplitude/periodicity/ phase shift displacement Circle Trigonometric (circular) functions – x, y,  (0  2  ) -inverse functions -primary and secondary functions Trigonometric functions- t,  Tables Reference angles Right triangle on the xy -plane 5/28/201017Transition to College Mathematics Vector on the xy -plane Domain

Vector on the xy -plane Domain Sinusoidal Waveform Domain Trigonometric Functions Domain (0  2  ) Trigonometric Ratios Domain (0  /2) Unit Circle Domain Figure 64. The representation network of trigonometry for Pathway 1. Right Triangle Domain labels - A, B, C, a, b, c labels - opp, adj, hyp, θ labels - opp, adj, hyp,  A Trigonometric functions - x, y, r,  Primary ratios -inverse ratios Secondary ratios -inverse ratios Secondary functions -inverse functions Primary functions - x, y, r,  - inverse functions amplitude/periodicity/ phase shift displacement Circle Trigonometric (circular) functions – x, y,  (0  2  ) -inverse functions -primary and secondary functions Trigonometric functions- t,  Tables Reference angles Right triangle on the xy -plane 5/28/201018Transition to College Mathematics

Vector on the xy -plane Domain Sinusoidal Waveform Domain Trigonometric Functions Domain (0  2  ) Trigonometric Ratios Domain (0  /2) Unit Circle Domain Figure 70. The representation network of trigonometry for Pathway 2. Right Triangle Domain labels - A, B, C, a, b, c labels - opp, adj, hyp, θ labels - opp, adj, hyp,  A Trigonometric functions - x, y, r,  Primary ratios -inverse ratios Secondary ratios -inverse ratios Secondary functions -inverse functions Primary functions - x, y, r,  - inverse functions amplitude/periodicity/ phase shift displacement Circle Trigonometric (circular) functions – x, y,  (0  2  ) -inverse functions -primary and secondary functions Trigonometric functions- t,  Tables Reference angles Right triangle on the xy -plane 5/28/201019Transition to College Mathematics Unit semi circle

Vector on the xy -plane Domain Sinusoidal Waveform Domain Trigonometric Functions Domain (0  2  ) Trigonometric Ratios Domain (0  /2) Unit Circle Domain Figure 75. The representation network of trigonometry for Pathway 3. Right Triangle Domain labels - A, B, C, a, b, c labels - opp, adj, hyp, θ labels - opp, adj, hyp,  A Trigonometric functions - x, y, r,  Primary ratios -inverse ratios Secondary ratios -inverse ratios Secondary functions -inverse functions Primary functions - x, y, r,  - inverse functions amplitude/periodicity/ phase shift displacement Circle Trigonometric (circular) functions – x, y,  (0  2  ) -inverse functions -primary and secondary functions Trigonometric functions- t,  Tables Reference angles Right triangle on the xy -plane 5/28/201020Transition to College Mathematics Unit semi circle Unit quarter circle

 Textbook portrayal of representations  The application of functions  Labelling and measurement  Pedagogical approaches 5/28/201021Transition to College Mathematics

 Disconnections to prior knowledge  Disruptions in secondary-college pathway networks  Differences among notation, conventions, and symbols  Different pedagogical approaches  Evidence of critical correspondences  Interconnected source domains 5/28/201022Transition to College Mathematics

 Curriculum coordination across secondary school and college sectors  Emphasis on appropriate course selection and secondary school pathway  Implementation of an investigative approach in college courses 5/28/201023Transition to College Mathematics