Transition to college mathematics

An investigation of trigonometric representations as a source of student difficulties Transition to college mathematics OCMA 30th Annual Conference May 28, 2010 Patricia (Trish) byers

Overview • Positioning the study • defining representations • The research questions • Phases of the study • developing the networks • Research results Transition to College Mathematics

Defining representations • “A representation is a configuration of signs, characters, icons, or objects that can somehow stand for, or “represent” something else ...” (Goldin, 2003, p. 276). 0 1  !  a + b = b + a Transition to College Mathematics

Applying representations • What everyday objects could be represented using a sphere? Transition to College Mathematics

Some trigonometric representations hypotenuse P(x, y) r opposite   adjacent P(cos , sin ) http://cda.morris.umn.edu/~mcquarrb/Precalculus /Animations/SineCosineAnim.html Transition to College Mathematics

Research questions • 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? • How can this research inform the teaching and learning of trigonometry in the secondary and college educational panels? Transition to College Mathematics

Selecting the college program • Electrical engineering technology program • College Mathematics Project results from Fall 2004 • MTCU Program standards – • General education • Generic • Vocational Transition to College Mathematics

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

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

Phases of the study • Phase 1: • Analysis of expert teacher interviews • Phase 2: • Analysis of postsecondary resource textbooks • Phase 3: • Analysis of secondary and college textbooks Transition 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. Secondary school teacher’s representation network • Begins with the right triangle • Linear, methodical, sequential steps suggesting a careful learning trajectory Transition to College Mathematics

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

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

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

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

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

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

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

Question 1 findings • Textbook portrayal of representations • The application of functions • Labelling and measurement • Pedagogical approaches Transition to College Mathematics

Question 2 findings • 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 Transition to College Mathematics

Recommendations • 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 Transition to College Mathematics

Transition to college mathematics