TRANSITION TO COLLEGE MATHEMATICS AN INVESTIGATION OF TRIGONOMETRIC

Overview � Positioning the study defining representations � The research questions � Phases of

Defining representations � “A representation is a configuration of signs, characters, icons, or objects

Applying representations � What everyday objects could be using a sphere? 5/28/2010 Transition to

Some trigonometric representations e te o p y h s u n P(x, y)

Research questions 1. What is the relationship between secondary and college technical mathematics treatment

Selecting the college program � Electrical engineering technology program � College Mathematics Project results

Pathways to college technology programs (2000) MCR 3 U Gr. 11 University Functions and

Pathway 1 MCR 3 U Gr. 11 University Functions and Relations MPM 1 D

Pathway 2 MCR 3 U Gr. 11 University Functions and Relations MPM 1 D

Pathway 3 MCR 3 U Gr. 11 University Functions and Relations MPM 1 D

Phases of the study � Phase 1: Analysis of expert teacher interviews � Phase

Secondary school teacher’s representation network Trigonometric ratios Tables Unit circle Sinusoidal waveform Right triangle

College teacher’s representation network Unit circle Trigonometric ratios Vector x = cos y =

Primary ratios -inverse ratios labels - opp, adj, hyp, θ Secondary ratios -inverse ratios

Question 1 findings � Textbook portrayal of representations � The application of functions �

Question 2 findings � Disconnections to prior knowledge � Disruptions in secondary-college pathway networks

Recommendations � Curriculum coordination across secondary school and college sectors � Emphasis on appropriate

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TRANSITION TO COLLEGE MATHEMATICS AN INVESTIGATION OF TRIGONOMETRIC REPRESENTATIONS AS A SOURCE OF STUDENT DIFFICULTIES OCMA 30 th 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 5/28/2010 Transition to College Mathematics 2

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). 1 0 ! x y -1 2 0 0 1 -2 a+b=b+a 5/28/2010 Transition to College Mathematics 3

Applying representations � What everyday objects could be using a sphere? 5/28/2010 Transition to College Mathematics represented 4

Some trigonometric representations e te o p y h s u n P(x, y) adjacent 5/28/2010 r opposite P(cos , sin ) Transition to College Mathematics t sin(t) 0 0 /2 1 0 3 /2 -1 http: //cda. morris. umn. edu/~mcquarrb/Precalculus /Animations/Sine. Cosine. Anim. html 5

Research questions 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? 5/28/2010 Transition to College Mathematics 6

Selecting the college program � Electrical engineering technology program � College Mathematics Project results from Fall 2004 � MTCU Program standards – General education Generic Vocational 5/28/2010 Transition to College Mathematics 7

Pathways to college technology programs (2000) MCR 3 U Gr. 11 University Functions and Relations MPM 1 D Gr. 9 Academic Principles of Mathematics MFM 1 P Gr. 9 Applied Foundations of Mathematics MPM 2 D Gr. 10 Academic Principles of Mathematics MFM 2 P Gr. 10 Applied Foundations of Mathematics MCF 3 M Gr. 11 University/ College Functions MBF 3 C Gr. 11 College Mathematics of Personal Finance MCT 4 C Gr. 12 College Mathematics for College Technology Mathematics MAP 4 C Gr. 12 College and Apprenticeship Mathematics Figure 5. Secondary school mathematics pathways (adapted from Ministry of Education, 2000 a). 5/28/2010 Transition to College Mathematics 8

Pathway 1 MCR 3 U Gr. 11 University Functions and Relations MPM 1 D Gr. 9 Academic Principles of Mathematics MFM 1 P Gr. 9 Applied Foundations of Mathematics MPM 2 D Gr. 10 Academic Principles of Mathematics MFM 2 P Gr. 10 Applied Foundations of Mathematics MCF 3 M Gr. 11 University/ College Functions MBF 3 C Gr. 11 College Mathematics of Personal Finance MCT 4 C Gr. 12 College Mathematics for College Technology Mathematics MAP 4 C Gr. 12 College and Apprenticeship Mathematics 1 Figure 6. Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000 a) highlighting Pathway 1. 5/28/2010 Transition to College Mathematics 9

Pathway 2 MCR 3 U Gr. 11 University Functions and Relations MPM 1 D Gr. 9 Academic Principles of Mathematics MFM 1 P Gr. 9 Applied Foundations of Mathematics MPM 2 D Gr. 10 Academic Principles of Mathematics MFM 2 P Gr. 10 Applied Foundations of Mathematics MCF 3 M Gr. 11 University/ College Functions MBF 3 C Gr. 11 College Mathematics of Personal Finance MCT 4 C Gr. 12 College Mathematics for College Technology Mathematics 2 MAP 4 C Gr. 12 College and Apprenticeship Mathematics Figure 7. Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000 a) highlighting Pathway 2. 5/28/2010 Transition to College Mathematics 10

Pathway 3 MCR 3 U Gr. 11 University Functions and Relations MPM 1 D Gr. 9 Academic Principles of Mathematics MFM 1 P Gr. 9 Applied Foundations of Mathematics MPM 2 D Gr. 10 Academic Principles of Mathematics MFM 2 P Gr. 10 Applied Foundations of Mathematics MCF 3 M Gr. 11 University/ College Functions MBF 3 C Gr. 11 College Mathematics of Personal Finance MCT 4 C Gr. 12 College Mathematics for College Technology Mathematics 3 MAP 4 C Gr. 12 College and Apprenticeship Mathematics Figure 8. Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000 a) highlighting Pathway 3. 5/28/2010 Transition to College Mathematics 11

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 5/28/2010 Transition to College Mathematics 12

Secondary school teacher’s representation network Trigonometric ratios Tables Unit circle Sinusoidal waveform Right triangle Figure 18. Conceptualization of the secondary school teacher’s network of trigonometric representations. • • 5/28/2010 Begins with the right triangle Linear, methodical, sequential steps suggesting a careful learning trajectory Transition to College Mathematics 13

College teacher’s representation network 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. • • 5/28/2010 Begins with the unit circle; no right triangle Non-linear, multiple connections suggesting an integrated approach Transition to College Mathematics 14

Primary ratios -inverse ratios labels - opp, adj, hyp, θ Secondary ratios -inverse ratios labels - opp, adj, hyp, A Trigonometric Ratios Domain (0 /2) labels A, B, C, a, b, c Right Triangle Domain Principles of Euclidean geometry Points/Line segments/Angles/ Triangles xy-plane Slope Transformations Functions/Relationships Accuracy & precision Proportionality Graphing Right triangle on the xy-plane Reference angles Primary functions - x, y, r, -inverse functions Vector on the xy-plane Domain Sinusoidal Waveform Domain Trigonometric functions - x, y, r, Secondary functions -inverse functions Trigonometric Functions Domain (0 2 ) Trigonometric (circular) functions – x, y, (0 2 ) -inverse functions -primary and secondary functions Tables Unit Circle Domain Circle amplitude/periodicity/ phase shift displacement Trigonometric functions- t, Figure 50. A general representation network of trigonometry. 5/28/2010 Transition to College Mathematics 15

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) labels A, B, C, a, b, c Right Triangle Domain Right triangle on the xy-plane Reference angles Secondary functions -inverse functions Trigonometric Functions Domain (0 2 ) Trigonometric (circular) functions – x, y, (0 2 ) -inverse functions -primary and secondary functions Tables Vector on the xy-plane Domain Unit Circle Domain Sinusoidal Waveform Domain Trigonometric functions - x, y, r, Circle amplitude/periodicity/ phase shift displacement Trigonometric functions- t, Building the representation network of trigonometry for pathway analysis. 5/28/2010 Transition to College Mathematics 16

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) labels A, B, C, a, b, c Right Triangle Domain Right triangle on the xy-plane Reference angles Secondary functions -inverse functions Trigonometric Functions Domain (0 2 ) Trigonometric (circular) functions – x, y, (0 2 ) -inverse functions -primary and secondary functions Tables Vector on the xy-plane Domain Unit Circle Domain Sinusoidal Waveform Domain Trigonometric functions - x, y, r, Circle amplitude/periodicity/ phase shift displacement Trigonometric functions- t, Figure 64. The representation network of trigonometry for Pathway 1. 5/28/2010 Transition to College Mathematics 18

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) labels A, B, C, a, b, c Right Triangle Domain Right triangle on the xy-plane Reference angles Secondary functions -inverse functions Trigonometric Functions Domain (0 2 ) Trigonometric (circular) functions – x, y, (0 2 ) -inverse functions -primary and secondary functions Tables Vector on the xy-plane Domain Unit Circle Domain Sinusoidal Waveform Domain Trigonometric functions - x, y, r, Circle Unit semi circle amplitude/periodicity/ phase shift displacement Trigonometric functions- t, Figure 70. The representation network of trigonometry for Pathway 2. 5/28/2010 Transition to College Mathematics 19

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) labels A, B, C, a, b, c Right Triangle Domain Right triangle on the xy-plane Reference angles Secondary functions -inverse functions Trigonometric Functions Domain (0 2 ) Trigonometric (circular) functions – x, y, (0 2 ) -inverse functions -primary and secondary functions Tables Vector on the xy-plane Domain Unit Circle Domain Sinusoidal Waveform Domain Trigonometric functions - x, y, r, Circle Unit semi circle amplitude/periodicity/ phase shift displacement Unit quarter circle Trigonometric functions- t, Figure 75. The representation network of trigonometry for Pathway 3. 5/28/2010 Transition to College Mathematics 20

Question 1 findings � Textbook portrayal of representations � The application of functions � Labelling and measurement � Pedagogical approaches 5/28/2010 Transition to College Mathematics 21

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 5/28/2010 Transition to College Mathematics 22

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 5/28/2010 Transition to College Mathematics 23