TRANSITION TO COLLEGE MATHEMATICS

AN INVESTIGATION OF TRIGONOMETRIC REPRESENTATIONS AS A SOURCE OF STUDENT DIFFICULTIES

OCMA 30th Annual ConferenceMay 28, 2010

Patricia (Trish) Byers

Transition to College Mathematics 2

Overview

Positioning the study defining representations

The research questions

Phases of the study developing the networks

Research results5/28/2010

Transition to College Mathematics 3

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).

x y

-1 2

0 0

1 -2

a + b = b + a

1!

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0

Transition to College Mathematics 4

Applying representations

What everyday objects could be represented using a sphere?

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Transition to College Mathematics 5

Some trigonometric representations

opposite

adjacent

hypo

ten

use

P(x, y)

r

P(cos , sin )

t sin(t)

0 0

/2 1

0

3/2 -1

http://cda.morris.umn.edu/~mcquarrb/Precalculus/Animations/SineCosineAnim.html

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Transition to College Mathematics 6

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?

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Transition to College Mathematics 7

Selecting the college program Electrical engineering technology program

College Mathematics Project results from Fall 2004

MTCU Program standards – General education Generic Vocational

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Transition to College Mathematics 8

Pathways to college technology programs (2000)

MFM1PGr. 9 Applied

Foundations of Mathematics

MPM1DGr. 9 Academic

Principles of Mathematics

MPM2DGr. 10 Academic

Principles of Mathematics

MCF3MGr. 11

University/College

Functions

MFM2PGr. 10 AppliedFoundations of Mathematics

MBF3CGr. 11 CollegeMathematics of

Personal Finance

MAP4CGr. 12 CollegeCollege and

Apprenticeship Mathematics

MCT4CGr. 12 College

Mathematics for College Technology

Mathematics

MCR3UGr. 11 University

Functions and Relations

Figure 5.  Secondary school mathematics pathways (adapted from Ministry of Education, 2000a).

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Pathway 1

MFM1PGr. 9 Applied

Foundations of Mathematics

MPM1DGr. 9 Academic

Principles of Mathematics

MPM2DGr. 10 Academic

Principles of Mathematics

MCF3MGr. 11

University/College

Functions

MFM2PGr. 10 AppliedFoundations of Mathematics

MBF3CGr. 11 CollegeMathematics of

Personal Finance

MAP4CGr. 12 CollegeCollege and

Apprenticeship Mathematics

MCT4CGr. 12 College

Mathematics for College Technology

Mathematics

MCR3UGr. 11 University

Functions and Relations

Figure 6.  Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000a) highlighting Pathway 1.

1

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Transition to College Mathematics 10

Pathway 2

MFM1PGr. 9 Applied

Foundations of Mathematics

MPM1DGr. 9 Academic

Principles of Mathematics

MPM2DGr. 10 Academic

Principles of Mathematics

MCF3MGr. 11

University/College

Functions

MFM2PGr. 10 AppliedFoundations of Mathematics

MBF3CGr. 11 CollegeMathematics of

Personal Finance

MAP4CGr. 12 CollegeCollege and

Apprenticeship Mathematics

MCT4CGr. 12 College

Mathematics for College Technology

Mathematics

MCR3UGr. 11 University

Functions and Relations

Figure 7.  Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000a) highlighting Pathway 2.

2

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Transition to College Mathematics 11

Pathway 3

MFM1PGr. 9 Applied

Foundations of Mathematics

MPM1DGr. 9 Academic

Principles of Mathematics

MPM2DGr. 10 Academic

Principles of Mathematics

MCF3MGr. 11

University/College

Functions

MFM2PGr. 10 AppliedFoundations of Mathematics

MBF3CGr. 11 CollegeMathematics of

Personal Finance

MAP4CGr. 12 CollegeCollege and

Apprenticeship Mathematics

MCT4CGr. 12 College

Mathematics for College Technology

Mathematics

MCR3UGr. 11 University

Functions and Relations

Figure 8.  Secondary school mathematics pathways to college courses (adapted from Ministry of Education, 2000a) highlighting Pathway 3.

3

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Transition to College Mathematics 12

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

textbooks5/28/2010

Transition to College Mathematics 13

Secondary school teacher’s representation network

Unit circleTrigonometric 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

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Transition to College Mathematics 14

College teacher’s representation network

x = cos y = sin (trigonometric circular functions)

VectorTrigonometric 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 approach5/28/2010

Vector on the xy-plane

DomainSinusoidal WaveformDomain

Trigonometric FunctionsDomain (02)

Trigonometric Ratios

Domain(0/2)

Unit CircleDomain

Figure 50. A general representation network of trigonometry.

Right TriangleDomain

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

shiftdisplacement

Circle

Trigonometric (circular) functions –

x, y, (02)-inverse functions

-primary and secondary functions

Trigonometric functions- t,

Tables

Reference angles

Right triangle on

the xy-plane

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Principles of Euclidean geometryPoints/Line segments/Angles/

Trianglesxy-planeSlope

TransformationsFunctions/Relationships

Accuracy & precisionProportionality

Graphing

Vector on the xy-plane

DomainSinusoidal WaveformDomain

Trigonometric FunctionsDomain (02)

Trigonometric Ratios

Domain(0/2)

Unit CircleDomain

Building the representation network of trigonometry for pathway analysis.

Right TriangleDomain

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

shiftdisplacement

Circle

Trigonometric (circular) functions –

x, y, (02)-inverse functions

-primary and secondary functions

Trigonometric functions- t,

Tables

Reference angles

Right triangle on

the xy-plane

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Vector on the xy-plane

DomainSinusoidal WaveformDomain

Trigonometric FunctionsDomain (02)

Trigonometric Ratios

Domain(0/2)

Unit CircleDomain

Building the representation network of trigonometry for pathway analysis.

Right TriangleDomain

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

shiftdisplacement

Circle

Trigonometric (circular) functions –

x, y, (02)-inverse functions

-primary and secondary functions

Trigonometric functions- t,

Tables

Reference angles

Right triangle on

the xy-plane

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Vector on the xy-plane

Domain

Vector on the xy-plane

DomainSinusoidal WaveformDomain

Trigonometric FunctionsDomain (02)

Trigonometric Ratios

Domain(0/2)

Unit CircleDomain

Figure 64. The representation network of trigonometry for Pathway 1.

Right TriangleDomain

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

shiftdisplacement

Circle

Trigonometric (circular) functions –

x, y, (02)-inverse functions

-primary and secondary functions

Trigonometric functions- t,

Tables

Reference angles

Right triangle on

the xy-plane

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Vector on the xy-plane

DomainSinusoidal WaveformDomain

Trigonometric FunctionsDomain (02)

Trigonometric Ratios

Domain(0/2)

Unit CircleDomain

Figure 70. The representation network of trigonometry for Pathway 2.

Right TriangleDomain

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

shiftdisplacement

Circle

Trigonometric (circular) functions –

x, y, (02)-inverse functions

-primary and secondary functions

Trigonometric functions- t,

Tables

Reference angles

Right triangle on

the xy-plane

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Unit semicircle

Vector on the xy-plane

DomainSinusoidal WaveformDomain

Trigonometric FunctionsDomain (02)

Trigonometric Ratios

Domain(0/2)

Unit CircleDomain

Figure 75. The representation network of trigonometry for Pathway 3.

Right TriangleDomain

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

shiftdisplacement

Circle

Trigonometric (circular) functions –

x, y, (02)-inverse functions

-primary and secondary functions

Trigonometric functions- t,

Tables

Reference angles

Right triangle on

the xy-plane

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Unit semicircle

Unit quarter circle

Transition to College Mathematics 21

Question 1 findings

Textbook portrayal of representations

The application of functions

Labelling and measurement

Pedagogical approaches

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Transition to College Mathematics 22

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

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Transition to College Mathematics 23

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

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