Transition to college mathematics
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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!
5/28/2010
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
5/28/2010
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?
5/28/2010
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
5/28/2010
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).
5/28/2010
Transition to College Mathematics 9
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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
5/28/2010
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