1. science
2. mathematics
3. geometry

Bridging Curriculum Concepts
through Trigonometric
Representations
OCMA 28th Annual Conference
Patricia (Trish) Byers
Georgian College
[email protected]
Meaningful research
Bridging Curriculum Concepts
through Trigonometric
Representations
Defining representations
Rationale for representations
Mapping representations through the
curriculum
Trigonometric representations –
preliminary findings & implications for
teaching
OCMA 2008
Focusing the analysis
Recent secondary school mathematics
curriculum changes;
Results from the College Mathematics
Project 2006;
Personal college experiences teaching
trigonometry and with student difficulties
learning representations.
OCMA 2008
College Mathematics Project 2006
Scope

more than 5000 students enrolled in 139
technology programs at 6 Ontario colleges
Steering Committee

representatives from the 6 participating
colleges, 9 partner school boards, SCWI-GTA,
ACAATO, MTCU, and the Ministry of
Education, YSIMSTE representatives
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College programs – A sample
Engineering technology programs


Architectural
Mechanical
tool & die
design




Construction
Electrical
Applied Science (e.g., Environmental)
Computer Science
OCMA 2008
Secondary school math curriculum 2007 –
Pathway 1
Grade 9
Academic
Grade 9
Applied
T
Gr 10
Academic
MPM2D
Grade 11 U
Functions
Grade 12 U
Advanced
Functions
Calculus and
Vectors
12U Course
Gr 10 Applied
MFM2P
Gr 11 U/C
Function
Applications
MCF3M
Grade 12 U
Data
Management
Gr 12 C
College
Technology
MCT4C
Gr 11 C
Foundations for
College Math
MBF3C
Gr 12 C
Foundations for
College Math
MAP4C
Secondary school math curriculum 2007 –
Pathway 2
Grade 9
Academic
Grade 9
Applied
T
Gr 10
Academic
MPM2D
Grade 11 U
Functions
Grade 12 U
Advanced
Functions
Calculus and
Vectors
12U Course
Gr 10 Applied
MFM2P
Gr 11 U/C
Function
Applications
MCF3M
Grade 12 U
Data
Management
Gr 12 C
College
Technology
MCT4C
Gr 11 C
Foundations for
College Math
MBF3C
Gr 12 C
Foundations for
College Math
MAP4C
Secondary school math curriculum 2007 –
Pathway 3
Grade 9
Academic
Grade 9
Applied
T
Gr 10
Academic
MPM2D
Grade 11 U
Functions
Grade 12 U
Advanced
Functions
Gr 10 Applied
MFM2P
Gr 11 U/C
Function
Applications
MCF3M
Grade 12 U
Data
Management
Gr 12 C
College
Technology
MCT4C
Calculus and
Vectors
12U Course
OCMA 2008
Gr 11 C
Foundations for
College Math
MBF3C
Gr 12 C
Foundations for
College Math
MAP4C
Results from CMP 2006
Summary of student data analysis
 The study found that 30% to 50% of all
students (all clusters) in all program areas
were at risk of failing or failing.
% At Ris k or Failing Firs t Se m e s te r Colle ge Math
90
80
70
60
%
50
%-ROG
40
%-*ROG
30
20
10
0
Mech
Const
Elect
App Sci
Clus te r
OCMA 2008
Comp
Results from CMP 2006
More than 700 students entering 1st year
technology programs at 6 Ontario colleges in
F04, but fewer than 25% had taken MCT4C
(Mathematics for College Technology).
69% of these students achieved an A, B or C
grade in their 1st semester college
mathematics course, with 31% obtaining a D,
F or withdrawal from the course.
OCMA 2008
Results from CMP 2006
By contrast, the Grade 12 mathematics course
taken by over half of the students was MAP4C
(College and Apprenticeship Mathematics).
Of this group, <35% achieved a good grade
(A, B or C) in first semester college
mathematics and 65% obtained a D, F or
withdrawal from the course.
OCMA 2008
Mathematics curriculum
engineering technology programs
Geometry

2- & 3-dimensions
Linear equations

Algebraic & graphic solutions
Trigonometry



Right angle trigonometry – acute & obtuse angles;
sine & cosine laws; working in all 4 quadrants, etc.
Sinusoidal waveforms & graphing
Vectors – resolving vectors; adding vectors; vectors in
rectangular & polar form
Complex numbers

rectangular, polar, exponential forms
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Representations and their role in
teaching & learning trigonometry
“The ways in which mathematical ideas are represented
is fundamental to how people can understand and use
those ideas” (NCTM, 2000, p 67).
The AMATYC Standards for Intellectual Development
(2006) refer to students learning through modeling,
linking multiple representations, and, selecting, using,
and translating among numerical, graphical, symbolic,
and verbal representations to organize and solve
problems (p. 5).
OCMA 2008
Representations and their role in
teaching & learning trigonometry
It is suggested that mathematical sophistication develops
out of a comprehensive cache of representations that
support deep conceptual understanding (Pritchard &
Simpson, 1999, p. 87).
Research in learning trigonometric functions reveals that
a key source of student difficulty is the lack of ability to
move from one representation to another.
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Focusing teaching & learning
To investigate whether trigonometric
representations are a source of difficulty as
students transition from secondary to college
mathematics.
OCMA 2008
Defining representations
“A representation is a configuration of signs,
characters, icons, or objects that can somehow
stand for, or “represent” something else ...
According to the nature of the representing
relationship, the term represent can be interpreted in
many ways, including the following (the list is not
exhaustive): correspond to, denote, depict, embody,
encode, evoke, label, mean, produce, refer to,
suggest, or symbolize (italics in the original)”
(Goldin, 2003, p. 276).
OCMA 2008
Systems of representations
External systems



Structured by the conventions underlying them
No longer arbitrary
Accepted by the mathematics community waiting
to be “discovered” by the student
Internal systems


Demonstrate how a student understands a
mathematical concept
Verbal/syntactic; Imagistic; Formal notational;
Affective
Dimensions in representations


Horizontal: between external systems
Vertical: with external & internal systems
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Ways to represent
a function
Stewart, Redlin, & Watson (2002, p. 150)
1.
2.
3.
4.
Verbally – in words
Algebraically – by an explicit formula
Visually – with a diagram or figure
Numerically – by a table of values
OCMA 2008
Ways to represent
a trigonometric function
Algebraic/symbolic

formulas for trigonometric ratios &
trigonometric functions
Numeric

tables
Visual

right triangle, circle, sinusoidal waveform
OCMA 2008
Ministry of Education
curriculum expectations
Gr. 10 Academic-Trigonometry
By the end of this course,
students will:
1.
2.
3.
4.
use their knowledge of ratio
and proportion to investigate
similar triangles and solve
problems related to similarity;
solve problems involving right
triangles, using the primary
trigonometric ratios and the
Pythagorean theorem;
solve problems involving
acute triangles, using the sine
law and the cosine law.
Under Analytic Geometry,
properties of the circle given
by the equation x2 + y2 = r2
Gr. 10 Applied-Measurement &
Trigonometry
By the end of this course,
students will:
1.
2.
3.
use their knowledge of ratio
and proportion to investigate
similar triangles and solve
problems related to similarity;
solve problems involving right
triangles, using the primary
trigonometric ratios and the
Pythagorean theorem;
solve problems involving the
surface areas and volumes of
three-dimensional figures,
and use the imperial and
metric systems of
measurement.
Ministry of Education
curriculum expectations
Gr. 11M-Trigonometric Functions
By the end of this course, students
will:
1.
2.
3.
solve problems involving
trigonometry in acute triangles
using the sine law and the cosine
law, including problems arising
from real-world applications;
demonstrate an understanding of
periodic relationships and the sine
function, and make connections
between the numeric, graphical,
and algebraic representations of
sine functions;
identify and represent sine
functions, and solve problems
involving sine functions, including
problems arising from real-world
applications.
Gr. 11C-Geometry &
Trigonometry
By the end of this course,
students will:
1.
2.
represent, in a variety of
ways, two-dimensional
shapes and threedimensional figures arising
from real-world applications,
and solve design problems;
solve problems involving
trigonometry in acute
triangles using the sine law
and the cosine law, including
problems arising from realworld applications.
Ministry of Education
curriculum expectations
Gr. 12(MCT) –Trigonometric
Functions
By the end of this course,
students will:
1.
2.
3.
determine the values of the
trigonometric ratios for angles
less than 360º, and solve
problems using the primary
trigonometric ratios, the sine
law, and the cosine law;
make connections between the
numeric, graphical, and
algebraic representations of
sinusoidal functions;
demonstrate an understanding
that sinusoidal functions can be
used to model some periodic
phenomena, and solve related
problems, including those
arising from real-world
applications.
Gr. 12C(MAP)-Geometry &
Trigonometry
By the end of this course, students
will:
OCMA 2008
1.
2.
solve problems involving
measurement and geometry and
arising from real-world applications;
explain the significance of optimal
dimensions in real-world
applications, and determine optimal
dimensions of two-dimensional
shapes and three-dimensional
figures;
solve problems using primary
trigonometric ratios of acute and
obtuse angles, the sine law, and the
cosine law, including problems
arising from real-world applications,
and describe applications of
trigonometry in various occupations.
Mapping the right triangle
representation
Right Triangle
(10 Applied &
Academic)
9 Applied & Academic:
Equivalent ratios
Ratios & proportion
Proportional reasoning
Pythagorean theorem
Interior & exterior angles of triangles
Angle measurement & polygons
10 Applied & Academic:
Similar triangles
Pythagorean theorem
10 Academic:
Proportional reasoning
Mapping the right triangle
representation
Right Triangle
Representation
(10 Applied &
Academic)
Sine & Cosine Law
for acute triangles
(10 Academic)
Sine & Cosine Law
for acute triangles (11
Applied & Mixed)
Sine & Cosine Laws for
oblique triangles
(12 Mixed)
OCMA 2008
Sine & Cosine Laws for
oblique triangles
Primary trig ratios of
obtuse angles
(12 College)
Preliminary findings
1. The mapping resembles a hypothetical
learning trajectory (HLT)
Components:

“the learning goal, the developmental progressions
of thinking and learning, and a sequence of
instructional tasks” (Clements & Sarama, 2004, p.
85).
OCMA 2008
A relationship with HLT
Learning goals
Progressions
of thinking &
learning
Primitive
characters or
signs
Configurations
Representations
OCMA 2008
Learning tasks
Preliminary findings
2. Representations are arbitrary but are
established through use becoming signs and
configurations for newly developing
representations.
OCMA 2008
Implications
Potential discrepancies exist between types of
trigonometric representations & depth to which
these are taught (hence, disruptions in a
hypothetical learning trajectory from secondary
school to college).
Potential student difficulties learning
trigonometric representations can be identified .
Strategies to help students with potential
difficulties learning trigonometric representations
need developing.
OCMA 2008
Implications
A point of focus to share various representations
used in the others’ classrooms of each
educational sector beginning the conversation
on student difficulties in college mathematics.
A point of departure to build a destination
bridging sequence to address representations
not taught in secondary school but required for
college studies.
Further research to unpack discrepancies in
other college mathematics concepts.
OCMA 2008
References
Clements, D.H., Sarama, J. (2004). Learning
trajectories in mathematics education. Mathematical
Thinking and Learning, 6(2), 81-89. Mahwah, NJ:
Lawrence Erlbaum Associates.
Goldin, G. (2003). Representation in school
mathematics: A unifying research perspective. In J.
Kilpatrick, W.G. Martin; D. Schifter (Eds.), A
Research Companion to Principles and Standards for
School Mathematics. Reston, VA: The National
Council of Teacher of Mathematics. pp. 275-284.
OCMA 2008
Bridging Curriculum Concepts
using Trigonometric
Representations
OCMA 28th Annual Conference
Patricia (Trish) Byers
Georgian College
[email protected]