This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Bridging Curriculum Concepts
through Trigonometric
Representations
Representations
OCMA 28th Annual Conference
Patricia (Trish) Byers
Georgian College
tbyers@georgianc.on.ca Meaningful research
Meaningful Bridging Curriculum Concepts
Bridging
through Trigonometric
Representations
Representations
Defining representations
Rationale for representations
Mapping representations through the
Mapping
curriculum
curriculum
Trigonometric representations –
Trigonometric
preliminary findings & implications for
teaching
teaching
OCMA 2008 Focusing the analysis
Recent secondary school mathematics
Recent
curriculum changes;
curriculum
Results from the College Mathematics
Results
Project 2006;
Project
Personal college experiences teaching
Personal
trigonometry and with student difficulties
learning representations.
learning OCMA 2008 College Mathematics Project 2006
Scope
Scope more than 5000 students enrolled in 139
more
technology programs at 6 Ontario colleges
technology Steering Committee representatives from the 6 participating
representatives
colleges, 9 partner school boards, SCWIcolleges,
GTA, ACAATO, MTCU, and the Ministry of
GTA,
Education, YSIMSTE representatives
Education,
OCMA 2008 College programs – A sample
Engineering technology programs Architectural
Architectural
Mechanical
Mechanical
tool & die
tool
design
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 Calculus and
Vectors
12U Course OCMA 2008 Gr 12 C
College
Technology
MCT4C 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 Risk or Failing First Sem ester College M ath
90
80
70
60 % 50 %ROG 40 %*ROG 30
20
10
0
Mech Const Elect
Cluster OCMA 2008 App Sci 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
Mathematics
engineering technology programs
engineering
Geometry 2 & 3dimensions Linear equations Algebraic & graphic solutions Trigonometry Right angle trigonometry – acute & obtuse angles;
Right
sine & cosine laws; working in all 4 quadrants, etc.
sine
Sinusoidal waveforms & graphing
Vectors – resolving vectors; adding vectors; vectors in
Vectors
rectangular & polar form
rectangular Complex numbers
Complex rectangular, polar, exponential forms
OCMA 2008 Representations and their role in
Representations
teaching & learning trigonometry
teaching
“The ways in which mathematical ideas are represented
The
represented
is fundamental to how people can understand and use
those ideas” (NCTM, 2000, p 67).
those
The AMATYC Standards for Intellectual Development
The
Standards
(2006) refer to students learning through modeling,
linking multiple representations, and, selecting, using,
multiple
and,
and translating among numerical, graphical, symbolic,
numerical graphical symbolic
and verbal representations to organize and solve
verbal
problems (p. 5).
OCMA 2008 Representations and their role in
Representations
teaching & learning trigonometry
teaching
It is suggested that mathematical sophistication develops
It
out of a comprehensive cache of representations that
representations
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.
move OCMA 2008 Focusing teaching & learning
To investigate whether trigonometric
To
representations are a source of difficulty as
students transition from secondary to college
mathematics.
mathematics. OCMA 2008 Defining representations
“A representation is a configuration of signs,
representation
characters, icons, or objects that can somehow
stand for, or “represent” something else ...
stand
According to the nature of the representing
According
relationship, the term represent can be interpreted in
relationship,
represent
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).
(Goldin,
OCMA 2008 Systems of representations
External systems Structured by the conventions underlying them
No longer arbitrary
No
Accepted by the mathematics community waiting
Accepted
to be “discovered” by the student
to Internal systems Demonstrate how a student understands a
Demonstrate
mathematical concept
mathematical
Verbal/syntactic; Imagistic; Formal notational;
Verbal/syntactic;
Affective
Affective Dimensions in representations Horizontal: between external systems
Vertical: with external & internal systems
OCMA 2008 Ways to represent
Ways
a function
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
Ways
a trigonometric function
trigonometric
Algebraic/symbolic
Algebraic/symbolic formulas for trigonometric ratios &
formulas
trigonometric functions
trigonometric Numeric tables Visual
Visual right triangle, circle, sinusoidal waveform OCMA 2008 Ministry of Education
Ministry
curriculum expectations
Gr. 10 AcademicTrigonometry
By the end of this course,
By
students will:
students
1. 2. 3. 4. use their knowledge of ratio
use
and proportion to investigate
similar triangles and solve
problems related to similarity;
problems
solve problems involving right
solve
triangles, using the primary
trigonometric ratios and the
Pythagorean theorem;
Pythagorean
solve problems involving
solve
acute triangles, using the sine
law and the cosine law.
law
Under Analytic Geometry,
Under
properties of the circle given
by the equation x2 + y2 = r2 Gr. 10 AppliedMeasurement
Gr.
& Trigonometry
Trigonometry
By the end of this course,
By
students will:
students
1. 2. 3. use their knowledge of ratio
use
and proportion to investigate
similar triangles and solve
problems related to similarity;
problems
solve problems involving right
solve
triangles, using the primary
trigonometric ratios and the
Pythagorean theorem;
Pythagorean
solve problems involving the
solve
surface areas and volumes of
threedimensional figures,
and use the imperial and
metric systems of
measurement.
measurement. Ministry of Education
curriculum expectations
Gr. 11MTrigonometric Functions
Gr.
By the end of this course, students
By
will:
will:
1. 2. 3. solve problems involving
solve
trigonometry in acute triangles
trigonometry in
using the sine law and the cosine
law, iincluding problems arising
law ncluding
from realworld applications;
from
demonstrate an understanding of
demonstrate
periodic relationships and the sine
function, and make connections
between the numeric, graphical,
and algebraic representations of
sine functions;
sine
iidentify and represent sine
dentify
functions, and solve problems
involving sine functions, including
involving sine
problems arising from realworld
applications.
applications. Gr. 11CGeometry &
Gr.
Trigonometry
Trigonometry
By the end of this course,
By
students will:
students
1. 2. represent, in a variety of
represent,
ways, twodimensional
shapes and threeshapes
dimensional figures arising
dimensional
from realworld applications,
and solve design problems;
and
solve problems involving
solve
trigonometry in acute
triangles using the sine law
triangles
and the cosine law, iincluding
ncluding
and
problems arising from realproblems
world applications. Ministry of Education
curriculum expectations Gr. 12(MCT) –Trigonometric
Gr.
Functions
Functions
By the end of this course,
By
students will:
students
1. 2. 3. determine the values of the
determine
trigonometric ratios for angles
less than 360º, and solve
problems using the primary
trigonometric ratios, the sine
law, and the cosine law;
law,
make connections between the
make
numeric, graphical, and
algebraic representations of
algebraic
sinusoidal functions;
sinusoidal
demonstrate an understanding
demonstrate
that sinusoidal functions can be
used to model some periodic
phenomena, and solve related
problems, including those
arising from realworld
applications.
applications.
OCMA 2008 Gr. 12C(MAP)Geometry &
Gr.
Trigonometry
Trigonometry
By the end of this course, students
By
will:
will:
1. 2. solve problems involving
solve
measurement and geometry and
arising from realworld applications;
explain the significance of optimal
dimensions in realworld
applications, and determine optimal
dimensions of twodimensional
shapes and threedimensional
figures;
figures;
solve problems using primary
solve
trigonometric ratios of acute and
obtuse angles, the sine law, and the
cosine law, iincluding problems
ncluding
cosine
arising from realworld applications,
and describe applications of
trigonometry in various occupations.
trigonometry Mapping the right triangle
representation
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
Mapping
representation
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
The
learning trajectory (HLT)
learning
Components: “the learning goal, the developmental progressions
the
of thinking and learning, and a sequence of
instructional tasks” (Clements & Sarama, 2004, p.
85).
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
Representations
established through use becoming signs and
configurations for newly developing
representations. OCMA 2008 Implications
Potential discrepancies exist between types of
Potential
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 .
trigonometric
Strategies to help students with potential
Strategies
difficulties learning trigonometric representations
need developing.
need
OCMA 2008 Implications
A point of focus to share various representations
point
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
bridging
not taught in secondary school but required for
college studies.
college
Further research to unpack discrepancies in
Further
other college mathematics concepts.
other
OCMA 2008 References
Clements, D.H., Sarama, J. (2004). Learning
trajectories in mathematics education. Mathematical
Thinking and Learning, 6(2), 8189. Mahwah, NJ:
Thinking
(2),
Lawrence Erlbaum Associates.
Lawrence
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
School
Reston,
Council of Teacher of Mathematics. pp. 275284.
Council OCMA 2008 Bridging Curriculum Concepts
using Trigonometric
Representations
Representations
OCMA 28th Annual Conference
Patricia (Trish) Byers
Georgian College
tbyers@georgianc.on.ca ...
View Full
Document
 Winter '08
 JARVIS
 Trigonometric Ratios

Click to edit the document details