Transcript Senior Mathematics Curriculum Revision

Senior Mathematics
Curriculum Revision
Supporting students and
teachers by keeping Ontario’s
K - 12
curriculum current and relevant
College Math Project Forum
June 15, 2006
Anthony Azzopardi
Curriculum and Assessment Policy Branch
Ministry of Education
4
What is Curriculum Review?
A staged process to review Kindergarten
to Grade 12 curriculum documents by
discipline area that:
• builds on the quality curriculum
currently in place
• ensures that the curriculum remains
current and relevant
Curriculum Review Process
• integrate review of elementary and
secondary curriculum policy documents
• have parallel revision processes for English
and French language curriculum
• involve teachers, principals, board staff,
subject experts, education stakeholders,
parents, students and sector
representatives.
Curriculum Review Process
Sept.
2004
Sept.
2003
Sept.
2005
Grades 1 - 10
Grade 12
Analysis and Synthesis
Revision and Feedback Consultation
Editing, Publication and Distribution
Mandatory Implementation
Sept.
2007
*
Grade 11
*
Sept.
2006
*
*
Curriculum Review Process
Subject /
Division
Associations
Technical
Analysis
Focus
Groups
Other
Consultations
and Input
Analysis / Synthesis
Achievement
Charts
Research
Revision / Feedback
Feedback
Consultations
Ontario Mathematics Curriculum
2000
Grade 9
Academic
Grade 9
Applied
Grade 10
Academic
Grade 10
Applied
Grade 11 U
Functions and
Relations
Grade 12U
Geometry and
Discrete
Grade 12U
Advanced
Functions
Grade 11 U/C
Functions
Grade 12U
Data
Management
Grade 11 C
Personal
Finance
Grade 12C
Math for College
Technology
Grade 12C
College and
Apprenticeship
Grade 11 W
Math for
Everyday Life
Grade 12 W
Math for
Everyday Life
Student Destinations
1999-2000 Cohort to Fall 2004
33%
to
University
19%
to
College
18%
OSSD
to Work
Grade 9 Enrolment
= 100%
30%
Leave
before
OSSD
Source: Alan King,
Double Cohort Study
2005
Double Cohort Study – Phase 4
Grade 11 Achievement
Grade 11
Courses
Functions and
Relations (U)
2001-2002
2002-2003
2003-2004
11.4% 11.0% 9.2%
Functions (U/C)
20.9% 19.7% 18.2%
Personal
Finance (C)
18.6% 17.3% 16.5%
Math For
Everyday Life
(W)
17.0% 15.8% 15.3%
Grade 11 Student Achievement
Marks Distribution (% Grade 11 Students 2003-2004)
% Students
20
15
10
5
0
<50
50
51-55
56-60
61-65
66-70
71-75
76-80
81-85
86-90
>90
Achievement
MEL 3E
MBF 3C
MCF 3M
MCR 3U
ENG 3U
Double Cohort Study: Phase 4, 2005
Grade 12 Student Achievement
Marks Distribution (% Grade 12 Students 2003-2004)
% Students
20
15
10
5
0
<50
50
51-55
56-60
61-65
66-70
71-75
76-80
81-85
86-90
>90
Achievement
MEL 4E
MGA 4U
MAP 4C
ENG 4U
MCT 4C
MDM 4U
MCB 4U
Double Cohort Study: Phase 4, 2005
PISA 2003: Indices of Student Engagement
In Mathematics (15 year olds)
Significantly
higher than
Canadian average
Interest and
enjoyment in
mathematics
Performing the
same as the
Canadian average
Significantly lower
than Canadian
average
ONTARIO
NFLD, PEI, NS,
NB, QU, MAN, SK,
AL
BC
Belief in
usefulness of
mathematics
NS, QU
NFLD, PEI, MAN,
SK, AL
ONTARIO
NB, BC
Mathematics
confidence
QU, AL
NFLD, BC
ONTARIO
PEI, NS, NB, MAN,
SK
Perceived ability
in mathematics
QU, AL
NFLD, PEI, NS,
NB, SK
ONTARIO
MAN, BC
Mathematics
anxiety
ONTARIO
NB, QU, MAN, SK,
AL, BC
NFLD, PEI, NS
Double Cohort Study – Phase 4
Grade 11 Enrolment
Grade 11
Courses
Functions and
Relations (U)
2001-2002
34.3%
2002-2003
2003-2004
28% 26.8%
Functions (U/C)
26.2% 27.4% 26.1%
Personal
Finance (C)
29.6% 32.8% 34.4%
Math For
Everyday Life
(W)
10%
11.7% 12.7%
Consultation with Colleges
•
•
•
•
•
•
•
•
Heads of Technology - Spring 2004
College Math Survey – April 2004
ACAATO consultation - June 2004
Colleges Gr 9/10 feedback – Nov 2004
Revision writing - July 2005
Feedback consultation Gr 11/12 - Nov 2005
Grade 11Consultations – Spring 2006
Revision writing – July 2006
ACAATO Recommendations 2004
• Create a clearer pathway from Grade 10
Applied to Grade 12 College Tech
• Revise Grade 11 Personal Finance course
to better prepare students for Grade 12C.
• Address overlap in 11U and 11M to ensure
11M is more appropriate for students
entering college tech programs.
ACAATO Recommendations 2004
• Grade 12 College Tech should be more
appropriate for college bound students;
• Improve how the curriculum helps
students develop concepts, basic numeric
and algebraic skills and the ability to apply
processes such as problem-solving,
estimation and communication.
ACAATO Research 2004
Math-related Program Clusters:
 Applied Arts
 Business
 Health Sciences
 Hospitality
 Human Services
 Technology
 Skilled Trades
Review Process: Synthesis
Revisions address:
• Curriculum Expectations
• Equity
• Learning
• Teaching
• Assessment and Evaluation
• Learning Tools
Goals Of Revision:
• Reduce the density of the curriculum
• Provide more opportunities for students
to develop and apply important life-long
process skills
• Provide clearer pathways
• Incorporate more grade and destination
appropriate topics and skills
Goals Of Revision:
• Enhance curriculum coherence and
concept development over the grades
• Improve student achievement and
graduation rates
• Improve access to higher mathematics,
attitudes towards mathematics, student
retention in mathematics
MATHEMATICAL
PROCESSES
INTRODUCTION
PATHWAYS
REVIEW
SAMPLE
PROBLEMS
EXAMPLES
STRANDS
Mathematics
Grades 11 and 12
SUBHEADINGS
ACHIEVEMENT
CHART
OVERALL/SPECIFIC
EXPECTATIONS
42
Review Process: Feedback Consultations
• feedback consultation on proposed revisions
to Grades 11 and 12 occurred in the fall of
2005

Day 1 - information provided on curriculum review
process and the proposed revisions in the draft
 Day 2 - participants share feedback on the draft of
proposed revisions gathered through a consultation
process within their board or organization
• information from the consultations and
feedback sessions informs further
revisions
Review Process: Feedback Consultations
Grade 11: Foundations for College Math
Strengths:
• destination appropriate
• expectations clearer; examples and sample
problems clarify the intended depth,
breadth, and level of difficulty
• better results; expanding on topics
introduced in grade 10 provides better
preparation for grade 12 College course
Review Process: Feedback Consultations
Grade 11: Foundations for College Math
Suggestions and Considerations:
• more examples and “Sample Problems”
• identify use of technology in more specific
places
• consider impact of availability of local
technology to support implementation
• students from 10 Applied without a strong
foundation may find this course
challenging
Review Process: Feedback Consultations
Grade 11: Functions and Applications
Strengths:
 provides good grounding for broad range of
math applications
 revised version is more destination
appropriate for college bound students
 many expectations call for investigation; the
smaller number of expectations should help
support this change
 clarity of expectations
Review Process: Feedback Consultations
Grade 11: Functions and Applications
Suggestions and Considerations:
• more examples and sample problems
• more examples of the use of technology
• should ‘radians’ have been removed?
• the course is better preparation for Gr. 12C
Math for College Tech than for the Gr. 12U
Data Management course
• students from 10 Applied may find the
course challenging
Media Response to Revision
Public response to the proposed
DRAFT Senior Mathematics
revisions focused primarily on
the issue of Calculus.
Questions Raised:
•
•
•
•
CURRICULUM
QUALITY
CURRICULUM
QUALITY
Does COMPLEX = DIFFICULT?
Does RIGOROUS = HARD ?
How many are served better by a DENSE CURRICULUM?
For whom is a DENSE CURRICULUM developmentally
appropriate?
• Who is marginalized by a DENSE CURRICULUM?
• What is the relationship between content density and
curriculum quality?
CONTENT DENSITY
CONTENT DENSITY
Review Process: Minister’s Task Force
• February 16, 2006 – Minister announces
extended review and organization of the Ministry
of Education’s first Curriculum Council: Task
Force on Senior High School Mathematics
• February/March 2006 – Task Force
Consultations
• April 2006 – Task Force submits report
• June 2006 – Task Force report released
(www.edu.gov.on.ca)
Task Force Recommendation:
•
That the Grade 12 courses Mathematics
for Work and Everyday Life, Foundations
for College Mathematics and
Mathematics for College Technology be
implemented essentially as currently
planned.
Key Message: Curriculum
The revised curriculum is more
coherent, focused on important
mathematics and well articulated
across the grades.
Summary of Key Changes
• address concerns regarding an overcrowded
curriculum: reduced the number of expectations
(e.g., removed “Conics ” strand from Grade 11U
Functions course)
• address high failure rates: (e.g., concepts
developed in a more developmentally appropriate
manner and link better with Grade 10)
• create clearer pathways to Grade 12 from
Grades 9 and 10 Applied Mathematics
courses ( e.g., revised 11U/C to articulate with
both 10 Academic and 10 Applied)
Summary of Key Changes
• create clearer pathways for students not entering
mathematics or science programs at university
(e.g., created a more focused pathway through Grade
11U/C Function Applications to Grade 12);
• revise expectations to reflect a better balance
between the development of procedural fluency,
deeper conceptual understanding and the ability to
apply key mathematical processes like problem
solving, communication and reasoning.
• improve curriculum coherence (e.g., reorganized
strands in college destination courses to improve
concept development);
Summary of Key Changes
• reduce or eliminate overlap (e.g., reduced overlap
between Grade 11U and Grade 11U/C mathematics
courses);
• engage students in a more relevant high school learning
experience by increasing emphasis on connections within
mathematics and between mathematics and the realworld (e.g., stronger connections between topics related
to functions from Grade 9 through to Grade 11, increased
career connections in the Grade 11 workplace course)
• encourage the use of a broad range of learning tools
to support meaningful student learning in mathematics
(e.g., revised specific expectations to include more
references to the use of technological tools like graphing
technology, calculators, statistical software)
Clear Pathways (DRAFT)
Grade 9
Academic
Grade 9
Applied
Grade 9
L.D.C.C.
T
Grade 10
Academic
Grade 11 U
Functions
Grade 12U
Advanced
Functions
Calculus and
Vectors
12U Course
Grade 10
Applied
Grade 11 M
Function
Applications
Grade 12U
Data
Management
Grade 12C
College
Technology
Grade 10
L.D.C.C.
Grade 11 C
Foundations for
College Math
Grade 11E
Work and
Everyday Life
Grade 12 C
Foundations for
College Math
Grade 12E
Work and
Everyday Life
Comparing Strands: Grade 11U
2000 Curriculum
Revised 2006 Curriculum
Financial Applications of Sequences and Series
•
–
–
–
•
–
–
–
–
•
•
–
–
–
Polynomials/Rational Expressions and
Exponential Expressions
Inverses/Transformations/Function Notation
Mathematical Reasoning
Loci and Conics
–
–
–
Loci
Equations
Solving Problems
Representing Exponential Functions
Connecting Graphs and Equations of Exponential
Functions
Solving Problems Involving Exponential
Functions
Discrete Functions
–
–
–
•
Representing Functions
Solving Problems Involving Quadratic Functions
Determining Equivalent Algebraic Expressions
Exponential Functions
–
–
Sine Law/Cosine Law for Oblique Triangles
Understanding and Applying Radian Measure
Graphs and Equations of Sinusoidal Functions
Models of Sinusoidal Functions
Tools for Operating and Communicating with
•
Functions
–
•
–
–
–
Arithmetic/Geometric Sequences and Series
Compound Interest and Annuity Problems
Financial Decision Making
Trigonometric Functions
Characteristics of Functions
Representing Sequences
Investigating Arithmetic and Geometric
Sequences and Series
Solving Problems Involving Financial
Applications
Trigonometric Functions
–
–
–
Determining and Applying Trigonometric Ratios
Connecting Graphs and Equations of Sinusoidal
Functions.
Solving Problems Involving Sinusoidal Functions
Revision Highlights: 11U
Increased focus on:
• characteristics of
functions;
• transformations;
• exponential functions;
• discrete functions;
• modelling;
• rate of change;
• radical expressions;
• reciprocal trig identities;
• periodic functions;
Decreased focus on:
• conics and loci;
• annuities and mortgages;
• solving exponential
equations;
• solving trig equations;
• complex roots;
• radians;
• tangent function;
• solving linear inequalities
Return
Comparing Strands: Grade 11C
2000 Curriculum
•
Models of Exponential Growth
–
–
–
•
•
–
–
Nature of Exponential Growth
Mathematical Properties of Exponential
Functions
Manipulating Expressions
–
–
•
Arithmetic/Geometric Sequences and
Series
Compound Interest and Annuity Problems
Effect of Compounding
Personal Financial Decisions
–
–
–
–
–
Owning/Operating A Vehicle
Renting/Buying Accommodation
Designing Budgets
Making Informed Decisions
Career Opportunities
Mathematical Models
–
Compound Interest/Annuities
–
•
Revised 2006 Curriculum
•
Personal Finance
–
–
–
Solving Problems Involving Compound
Interest
Comparing Financial Services
Owning/Operating A Vehicle
Geometry and Trigonometry
–
–
•
Connecting Graphs and Equations of Quadratic
Relations
Connecting Graphs and Equations of
Exponential Relations
Solving Problems Involving Exponential
Relations
Representing Two-Dimensional Shapes and
Three-Dimensional Figures
Applying the Sine Law and the Cosine Law in
Acute Triangle
Data Management
–
–
Working With One-Variable Data
Applying Probability
Revision Highlights: 11C
Increased focus on:
• quadratic relations;
• modelling;
• exponents;
• two-dimensional
shapes;
• three-dimensional
figures;
• sine and cosine laws;
• one variable statistics;
• probability;
Decreased focus on:
• sequences and series;
• annuities and
mortgages;
• financial decision
making;
• career opportunities;
Return
Comparing Strands: Grade 11E
2000 Curriculum
Revised 2006 Curriculum
•
•
Earning and Purchasing
– Earning
– Describing Purchasing Power
– Purchasing
•
Saving, Investing and Borrowing
– Comparing Financial Services
– Saving and Investing
– Borrowing
•
Transportation and Travel
– Owning and Operating a
– Travelling by Automobile
– Comparing Modes of Transportation
Earning, Paying Taxes and Purchasing
– Earning Money
– Describing Forms of Taxation
– Purchasing Items
•
Saving, Investing and Borrowing
– Calculating Simple and Compound
Interest
– Understanding Saving and Investing
– Understanding Borrowing
•
Transportation and Travel
– Understanding the Costs of Owning and
Operating a Vehicle
– Understanding the Costs of Travelling
by Automobile
– Comparing Travel Costs
Revision Highlights: 11E
Increased focus on:
• connections to
workplace;
• gathering and
interpreting
information;
Decreased focus on:
• personal income
tax;
• monitoring value of
investments;
Return
Comparing Strands: Grade 11M
2000 Curriculum
Revised 2006 Curriculum
•
•
Financial Applications of Sequences and
Series
–
–
–
•
–
–
Arithmetic/Geometric Sequences and Series
Compound Interest and Annuity Problems
Financial Decision Making
–
Solving Quadratic Equations
Connecting Graphs and Equations of
Quadratic Functions
Solving Problems Involving Quadratic
Functions
Trigonometric Functions
–
–
–
–
•
Quadratic Functions
Sine Law/Cosine Law for Oblique Triangles
Understanding and Applying Radian Measure
Graphs and Equations of Sinusoidal Functions
Models of Sinusoidal Functions
•
–
–
Tools for Operating and
Communicating with Functions
–
–
–
Polynomials/Rational Expressions and
Exponential Expressions
Inverses/Transformations/Function Notation
Mathematical Reasoning
Exponential Functions
–
•
Connecting Graphs and Equations of
Exponential Functions
Solving Problems Involving Exponential
Functions
Solving Financial Problems Involving
Exponential Functions
Trigonometric Functions
–
–
–
Applying the Sine Law and the Cosine Law
in Acute Triangles
Connecting Graphs and Equations of Sine
Functions
Solving Problems Involving Sine Functions
Revision Highlights: 11M
Increased focus on:
• characteristics of
functions;
• quadratic functions;
• exponential functions;
• modelling;
• rate of change;
• periodic functions;
Return
Decreased focus on:
• sequences and series;
• rational expressions;
• annuities and mortgages;
• solving exponential
equations;
• solving trig equations;
• complex roots;
• radians;
• cosine/tangent functions;
• rational expressions;
• inverse functions;
• transformations;
• solving linear inequalities;
Grade 11M: Functions and Applications
Connections to Other Courses
MCF3M
MCR3U
MBF3C
MPM2D
MFM2P
Concept Development:
Looking at Financial Concepts
Concept
11E
Earnings


Purchasing


Financial
Services


Investing and
Borrowing

Simple Interest
Compound
Interest
Annuities
11C
11M
11U













DRAFT: Grade 11 Mathematics Training – Spring 2006: Day 2
Quadratic
Exponential
Trig
Polynomial
Rational
Algebraic
Representation
(e.g., Solving
Equations)
9
9
10
Inverse
Transformations
Domain
and
Range
Function
Graphical
Representation
(e.g., Zeros of
Function)
Linear
Numerical
Representation
(e.g., Finite
Differences)
Relation
EARNING ACTIVITY: FUNCTIONS
Concept Development:
Looking at Functions
Revising the Expectations
• some expectations were revised by:
- combining similar expectations
- folding expectations into processes
- reducing overlap of content among
expectations
- removing inappropriate expectations
• some expectations were expanded for
clarity
Eliminating Redundancy
2006 REVISED CURRICULUM
Grade 11U: Functions
2006 REVISED CURRICULUM
Grade 11M: Functions and
Applications
• Understanding
• Quadratic Functions
Functions
• Exponential Functions
• Exponential Functions • Trigonometric
• Discrete Functions
Functions
• Trigonometric
Functions
Improving Clarity
2000 CURRICULUM
Grade 11E: Mathematics for
Everyday Life
•calculate compound
interest by using the
simple-interest formula
and a given spreadsheet
template;
2006 REVISED CURRICULUM Grade
11E: Mathematics for Work and
Everyday Life
•determine, through investigation
using technology, the compound
interest for a given investment,
using repeated calculations of
simple interest for no more than six
compounding periods. (Sample
problem: Someone deposits $5
000 at 4% interest per annum,
compounded semi-annually. How
much interest accumulates in 3
years? );
Real-world Connections
2006 DRAFT REVISED CURRICULUM Grade 11M:
Functions and Applications
solve problems arising from real-life situations, given the
algebraic representation of quadratic relationship (e.g., given the
equation of a quadratic function representing the height of a ball
over an elapsed time, answer questions that involve finding the
maximum height of the ball, the length of time needed for the ball
to touch the ground, and the time interval when the ball is higher
than a given measurement) (Sample problem: The relationship
between power dissipated in a load resistor, P (in Watts, W),
electrical potential (in Volts, V), current (in amperes, A) and
resistance , R (in Ohms, Ω) is described by the formula P = EI –
I2R. If the electrical potential is fixed at 24 V, and the resistance
is fixed at 1.5 Ω , determine graphically and algebraically the
current that results in the maximum power dissipated.) < NEW >
Real-world Connections
2006 REVISED CURRICULUM Grade 11M:
Functions and Applications
collect data arising from applications that can be modelled
as an exponential relation, through investigation with and
without technology, from primary sources using a variety of
tools (e.g., concrete materials; measurement tools such as
electronic probes) or from secondary sources (e.g., web
sites such as Statistics Canada, E-STAT), and graph the
data (Sample problem: Collect data and graph the cooling
curve representing the relationship between temperature
and time for hot water cooling in a porcelain mug. Predict
the shape of the cooling curve when hot water cools in an
insulated mug. Test your prediction.)
Real-world Connections
There was a time when some said the national debt increased
exponentially. Determine if there is a domain over which the graph of the
National Debt could be modelled by an exponential curve.
National Debt (1867-2005)
700000
Debt in Millions
600000
500000
400000
Year
Debt
300000
200000
100000
0
1
11 21 31 41 51
61 71 81 91 101 111 121 131
Year
Real-world Connections
More Examples
2000 CURRICULUM
Grade 11E: Mathematics for
Everyday Life
< NEW >
2006 REVISED CURRICULUM Grade
11E: Mathematics for Work and
Everyday Life
•describe the effects of different
remuneration methods (e.g., hourly
rate, overtime rate, job or project
rate, commission, salary, gratuities)
and remuneration schedules (e.g.,
weekly, biweekly, semi-monthly,
monthly) on decisions related to
personal spending habits (e.g., the
timing of a major purchase, the
scheduling of mortgage payments
and other bill payments.);
Key Message: Equity
The revised curriculum supports
equity by promoting excellence in
mathematics education for all
students.
Equity – NCTM Perspective
• All students, regardless of their personal
characteristics, backgrounds, or physical
challenges, must have opportunities to study and
support to learn mathematics
• All students need access each year they are in
school to a coherent, challenging mathematics
curriculum taught by competent and wellsupported mathematics teachers.
• Too many students, especially students who are
poor, not native speakers of English, disabled,
female, or members of minority groups, are
victims of low expectations in mathematics.
Equity – Feedback
• Revisions must meet the needs of the
students entering mathematics-related
university programs.
• Equal access to senior mathematics
courses across the province is very
important.
• The current curriculum is too dense
resulting in a reduction of students
engaging in senior mathematics and a
decrease in the chance of success for
some students.
What Factors Contribute Most To
Students’ Success in Mathematics?
• active participation in meaningful mathematics;
• in-depth understanding of mathematics is
supported by active involvement in mathematical
modelling, problem solving and reasoning through
application
• ample time to perform investigations and to revise
work;
• classroom practices that encourage discussion
among students and between students and
teachers;
• student reflection on their work;
• an appreciation of student diversity.
Ed Thoughts 2002 – Research and Best Practice.
What Factors Contribute Most To
Students’ Success in Mathematics?
• learning experiences that involve a range of
activity from short whole-group instruction to
longer times engaged in problem solving
• positive student-teacher relationships
• “user-friendly” classroom environments in
which prior knowledge is identified and built
upon, and where instruction is developmentally
appropriate
Ed Thoughts 2002 – Research and Best Practice.
Equity: Developmentally Appropriate
A developmentally appropriate curriculum
• is challenging but attainable for most
students of a given age group preparing
for a given destination
• allows enough flexibility to respond to
inevitable individual variation
• is consistent with the students’ ways of
thinking and learning
(Adapted from Clements, Sarama & DiBiase, 1997)
How do Students’ Attitudes Affect Their
Performance and Future Opportunities?
Students’ attitudes toward mathematics have a
great effect on student achievement.
• Students who enjoy mathematics tend to
perform well in their mathematics course work
and are more likely to enrol in the more
advanced mathematics courses.
• Students who dislike mathematics tend not to
do well in these classes, and/or do not attempt
the more advanced mathematics classes in
secondary school.
Ed Thoughts 2002 – Research and Best Practice
How do Students’ Attitudes Affect Their
Performance and Future Opportunities?
Students develop positive attitudes when they
• make mathematical conjectures;
• make breakthroughs as they solve
problems;
• see connections between important ideas.
Ed Thoughts 2002: Research and Best Practice
How do Students’ Attitudes Affect Their
Performance and Future Opportunities?
Students with a productive attitude
• find sense in mathematics,
• perceive it as both useful and
worthwhile,
• believe that steady effort in learning
mathematics pays off
• view themselves as effective learners
and doers of mathematics.
Ed Thoughts 2002: Research and Best Practice
How do Students’ Attitudes Affect Their
Performance and Future Opportunities?
Students experience frustration when they
are not making progress towards solving a
problem. Therefore, it is important that
instruction provide appropriately challenging
problems so students can learn and
establish the norm of perseverance for
successful problem solving.
Ed Thoughts 2002: Research and Best Practice
Equity
Students can be considered to
be “at-risk” when they are in
peril of not reaching their
learning potential.
CMESG Work Group
Personal Reflection
Reflection:
Most students who take mathematics do not
pursue post secondary destinations that have an
emphasis on mathematics. What are the important
skills you believe these students should develop
through senior mathematics?
Key Message: Learning
The revised curriculum supports
students learning mathematics
with understanding and actively
building new knowledge from
experience and prior knowledge.
Developing Understanding
We use the ideas
we already have
(blue dots) to
construct new
ideas (red dot).
The more ideas
we use and the
more connections
we make, the better
we understand.
John Van de Walle
Conceptual Understanding
• Conceptual understanding supports
retention. When facts and procedures are
learned in a connected way, they are
easier to remember and use and can be
reconstructed when forgotten.
Hiebert and Wearne 1996; Bruner 1960, Katona 1940
Improving Articulation Across
The Grades
Academic Pathway
Applied Pathway
Grade 9
•Linear Relations
•Linear Relations
Grade 10
•Quadratic Relations
•Modeling Linear Relations
•Quadratic Relations
Draft
Grade 11
•Understanding Functions
•Exponential Functions
•Discrete Functions
•Trigonometric Functions
•Mathematical Models
–Quadratic Relations
–Exponential Relations
Proposed
Grade 12
(Nov
2005)
•Polynomial Functions
•Trigonometric,
Exponential and
Logarithmic Functions
•Rates of Change
Mathematical Models:
–Solving Exponential
Equations
–Interpreting and Analyzing
Graphical Representations
–Interpreting and Analyzing
Algebraic Representations
Improving Articulation Across
The Grades
Draft Revised Gr. 11 Foundations
Proposed Draft Gr. 12 C (Nov 2005)
Mathematical Models
–Investigating Graphs and Equations of
Quadratic Relations
–Understanding Exponential Growth and Decay
–Investigating Graphs and Equations of
Exponential Relations
Mathematical Models
–Solving Exponential Equations
–Interpreting and Analyzing Graphical
Representations
–Interpreting and Analyzing Algebraic
Representations
Personal Finance
–Solving Problems Involving Compound Interest
–Investing and Borrowing
–Owning and Operating A Vehicle
Personal Finance
–Understanding Annuities
–Renting/Buying Accommodation
–Designing Budgets
Measurement and Trigonometry
–Representing Two-Dimensional Shapes and
Three Dimensional Figures
–Applying the Sine Law and the Cosine Law in
Acute Triangles
Measurement and Trigonometry
–Optimization Problems
–Solving Problems Involving
Trigonometry
Reasoning With Data
–Working with One Variable Data
-Applying Probability
Reasoning With Data
–Two Variable Analysis
–Evaluating Validity
Developing Concepts Through
Investigation
2000 CURRICULUM
Grade 11M: Functions
•define the term
function;
2006 DRAFT REVISED CURRICULUM
Grade 11U: Functions and
Applications
• explain the meaning of the term
function, through investigation of
linear and quadratic relations
using a variety of representations
(i.e., tables of values, mapping
diagrams, graphs, functions
machines) (Sample problem: give
examples of linear and quadratic
relations that are functions and
that are not functions using a
variety of representations);
Representations
Graphical
Representation
Algebraic
Representation
f(x) = 2x - 1
Numerical
Representation
Concrete
Representation
Culminating With Solving Problems
2000 CURRICULUM
Grade 11: Mathematics
of Personal Finance
< NEW >
2006 DRAFT REVISED CURRICULUM Grade
Grade 11: Foundations for College
Mathematics
•solve design problems that satisfy given
constraints (e.g., design a rectangular
berm that would hold all the oil that could
leak from a cylindrical storage tank),
using physical models (e.g., built from
popsicle sticks, cardboard, duct tape) or
drawings (e.g., made using design
software) (Sample problem: Design and
construct a model boat that can carry the
most pennies, using one sheet of 8 ½” x
11” card stock and no more than five
popsicle sticks)
Balancing Conceptual and Procedural
Learning
Reflection:
• Does the balance vary depending on the students?
• Does the balance vary depending on the course?
• Is there an order?
• Does the balance vary depending on whether
the concept is new or an extension?
Personal Reflection
Balanced Activity Reflection:
What does an appropriate balance mean to you
and how does this impact on your students’ long
term success in senior mathematics?
Key Message: Teaching
The revised curriculum supports
effective mathematics teaching that
requires understanding what
students know and need to learn
and do.
Teaching
Learning mathematics … requires
understanding and being able to apply
procedures, concepts and processes. In
the twenty-first century, all students should
be expected to understand and be able to
apply mathematics.
NCTM, Principles and Standards, 2000.
Mathematical Processes: Research
Mathematical proficiency, as we see it, has five
components, or strands:
• procedural fluency—skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately
• conceptual understanding—comprehension of
mathematical concepts, operations, and relations
• strategic competence—ability to formulate,
represent, and solve mathematical problems
• adaptive reasoning—capacity for logical thought,
reflection, explanation, and justification
• productive disposition—habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled
with a belief in diligence and one’s own efficacy.
(Kilpatrick, Swafford, &Findell, 2001)
Mathematical Processes
Problem Solving
Reasoning and Proving
Reflecting
Selecting Tools and Computational Strategies
Connecting
Representing
Communicating
Mathematical Processes:
• the Actions of Mathematics
• ways of acquiring and using the content,
knowledge and skills of mathematics
• interconnected
• not New !!
Mathematical Processes
Representing
pose and solve problems related to
“pose
models of sinusoidal functions drawn from
applications and communicate
a variety of applications,
justification
the solution with clarity and justification,
using appropriate mathematical forms …
pp. 24, Gr 11, 1999
Connecting
Mathematical Processes
Representing
Problem Solving
Reflecting
Communicating
Reasoning and Proving
Connecting
Selecting Tools and
Computational
Strategies
Mathematical Proficiency
Mathematical Processes
Key Message:
Assessment and Evaluation
The revised curriculum supports
assessment for the learning of
important mathematics and to
furnish useful information to both
teachers and students.
Assessment and Evaluation
• Do overall expectations
have to be evaluated?
YES
• Do all specific
expectations have to be
evaluated?
NO
• Do all specific
expectations have to be
taught?
YES
Knowledge and Understanding
• Factual/Procedural Knowledge
• Relationships (e.g. Pythagorean
Relationship)
• Procedural Fluency (e.g. multi-digit
computation)
• Meanings of terms in mathematics (e.g.,
property, parallelogram)
• Conceptual Understanding
• Reflecting an understanding of
mathematical concepts (e.g. place value,
area, rate)
Thinking
Use of planning skills
• understanding the problem (e.g., formulating and
interpreting the problem, making conjectures)
• making a plan for solving the problem
Use of processing skills
• carrying out a plan (e.g., collecting data, questioning,
testing, revising, modelling, solving, inferring, forming
conclusions)
• looking back at the solution (e.g., evaluating
reasonableness, making convincing arguments,
reasoning, justifying, proving, reflecting)
Use of critical/creative thinking processes (e.g.,
problem-solving, inquiry)
Application
• Application of knowledge and skills in familiar
contexts
• Transfer of knowledge and skills to new contexts
• Making connections within and between various
contexts (e.g., connections between concepts,
representations, and forms within mathematics;
connections involving use of prior knowledge and
experience; connections between mathematics,
other disciplines, and the real world)
Communication
• Expression and organization of ideas and
mathematical thinking using oral, visual and
written forms
• Communication for different audiences and
purposes in oral, visual, and written forms
• Use of conventions, vocabulary, and terminology
of the discipline in oral, visual, and written forms
Key Message: Learning Tools
The revised curriculum
supports the use of
technology and manipulatives
as tools for teaching and
learning mathematics.
Learning Tools:
Dynamic Geometry Software/Spreadsheets
2000 CURRICULUM
Grade 11M:
Functions
<NEW>
2006 REVISED CURRICULUM
Grade 11M:
Functions and Applications
•verify, through investigation using
technology (e.g., dynamic
geometry software, spreadsheets)
the sine law and the cosine law
(e.g., compare, using dynamic
geometry software, the ratios of
a/sin A, b/sin B and c/sin C in
triangle ABC, while dragging one
of the vertices);
Learning Tools:
Dynamic Statistics Software/Spreadsheets
2000 CURRICULUM
Grade 11C:
Personal Finance
< NEW>
2006 REVISED CURRICULUM
Grade 11C: Foundations of
Mathematics
•collect one-variable data from
secondary sources (e.g., internet
databases) and organize and
store the data using a variety of
tools (e.g., spreadsheets,
dynamic statistical software);
Learning Tools:
Calculators and Manipulatives
2000 CURRICULUM
Grade 11C: Mathematics
of Personal Finance
2006 CURRICULUM Grade 11C:
Foundations of Mathematics
•expand and simplify
polynomial expressions
involving the multiplying
and squaring of
binomials;
• expand and simplify, using
a variety of tools (e.g., paper
and pencil, algebra tiles,
computer algebra systems)
quadratic expressions in onevariable, involving multiplying
and squaring of binomials (e.g.,
½ x + 1)(3x – 2) or 5(3x – 1)2)
Learning Tools:
Cooling Curve
Learning Tools:
Fuel Consumption Calculator
Learning Tools:
Algebra Tiles:
a Square
Completing the
Learning Tools:
TVM Solver:
Doubling Time
Learning Tools:
Half-Life Activity
Next Steps
1
Working Toward Alignment
INTENDED
CURRICULUM
Ministry
Curriculum
Expectations
DELIVERED
CURRICULUM
Instructional
Program
In The
Classroom
ACHIEVED
CURRICULUM
What Is
Being
Assessed