Transcript Teaching Lense - Upper Canada District School Board

Sustaining
Quality
Curriculum
Grades 7 - 10
Teaching
REVISED
TEACHING
DISTRICT
TRAINING
SESSION
Key Revision Message: Teaching
The revised curriculum supports
effective mathematics teaching that
requires understanding what students
know and need to learn and do.
Clarity
1997 CURRICULUM
Grade 7
• recognize patterns and
use them to make
predictions;
SPRING 2005 DRAFT
Grade 7
• make predictions, by extending
patters, through investigations
with concrete materials; Sample
Problem: Investigate the surface
area of cube towers to predict the
surface area of a cube tower that
is fifty cubes high.
Clarity
1999 CURRICULUM
Grade 9 Academic
• expand and simplify
polynomial expressions
involving one variable
SPRING 2005 DRAFT
Grade 9 Academic
•multiply a polynomial by a
monomial in the same variable to
give results up to degree three
(e.g….) using a variety of tools…
Use of “Through Investigation”
1997 CURRICULUM
Grade 8
•
investigate the
Pythagorean
relationship using
area models and
diagrams;
SPRING 2005 DRAFT
Grade 8
•determine, through
investigation, the
Pythagorean relationship,
using a variety of tools and
strategies (e.g., dynamic
geometry software, paper
and scissors, geoboards)
Use of “Through Investigation”
1999 CURRICULUM
Grade 10 Academic
SPRING 2005 DRAFT
Grade 10 Academic
• describe the nature
of change in a
quadratic function
using finite differences
in tables of values, and
compare the nature of
change in a quadratic
function with the nature
of change in a linear
function
•Determine, through investigation,
that a quadratic relation of the form
y = ax2 + bx + c can be graphically
represented as a parabola, and that
the table of values yields a constant
second difference (Sample problem:
Graph the relation…observe the
shape…calculate first and second
differences…Describe….)
More References to Solving Problems
1997 CURRICULUM
Grade 8
• apply the
Pythagorean
relationship to
numerical problem
involving area and
right triangles;
SPRING 2005 DRAFT
Grade 8
• solve problems
involving right triangles
geometrically, using the
Pythagorean relationship;
More References to Solving Problems
1999 CURRICULUM
Grade 9 Applied
and Academic
• calculate sides in
right triangles, using
the Pythagorean
theorem, as required
in topics throughout
the course (e.g.,
measurement)
SPRING 2005 DRAFT
Grade 9 Applied and
Academic
•solve problems using the
Pythagorean Theorem in
applications (e.g., Calculate
the perpendicular height of
a cone given the radius and
the slant height for the
purpose of finding the
volume of a cone);
More Examples and Sample Problems
1997 CURRICULUM
Grade 8
• investigate
measures of
circumference using
concrete materials
(e.g., use string to
measure the
circumference of
cans or bottles);
SPRING 2005 DRAFT
Grade 8
•
measure the
circumference, radius and
diameter of circular
objects, using concrete
materials; Sample
Problem: Use string to
measure the
circumferences of different
circular objects
More Examples and Sample Problems
1999 CURRICULUM
Grade 10 Academic
and Applied
• solve problems
involving similar
triangles in realistic
situations (e.g….)
SPRING 2005 DRAFT
Grade 10 Academic and
Applied
• solve problems involving
similar triangles in realistic
situations (e.g…) (Sample
problem: Use a metre stick
to find the height of a tree,
by means of the similar
triangles formed by the tree,
the metre stick, and their
shadows)
Mathematical Process Expectations
• apply developing problem-solving strategies as
they pose and solve problems and conduct
investigations, to help deepen their mathematical
understanding
•demonstrate that they are reflecting on and
monitoring their thinking to help clarify their
understanding as they complete an investigation
or solve a problem (e.g., by explaining to others
why they think their solution is correct.)
•select and use a variety of concrete, visual, and
electronic learning tools and appropriate
computational strategies to investigate
mathematical ideas and to solve problems
Mathematical Processes:
• The actions of mathematics
• Ways of acquiring and using the
content, knowledge and skills of
mathematics
Teaching Activity – Part A
• Select one statement slip.
• Everyone in your group has a statement relating
to a different process.
• Which process does your statement relate to?
• Share your statement with the group.
Teaching Activity – Part A
Identify your mathematical process
Reasoning and
Proving
Representing
Reflecting
Problem Solving
Communicating
Connecting
Selecting Tools
and Strategies
How Did You Do?
Reasoning and
Proving
Communicating
Problem Solving
Connecting
x
x
x
Representing
Reflecting
Selecting Tools and
Computational
Strategies
Teaching Activity – Part A
• Sequence the statements into a
meaningful message about your process.
• Use the stem statement below to
summarize your process.
The important thing about (insert
process here) is (insert a statement that
captures the essence of the process).
Teaching Activity – Part A
Consider the following expectation:
Determine the relationships among units and
measurable attributes, including the area of a
circle and volume of a cylinder.
Can students develop and demonstrate your
group’s mathematical process by addressing the
expectation above?
Teaching Activity – Part B
• With a partner, read the cluster of
expectations for Grade 8 Measurement
Relationships
• Identify those expectations which provide
an opportunity for students to develop and
demonstrate your groups process
• Share your thoughts with others at your
table.
• Place the large coloured dots/X’s beside
the expectations on the wall chart.
What Do You Notice About The Distribution?
Reasoning and
Proving
Communicating
Problem Solving
Connecting
x
x
x
Representing
Reflecting
Selecting Tools and
Computational
Strategies
Messages from Focus Groups
Teaching must include providing
opportunities for students to develop and
demonstrate essential mathematical
processes such as problem solving and
communication.
Investigation and modelling play a critical
role in learning mathematics through
problem solving.
Existing Process Expectations
• judge the reasonableness of answers produced
by a calculator, a computer, or pencil and paper,
using mental mathematics and estimation.
From Grade 9 Measurement and Geometry
• communicate solutions to problems in
appropriate mathematical forms (e.g. written
explanations, formulas, charts, tables, graphs) and
justify the reasoning used in solving problems
From Grade 9 Number Sense and Algebra
Research on Mathematical
Processes
Mathematical proficiency, as we see it, has five
components, or strands:
•
•
•
•
•
conceptual understanding—comprehension of mathematical
concepts, operations, and relations
procedural fluency—skill in carrying out procedures flexibly,
accurately, efficiently, and appropriately
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: Other References
• NCTM
• Quebec Curriculum
• Singapore Curriculum
• Danish Curriculum
Relationships
Reasoning and
Proving
•
CONCEPTS
SKILLS
FACTS
Representing
Connecting
FACTS
SKILLS
CONCEPTS
FACTS
SKILLS
Reflecting
Problem Solving
CONCEPTS
Communicating
PRIOR KNOWLEDGE AND
UNDERSTANDING
Selecting Tools
and Strategies
Balancing Mathematical
Processes
Mathematical
Concepts, Facts
and Procedures
KNOWING
INSTRUCTION
ASSESSMENT
EXPECTATIONS
CATEGORIES
Mathematical
Processes
DOING
Connecting Achievement Chart and
Mathematical Processes
MATHEMATICS
ARTS
Knowledge and UnderstandingSCIENCE
Thinking
Communication
SOCIAL
STUDIES
Application
PHYSICAL
EDUCATION
LANGUAGE
ARTS
Connecting Achievement Chart and
Mathematical Processes
Knowledge and Understanding
Procedural Knowledge
Conceptual Understanding
Mathematical
ThinkingProcesses
Problem Solving
Reflecting Reasoning and Proving
Communication
Communicating
Representing
Application
Selecting Tools and Strategies
Connecting
Teaching
The revised curriculum supports effective
mathematics teaching that requires
understanding what students know and
need to learn and do.
Impact on Teaching
How do the revisions that
focus on mathematical
processes impact on
teaching?
Introductory Task
Teaching Connection
Which mathematical
processes could this
task help students to
develop and
demonstrate?
Where? Why? How?