Transcript Learning - Upper Canada District School Board

REVISED
LEARNING
DISTRICT
TRAINING
SESSION
Learning
Learning
The revised curriculum supports
students learning mathematics
with understanding and actively
building new knowledge from
experience and prior knowledge.
Conceptual Understanding
• Conceptual understanding refers to an
integrated and functional grasp of
mathematics.
• It is more than knowing isolated facts and
procedures.
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
Conceptual Understanding
• Knowledge that has been learned with
understanding provides the basis for
generating new knowledge and for solving
new and unfamiliar problems.
Bransford, Brown and Cocking 1998
Activate Your Memory
Try This!
Write
DIG
Write
HAD
Write
AGE
How
Did
You
Do?
Making a Connection!
A
B
C
D
E
F
G
H
I
Developing Understanding
We use the ideas
we already have
(blue dots) to
construct a new
idea (red dot).
The more ideas
used and the
more connections
made, the better
we understand.
John Van de Walle
Making Connections
COURSE: GRADE 9 Applied and Academic
1999 Curriculum
Revision
-determine, from examination
of patterns, the exponent rules
for multiplying and dividing
monomials and the exponent
rule for the power of a power,
and apply….
-determine the meaning of
negative exponents and of
zero as an exponent from
activities involving graphing,
using technology, and from
activities involving patterning
-describe the relationship
between the algebraic and
geometric representations of
a single variable term up to
degree three ( i.e., length,
which is one dimensional,
can be represented by x;
area, which is two
dimensional can be
represented by x^2, and
volume, which is three
dimensional can be ….
“Education that consists in
learning things and not the
meaning of them is like feeding
upon husks and not the corn.”
Mark Twain
Developmental Continuum
COURSE: GRADE 9 Applied to 10 Applied
1999 Curriculum
Grade 9 Applied:
Linear relationships are
generalised as analytic
geometry (linear modelling)
Revision
Grade 9 Applied:
Linear relationships
(understanding of, and
applications of “real life”
examples)
Grade 10 Applied:
Linear relationships are
generalized as analytic
geometry, spreading this
concept over 2 years
Continuum of Learning:
Support Resource for Draft Revision Spring 2005
Number Sense and Numeration
GRADE 1: read, represent, order, and
compare whole numbers to 50, and
investigate money amounts and fractions;
GRADE 5: read, represent, order, and
compare whole numbers to 100 000,
decimal numbers to hundredths,
proper and improper fractions, and
mixed numbers;
GRADE 2: read, represent, order, and
compare whole numbers to 100, and
represent money amounts and fractions
using concrete materials
GRADE 6: read, represent, order, and
compare whole numbers to 1 000 000,
decimal numbers to thousandths,
proper and improper fractions, and
mixed numbers
GRADE 3: read, represent, order, and
compare whole numbers to 1000, and
demonstrate their understandings about
money and fractions
GRADE 7: represent, order, and
compare numbers, including integers
GRADE 4: read, represent, order, and
compare whole numbers to 10 000, decimal
numbers to tenths, and simple fractions, and
expand their understandings about money
GRADE 8:represent, order, and
compare equivalent representations
of numbers including those involving
exponents
Developing Concepts Across the Grades
1997 Curriculum
Proportional Reasoning
Draft Revision Spring 2005
Proportional Reasoning
Grade 7: No Specific Reference
Grade 7:
Number Sense and Numeration
Proportional Relationships
Grade 8: Under Applications
Grade 8:
Number Sense and Numeration
Proportional Relationships
Grade 9 Applied:
Number Sense and Algebra
Solving Numerical Problems
Grade 9 Applied:
Number Sense and Algebra:
Proportional Reasoning
Grade 10 Applied:
Proportional Reasoning
Grade 10 Applied:
Measurement and Trigonometry:
Solving Problems Involving
Similar Triangles
How the Revised Curriculum
fits together…
It fits like a jigsaw puzzle…
Understanding Exponents
Gr 7
Gr 8
Gr 9
Gr 10
Explain the Express
faSubstitute
x fb = f a+b
relationship Represent
repeated whole Derive through Determine the
Extend
the
into
and
numbers
in
between
multiplication investigation the meaning of zero
multiplication
aevaluate
b = f a-b
f
/f
expanded
form
exponent
using
exponent
rules and negative
rule
to
derive
347
=
3x10²+4x10+7
algebraic
powers for multiplying exponents
notation and using
exponential
and
understand
expressions
4
a
b
axb
the
notation
2x2x2x2=2
(fand
)
=
f
dividing
347
=
the
power of a
involving
measurement 3x10²+4x10+7
monomials
power
rule
2x2x2x2=24
exponents
of area and
volume
Developing Concepts - Volume
Grade 4
Grade 5
Grade 6
Grade 7
Measure Volume
Gr 8
Grade 9
Grade 9
Grade 10
Academic
Applied
Applied
Solve problems
Develop formulas
Involving optimal
for the volumes of involving volumes
Solve problems
volume. Solve the pyramids, cones
of prisms, pyramids
max and min vol
cylinders, cones,
and spheres.
pyramids, cones
spheres, and a
and spheres.
combination.
Gr 9
Ratio, Rate and Proportion
Grade 9 – the seven specific expectations are:
1) Perform operations with rational numbers, as necessary to support other topics of
this course (e.g., rate of change, proportionality, measurement, percent)
2) Illustrate equivalent ratios using a variety of tools
3) Represent directly proportional relationships with equaivalent ratios and
proportions, arising from realistic situations (Sample problem: You are building a
skateboard ramp whose ratio of height to base must be 2:3. Write a proportion that
could be used to find the base if the height is 4.5 m)
4) Solve for the unknown value in a proportion
5) Make comparisons using unit rate
6) Solve problems involving ratios, rates, and directly proportional relationships in
various contexts
7) Solve problems requiring the expression of percents, fractions, and decimals in
their equivalent forms (e.g., calculating simple interest and sales tax, analysing data
(Sample problem: Of the 29 students in a Grade 9 Math class, 13 are taking science
this semester. If this class is representative of all the Grade 9 students in the
school, what percent of the 236 Grade 9 students are taking science this semester?
How many grade 9 students does this percent represent?)
Ratio, Rate and Proportion
Gr 7
Gr 8
Your turn to find
the expectations
for gr 7 and 8
Gr 9
There are 7
specific
expectations
under the
Overall expec:
solve problems
involving
proportional
reasoning
Gr 10
Use their
knowledge of
ratio and
proportion … and
solve problems.
Determine the
lengths of sides of
similar triangles
…
Ratio, Rate and Proportion
Grade 7
Grade 8
1)Solve problems that involve determining
whole number percents (Sample problem:
If there are 5 blue marbles in a bag of 20
marbles, what percentage of the marbles
are not blue?)
1)Solve problems involving rates (Sample
problem: A pack of 24 CD’s costs $7.99. A
pack of 50 CD’s costs $10.45. What is the
most economical way to purchase 130 CD’s?
2)Recognize and describe real-life situations
2)Define rate as a comparison of two
involving two quantities that are directly
quantities with different units (e.g., speed is proportional (e.g., number of servings and
a rate that compares distance to time)
quantities in a recipe, mass and volume,
circumferences and diameters of circles)
3)Determine through investigation, the
relationship amoung fractions, decimals,
percent and ratios
3) Solve problems involving percent arising
from real-life contexts (e.g., discounts, sales
4)Solve problems involving the calculation tax, simple interest)
of unit rates (Sample Problem: You go
4) Solve problems involving proportions using
shopping and notice that 25 kg of Carol’s
concrete materials, drawings, and variables
Famous Potaotoes costs $12.95. And 10 kg (Sample Problem. The ratio of stone to sand in
of Gillian’s Potatoes costs $5.78. Which is HardFast Concrete is 2 to 3. How much stone
is needed if 15 bags of sand are used?)
the better deal?
Your turn to put the puzzle together
Gr 7
Gr 8
Gr 9
Gr 10
Your turn to investigate expectations from
grade 7, 8, 9 and 10 that build on one another.
Concept Development
• Expectations that introduce and develop a
concept often include the phrase “through
investigation”.
• Expectations that involve concepts that
give rise to procedural learning and
require some level of proficiency often
include the phrase “solve problems”.
Through Investigation
COURSE: GRADE 9 Applied and Academic
1999 Curriculum
Revision
-solve simple problems, using
the formulas for the surface
area of prisms and cylinders
and for the volume of prisms,
cylinders, cones and spheres
-develop through
investigation (e.g. using
concrete materials) the
formula for the volume of a
pyramid, a cone, and a
sphere (e.g., use 3
dimensional figures to show
that the volume of a pyramid
(cone) is one third the
volume of a prism (or
cylinder) with the same base
and height
-solve problems…
Through Investigation
Grades 1 - 8
1997 CURRICULUM
Grade 7
• describe data using
measures of central
tendency (mean, median
and mode);
SPRING 2005 DRAFT
Grade 7
•compare, through
investigation, how the data
values affect the median and
the mean (e.g., changing the
value of an outlier can have a
significant effect on the mean
and no effect on the median);
Through Investigation
COURSE: GRADE 10 Applied and Academic
1999 Curriculum
Revision
-define the formulas for the
sine, the cosine, and the
tangent of angles, using the
ratios of sides in right
triangles
-determine through
investigation (e.g., using DGS,
concrete materials), the
relationship between the ratio
of two sides in a right triangle
and the ratio of the two
corresponding sides in a similar
right triangle, and define the
sine, cosine, and tangent ratios
(e.g.,
sinA=opposite/hypontenuse)
Culminating With Solving Problems
Learning should culminate with
the application of knowledge and
skills to solve problems.
Culminating with Solving Problems
1997 CURRICULUM
Grade 8
• multiply and divide
integers;
FEBRUARY 2005 DRAFT
Grade 8
• solve problems involving
operations with integers, using
a variety of tools;
Culminating With Solving Problems
COURSE: GRADE 10 Academic
1999 Curriculum
-solve quadratic equations
using the quadratic formula
Revision
-explore the algebraic
development of the quadratic
formula (e.g., given the
algebraic development,
connect the steps to a
numerical example; follow a
demonstration of the
algebraic development
(student reproduction of the
development of the general
case is not required)
-solve quadratic equations…
Procedural Fluency
Procedural Fluency refers to knowledge of
procedures, knowledge of when and how
to use them appropriately, and skill in
performing them flexibly, accurately and
efficiently.
Kilpatrick et al, 2001
Yours is not to
reason why,
just invert and
multiply!
Balancing Conceptual Understanding and
Procedural Fluency
• Pitting procedural fluency against conceptual
understanding creates a false dichotomy.
• Understanding makes learning skills easier, less
susceptible to common errors and less prone to
forgetting.
• Also, a certain level of skill is required to learn
many mathematical concepts with
understanding
Hiebert and Carpenter 1992
Procedural Fluency
• A good conceptual understanding of place
value supports the development of fluency
in multidigit computation.
Heibert, Carpenter et al 1997; Resnick and Omanson 1987
Procedural / Conceptual
COURSE: GRADE 10 Academic
1999 Curriculum
Revision
-determine, through
investigation, the relationships
between the angles and sides
in acute triangles (e.g., the
largest angle is opposite the
longest side; the ratio of the
sines of the opposite angles),
using DGS
-explore the development of the
cosine law within acute
triangles (e.g., use DGS to
verify the cosine law; follow the
algebraic development of the
cosine law and identify its
relationship to the Pythagorean
theorem and the cosine ratio
(student reproduction of the
development of the formula is
not required)
-solve problems …
Connecting to the problem
• What concepts are incorporated into this
problem?
• Are there aspects of the problem that
begin at the conceptual stage and move
towards the procedural?