Transcript Document

Presented by
Taryn DiSorbo and Kim Vesper
 Developing
students’ capacity to engage in
the mathematical practices specified in the
Common Core State Standards will ONLY be
accomplished by engaging students in solving
challenging mathematical tasks, providing
students with tools to support their thinking
and reasoning, and orchestrating
opportunities for students to talk about
mathematics and make their thinking public.
It is the combination of these three
dimensions of classrooms, working in unison,
that develop students habits of mind and
promote understanding of mathematics.
 The
tasks or activities in which students
engage should provide opportunities for them
to “figure things out for themselves.” (NCTM,
2009, pg. 11), and to justify and
communicate the outcome of their
investigation;
 Tools (i.e. language, materials, and
symbols) should be available to provide
external support for learning. (Hiebert, et al,
1997); and
classroom talk should make
students’ thinking and reasoning public so
that it can be refined and/or extended.
(Chapin, O’Conner, & Anderson, 2009).
 Productive







Significant Content (i.e. they have the potential to
leave behind important residue)(Hiebert et al, 1997).
High Cognitive Demand (Stein et. al, 1996; Boaler &
Staples 2008)
Multiple ways to enter the task and show competence
(Lotan, 2003)
Require justification or explanation (Boaler & Staples
2008)
Make connections between different representations
(Lesh, Post, & Behr, 1988)
Provide a context for sense making (Van De Walle,
Karp, & Bay-Williams, 2013)
Provide opportunities to look for patterns, make
conjectures, and form generalizations. (Stylianides,
2008; 2010)
Teacher Actions
Practice
Task Features
Tools
1
Make sense of problems
and persevere in solving
them
• High Cognitive
Demand
• Multiple Entry Points
• Requires Explanation
2
Reason Abstractly and
quantitatively
Contextual
3
Construct Viable
arguments and critique the
reasoning of others
Requires justification or
proof
4
Model with mathematics
Contextual
5
Use appropriate tools
strategically
6
Attend to precision
7
Look for and make use of
structure
Opportunity to look for
patterns and make
conjectures
8
Look for and express
regularity in repeated
reasoning
Opportunity to make
generalizations
Talk
Teacher Actions
Practice
Task Features
Tools
1
Make sense of problems and
persevere in solving them
• High Cognitive
Demand
• Multiple Entry Points
• Requires Explanation
Make resources available
that will support entry and
engagement
2
Reason Abstractly and
quantitatively
Contextual
Encourage use of different
representational forms
3
Construct Viable arguments
and critique the reasoning
of others
Requires justification or
proof
4
Model with mathematics
Contextual
5
Use appropriate tools
strategically
Can be solved using
different tools
6
Attend to precision
7
Look for and make use of
structure
Opportunity to look for
patterns and make
conjectures
8
Look for and express
regularity in repeated
Opportunity to make
generalizations
Make tools available that
will support entry and
engagement
Talk
 Students
must talk, with one another as well
as in response to the teacher. When the
teacher talks most, the flow of ideas and
knowledge is primarily from teacher to
student. When students make public
conjectures and reason with others about
mathematics, ides, and knowledge are
developed collaboratively, revealing
mathematics as constructed by human beings
within an intellectual community.
 NCTM, 1991, p. 34
Teacher Actions
Practice
Task Features
Tools
1
Make sense of problems and
persevere in solving them
• High Cognitive Demand
• Multiple Entry Points
• Requires Explanation
2
Reason Abstractly and
quantitatively
Contextual
3
Construct Viable arguments
and critique the reasoning of
others
Requires justification or
proof
4
Model with mathematics
Contextual
5
Use appropriate tools
strategically
Can be solved using
different tools
6
Attend to precision
7
Look for and make use of
structure
Opportunity to look for
patterns and make
conjectures
8
Look for and express regularity
in repeated reasoning
Opportunity to make
generalizations
Make resources
available that will
support entry and
engagement
Talk
• Ask students to
explain their
thinking
• Ask questions that
assess and advance
student
understanding
Teacher Actions
Practice
Task Features
Tools
Talk
1
Make sense of problems and
persevere in solving them
• High Cognitive
Demand
• Multiple Entry Points
• Requires Explanation
Make resources
available that will
support entry and
engagement
• Ask students to explain their
thinking
• Ask questions that assess and
advance student understanding
2
Reason Abstractly and
quantitatively
Contextual
Encourage use of
different
representational
forms
Prompt students to make
connections between symbols and
what they represent in context
3
Construct Viable arguments
and critique the reasoning of
others
Requires justification or
proof
Ask students to argue for their point
of view and evaluate and make
sense of the reasoning of their peers
4
Model with mathematics
Contextual
Ask students to justify their models
5
Use appropriate tools
strategically
Can be solved using
different tools
6
Attend to precision
7
Look for and make use of
structure
Opportunity to look for
patterns and make
conjectures
Encourage students to look for
patterns
8
Look for and express
regularity in repeated
reasoning
Opportunity to make
generalizations
Prompt students to consider the
connections between the current
task and prior tasks
Make tools
available that will
support entry and
engagement
Encourage students to be clear and
use appropriate terminology
No matter where you live-Pennsylvania,
California, Virginia, or Canada-students need
opportunities to engage in the habits of practice
embodied in the Standards for Mathematical
Practices in CCSSM
 The features of the tasks that you select for
students to work on set the parameters for
opportunities they have to engage in these
practices-high-level tasks are necessary but not
sufficient conditions.
 The way in which you support students-the
questions you ask and the tools you provide-help
good tasks live up to their potential.

 According
to Webster, RIGOR is strict
precision or exactness.
 According to mathematicians RIGOR is having
theorems that follow from axioms by means
of systematic reasoning.
 According to most people RIGOR means “too
difficult” and “only limited access is
possible.”
 Also they believe RIGOR is an excuse to avoid
high quality math teaching and learning
 Rigor
is teaching
and learning that
is active, deep,
and engaging.
ACTIVE
ENGAGING
DEEP
 Active
learning involves conversation and
hands-on/minds-on activities. For example,
questioning and discovery learning goes on.
ACTIVE
ENGAGING
DEEP

Deep learning is
focused, attention
given to details and
explanations, via
problem solving or
projects. Students
concentrate on the
intricacies of a skill,
concept, or activity.
ACTIVE
ENGAGING
DEEP
 When
learning is engaging, students make a
real connection with the content. There is a
feeling that, although learning may be
challenging, it is satisfying.
ACTIVE
ENGAGING
DEEP
 Rigor
is a process—not a problem
ACTIVE
ENGAGING
DEEP
USING THE NCTM PRACTICE STANDARDS AND
THE CCSSM
HOW DO WE GET
STUDENTS ACTIVE AND
ENGAGED, THINKING
DEEPLY ABOUT MATH?
1.
2.
E
D
F
C
3.
4.
A
B
5.
6.
Name the polygon
Describe the polygon using
the following terms:
congruent, parallel,
perpendicular, angle,
measure, base, height,
sides.
Label the vertices using
the letters A-F
Describe the relationship
between 𝐴𝐵 and 𝐸𝐷
Identify congruent sides
using appropriate notation.
For each angle, provide an
estimate, with
justification, of its
measure.
7.
E
D
F
8.
C
A
B
9.
10.
Is this a regular or irregular
polygon? Write a
descriptive paragraph to
support your answer.
Include diagram.
Explain a method you
would use to find the
perimeter of the polygon.
Using a ruler, determine
the perimeter to the
nearest centimeter.
Describe a method to
describe the area. Label
your steps in sequential
order. Use pictures to
describe your steps if you
want.
11.
E
12.
D
13.
F
C
14.
A
B
15.
Formulate an expression
that represents the area of
the polygon.
Implement your method to
solve for the area.
If the lengths of the sides
were doubled predict how
that would affect the
perimeter of the figure.
If the lengths of the sides
were doubled predict how
that would affect the area
of the figure.
If the measures of some
angles increased, how
would the lengths of the
sides change? Justify your
answer.
16.
E
D
F
C
A
17.
B
18.
Measure each angle and
find the sum of the angle
measures. Compare the
sum of the angle measures
to the sum of the angle
measures in a triangle, a
quadrilateral, and a
pentagon. What pattern
do you notice?
If the polygon were the
base of a 3 dimensional
figure, what type of figure
could it be? Explain your
answer.
If the polygon is the base
of a hexagonal prism, what
would its sides look like?
19.
E
20.
D
F
C
A
B
21.
22.
23.
How many faces, vertices,
and edges would the
hexagonal prism have?
Explain how you could
determine the volume of
the hexagonal prism.
Compare your method to a
classmate’s. How are the
two methods alike? How
are the two methods
different?
How many lines of
symmetry can you draw in
the polygon?
Name a line segment that
shows a line of symmetry.
Use mathematical notation
to identify parallel sides.
24.
E
25.
D
26.
F
C
A
B
27.
28.
Draw the polygon in
Quadrant I of a coordinate
plane.
Identify the coordinate
pairs of each vertex of the
polygon.
If you translated the
polygon 2 units to the right
and 3 units down, what
would the new coordinate
pairs be for each vertex?
If you rotate the polygon
90o, in which quadrant
would it be located?
Draw a 90o rotation.
29.
E
30.
D
F
C
A
B
31.
32.
Reflect the original polygon in
Quadrant I over the x-axis.
Identify the coordinate pairs of
the image polygon.
What type of transformation
would have occurred if the
image of the original polygon in
Quadrant I were in Quadrant 3?
Illustrate your answer.
If the original polygon in
Quadrant I were dilated by a
scale factor of ½, what would
the coordinate pairs of the new
polygon be?
Draw a similar figure and write a
proportion that shows the scale
factor.
Overwhelming evidence suggests that
we have greatly underestimated
human ability by holding expectations
that are too low for too many
children, and by holding differential
expectations where such
differentiation is not necessary.
 Say
“Hi” to everyone at your table
and shake hands.
 How
many handshakes were exchanged
at your table?
 If we exchange handshakes for the
entire room, how many handshakes
would there be?
We could actually have the 25
people perform the activity of
shaking each other’s hands and
keep track!
Number of
Students, n
Number of
handshakes,
Hn
2
1
3
3
4
6
5
10
6
15
7
21
8
28
9
36
10
45
.
.
.
.
.
.
H4=H3+3=3+3=6
H6=H5+5=10+5=15
Hn=Hn-1+(n+1)
 1st
student shakes hands with 24 other
students.
 2nd student shakes hands with 23 other
students.
 3rd student shakes hands with 22 other
students.
 And so on…
 Therefore, the total number of handshakes
will be…
24+23+22+…+3+2+1=300
𝑛 𝑛−1
𝐻𝑛 =
2
A
Geometric/spatial
representation of
the problem
 4+3+2+1=10
Triangular numbers can be
Represented by dots arranged
In a triangle
Triangular
number
1
2
3
4
…
Number of
dots
1
3
6
10
…
Create instructional strategies that will address:
1. common misconceptions
2. errors
3. differentiation of instruction
4. student engagement
5. reflection opportunities
6. mathematical communication
7. vocubalury
8. multiple representations of
mathematical concepts.
PRO EQUITY MODEL
1.
2.
3.
4.
5.
6.
7.
8.
Identity Property of Addition (0+n=n+0=n)
Zero Property of Multiplication
Identity Property of Multiplication
(1xn=nx1=n)
Golden Rule of Equations
The Distributive Property of Multiplication
over Addition
Commutative Property of Addition and
Multiplication
Associative Property of Addition and
Multiplication.
Isolate the Unknown
 Solve:
x+4=16
x+4=16
-4 -4
x+0=12
x=12
STATEMENTS
1. Solve: x+4=16
2. x+4-4=16-4
3. x+0=12
4. x=12
REASONS
1. Given
2.
Golden Rule of Eq
3.
Simplify (CLT)
Identity property
of addition
4.
STATEMENTS
1. Solve: 3x=15
2. 3x÷3=15÷3
3. 1x=5
4. x=5
REASONS
1. Given
2.
Golden Rule of Eq
3.
Simplify (divide)
Identity property
of multiplication
4.
STATEMENTS
1. Solve: 3x+4=16
2. 3x+4-4=16-4
3. 3x+0=12
4. 3x=12
5.
6.
7.
3x÷3=12÷3
1x=4
x=4
REASONS
1. Given
2. Golden Rule of Eq
3. Simplify (CLT)
4. Identity property
of addition
5. Golden Rule of Eq
6. Simplify (divide)
7. Identity property
of Multiplication
STATEMENTS
1.
Solve: 8(x-2)=3x+4
2.
8x-16=3x+4
3.
8x+16-16=3x+4+16
4.
8x+0=3x+20
5.
8x=3x+20
6.
7.
8.
9.
10.
11.
8x-3x=3x+20-3x
5x=20+0
5x=20
5x÷5=20÷5
1x=4
X=4
REASONS
1.
Given
2.
Simplify (distribute)
3.
Golden Rule of Eq
4.
Simplify (CLT)
5.
Identity Property of
addition
6.
Golden Rule of Eq
7.
Simplify (CLT)
8.
Identity Property of
addition
9.
Golden Rule of Eq
10.
Simplify (divide)
11.
Identity Property of
Multiplication
Peg Smith (2013, August). Tasks, Tools,
and Talk: A Framework for Enacting
Mathematical Practices.
Taryn DiSorbo
http://www.nctm.org/uploadedFiles/Prof
essional_Development/Institutes/High_Sc
hool_Institute/2013/Workshop_Materials/
Smith-Tasks,%20Tools,%20Talk.pdf
Kim Vesper
Lee V. Stiff, North Carolina State
University and EDSTAR Analytics, Inc.
http://www.nctm.org/uploadedFiles/Prof
essional_Development/Institutes/High_Sc
hool_Institute/2013/Workshop_Materials/
NCTM_HS_Institute_Presentation_DC_Augu
st2013_PDFVersion.pdf
[email protected]
[email protected]