Transcript No Slide Title

Math Standards and the
Importance of
Mathematical Knowledge
in Instructional Reform
Cheryl Olsen
Visiting Associate Professor, UNL
Associate Professor, Shippensburg
University, Pennsylvania
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Why Principles & Standards?
The Case Is Straightforward
 The world is changing.
 Today’s students are different.
 School mathematics is not working well
enough for enough students.
Therefore, school mathematics
must continue to improve.
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Principles and Standards for
School Mathematics
 A comprehensive and
coherent set of goals for
improving mathematics
teaching and learning in
our schools.
“Higher Standards
for Our Students...
Higher Standards
for Ourselves”
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The Standards
Content
Process
 Number and Operations  Problem Solving
 Algebra
 Geometry
 Measurement
 Data Analysis and
Probability
 Reasoning and
Proof
 Communication
 Connections
 Representation
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Emphasis Across the Grades
Pre-K–2
3–5
6–8
9–12
Number
Algebra
Geometry
Measurement
Data Analysis
and Probability
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Reasoning and Proof Standard
Instructional programs from prekindergarten
through grade 12 should enable all students to—
 recognize reasoning and proof as fundamental
aspects of mathematics;
 make and investigate mathematical
conjectures;
 develop and evaluate mathematical
arguments and proofs;
 select and use various types of reasoning
and methods of proof.
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Middle-grades students are
drawn toward mathematics
if they find both challenge
and support in the
mathematics classroom.
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More and Better Mathematics
Grades 6–8
 More understanding and flexibility with
rational numbers
 More algebra and geometry
 More integration across topics
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More flexibility
 Imagine you are working with your class on multiplying
large numbers. Among your students’ papers, you
notice that some have displayed their work in the
following ways:
 Which student(s) would you judge to be using a
method that could be used to multiply any two whole
numbers?
Student A
Student B
Student C
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35
35
 25
 25
 25
125
175
25
700
150
875
100
75
875
600
875
Ball & Hill
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Flexible Use of
Rational Numbers
A group of students has $60 to spend on dinner.
They know that the total cost, after adding tax and
tip, will be 25 percent more than the food prices
shown on the menu. How much can they spend on
the food so that the total cost will be $60?
$60
Cost of Food
Tax
and Tip
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Interplay between
Algebra and Geometry
Explain in words, numbers, or tables
visually and with symbols the number of
tiles that will be needed for pools of
various lengths and widths.
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Student Responses
Pool
Width
Number
of Tiles
1
1
8
2
1
10
3
1
12
3
2
14
3
3
16
3
4
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Width
Length
Pool
Length
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Student Responses
1) T = 2(L + 2) + 2W
2) 4 + 2L + 2W
3) (L + 2)(W + 2) – LW
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Stronger Basics
Increasing students’ ability to
understand and use—
 rational numbers
 linear functions
 proportionality
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Understanding of
Rational Numbers
This strip
3/4 of the whole.
represents
Draw the fraction strip that shows
1/2, 2/3, 4/3, and 3/2.
Be prepared to justify your
answers.
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Understanding the Division of
Rational Numbers
If 5 yards of ribbon are cut into pieces that
are each 3/4 yard long to make bows,
how many bows can be made?
Number of Bows
1
0
2
1
3
2
4
5
3
6
4
2
3
5
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A Middle Grades Lesson
Do 3 tubes with the same surface
area have the same volume?
Note: The tubes are not drawn to scale.
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What Next
Questions for students:
 Will all the cylinders hold the same
amount? Explain your reasoning.
 How does changing the height of
the cylinder affect the
circumference?
 How does this affect the volume?
Explain.
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Making a Discovery and the
Mathematics of the Solution
 Fill the tube (tallest one first) and then
remove it, emptying the contents into the
tube with twice the circumference.
 What is the next step of the lesson?
 What do the students know about the
tubes? How does the volume change in
comparison to the changes in the height?
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Qualities of the Lesson
 A question about an important mathematics
concept was posed.
 Students make conjectures about the problem.
 Students investigate and use mathematics to
make sense of the problem.
 The teacher guides the investigation through by
questions, discussions and instruction.
 Students expect to make sense of the problem.
 Students apply their understanding to another
problem or task involving these concepts.
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Linear Functions
Keep-in-Touch
ChitChat
$20 per month
Only 10¢ for
each minute
45¢ per
minute
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A Student’s Solution
No. of minutes
Keep in Touch
ChitChat
0
10
20
30
40
50
$20.00 $21.00 $22.00 $23.00 $24.00 $25.00
$0.00
$4.50
$9.00 $13.50 $18.00 $22.50
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Keep in touch
y = 20 + .10x
cost
Other Approaches
Chit chat
y = .45x
# of minutes
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Understanding Proportions
Which is the better buy?
12 tickets for $15.00
or
20 tickets for $23.00
Solve by scaling:
12 tickets for $15
60 tickets for $75.
20 tickets for $23
60 tickets for $69.
Solve by unit-rate:
$15 for 12 tickets
$1.25 for 1 ticket
$23 for 20 tickets
$1.15 for 1 ticket
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Developing Flexible Problem Solvers
 Builds on and helps build “more and better
mathematics”
 Builds on and helps strengthen “stronger/bolder
basics”
 Builds on and enhances flexible use of
representations
 Builds on and deepens UNDERSTANDING of
mathematical ideas
 Develops through regular experience with
interesting, challenging problems
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Dynamic Pythagorean
Relationships
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Flexible Use of Proportions
A baseball team won 48 of its first 80 games. How
many of its next 50 games must the team win in
order to maintain the ratio of wins to losses?
Ratio
48:32 — simplify to 3:2
Proportion
48/80 = x/50
Percents - Decimals
48/80 — ratio = 60%; find 60% of 50 games;
represent as 0.600
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Problems That Require Students to
Think Flexibly about Rational Numbers
Using the points you are given on the
number line above, locate 1/2, 2 1/2, and
1/4. Be prepared to justify your answers.
1
1
12
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Problems That Require Students to
Think Flexibly about Rational Numbers
Use the drawing to justify as many different
ways as you can that 75% = 3/4. You may
reposition the shaded squares if you wish.
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Locating Square Roots
27
0
1
2
3
4
5
99
6
7
8
9 10
27 is a little more than 5 because 52 = 25
99 is a little less than 10 because 102 = 100
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How Can Administrators Make
a Difference?
 Setting high expectations for student
achievement
 Supporting teachers
 Having conferences with teachers and
supervising instruction
 Asking questions
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Process of Moving Forward
What Does It Take?
 Participation of all
constituencies
 Ongoing examination of
the vision of school
mathematics
 High-quality instructional
materials
 Assessments aligned with
curricular goals
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Principles and Standards
Web Site
standards.nctm.org
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