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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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Therefore, school mathematics must continue to improve.

Therefore, school mathematics must continue to improve.

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.

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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.

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“Higher Standards for Our Students...

Higher Standards for Ourselves”

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The Standards

Number and Operations

Algebra

Geometry

Measurement

Data Analysis and Probability

Problem Solving

Reasoning and Proof

Communication

Connections

Representation

Content Process

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Emphasis Across the Grades

Number

Algebra

Geometry

Measurement

Data Analysis and Probability

Pre-K–2 3–5 6–8 9–12

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Reasoning and Proof Standard

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.

Instructional programs from prekindergarten through grade 12 should enable all students to—Instructional programs from prekindergarten through grade 12 should enable all students to—

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Middle-grades students are

drawn toward mathematics

if they find both challenge

and support in the

mathematics classroom.

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

More understanding and flexibility with rational numbers

More algebra and geometry

More integration across topics

Grades 6–8Grades 6–8

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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 C35

25

125

75

875

35

25

175

700

875

35

25

25

150

100

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?

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 Taxand 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.

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 LengthPool

LengthPool

WidthPool

Width

1

2

3

3

3

3

1

2

3

3

3

3

1

1

1

2

3

4

1

1

1

2

3

4

8

10

12

14

16

18

8

10

12

14

16

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Number of TilesNumber of Tiles Width Width

Len

gth

L

eng

th

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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

rational numbers

linear functions

proportionality

Increasing students’ ability to understand and use—Increasing students’ ability to understand and use—

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Understanding of Rational Numbers

This strip represents 3/4 of the whole.

Draw the fraction strip that shows

1/2, 2/3, 4/3, and 3/2.

Be prepared to justify your answers.

This strip represents 3/4 of the whole.

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

0 1 2 3 4 5

11 22 33 44 55 66 2233

Number of Bows

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?

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?

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A Middle Grades Lesson

Do 3 tubes with the same surface area have the same volume?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

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.

Questions for students:Questions for students:

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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

ChitChatChitChatKeep-in-TouchKeep-in-Touch

$20 per month NO monthly fee

NO monthly fee

45¢ per minute45¢ per minuteOnly 10¢ for

each minuteOnly 10¢ for each minute

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A Student’s Solution

No. of minutesNo. of minutes

Keep in TouchKeep in Touch

ChitChatChitChat

$20.00$20.00

00

$0.00$0.00

$21.00$21.00

1010

$4.50$4.50

$22.00$22.00

2020

$9.00$9.00

$23.00$23.00

3030

$13.50$13.50

$24.00$24.00

4040

$18.00$18.00

$25.00$25.00

5050

$22.50$22.50

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Other Approaches

Keep in touch y = 20 + .10x

Chit chat y = .45x

Keep in touch y = 20 + .10x

Chit chat y = .45x

cost

# of minutes

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Understanding Proportions

12 tickets for $15.0012 tickets for $15.00

Which is the better buy?Which is the better buy?

20 tickets for $23.0020 tickets for $23.00oror

Solve by unit-rate:Solve by unit-rate:

$15 for 12 tickets $1.25 for 1 ticket$15 for 12 tickets $1.25 for 1 ticket

$23 for 20 tickets $1.15 for 1 ticket$23 for 20 tickets $1.15 for 1 ticket

12 tickets for $15 60 tickets for $75.12 tickets for $15 60 tickets for $75.

20 tickets for $23

60 tickets for $69.

20 tickets for $23

60 tickets for $69.

Solve by scaling:Solve by scaling:

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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

Developing Flexible Problem SolversDeveloping Flexible Problem Solvers

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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?

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?

Ratio48:32 — simplify to 3:2

Ratio48:32 — simplify to 3:2

Proportion 48/80 = x/50Proportion 48/80 = x/50

Percents - Decimals48/80 — ratio = 60%; find 60% of 50 games;

represent as 0.600

Percents - Decimals48/80 — 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.

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.

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

2 3 4 5 6 7 8 9 100 1

27 99

27 is a little more than 5 because 52 = 25

is a little less than 10 because 102 = 10099

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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