![Page 1: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/1.jpg)
1
Math Standards and the Importance of
Mathematical Knowledge in Instructional Reform
Cheryl Olsen
Visiting Associate Professor, UNL
Associate Professor, Shippensburg University, Pennsylvania
![Page 2: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/2.jpg)
2
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.
![Page 3: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/3.jpg)
3
Principles and Standards for School Mathematics
A comprehensive and coherent set of goals for improving mathematics teaching and learning in our schools.
3
“Higher Standards for Our Students...
Higher Standards for Ourselves”
![Page 4: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/4.jpg)
4
The Standards
Number and Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
Problem Solving
Reasoning and Proof
Communication
Connections
Representation
Content Process
![Page 5: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/5.jpg)
5
Emphasis Across the Grades
Number
Algebra
Geometry
Measurement
Data Analysis and Probability
Pre-K–2 3–5 6–8 9–12
![Page 6: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/6.jpg)
6
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—
![Page 7: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/7.jpg)
7
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.
![Page 8: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/8.jpg)
8
More and Better Mathematics
More understanding and flexibility with rational numbers
More algebra and geometry
More integration across topics
Grades 6–8Grades 6–8
![Page 9: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/9.jpg)
9
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
![Page 10: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/10.jpg)
10
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
![Page 11: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/11.jpg)
11
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.
![Page 12: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/12.jpg)
12
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
18
Number of TilesNumber of Tiles Width Width
Len
gth
L
eng
th
![Page 13: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/13.jpg)
13
Student Responses
1) T = 2(L + 2) + 2W
2) 4 + 2L + 2W
3) (L + 2)(W + 2) – LW
![Page 14: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/14.jpg)
14
Stronger Basics
rational numbers
linear functions
proportionality
Increasing students’ ability to understand and use—Increasing students’ ability to understand and use—
![Page 15: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/15.jpg)
15
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.
![Page 16: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/16.jpg)
16
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?
![Page 17: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/17.jpg)
17
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.
![Page 18: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/18.jpg)
18
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:
![Page 19: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/19.jpg)
19
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?
![Page 20: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/20.jpg)
20
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.
![Page 21: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/21.jpg)
21
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
![Page 22: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/22.jpg)
22
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
![Page 23: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/23.jpg)
23
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
![Page 24: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/24.jpg)
24
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:
![Page 25: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/25.jpg)
25
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
![Page 26: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/26.jpg)
26
Dynamic Pythagorean Relationships
![Page 27: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/27.jpg)
27
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
27
![Page 28: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/28.jpg)
28
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
![Page 29: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/29.jpg)
29
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.
![Page 30: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/30.jpg)
30
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
![Page 31: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/31.jpg)
31
How Can Administrators Make a Difference?
Setting high expectations for student achievement
Supporting teachers
Having conferences with teachers and supervising instruction
Asking questions
![Page 32: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/32.jpg)
3232
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
![Page 33: 1 Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor,](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649d4c5503460f94a2b181/html5/thumbnails/33.jpg)
3333
Principles and Standards Web Site
standards.nctm.org