Math Standards and the Importance of Mathematical Knowledge in Instructional Reform

1. Math Standards and the Importance of Mathematical Knowledge in Instructional Reform Cheryl Olsen Visiting Associate Professor, UNL Associate Professor, Shippensburg University, Pennsylvania

2. 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.

3. 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” 3

4. Number and Operations Algebra Geometry Measurement Data Analysis and Probability Problem Solving Reasoning and Proof Communication Connections Representation The Standards Content Process

5. Number Emphasis Across the Grades Pre-K–2 3–5 6–8 9–12 Algebra Geometry Measurement Data Analysis and Probability

6. 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.

7. Middle-grades students are drawn toward mathematics if they find both challenge and support in the mathematics classroom.

8. Grades 6–8 More and Better Mathematics • More understanding and flexibility with rational numbers • More algebra and geometry • More integration across topics

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? Ball & Hill

10. $60 Cost of Food Tax and Tip 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?

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.

12. Pool Length Pool Width Number of Tiles 1 2 3 3 3 3 1 1 1 2 3 4 8 10 12 14 16 18 Student Responses Width Length

13. Student Responses 1) T = 2(L + 2) + 2W 2) 4 + 2L + 2W 3) (L + 2)(W + 2) – LW

14. Stronger Basics • rational numbers • linear functions • proportionality Increasing students’ ability to understand and use—

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.

16. 2 3 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 2 3 4 5 6 0 1 2 3 4 5

17. A Middle Grades Lesson Do 3 tubes with the same surface area have the same volume? Note: The tubes are not drawn to scale.

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:

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?

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.

21. Only 10¢ for each minute 45¢ per minute Linear Functions Keep-in-Touch ChitChat NO monthly fee $20 per month

22. A Student’s Solution No. of minutes 0 10 20 30 40 50 Keep in Touch $20.00 $21.00 $22.00 $23.00 $24.00 $25.00 ChitChat $0.00 $4.50 $9.00 $13.50 $18.00 $22.50

23. Keep in touch y = 20 + .10x Chit chat y = .45x Other Approaches cost # of minutes

24. 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 Understanding Proportions Which is the better buy? 12 tickets for $15.00 or 20 tickets for $23.00

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 Solvers

26. Dynamic Pythagorean Relationships

27. 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 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? 27

28. 1 2 1 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

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.

30. 27 99 Locating Square Roots 27 99 0 1 2 3 4 5 6 7 8 9 10 is a little more than 5 because 52 = 25 is a little less than 10 because 102 = 100

31. How Can Administrators Make a Difference? • Setting high expectations for student achievement • Supporting teachers • Having conferences with teachers and supervising instruction • Asking questions

32. 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 32

33. Principles and StandardsWeb Site standards.nctm.org 33