1 Montana Mathematics Education Forum September 21, 2009 Dr. Eric Milou Rowan University Department of Mathematics x3876 K-12 Standards Based Mathematics: Debate, Dialogue, and Direction

2 Overview  Major issues & debates in mathematics education  Reports/Research  Common core standards initiative  National Math Panel report  NCTM focal points  Debate  Concepts vs. Skills Debate  Why Don’t My Students Have Number Sense?  The Role of Technology

Common Core Standards   The Council of Chief State School Officers and The National Governors Association Center for Best Practices  Fewer, clearer, and higher, to best drive effective policy and practice;  Aligned with college and work expectations, so that all students are prepared for success upon graduating from high school;  Inclusive of rigorous content and application of knowledge through high-order skills, so that all students are prepared for the 21 st century;  Internationally benchmarked, so that all students are prepared for succeeding in our global economy and society; and  Research and evidence-based. 3

The Process  Develop End-of-High-School Expectations  30 day public comment period starting today  National Validation Committee  Develop K-12 Standards in Language Arts and Math  Adoption  Each state adopting the common core either directly or by fully aligning its state standards may do so in accordance with current state timelines for standards adoption not to exceed three (3) years.  States that choose to align their standards to the common core standards agree to ensure that the common core represents at least 85 percent of the state’s standards in English language arts and mathematics. 4

Race to the Top Fund  States will soon have a strong incentive to take the Common Core standards process seriously.  The Obama administration has proposed giving states that adopt common standards a competitive advantage in seeking federal aid, as part of the $4 billion Race to the Top Fund, a pool of economic- stimulus money. 5

Mathematical Principles  Mathematical Practice  Number  Expressions  Equations  Functions  Quantity  Modeling  Shape  Coordinates  Probability  Statistics 6

Equations  Core Concepts  An equation is a statement that two expressions are equal.  The solutions of an equation are the values of the variables that make the resulting numerical statement true.  The steps in solving an equation are guided by understanding and justified by logical reasoning.  Equations not solvable in one number system might be solvable in a larger system.  Core Skills  Understand a problem and formulate an equation to solve it.  Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.  Rearrange formulas to isolate a quantity of interest.  Solve systems of equations.  Solve linear inequalities in one variable and graph the solution set on a number line.  Graph the solution set of a linear inequality in two variables on the coordinate plane. 7

Common Core Concerns  “Career/workplace readiness” = “College readiness” ?  The role of technology  Missing is the math used in finance, medicine, and other areas.  Zalman Usiskin: “Completely missing is the whole notion of literacy and citizenship and solving problems you see in life, rather than just mathematical problems. Everyday math applications are just not there.”  My greatest concern resides in the next step—the development of assessments based on these standards 8

Core Skill?  Which of (i)–(iv) would be the most productive first step in factoring expressions (a)–(f)?  (i) Finding a common factor  (ii) Expressing as a difference of two squares  (iii) Expressing as a perfect square  (iv) None of the above.  (a) 4x 2 + 5x  (b) 36x 2 – 25y 2  (c) x 4 – y 4  (d) x 2 + 6xy + 9y 2  (e) x 2 + 5x + 4  (f) 0.5x 2 – 2.5x - 7 9

10 National Math Panel (NMP)  Any approach that continually revisits topics year after year without closure is to be avoided.  High-quality research does not support the contention that instruction should be either entirely “student centered” or “teacher directed.”  A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

11 A NMP train…

12 NCTM Focal Points  The Curriculum Focal Points are the most important mathematical topics for each grade level. They comprise related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning.  The focal points are not about the basics; they are about important foundational topics.  Students should learn and be able to recall basic facts and become computationally fluent, but such knowledge and skills should be acquired with understanding.

Teaching Gap, Stigler and Hiebert, 1999 Stated vs Developed

14 We need a BALANCE  Balance  Direct Instruction  Constructivism  Balance  Conceptual Understanding  Algorithmic Proficiency

15 Your Test! 4 x 9 x 25 How many ounces are in a gallon? 50 ÷ 1/2

16 Conceptual Understanding  24 ÷ 4 = 6  24 ÷ 3 = 8  24 ÷ 2 =12  24 ÷ 1 = 24  24 ÷ 1/2 = ??

17 Fractions - Conceptually More than 1 or Less than 1 Explain your reasoning The F word

18 Which is larger?  (2/3 + 3/4 + 4/5 + 5/6) OR 4  12.5 x 45 OR 4.5 x 125  (1/3 + 2/4 + 2/4 + 5/11) OR 2

19 Computation

20 Active Computation  Fifty (1, 2, 3, 4, 5, 6 and addition)  Buzz (3)  Product Game 

21 Conceptual & Contextual  = ?  How do we teach this? xxx x xx x x x x x x x x x x x

Best Friends

= ? = 15

= / / > --> 10 --> --> --> --> --> --> --> 17

25 How Many Circles? 50

= ? = = = 1000

27 High School Example  Not more traditional word problems  Placing mathematical lessons into settings  Giving students a reason to learn the skill

28 Drug Steady State  10 mg Zrytec daily  50% of Zrytec is eliminated daily  What % is in the body  after 7 days  10 days  n days?

29 10mg Daily of Zrytec DayMg. of Zyrtec

30 Representations

Technology  Teachers are able to engage and motivate students with technology.  A review of 11 studies that met the National Math Panel’s rigorous criteria found limited or no impact of calculators on calculation skills, problem solving, or conceptual development. 31

Polygons 32

Engage Students to Use Technology 33

34 Fact #1 A

35 Fact #2 B

36 Fact #3 C

37 Fact #4 D

38 Fact #5 E

39 Fact #6 F

40 Fact #7 G

41 Fact #8 H

42 Fact #9 I

43 What is this?

44 What is this? F A C E

45 What If? AB C DE F GH I

46 Try Again

47 Try Again D E C A D E

48 What’s the Point?  Isolated Basic Skills  Less likely to retain information  Connected Facts, Use of Patterns, Facts in Context  More likely to retain information

Standards Based Mathematics  All students have a chance to succeed and be based on a conception of mathematics that:  balances the development of skills and concepts that empower students to solve practical and real-world problems;  makes judicious use of technology, like every other productive segment of our society, to enhance learning and preclude the waste of precious time devoted to pencil and paper skills that are no longer used in the workplace;  builds coherently over time when students are confronted with interesting and complex problems and expected to struggle as part of the learning process.

50 Thank You Dr. Eric Milou Rowan University