Common Core State Standards for Mathematics

1. Common Core State Standards for Mathematics Bernie Madison, University of Arkansas Reviewer for ACE Reviewer for MAA Member of writing team

2. Phil Daro, one of the writers of the CCSS, has described the goal of the standards as answering the question, “What is the math I want students to walk away with?” The three principal writers were Phil Daro, Bill McCallum, and Jason Zimba

3. Daro identified some of the key assumptions held by the writers, including the view that (a) the CCSS were written to assume 100% mastery, in any given year, of the preceding year’s standards; (b) standards are high points, not finish lines or curriculum; and (c) the grain size for effective change should be at the chapter or unit rather than at the lesson level.

4. Daro reviewed the objectives in writing the standards: they were to be “higher, clearer, and fewer,” “benchmarked to those of high-performing countries,” “oriented to college and career readiness,” and “evidence-based.” Daro emphasized the importance of mathematical practices as a foundation across the K–12 standards, the focus on understanding core concepts, and the importance of fluency with core skills.

5. A number of critical elements are coalescing to change the face of K–12 mathematics education in the U.S. These elements include the broad level of state support for the CCSS and collaboration among states in developing new, common forms of assessment. Also shared is a common goal of bringing greater focus and coherence to K–12 mathematics, an emphasis on mathematical practices, and attention to developing understanding of concepts and skills. JereConfrey & Erin Krupa, Summary of CSMC Conference on CCSSM, August 2010 This conference of 43 leaders in design of mathematics curricula issued 16 recommendations in response to CCSSM. Three of these were: Ensure that CCSSM is a living document. Recognize the imperative to help teachers and schools interpret the Standards. Lead with the mathematical practices.

6. Omission of “citizenship” in the heading only flags the problem. The standards themselves reflect this omission with only periodic passing references to solving problems concerning situations in the contemporary (really) real world. Missing the opportunity to make this critical adjustment in US K-12 mathematics and statistics will be rather tragic. The treatment of measurement, quantity, and modeling is too focused on geometric measurement and calculations. Measurements of attributes such as temperature, weight, speed, capacity, and brightness are only mentioned in “solve problems on” statements. Reasonableness of solutions and estimations are mentioned, but not development of personal quantitative benchmarks by students. Students must have some criteria for judging reasonableness. Even for measuring by counting, there is a need for benchmarks for attributes such as populations and amounts of money. B. Madison, MAA Review of CCSSM

7. Some differences because of CCSSM • More focused/coherent – focus on practices! • Teach to mastery – maybe. Mastery requires continued practice. • Fractions earlier • Algebra throughout • Probability & statistics later • Logical argument (aka proofs) throughout and not just in geometry.

8. Overall Structure • Standards for Mathematical Practice • Grades K-8 (Standards for each grade) • Progression of learning in grade • Overview • Standards • Understand • Do • High School (Standards by content domain)

9. Overall Structure (continued) • High School • Introduction • Number and quantity • Algebra • Functions • Modeling • Geometry • Statistics and probability

10. Standards for Mathematical Practice • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

11. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel in the expansions of (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work through the solution to a problem, proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

12. Standards of Mathematical Practice Sources • Mathematical proficiency (Adding It Up) • Procedural fluency • Conceptual understanding • Strategic competence • Adaptive reasoning • Productive disposition • NCTM process standards • Problem solving • Reasoning and proof • Representations • Communication • Connections

13. Content Standards Practice Standards Understanding Expectations Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Look for and make use of structure.

14. Development of Algebraic Thinking Grade 1 Sample Grade 2 Sample Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. If 18 apples are to be packed 6 to a bag, then how many bags are needed? • Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. • 9 = 6 +

15. Development of Algebraic Thinking Grade 3 Sample Grade 4 Sample Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Twenty apples are to be shared fairly by six people. How many apples will each person get and how many apples will be left over? • Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. • A red hat costs $18 and that is three times what a blue hat costs. How much does a blue hat cost?

16. Development of Algebraic Thinking Grade 5 Sample Grade 6 Sample Write, read, and evaluate expressions in which letters stand for numbers. Write expressions that record operations with numbers and with letters standing for numbers. Express the calculation “Subtract y from 5 and get 7.” • Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. • What is the value of 3(x - 2y) if x is 4 and y is 7?

17. Development of Algebraic Thinking Grade 7 Sample Grade 8 Sample Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. • Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. • If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50.

18. To Be Determined • Curriculum • Pedagogy • Assessments • Explanatory problems • Professional development

19. Sample Teachers’ Views • Larger grain size in CCSSM • More interpretation, classify, understand • More algebra in 8th grade • Transformation approach in geometry • Depth greater in CCSSM • Pre-calculus part in much larger

20. Sample Questions from Teachers • How will they (CCSSM) be phased in? When? • Who will develop curricula? How? • Will sequencing be standardized among states? • Will there be a national test? • Worry about non-college-bound? • Professional development • Can elementary teachers handle the algebraic thinking at lower grades?