NCTM Curriculum Focal Points and State Standards Center for the Study of Mathematics Curriculum Barbara Reys, University of Missouri

Review the influence of NCTM’s curriculum recommendations on state standards. Examine the alignment of topics designated as CFPs with current state standards. Overview

Influence of NCTM standards NCTM’s entry into articulating mathematics standards (1989) coincided with an increased role by states in providing leadership for standards (Long, 2003) During this period the role, authority, and specification of state standards varied (CCSSO, 1995 and 1997). Curriculum and Evaluation Standards for School Mathematics (1989) and Principles and Standards for School Mathematics (2000) had a significant impact on the content of state standards (Martin, Stein, & Ferrini-Mundy, 2002)

No Child Left Behind (2001) Each state is required to: adopt challenging academic content standards that will be used by the State, its local educational agencies, and its schools. measure the achievement of students in mathematics against the standards in each of grades 3 through 8.

Prior to NCLB, many states did not have content standards that specified the mathematics that students should learn at each grade level. As of 2006, 42 states published GLEs. - Most common: K-8 (37 states) - Others: K-7, 3-8 or 3-10 (5 states) Grade-Level Learning Expectations (GLEs)

Increased Specificity, Authority, and Influence For many states, GLEs represented increased specificity compared to previous standards. GLEs carry additional “weight” or influence since they are tied to NCLB-mandated annual assessments. Teachers and state department leaders acknowledge increased influence of GLEs at the classroom level.

Influence of NCTM Standards In a 2005 survey of ASSM members (47 states responding), PSSM was noted as the most influential reference document in the development of the state standards - 80% of respondents indicated PSSM was a strong influence (NAEP framework cited as the next most influential document by 40% of respondents). In the same survey 85% of respondents indicated that national leadership for mathematics curriculum articulation is needed.

To what extent is the grade placement of NCTM’s Curriculum Focal Points consistent with state GLEs? To what extent is the grade placement of K-8 GLEs similar across states?

Selected topics within the number and operation strand: Fluency with basic number combinations Fluency with whole number computation Fluency with fraction computation Reviewed GLEs from 42 state documents. Compiled and compared grade placement of specific GLEs. When are particular topics introduced and when is proficiency expected?

Differences in GLE Documents Language Level of specificity or grain size Grade placement of key topics Trajectory of development of topics

Example of Variation of GLEs (Basic Number Combinations) Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory. (CA, gr. 1) States and uses with efficiency and accuracy basic addition facts with sums from 0 to 20 and corresponding subtraction facts. (KS, gr. 2) Recall basic addition and subtraction facts through 18. (ID, gr. 3)

Level of Specificity Demonstrate computational fluency for basic addition and subtraction facts with sums through 18 and differences with minuends through 18, using horizontal and vertical forms. (AL, gr. 2) vs. State addition and subtraction facts. (AZ, gr. 2)

Fluency with Fraction Computation

Progression of GLEs (FL) (addition/subtraction of fractions) Explains and demonstrates the addition and subtraction of common fractions using concrete materials, drawings, story problems, and algorithms. (Gr. 4) Solves real-world problems involving the addition or subtraction of decimals (to hundredths) or common fractions with like and unlike denominators. (Gr. 4) Solves real-world problems involving addition, subtraction, multiplication, and division of whole numbers, and addition, subtraction, and multiplication of decimals, fractions, and mixed numbers using an appropriate method (for example, mental math, pencil and paper, calculator). (Gr. 5)

Progression of GLEs (AZ) (addition/subtraction of fractions) Demonstrate addition and subtraction of fractions with like denominators (halves) using models. (gr. 1) Demonstrate addition and subtraction of fractions with like denominators (halves and fourths) using models. (gr. 2) Add or subtract fractions with like denominators (halves, thirds, fourths, eighths, and tenths) appropriate to grade level. (gr. 3) Add or subtract fractions with like denominators, no regrouping. (gr. 4) Add or subtract proper fractions and mixed numbers with like denominators with regrouping. (gr. 5) Add or subtract proper fractions and mixed numbers with unlike denominators with regrouping. (gr. 6)

Addition and Subtraction of Fractions

Curriculum Focal Point Developing an understanding of and fluency with addition and subtraction of fractions and decimals. (Grade 5)

Addition and Subtraction of Fractions CFP

Addition and Multiplication of Basic Number Combinations

Addition of Basic Number Combinations

Curriculum Focal Point Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts. (Grade 1) Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction. (Grade 2)

Addition of Basic Number Combinations CFP

Fluency with Multiplication of Basic Number Combinations CFP

Curriculum Focal Point Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts. (Grade 3) Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication (Grade 4)

Fluency with Multiplication of Basic Number Combinations CFP

Multi-digit Computation: Addition and Subtraction

Progression of GLEs (CA) (Multi-digit Whole Number Addition) Solve addition and subtraction problems with one- and two-digit numbers (Grade 1) Find the sum or difference of two whole numbers up to three digits long (Grade 2) Use mental arithmetic to find the sum or difference of two two-digit numbers (Grade 2) Find the sum or difference of two whole numbers between 0 and 10,000 (Grade 3) Demonstrate an understanding of, and the ability to use, standard algorithms for the addition and subtraction of multidigit numbers (Grade 4)

Fluency with Addition of Multi-Digit Whole Numbers

Curriculum Focal Point Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction. (Grade 2)

Fluency with Addition of Multi-Digit Whole Numbers CFP

Multi-digit Computation: Multiplication

Fluency with Multiplication of Multi-Digit Whole Numbers

Curriculum Focal Point Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication (Grade 4)

Multiplication of Multi-Digit Whole Numbers CFP

Across state GLEs, for most topics, there is considerable variation with regard to the grade placement (introduction, development, proficiency/fluency). Within state GLEs, development of topics such as computational fluency are spread across several grades rather than serving as a “focal point” in any one grade.

The idea of organizing learning goals according to “focal points” is not a common concept in the U.S., and not represented in most state standards. There is considerable work to be done to realize the vision of “curriculum focal points” –Not just about matching the grade level to CFP. –Major shift in how curriculum is organized. –Look to standards and curriculum from other countries for some models.

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PSSM (Grades 3-5): develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students experience. use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals. PSSM (Grades 6-8): develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use. develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.