Analysis of the Intended Mathematics Curriculum as Represented in State-Level Standards: Consensus or Confusion?

Barbara J. Reys Center for the Study of Mathematics Curriculum

The intended curriculum: What mathematics should students learn and when should they learn it?

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 curriculum standards that specified mathematics that students should learn (and what teachers should teach) at each grade level.

Publication of State-Level Mathematics Curriculum Standards

(as of May 2006) 2006 2005 2004 2003 2002 2001 2000 pre-2000 4 states 9 states 13 states 8 states 4 states 4 states 2 states 7 states (FL, 1999)

Increased Specificity, Authority, and Influence

For many states, their most recent curriculum standards represent increased specificity of learning expectations compared to previous standards.

The standards carry additional “weight” or influence since they are tied to NCLB-mandated annual assessments in grades 3-8.

Teachers and state department leaders acknowledge the increased influence of state standards in determining curriculum focus at the classroom level.

http://www.mathcurriculumcenter.org/states.php

Grade-Level Learning Expectations (GLEs)

• GLE documents describe mathematics learning expectations for specific grades • 42 states have GLE documents – Most common grades: K-8 (37 states) – Others: K-7, 3-8 or 3-10 (5 states)

To what extent are the elementary and middle (K-8) grade-level learning expectations described in state-level mathematics curriculum standards similar in terms of content and grade placement?

Analysis of State GLEs

• Elementary and middle school documents produced by 42 states.

• Not a comprehensive analysis.

• Not evaluative.

• Descriptive.

• Chose particular topics within specific strands for analysis (number, algebra, reasoning).

• Utilized an “organic” or bottom-up approach with “learning expectations” as the unit of analysis.

Differences in GLE Documents

• Organization of GLEs • Language used to describe learning expectations.

• Level of specificity or grain size of learning expectations.

• Grade placement of key 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) • Recalls (from memory) the addition facts and corresponding subtraction facts. (FL, gr. 2) • Recall basic addition and subtraction facts through 18. (ID, gr. 3)

CA FL MO MN NY KS

Example of Variation in Number of GLEs (grain size)

1st 25 78 20 18 56 57 2nd 31 84 27 26 45 59 3rd 38 88 31 26 52 57 4th 43 89 33 25 56 56 5th 27 77 34 26 67 60 6th 36 78 38 30 64 69 7th 40 89 34 27 63 74 Mean 34.3

83.3

31.0

26.3

56.4

61.7

Mean number of GLEs by grade level across all 42 state documents: 47

Presentation of Selected Findings

• • • • Grade placement variation regarding – Whole number computation – Fraction Computation Emphasis on calculators/technology The “national” 4th grade mathematics curriculum Recommendations

Multi-digit Whole Number Computation

Example GLEs

(Multi-digit Whole Number Addition)

• Using pictures, diagrams, numbers or words, demonstrate addition and subtraction of whole numbers with 2-digit numbers (CO, gr. 3) • Add and subtract two three-digit whole numbers(AZ, gr. 3) • Explains and demonstrates the addition and subtraction of whole numbers (up to three digits or more) using concrete materials, drawings, symbols, and algorithms. (FL, gr. 3) • The student will solve problems involving the sum or difference of two whole numbers, each 9,999 or less, with or without regrouping, using various computational methods, including calculators, paper and pencil, mental computation, and estimation (VA, gr. 3)

3 2 5 4 7 6

Addition of Multi-Digit Whole Numbers

1 0 SC GA NC WV VT LA N H I /R o D D EA IN MO NV OH OK OR SD WA NJ AZ CA DC FL MI NM NY TX MD MN

States

UT VA CO ME WY TN AL KS MS ND AR ID Culminating Learning Expectation Intermediate Expectations Initial Learning Expectation Repeated Expectation AK HI

Multiplication of Multi-Digit Whole Numbers

5 4 7 6 3 2 1 0 o D D A E SC AL CA DC FL NJ OH OR SD VA WA WV GA ID IN LA ME MN MO N H /R I

States

VT MI MD MS ND NM TX AZ KS NC NV NY OK TN UT WY AR AK CO HI Culminating Learning Expectation Intermediate Expectations Initial Learning Expectation Repeated Expectation

Grade Placement of Culminating GLE for Whole Number Computation

Operation Addition Subtraction Multiplication Division Grade Placement (culminating GLE) 3 rd 4 th 3 rd 4 th 4 th 5 th 4 th 5 th Number of states 13 16 15 15 21 15 12 23 Range 1 st – 6 th 1 st – 6 th 3 rd – 6 th 4 th – 6 th

Fluency with Fraction Computation

Example GLEs

• Add and subtract common fractions and mixed numbers with unlike denominators. (GA, gr. 5) • 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). (FL, gr. 5) • Demonstrate computational fluency with addition, subtraction, multiplication, and division of decimals and fractions. (AL, gr. 6)

Addition and Subtraction of Fractions

9 8 7 6 5 4 3 2 1 0 HI GA IN MS ND VA WV AL o D D EA FL MI N H I /R NJ OK VT AZ CO CA AK DC ID KS

States

MD NM NY OH OR TN WA AR LA NV TX UT MO NC ME MN SC WY SD Culminating Learning Expectation Intermediate Expectations Initial Learning Expectation Repeat and/or Extension Expectations

Progression of GLEs (FL)

• 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)

When do states expect students to proficiently add, subtract, multiply and divide fractions?* 4 th grade 5th grade 6 th grade 7 th grade 8 th grade None Addition of fractions 1 state 15 states 20 states 6 states Subtraction of fractions 1 state 15 states 20 states 6 states Multiplication of fractions 2 states 25 states 13 states 1 state Division of fractions 1 state 24 states 14 states 1 state 1 state

*For this summary, we used the culminating learning expectation that indicated students were working with common and uncommon denominators when adding and subtracting fractions.

General Finding:

There exists considerable variation across the state curriculum standards with regard to the grade placement of key number and operation learning expectations.

What messages regarding calculators and technology are conveyed within the state standards documents?

Fordham Foundation (

The State of Math Standards

, 2005) Calculators . “One of the most debilitating trends in current state math standards is their excessive emphasis on calculators. Most standards documents call upon students to use them starting in the elementary grades, often beginning with Kindergarten.”

Messages regarding calculators and technology

Searched all state-level elementary and middle grades GLE documents: • 20 states include a statement regarding the role of calculators/technology within the introductory material of their GLE document.

• 32 states mention “calculator” or “technology” within specific GLEs. (FL) • 17 states include some attention to calculators/technology in both the introductory material and within specific GLEs.

Introductory Comments

• Technology will be a fundamental part of mathematics teaching and learning. (KS) • Extensive reliance on calculators runs counter to the goal of having students practice [computational and procedural skills]. More to the point, it is imperative that students in the early grades be given every opportunity to develop a facility with basic arithmetic skills without reliance on calculators . . . It should not be assumed that caution on the use of calculators is incompatible with the explicit endorsement of their use when there is a clear reason for such an endorsement. Once students are ready to use calculators to their advantage, calculators can provide a very useful tool not only for solving problems in various contexts but also for broadening students’ mathematical horizons. (CA)

Common Messages Within Introductory Comments

• Appropriate use of calculators/technology is encouraged.

• The existence of calculators/technology does not diminish the need for computational fluency.

• Calculators/technology can support increased understanding of mathematics.

• Calculators/technology can support effective teaching.

• Calculators/technology are commonly used in the workplace, therefore students should learn to use these tools to solve problems.

• Teachers are responsible for appropriate and effective use of calculators/technology.

Review of GLEs referring to calculators/technology

• Compiled a set of

451

GLEs from 32 state documents that include “calculator” or “technology” or both (about 3% of all GLEs) –

21

GLEs from 7 states indicate that students should

NOT

use calculators –

34

GLEs focused on computer technology (software) rather than calculators.

–

396

GLEs were used for this analysis • The mean number of GLEs referencing calculators was

12.4

per state document or about

1.4

per grade.

Example GLEs

(reference to calculators and/or technology)

• • • •

Use technology, including calculators, to understand quantitative relationships, e.g., for skip counting and pattern exploration. (NY; gr. K,1,2,3,4) Counts to 1000 or more by 2s, 3s, 5s, 10s, 25s, 50s and 100s using a variety of ways, such as mental mathematics, paper and pencil, hundred chart, calculator, and coins in various increments . (FL, gr. 2) Solve problems using the four operations with whole numbers, decimals, and fractions. Determine when it is appropriate to use estimation, mental math strategies, paper and pencil, or a calculator. (UT; gr. 5,6) Use appropriate technology to gather and display data sets and identify the relationships that exist among variables within the data set. (ID, gr. 7)

References to “calculators” and/or “technology”

Grade K 1 2 3 4 5 6 7 8 Mean # of GLEs referencing "Calculator" or "Technology" 0.26

0.65

0.87

1.16

1.42

1.61

1.90

2.13

2.77

Role of calculator/technology within GLE

Purpose % of GLEs Solve problems or equations (N = 396) 33 Represent/model Compute or estimate Develop or demonstrate conceptual understanding Describe, explain, justify, or reason Analyze 27 20 16 16 13

Our analysis of the state GLE documents does not support the finding of the Fordham Foundation report regarding emphasis on calculators.

What mathematics are fourth graders in the U.S. expected to learn?

Analysis of 4th Grade GLEs

• Goal was to document the level of consensus regarding mathematics GLEs at one grade level (we chose 4th grade).

• Focused on GLE documents from the ten most populous states that publish such documents (CA, TX, NY, FL, OH, MI, NJ, NC, GA, VA)

1.

2.

3.

4.

5.

Method

Collected the 10 state documents, combined and sorted all GLEs (

492

total) by content strand.

Searched for common themes across GLEs and developed list of “substrands”.

Sorted all GLEs into substrands and eliminated duplicates.

Developed list of “distinct” GLEs ( coded all 4th grade GLEs to determine commonality across the 10 states.

108

total) and Summarized findings by content strand.

Distinct Set of 4th Grade GLEs

(with duplicates removed) Total: 108

Common GLEs across all 10 documents

(4 of 108) • • • •

Read, write, compare, and order whole numbers.

Read, write, compare and order decimals.

Add and subtract decimals.

Solve problems involving whole number multiplication and division.

• • •

Unique GLEs - in only one of ten documents

( 28 of 108 GLEs)

Use concrete materials and symbolic notation to represent numbers in bases other than base ten, such as base five. Compare decimal number system to the Roman numeral system (using the Roman numerals I, V, X, L, C, D, and M.) Use models to identify perfect squares to 100.

GLEs common to at least 6 of the 10 states

What are the consequences of differences in the grade placement of learning expectations across states?

- For teacher preparation and professional development?

- For development of textbooks?

- For comparisons of student performance?

Recommendations Regarding the Specification of Learning Expectations

Identify a small set of primary goals for each grade level.

At each grade, we recommend a general statement of major goals for the grade. These general goals may specify emphasis on a few strands of mathematics or a few topics within strands. These general goals should be coordinated across all grades, K-8, to ensure curricular coherence and comprehensiveness.

Limit the number of grade-level learning expectations to focus instruction and deepen learning

.

The set of learning expectations per grade-level should be manageable given the school year. Along with the statement of general goals and priorities for a particular grade, we suggest that the set of learning expectations per grade be limited to 20-25.

Develop clear statements of learning expectations focusing on mathematics to be learned

.

We recommend that learning expectations be expressed succinctly, coherently, and with optimum brevity, limiting the use of educational terms that may not communicate clearly to the intended audience of teachers, school leaders, and parents.

Be clear about the role of technology.

Provide guidance within particular learning goals or as part of an overall philosophical statement regarding the role of technology - specifying when it is an appropriate tool for computing and/or developing or representing mathematical ideas.

Collaborate to promote consensus

.

Fifty states with 50 state standards documents increases the likelihood of large textbooks that treat many topics superficially. In order to increase the likelihood of focused curriculum materials, states will need to work together to create some level of consensus about important learning goals and expectations at each grade.

Don’t reinvent the wheel

.

Variation in learning goals across states directly influence the quality and coherence of commercially developed textbook materials. It also limits our ability to develop and provide focused professional development for teachers.

Many models of curriculum standards exist for review, refinement and/or adoption.

Recent Work of National Organizations

• College Board (standards, grades 7-12) • Achieve, Inc. (standards for high school) • NCTM (Focal points, grades K-8) • ASA (K-12 standards for statistics)

Full report of the study will be available in October 2006.

Information Age Publishing Company:

http://www.infoagepub.com/

http://mathcurriculumcenter.org

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