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 4 states2005 9 states2004 13 states2003 8 states2002 4 states

2001 4 states2000 2 states

pre-2000 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

• 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)

Grade-Level Learning Expectations (GLEs)

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)

1st 2nd 3rd 4th 5th 6th 7th MeanCA 25 31 38 43 27 36 40 34.3FL 78 84 88 89 77 78 89 83.3MO 20 27 31 33 34 38 34 31.0MN 18 26 26 25 26 30 27 26.3NY 56 45 52 56 67 64 63 56.4KS 57 59 57 56 60 69 74 61.7

Example of Variation in Number of GLEs (grain size)

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)

0

1

2

3

4

5

6

7

States

Gra

de

Le

ve

l

Culminating Learning ExpectationIntermediate ExpectationsInitial Learning ExpectationRepeated Expectation

Addition of Multi-Digit Whole Numbers

Multiplication of Multi-Digit Whole Numbers

0

1

2

3

4

5

6

7

States

Gra

de L

evel

Culminating Learning ExpectationIntermediate ExpectationsInitial Learning ExpectationRepeated Expectation

Grade Placement of Culminating GLE for Whole Number Computation

OperationGrade Placement (culminating GLE)

Number of states

Range

Addition3rd

4th

13

161st – 6th

Subtraction3rd

4th

15

151st – 6th

Multiplication4th

5th

21

153rd – 6th

Division4th

5th

12

234th – 6th

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

0

1

2

3

4

5

6

7

8

9

States

Gra

de

Lev

el

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?*

Addition of fractions

Subtraction of fractions

Multiplication of fractions

Division of fractions

4th grade 1 state 1 state

5th grade 15 states 15 states 2 states 1 state

6th grade 20 states 20 states 25 states 24 states

7th grade 6 states 6 states 13 states 14 states

8th grade 1 state 1 state

None 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

• 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.

Searched all state-level elementary and middle grades GLE documents:

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”

GradeMean # of GLEs referencing

"Calculator" or "Technology"

K 0.26

1 0.65

2 0.87

3 1.16

4 1.42

5 1.61

6 1.90

7 2.13

8 2.77

Role of calculator/technology within GLE

Purpose % of GLEs

(N = 396)

Solve problems or equations 33

Represent/model 27

Compute or estimate 20

Develop or demonstrate conceptual understanding

16

Describe, explain, justify, or reason 16

Analyze 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)

Method1. Collected the 10 state documents, combined

and sorted all GLEs (492 total) by content strand.

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

3. Sorted all GLEs into substrands and eliminated duplicates.

4. Developed list of “distinct” GLEs (108 total) and coded all 4th grade GLEs to determine commonality across the 10 states.

5. 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

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.

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

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/

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http://mathcurriculumcenter.org