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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)
Example of Variation in
Number of GLEs (grain size)
CA
FL
MO
MN
NY
KS
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 number of GLEs by grade level across all 42
state documents: 47
Mean
34.3
83.3
31.0
26.3
56.4
61.7
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)
Addition of Multi-Digit
Whole Numbers
7
6
4
3
2
1
VT
LA
NH/R
I
DoD
EA
IN
MO
NV
OH
OK
OR
SD
WA
NJ
AZ
CA
DC
FL
MI
NM
NY
TX
MD
MN
UT
VA
CO
ME
WY
TN
AL
KS
MS
ND
AR
ID
AK
HI
0
SC
GA
NC
WV
Grade Level
5
States
Culminating Learning Expectation
Intermediate Expectations
Initial Learning Expectation
Repeated Expectation
Multiplication of Multi-Digit
Whole Numbers
7
6
4
3
2
1
States
Culminating Learning Expectation
Intermediate Expectations
Initial Learning Expectation
Repeated Expectation
HI
CO
AK
AR
UT
WY
TN
NY
OK
NC
NV
KS
TX
AZ
ND
NM
MI
MD
MS
VT
MO
NH /R
I
MN
ME
IN
LA
ID
WV
GA
VA
WA
SD
NJ
OH
OR
FL
AL
CA
DC
A
SC
0
Do D
E
Grade Level
5
Grade Placement of Culminating
GLE for Whole Number Computation
Grade Placement
(culminating GLE)
Number of
states
Range
Addition
3rd
4th
13
16
1st – 6th
Subtraction
3rd
4th
15
15
1st – 6th
Multiplication
4th
5th
21
15
3rd – 6th
Division
4th
5th
12
23
4th – 6th
Operation
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
5
4
3
2
1
0
HI
GA
IN
MS
ND
VA
WV
AL
DoD
EA
FL
MI
NH/R
I
NJ
OK
VT
AZ
CO
CA
AK
DC
ID
KS
MD
NM
NY
OH
OR
TN
WA
AR
LA
NV
TX
UT
MO
NC
ME
MN
SC
WY
SD
Grade Level
6
States
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
4th grade
1 state
1 state
5th grade
15 states
6th grade
7th grade
8th grade
None
Multiplication of
fractions
Division of
fractions
15 states
2 states
1 state
20 states
20 states
25 states
24 states
6 states
6 states
13 states
14 states
1 state
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
Mean # 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
Solve problems or equations
Represent/model
Compute or estimate
Develop or demonstrate conceptual
understanding
Describe, explain, justify, or reason
Analyze
% of GLEs
(N = 396)
33
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)
Method
1.
2.
3.
4.
5.
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 (108 total) and
coded all 4th grade GLEs to determine
commonality across the 10 states.
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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