Content Standards in an
International Context
William H. Schmidt
Michigan State University
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
Mean task input as percentiles of the 1960 task distribution
The Economy Has Changed…Have Our Schools?
Skills Demanded by the U.S. Economy
65
Non-routine
Interactive
Non-routine
Analytic
60
55
50
Routine
Manual
45
Routine
Cognitive
40
1960
1970
1980
1990
2002
Sources: Levy and Murnane; Schleicher, 2007
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
Non-routine
Manual
What Americans Value
U.S.
Rankings
Olympics
U.S.
Rankings
onon
PISA
Swimming
Science
Basketball
Reading
1
Finland
546
2
Canada
534
3
New Zealand
529
4
Australia
528
5
Ireland
527
6
Korea
525
7
United
Kingdom
523
1
Finland
563
1
2
Canada
534
3
Japan
531
4
New Zealand
530
4
5
Australia
527
6
Netherlands
525
7
Korea
522
7
8
Germany
516
8
Problem
Rowing
Solving
Triathlon
Math
Korea
550
1
Finland
548
2
Finland
548
2
Korea
547
3
Japan
547
3
Netherlands
531
New Zealand
533
4
Switzerland
530
5
Australia
530
5
Canada
527
6
Canada
529
6
Japan
523
Belgium
525
7
New Zealand
522
Switzerland
521
8
Belgium
520
Australia
520
8
Japan
522
9
United Kingdom
515
9
Netherlands
520
9
9
Sweden
516
10
Czech Republic
513
10
France
519
10
Denmark
513
10
Austria
507
11
Switzerland
512
11
Denmark
517
11
Czech Republic
510
Austria
511
12
Czech Republic
516
12
Iceland
506
Belgium
510
13
Germany
514
13
Austria
505
14
Sweden
509
14
Germany
504
15
Sweden
502
16
Ireland
501
17
France
496
18
United Kingdom
495
19
Poland
495
20
Slovak Republic
492
21
Hungary
491
22
Luxembourg
490
23
Norway
490
24
Spain
480
25
United States
474
11
Belgium
507
12
12
Iceland
507
13
13
Norway
505
14
Ireland
508
505
15
Hungary
504
Sweden
503
14
France
15
United States
504
16
16
Denmark
497
17
Poland
498
18
Denmark
496
19
France
495
20
Iceland
491
U.S. Ranks 15th
17
Switzerland
494
18
Spain
493
19
Czech Republic
492
20
Italy
487
21
Germany
484
22
Hungary
480
23
Poland
479
24
Portugal
479
25
Greece
474
26
Luxembourg
441
27
Mexico
422
Sources: PISA 2004, 2007
U.S. Ranks 21st
21
United States
489
22
Slovak Republic
488
23
Spain
488
24
Norway
487
25
Luxembourg
486
26
Italy
475
27
Portugal
474
15
Austria
506
16
Iceland
505
17
Hungary
501
18
Ireland
499
19
Luxembourg
494
20
Slovak Republic
492
21
Norway
490
22
Poland
487
23
Spain
482
24
United States
477
25
Portugal
470
U.S. Ranks 24th
26
Italy
470
27
Greece
449
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
U.S. Ranks 25th
26
Portugal
466
27
Italy
462
Elementary Students’ Performance on a Fractions
Problem
4.
Which fraction is located between
A.
1
4
B.
1
2
C.
8
8
D.
10
8
Mi PROM/SE
3
7
and
on the number line?
8
8
Grade 3
Grade 4
Grade 5
19.1
39.8
50.8
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
Elementary & Middle School Students’ Performance
on a Fractions Problem
4 1 1
 
20.
What is the value of
?
5 3 15
A.
1
5
B.
2
5
C.
7
Mi 15
PROM/SE
D.
3
4
E.
4
5
Grade 3
Grade 4
Grade 5
Grade 6
Grade 7
Grade 8
14.1
21.2
16.7
28.4
37.7
51.2
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
High School Students’ Performance on a
Mathematics Literacy Problem
29.
Stu wants to wrap some ribbon around a box as shown below and have 25
centimeters left to tie a bow.
How long a piece of ribbon does he need?
A.
46 cm
B.
77 cm
C.
65 cm
D.
71 cm
Mi PROM/SE
Grade 9
Grade 10
Grade 11
Grade 12
38.0
41.0
43.3
50.2
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
High science performance
565
Finland
Average performance
of 15-year-olds in
science – extrapolate
545
Hong Kong-China
and apply
Canada
Japan
New Zealand
Australia
525
Netherlands
Liechtenstein
Korea
Slovenia
United Kingdom Germany
Czech Republic Switzerland
Macao-China Austria
Belgium
Ireland
505
Hungary
Sweden
Poland
France Denmark
Iceland Croatia
United States Latvia
Slovak Republic, Spain, Lithuania
Norway 485
Luxembourg
Chinese Taipei
Estonia
Portugal
Russian Federation
Italy
Greece
465
Israel
445
16
… 18 countries perform below this line
Low science performance
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
6
Instructional Content Constructs
Curricular Coherence
Curricular Structure
Curricular Focus
Exposure Time (OTL)
Curricular Rigor
Level of Cognitive Complexity
Top Achieving Countries’ Mathematics Curriculum
Grade
Topic
Whole Number: Meaning
Whole Number: Operations
Measurement Units
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5
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Common Fractions
Equations & Formulas
Data Representation & Analysis
2-D Geometry: Basics
2-D Geometry: Polygons & Circles
Measurement: Perimeter, Area & Volume
Rounding & Significant Figures
Estimating Computations
Whole Numbers: Properties of Operations
Estimating Quantity & Size
Decimal Fractions
Relation of Common & Decimal Fractions
Properties of Common & Decimal Fractions
Percentages
Proportionality Concepts
Proportionality Problems
2-D Geometry: Coordinate Geometry
Geometry: Transformations
Negative Numbers, Integers, & Their Properties
Number Theory
Exponents, Roots & Radicals
Exponents & Orders of Magnitude
Measurement: Estimation & Errors
Constructions Using Straightedge & Compass
3-D Geometry
Geometry: Congruence & Similarity
Rational Numbers & Their Properties
Patterns, Relations & Functions
Proportionality: Slope & Trigonometry
Real Numbers, Their Subsets & Properties
Validation & Justification
© 2006 Michigan State University,
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n Intended by all of the top-achieving countries.
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21 States’ Mathematics Standards
State A
State K
State F
Topics
G1
G2
G3
G4
G5
G6
G7
G8
Topics
G1
G2
G3
G4
G5
G6
G7
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Topics
Whole Number Meaning
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Whole Number Meaning
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Whole Number Meaning
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Whole Number Operations
Measurement Units
Common Fractions
Equations & Formulas
Data Representation & Analysis
2-D Geometry: Basics
Polygons & Circles
Perimeter, Area & Volume
Properties of Whole Number Operations
Proportionality Concepts
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Data Representation & Analysis
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Perimeter, Area & Volume
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Rounding & Significant Figures
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Estimating Computations
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Decimal Fractions
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Estimating Quantity & Size
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Exponents & Orders of Magnitude
Measurement Estimation & Errors
Constructions w/ Straightedge & Compass
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Congruence & Similarity
Rational Numbers & Their Properties
Patterns, Relations & Functions
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152Proportionality Problems
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141Geometry: Transformations
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Negative Numbers, Integers & Their Properties
Number Theory
Exponents, Roots & Radicals
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Measurement Estimation & Errors
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G6
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G2
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Percentages
Common Fractions
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State D
G1
Whole Number Meaning
G4
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Measurement Units
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Common Fractions
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Equations & Formulas
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Data Representation & Analysis
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2-D Geometry: Basics
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Polygons & Circles
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Perimeter, Area & Volume
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Rounding & Significant Figuresl
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Estimating Computations
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Properties of Whole Number Operations
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Estimating Quantity & Size
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Decimal Fractions
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Relationship of Common & Decimal
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Properties of Common & Decimal
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Whole Number Operations
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State Q
G1
Whole Number Operations
Whole Number Meaning
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Exponents & Orders of Magnitude
State M
G1
G8
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Slope & Trigonometry
Proportionality Concepts
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G7
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Percentages
State L
Topics
G6
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Properties of Common & Decimal Fractions
Negative Numbers, Integers & l
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Properties
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Slope & Trigonometry
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2-D Coordinate Geometry
G8
G3
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Congruence & Similarity
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Proportionality Problems
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G2
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Measurement Units
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Common Fractions
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Equations & Formulas
l
Data Representation & Analysis
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2-D Geometry: Basics
l
Polygons & Circles
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Perimeter, Area & Volume
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Rounding & Significant Figuresl
G5
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Estimating Quantity & Size
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Decimal Fractions
l Fractions
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Relationship of Common & Decimal
Proportionality Concepts
G7
G1
Whole Number Operations
3-D Geometry
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Percentages
3-D Geometry
G4
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Estimating Computations
Constructions w/ Straightedge & Compass
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G3
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Properties of Whole Number Operations
Decimal Fractions
Geometry: Transformations
G2
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Measurement Units
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Common Fractions
l
Equations & Formulas
l
Data Representation & Analysis
l
2-D Geometry: Basics
l
Polygons & Circles
l
Perimeter, Area & Volume
Whole Number Operations
Rounding & Significant Figures
l
Exponents & Orders of Magnitude
Whole Number Meaning
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Estimating Computations
State U
G1
Whole Number Meaning
Relationship of Common & Decimal Fractions
State H
G1
Proportionality Concepts
3-D Geometry
l
Slope & Trigonometry
Properties of Common & Decimal Fractions
Number Theory
l
Topics
Slope & Trigonometry
Congruence & Similarity
Relationship of Common & Decimal Fractions
Proportionality Problems
l
l
l
l
Constructions w/ Straightedge & Compass
Decimal Fractions
Percentages
l
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Exponents & Orders of Magnitude
Rounding & Significant Figures
Properties of Whole Number Operations
l
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Properties of Common & Decimal Fractions
2-D Coordinate Geometry
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G8
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G6
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G7
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G6
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G5
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G5
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Estimating Computations
l
Properties of Whole Number Operations
l l l l
Estimating Quantity & Size
l
Decimal Fractions
l
Relationship of Common & Decimal Fractions l
Properties of Common & Decimal Fractions
l
Percentages
l
Proportionality Concepts
l
Whole Number Meaning
l
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G4
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G4
Topics
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2-D Geometry: Basics
Estimating Computations
Patterns, Relations & Functionsl
G8
State C
Perimeter, Area & Volume
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Rational Numbers & Their Properties
l
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l
Slope & Trigonometry
Polygons & Circles
Congruence & Similarity
G7
Rational Numbers & Their Properties
Data Representation & Analysis
l
l
3-D Geometry
l
l
l
l
l
l
l
l
l
Congruence & Similarity
Equations & Formulas
l
Measurement Estimation & Errors
G6
Constructions w/ Straightedge & Compass
Common Fractions
l
l
Exponents & Orders of Magnitude
l
l
l
l
l
l
l
l
l
Exponents, Roots & Radicals
Measurement Units
Exponents, Roots & Radicals
G5
Number Theory
Whole Number Operations
l
l
Number Theory
G3
l
l
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l
l
l
l
l
l
Properties of Whole Number Operations
Negative Numbers, Integers & Their Properties
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
Negative Numbers, Integers & Their Properties
Whole Number Meaning
l
l
Geometry: Transformations
G4
2-D Coordinate Geometry
Topics
l
Proportionality Concepts
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
Proportionality Problems
Patterns, Relations & Functions
l
l l l l
l l l l
Decimal Fractions
l Fractions
l l l
Relationship of Common & Decimal
G3
Proportionality Concepts
3-D Geometry
l
Estimating Quantity & Size
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
Percentages
Measurement Estimation & Errors
l
l
Properties of Whole Number Operations
2-D Coordinate Geometry
G2
l
l
l
Measurement Units
l
Common Fractions
l
Equations & Formulas
l
Data Representation & Analysis
l
2-D Geometry: Basics
l
Polygons & Circles
l
Perimeter, Area & Volume
Rounding & Significant Figures
l
Estimating Computations
G2
Properties of Common & Decimal Fractions
Exponents & Orders of Magnitude
l
l
l
l
l
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State G
G1
Whole Number Meaning
Geometry: Transformations
l
l
l
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l
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l
Slope & Trigonometry
State B
Topics
l
l
l
Perimeter, Area & Volume
l
l
l
l
Constructions w/ Straightedge & Compass
Rational Numbers & Their Properties
Patterns, Relations & Functions
Polygons & Circles
Percentages
Constructions w/ Straightedge & Compass
3-D Geometry
l
l
2-D Geometry: Basics
Proportionality Problems
l
l
Negative Numbers, Integers & Their Properties
Exponents & Orders of Magnitude
l
Data Representation & Analysis
l
l
l
l
Properties of Common & Decimal Fractions
Proportionality Problems
Number Theory
Equations & Formulas
State P
G1
Whole Number Operations
Rounding & Significant Figures
l
l
l
Estimating Quantity & Size
l
Decimal Fractions
l
Relationship of Common & Decimal Fractions
l
Properties of Common & Decimal Fractions
l
Percentages
Estimating Computations
Geometry: Transformations
Measurement Units
Common Fractions
Rounding & Significant Figures
2-D Coordinate Geometry
Whole Number Operations
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
Percentages
Proportionality Concepts
l
l
l
Proportionality Problems
l
l
l
Exponents, Roots & Radicals
l l l
Geometry: Transformations l
l l
Negative Numbers, Integers & l
Their l
Properties
l
Number Theory
l l l
2-D Coordinate Geometry
l
l
l
l
Exponents & Orders of Magnitude
Measurement Estimation & Errors
l
l
l
l
Constructions w/ Straightedge & Compass
l
l
3-D Geometry
Congruence & Similarity
l
l
l
l
l
l
l
l
l
l
Rational Numbers & Their Properties
l
Patterns, Relations & Functionsl
Slope & Trigonometry
l
Common Core Mathematics Standards
Grade
Topic
Whole Number: Meaning
Whole Number: Operations
Whole Numbers: Properties of Operations
Common Fractions
Measurement Units
2-D Geometry: Polygons & Circles
Data Representation & Analysis
3-D Geometry
Measurement: Estimation & Errors
Number Theory
2-D Geometry: Basics
Rounding & Significant Figures
Relation of Common & Decimal Fractions
Estimating Computations
Measurement: Perimeter, Area & Volume
Equations & Formulas
Decimal Fractions
Patterns, Relations & Functions
Geometry: Transformations
Properties of Common & Decimal Fractions
Exponents & Orders of Magnitude
2-D Geometry: Coordinate Geometry
Exponents, Roots & Radicals
Percentages
Negative Numbers, Integers, & Their Properties
Proportionality Concepts
Proportionality Problems
Rational Numbers & Their Properties
Constructions Using Straightedge & Compass
Systematic Counting
Uncertainty & Probability
Real Numbers, Their Subsets & Properties
Geometry: Congruence & Similarity
Proportionality: Slope & Trigonometry
Validation & Justification
1
2
3
4
5
6
7
8
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© 2006 Michigan State University, Center for Research in Mathematics and Science Education
State Level Agreement
Grade
Topic
1
Whole Number: Meaning
Whole Number: Operations
Measurement Units
Common Fractions
Equations & Formulas
Data Representation & Analysis
2-D Geometry: Basics
2-D Geometry: Polygons & Circles
Measurement: Perimeter, Area & Volume
Rounding & Significant Figures
Estimating Computations
Whole Numbers: Properties of Operations
Estimating Quantity & Size
Decimal Fractions
Relation of Common & Decimal Fractions
Properties of Common & Decimal Fractions
Percentages
Proportionality Concepts
Proportionality Problems
2-D Geometry: Coordinate Geometry
Geometry: Transformations
Negative Numbers, Integers, & Their Properties
Number Theory
Exponents, Roots & Radicals
Exponents & Orders of Magnitude
Measurement: Estimation & Errors
Constructions Using Straightedge & Compass
3-D Geometry
Geometry: Congruence & Similarity
Rational Numbers & Their Properties
Patterns, Relations & Functions
Proportionality: Slope & Trigonometry
2
3
4
5
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6
7
8
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m
m
m
m
m
m
m
m
m
m
m
m
m
m
Intended by all states
m
m
m
m
m
Intended by none of the states
Intended by more than half of the top-achieving countries
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
l
m
Instructional
Days Spent on Selected Areas of
Instructional Days Spent on Selected Areas of Mathematics at Elementary and Middle Grades
Mathematics at Elementary and Middle Grades
180
160
140
Instructional Days
120
100
80
60
40
20
0
Grade 2: Arithmetic
N=552
Grade 4: Arithmetic
N=542
Grade 6: Transition Arithmetic
N=268
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
Grade 8: Algebra
N=254
Are All Algebra Courses the Same?
•
•
•
•
•
•
•
•
•
•
•
•
Fundamentals of Algebra
Advanced Algebra
Algebra I
Algebra IA
Algebra IB
Algebra IC
Algebra I Hons.
Basic Algebra I
Elementary Algebra
Essentials of Algebra
Introduction to Algebra
Enrich. Algebra
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
What Mathematics is Covered in These Courses?
•
•
•
•
•
•
•
•
•
•
•
Academic Math I
Accel. Math I
Alt. Math
Basic Math
Consumer Math
Contemporary Math
Corrective Math
Fundamentals of Math
General Math
Math Skills
Life Skills Math
•
•
•
•
•
•
Math Proficiency
Transitional Math
Unified Math’
Tech. Math
Voc. Math
Senior Math
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
Relationship between Parents’ Education and
Student Achievement
1999 TIMSS-R Districts
600
y = 2.4911x + 402.64
R2 = 0.7409
580
Mathematics Achievement
560
540
520
Districts
US
Districts
500
480
460
440
420
400
15
20
25
30
35
40
45
50
55
60
65
Percent of Students with Either Parent Finishing University
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
70
75
Relationship between Algebra Instruction and
Student Achievement
1999 TIMSS-R Districts
600
y = 1.3545x + 463.53
R2 = 0.7509
580
Mathematics Achievement
560
540
520
500
480
460
Districts
US
Districts
440
420
400
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
Percent of Students Whose Teachers reported Emphasized
Mostly Algebra in Mathematics Class
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
95 100
Relationship between Parents’ Education and
Algebra Instruction
1999 TIMSS-R Districts
100
Percent of Students Whose Teachers reported
Emphasized
Mostly Algebra in Mathematics Class
90
80
y = 1.3034x - 18.195
R2 = 0.7048
70
60
Districts
US
Districts
50
40
30
20
10
0
15
20
25
30
35
40
45
50
55
60
65
Percent of Students with Either Parent Finishing University
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
70
75
Dot Plot of Institution Mean Lower Secondary MCK
Scores by Country
750
700
600
550
500
450
400
350
300
e
bl
ic
U
SPu
SPr
iv
at
U
ila
Th
a
itz
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
nd
nd
er
la
ai
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or
ap
Sp
Sw
Fe
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us
C
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a
Si
ng
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Po
ilip
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ay
Ph
or
w
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an
si
a
M
al
ay
an
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G
er
m
ia
rg
ip
Ta
ne
se
hi
G
eo
ei
le
hi
C
ts
w
an
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250
Bo
Mathematics Knowledge Scale Score
650
As Others Rise to the Challenge, U.S. Advantage Drops
Percent of Population with HS Degree or Equivalent
13
1
%
1990s
1980s
1970s
1960s
1
27
Notes: 1) Excluding ISCED 3C short programmes;
3) Including some ISCED 3C short programmes;
Source: Schleicher 2007
2) Year of reference 2004;
4) Year of reference 2003.
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
Even in Higher Education, the World is Catching up…
College Graduation Rates
%
15
2
Note: Net graduation rate is calculated by summing the graduation rates by single year of age in 2005. Year of reference 2004.
Source: OECD 2007
© 2006 Michigan State University, Center for Research in Mathematics and Science Education
2005
1995