Content and School Effectiveness: Role and Impact

December 12, 2006

William H. SchmidtMichigan State University

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Instructional Content Constructs

Curricular CoherenceCurricular Structure

Curricular FocusExposure Time (OTL)

Curricular RigorLevel of Cognitive Complexity

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Coherence Defined

We define content standards, in the aggregate, to be coherent if they are articulated over time as a sequence of topics and performances consistent with the logical and, if appropriate, hierarchical nature of the disciplinary content from which the subject-matter derives. This is not to suggest the existence of a single coherent sequence, only that such a sequence reflect the inherent structure of the discipline.

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

1 Numbers

2 Measurement

3 6.2.1 Representation of numerical situations

6.2.2 Informal solution of simple equations

4 6.2.3 Operations with expressions

6.2.4 Equivalent expressins

5 Proportionality 6.2.5 Linear equations: solutions

6 6.2.6 Quadratic equations: solutions

6.2.7 Polynomial equations: solutions

7 6.2.8 Trig identities and equations: solutions

6.2.9 Logs and exponential equ'ns: solu'ns

8 Elementary Analysis 6.2.10 Solution of more complex equations

9 Validation and Structure 6.2.11 Other solution methods for equations

10 Other Content 6.2.12 Inequalities: solution and graphs

6.2.13 Systems of equations: solutions

6.2.14 Systems of inequalities: solu'ns, graphs

6.2.15 Substituting into/rearranging formulas

6.2.16 General equation of the second degree

Geometry: Positions, Visualizations and Shape

Geometry: Symmetry, Congruence and Similarity

Functions, Relations and Equations 6.1

Patterns, Relations, and Functions

Data Representation, Probability and Statistics

6.2Equations and Formulas

TIMSS Mathematics Framework – An Example

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Congruence and Similarity Equations and Formulas

Ages Ages

6 7 8 9 10 11 12 13 14 15 16 17 18 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Argentina + Argentina + + -Canada - - - - - - - - - - - - Canada - - - - + + + + + -Cyprus + - + Cyprus - - - - - - - + + + + +Denmark + - - + + - Denmark - - - + - - + + + + +Hungary - - - - - + - - + - Hungary - - + - - + - - +Iceland + - - + - Iceland - - + - +Iran + - - + - - Iran + - - - +Ireland + + + - + Ireland - - - - - - - - + + + +Japan + + + - Japan + + + + + + + + + +New Zealand - - - - - New Zealand - - - - - - - - - + + + +Spain + + Spain - - - - - -Tunisia + - - - - + Tunisia - + + +USA - - + - + USA - - - - - + + + - + +

These data are typical topic trace maps for a sample of countries selected to show representative diversity. The results are typical of those for other topics and countries.

Two Topic Trace Maps

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Top Achieving Countries’ Mathematics Curriculum

s I ntended by 4 out of the 6

top- achieving countries

l I ntended by all but one of the

top- achieving countries (5 out of 6).

n I ntended by all of the top- achieving countries.

Grade

Topic 1 2 3 4 5 6 7 8 9 10 11 12

Whole Number: Meaning n n n l lWhole Number: Operations n n n n lMeasurement Units s n n n n n lCommon Fractions s n n lEquations & Formulas s l l l n n n n n nData Representation & Analysis s s l l s s s s s2-D Geometry: Basics s l l l n n l2-D Geometry: Polygons & Circles s l l n n n lMeasurement: Perimeter, Area & Volume l l l l s sRounding & Significant Figures l lEstimating Computations l l lWhole Numbers: Properties of Operations l lEstimating Quantity & Size s sDecimal Fractions l n lRelation of Common & Decimal Fractions s n lProperties of Common & Decimal Fractions l lPercentages l lProportionality Concepts l l l sProportionality Problems l l n n s s2-D Geometry: Coordinate Geometry s s l l n l lGeometry: Transformations l l l s s sNegative Numbers, I ntegers, & Their Properties s lNumber Theory l s sExponents, Roots & Radicals l l n lExponents & Orders of Magnitude s s sMeasurement: Estimation & Errors sConstructions Using Straightedge & Compass n s3-D Geometry l n n l n sGeometry: Congruence & Similarity n lRational Numbers & Their Properties s sPatterns, Relations & Functions s n n n nProportionality: Slope & Trigonometry s l lReal Numbers, Their Subsets & Properties lValidation & J ustification l n l lStructuring & Abstracting sUncertainty & Probability s n lComplex Numbers & Their Properties s sI nfinite Processes n nChange n nVectors n sSystematic Counting s s

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Mathematics Topics Intended at Each Grade by 1989 NCTM StandardsGrade

Topic 1 2 3 4 5 6 7 8

Whole Number Meaning

Whole Number Operations

Measurement Units

Common Fractions

Equations & Formulas

Data Representation & Analysis

2-D Geometry: Basics

Polygons & Circles

Perimeter, Area & Volume

Rounding & Significant Figures

Estimating Computations

Properties of Whole Number Operations

Estimating Quantity & Size

Decimal Fractions

Relationship of Common & Decimal Fractions

Properties of Common & Decimal Fractions Percentages Proportionality Concepts Proportionality Problems 2-D Coordinate Geometry

Geometry: Transformations

Negative Numbers, Integers & Their Properties Number Theory

Exponents, Roots & Radicals Exponents & Orders of Magnitude

Measurement Estimation & Errors

Constructions w/ Straightedge & Compass

3-D Geometry

Congruence & Similarity

Rational Numbers & Their Properties Patterns, Relations & Functions

Slope & Trigonometry Number of additional topics intended by the standards

to complete the curriculum at each grade level.70 0 1 1 2 3 5

A+ Profile

Intended Topic

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Mathematics Topics Intended at Each Grade by 2000 NCTM StandardsGrade

Topic 1 2 3 4 5 6 7 8

Whole Number Meaning l l l l lWhole Number Operations l l l l lMeasurement Units l l l l l l l lCommon Fractions l l l l l l l lEquations & Formulas l l l l l lData Representation & Analysis l l l l l l l l2-D Geometry: Basics l l l l l l l lPolygons & Circles l l l l l l l lPerimeter, Area & Volume l l l l l l l lRounding & Significant Figures l l lEstimating Computations l l l l l l l lProperties of Whole Number Operations l l l l lEstimating Quantity & Size l l lDecimal Fractions l l l l l lRelationship of Common & Decimal Fractions l l l l l lProperties of Common & Decimal Fractions l l lPercentages l l l l l lProportionality Concepts l l lProportionality Problems l l l2-D Coordinate Geometry l l l l l l l lGeometry: Transformations l l l l l l l lNegative Numbers, Integers & Their Properties l l l l l lNumber Theory l l lExponents, Roots & Radicals l l lExponents & Orders of Magnitude l l lMeasurement Estimation & Errors l l l l l l l lConstructions w/ Straightedge & Compass l l l l l l l l3-D Geometry l l l l l l l lCongruence & Similarity l l l l l l l lRational Numbers & Their Properties l l lPatterns, Relations & Functions l l l l l l l lSlope & Trigonometry l l l

Top Achieving Countries' Profile

Topic Intended l

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

State K State PTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning Whole Number Operations Whole Number Operations Measurement Units Measurement Units Common Fractions Common Fractions Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures

Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions

Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass 3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry

State L State QTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning 1111

Whole Number Operations Whole Number Operations 1112

Measurement Units Measurement Units 121

Common Fractions Common Fractions 1121

Equations & Formulas Equations & Formulas 162

Data Representation & Analysis Data Representation & Analysis 171

2-D Geometry: Basics 2-D Geometry: Basics 132

Polygons & Circles Polygons & Circles 133

Perimeter, Area & Volume Perimeter, Area & Volume 122

Rounding & Significant Figures Rounding & Significant Figures 1152

Estimating Computations Estimating Computations 1153

Properties of Whole Number Operations Properties of Whole Number Operations 1113

Estimating Quantity & Size Estimating Quantity & Size 1151

Decimal Fractions Decimal Fractions 1122

Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions 1123

Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions 1125

Percentages Percentages 1124

Proportionality Concepts Proportionality Concepts 151

Proportionality Problems Proportionality Problems 152

2-D Coordinate Geometry 2-D Coordinate Geometry 131

Geometry: Transformations Geometry: Transformations 141

Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties 1131

Number Theory Number Theory 1144

Exponents, Roots & Radicals Exponents, Roots & Radicals 1142

Exponents & Orders of Magnitude Exponents & Orders of Magnitude 1154

Measurement Estimation & Errors Measurement Estimation & Errors 123

Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass 143

3-D Geometry 3-D Geometry 134

Congruence & Similarity Congruence & Similarity 142

Rational Numbers & Their Properties Rational Numbers & Their Properties 1132

Patterns, Relations & Functions Patterns, Relations & Functions 161

Slope & Trigonometry Slope & Trigonometry 153

State A State FTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning Whole Number Operations Whole Number Operations Measurement Units Measurement Units Common Fractions Common Fractions Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures

Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations

Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions

Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude

Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass

3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry

State B State GTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning Whole Number Operations Whole Number Operations Measurement Units Measurement Units Common Fractions Common Fractions Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass 3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry

State U State DTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning Whole Number Operations Whole Number Operations Measurement Units Measurement Units Common Fractions Common Fractions Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions

Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass 3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry

State E State ITopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning 1111 Whole Number Meaning 1111

Whole Number Operations 1112 Whole Number Operations 1112

Measurement Units 121 Measurement Units 121

Common Fractions 1121 Common Fractions 1121

Equations & Formulas 162 Equations & Formulas 162

Data Representation & Analysis 171 Data Representation & Analysis 171

2-D Geometry: Basics 132 2-D Geometry: Basics 132

Polygons & Circles 133 Polygons & Circles 133

Perimeter, Area & Volume 122 Perimeter, Area & Volume 122

Rounding & Significant Figures 1152 Rounding & Significant Figures 1152

Estimating Computations 1153 Estimating Computations 1153

Properties of Whole Number Operations 1113 Properties of Whole Number Operations 1113

Estimating Quantity & Size 1151 Estimating Quantity & Size 1151

Decimal Fractions 1122 Decimal Fractions 1122

Relationship of Common & Decimal Fractions 1123 Relationship of Common & Decimal Fractions 1123

Properties of Common & Decimal Fractions 1125 Properties of Common & Decimal Fractions 1125

Percentages 1124 Percentages 1124

Proportionality Concepts 151 Proportionality Concepts 151

Proportionality Problems 152 Proportionality Problems 152

2-D Coordinate Geometry 131 2-D Coordinate Geometry 131

Geometry: Transformations 141 Geometry: Transformations 141

Negative Numbers, Integers & Their Properties 1131 Negative Numbers, Integers & Their Properties 1131

Number Theory 1144 Number Theory 1144

Exponents, Roots & Radicals 1142 Exponents, Roots & Radicals 1142

Exponents & Orders of Magnitude 1154 Exponents & Orders of Magnitude 1154

Measurement Estimation & Errors 123 Measurement Estimation & Errors 123

Constructions w/ Straightedge & Compass 143 Constructions w/ Straightedge & Compass 143

3-D Geometry 134 3-D Geometry 134

Congruence & Similarity 142 Congruence & Similarity 142

Rational Numbers & Their Properties 1132 Rational Numbers & Their Properties 1132

Patterns, Relations & Functions 161 Patterns, Relations & Functions 161

Slope & Trigonometry 153 Slope & Trigonometry 153

State STopics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Operations Measurement Units Common Fractions Equations & Formulas Data Representation & Analysis 2-D Geometry: Basics Polygons & Circles Perimeter, Area & Volume Rounding & Significant Figures Estimating Computations Properties of Whole Number Operations Estimating Quantity & Size Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Percentages Proportionality Concepts Proportionality Problems 2-D Coordinate Geometry Geometry: Transformations Negative Numbers, Integers & Their Properties Number Theory Exponents, Roots & Radicals Exponents & Orders of Magnitude Measurement Estimation & Errors Constructions w/ Straightedge & Compass 3-D Geometry Congruence & Similarity Rational Numbers & Their Properties Patterns, Relations & Functions Slope & Trigonometry

State OTopics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Operations Measurement Units Common Fractions Equations & Formulas Data Representation & Analysis 2-D Geometry: Basics Polygons & Circles Perimeter, Area & Volume Rounding & Significant Figures

Estimating Computations

Properties of Whole Number Operations Estimating Quantity & Size

Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Percentages Proportionality Concepts Proportionality Problems 2-D Coordinate Geometry Geometry: Transformations Negative Numbers, Integers & Their Properties Number Theory Exponents, Roots & Radicals Exponents & Orders of Magnitude

Measurement Estimation & Errors Constructions w/ Straightedge & Compass 3-D Geometry Congruence & Similarity Rational Numbers & Their Properties Patterns, Relations & Functions Slope & Trigonometry

State C State H State MTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning Whole Number Meaning

Whole Number Operations Whole Number Operations Whole Number Operations Measurement Units Measurement Units Measurement Units Common Fractions Common Fractions Common Fractions Equations & Formulas Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures Rounding & Significant Figures Estimating Computations Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations Properties of Whole Number Operations Estimating Quantity & Size Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions

Percentages Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry 2-D Coordinate Geometry Geometry: Transformations Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude Exponents & Orders of Magnitude Measurement Estimation & Errors Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass 3-D Geometry 3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry Slope & Trigonometry

State R State JTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning

Whole Number Operations Whole Number Operations Measurement Units Measurement Units Common Fractions Common Fractions Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry

Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass 3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry

State N State TTopics G1 G2 G3 G4 G5 G6 G7 G8 Topics G1 G2 G3 G4 G5 G6 G7 G8

Whole Number Meaning Whole Number Meaning

Whole Number Operations Whole Number Operations Measurement Units Measurement Units Common Fractions Common Fractions Equations & Formulas Equations & Formulas Data Representation & Analysis Data Representation & Analysis 2-D Geometry: Basics 2-D Geometry: Basics Polygons & Circles Polygons & Circles Perimeter, Area & Volume Perimeter, Area & Volume Rounding & Significant Figures Rounding & Significant Figures Estimating Computations Estimating Computations Properties of Whole Number Operations Properties of Whole Number Operations Estimating Quantity & Size Estimating Quantity & Size Decimal Fractions Decimal Fractions Relationship of Common & Decimal Fractions Relationship of Common & Decimal Fractions Properties of Common & Decimal Fractions Properties of Common & Decimal Fractions

Percentages Percentages Proportionality Concepts Proportionality Concepts Proportionality Problems Proportionality Problems 2-D Coordinate Geometry 2-D Coordinate Geometry Geometry: Transformations Geometry: Transformations Negative Numbers, Integers & Their Properties Negative Numbers, Integers & Their Properties Number Theory Number Theory Exponents, Roots & Radicals Exponents, Roots & Radicals Exponents & Orders of Magnitude Exponents & Orders of Magnitude Measurement Estimation & Errors Measurement Estimation & Errors Constructions w/ Straightedge & Compass Constructions w/ Straightedge & Compass

3-D Geometry 3-D Geometry Congruence & Similarity Congruence & Similarity Rational Numbers & Their Properties Rational Numbers & Their Properties Patterns, Relations & Functions Patterns, Relations & Functions Slope & Trigonometry Slope & Trigonometry

21 U.S. States’ Mathematics Standards

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Model

Predictor

Number of Topics Aligned with Ideal Scenario

9.94 ( 7.38 , 0.196 ) 7.48 ( 4.04 , 0.081 ) 3.57 ( 1.48 , 0.023 ) 2.83 ( 1.12 , 0.017 )

Total Number of Topics in Curriculum

-2.42 ( 1.13 , 0.047 ) -2.58 ( 1.05 , 0.025 ) -1.64 ( 0.57 , 0.008 ) -1.39 ( 0.45 , 0.004 )

Model Fit

R-Square 0.2191 0.2574 0.2205 0.2490

Residual Mean Sq. 1652.9 1489.0 1476.0 1331.4

p< 0.1222 0.0687 0.0270 0.0136

No. of Countries 20 21 32 33Standardized Coefficient

Number of Topics Aligned with Ideal Scenario

Total Number of Topics in Curriculum

-0.6969 -0.9703 -1.0044 -1.0204

0.4374 0.7331 0.8481 0.8240

Estimate ( StdError, p )Estimate ( StdError, p )Estimate ( StdError, p )Estimate ( StdError, p )

Grade 3 Grade 4 Grade 7 Grade 8

Table 1

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

RANOMIZED STUDY

Design2X2X2X2X2X2 randomized block

Unit of analysisDistrict(62)

OutcomesStudent achievement and content coverage

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Code Description IGP Index1111 Whole Number: Meaning 1.81122 Decimal Fractions 4.61141 Binary Arithmetic &/or Other Number Bases 6.61143 Complex Numbers & Their Properties 10.7141 Geometry: Transformations 7.1151 Proportionality Concepts 6.4161 Patterns, Relations & Functions 9.0162 Equations & Formulas 7.0171 Data Representation & Analysis 7.4172 Uncertainty & Probability 9.7

Examples of the IGP Index

International Grade Placement of Curriculum Content Across Districts

Gra

de P

lace

men

t of

Cur

ricul

um C

onte

nt

Grade© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Teachers Coverage of Mathematics Topics in Two Types of SchoolsHigh SES Schools Low SES Schools

Arithematic

Geometry

Algebra

Other

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Variation in the mathematics content index (IGP) in schools having multiple tracks and schools having single tracks

Track

School

Class

Single-Track Schools Multiple-Track Schools

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Relationship between Parents’ Education and Student Achievement1999 TIMSS-R Districts

y = 2.4911x + 402.64

R2 = 0.7409

400

420

440

460

480

500

520

540

560

580

600

15 20 25 30 35 40 45 50 55 60 65 70 75

Percent of Students with Either Parent Finishing University

Ma

the

ma

tic

s A

ch

iev

em

en

t

Districts

US

Districts

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Relationship between Parents’ Education and Algebra Instruction

1999 TIMSS-R Districts

y = 1.3034x - 18.195

R2 = 0.7048

0

10

20

30

40

50

60

70

80

90

100

15 20 25 30 35 40 45 50 55 60 65 70 75

Percent of Students with Either Parent Finishing University

Per

cen

t o

f S

tud

ents

Wh

ose

Tea

cher

s re

po

rted

E

mp

has

ized

M

ost

ly A

lgeb

ra in

Mat

hem

atic

s C

lass

Districts

US

Districts

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

Relationship between Algebra Instruction and Student Achievement

1999 TIMSS-R Districts

y = 1.3545x + 463.53

R2 = 0.7509

400

420

440

460

480

500

520

540

560

580

600

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Percent of Students Whose Teachers reported Emphasized Mostly Algebra in Mathematics Class

Ma

the

ma

tic

s A

ch

iev

em

en

t

Districts

US

Districts

Results of Fitting the Hiarchical Linear Model: Levels 3 Fixed Effects

Fixed Effect EstimateStandard

Error t-ratioApprox.

d.f. p-value

INTRCRT3, G000 164.64 78.65 2.09 19 0.05

LUNCH, G001 -0.86 0.13 -6.46 19 0.00

IGP, G002 30.20 11.87 2.54 19 0.02

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

© 2006 Center for Research in Mathematics and Science Education, Michigan State University

NON EXPERIMENTAL STUDIES

Bias is almost always inherent in school effectiveness studies when content is not adequately taken into account

Percent of U.S. Students Enrolled in Each Type of Math Course Using Each Type of Textbook

© 2005 MSU PROM/SE Promoting Rigorous Outcomes in Mathematics and Science Education, Supported by NSF Cooperative Agreement EHR-0314866

Textbook Type Course Type Regular Pre-Algebra Algebra

Remedial 91 (8.5) 9.1 (8.5)

Regular 74 (6.0) 21 (5.1) 4.5 (1.9)

Enriched 74 (15.8) 14 (8.7) 13 (3.8)

Pre-Algebra 30 (8.5) 59 (10.4) 11 (6.8)

Algebra I 25 (9.1) 5.7 (4.0) 70 (9.6)