content and school effectiveness: role and impact december 12, 2006 william h. schmidt michigan...
TRANSCRIPT
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)