International Benchmarking Plenary (Bill Schmidt)
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Instructional Content Constructs Curricular Coherence Curricular Structure Curricular Focus Exposure Time (OTL) Curricular Rigor Level of Cognitive Complexity © 2009 Michigan State University, Center for Research in Mathematics and Science Education Top Achieving Countries’ Mathematics Curriculum Grade Topic Whole Number: Meaning Whole Number: Operations Measurement Units 1 2 3 4 5 n n s n n n n n n s s s s l n n n l s l s l l l l s l s l l n n l l l l l l l l s n n l l l l s 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 © 2009 Michigan State University, 6 7 n l l l l l l l n n n l 8 9 10 11 12 n s n n s n s l n ss n s n s n s l Intended by 4 out of the 6 top-achieving countries l Intended by all but one of the l l l l l l l s l s top-achieving countries (5 out of 6). n Intended by all of the top-achieving countries. l n l l l l l s s n l s n l l s n s s l s s l s s n s l s n n s s s n l s n l l l Center for Research in Mathematics and Science Education l s l n s n l n n n l l 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 G8 Topics Whole Number Meaning l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Whole Number Meaning l l l l l l l l l l l l l l l l l l l l Whole Number Meaning l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 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 l l l l l l l l l l l l l l l Exponents, Roots & Radicals Measurement Estimation & Errors Congruence & Similarity l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Slope & Trigonometry 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 l Rounding & Significant Figures l Estimating Computations l Properties of Whole Number Operations l Estimating Quantity & Size l Decimal Fractions l Relationship of Common & Decimal Fractions l Whole Number Operations l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Estimating Quantity & Size l l l l l l l l l l l l l l l l l l l l G2 G3 G4 G5 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 2-D Coordinate Geometry Geometry: Transformations Negative Numbers, Integers & Their Properties l l l l l Exponents, Roots & Radicals Exponents & Orders of Magnitude Measurement Estimation & Errors Constructions w/ Straightedge & Compass l l l l l Congruence & Similarity Rational Numbers & Their Properties Patterns, Relations & Functions Slope & Trigonometry l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l G6 G7 G8 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Estimating Quantity & Size l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Number Theory l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Exponents, Roots & Radicals l Measurement Estimation & Errors l l l Congruence & Similarity l l l l l l l l l l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl Whole Number Meaning l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Proportionality Problems l l l l l l Negative Numbers, Integers & l Their l Properties l Number Theory l l l l l l l l l l l l l l l l l l l l l l l l l l l l Geometry: Transformations Exponents, Roots & Radicals Measurement Estimation & Errors l l l l 3-D Geometry l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl l l l l l l l l l l l Topics G2 G3 l Whole Number Operations l Measurement Units l Common Fractions l Equations & Formulas l Data Representation & Analysis l 2-D Geometry: Basics l Polygons & Circles l l l l l l l l l l l l l l l l l Perimeter, Area & Volume l l Rounding & Significant Figures l l Estimating Computations l l l Properties of Whole Number Operations l l l Estimating Quantity & Size l l Decimal Fractions l l l Relationship of Common & Decimal l Fractions l l Properties of Common & Decimal Fractions l l Percentages l l Proportionality Concepts G4 G5 G6 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Proportionality Problems 2-D Coordinate Geometry Geometry: Transformations l l l l l Negative Numbers, Integers & Their Properties Number Theory Exponents, Roots & Radicals l l l l Exponents & Orders of Magnitude Measurement Estimation & Errors Constructions w/ Straightedge & Compass l l l l l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl l l l l l l l l l l l l l l l l l l l l l G7 G8 G2 G3 l l l l l l l l l l l l l l l l l l Proportionality Concepts l Proportionality Problems l 2-D Coordinate Geometry l l l Geometry: Transformations l Negative Numbers, Integers & Their Properties l l l l l Number Theory l Exponents, Roots & Radicals l l l l l l l l l l l l l l l l l l l l l l l Decimal Fractions Percentages l l l l l l l l l l l l l l l l l l l l l G2 G3 G4 G5 G6 G7 l l l l l l l l l l l l l l l l l l l l l l l l l l l Negative Numbers, Integers & l Their l Properties l Number Theory l l l Exponents, Roots & Radicals l l l Exponents & Orders of Magnitude l l l Measurement Estimation & Errors l l l Constructions w/ Straightedge l & Compass l l 3-D Geometry l l l Congruence & Similarity l l l Rational Numbers & Their Properties l l l Patterns, Relations & Functionsl l l Slope & Trigonometry l l l Rounding & Significant Figuresl l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Percentages Proportionality Concepts 2-D Coordinate Geometry Geometry: Transformations l l l l l l l l Number Theory l l l l l l l l l l l l l l l l l l l l l l l l l Measurement Estimation & Errors l l l Constructions w/ Straightedge l & Compass l l 3-D Geometry l l l Congruence & Similarity l l l l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl 3-D Geometry G6 G7 G8 Topics l l l l l l Whole Number Meaning Data Representation & Analysis l l l l Polygons & Circles l l l Perimeter, Area & Volume l l Rounding & Significant Figures l Estimating Computations l Properties of Whole Number Operations l l l Estimating Quantity & Size l l Decimal Fractions l l l l l l l l l l l l l l l l l l l l l l l l l l Equations & Formulas 2-D Geometry: Basics Whole Number Operations l l l l l l l l l l l l l l Topics l l l l l l l l l l l l l l l l l l l l l l Proportionality Problems l 151Proportionality Concepts 152Proportionality Problems l l l l l l l l 1312-D Coordinate Geometry 141Geometry: Transformations l l l l l l Negative Numbers, Integers & Their Properties Number Theory Exponents, Roots & Radicals Exponents & Orders of Magnitude Measurement Estimation & Errors l Constructions w/ Straightedge & Compass l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Measurement Units Equations & Formulas l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l G5 G6 G7 G8 Topics l l l l l l l Whole Number Meaning G2 l l l l l l l l l l l l l l l l l l l l l G4 G5 G6 G7 G8 l l l l l l l l l l l l l G3 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Decimal Fractions Relationship of Common & Decimal l Fractions l l l l l l l l Proportionality Problems l l l 2-D Coordinate Geometry l l l Geometry: Transformations l l l Negative Numbers, Integers & l Their l Properties l Percentages l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Exponents & Orders of Magnitude Measurement Estimation & Errors l l l l l l l l l 2-D Coordinate Geometry l l l l l l l l l l l l l l l l l l l l l l l l l l l l Congruence & Similarity Rational Numbers & Their Properties Patterns, Relations & Functionsl l l l l l l l l l l l l l l l l l l l Measurement Units Common Fractions l l l l l G2 G3 G4 G5 G6 G7 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Data Representation & Analysis l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Whole Number Operations l l l l l l l l l l l l l l l l l l l Measurement Units Common Fractions Equations & Formulas 2-D Geometry: Basics Polygons & Circles Perimeter, Area & Volume l l l l Exponents, Roots & Radicals l l l l l l l l l l l l l l l l l l l l l l l l Rounding & Significant Figures Estimating Computations 3-D Geometry l l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl l l l l l l l l l l l l l l l l Slope & Trigonometry Topics 1121 l 162 171 l 132 133 122 1153 l l l l l l l l l 1122 1123 G3 Decimal Fractions G5 G6 G7 G8 l l l l l l l l l l l l l l Relationship of Common & Decimal Fractions l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 131 2-D Coordinate Geometry 141 Geometry: Transformations l l l l l l l l l l l l l l l l l l l l l l 1131 Negative Numbers, Integers & Their Properties 1144 Number Theory l l l l l l l l 1142 Exponents, Roots & Radicals 1154 Exponents & Orders of Magnitude 123 Measurement Estimation & Errors l l l l l l 134 3-D Geometry 142 Congruence & Similarity l l l l 1132 Rational Numbers & Their Properties l l 161 Patterns, Relations & Functionsl l l 153 Slope & Trigonometry G3 G4 G5 G6 G7 G8 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Percentages Topics Whole Number Meaning Whole Number Operations Measurement Units G2 G3 G4 G5 G6 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Common Fractions Equations & Formulas l Data Representation & Analysis l 2-D Geometry: Basics Polygons & Circles l l l l l Perimeter, Area & Volume l Estimating Computations Properties of Whole Number Operations l Estimating Quantity & Size l l l l Proportionality Concepts Proportionality Problems l l l l l l l l l l l l l l l l l l Decimal Fractions l l l l l l l l l l l l l l l l l l l l l Exponents, Roots & Radicals Exponents & Orders of Magnitude l Measurement Estimation & Errors l l l l l l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl l l l l l l l l l l l l l l l l l l l l l l l l l l G7 l Measurement Estimation & Errors 3-D Geometry Congruence & Similarity l l l l l l l l l l Rational Numbers & Their Properties Patterns, Relations & Functionsl l Slope & Trigonometry l Topics G1 G2 G3 G4 G5 G6 G7 G8 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 1131 Negative Numbers, Integers & Their Properties l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 1111 Whole Number Meaning 1112 Whole Number Operations 121 Measurement Units 1121 Common Fractions 162 Equations & Formulas 171 Data Representation & Analysis l 132 2-D Geometry: Basics 133 Polygons & Circles Geometry: Transformations l l 122 Perimeter, Area & Volume 1152 Rounding & Significant Figures 1153 Estimating Computations 1113 Properties of Whole Number Operations l l 1151 Estimating Quantity & Size 1122 Decimal Fractions 1123 Relationship of Common & Decimal Fractions 1125 Properties of Common & Decimal Fractions 1124 Percentages 151 Proportionality Concepts 152 Proportionality Problems 131 2-D Coordinate Geometry 141 Geometry: Transformations l l l l l l l l 1144 Number Theory 1142 Exponents, Roots & Radicals 1154 Exponents & Orders of Magnitude 123 Measurement Estimation & Errors l 143 Constructions w/ Straightedge & Compass 134 3-D Geometry l l l 142 Congruence & Similarity 1132 Rational Numbers & Their Properties 161 Patterns, Relations & Functionsl l l l l l l l l l l l l l l 153 Slope & Trigonometry l l l l l l l Negative Numbers, Integers & Their l Properties l l Exponents, Roots & Radicals l l l Exponents & Orders of Magnitude Measurement Estimation & Errors l l Constructions w/ Straightedge & Compass l Congruence & Similarity l l l l l l l l l l Rational Numbers & Their Properties Patterns, Relations & Functions l Topics G1 G2 G3 G4 G5 G6 G7 G8 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 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Whole Number Meaning Whole Number Operations l l l l l l l l l l l l l l l l l l l l l l Relationship of Common & Decimal l Fractions l l Estimating Computations l Decimal Fractions l l l l l Properties of Common & Decimal Fractions l Proportionality Problems © 2009 Michigan State University, Center for Research in Mathematics and Science Education l l Properties of Whole Number Operations l l l l Proportionality Concepts Slope & Trigonometry l l l l l Constructions w/ Straightedge l & Compass Rounding & Significant Figures Percentages 3-D Geometry l Exponents & Orders of Magnitude G8 Properties of Common & Decimal Fractions Number Theory l l l l l l l l l Estimating Quantity & Size Relationship of Common & Decimal Fractions 2-D Coordinate Geometry l l State T G1 Rounding & Significant Figures l l l l l Exponents, Roots & Radicals State N G2 l l l l l l l Proportionality Concepts l Proportionality Problems l l l 2-D Coordinate Geometry l l Geometry: Transformations l l Negative Numbers, Integers & Their Properties l Number Theory Percentages l l l l l l l l l l l l l G4 l l l l l l l l l l l l l l 152 Proportionality Problems G1 l l l l l Polygons & Circles l l l Perimeter, Area & Volume l l l Rounding & Significant Figuresl l l Estimating Computations l l l Properties of Whole Number Operations l l l Estimating Quantity & Size l l l Decimal Fractions l l l Relationship of Common & Decimal l Fractions l l Geometry: Transformations G2 l l l l Measurement Units l l Common Fractions l l Equations & Formulas l l Data Representation & Analysis l l 2-D Geometry: Basics l l Polygons & Circles l l Perimeter, Area & Volume l l Rounding & Significant Figuresl l Estimating Computations l l Properties of Whole Number Operations l l Estimating Quantity & Size l l Decimal Fractions l l 151 Proportionality Concepts l l l l l l l l l l State O G1 l 1124 Percentages l l G8 l l Properties of Whole Number Operations l Fractions l Relationship of Common & Decimal l l l l l Exponents & Orders of Magnitude l Measurement Estimation & Errors l l l Estimating Quantity & Size Number Theory 1151 l 2-D Geometry: Basics Slope & Trigonometry G1 Whole Number Meaning Negative Numbers, Integers & Their Properties 121 l l l Data Representation & Analysis l Congruence & Similarity Topics l 143 Constructions w/ Straightedge & Compass Equations & Formulas 3-D Geometry l Geometry: Transformations 1152 l Constructions w/ Straightedge & Compass l l 1112 Whole Number Operations l Negative Numbers, Integers & Their Properties Constructions w/ Straightedge & Compass G8 l 1125 Properties of Common & Decimal Fractions Number Theory Exponents, Roots & Radicals G7 l l l l l l l l l l l l l Fractions l Properties of Common & Decimal Proportionality Problems 2-D Coordinate Geometry l G6 l l l l l l l l l l l l Proportionality Concepts l 1113 l Properties of Common & Decimal Fractions Number Theory G5 l l l l l l l l l l l l Percentages 1111 Whole Number Meaning l l Whole Number Operations l l l l l l l l l G4 Properties of Common & Decimal Fractions State J G4 G3 State I G1 161Patterns, Relations & Functionsl 153Slope & Trigonometry l l l l l l l l l l l l Congruence & Similarity l 142Congruence & Similarity 1132Rational Numbers & Their Properties l G2 l l Constructions w/ Straightedge & Compass 143Constructions w/ Straightedge & Compass 1343-D Geometry l l l G3 l Polygons & Circles l Perimeter, Area & Volume l Rounding & Significant Figuresl Estimating Computations l Slope & Trigonometry l 1154Exponents & Orders of Magnitude 123Measurement Estimation & Errors l l l l Data Representation & Analysis l 3-D Geometry l l l 1131Negative Numbers, Integers & Their Properties l l 1144Number Theory l l 1142Exponents, Roots & Radicals G2 Proportionality Concepts l l l l l l l l l l l l l l l l l l 1125Properties of Common & Decimal Fractions 1124Percentages Properties of Common & Decimal Fractions l l l l l l Proportionality Concepts l l l 121Measurement Units l 1121Common Fractions l 162Equations & Formulas l 171Data Representation & Analysis l l l 1322-D Geometry: Basics l 133Polygons & Circles l l l 122Perimeter, Area & Volume l 1152Rounding & Significant Figuresl l 1153Estimating Computations l l l 1113Properties of Whole Number Operations l l 1151Estimating Quantity & Size l 1122Decimal Fractions l l 1123Relationship of Common & Decimal Fractions l l l l 2-D Geometry: Basics l l l l l l l Slope & Trigonometry Properties of Whole Number Operations l l l l Patterns, Relations & Functionsl Estimating Quantity & Size l l l Rational Numbers & Their Properties G1 Common Fractions l l l Congruence & Similarity State R G5 l State S G1 l l l Measurement Units l l Common Fractions l l Equations & Formulas l l Data Representation & Analysis l l 2-D Geometry: Basics l l Polygons & Circles l l Perimeter, Area & Volume l Rounding & Significant Figures l l l Estimating Computations l Properties of Whole Number Operations l l Estimating Quantity & Size l Decimal Fractions l Relationship of Common & Decimal Fractions Whole Number Operations Constructions w/ Straightedge & Compass Slope & Trigonometry l l l l l l Measurement Estimation & Errors l l l l l l Exponents & Orders of Magnitude G4 l Exponents & Orders of Magnitude l l l l l l Exponents, Roots & Radicals l l l l l Topics Whole Number Meaning Exponents, Roots & Radicals 1111Whole Number Meaning 1112Whole Number Operations Negative Numbers, Integers & Their Properties G3 l l l l l l l l l l l l l l l Geometry: Transformations l l l Negative Numbers, Integers & l Their l Properties Proportionality Problems l l l l Geometry: Transformations l l l l Properties of Common & Decimal Fractions G2 2-D Coordinate Geometry l l l l l l l Estimating Quantity & Size l l l Decimal Fractions l l l Relationship of Common & Decimal Fractions l l l l l l l Measurement Units l l l l l l Perimeter, Area & Volume G1 Topics l Data Representation & Analysis l Properties of Whole Number Operations l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Common Fractions Estimating Computations G8 l l l l l l l l State E G1 Whole Number Meaning Polygons & Circles G7 l l l l l l l l 2-D Coordinate Geometry G8 Topics 2-D Geometry: Basics G6 l l l l l l l l Number Theory l l l l l l l l l l l Equations & Formulas G5 Proportionality Problems l l l l l l l l l G8 Measurement Units l l l l l l l l l l l l Proportionality Concepts l l l l l l l l l l l l l l l l l l l l Whole Number Operations G4 l l l l l l l l l l l l l Fractions l l Relationship of Common & Decimal G7 Geometry: Transformations Slope & Trigonometry l l l l l l l l l l l l 2-D Coordinate Geometry Congruence & Similarity l G6 Proportionality Problems 3-D Geometry l Rational Numbers & Their Properties Patterns, Relations & Functionsl G3 l Fractions l l Properties of Common & Decimal l l l l Constructions w/ Straightedge l & Compass l l l 3-D Geometry l l l Congruence & Similarity l l l l G2 l l l l l l Measurement Units l l Common Fractions l l Equations & Formulas l l Data Representation & Analysis l l 2-D Geometry: Basics l l Polygons & Circles l l Perimeter, Area & Volume l Rounding & Significant Figuresl l l Estimating Computations l l Properties of Whole Number Operations Whole Number Operations Estimating Quantity & Size l l l l l l l l l l l Percentages l l l l l l l l Measurement Estimation & Errors Properties of Common & Decimal Fractions l l l l l l l l l l l l l l l G5 l Relationship of Common & Decimal Fractions l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Percentages Common Fractions l l l l l l l State D G1 Whole Number Meaning G4 l l l l l l Measurement Units l l l Common Fractions l l l Equations & Formulas l l l Data Representation & Analysis l l l 2-D Geometry: Basics l l l Polygons & Circles l l l Perimeter, Area & Volume l l l Rounding & Significant Figuresl l l Estimating Computations l l l Properties of Whole Number Operations l l l Estimating Quantity & Size l l l Decimal Fractions l l l Relationship of Common & Decimal l Fractions l l Properties of Common & Decimal l Fractions l l Whole Number Operations l l l l l l Topics State Q G1 Whole Number Operations Whole Number Meaning l l l l l l l l l Exponents & Orders of Magnitude State M G1 G8 l l l l l l l l Slope & Trigonometry Proportionality Concepts l l l l l G7 l l l l l l l l l Percentages State L Topics G6 l l l l l l l l l Properties of Common & Decimal Fractions Negative Numbers, Integers & l Their l Properties l l l l l l l l l l l l l l l l l l l l l l Slope & Trigonometry l l 2-D Coordinate Geometry G8 G3 l l l l l l l l l l Congruence & Similarity l Proportionality Problems l l l l l l l l l l l l G2 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 l Rounding & Significant Figuresl G5 l l l l l l l l l l l l l l Estimating Quantity & Size l l l Decimal Fractions l Fractions l l Relationship of Common & Decimal Proportionality Concepts G7 G1 Whole Number Operations 3-D Geometry l Percentages 3-D Geometry G4 l l l l l l l l l Estimating Computations Constructions w/ Straightedge & Compass l l G3 l l l l l l l l l l l Properties of Whole Number Operations Decimal Fractions Geometry: Transformations 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 Whole Number Operations Rounding & Significant Figures l Exponents & Orders of Magnitude Whole Number Meaning l l l l l l l l l l l l l 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 l l l l Exponents & Orders of Magnitude Rounding & Significant Figures Properties of Whole Number Operations l l l l l l l l l l l l l l l l l l l l l l l l Properties of Common & Decimal Fractions 2-D Coordinate Geometry l G8 l l l l l l l l G6 l l l l l l l l G7 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l G6 l l l l l l l l l G5 l l l l l l l l G5 l l l l l l l l l l l l l l l l l l l 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 l l l l l l G4 l l l l l l l l l G4 Topics l l l l l l l l l 2-D Geometry: Basics Estimating Computations Patterns, Relations & Functionsl G8 State C Perimeter, Area & Volume l l l l l l l l l l Rational Numbers & Their Properties l l l l l l l l 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 l 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 l l State G G1 Whole Number Meaning Geometry: Transformations l l l l l l 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 Topics for Which All States Intend or Don’t Intend Coverage Grade Topic 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 1 2 3 m m m m m m m m m m m m m m m m 4 5 6 7 8 Intended by all the States l Intended by none of the states m m m m © 2009 Michigan State University, Center for Research in Mathematics and Science Education 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 © 2009 Michigan State University, Center for Research in Mathematics and Science Education Grade 8: Algebra N=254 Teachers Coverage of Mathematics TopicsSchool in Two 4 Types of School 1 Schools Low SES Schools High SES Schools Arithematic School 1 Geometry Algebra Other Arithematic Geometry Algebra Other © 2009 Michigan State University, Center for Research in Mathematics and Science Education Diagram of Mathematics Course-Taking Sequences in District F Calculus I CP Analytic. Geom. CP Trig. I CP Discrete Math Statistics CP Geometry 2 CP Geometry 1 CP Algebra 1 Calculus II CP CP Algebra 3 CP Algebra 2 Basic Geometry 1 CP Algebra 4 Basic Geometry 2 Basic Algebra 4 Basic Algebra 3 Basic Algebra 2 Basic Algebra 1 Integrated Math 4 Integrated Math 3 Integrated Math 2 Integrated Math 1 General Math 4 General Math 1 General Math 2 General Math 3 © 2009 Michigan State University, Center for Research in Mathematics and Science Education Minimal Standards? • One course in each: Mathematics, Biology, Chemistry, Physics • Algebra II/Biology/Chemistry/Physics • Less than 30 % of ALL Students • Less than 5 % of Voc/Tech Students • About 15 % of General Academic Students • About 40 % of College Preparatory Students © 2009 Michigan State University, Center for Research in Mathematics and Science Education Are All Algebra I Courses the Same? • • • • • • • • • • • • • Fundamentals of Algebra Advanced Algebra Algebra Advanced Honors Algebra I Honors Algebra I Algebra I CP Algebra IA Algebra IB Algebra IC Basic Algebra I Basic Algebra IA Basic Algebra IB Basic Algebra IC • Adult Ed. Algebra • Adult Ed. Adv. Algebra • Applied Algebra • Elementary Algebra • Essentials of Algebra • Introduction to Algebra • Enrich. Algebra • General Algebra • Modern Algebra © 2008 MSU PROM/SE Promoting Rigorous Outcomes in Mathematics and Science Education, Supported by NSF Cooperative Agreement EHR-0314866 © 2009 Michigan State University, Center for Research in Mathematics and Science Education Are All General Mathematics/PreAlgebra Courses the Same? • • • • • • • • • • • 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 Last Math © 2008 MSU PROM/SE Promoting Rigorous Outcomes in Mathematics and Science Education, Supported by NSF Cooperative Agreement EHR-0314866 © 2009 Michigan State University, Center for Research in Mathematics and Science Education Relative Emphasis in Mathematics, Math Ed, and General Pedagogy Weighted by Time in Teacher Preparation Taiwan USA 1900 Germany 2000 Mathemathics Mathematics Education General Pedagogy © 2009 Michigan State University, Center for Research in Mathematics and Science Education 2700
