Title page
Contents
Introduction to Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades 8
Recommendation 1: Systematic Instruction 12
Provide systematic instruction during intervention to develop student understanding of mathematical ideas 12
Recommendation 2: Mathematical Language 18
Teach clear and concise mathematical language and support students' use of the language to help students effectively communicate their understanding of mathematical concepts 18
Recommendation 3: Representations 28
Use a well-chosen set of concrete and semi-concrete representations to support students' learning of mathematical concepts and procedures 28
Recommendation 4: Number Lines 36
Use the number line to facilitate the learning of mathematical concepts and procedures, build understanding of grade-level material, and prepare students for advanced mathematics 36
Recommendation 5: Word Problems 47
Provide deliberate instruction on word problems to deepen students' mathematical understanding and support their capacity to apply mathematical ideas 47
Recommendation 6: Timed Activities 58
Regularly include timed activities as one way to build students' fluency in mathematics 58
Glossary 63
Appendix A: Postscript from the Institute of Education Sciences 65
Appendix B: Methods and Processes for Developing This Practice Guide 69
Appendix C: Rationale for Evidence Ratings 72
Appendix D: About the Panel and Key WWC Staff 138
Appendix E: Disclosure of Potential Conflicts of Interest 142
References 143
Notes 151
Table 1. Recommendations and corresponding levels of evidence 10
Table 2.1. Example word list that can be used across settings in grades K-6 by all teachers in the school 21
Table 2.2. A mathematical language chart that supports early elementary (grade K-2) students as they use mathematical language to present their thinking 26
Table 2.3. A mathematical language chart that supports upper elementary (grade 3-6) students as they use mathematical language to present their thinking 27
Table 3.1. Examples of common concrete and semi-concrete representations that can be used for a sample of mathematics concepts and procedures 30
Table 5.1. Clarify words presented in word problems prior to students solving the problem 55
Table 5.2. Examples of key words matched to an operation and why they fail 57
Table 6.1. Examples of activities that can support fluency for various intervention topics 59
Boxes
Box 1. Levels of evidence 9
Examples
Example 1.1. Putting together the steps of Recommendation 1 15
Example 2.1. Graphic organizer that depicts a student-friendly definition, characteristics, examples, and non-examples for the term unit fraction 19
Example 2.2. Concrete representation used to build students' understanding of the meaning of equal and the equal sign symbol in early elementary school (grades K-2) 20
Example 2.3. Role-playing with hand gestures that teach the meaning of mathematical ideas or vocabulary 21
Example 2.4. Teacher using mathematical vocabulary when thinking aloud during mathematics intervention in upper elementary (grades 3-6) 22
Example 2.5. Teacher leads an instructional activity to broaden students' understanding of the terms factor and product 24
Example 2.6. Teacher prompts students to use mathematical terminology in their explanations 25
Example 3.1. Teacher represents the addition problem with base 10 blocks, which are proportional for showing place value and regrouping concepts 31
Example 3.2. Teacher shows how combining two groups (a group of 4 and a group of 5) relates to concrete and semi-concrete representations and to an equation 32
Example 3.3. Teacher explains how to use base 10 blocks, with which the students are already familiar, to solve addition and subtraction problems with decimals 33
Example 4.1. Number line representing magnitudes of whole, positive, negative, rational, and irrational numbers 36
Example 4.2. Connecting individual concrete units to a number line to represent positive whole numbers 37
Example 4.3. Number line with halves, fourths, fifths, and eighths 39
Example 4.4. Fractions equal to, greater than, and less than 1 39
Example 4.5. Equivalent fractions are positioned at the same point on the number line 40
Example 4.6. Connecting a concrete representation of a length to a number line 40
Example 4.7. Label tick marks that represent the same equivalences vertically at the same position on the number line, rather than side by side 41
Example 4.8. Use number lines to teach the relative magnitude of whole numbers in early elementary (grades K-2) 41
Example 4.9. Students estimate the location of four fractions using benchmark numbers and places the flashcards on the 0-1 number line 42
Example 4.10. Show early elementary (grades K-2) students how to use number lines to add and subtract whole numbers 43
Example 4.11. Use the number line to show students fraction addition 44
Example 4.12. Multiplication with a fraction and a whole number 45
Example 4.13. Division with a fraction and a whole number 45
Example 5.1. Introducing a Change problem 49
Example 5.2. Upper elementary (grade 3-6) teacher thinking aloud how she sets up and solves an Equal Groups problem using a prompt card 50
Example 5.3. Problem types with less familiar features 52
Example 5.4. Teacher guides students through identifying relevant information and using a concrete representation to visualize the story 53
Example 6.1. Graph tracking scores for timed fluency activities 61
Table A.1. IES levels of evidence for What Works Clearinghouse practice guides 67
Table C.1. Mapping between studies and recommendations 72
Table C.2. Relevant domains for each recommendation 74
Table C.3. Domain-level effect sizes across the 43 studies supporting Recommendation 1 76
Table C.4. Studies providing evidence for Recommendation 1: Provide systematic instruction during intervention to develop student understanding of mathematical ideas 78
Table C.5. Domain-level effect sizes across the 16 studies supporting Recommendation 2 93
Table C.6. Studies providing evidence for Recommendation 2: Teach clear and concise mathematical language and support students' use of the language to help students effectively communicate... 95
Table C.7. Domain-level effect sizes across the 28 studies supporting Recommendation 3 100
Table C.8. Studies providing evidence for Recommendation 3: Use a well-chosen set of concrete and semi-concrete representations to support students' learning of mathematical concepts and... 102
Table C.9. Domain-level effect sizes across the 14 studies supporting Recommendation 4 111
Table C.10. Studies providing evidence for Recommendation 4: Use the number line to facilitate the learning of mathematical concepts and procedures, build understanding of grade-level material,... 113
Table C.11. Domain-level effect sizes across the 18 studies supporting Recommendation 5 118
Table C.12. Studies providing evidence for Recommendation 5: Provide deliberate instruction on word problems to deepen students' mathematical understanding and support their capacity to apply... 120
Table C.13. Domain-level effect sizes across the 27 studies supporting Recommendation 6 127
Table C.14. Studies providing evidence for Recommendation 6: Regularly include timed activities to build students' retrieval of basic facts and fluent use of critical steps for more complex mathematics 129
Figure B.1. Studies identified, screened, and reviewed for this practice guide 70