1. For more details see Chapter 2 “The State of School Mathematics in the United States” (pp. 31– 70) in: National Research Council. (2001). *Adding it up: Helping children learn mathematics**.* Mathematics Learning Study Committee, J.Kilpatrick, J.Swafford, B.Findell, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

2. For information on the Third International Mathematics and Science Study benchmarking studies, visit http://www.nces.ed.gov/timss/timss95/benchmark.asp [September 25, 2001].

*See also* Reese, C.M., Miller, K.E., Mazzeo, J., & Dossey, J.A. (1997). *NAEP 1996 mathematics report card for the nation and the states* (NCES 97–488). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97488.shtml [September 25, 2001].

*See also* Campbell, J.R., Voelkl, K.E., & Donahue, P.L. (2000). *NAEP 1996 trends in academic progress* (NCES 97–985r). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97985r.shtml [September 25, 2001].

3. In the full report, the five strands are labeled somewhat differently: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Although the labels have been simplified in this report, their descriptions reflect those in the full report.

4. For a fuller discussion of this research, see the section on Calculators (pp. 354–356) in Chapter 9 “Teaching for Mathematical Proficiency” in *Adding It Up*.

5. See the section on Subtraction Algorithms (pp. 204–206) in Chapter 6 “Developing Proficiency with Whole Numbers” in *Adding It Up*.

6. For a review of the research and a more detailed discussion of how children learn basic number combinations, see Chapter 6 “Developing Proficiency with Whole Numbers” (pp. 181–229) in *Adding It Up*.

7. Ben-Chaim, D., Fey, J.T., Fitzgerald, W.M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences. *Educational Studies in Mathematics**,* *36**,* 247–273.

8. Langrall, C.W., & Swafford, J.O. (2000). Three balloons for two dollars: Developing proportional reasoning. *Mathematics Teaching in the Middle School**,* *6**,* 254–261.

9. National Research Council. (1998). *Preventing reading difficulties in young children**.* Committee on the Prevention of Reading Difficulties in Young Children, C.E.Snow, M.S.Burns, and P.Griffin, Editors. Commission on Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/6023.html

*See also* Silver, E.A., & Kenney, P.A. (2000). *Results from the seventh mathematics assessment of the National Assessment of Educational Progress**.* Reston, VA: National Council of Teachers of Mathematics.

*See also* Thorndike, E.L. (1922). *The psychology of arithmetic**.* New York: Macmillan.

10. Campbell, J.R., Voelkl, K.E., & Donahue, P.L. (2000). *NAEP 1996 trends in academic progress* (NCES 97–985r). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97985r.shtml [September 25, 2001].

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NOTES
1. For more details see Chapter 2 “The State of School Mathematics in the United States” (pp. 31– 70) in: National Research Council. (2001). Adding it up: Helping children learn mathematics. Mathematics Learning Study Committee, J.Kilpatrick, J.Swafford, B.Findell, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
2. For information on the Third International Mathematics and Science Study benchmarking studies, visit http://www.nces.ed.gov/timss/timss95/benchmark.asp [September 25, 2001].
See also Reese, C.M., Miller, K.E., Mazzeo, J., & Dossey, J.A. (1997). NAEP 1996 mathematics report card for the nation and the states (NCES 97–488). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97488.shtml [September 25, 2001].
See also Campbell, J.R., Voelkl, K.E., & Donahue, P.L. (2000). NAEP 1996 trends in academic progress (NCES 97–985r). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97985r.shtml [September 25, 2001].
3. In the full report, the five strands are labeled somewhat differently: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Although the labels have been simplified in this report, their descriptions reflect those in the full report.
4. For a fuller discussion of this research, see the section on Calculators (pp. 354–356) in Chapter 9 “Teaching for Mathematical Proficiency” in Adding It Up.
5. See the section on Subtraction Algorithms (pp. 204–206) in Chapter 6 “Developing Proficiency with Whole Numbers” in Adding It Up.
6. For a review of the research and a more detailed discussion of how children learn basic number combinations, see Chapter 6 “Developing Proficiency with Whole Numbers” (pp. 181–229) in Adding It Up.
7. Ben-Chaim, D., Fey, J.T., Fitzgerald, W.M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences. Educational Studies in Mathematics, 36, 247–273.
8. Langrall, C.W., & Swafford, J.O. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in the Middle School, 6, 254–261.
9. National Research Council. (1998). Preventing reading difficulties in young children. Committee on the Prevention of Reading Difficulties in Young Children, C.E.Snow, M.S.Burns, and P.Griffin, Editors. Commission on Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/6023.html
See also Silver, E.A., & Kenney, P.A. (2000). Results from the seventh mathematics assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.
See also Thorndike, E.L. (1922). The psychology of arithmetic. New York: Macmillan.
10. Campbell, J.R., Voelkl, K.E., & Donahue, P.L. (2000). NAEP 1996 trends in academic progress (NCES 97–985r). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97985r.shtml [September 25, 2001].

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11. Knapp, M.S., Shields, P.M., & Turnbull, B.J. (1995, June). Academic challenge in high-poverty classrooms. Phi Delta Kappan, 76, 770–776.
12. See pages 315–333 of Chapter 9 “Teaching for Mathematical Proficiency” in Adding It Up for four classroom vignettes illustrating different images of mathematics instruction.
13. See the discussion of Communities of Learners (pp. 344–345) in Chapter 9 “Teaching for Mathematical Proficiency” in Adding It Up.
14. For a fuller discussion of the research, see the section on Cooperative Groups (pp. 348–349) in Chapter 9 “Teaching for Mathematical Proficiency” in Adding It Up.
15. For a fuller discussion of the research, see the section on Grouping (pp. 346–348) in Chapter 9 “Teaching for Mathematical Proficiency” in Adding It Up.
16. See the section on Instructional Programs and Materials (pp. 36–39) in Chapter 2 “The State of School Mathematics in the United States” in Adding It Up.
17. See the section on Manipulatives (pp. 353–354) in Chapter 9 “Teaching for Mathematical Proficiency” in Adding It Up.
18. See the section on Assessments (pp. 39–44) in Chapter 2 “The State of School Mathematics in the United States” in Adding It Up.
19. See the section on Knowledge of Mathematics (pp. 372–378) in Chapter 10 “Developing Proficiency in Teaching Mathematics” in Adding It Up.
20. See the section on Programs to Develop Proficient Teaching (pp. 385–397) in Chapter 10 “Developing Proficiency in Teaching Mathematics” in Adding It Up for a description of four different approaches to professional development and teacher education.

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