**Suggested Citation:**"Notes and Credits." National Research Council. 2002.

*Helping Children Learn Mathematics*. Washington, DC: The National Academies Press. doi: 10.17226/10434.

**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].

**Suggested Citation:**"Notes and Credits." National Research Council. 2002.

*Helping Children Learn Mathematics*. Washington, DC: The National Academies Press. doi: 10.17226/10434.

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.

**Suggested Citation:**"Notes and Credits." National Research Council. 2002.

*Helping Children Learn Mathematics*. Washington, DC: The National Academies Press. doi: 10.17226/10434.

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