Perimeter, Area, Volume, and All That: A Study of Measurement

Endnotes

1. John A. Van de Walle, Karen S. Karp, Jennifer M. Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally (Boston: Pearson Education, Inc., 2001), 306.
2. Elida V. Laski, Jamilah R. Joedan, Crolyn Daust, and Angela K. Murray, “What Makes Mathematics Manipulatives Effective? Lessons from Cognitive Science and Montessori Education,” SAGE Open (April 2015): 1-4.
3. Alexander Karp and Nicholas Wasserman, Mathematics in Middle and Secondary School: A Problem Solving Approach (Charlotte, NC: Information Age Publishing, 2014), 319.
4. Sybilla Beckmann, Mathematics for Elementary Teachers (Boston: Pearson Education, Inc., 2017), 5: 492.
5. Kira J. Carbonneau, Scott Marley, and James P. Selig, “A Metanalysis of the Efficacy of Teaching Mathematics with Concrete Manipulatives,” Journal of Educational Psychology 105, no. 2 (2013): 380-400.
6. Virginia Board of Education, Mathematics 2016 Standards of Learning: Grade 6 Curriculum Framework (Virginia Board of Education, 2016), np.
7. Roger Howe, “Yale National Institute Lecture: Measurement, Scaling, and Dimension” (lecture given at the Yale National Institute, Yale University, New Haven, CT, July 15, 2019).
8. Aryn A. Siegel and Enrique Ortiz, “Perimeter and Beyond,” Teaching Children Mathematics, 19, no. 1 (August 2012): 39, National Council of Teachers of Mathematics.
9. Siegel and Ortiz, 40-41.
10. Ibid.
11. Van de Walle and Lovin, 283.
12. Ibid.
13. Ibid.
14. Ibid.
15. Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry (New York: Dover, 2004), 133.
16. Eun Mi Kim et al., “A Learning Progression for Geometrical Measurement in One, Two, and Three Dimension,” Research Report (December 2017): 2.
17. Ibid.
18. Kim et al., 2.
19. Kim et al., 14.
20. Kira J. Carbonneau, Scott C. Marley, and James P. Selig, “A Metaanalysis of the Efficacy of Teaching Mathematics with Concrete Manipulatives,” Journal of Educational Psychology 105 (2013): 380-400.
21. Jo Boaler, Lang Chen, Cathy Williams and Montserrat Cordero, “Seeing as Understanding: The Importance of Visual Mathematics for Our Brain and Learning,” Journal of Applied and Computational Mathematics, 5 (2016): 1.
22. Jo Boaler, Lang Chen, Cathy Williams and Montserrat Cordero, “Seeing as Understanding: The Importance of Visual Mathematics for Our Brain and Learning,” Journal of Applied and Computational Mathematics, 5 (2016): 1.
23. Laski et al.
24. Ibid.
25. Elida V. Laski, Jamilah R. Jordan, Carolyn Daoust, and Angela K. Murray, “What Makes Mathematics Manipulatives Effective? Lessons from Cognitive Science and Montessori Education,” SAGE Open, (April 2015): 1.
26. For polyomino activities related specifically to perimeter and area, see Henri Piccioto’s section “Perimeter and Area” in Polyomino Lessons (1986) at www.MathEducationPage.org.
27. In a draft of this unit with his comments within, Roger Howe provided the definition quoted here. This definition is derived from Golomb’s 1965 work Polyominoes. Roger Howe, email correspondence with the author, August 3, 2019. See generally Solomon W. Golomb, Polyominoes (New York: Scribners, 1965).
28. Y. Yuan, Chun-Yi Lee, and C-H Wang, “A Comparison Study of Polyominoes Explorations in a Physical and Virtual Manipulative Environment,” Journal of Computer Assisted Learning 26, no. 4 (2010): 307-316.
29. Stephanie Bell, “Project-Based Learning for the 21^st Century: Skills for the Future,” The Clearing House: A Journal of Educational Strategies, 83, no. 2 (2010): 39-43.
30. Dorothy Varygiannes, "The Impact of Open-Ended Tasks," Teaching Children Mathematics 20, no. 5 (2014): 277-80. 
31. Varygiannes, 278.
32. Aryn Siegel and Enrique Ortiz, “Perimeter and Beyond,” Teaching Children Mathematics 19, no. 1 (August 2012): 39.
33. Kim et al., 14.