Definitions of Volume from Math Ed Textbooks

In a side project of Strengthening Tomorrow’s Education in Measurement data, I analyzed definitions from elementary education mathematics content textbooks. I began with the list of content textbooks from McCrory (2006) and found updated versions when possible – some textbooks were out of print and others were not accessible to me. (see references section for list of textbooks – I was able to find copies of the textbooks with a number in brackets on the left side, and others are greyed out.) Definitions across textbooks varied, especially the level and focus.

Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.

• p. 608 – Over time, two related systems of measure for what we call volume evolved, depending on whether the thing being measured was dry or wet.
• p. 612 – volume (dry) and capacity (liquid);
• p. 637- the amount of space contained within that object
• p. 641 – volume reflects how much it will take to fill an object

Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.

• p. 153 – The volume, in cubic units, of a box or box shape is just the number of 1-unit-by-1-unit-by-1-unit cubes that it would take to fill the box or make the box shape (without gaps or overlaps).
• p. 483 – A volume describes the size of an object (or a part of an object) that is three-dimensional; the volume of that three-dimensional object is how many of a chosen unit of volume (such as cubic inches, cubic centimeters, etc.) it takes to fill the object without gaps or overlaps, where it is understood that we may use parts of a unit, too. Roughly speaking, an object is three-dimensional if at each location there are three independent directions along which to move within the object.

Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math.

• p. 701 – To measure the amount of space in tanks, buildings, refrigerators, cars and other three-dimensional figures, we need units of measure that are also three-dimensional figures. The number of such units needed to fill a figure is its volume. Cubes are convenient because they pack together without gaps or overlapping.

Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.

• p. 906 – volume describes how much space a three-dimensional figure contains. The unit of measure for volume must be a shape that tessellates space. Cubes tessellates space; that is, they can be stacked so that they leave no gaps and fill space. Standard units of volume are based on cubes and are cubic units. A cubic unit is the amount of space enclosed within a cue that measures 1 unit on a side.

Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.

• p. 558 – Any real object takes up space. Even a scrap of paper, a speck of dust, or a strand of cobweb takes up some space. The quantity of space occupied is called the volume. As with the terms length and area, the term volume is sometimes used for the attribute as well as the measurement of the attribute.
• p. G-16 – the number of 3D units that could fit inside the region, or that could be used to make an exact model of the region. The units are usually cubic regions (e.g., cm^3).

Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.

• p. 530 – Volume is the measure of space taken up by a solid three-dimensional space. The unit… is the volume of a cube whose side length is one of the standard units of length.

Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.

• p. 680 – The volume of the vase would be the amount of material comprising the vase itself.

Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.

• p. 193 – volume of a solid is the number of unit cubes needed to fill the solid
• p. 193 – Volume measurements come in three different forms: liquid, solid, and air/space.  Each has unique features that make it easy or hard to measure.
1. The volume of a Liquid is easily measured – use a measuring cup.
2. The volume of a Solid is easily measured for rectangular solids, but volume is not intuitive or easy to measure for irregular solids.
3. The volume of Air/Space is neither intuitive nor easily measured.

p. 198 – [image provided illustrating the 5 types of measurement that is usually studied in elementary school and their relationships – I just thought the Volume/Capacity is nicely shown]

Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.

• p. 571 – Volume is the amount of space occupied by a three-dimensional figure. Volume is usually measured in cubic units. Whereas surface area is the total area of the faces of a solid, volume is the capacity of a solid.

Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education, Inc.

• p. 391 – Volume and capacity are both terms for measures of the “size” of three-dimensional regions…The term volume can be used to refer to the capacity of a container but is also used for the size of solid objects. Standard units of volume are expressed in terms of length units, such as cubic inches or cubic centimeters.

References

McCrory, R. (2006). Mathematicians and mathematics textbooks for prospective elementary teachers. Notices of the AMS, 51(1), 20–29.
List of Content Textbooks

[1] Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.

[2] Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.

[3] Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math.

[4] Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.

[5] Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.

Darken, B. (2003). Fundamental mathematics for elementary and middle school teachers: Kendall/Hunt.

Devine, D. F., Olson, J., & Olson, M. (1991). Elementary mathematics for teachers. New York, NY: Wiley. (NO LONGER IN PRINT)

Jensen, G. R. (2003). Arithmetic for teachers: With applications and topics from geometry: American Mathematical Society.

Jones, P., Lopez, K. D., & Price, L. E. (1998). A mathematical foundation for elementary teachers. New York: Addison Wesley Higher Education.

Krause, E. F. (1991). Mathematics for elementary teachers: A balanced approach. Lexington, MA: D.C. Heath and Company. (NO LONGER IN PRINT)

[6] Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.

Masingila, J. O., Lester, F. K., & Raymond, A. M. (2002). Mathematics for elementary teachers via problem solving: Prentice Hall.

[7] Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.

O’Daffer, P. G. (1998). Mathematics for elementary school teachers. Reading, Mass.: Addison-Wesley.

[8] Parker, T. H., & Baldridge, S. J. (2004). Elementary mathematics for teachers (Volume 1). Okemos, MI: Sefton-Ash Publishing. [Note: FIRST VOLUME, PRIMARILY NUMBER AND OPERATIONS.]

[8a] Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.

Sgroi, R. J., & Sgroi, L. S. (1993). Mathematics for elementary school teachers: Problem-solving investigations. Boston: PWS Publishing Company. (NO LONGER IN PRINT)

[9] Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.

Wheeler, E., & Brawner, J. (2005). Discrete mathematics for teachers (Preliminary ed.): Houghton Mifflin. (THIS BOOK FOCUSES ON DIFFERENT CONTENT THAN THE OTHERS.)

Wheeler, R. E., & Wheeler, E. R. (2002). Modern mathematics (Eleventh ed.): Kendall/Hunt Publishing Company. (THIS BOOK IS USED FOR GENERAL MATHEMATICS CLASSES AS WELL AS CLASSES FOR ELEMENTARY TEACHERS.)

[10] Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education,