# Definitions of Length 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. 607 – refers to length as a “linear measure”

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

• 481 – A length describes the size of something (or a part of something) that is one-dimensional; the length of that one-dimensional object is how many of a chosen unit of length (such as inches, centimeters, etc.) it takes to cover the object without gaps or overlaps, where it is understood that we may use parts of a unit, too. Roughly speaking, an object is one-dimensional if at each location, there is only one 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. 655 – 1 length (or linear) unit

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. 838 – linear measure

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

• 528 – We speak of the length of a piece of wire or a rectangle in two ways. The term length might refer to the quality or attribute we are focusing on, or it might refer to the measurement of that quality. The context usually makes clear which reference is intended.
• p. G-7 – the characteristic of one-dimensional shapes that is measured with a ruler; for example, width, height, depth, thickness, perimeter, and circumference all refer to the same characteristic

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

• [none]

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 – Regardless, in every case, we can select some appropriate unit and determine how many units are needed to span the object. This is an informal measurement method of measuring length, since it involves naturally occurring units and is done in a relatively imprecise way.

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

• 5 – In geometry one also has a notion of length and distance. The distance between two points A and B is the length of the segment ; in this book we will denote this length by AB (some textbooks use a different notation for length). Note that AB is a number whereas is a segment.
• p. 6 – Lengths are not numbers because any measurement of length involves a two-step process:
• Choose a unit length. [picture omitted – shows example of 1 unit]
• Express other lengths as multiples of that unit. [picture omitted – shows multiple of unit “The bar is 4 units long.”]]

The resulting length is then a quantity; a number times a unit. [picture omitted – shows that in 48 inches, 48 is the number and inches is the unit]

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

• [none]

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. 382 – There are four important principles of iterating units of length, whether they are nonstandard or standard (Dietiker, Gonulates, Figueras, & Smith, 2010, p. 2): *All units must have equal length – if not, you cannot accumulate units by counting. *All units must be placed on the path being measured – otherwise, a different quantity is being measured. *The units must be without gaps – if not, part of the quantity is not being measured. *The units must not overlap – otherwise, part of the quantity is measured more than once.

Wu, H. H. (2011). Understanding Numbers in Elementary School Mathematics. AMS.

• [none]

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, Inc.

[11] Wu, H. H. (2011). Understanding Numbers in Elementary School Mathematics. AMS.