Definitions of Area 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. 621 – Questions about area generally deal with “how much” it takes to cover an object – for example, how much fertilizer to cover a lawn, how much material to cover a bed. In order to answer area questions, we have to select an appropriate unit, and thus the answer takes the form of how many of those units.;
  • p. 637 – surface area is  the area needed to cover all the faces of a three-dimensional object

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

  • p. 151 – In general, for any unit of length, one square unit is the area of of a square that is 1 unit wide and 1 unit long. The area of a region, in square units, is the number of 1-unit-by-1-unit squares it takes to cover the region without gaps or overlaps (where square may be cut apart if necessary)
  • p. 482 – An area or a surface area describes the size of an object (or a part of an object) that is two-dimensional; the area of that two-dimensional object is how many of a chosen unit of area (such as square inches, square 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. The surface area of a solid shape is the total area of its outside surface. Roughly speaking, an object is two-dimensional if, at each location, there are two 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. 676 – To measure the sizes of plots of land, panes of glass, floors, walls, and other such surfaces, we need a new type of unit, one that can be used to cover a surface. The number of units it takes to cover a surface is called its area.Squares have been found to be the most convenient shape for measuring area.
  • p. 704 – Another important measure associated with objects in space is their amount of surface. Surface area is expressed as the number of unit squares needed to cover the surface of a three-dimensional figure.

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. 854 – quote from “Focal Points”: “Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square unit that is 1 unit on a side is the standard unit for measuring area. “
  • p. 854 – Area is measured using square units and the area of a region is the number of nonoverlapping square units that covers the region.

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

  • p. 547 – We speak of the area of a field, a lake, a country, a geometric shape, a wall, or your body, all of which are examples of surfaces or regions – area is a characteristic of surfaces or regions. As with length, the term area is used to refer both to the attribute (“The wolf wandered over a wide area”) and to the measurement (“The area of the field is 15 acres”).
  • p. G-2 – the number of square units that would be required to cover the region. the region could be a 2D region, or it could refer to all the surfaces of a 3D figure, in which case it is called the surface area.

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

  • p. 530 – Area is a measure of the region bounded by a closed plane curve. Any shape could be chosen as a unit, but the square is the most common. The size of the square is arbitrary, but it is natural to choose the length of a side to correspond to a unit measure of length. Areas are therefore usually measured in square inches, square feet, and so on.

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 – To measure the area of a region informally, we select a convenient two-dimensional shape as our unit and determine how many such units are needed to cover the region.

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

  • p. 107 – 108 – A portion of the plane is called a region. A triangular region is a triangle together with its interior, and a polygonal region is a polygon together with its interior. A circular region is a disk. A region formed from several pieces is a composite region.
  • [image omitted – examples of triangle, triangular region, rectangular region, disk or circular region, composite region]
  • Area is a way of associating to each region R a quantity Area(R) that reflects our intuitive sense of “how big” the region is without reference to the shape of the region. Area is defined by the same two-step scheme used to define length, weight, and capacity:
    • Choose a “unit region” and declare its area to be 1 unit of area
    • Express the areas of other regions as multiples of this unit area.
  • p. 110 – Definition 1.2 (School Definition).  The area of a region tiled by unit squares is the number of squares it contains.
  • p. 193 – area is the number of unit squares needed to cover a region

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

  • p. 531 – If you want to know the size of the interior of a field, or which package of gift wrap is a better buy, you measure the area. Area is the measure of a closed, two-dimensional region.
  • p. 564: The total surface area of a closed space figure is the sum of the areas of all its surfaces. A surface area in square units indicates how many squares it would take to cover the outside of a space figure.

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. 384 – Area  is the two dimensional space inside a region.

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

  • p. 45 – We begin with a definition of area that is adequate for the present need. A square with length 1on each side is called a unit square. The area of the unit square is by definition equal to 1. We say a collection or rectangles {R_j} tile or pave a given rectangle R, if, by combining the R_j’s together we get the whole rectangle R, and if the R_j’s intersect at most along their boundaries. With all this terminology in place, the area of a general rectangle is by definition the number of unit squares required to pave that rectangle. (Remember that we are dealing only with whole numbers at this point and therefore the lengths of all rectangles are whole numbers.)
  • p. 191 – Because we will have to use the concept of area in a more elaborate fashion, let us first give a more detailed discussion of the basic properties of area. The basic facts about area that we need are rather mundane and are summarized below.
    • The area of a planar region is always a number.
    • The area of the unit square is by definition the number 1.
    • If two regions are congruent, then their areas are equal.
    • If two regions have at most (part of) their boundaries in common, then the area of the region obtained by combining the two is the sum of their individual areas.

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.

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