Supporting Spatial Reasoning: Identifying Aspects of Length, Area, and Volume in Textbook Definitions

Length, area, and volume share structural similarities enabling flexibility in reasoning for real-world applications. Deep understanding of structures can help teachers connect these concepts to support their students’ mathematical reasoning and practices involving real-world situations. In mathematics textbooks designed for future teachers, definitions of length, area, and volume vary from procedural (e.g., use a ruler to measure side lengths, use formulas to calculate measures) to conceptual (e.g., construct appropriate n-dimensional units that tessellate the n-dimensional space) to formal (e.g., construct a function mapping qualitative size to a quantity of appropriate units). Most textbooks describe length, area, and volume as quantitative measurements and provide examples of standard units. Definitional aspects such as describing size as an attribute or measurement, identifying dimensionality of a space, or constructing appropriate nonstandard units are inconsistently acknowledged across textbooks. Attending to definitional aspects of spatial attributes and their quantification can open conversations about the structure and essential meanings of length, area, and volume.