Standard C.4. Exploring, Examining, and Enumerating Space

Notably, a focus on spatial reasoning provides multiple entry points and equitable access to mathematics. Geometric work can support culturally relevant connections as students look for and make use of their spatial reasoning to understand their world and to participate in mathematics activities that contribute to their environments and communities both in and outside of school (e.g., determining where the library bus can park to serve more families, designing dog pens for a new animal shelter)(46). 

EMSs understand that students develop the language of geometry through experiences that allow them to observe, touch, sort, and create a variety of shapes. They also recognize the multifaceted nature of how children communicate their spatial ideas (47). Precision of geometric language develops as students describe and compare shapes. Support for multilingual students that includes the use of visuals, rehearsals, and language cognates should be embedded in activities (48). EMSs understand that the visualization of shapes in two and three dimensions requires that students view shapes from different viewpoints and orientations. EMSs recognize that it is important for students to work with a variety of shapes so that they do not develop restricted views of shapes that must later be unlearned (e.g., all triangles look like equilateral triangles and sit on a base). EMSs recognize that students work with different aspects of shapes at different grade levels and work with increasingly more complex shapes.  

C.4.b. Composing, decomposing, and understanding space 

At first, young students compose with different shapes to create pictures (e.g., a square and a triangle can make a house) or cover pictures made up of different shapes (sometimes by trial and error) by matching smaller shapes within an existing larger shape (e.g., tangrams). Students then begin to use defining attributes of shapes to compose new shapes, such as putting together two triangles with same side lengths or two right triangles to make a square. Students can discuss the parts and totals of the shapes they compose and decompose and use a new composed shape as a unit to make other shapes. Students’ development of part-whole relationships plays a critical role as they decompose and recompose space. Ideas about congruence and similarity are developed as students manipulate and compare shapes. Key geometric ideas of equivalence include understanding that any two objects of the same size that occupy the same space and have different shapes are considered equivalent. 

Students learn about two and three dimensional shapes by composing and decomposing them. Through these transformations they learn about the components and properties of shape. Their ability to visualize, describe, and transform geometric regions develops alongside their ability to recognize and use iteration of measurement units, the construction of patterns, the decomposition and composition of numbers, including fractions. 

EMSs understand that the compositions and decompositions of regions are important for solving a wide variety of area problems. For example, the area of a triangle can be determined by composing a rectangle made of two of those triangles as shown in figure 7a. Also, knowledge of finding the area of rectangles and triangles can be extended to find the area of any two dimensional shape, essentially composing, decomposing, and rearranging through rotations, reflections, and translations to find the known shape (see figure 7b for an example of how this thinking is extended to find the area of trapezoids).

EMSs support this area of work by encouraging students to analyze and make use of structure. EMSs know how to develop experiences that allow students to engage in the creating, composing, and decomposing of units and higher order units. They understand that students move from pre-composer level to piece assembler to picture-maker to shape composers (49). EMSs know that observing students as they compose and decompose shapes offers important information about how a student is thinking about space. Students might begin by using trial and error to cover a shape composed of individual shapes. Later on, students are able to visualize blank spaces as composed of individual shapes and can mentally manipulate shapes. EMSs are able to draw upon students’ ideas about the composition and decomposition of shape as students develop strategies and their own formulas for finding the area and volume of a variety of shapes. EMSs understand that students’ strategies are later extended to shapes that may have side lengths or units that are fractional or decimal amounts.

C.4.c.  Understanding spatial relationships and spatial structuring

Young students develop spatial reasoning through puzzle play, board games, or building with blocks or geometric shapes (50). Early work with composing and decomposing lays the foundation for spatial structuring. Positional words and phrases such as above, below, next to, and inside of are used to describe spatial relations first informally, and then later on in more formal ways. Additional visualization work can occur through experiences with picture books, dot images, or shape images. Spatial structuring involves the mental operation of constructing an organized form of an object or set of objects. Students can use structures such as arrays to move from seeing discrete objects to a set of squares in rows and columns. These structures are used first with two dimensional shapes and then the arrays can be layered to measure the volume of the three dimensional shapes (see figure 8).

Spatial reasoning plays a fundamental role in our everyday lives and also plays a vital role across all mathematical domains. Researchers have found connections between spatial reasoning and the understanding of number, measurement, and problem solving (51). The National Research Council (52) has identified three aspects of spatial reasoning: (a) concepts of space; (b) tools of representation; and (c) processes of reasoning. Understanding concepts of space includes understanding relative distance or size as well as aspects related to continuity and dimensionality. 

EMSs recognize the important role of viable explanations and justifications as students solve problems related to spatial relationships and structures. They attend to and support students’ cultural competencies, including the use of gestures and developing language as strategies for describing spatial relationships. EMSs understand and leverage the strong relationship between number and spatial reasoning by encouraging the use of varied representations (e.g., concrete objects, drawings, area formulas) and the manipulation of shapes and structures to explain and justify, for example, properties of the operations, such as using an array to develop a justification for the commutative property. EMSs are aware that the segmenting of space can present students with instances in which there is a need to use fractional units. They recognize the importance of students having opportunities to segment space in units that may be expressed as whole numbers, fractions, or decimals. These experiences support understandings related to number line diagrams (e.g., recognizing fractional amounts as a point on a number line) and coordinate axes (e.g., graphing points on the coordinate plane).

C.4.d. Enumerating space through geometric measurement

Children bring many informal measurement experiences from everyday life to their more formal school-based encounters. They use informal words such as tall or big to describe many dimensions of objects. Young students begin this work by using informal methods of measuring and comparing objects. At first, they have to identify what attribute of an object they are going to measure, and then decide what tools and strategies they will use to do so. The use of direct comparisons, that is, comparing an attribute of two objects without measurement (e.g., two students stand back to back to see who is taller) is number free and allows for a focus on the attribute that is being measured (53). The use of indirect comparisons (i.e., using a third object to compare two objects) occurs as students look for and make use of measurement benchmarks such as a doorway or a pencil. During the elementary grades, qualitative perceptions progress to more quantitative descriptions. Later on, these quantita- tive descriptions can become numeric comparisons between an object and a specific unit (54).

Measurement of space is a contextual application of number sense and spatial reasoning. Geometry and measurement provide important contexts that develop, deepen, and allow for application of whole number and fraction knowledge and related understandings. Many specific aspects of an object can be measured such as dimension (e.g., length, diameter), area, volume, angle size, etc. Measuring any attribute requires an understanding of that attribute as continuous that can be subdivided into smaller iterations or units, rather than as a count of discrete objects. This continuous measurement unit is in contrast to cardinality, which involves a discrete attribute.  

For example, an area can be found using an array (a collection of discrete units arranged in rows and columns) or an area model (a continuous measure, multiplying the side lengths). Geometric measurement spans across three dimensions of space (i.e., 1D, 2D, 3D), includes both regular and irregular shapes, and attends to the measurable aspects of those shapes (i.e., length, area, volume, angle). In later grades, students apply their understanding about features of shape, transformation, and equivalence to make generalizations about measurement formulas and their use (see figure 9). 

EMSs recognize the role of reasoning and problem solving in geometric measurement, and the importance of regular problem solving opportunities that foster geometric thinking. They understand that the use of measurement formulas without conceptual understanding negatively impacts students’ progress toward developing deep understandings of geometric measurement. EMSs support the use of non-standard and standard units through explorations that lead to the need for more precise measurements (i.e., measuring all the space, no gaps or overlaps). They design and implement tasks that allow for the internalization of units and measurement processes. EMSs understand that geometric measurement involves a continuous property of space (i.e., length, area, volume) and that the iteration and counting of units requires the use of spatial structuring. They understand the value of students engaging in three dimensional problems that involve geometric relationships, such as tasks that involve nets and thus provide contexts linking two dimensional and three dimensional relationships (55). 

EMSs consider the complexity of ideas related to units and unit iteration. They foster understanding of these ideas for teachers and students by developing measurement experiences that illuminate the following: the selection of a unit must match the attribute of the object being measured; the iterated unit must be a constant size (and orientation if a non-square or non-cubical unit); the need to measure the entire space of focus without gaps or overlaps; and an understanding that the smaller the unit is, the more units are needed and the more precise the measure. However, there always will be some degree of error.