# EC.1. Deep Understanding of Early Mathematics

Although the fact that all teachers need to understand the mathematics they are to teach seems obvious, many have not had experiences in the U.S. educational system to understand key conceptual understandings of the major topics of early mathematics (Ma, 1999; NRC, 2009). Effective teachers at the early childhood level hold deep conceptual understandings of the mathematics they teach as well as knowledge of how these foundational mathematical ideas connect to subsequent learning on the mathematical horizon.

The depth of the mathematics in the early years is often misunderstood (Lee & Ginsburg, 2009):

Mathematical ideas that are suitable for preschool and the early grades reveal a surprising intricacy and complexity when they are examined in depth. At the deepest levels, they form the foundations of mathematics that have been studied extensively by mathematicians over centuries … and remain a current research topic in mathematics. (NRC, 2009, p. 21)

As an example, teachers need to understand how counting relates to place value, in that only 10 digits are needed to write any counting number by creating larger and larger units (which are the values of places in a written numeral) by taking the value of each place to be equal to 10 of the place to its right (e.g., the 1 in 125 is equal to 10 tens). In this way, every counting number can be expressed in a unique way as a numeral made of a string of digits. These ideas connect the study of counting and place value and are essential in the development of arithmetic, such as for understanding that 63 + 10 is 73 without having to count by ones. Full appreciation of the multiplicative relationship in place value (taking the value of each place to be 10 times the value of the previous place to its right), developed over time, is critical for such further mathematics learning as understanding decimal numbers.

High-quality programs prepare candidates with broad and deep understandings of fundamental mathematics. Most important is the domain of number and the related concepts of quantity and relative quantity, counting, and arithmetical operations. Also critical are the domains of geometry and measurement, through which people mentally structure the spaces and objects around them. Connections and coherence among mathematical ideas are enriched when candidates apply number concepts and processes to these spatial structures. In addition, these domains provide rich contexts to further develop the ability to reason mathematically.

In brief summaries below, we highlight the important mathematical concepts and practices that a well-prepared beginning teacher of mathematics must know to be able to support learners in Pre-K through Grade 2. The summaries include specific, but not exhaustive, examples of what understanding and being ready to teach this content entail. These concepts connect to and reflect The Mathematical Education of Teachers II (CBMS, 2012) and the research in mathematics teacher education. They are focused on counting and cardinality, number and operations in base ten, operations and algebraic thinking, geometry, measurement, and data. The mathematical ideas at the early childhood level form the subtle and complex foundation of school mathematics. Prospective early childhood teachers and the programs that prepare them well do not dismiss these foundational ideas as simple but rather treat them with the mathematical respect that a careful and sustained examination affords and consider appropriate placement of these topics in either content or methods courses. These ideas are closely connected to their upper elementary successors (as addressed in Chapter 5), and programs preparing early childhood teachers ensure that they offer candidates opportunities to examine the full spectrum of mathematical connections across the preschool through Grade 5 span.

This examination of foundational mathematics includes more than learning appropriate content. Well-prepared early childhood teachers continually engage in the mathematical practices and processes of reasoning, sense making, and problem solving. They use mathematics to model, understand, and analyze real-world situations, and they understand that using manipulatives is not the same as mathematical modeling, which would entail using number sentences to mathematize a real-world, contextual situation (Consortium for Mathematics and Its Applications & Society for Industrial and Applied Mathematics [COMAP & SIAM], 2016).

Counting and Cardinality. Well-prepared beginners understand the fundamental ideas and nuances of counting and cardinality. Cardinality means how many things are in a set. Young children determine cardinality by perceptual subitizing (immediately recognizing small numbers of objects, up to about 4) or conceptual subitizing (using a number composition/decomposition for larger numerosities), counting, or matching. Teachers of mathematics must understand each of these ways, how they are related, and how they develop. All three can be used to establish quantity—here, the number of things in a set (another type of quantity is the amount of matter that can be measured—continuous quantity). No matter which of the three methods one uses, the number—the cardinality—will be the same. Further, number is an abstraction, and the same number quantifies many different collections, such as three trees, three cookies, or three people. At the core of this commonality is the notion of one-to-one correspondence. Any set of three can be placed in one-to-one correspondence with any other set of three. That is, each member of a set of three cookies can be matched with one and only one member of the set of three people. Thus, any method correctly applied—subitizing, counting, or matching—will result in the same number in a given set. Because of the infinite number list, counting can be used to quantify any discrete set and so is particularly important. Placing the number list in one-to-one correspondence with a set tells how many are in that set; the last number so placed is the cardinal number of the set. We do not have to memorize an infinitely long number list because of our use of the base-ten system, which is discussed below (for more details, see NRC, 2009).

Number and Operations in Base Ten. Well-prepared beginners understand that “comparison of quantities and less-than and greater-than relationships is an early step toward decomposing and composing numbers in ways that are necessary in common addition, subtraction, multiplication, and division procedures” (Dougherty, Flores, Louis, & Sophian, 2010, p. 1). They recognize the importance of the benchmarks of 5 and 10 as support for seeing numbers as combinations of other numbers, such as 7 as the combination of 5 and 2 more; 7 is also 3 away from 10.

Table 4.2 lists items from the MET II report (CBMS, 2012) related to number and numeration in Early Childhood.

### Table 4.2. Connections to MET II (CBMS, 2012) Related to Number and Numeration in Early Childhood

Counting and Cardinality

MET II describes the following essential ideas related to counting and cardinality:

• “The intricacy of learning to count, including the distinction between counting as a list of numbers in order and counting to determine a number of objects.” (p. 25)

Number and Operations in Base Ten

MET II describes the following essential ideas related to number and operations in base ten:

• “How the base-ten place-value system relies on repeated bundling in groups of ten and how to use objects, drawings, layered place-value cards, and numerical expressions to help reveal base-ten structure. Developing progressively sophisticated understandings of base-ten structure as indicated by these expressions:

357 = 300 + 50 + 7

= 3 × 100 + 5 × 10 + 7 × 1

= 3 × (10 × 10) + 5 × 10 + 7 × 1

= 3 × 102 + 5 × 101 + 7 × 100.

• How efficient base-ten computation methods for addition and subtraction rely on decomposing numbers represented in base ten according to the base-ten units represented by their digits and applying (often informally) properties of operations, including the commutative and associative properties of addition, to decompose the calculation into parts. How to use mathematical drawings or manipulative materials to reveal, discuss, and explain the rationale behind computation methods.” (p. 27)

Operations and Algebraic Thinking. Well-prepared beginners need experiences with the varied arithmetic problem types, such as joining, separating, and comparing problems with different parts of a problem situation unknown (NGA & CCSSO, 2010). For example, a separate, result-unknown is the typical take-away problem, but a separate start-unknown is less familiar to many (e.g., “Nita had some cars. She gave three to Kerri and now she has eight. How many did she have to start with?”). Such problems can be used to develop intuitions about the meanings of operations and their relationships to one another. Well-prepared beginners understand the importance of using varied problem situations, recognizing the unique complexities of shifting the location of the unknown and characterizing young children’s thinking relative to their solution strategies (Carpenter, Fennema, Franke, Levi, & Empson, 2014). This knowledge enables them to teach in ways that are responsive to their children's thinking while providing a framework to build upon that thinking.

The concepts of addition and subtraction are deeply understood by well-prepared beginning Pre-K to Grade 2 mathematics teachers; they understand multiple representations of the concepts as well as how to sequence and teach this content to students. They recognize the relationship between the content that precedes addition and subtraction (e.g., counting and cardinality) and the content that follows (e.g., multiplication and division).

Well-prepared beginners of early childhood mathematics understand that the “mathematical foundations for understanding computational procedures for addition and subtraction of whole numbers are the properties of addition and place value” (Caldwell, Karp, & Bay-Williams, 2011, p. 28). The commutative and associative properties support flexibility with computations. For example, when asked to find the sum of 4, 5, and 6, one may recognize that 4 and 6 make 10 then combine 10 and 5 to make 15. The associative property allows for the regrouping of addends without changing the sum. Combining the “notion of decomposing a number with the commutative and associative properties is foundational to most addition and subtraction fact strategies” (Caldwell et al., 2011, p. 29).

Finally, well-prepared beginners meaningfully use symbols such as the equal sign. They understand that the equal sign denotes that two expressions have the same value, and they avoid the common misconception of the equal sign as merely an indication that the answer comes next (Falkner, Levi, & Carpenter, 1999). Well-prepared beginners challenge children's conceptions by purposefully recording equations that place the equal sign in varied locations, such as 5 = 2 + 3, 2 + 3 = 5, and 5 = 5.

Table 4.3 lists items from the MET II report (CBMS, 2012) related to operations and algebraic thinking in early childhood.

### Table 4.3. Connections to MET II (CBMS, 2012) Related to Operations and Algebraic Thinking in Early Childhood

MET II describes the following essential ideas related to operations and algebraic thinking:

• “The different types of problems solved by addition, subtraction, multiplication, and division, and meanings of the operations illustrated by these problem types.
• Teaching–learning paths for single-digit addition and associated subtraction, including the use of properties of operations….
• Recognizing the foundations of algebra in elementary mathematics, including understanding the equal sign as meaning ‘the same amount as’ rather than a ‘calculate the answer’ symbol.” (p. 26)

Geometry. Well-prepared beginning teachers of mathematics in Pre-K to Grade 2 possess strong understandings of geometry, including spatial relationships. Geometry is the study of shapes and space, including two-dimensional (2D) and three-dimensional (3D) spaces. Although all objects in the world have shape, early 2D geometry is focused on shapes with basic attributes, such as circles, triangles, rectangles, squares, rhombuses, trapezoids, hexagons, and other polygons. Most objects that people make can be modeled with such shapes, especially when they decompose or compose such shapes into more complicated forms. Teachers need to know far more than the names of such shapes. They need to know the components (parts) of the shapes (sides and angles, for examples), the properties of shapes (properties are not components of shapes but rather are the relationships between two or more components, such as equal-length sides, parallel sides, or right angles). Thus, the study of geometry is not only naming shapes as wholes; it is also about finding and analyzing the components and properties of shapes. Discussing the definition of shapes is a deep and fundamental mathematical practice, and teachers need to understand geometry deeply enough to guide discussions of different informal definitions. For example, beginning teachers need to understand why the all-too-common pseudodefinition of rectangles as having “two long sides and two short sides” is completely inadequate (i.e., many shapes, such as nonrectangular parallelograms fit that description; further, squares do not fit that description but are rectangles. Rectangles are better defined as four-sided polygons with all right angles).

Common 3D shapes include cubes, prisms, cylinders, pyramids, cones, and spheres. Many common objects are approximate versions of these ideal shapes, such as filing cabinets described as rectangular prisms and certain ice cream cones described as cones. As with the study of 2D shapes, the study of 3D shapes is not only about naming these shapes as wholes and learning their names but also entails finding and analyzing their components (e.g., faces, edges, and curved surfaces) and their properties.

Similar to the way that 10 single units can be composed to make a unit of 10, shapes can be composed to make a unit of unit shapes or decomposed into smaller shapes. Such compositions and decompositions are important in the study of geometry, and in many other areas in mathematics such as fractions, but also in other subjects, from the sciences to the arts. One special composition is composing squares into rectangular regions—the beginning of spatial structuring of those regions into rows and columns that underlies both coordinate geometry and area measurement. Similarly, composing and decomposing 3D shapes is an important foundation for understanding volume in later grades.

Thus, learning geometry is a central topic for its own sake and for its contributions to learning other topics and other subjects. Further, geometry learning provides opportunities to develop ability to reason mathematically.

Measurement. Well-prepared beginners possess strong understandings of measurement, including quantifying two-dimensional (2D) and three-dimensional (3D) spaces. Measurement is the process of determining the size of an object. However, multiple ways exist for determining size, depending on the attribute one chooses. For example, a room may be described as having certain length, width, and height (each 1D) or a certain floor area (2D), or a volume (3D). These measurable geometric attributes are the most important to mathematics (although many more, such as the temperature of the room, are important in science). To measure a quantity, one must choose a unit appropriate to the attribute being measured, and then the size of an object is the number of those units needed to quantify that attribute (e.g., to fill the space or cover the area). Core measurement concepts include iteration, conservation, and origin, and provide a framework for connecting linear measurement with measures of area and volume (Clements & Sarama, 2014). (For more details, see NRC, 2009.)

Data. Well-prepared beginners of mathematics in Pre-K to Grade 2 understand that the foundations of statistical reasoning begin with collecting and organizing data to answer a question about our world and then examining the variability of that situation The question is posed, a plan is made to collect data that will address the question, and the data are classified into different categories. The categorized data are usually displayed graphically to describe or compare the categories. Because the process of describing or comparing categories usually involves number or measurement, number and measurement are central to data, and data analysis provides a context in which number and measurement are used (for more details, see National Research Council, 2009).

Table 4.4 lists items from the MET II report (CBMS, 2012) related to geometry, measurement, and data in early childhood.

### Table 4.4. Connections to MET II (CBMS, 2012) Related to Geometry, Measurement, and Data in Early Childhood

Geometry

MET II describes the following essential ideas related to geometry:

• “Understanding geometric concepts of angle, parallel, and perpendicular, and using them in describing and defining shapes; describing and reasoning about spatial locations (including the coordinate plane).
• Classifying shapes into categories and reasoning to explain relationships among the categories." ( p. 30)

Measurement and Data

MET II describes the following essential ideas related to measurement and data:

• “The general principles of measurement, the process of iterations, and the central role of units: that measurement requires a choice of measureable attribute, that measurement is comparison with a unit and how the size of a unit affects measurements, and the iteration, additivity, and invariance used in determining measurements.
• How the number line connects measurement with number through length ….
• Using data displays to ask and answer questions about data, and analyzing data with attention to the shape and spread. Examine the way data is [sic] collected and what analyses of the data mean about the situation.” (p. 29)