Standard C.1. Understanding Number Relationships and Structure

Developing a rich, flexible understanding of what numbers are, how they can be combined and separated in different ways, their relative magnitude and relationship to benchmark numbers, and the structure of our base ten number system is a central goal of learning and teaching mathematics at the elementary school level (13). This standard focuses on developing robust understandings of numbers and the number system. Supporting the development of number and number sense has far-reaching implications for learning, with research showing that young children’s understanding of counting and early number are the strongest predictors of fifth-grade mathematics achievement (14).

Elementary Mathematics Specialists (EMSs) understand the fundamental role of counting in making sense of number and in providing a foundation for understanding the operations. Counting also makes it possible to compare and order quantities, and to derive meaning of the base ten number system. As the number system is extended to include rational numbers, EMSs understand that fractions and decimals are also numbers and composed of countable units, allowing for additional work with counting, comparing, ordering, and relating these amounts to benchmarks and estimation. EMSs work to build robust and flexible understandings of what numbers are throughout the elementary grades by supporting coherence across counting, the base ten number system, and rational numbers.

C.1.a. Building connections between counting and cardinal understanding of number

C.1.b. Making sense of number composition and the base ten number system

C.1.c. Understanding fractions and decimals as an extension of the number system

C.1.a. Building connections between counting and cardinal understanding of number

Counting provides a foundation for children to make sense of what numbers are and how they relate to one another. While among the first mathematics opportunities with which young children are likely to encounter and engage, counting objects and understanding the unit is a complex, interconnected process. For young children, learning to count involves learning to coordinate each individual object or thing being counted with the sequence of number words, and coming to understand that the final number assigned when counting reflects a particular property of the entire set—its quantity or cardinal value. Even very young children often display an emerging, intuitive sense of fundamental counting principles. For example, without being instructed to do so, children will use a variety of methods for keeping track of their counting so that each object is counted once and only once (e.g., by standing up objects as they count, moving objects, or lining them up). Later, as they attempt to count beyond ten, children’s invented number words (five-teen; twenty-ten, twenty-eleven) and sequences (28, 29… 40, 41, 42) often reveal an awareness of the patterns and underlying structure of the base ten number system. Regular and varied opportunities to count collections of objects allow children to grapple with and make meaning of the relationship between the process of counting and its outcome. Allowing children to count in ways that they choose (e.g., rather than requiring that children place objects in a line before counting) supports them in developing flexibility when determining what must always be the same when counting (the stable order of the number word sequence) and what can be altered (objects can be counted in whatever order you wish–order irrelevance). Collections provide opportunities to compare and reason about subsets of objects (e.g., “Do you have more yellow bears or blue bears?”), and can provide an entry point into problem solving (e.g., “If all the green bears walked away, how many bears would you have then?”). Counting progressively larger collections provides a natural invitation to organize and create groups, supporting children to connect skip counting sequences with the equal groups they represent (15).

Building deep understanding of how counting relates to number is a central theme and foundational element of elementary mathematics. In essence, numbers themselves are “ideas—abstractions that apply to a broad range of real or imagined situations.” (16) Numbers are encountered in different contexts that reflect different meanings. Most often, numbers are used to indicate amounts, or cardinal values (e.g., three cookies, 15 minutes). However, numbers can also hold ordinal (e.g., third in line, second floor) or nominal (uniform number 22, 6475 Alvarado Road) meanings. Cardinal understanding allows children to use the ordinal nature of the number word sequence to reason about number relationships. For example, quantities can be compared by counting each set or by knowing what number comes before or after another number in the sequence (e.g., six is more than four because it comes after). Cardinal understanding also allows children to make sense of how, when counting: (a) the next number is one more than the previous number (e.g., 6 is one more than 5–sometimes called the successor function); and (b) that each new number also includes the previous quantities in the sequence (e.g., there is a 4 inside of 5–sometimes referred to as hierarchical inclusion).

The mathematical relationships, processes, and practices children engage in through counting are developed and expanded upon throughout the elementary grades and beyond. EMSs understand that learning to count with understanding is complex. Children’s grasp of the number word sequence, one-to-one correspondence, and cardinality develop concurrently rather than sequentially and in different ways for different children. EMSs understand the ways in which counting and cardinality provide a foundation for the operations and how they relate to one another. For example, creating, acting upon, and counting sets of objects allow children to begin to represent and solve many different kinds of problems. EMSs support the use of objects, fingers, sounds/actions, tallies, and drawings and recognize the importance of connecting written methods to manipulatives and the actions of counting and representing collections. They understand that extending the number sequence beyond single digits involves making use of the base ten structure of the number system as well as correspondences between units. Counting ideas will be extended to counting by groups (e.g., 2s, 5s, 10s), unit fractions, and decimals.

C.1.b. Making sense of number composition and the base ten number system

At the same time children build connections between counting and number, they begin to notice how amounts are grouped together in everyday life (e.g., my hand has 5 fingers, there are 8 crayons in 1 box), and how quantities can be thought of as composed of smaller groups. For example, a child might remark that within an arrangement of 5 blueberries on a plate, there is a set of 2 blueberries and another set of 3 blueberries. This ability to perceive small quantities as total units in and of themselves (without counting) is called subitizing (17). Flexibility in composing and decomposing quantities in different ways (e.g., 5 = 2 + 3 = 1 + 4 = 1 + 2 + 2) is essential in allowing children to make use of the structure of numbers and the number system to reason about and solve problems. A critical phase of development is when children begin to connect the patterns of the number word sequence with the base ten structure of the number system. For example, children are often able to count using the final numbers in the teen sequence (e.g., 16 through 19 where the naming structure is more transparent, but omitting 13 or 15) before accurately using the entire sequence of teen number words (18). Similar to other numbers, at first children understand 10 as composed of 10 ones. Over time, however, children will begin to work additively with 10 as a unit in and of itself, counting by 10s or adding and subtracting 10 at a time. Eventually, children begin to understand that the number names (and written numerals) reflect counts of different base ten units (89 is 8 tens and 9 ones) and use this understanding of the base ten structure of the number system to reason about and solve problems (19).

Noticing and Using Mathematical Structures. When students have opportunities to consider varied ways of grouping collections, and recording the results, they notice the special relationship between the number of total units and the number of groups and number of ones when creating groups of ten. For example 89 units can be grouped by fives totalling 17 groups of 5 and 4 ones; or groups by twos totalling 44 groups of 2 and 1 one; but when grouping by tens there are 8 tens and 9 ones, corresponding to the 89 units.

During elementary school, work with number shifts from working with collections of individual units (24 = 1 + 1 + 1 + 1…), to additive compositions of different parts numbers (24 = 12 + 12 or 10 + 10 + 4), to multiplicative compositions (24 = 4 • 6) or a combination of the two (24 = (2 • 10) + 4). Similarly, models and representations of these ideas shift from concrete representations of individual objects (e.g., counters or drawings), to composing or decomposing numerals, to using formal mathematical symbols within expressions and equations. In making connections between these varied representations of number, it is critical to understand the equal sign as denoting a relationship of equivalence, rather than merely signaling the outcome of performing an operation. Overcoming the ways children and adults are conditioned to expect equations in the form of a + b = c often requires explicit conversations and consistent use of the conventions of what is “allowed” in equation formats when using the equal sign (20). The idea that the value of a number corresponds with the sum of the base ten value of each digit is central to this work. Within whole numbers, the value of the place farthest to the right is one; the value of all other places is ten times the value of the place to its right. This idea is then extended to decimal numbers when numbers can be described as being 1/10, 1/100, or 1/10^n of a whole. Patterns in the number of zeros of the product occur when multiplying a whole number by powers of 10. Similarly, patterns in the placement of the decimal point occur when a decimal is multiplied or divided by a power of 10. Student generalizations that “you can add zeros” to whole numbers or that “the decimal point moves,” in one direction or the other while appearing to be a mathematical shortcut, can emerge from students as a conjecture and be defended mathematically, through problem solving. It should not be taught as a rule.

EMSs understand that the idea of ten ones being simultaneously thought of as one ten is a complex idea. They recognize the importance of experiences that allow students to represent the base ten structure of numbers (e.g., piles of ten, ten frames, cubes connected in towers of ten) before they work with pre-structured proportional representations (e.g., base ten blocks). EMSs understand that the relationship between places in a multi-digit number is multiplicative, and that the multiplicative reasoning needed to understand base ten relationships develops over time. EMSs also understand that students often begin working with the principles of place value before they learn to multiply, and that understanding connections between multiplication by a multiple of ten and place value can deepen the sense making of place value. It is critical that EMSs recognize what is right about a child’s thinking, even when they express their ideas in ways that might at first appear to be incorrect or incomplete. For example, an EMS considers the base ten understanding of a child who writes 100604 after counting a collection of 164 colored popsicle sticks knowing that written and spoken numbers are different and related systems. They understand zero is complex and has a role in place value (i.e., as a placeholder) that is different from the roles zero has in the number system (i.e., the boundary between negative and positive numbers) and computation (e.g., identity element for addition: zero added to any number results in that number; zero property: any number multiplied by zero equals zero). They recognize that as students consider the regularity of the progression of products when multiplying by powers of ten (for example 53, 530, 5,300, etc.), and describe and explain why this regularity occurs, students are extending the ideas of the base ten structure to larger numbers. EMSs support work in this area that goes beyond rote understandings that simply involve stating “the number of tens'' and “the number of ones” (e.g., understanding that the number of tens in 385 is not 8 but is 38 tens). They understand the place-value system is based on powers of ten and multi-digit numbers can be expressed using expanded notation with the digits 0 through 9 multiplied by the appropriate power of 10 and connected to a count of base ten blocks. With this knowledge, EMSs develop strategies for engaging others in understanding the multiple ways a number can be decomposed and how conceptual understanding precedes computational proficiency (21).

C.1.c. Understanding fractions and decimals as an extension of the number system

Young children can talk about, represent, and reason about fractional amounts prior to formal instruction. In children’s everyday experiences, the word half might refer informally to parts or pieces of something that was once whole (e.g., “I broke the graham cracker into 3 halves”), or to a set (e.g., “so far I’ve only eaten half of my candy”). Children may also associate terms such as quarters with particular amounts (i.e., a quarter is 25 cents, but a quarter hour is 15 minutes) rather than as a fractional relationship to a whole (i.e., one-fourth of the whole). However, children are able to build from these everyday ideas despite their informality and lack of precision. Young children can tell you they would rather be a child sitting at the table where three people are sharing one cookie than at the table where four people are sharing one cookie. They can also begin to create drawings that represent fractional quantities through solving problems involving whole numbers where the remainders can be partitioned or shared (see the operations section). Children will often describe the amounts they create using informal language. For example, when asked how 4 people could share 10 submarine sandwiches, a child split each sandwich into four parts (so that each person would get 14 of each sandwich), but described her answer as “10 pieces” (see figure 2). Equal sharing tasks like this example allow children to build from what they know about a whole to begin to represent and describe fractional amounts, providing a foundation for introducing formal fraction terms and symbols (22). They also provide opportunities for children to wrestle with counting and combining unit fractions (1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 10 x 1/4 = 10/4) or to explore questions about equivalence that emerge from comparing different solution paths (e.g., is 10/4 the same as two whole sandwiches and 1/2 of a sandwich?).

Figure 2

Child’s Solution for How 4 People Could Share 10 Submarine Sandwiches

Fractions can hold different meanings depending on the context in which they are used (23). A critical concept that underlies all of these meanings is that fractions are numbers. Fractions can be used to describe an amount that reflects a part-whole relationship (3/4 of a brownie). Fractions can also describe the result of division or a quotient (when 4 people share 3 brownies equally, each person receives 3/4 of a brownie). A ratio interpretation of a fraction could describe the relationship between types of brownies (at the party there are 3 cream cheese brownies for every 4 chocolate chip brownies) or between a type of brownie and the total number of brownies (there are 3 cream cheese brownies for every 4 brownies), with only the later equivalent to 3/4 when interpreted using different meanings involving the same sized whole. Fractions, ratios, and rates are connected by unit rates, with unit rates being used to solve problems and formulate equations for proportional relationships (24). Fractions can describe measures, such as 3/4 cup of chocolate chips for each batch of brownies. Finally, fractions can be used as operators, indicating that an amount is scaled or changed in a particular way (e.g., if you only want to make 3/4 of a batch, take 3/4 of the amount listed for each ingredient). In each of these interpretations, defining what constitutes the whole in a given problem or situation is fundamental to the meaning of fractions. For example, consider a situation where there are 2 boxes of granola bars with 4 granola bars in each box. The meaning of 1/4 is dependent upon what is identified as the whole. If the whole is both boxes, 1/4 would mean 2 granola bars. If the whole is one box, 1/4 would mean 1 granola bar. If the whole is one granola bar, 1/4 would mean 1/4 of a granola bar.

Representing and Connecting.

As students explore the multiple meanings of fractions in varied contexts, it can be useful to examine and make connections to what both the numerator and denominator mean within and across a particular context. For a part-whole interpretation, 3/4 of a brownie suggests a brownie has been cut into 4 equal parts (denominator) and three of the parts are counted (numerator). For a quotient interpretation, 3/4 is the size of the resulting piece when 3 brownies (numerator) were cut into 4 parts (denominator). In the case of the ratio interpretation, the 3 is the count of a particular brownie type while the meaning of the denominator changes depending on whether the ratio is part-part or part-whole.

While student intuition is powerful for early understandings and making meaning of rational numbers, representing them symbolically is complex and often counterintuitive. For example, one-fourth can be represented symbolically in multiple ways as a fraction in the form a/b (1/4, 2/8, 25/100) or as a decimal fraction (0.25, 0.250, 0.25000000). EMSs recognize that earlier work with whole numbers can be overgeneralized (e.g., 4 has a greater value than 2 so 1/4 must have a greater value than 1/2) so students need opportunities to work with a variety of representations of fractions. These include linear, set, and region models to help students develop understandings of fractions as numbers. Additionally and importantly, just as students use manipulative materials and drawings to help anchor a mental image of a whole number, they can use the number line to show how a fraction (or decimal or percent) can be inserted between any two fractions. The number line provides a visual representation of fractions and decimals, and also serves as a measurement or iteration model for computation (25). This visual representation also supports “seeing” the density of rational numbers, which is particularly important at the middle school level and beyond (26). EMSs understand how fraction equivalence and magnitude, that is being able to determine if a number is greater or less than another number (e.g., 4/5 is close to but less than 1), is foundational for extending work with rational numbers. Developing flexibility in creating equivalent fractions and using fractions such as 1/2 or 1/4 as “benchmarks” allows learners to informally extend their fraction understandings to include comparing and ordering fractions (27). This flexibility should be extended to representing, comparing, and ordering equivalent fractions, decimals, and common percents (e.g., 1/2 = 5/10 = 0.5 = 50%, 0.9 > 7/8) at the appropriate grade levels.

Table 2

Mathematical Storyline for Number Relationships and Structure

(13) National Council of Teachers of Mathematics [NCTM], 2000. (14) Nguyen et al., 2016. (15) Carpenter et al., 2017; Franke et al., 2018.(16) National Research Council [NRC], 2001, p. 72.(17) Clements, 1999. (18) Gould, 2017; McMillan et al., 2023. (19) Carpenter et al., 2015; Jaslow & Jacobs, 2009.(20) Carpenter et al., 2003; Rittle-Johnson et al., 2011.(21) NCTM, 2023a. (22) Empson & Levi, 2011; Siegler et al., 2010. (23) Barnett-Clarke et al., 2010; Behr et al., 1983; Kieren, 1992; NRC, 2005. (24) CBMS, 2012.(25) Fennell, 2007. (26) Saxe et.al., 2007. (27) Bray & Abreu-Sanchez, 2010.