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Providing opportunities for children to solve a variety of problems using their own strategies supports them in making sense of what the operations are and how they work. This standard focuses on how understanding of the operations develops and is extended. EMSs understand how children’s informal ways of approaching problem solving situations relate to what adults understand as the more formally defined operations of addition, subtraction, multiplication, and division. They also understand the ways children’s intuitive problem solving strategies can be leveraged to illuminate relationships and connections among the operations, and how these relationships and connections are extended across the number system, including multi-digit whole numbers, fractions, and decimals. Deep knowledge of the operations is critically important for later learning, with research showing that first-graders’ knowledge of arithmetic is predictive of their ability to operate with fractions in middle school (28).
Elementary Mathematics Specialists (EMSs) understand the algebraic underpinnings of computation across the number system, including how fundamental properties (e.g., commutativity) of number and operations are used naturally by children in their strategies for solving problems. Rather than isolating the operations and domains of number from one another, EMSs work to ensure that instruction emphasizes connections and coherence between whole numbers, fractions, and decimals. In doing so, EMSs create opportunities for learners to develop, articulate, and justify generalizations about how the operations work, and how these generalizations relate to a range of computational strategies and help to define a student’s sense of number.
C.2.a. Building understandings of the operations through contextualized situations
C.2.b. Fostering connections among the operations
C.2.c. Extending understandings of the operations to multi digit whole numbers
C.2.d. Extending understandings of the operations to all rational numbers
C.2.a. Building understandings of the operations through contextualized situations
Children enter school with a great deal of intuitive and informal knowledge of mathematics that can be built upon to make meaning of what the operations are and how they work. Without being told or shown how to solve problems, children as young as preschoolers will draw on their knowledge of real or imagined story contexts to generate solution methods that model the actions and relationships between quantities within the problem. Similarly, older children’s work with operations may be inspired by playing games or constructing mathematical models that involve some form of data collection (29). Collectively, these intuitive ways of reasoning about, representing, and solving problems allow children to mathematize (i.e., decontextualize by translating from contextualized situations to mathematics) and construct viable solutions to a wide range of problems, recontextualizing their work to ensure strategies and solutions address the problem posed. Accordingly, the “operations of addition, subtraction, multiplication, and division can be defined in terms of these intuitive problem solving processes, and symbolic procedures can be developed as extensions of them (30).”
Developing robust, interrelated understandings of the operations involves making sense of a wide range of problem structures. For example, young children can begin to construct ideas about addition and subtraction by posing and solving problems that describe joining and separating situations with an unknown result. Young children can also begin to develop foundational meanings of multiplication and division by solving problems involving equal groups. Over time, children extend their views of addition and subtraction by solving problems that do not contain explicit actions, such as comparison or part-part-whole situations, or problems that are not easily solved by directly modeling a problem’s sequence of actions (e.g., start-unknown situations). Children also learn to build upon their understandings of equal groups situations to make meaning of multiplication and division situations involving arrays, area, and multiplicative comparisons, with multiplicative comparison paving the way to understanding problems involving rates, ratios, and scaling at the middle school level and beyond. During elementary school, work with operations is extended to multi-digit whole numbers, fractions, and decimals, and includes multi-step problems involving more than one operation.
EMSs should understand the range of problem structures for addition, subtraction, multiplication, and division; how children (rather than adults) are likely to make sense of and represent these situations; the relative difficulty of modeling the actions and/or relationships of different problem structures; and potentially productive (or problematic) ways various problem structures can be incorporated into instructional sequences and instructional materials. EMSs understand that the contextualizing, decontextualizing, and recontextualizing of expressions and equations is an important part of making sense of the quantities and relationships represented by symbolic notations. EMSs recognize that when students construct their own strategies for solving problems, they develop an intuitive understandings of the properties of operations and the relationships among operations. Relatedly, building understandings of the operations through contextual situations can also provide an entry point into “reading and writing the world with mathematics” to interrogate social and community issues and inequities (e.g., from school-community issues of distributing snacks fairly and sharing school supplies to broader world issues of protecting salmon and accessing clean water)(31).
Contextualizing and Decontextualizing.
Students develop deep understandings of mathematics as move between the real world and related mathematics representations (i.e., model with mathematics). Given a real world situation, students can be asked to pose and solve problems using varied mathematics, or decontextualize the situation. Once they have a model that answers their question, which may require iteration, they recontextualize as they return the real world to verify their model is effective. Students may also enter this cycle when given a decontextualized situation and they create a context that connects to the real world.
C.2.b. Fostering connections among the operations
Given opportunities to build from their intuitive strategies and to communicate with others about their ideas, children’s natural approaches to solving problems will become increasingly abstract over time. For example, a kindergartener might initially solve a multiplication story about 3 packages of gum with 6 pieces of gum in each package by representing three packages, each of which contains 6 individual cubes, and counting all the cubes (by ones) to equal 18. Later, a child might skip count by 6 three times, or derive the solution by knowing that 3 packages with 5 pieces of gum would equal 15, and one more piece for each package is 3 more, so 15 + 3 equals 18 (32). Providing students with opportunities to wrestle with the similarities, differences, connections between, and varied representations of these approaches helps to build understandings of the relationships between and among the operations and develop skills in selecting the solution strategy best for them and the particular problem.
Representing and Connecting.
As students are provided opportunities to engage in making sense of and solving problems then recording their thinking in ways that are natural to them, they come to see connections both within and across contextual, verbal, physical, visual, and symbolic representations (e.g., models and drawings of 3 packages each with 6 pieces of gum with connected equations) as well as the similarities and differences among the operations (i.e., 6 + 6 + 6 = 3 × 6).
As the meanings of the operations develop, children’s solution strategies and ways of representing their ideas become increasingly abstract. Initially, many children will use fingers, cubes, or drawings to represent quantities as composed of individual units, and then count (by ones) to operate on the quantities they have created. In this way counting both provides access to and unites the operations. For example, as children begin to count some quantities as composed units themselves, they might solve a story problem asking about the number of cars that will be needed to take 30 children on a field trip if 5 children can fit in each car by (a) finding the number of times 5 can be added to reach 30 by skip counting, or (b) the number of times 5 can be subtracted from 30 by repeated subtraction. In this case, the richness of children’s varied approaches supports them in grappling with relationships between division, addition, and subtraction.
EMSs understand and support others in understanding the patterns of development in student-generated strategies, how students’ solution paths are influenced by and related to different problem structures, and the connecting threads between concrete, semi-concrete, early deriving strategies, and those that are more abstract. In their work with students, EMSs support these connections by creating classroom environments in which students explain the details of their mathematical ideas and engage with their classmates’ reasoning by drawing attention to similarities, differences, and connections. As they support students in noticing these patterns, they draw attention to the importance of selecting a strategy that makes sense for the problem and to them, recognizing that the most efficient and accurate strategy for a student may or may not be the standard algorithm. EMSs also understand how the properties of operations (e.g., commutative, associative, distributive) often underpin students’ invented approaches, and the different ways students and teachers can represent varied solution paths. For example, a child might derive a solution for a join, change unknown story problem (7 + __ = 12) by recalling that 7 and 3 more equals 10, and 2 more equals 12, so 3 more and 2 more equals 5 (implicitly drawing on the associative property of addition, see figure 3). EMSs understand the importance of students communicating and justifying their use of the properties of the operations in varied ways. Students might use pictures, gestures, story contexts, or natural language to communicate the generalizations they are noticing about the operations (33). For example, they might use two stacks of cubes to demonstrate the commutative property of addition, by “switching them” the placement of two stacks to show that the sum remains the same even if the order of the two addends is changed.
7 + __ = 12
(7 + 3) + 2 = 12
7 + (3 + 2) = 12
3 + 2 = 5
Figure 3
Illustration of a Child’s Informal Use of the Associative Property of Addition within a Mental Solution Strategy
C.2.c. Extending understandings of the operations to multi-digit whole numbers
A rich foundation of reasoning and problem solving experiences supports children as they extend their ideas of how the operations behave as the number system expands to include multi-digit whole numbers. Counting (by ones) allows young children to solve problems involving multi-digit numbers before they understand the relationship between tens and ones or work with units of ten. Multiplication and measurement (also known as quotative) division problems involving groups of 10 are especially powerful as they invite children to create groups of ten from collections of individual units or to find the total created by iterating groups of 10. Children’s strategies for solving these problems progress to reveal an increasing grasp of operating on 10 as a unit and of the relationships between tens and the base ten number system (34). As children continue to solve more sophisticated problems, their expanding understanding of the base ten system allows them to solve multi-digit problems in ways that are closely related to single-digit strategies such as make ten. Recalling the child’s single-digit strategy from above, a related strategy for solving a multi-digit join change unknown problem (70 + __ = 128) might be that 70 and 30 more equals 100, and 28 more equals 128, then 30 more and 28 more equals 58 more, so 58.
C.2.d. Extending understandings of the operations to all rational numbers
Young children can also begin to develop understandings of fractional amounts through solving equal sharing story contexts in which the leftover amounts can be partitioned and acted on in different ways depending on the context (35). In solving these problems, children create informal representations of fractional amounts that can serve as the basis for developing formal understandings of fraction terms and symbols. For example, in attempting to solve a problem such as If 4 friends want to share 6 sandwiches so that each person gets the same amount, how much of the sandwiches should each person get?, children will often draw sandwiches and partition the leftovers into parts that represent halves or fourths before they can formally name the fractional amounts created through their solutions (see Standard C.1.c).
Once children can connect their concrete and semi-concrete representations of fractional amounts with the corresponding abstract terms and symbols, they can begin to make sense of whole numbers as composed of unit fractions (2 = 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4), and of unit fractions as the building blocks for other fractions (3/4 = 1/4 + 1/4 + 1/4). Similar to previous work involving groups of 10, multiplication and measurement division problems involving groups of same-sized unit fractions are especially powerful in providing opportunities for children to iterate unit fractions and decompose wholes into unit fractions and to reason about these relationships (36). For example, to find the number of days a guinea pig could be fed 1/4 of a carrot each day if there were 6 carrots, a child represented each carrot as composed as 4 fourths, and found the total number of fourths within their drawing to determine the number of days the guinea pig could be fed (see figure 4). Because the strategies children develop for solving problems can involve drawing, skip counting, adding, and subtracting fractional amounts, children’s approaches to solving what adults might recognize as multiplication and division problems connect operating on fractions, and can precede formal work with addition and subtraction (e.g., 2/4 + 1/4 = __).
Figure 4
A Child’s Solution for Finding How Many Days 6 Carrots Will Last if a Guinea Pig is Fed 1/4 of a Carrot Each Day
EMSs understand the connections and relationships between children’s strategies for operating on whole numbers and the ways that children’s work within earlier number domains can provide a foundation for work with multi-digit whole numbers, fractions, and decimals. EMSs attend to the central role of both formal and informal representations in allowing children to both establish and extend their understandings of the operations and a variety of invented and standard algorithms across number domains. EMSs also understand how to build from children’s understandings of fractions as quantities to make sense of fractions (including decimal fractions) as quotients, operators, and ratios. Importantly, EMSs know that students having a thorough understanding of PK-8 mathematics includes: robust interconnections of conceptual understanding of numbers and operations; flexibility and fluency with procedures; and the mathematical agency to select the best strategy for the given context (27).
(28) Bailey et al., 2014.(29) McClain et al., 2000; Sembiring et al, 2008.(30) Carpenter et al., 2015, p. 4(31) Arnold et al., 2021; Bartell et al., 2022; Chao & Jones, 2016; Gutstein, 2006; Koestler et al., 2022.(32) Carpenter et al., 1993; Celedón-Pattichis & Ramirez, 2012; Turner & Celedón-Pattichis, 2011. (33) Schifter & Russell, 2022.(34) Bray & Blais, 2017.(35) Empson & Levi, 2011; Siegler et al., 2010. (36) Empson et al., 2020. (37) NCTM, 2023c.
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