Our Instructional Approach and the Reserach that guides it
Introduction
The Primary Numeracy Assessment is a groundbreaking, standards-based tool developed by Numeracy Consultants to support educators in identifying key numeracy standards where students may struggle within their state-mandated curriculum. It is the core of the Primary Numeracy Intervention Program. Built on extensive research and the latest understanding of numeracy development, this assessment filters through a wide array of standards to focus on those most critical to early mathematical success.
Covering a broad spectrum of foundational numeracy domains, the Primary Numeracy Assessment targets essential skills such as forward and backward counting, skip counting by tens, the strategic use of addition and subtraction, place value understanding, numeral recognition, and the concept of part-part-whole relationships. These core competencies form the basis of a student's long-term mathematical development, supporting both academic progress and real-world problem-solving abilities.
At the heart of our instructional framework, the Primary Numeracy Assessment provides the data needed to close learning gaps and refine teaching strategies. Its insights drive a focused, evidence-based approach to instruction, empowering educators to respond to students’ individual needs with accuracy and confidence.
Forward Counting: A Precursor to Addition
Forward counting serves as an essential stepping stone toward understanding numbers and performing basic mathematical operations. It is a skill that typically develops before children can confidently tackle more complex mathematical concepts such as addition. Research by Fuson and Hall (1983) highlights the importance of forward counting as a foundation for addition and subtraction. They argue that a strong ability to count forward contributes to children's early numerical sense, which is vital for understanding mathematical relationships.
Research Findings
Fuson and Hall’s findings suggest a close link between forward counting fluency and a child’s ability to conceptualize addition. Fluency in counting forward helps children internalize numerical sequences—an understanding that underpins the additive process, which involves moving forward along a number line (Fuson & Hall, 1983).
Douglas Steffe’s work (1992) further supports the cognitive significance of forward counting in the development of number sense. Steffe emphasizes that forward counting is not simply a memorization task, but a mental activity that supports children in recognizing numerical relationships and patterns. As such, forward counting becomes a key building block for numeracy, laying the groundwork for later mathematical problem-solving (Steffe, 1992).
Backward Counting: A Foundational Support for Subtraction
Backward counting, often overshadowed by its more frequently practiced counterpart—forward counting—is a foundational skill in numeracy development. This paper explores the significance of backward counting, emphasizing its role in supporting children’s understanding of the inverse relationship between addition and subtraction. Drawing on the work of Baroody, Gannon, and Douglas Steffe, we examine how proficiency in backward counting contributes to subtraction competence
Backward Counting and the Inverse Relationship
Complementary to forward counting, backward counting helps children understand that subtraction is the inverse of addition. Baroody and Gannon (1984) highlighted a positive correlation between children's ability to count backward and their success in subtraction tasks. By learning to reverse the counting sequence, children begin to conceptualize subtraction as "taking away" or "moving back" on the number line—key ideas in early arithmetic.
Research Findings
Baroody and Gannon's research suggests that children who are proficient in backward counting are more likely to understand and apply subtraction strategies effectively. Their work emphasized that backward counting is not merely a memorized routine, but a tool that enables children to navigate subtraction problems with greater confidence (Baroody & Gannon, 1984).
Douglas Steffe’s developmental research (1994) also underscores the role of counting, including backward counting, in building number sense. Steffe found that children who construct an understanding of number sequences—both forward and backward—tend to perform better in arithmetic tasks, particularly those involving subtraction. His work emphasizes that backward counting supports the conceptual development of subtraction as an operation that undoes addition, thereby strengthening mathematical fluency.
Conclusion
Backward counting is a crucial, though sometimes underemphasized, skill in early numeracy development. Research by Baroody, Gannon, and Steffe shows that it plays a significant role in helping children understand subtraction and its inverse relationship with addition. When children can confidently count backward, they are better equipped to solve subtraction problems and reason mathematically. For this reason, both educators and parents should prioritize backward counting as a meaningful component of early math instruction.
Counting by Tens: A Foundation for Place Value Understanding
Counting by tens is a foundational numeracy skill that significantly supports children’s understanding of place value—an essential concept in early mathematics. Place value refers to the value of digits in a number based on their position, and skip-counting by tens helps children intuitively grasp this structure by reinforcing grouping and numerical magnitude. Research from Fuson, Baroody, Steffe, White, and Wright consistently underscores the importance of this skill in the development of number sense.
Research Findings
Fuson and Briars (1990) emphasize that counting by tens provides children with a structured method for managing larger quantities, while also introducing them to the concept that each digit in a number represents a specific power of ten. This repeated exposure lays a critical foundation for understanding how numbers are composed and decomposed in our base-ten system.
Building on this, Baroody and Ginsburg (1983) highlight how skip-counting supports children’s understanding of grouping—particularly the grouping of quantities into sets of ten. This understanding is central to grasping place value, where each digit’s position corresponds to a multiple of ten. By counting in tens, children begin to view numbers as composed of units, tens, hundreds, and beyond.
Steffe (1992) extends this perspective by arguing that counting by tens is not merely a rote procedure, but a cognitive process that helps children build deeper numerical relationships. As they count in tens, children begin to see how the placement of digits contributes to the overall value of numbers, enhancing their conceptual understanding of place value and its role in arithmetic operations.
Similarly, Wright (1984) stresses that counting by tens serves as a bridge between basic counting strategies and more structured mathematical reasoning. His research shows that students who can skip-count effectively are better prepared to grasp base-ten representations and apply regrouping strategies in addition and subtraction.
Finally, White (1982) and colleagues demonstrated that counting by tens helps children connect symbolic numerals with real quantities. This ability to link the abstract and the concrete is essential for transitioning from counting physical objects to understanding numerical magnitude and the structure of the base-ten system at a more advanced level.
Conclusion
Counting by tens plays a central role in numeracy development and serves as a critical stepping stone to understanding place value. Research across decades affirms its value in fostering grouping, symbol-quantity correspondence, and structured numerical thinking. Educators and caregivers should intentionally include skip-counting activities in early math instruction to support children's deeper understanding of the base-ten system and prepare them for more complex operations.
Arithmetic Thinking Strategies: From Counting to Conceptual Fluency
Early arithmetic, particularly addition and subtraction—forms the bedrock of a child’s mathematical development. As children build fluency, they adopt various strategies to solve problems, ranging from simple counting to more efficient mental computation. These strategies evolve along a developmental continuum, gradually shifting from concrete to abstract thinking (Baroody & Dowker, 2003). Understanding this progression allows educators to better support children’s mathematical growth and guide them toward fluency and flexibility in problem-solving.
Recounting or "Counting All"
This is one of the earliest, and least efficient, strategies children use when learning addition. In this approach, they begin counting from one and continue all the way to the total. For example, when solving 6 + 4, a child might count, “1, 2, 3, 4, 5, 6,” then “7, 8, 9, 10.” Although typical for beginners, this method is slow and can impede the development of mental math skills if relied on for too long (Fuson & Briars, 1990).
Counting On
As children progress, they often adopt the more efficient strategy of counting on. Instead of starting from one, they begin with the larger addend and count up by the value of the smaller one. For instance, to solve 6 + 4, a child starts at 6 and counts “7, 8, 9, 10.” This shift represents a significant developmental milestone—from counting individual objects to performing mental addition—and supports the growth of number sense (Fuson & Briars, 1990; Fuson, 1992).
Make Ten (Decomposing and Recomposing)
This strategy builds on the understanding of ten as a key benchmark number. Students break apart one of the addends to create a sum of ten with the other, then add what's left. For example, in 7 + 8, a child might decompose 8 into 3 and 5, add 7 + 3 to make 10, and then add the remaining 5 to reach 15. Often referred to as “friends of ten,” this strategy promotes flexible thinking and strengthens mental calculation (Carpenter & Moser, 1984).
Doubles and Doubles Plus or Minus
Recognizing doubles (e.g., 6 + 6) provides a useful anchor for related addition facts. To solve 6 + 7, a child might recall that 6 + 6 = 12, then add 1 more to reach 13. This approach not only enhances arithmetic fluency but also deepens understanding of number relationships and fact families (Baroody & Dowker, 2003).
Instructional Implications
As children mature mathematically, they should be encouraged to transition from inefficient strategies such as "counting all" to more advanced and flexible ones like “make ten” or “doubles plus one.” According to Baroody and Dowker (2003), over-reliance on early-stage strategies can lead to slower calculation, reduced fluency, and difficulty with more complex mathematical tasks. Educators play a vital role in guiding this transition, using targeted instruction and practice to foster strategic thinking and conceptual understanding.
Conclusion
The progression from basic counting to sophisticated mental strategies is central to arithmetic learning. While early strategies like recounting may serve as initial stepping stones, it is critical for educators to promote more efficient and conceptually rich strategies. Doing so not only enhances fluency but also deepens children’s understanding of numbers, operations, and the structure of our number system.
Understanding Place Value: A Cornerstone of Early Numeracy
Place value is a foundational concept in early mathematics education, especially during the primary grades. It refers to the understanding that the value of a digit is determined by its position within a number. This understanding is essential for developing number sense, performing arithmetic operations, and progressing toward more advanced mathematical ideas. The significance of place value in early numeracy development is well-documented in both research and educational standards.
The Role of Place Value in Building Number Sense
A strong grasp of place value is critical to developing number sense, which includes understanding numerical relationships, estimating, and recognizing patterns. The National Research Council (2001), in its landmark report Adding It Up: Helping Children Learn Mathematics, identifies number sense as a core strand of mathematical proficiency and highlights place value as a key component in cultivating it. When students comprehend that digits represent different values depending on their position, they become more adept at making sense of large numbers and identifying numerical patterns, thus laying the groundwork for mental math and flexible thinking.
Place Value and Operational Fluency
Understanding place value is also vital for developing mathematical fluency. Fuson (1990) describes how conceptual structures for multi-unit numbers—such as tens, hundreds, and thousands—enable students to perform addition and subtraction more efficiently. With a firm grasp of place value, children are better equipped to regroup numbers, carry or borrow in multi-digit operations, and solve problems with confidence. This fluency translates into greater flexibility when applying arithmetic strategies and solving word problems.
A Prerequisite for Decimals and Fractions
Place value knowledge is not only essential for whole number operations but also forms the basis for understanding decimals and fractions. According to the Common Core State Standards for Mathematics (2010), place value concepts extend naturally into decimal notation and support students' ability to compare and compute with fractional values. Without a solid foundation in place value, students may struggle to comprehend how numbers smaller than one fit into our base-ten system.
Conclusion
Place value is a cornerstone of numeracy that supports the development of number sense, fluency, and problem-solving skills. It is also a critical stepping stone to understanding more complex mathematical concepts, such as decimals and fractions. Both research literature and national standards recognize the central role place value plays in shaping students’ mathematical competence. As such, intentional instruction in place value should remain a priority in early mathematics education.
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