Beyond Borrowing: What Effective Teachers Know About Subtraction

This research synthesizes key insights on What Effective Teachers Know About Subtraction

Introduction:

What does it mean for students to understand subtraction, not just perform the standard algorithm, but explain why regrouping works, flexibly select strategies, and connect subtraction to broader mathematical ideas? When should students be encouraged to use their strategies, and when should a particular method be emphasized? How do students’ conceptions of subtraction evolve, and what views and language best support their growth? These questions are essential for teaching subtraction in a way that builds deep, flexible mathematical thinkers.

Subtraction: Removal and Distance Views

How do subtraction's removal and distance views differ, and why must teachers understand both?
Subtraction can be understood through two principal views: the removal (take-away)
view and the distance (difference) view. In the removal view, subtraction is seen as removing the subtrahend from the minuend—taking a quantity away from a starting amount. This view aligns with the standard algorithm and is often represented with base-10 blocks. In the distance (difference) view, subtraction is understood as finding how far apart two numbers are on the number line, or the difference between two quantities. This perspective is often visualized with number lines or bar models and emphasizes the concept of distance between values, not just removal. Both views are essential for building a flexible understanding of subtraction, and students should be able to move between them depending on the context of the problem (National Council of Teachers of Mathematics [NCTM], 2016).

Language and Terminology in Subtraction

How does the vocabulary used in subtraction shape student thinking and misconceptions?
The vocabulary of subtraction shapes how students conceptualize the operation. In early grades, terms like “take away,” “how many left,” and “minus” are common for separation contexts. Students encounter words like “difference” and “compare” to describe measurement and comparison problems as they progress. In advanced stages, language such as “regroup,” “exchange,” “partition,” “compose,” “decompose,” “minuend,” and “subtrahend” becomes essential for understanding place value and the structure of subtraction (Ma, 1999; Fuson, 1992). The etymology of “subtract,” from the Latin subtrahere (“to draw from below”), reinforces the concept of removal or reduction. Using precise language, such as “regroup” instead of “borrow,” clarifies procedures and supports readiness for algebra (Yackel & Cobb, 1996).

There are two principal subtraction views: the removal view, where one quantity is removed from another, and the distance (difference) view, where the difference between two quantities is found. Both perspectives are essential for building a flexible understanding. As students encounter multi-digit subtraction, particularly in the removal view, teachers should prompt them to ask, “Do we have enough ones (or the relevant place value) to subtract?” If not, students must learn to partition each greater place value—breaking down hundreds into tens, tens into ones, etc.—to obtain sufficient units for subtraction. This recursive partitioning is at the heart of regrouping and deepens the understanding of place value structure (Brendefur et al., 2021).

The Cognitive Journey: From Fingers to Algorithms.

How do children’s earliest subtraction strategies shape their future mathematical thinking?
Subtraction is often viewed as a basic arithmetic operation, but research reveals it is a
complex cognitive process that develops through distinct stages, each with its challenges and opportunities. Before formal instruction, children naturally invent subtraction strategies—using fingers, objects, or counting up from the subtrahend. Carpenter’s research shows these invented strategies are not just temporary supports but powerful tools that, when encouraged, lead to lasting understanding and mathematical confidence (Carpenter et al., 1999). However, traditional instruction often replaces these approaches with a single algorithm, such as the standard “borrowing” method, which can stifle flexibility and adaptability (Fuson, 1992; Siegler, 1996). Siegler’s “overlapping waves” theory, closely related to progressive formalization, demonstrates that children naturally cycle through various strategies, gradually adopting more efficient ones through experience, not rote memorization (Siegler, 1996).

Persistent Misconceptions and Their Origins

Why do so many students make the same subtraction mistakes year after year—and how can we break the cycle?
Persistent misconceptions about subtraction often stem from overgeneralized rules or
incomplete mental models. Many students believe subtraction always makes numbers
smaller, a misconception that breaks down when encountering negative numbers (Ma, 1999).Place value errors are common, such as subtracting digits independently by column (e.g., solving 43 – 29 as 4 – 2 and 3 – 9 to get 24), reflecting a shaky grasp of positional notation (Battista, 2007). Misunderstandings also arise when students are unclear whether a problem asks for removal or distance (difference), especially if the language and models are not well defined in instruction. Classroom language can reinforce misconceptions; phrases like “take away” oversimplify subtraction and ignore contexts such as comparison or missing addend problems (Yackel & Cobb, 1996). Research by Hiebert, Battista, and Ma has shown that these misconceptions are persistent, especially when instruction focuses on speed and rote procedures over understanding (Hiebert & Wearne, 1986; Ma, 1999). Diagnostic tasks, such as asking students to explain why 32 – 18 is not 26, can reveal hidden gaps and guide targeted intervention (Carpenter et al., 1999).

Instructional Models: Building Conceptual Bridges

What happens when students use real-world models before memorizing algorithms?
Effective subtraction instruction, research shows, requires a careful balance of enactive, iconic, and symbolic learning experiences. The Dutch Realistic Mathematics Education (RME) framework advocates teaching subtraction through meaningful contexts (Gravemeijer,1994; Treffers, 1991). Instead of symbolic drills, students are presented with real-world problems and encouraged to invent and compare strategies. Instruction progresses from physical experiences with manipulatives to visual models such as number lines, bar models, and area models, and finally to symbolic representations and algorithms (Gravemeijer, 1994).The number line is a particularly versatile tool, as it can effectively support both the removal and distance (difference) perspectives of subtraction depending on its use. Like the base-10 model, the area model is especially effective for illustrating regrouping and place value concepts in the removal view of subtraction. In contrast, the bar model is beneficial for
representing the distance (difference) view, making the relationship between quantities visually explicit.

Grade-by-Grade Model Progression for Subtraction (K–6).

How do subtraction models evolve from cubes in kindergarten to decimals and fractions in sixth grade?
As students move through multiple grade levels, beginning in kindergarten and eventually moving into sixth grade, their understanding of subtraction is built through a progression of hands-on, visual, and symbolic models. In the earliest grades, children use concrete tools like cubes, counters, and fingers to model “taking away” and comparing quantities physically. Visual tools such as number lines and bar models help them see subtraction as removal and comparison, while simple equations connect these experiences to symbols and language. As students progress, subtraction problems increase in complexity, and the models evolve. By second and third grade, students use base-10 blocks and open number lines to explore regrouping and place value. In contrast, area and bar models support multi-digit and distance problems. In upper elementary, the focus shifts to the standard algorithm. However, visual representations of place value units, open number lines, and place value charts remain
essential, especially as students tackle decimals and fractions. This progression ensures that students can flexibly choose and explain efficient subtraction strategies by sixth grade, and have knowledge grounded in a deep understanding of number and place value (Carpenter et al., 1999).

Implications for Teaching and Curriculum.

What could you subtract from your math curriculum to make room for deeper
understanding?
The call to action for educators is clear: improving subtraction outcomes often requires subtracting, not adding. Schools should audit their curricula to eliminate redundant activities, such as excessive timed drills, to create space for deeper learning and sense-making. Teachers should be empowered through professional development emphasizing iconic models and conceptual understanding, not just procedural fluency. Classroom language—including terms like units, composing, decomposing, and partitioning—should be precise and consistently connected to visual and symbolic representations at every stage of learning (Ma, 1999; Gravemeijer, 1994). Above all, clarity in the distinction between removal and distance views and consistent use of precise language are essential for effective subtraction instruction and for preventing misconceptions.

Conclusion: Rewriting the Subtraction Narrative

How might subtraction become a tool for building reasoning and confidence, not just getting the right answer?
Subtraction is not just a rote procedure but a powerful context for developing mathematical reasoning, flexibility, and confidence. By grounding instruction in research, embracing both removal and distance views, prioritizing precise language, and addressing misconceptions head-on, educators can transform subtraction into a launchpad for deeper mathematical thinking as Carpenter reminds us, “Children’s informal strategies are not obstacles to overcome, but foundations to build upon” (Carpenter et al., 1999, p. 45). The challenge is not to teach harder but smarter—creating classrooms where subtraction is a site for reasoning, dialogue, and discovery.

References

Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Information Age.

Brendefur, J.L., Strother, S. Ismail, J., & Krone, K. (2021). Developing Mathematical Thinking for Number and Early Algebra Workbook: Number. Developing Mathematical Thinking Institute, Boise, ID.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Heinemann.

Fuson, K. C. (1992). Children's counting and concepts of number. Springer-Verlag.
Gravemeijer, K. (1994). Developing realistic mathematics education. Freudenthal Institute.

Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal
number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199–223). Lawrence Erlbaum.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum.

National Council of Teachers of Mathematics. (2016). Relational thinking: What’s the
difference? Teaching Children Mathematics, 23(5), 304–311.

Siegler, R. S. (1996). Emerging minds: The process of change in children's thinking. Oxford University Press.

Treffers, A. (1991). Didactical background of a mathematics program for primary education. In L. Streefland (Ed.), Realistic mathematics education in primary school (pp. 21–56). Freudenthal Institute.

Yackel, E., & Cobb, P. (1996). Socio mathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.

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Want to help students truly understand subtraction?
Research shows that using visual tools—like number lines and bar models—alongside precise language helps students grasp subtraction as both removal (taking away)
and distance (finding the difference between numbers). These representations connect math to real life, support flexible thinking, and make abstract ideas concrete.
Key vocabulary—such as “difference,” “distance,” “partition,” “compose,” and “decompose”— clarifies the structure of subtraction, especially when students need to partition each greater place value to regroup and solve multi-digit problems.
By combining hands-on models with explicit language, educators empower students to move confidently from concrete strategies to formal algorithms.

Let’s transform subtraction from a rote skill into a foundation for reasoning and discovery!