Few topics in the K–8 curriculum carry as much long-term weight, or cause as much difficulty, as fractions. Fractions are the first sustained encounter students have with numbers that do not behave like the counting numbers, and they mark the point at which many students’ mathematical confidence begins to erode. Yet the stakes could hardly be higher. Fraction knowledge in elementary school is one of the strongest predictors of algebra achievement and overall mathematics attainment years later, even after accounting for whole-number skill, IQ, working memory, and family background (Siegler et al., 2012; Bailey, Hoard, Nugent, & Geary, 2012). The National Mathematics Advisory Panel (2008) concluded that difficulty with fractions is pervasive and constitutes a major obstacle to further progress in mathematics, including algebra.

From a mathematics education perspective, fractions are not a single idea but a family of related meanings—part-whole, measure, quotient, operator, and ratio—that students must gradually coordinate (Kieren, 1976; Behr, Lesh, Post, & Silver, 1983). From a cognitive perspective, learning fractions requires a substantial reorganization of what a number is: students must extend their concept of number beyond counting to include quantities that can be infinitely partitioned and ordered by magnitude (Siegler, Thompson, & Schneider, 2011). Understanding both of these demands—the mathematical structure and the cognitive reorganization—is essential for learning and teaching fractions well. Simon (2006) termed these reorganizations key developmental understandings—conceptual advances that make later fraction concepts and procedures learnable rather than merely memorable.

Key Idea • Fraction knowledge is among the strongest predictors of later algebra and mathematics achievement, yet fractions demand a reorganization of what a number is, which is exactly why they are both so consequential and so difficult.

What Is a Fraction? The Subconstructs of Rational Number

A central insight from decades of research is that the symbol ¾ does not have a single meaning. Building on Kieren’s (1976) foundational analysis, Behr, Lesh, Post, and Silver (1983) described rational number as comprising several interrelated subconstructs: part-whole (3 of 4 equal parts of a whole), measure (a distance of ¾ along a number line, or ¾ as an iteration of the unit ¼), quotient (3 divided by 4, the result of sharing 3 items among 4 people), operator (a function that scales a quantity, as in ¾ of 12), and ratio (a multiplicative comparison of two quantities, 3 to 4). Competent understanding requires students to recognize these meanings, distinguish among them, and move flexibly among them. The symbol ¾ does not have a single meaning.

Instruction in the United States has traditionally overemphasized the part-whole subconstruct—typically shaded regions or “pieces of a pizza”—at the expense of the others (Charalambous & Pitta-Pantazi, 2007). While part-whole provides an accessible entry point, it is a limited foundation. It does not readily explain improper fractions (how does one shade 7 of 4 parts?), it obscures the idea that a fraction is itself a number with a location and a magnitude, and it does not connect naturally to division or measurement. Empirical work confirms that the part-whole and measure subconstructs play a particularly important role, and that the measure subconstruct in particular supports operations and equivalence (Charalambous & Pitta-Pantazi, 2007). A robust fraction curriculum therefore develops all of the subconstructs deliberately, with special attention to fractions as measured magnitudes.

Key Idea • A fraction is not one idea but several, part-whole, measure, quotient, operator, and ratio, and instruction should develop all of them, with particular attention to fractions as measured magnitudes rather than only shaded parts.

Organizing a Quantity, or Preserving It: The Shift a Model Asks

Visual models are critical to understanding and do different kinds of mathematical work. A ten frame is limited to organizing small quantities around five and ten, and a number bond shows how quantities can be composed into a total. However, there are other models that are more powerful and represent additional relationships. A bar model uses length to represent relative quantity, and a number line represents magnitude, distance, order, and measurement through position, both of which build a deep understanding of mathematics and can be used at every level of mathematics.

An early model may help students organize a small quantity. A more durable model also supports reasoning about magnitude, distance, units, and how quantities change in relation to one another. Dual coding theory suggests that learning is strengthened when verbal and visual representations work together (Clark & Paivio, 1991). But the visual representation still needs to show the mathematical structure students are expected to understand.

Key Idea • Some models organize a quantity; others preserve its magnitude, distance, and relationship. The shift is in what the model asks the student to reason about.

The Whole Number Bias: Reorganizing the Concept of Number

Perhaps the most documented obstacle in fraction learning is whole number bias: students’ tendency to apply the properties of whole numbers inappropriately to fractions (Ni & Zhou, 2005). Because students spend their early years reasoning exclusively about counting numbers, they build strong, well-practiced intuitions—larger numerals mean larger quantities, every number has a unique next number, multiplication makes bigger, and division makes smaller. Each of these intuitions fails for fractions.

Every upper-elementary teacher knows the consequences. Students judge that ⅛ is greater than ⅓ “because 8 is larger than 3.” They add ½ + ⅓ and get ⅖ by operating on numerators and denominators separately, as though each were an independent whole number. They are surprised that multiplying by ¾ yields a smaller result, or that there are infinitely many numbers between 0 and 1. The language of fractions can reinforce the bias: naming ⅓ as “one out of three” frames it as a count of parts and obscures that it is a single number, whereas naming it “one-third,” like “one-half,” treats the fraction as one quantity (Paik & Mix, 2003). Ni and Zhou (2005) argue that whole-number bias has both a natural cognitive origin—count-based number sense emerges early and is heavily reinforced—and an instructional origin, in curricula that introduce fraction procedures before students have reconceived what these new numbers are. From this perspective, the aim of assessment and instruction together is not merely to add new rules but to help students reorganize their concept of number so that whole numbers and fractions are understood as points on a common continuum; a diagnostic that surfaces where that reorganization has stalled makes the learning teachable.

Key Idea • The whole number bias, importing whole-number rules into fractions, is the central obstacle in fraction learning; the aim of diagnosis and instruction together is not more rules but a reorganization of number so that whole numbers and fractions belong to a common continuum.

CLASSROOM SNAPSHOT

Teacher: “Which is larger, ⅓ or ¼?”
Student A: “¼, because 4 is larger than 3.”
Student B: “⅓. If you cut one into 3 equal parts, each part is larger than if you cut it into 4.”
Teacher: “How could we settle it?”
Student B: “Put both on a number line from 0 to 1. ⅓ is farther from 0.”

Both students can compute; they differ in whether they reason about the size of the unit fraction or the size of the denominator—the center of the whole-number bias.

Magnitude and the Number Line: An Integrated Theory of Numerical Development

The most influential recent framework here is Siegler’s integrated theory of numerical development (Siegler, Thompson, & Schneider, 2011; Siegler, Fazio, Bailey, & Zhou, 2013). Its central claim is simple: what unifies whole numbers, fractions, decimals, and negatives is that they all have magnitudes that can be located on a number line. Numerical development, on this account, is the process of progressively broadening the set of numbers whose magnitudes a learner can represent and order, and of coming to understand that all real numbers share this property.

This theory reframes fractions as the decisive test case for theories of numerical development, because fractions are precisely where the count-based conception of number breaks down and a magnitude-based conception must take over (Siegler et al., 2013). Empirically, the accuracy with which students place fractions on a number line is strongly associated with overall fraction knowledge and with broader mathematics achievement, often more strongly than part-whole or procedural measures (Siegler et al., 2011). The number line is powerful precisely because it represents the measure subconstruct directly: it shows that ¾ is a single number with a definite size, that fractions can be compared as lengths, and that whole numbers and fractions live on the same continuum. For testing and instruction alike, this implies that the number line should be a central, not incidental, representation in fraction learning. Benchmarks of 0, ½, and 1 serve as anchors for this reasoning: a student who knows that ¾ lies nearer 1 than ½, or that ⅛ lies nearer 0, has a rough sense of size that constrains finer comparisons and flags an impossible result such as ½ + ⅓ = ⅖. Consistent with this, in a longitudinal study of 357 children, number-line estimation emerged as the strongest independent predictor of later fraction concepts and procedures among the cognitive and mathematical variables examined (Jordan et al., 2013).

Key Idea • What unifies whole numbers and fractions is that both have magnitudes locatable on a number line; number-line placement is among the strongest predictors of fraction knowledge, which is why the number line belongs at the center of learning and instruction.

Developmental Foundations: Equal Sharing, Partitioning, and Unitizing

The number line makes a fraction’s magnitude visible, but students can use it meaningfully only after they build the understandings beneath it, especially partitioning and unitizing. Long before formal fraction symbols appear, young children possess intuitive resources for reasoning about fair shares. Research in the tradition of Cognitively Guided Instruction shows that children as young as first grade can solve equal-sharing problems, dividing, say, several apples among a group of people, and that these informal partitioning strategies form a productive foundation for fraction concepts (Empson, 1999). In Empson’s (1999) study, sustained work with equal-sharing tasks helped young students develop meaning for fractional quantities and for the crucial idea that the units must be equal.

Two developmental ideas are central. The first is partitioning: dividing a quantity into equal units and coordinating the number of units with their size, so that students come to see that more units (parts) means smaller units (parts). The second is unitizing: understanding what counts as “one” and recognizing that the unit can be flexibly redefined—that ¾ can be seen as three iterations of the unit ¼, or that a group of objects can itself be treated as a single unit (Behr et al., 1983; Lamon, 2007). Analyses of children’s fraction schemes describe how learners construct increasingly sophisticated conceptions of the unit and of iteration over time, and how these schemes underpin later reasoning about equivalence and operations (Steffe & Olive, 2010). Realistic Mathematics Education similarly emphasizes building fractions from meaningful contexts of fair sharing and measurement before formalizing them (Streefland, 1991). Learning-trajectory research in the measurement tradition places this unit-and-iteration relationship at the very center of early fraction development, treating fraction-as-measure, rather than shaded parts of objects, as a foundational understanding (Simon, 2006; Simon, Placa, Avitzur, & Kara, 2018). The instructional implication is that fraction understanding should be built from partitioning and sharing activity, not introduced as ready-made symbolic rules.

Key Idea • Fractions should be built from children’s own partitioning and equal-sharing, establishing the unit, then iterating it, rather than introduced as ready-made symbolic rules.

Cognitive Foundations: Conceptual and Procedural Knowledge

Why is fraction arithmetic, in particular, so persistently difficult? Lortie-Forgues, Tian, and Siegler (2015) offer a detailed analysis pointing to both inherent and culturally contingent causes. Some difficulty is inherent: fraction operations violate whole-number expectations (multiplication can make smaller), the algorithms are notational and non-transparent (invert-and-multiply has no obvious meaning), and different operations require different treatments of the denominator. Other difficulties are contingent on how fractions are taught, including limited emphasis on magnitude and heavy reliance on memorized procedures.

This connects to a broader theme in mathematics learning: the relationship between conceptual knowledge (understanding of principles and relationships) and procedural knowledge (knowledge of steps and algorithms). Both matter, and in fractions they appear to develop in a mutually reinforcing, bidirectional relationship, with conceptual understanding of magnitude supporting the meaningful use and retention of procedures (Hecht & Vagi, 2010). When procedures are learned in isolation from magnitude—when ½ + ⅓ is a symbol-manipulation task disconnected from any sense of size—students have no way to detect that an answer of ⅖ is impossible, because it is smaller than one of the addends. Similarly, when asked to estimate fraction sums, a large share of students in Grades 4–8 produce answers smaller than one of the addends being combined—direct evidence that fraction procedures can develop without magnitude-based constraints (Braithwaite, Tian, & Siegler, 2018). Structured, magnitude-grounded representations reduce this kind of error by giving students a conceptual check on procedural work, echoing the well-established finding that learning organized around underlying structure transfers more readily than learning organized around surface features (Ni & Zhou, 2005; Lortie-Forgues et al., 2015).

Key Idea • Conceptual and procedural knowledge grow together; when procedures are learned apart from magnitude, students cannot tell an impossible answer from a correct one, which is why magnitude must anchor computation.

Building the Fraction Math Learning Profile: What the Diagnostic Should Measure

The research reviewed here leads to a practical question: if understanding fractions is this consequential and this fragile, what should a diagnostic actually measure? A Math Learning Profile (MLP) for fractions is not a score on a page of computation items; it is a map of the developmental understandings a student has, and has not yet, constructed, built so that a teacher can see where a student’s reasoning begins to break down. The literature points to a small set of constructs that predict later fraction learning and that together define what such a profile should assess.

Five understandings are primary, because they are the ideas that fraction concepts and operations rest upon. The first is the unit of one: whether a student recognizes what has been defined as the unit of one, understands that every fraction is named in relation to it, and can reconstruct the unit of one when only a fractional part is shown, for example, rebuilding one from three-fourths (Lamon, 2007). The second is partitioning: whether a student can partition the unit of one into equal-sized fractional units and reason that the number of equal partitions determines the size of each unit, so that more partitions produce smaller units (Lamon, 2007). The third is iterating: whether a student understands 1/b as a fractional unit made by partitioning and can iterate it to build a/b, coordinating the denominator as the unit size and the numerator as the count, including quantities greater than one (Behr, Lesh, Post, & Silver, 1983). The fourth, and the most predictive, is fraction magnitude: whether a student treats a fraction as a single number with a size, can place it on a number line, and can compare it against benchmarks such as 0, ½, and 1 (Siegler, Thompson, & Schneider, 2011; Jordan et al., 2013). The fifth is equivalence: whether a student recognizes the same magnitude expressed with different-sized fractional units, coordinating a change in unit size with the corresponding change in the number of units, rather than as the output of a rule applied to numerator and denominator (Charalambous & Pitta-Pantazi, 2007). Equivalence coordinates the change in unit size with the change in the number of units, so that as fourths become eighths the three units become six, and it is reversible: a student can partition back into larger units to recover ¾ from ⁶⁄₈ (Behr, Lesh, Post, & Silver, 1983).

Two further understandings are drawn from neighboring constructs and assumed rather than taught here: the quotient meaning of a fraction, which grows out of equal-sharing and belongs to work on division (Empson, 1999), and the operator and ratio meanings, which draw on multiplicative comparison and belong to Scaling and Comparing. A diagnostic built on these constructs should probe each in more than one representation, region, length, and number line, and should surface the whole number bias directly, because a student who places ⅛ beyond ⅓ has revealed a specific, addressable misconception rather than a wrong answer. Read across the developmental trajectory, these constructs locate not merely whether a student can do fractions today, but which understanding, if strengthened, would change what the student can do next.

Key Idea • A fraction Math Learning Profile should be built on the developmental understandings that predict later learning, the unit of one, partitioning, iterating, fraction magnitude, and equivalence, each probed across representations, so that the diagnostic identifies which understanding, not merely which answer, a student is still constructing.

Conclusion

Fractions occupy a pivotal place in the K–8 curriculum because they demand something new: a reorganization of the concept of number to include quantities that are partitioned, ordered by magnitude, and endowed with multiple interrelated meanings. From a mathematics education perspective, helping students learn fractions well means developing the full family of subconstructs—part-whole, measure, quotient, operator, and ratio—with particular emphasis on fractions as measured magnitudes. From a cognitive perspective, it means helping students overcome whole-number bias and locate fractions on a shared number line, so conceptual understanding of magnitude can anchor and give meaning to procedures.

Despite decades of research, fraction teaching in many classrooms remains heavily procedural and centered on part-whole, and the resulting difficulty persists into algebra and beyond; this is not inevitable but a solvable instructional-design problem. The payoff is substantial. Because early fraction knowledge is among the strongest predictors of later algebra and mathematics achievement, sustained investment in conceptual, magnitude-grounded fraction learning and instruction in the elementary and middle grades is time well spent. When students come to see a fraction as a number with a size—located on the same continuum as the whole numbers they already know—they gain not only competence with fractions but a foundation for the mathematics that follows.

Key Idea • Fractions reorganize number from counting to magnitude, and a wrong answer tells us less than which understanding a student is still constructing.

References

Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447–455.

Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational-number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–126). Academic Press.

Braithwaite, D. W., Tian, J., & Siegler, R. S. (2018). Do children understand fraction addition? Developmental Science, 21(4), e12601.

Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316.

Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 283–342.

Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116(1), 45–58.

Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101–144). ERIC/SMEAC.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age.

Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221.

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.

Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.

Paik, J. H., & Mix, K. S. (2003). U.S. and Korean children’s comprehension of fraction names: A reexamination of cross-national differences. Child Development, 74(1), 144–164.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697.

Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13–19.

Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296.

Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359–371.

Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the Learning Through Activity research program. The Journal of Mathematical Behavior, 52, 122–133.

Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer.

Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Kluwer.

Social Media

How can a student rename ¾ as ⁶⁄₈, and still not know whether ¾ is greater than ½?

Because renaming a fraction is not the same as understanding its size. For years, students read a numerator and a denominator as two separate whole numbers—long before they come to see a fraction as a single quantity with a place on the number line, so they judge ⅛ to be greater than ⅓ and add ½ and ⅓ to get ⅖. Our new DMT Insight explains the developmental understandings beneath fraction concepts and operations, why they are among the strongest predictors of later achievement, and what a diagnostic built to measure them should look for.

Inside this Insight, you will learn:

• Why whole-number reasoning quietly carries over into fractions, and how to surface it

• How a fraction is built by partitioning one into equal units and iterating a unit fraction

• Why the number line represents a fraction as a magnitude and predicts later achievement

• Which key constructs and predictors a fraction Math Learning Profile should be built upon

For more information, contact@dmtinstitute.com, or follow us:

● Facebook facebook.com/dmtinstitute

● Instagram instagram.com/dmtinstitute

● X x.com/dmtinstitute

● YouTube youtube.com/dmtinstitute

● LinkedIn linkedin.com/company/dmt-institute

● Bluesky bsky.app/profile/dmtinstitute.bsky.social

● TikTok tiktok.com/@math.success

● Substack mathsuccess.dmtinstitute.com

Visit us at mathsuccess.io and dmtinstitute.com.

A student can compute 503 − 297 correctly, line up the digits, regroup across the zero, and arrive at 206, yet still have little sense of the size of the answer. The procedure is correct, but the student may not understand the quantities or their relationship. This is one reason the choice of model matters.

Students do not learn mathematics only by hearing words or memorizing symbols. They develop mental representations of quantities, structures, and relationships that help them make sense of symbols. The model a student uses can strengthen that understanding, but only if it represents the mathematics we want students to notice and learn. Some models work well with small quantities and early ideas but quickly become useless as numbers and relationships become more complex. The important question is not whether a model works for one lesson, but which mathematical models are durable and will grow with students and the mathematics over time.

CLASSROOM SNAPSHOT

Teacher: How much is 503 minus 297?
Maya: 206. I lined up the digits and regrouped.
Teacher: About how large should the answer be?
Maya: I’m not sure. I’d have to do the steps again.
Diego: A little more than 200. 297 is close to 300, and 300 to 503 is about 203.
Teacher: How did you see that?
Diego: I thought about how far apart they are on the number line.

Maya’s answer is correct. But Diego is reasoning about the magnitude of the numbers and the distance between them. The number line supports that reasoning.

Key Idea • A correct answer can hide a missing sense of quantity. The model a student reasons on decides whether number sense is built, and a weak model hides understanding.

Organizing a Quantity, or Preserving It: The Shift a Model Asks

Visual models are critical to understanding and do different kinds of mathematical work. A ten frame is limited to organizing small quantities around five and ten, and a number bond shows how quantities can be composed into a total. However, there are other models that are more powerful and represent additional relationships. A bar model uses length to represent relative quantity, and a number line represents magnitude, distance, order, and measurement through position, both of which build a deep understanding of mathematics and can be used at every level of mathematics.

An early model may help students organize a small quantity. A more durable model also supports reasoning about magnitude, distance, units, and how quantities change in relation to one another. Dual coding theory suggests that learning is strengthened when verbal and visual representations work together (Clark & Paivio, 1991). But the visual representation still needs to show the mathematical structure students are expected to understand.

Key Idea • Some models organize a quantity; others preserve its magnitude, distance, and relationship. The shift is in what the model asks the student to reason about.

Iconic and Symbolic: What a Durable Model Must Hold

A durable model should represent several important mathematical ideas. It should show magnitude, so students can reason about how large a quantity is; distance, so they can compare how far apart quantities are; and the unit, so quantities can be composed through iteration and decomposed through partitioning. It should also support comparison and proportional reasoning. Models that represent these ideas can be used across a wider range of mathematics.

Models also differ in what they represent. Bruner (1966) distinguished iconic representations, which represent aspects of a quantity, from symbolic representations, which rely more heavily on convention. A ten frame, bar model, and number line provide visual information about quantity. A number bond is closer to a symbolic representation because the circles themselves do not show magnitude. A circle labeled 3 can be the same size as one labeled 30. Both models can show composition, but they do not provide students with the same quantitative information.

Key Idea • A durable model holds magnitude, distance, the unit, comparison, and proportion. An iconic model shows a quantity; a symbolic one only stands for it, and that difference sets their reach

Two Pathways: Why the Right Model Lightens the Load

From a cognitive psychology perspective, a model can reduce what students have to hold in working memory. Dual coding provides both visual-spatial and symbolic representations, so a position on a number line and its numeral can support the same idea (Clark & Paivio, 1991). When order, distance, and scale are visible in the model, students can devote more attention to the mathematical relationship they are trying to understand.

A consistent model can also help students develop a schema across related ideas. For example, aligned representations can help students see 3/4, 0.75, and 75% as different representations of the same magnitude rather than as three unrelated procedures. From a mathematics education perspective, this is important because students can continue to use the model to reason as the content becomes more complex. The model is useful when it makes the underlying mathematical structure visible. So, a ten-frame model is limited to numbers ten or less, and a number bond can help with composing and decomposing but also can lead students to misconceptions because the model does not help with magnitude or proportion. The number line and bar model are much more durable over time.

Key Idea • The right model gives a quantity two pathways and holds its structure on the page. Working memory is freed, and a stable schema can form.

Low Ceiling and High Ceiling: Where Each Model Stops

Using these criteria, the four common models have different levels of reach. Ten frames and number bonds have a relatively low ceiling. Bar models and number lines have a higher ceiling because they continue to represent important relationships as the mathematics becomes more complex. This is a difference in purpose and reach, not simply a judgment that one model is good and another is bad.

Ten frames are useful for early number. They support subitizing, recognizing a small quantity without counting each item, and help students use five and ten as benchmarks. Clements (1999) describes the importance of seeing small quantities as organized groups rather than counting each item. The limitation is that a ten frame is fixed at ten and uses discrete cells. It can show that 8 is two less than 10, but it does not directly represent distance, scale, or proportional relationships such as the connection among 8/10, 0.8, and 80%.

Number bonds show the composition and decomposition of a total and can be useful for reasoning about a missing quantity. Their limitation is that the common circle-and-line format does not represent magnitude. A circle labeled 3 looks the same size as a circle labeled 30, so the quantitative information comes from the numerals rather than the model itself. Number bonds can organize part-total relationships, but they provide little information about distance, scale, or proportion. There is also very little research examining number bonds as a distinct instructional model.

Bar models represent quantity with length. This allows students to use the same model for total-and-quantity relationships, comparison, equal groups, and ratio. The structure of the situation can be represented before students choose an operation. Ng and Lee (2009) found that bar diagrams supported children in representing and solving algebraic word problems before they used formal symbolic algebra. This is one reason the model has greater reach: the same basic representation can support whole-number comparison in elementary school and more complex relationships in later grades.

Number lines have the strongest research base of the four. Siegler, Thompson, and Schneider (2011) place knowledge of numerical magnitude at the center of numerical development and describe the number line as a single structure that encodes whole numbers, fractions, decimals, and negative numbers as positions along a continuous magnitude. The National Mathematics Advisory Panel (2008) identified proficiency with fractions as foundational for algebra and highlighted the need to locate and compare fractions, decimals, and percents by magnitude. Resnick, Newcombe, and Goldwater (2023) found that reasoning about fraction and decimal magnitudes accounted for the relation between proportional reasoning and computation, and they read their results as supporting measurement models and spatial scaling. In the largest test of the idea to date, Nuraydin, Stricker, Ugen, Martin, and Schneider (2023) assessed nearly an entire national cohort of ninth-graders (N = 6,484) on a single number-line estimation task and found accuracy strongly associated with a standardized measure of mathematical achievement, with placement of fractions more closely related to achievement than placement of whole numbers.

Key Idea • Ten frames and number bonds organize early number but hold no magnitude. Bar models and number lines encode magnitude, units, comparison, and proportion, which is why they keep working.

A Time-Limited Job: A Progression That Scales

A strong progression does not require abandoning ten frames and number bonds. It means using them for the ideas they represent well and then moving students toward models with greater mathematical reach. Ten frames are particularly useful for subitizing and benchmark reasoning around five and ten, while number bonds can support early composition and decomposition. As students begin reasoning more explicitly about magnitude, comparison, units, fractions, decimals, multiplication, division, and ratio, bar models and number lines should take a more central role. Double number lines, ratio bars, area models, and equations can then extend this reasoning into middle school and beyond.

There is also a mismatch between common classroom use and the research base. Number lines have a substantial research base, and bar models have research supporting their use for representing mathematical relationships. Ten frames and especially number bonds have a more limited dedicated research base. Lesh, Post, and Behr (1987) emphasize the importance of students moving among representations and translating between them. For that work to be productive, the representations need to make important mathematical relationships visible.

Key Idea • Give the early models a short, defined job, and give bar models and number lines the central, sustained role. The durable models are the least used and the best supported.

Conclusion

The four models are not interchangeable. Ten frames and number bonds are useful for particular early ideas, but they do not directly represent magnitude, distance, scale, or proportional relationships. Bar models and number lines illustrate this structure more clearly. They can support reasoning about quantity, units, comparison, iteration, partitioning, and scaling across a broader range of mathematical concepts.

For teachers, coaches, and leaders, the implication is straightforward: choose a model based on the mathematical ideas it represents. Ask what students can see and reason about with the model, what the model does not show, and whether it will continue to be useful as the mathematics develops. From this perspective, number lines and bar models should have a central role from early number through fractions, proportional reasoning, and algebra.

Key Idea • Choose models for what they hold. The number line and the bar model preserve magnitude and relationships from early number to algebra, and that is the structure on which a coherent pathway is built.

References

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Clark, J. M., & Paivio, A. (1991). Dual coding theory and education. Educational Psychology Review, 3(3), 149–210.

Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5(7), 400–405.

Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Erlbaum.

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.

Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313.

Nuraydin, S., Stricker, J., Ugen, S., Martin, R., & Schneider, M. (2023). The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6,484 Luxembourgish ninth-graders. Journal of Experimental Child Psychology, 225, 105521.

Resnick, I., Newcombe, N., & Goldwater, M. (2023). Reasoning about fraction and decimal magnitudes, reasoning proportionally, and mathematics achievement in Australia and the United States. Journal of Numerical Cognition, 9(1), 222–239.

Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296.

Social Media

Not all math models deserve the same amount of instructional time.

Ten frames and number bonds can be useful tools in early number work. They can help young students see small quantities, compose and decompose numbers, and make basic part–whole relationships visible.

But they have limits.

As numbers become larger and students encounter subtraction, fractions, decimals, ratios, and algebraic relationships, those models often stop revealing the mathematics students need to see. A ten frame is constrained by its fixed structure. A number bond can show that a quantity is made of parts, but it does not show magnitude, distance, comparison, or the size of a relationship.

That is why the number line and the bar model should play the central, sustained role in mathematics instruction.

The number line makes number magnitude visible. It represents:

• The size and order of numbers

• Distance and difference

• Benchmark numbers and estimation

• Fractions and decimals as numbers with locations and size

• Integers and distance from zero

• Multiplicative and proportional relationships

The bar model makes relationships between quantities visible. It supports students in representing:

• Parts and wholes

• Comparisons between quantities

• Unknown quantities

• Multiplication and division situations

• Fractions, ratios, and percent

• Equations and early algebraic reasoning

These are not just elementary-school visuals. They are durable models that grow with students. They offer a coherent visual language for reasoning across grades rather than a collection of disconnected tools for individual lessons.

Our new DMT Insight explains why schools should be intentional about which models become central in the curriculum.

For more information, contact@dmtinstitute.com, or follow us:

● Facebook facebook.com/dmtinstitute

● Instagram instagram.com/dmtinstitute

● X x.com/dmtinstitute

● YouTube youtube.com/dmtinstitute

● LinkedIn linkedin.com/company/dmt-institute

● Bluesky bsky.app/profile/dmtinstitute.bsky.social

● TikTok tiktok.com/@math.success

● Substack mathsuccess.dmtinstitute.com

Visit us at mathsuccess.io and dmtinstitute.com.

Students today have more access to mathematical support tools than at any point in history. Calculators solve computations instantly. Adaptive software guides students step by step through procedures. Artificial intelligence systems generate explanations, hints, and complete solutions within seconds. In many classrooms, students appear highly successful while using these supports. Yet teachers across all grades continue to report the same concerns: students struggle to retain learning, transfer ideas to new contexts, explain relationships, or reason flexibly when supports are removed.

The issue is not simply technology itself. The deeper issue is cognitive outsourcing — when tools begin performing the very mental work students still need to develop internally. Recent work in cognitive psychology, neuroscience, and mathematics education suggests that lasting mathematical understanding develops through effortful cognitive activity: organizing relationships, constructing models, coordinating units, retrieving connected ideas, and reasoning across multiple representations.

Horvath’s recent work in The Digital Delusion (Horvath, 2024) argues that modern digital environments often prioritize speed, convenience, and task completion over understanding and memory formation. This is not an argument against all technology. Adaptive practice that requires active recall and student-constructed responses is different from passive, recognition-based, or step-by-step scaffolding that bypasses cognitive effort. Mathematics may be especially vulnerable because mathematics is cumulative and hierarchical. Students cannot reason proportionally if multiplicative relationships remain fragile. They cannot solve equations flexibly if equivalence and operations overload working memory. They cannot model relationships if quantities and units are not meaningful.

For teachers, the central question is not whether technology should exist in your classroom. The question is: What kinds of mathematical experiences actually help students build connected mathematical understanding and how can a workbook, pencil, and student-drawn models play a non-negotiable role in that process?.

Mathematical Learning Depends on Connected Structures

For many years, mathematics instruction was often framed as a tension between “understanding” and “memorization.” Cognitive science research now suggests this is a false distinction. Understanding itself depends heavily on students building connected structures of meaning over time.

Working memory refers to the limited amount of information the brain can actively process at one time (Baddeley, 1992). Cognitive Load Theory suggests that when too many elements must be coordinated simultaneously, students experience overload and performance declines (Sweller, 1988). Mathematics places particularly heavy demands on working memory because students must coordinate quantities, operations, symbols, units, and relationships simultaneously.

Consider a student solving the equation ¾x + 5 = 17. A student must coordinate the magnitudes of fractions, fraction equivalence, inverse operations, symbolic structure, and procedural reasoning simultaneously. A proportional bar model can help students see the structure of the equation before manipulating symbols procedurally. In the first bar, the full value of x is partitioned into four equal ¼-units, marked by small ticks above. The 17 spans the right three ¼-units of x together with the 5, showing that ¾x + 5 = 17 — and therefore that ¾x must equal 12.

Figure 1. The full value of x is partitioned into four equal ¼-units. The 17 spans the right three ¼-units together with the 5.

Once students see that ¾x = 12, the next move follows naturally: ¾x represents three equal ¼-units of x, so each ¼-unit must equal 4. The second bar isolates the full value of x and brackets the same three ¼-units, this time labeled 12. From there, students can see that the fourth ¼-unit must also equal 4, so the whole x equals 16.

Figure 2. Isolating the full value of x. The same three ¼-units are now labeled 12, revealing that each ¼-unit must equal 4, and the whole x must equal 16.

This type of progression helps students reason structurally rather than memorize isolated steps. Students see that ¾x + 5 = 17 leads to ¾x = 12, that ¼x = 4, and that x = 16 — not as four separate procedural moves, but as a single connected story about how parts compose a whole.

This issue extends far beyond basic fact fluency. Successful mathematical reasoning depends on students building connected understanding of place-value relationships, fraction magnitude, multiplicative structures, proportional relationships, geometric composition, and symbolic equivalence.

Mathematical understanding develops more like weaving than stacking. Students do not simply place one skill on top of another; they return to important ideas, connect new strands, tighten relationships, and gradually create a stronger structure. Learning is recursive. Concepts are revisited and reorganized through repeated opportunities to model, explain, compare, discuss, and generalize.

Students do not learn mathematics deeply through exposure alone. They learn by organizing and reorganizing meaningful relationships over time.

Consider what happens when a fifth grader first encounters equivalent fractions. A student who has only watched examples on a screen may recognize that ½ and 2/4 are equivalent because they have seen the pairing many times. However, a student who has partitioned bars, folded strips of paper, drawn number lines, and explained why both fractions name the same point on a number line has built a connected schema. When that student later meets 3/6, 4/8, or 50/100, they do not need to memorize each case they recognize the underlying multiplicative structure. This is why your students forget by Friday if they only watched on a screen. Cognitive psychologists describe this as the development of relational understanding — the ability to see how ideas connect rather than treating each procedure as an isolated rule (Hiebert & Carpenter, 1992; Skemp, 1976).

Key Idea • Mathematical understanding develops when students build connected structures of meaning that support reasoning, transfer, and flexible problem solving.

Why Representation Matters in Mathematics

Mathematics is abstract, but students do not learn abstraction directly. Understanding develops gradually as students move from action and experience toward increasingly sophisticated representations.

Jerome Bruner (1966) described learning as progressing through three forms of representation: enactive (acting), iconic (visual representation), and symbolic (abstract notation). This progression is especially important in mathematics because symbols alone often conceal structure. Students may manipulate symbols procedurally without understanding the relationships those symbols represent.

Math models help bridge this gap. Number lines, bar models, area models, and partitioned visual structures allow students to see mathematical relationships before expressing them symbolically. These models are not decorative supports. They are cognitive tools that reduce abstraction while preserving mathematical structure.

Here is what the research says for your teaching: Students benefit most when they physically construct these models themselves — not when they watch you draw them, not when a screen animates them. Research comparing handwriting and drawing with typing suggests that physically constructing representations activates broader neural systems involving language, motor processing, spatial reasoning, attention, and memory encoding (Horvath, 2024). Drawing number lines and proportional bar models requires students to decide scale, partition space, coordinate quantities proportionally, and organize relationships visually.

This physical construction process supports attention, strengthens encoding, links motor activity with visual-spatial reasoning, and reduces passive recognition. Students actively organize relationships rather than simply selecting answers on a screen.

When a student draws a flawed number line — uneven spacing, mislabeled partitions — that mistake is a gift. You can see their thinking. You can intervene. Digital tools often hide these errors behind a correct “final answer.”

Students who never internalize multiplicative relationships, the magnitude of equivalent fractions, or symbolic structure often face major difficulties later, even if earlier digital performance appeared successful. This is one reason students need to draw number lines and proportional bar models themselves; the act of constructing these models supports reasoning, helps students coordinate quantities, and builds structural understanding.

Paivio’s Dual Coding Theory (1990) further suggests that learning is strengthened when information is represented both verbally and visually. In mathematics, students who coordinate symbolic notation with drawing representations build richer and more connected mathematical understanding over time.

Key Idea • Drawing representations and math models help students construct, organize, and internalize mathematical relationships.

Retrieval, Recall, and the Development of Transfer

One of the strongest findings in cognitive psychology is that retrieval strengthens learning. Students do not build mathematical understanding simply by seeing information repeatedly. Learning strengthens when students actively retrieve relationships, reconstruct ideas, and apply knowledge across contexts.

Many digital learning environments emphasize recognition rather than recall. Students identify correct answers from choices, follow guided prompts, or receive immediate scaffolding that reduces cognitive demand. These supports can improve short-term performance while simultaneously reducing the retrieval effort necessary for long-term understanding.

Here is the hard truth for teachers: If a student can solve a problem with step-by-step digital hints but cannot solve a similar problem on paper the next day, they have not learned mathematics. They have learned to follow a script.

Students need opportunities to explain relationships, reconstruct math models, write equations, compare strategies, justify reasoning, and solve problems without immediate digital support. Repeated retrieval strengthens the organization of mathematical ideas and supports transfer to new situations.

This is where a well-designed workbook matters. A workbook that only asks for final answers in blank boxes is not the same as one that requires students to draw, label, and explain. Avoid pages that reduce mathematics to fill-in-the-blank recognition. Varied practice worksheets — the kind where students draw, label, write equations, and explain are not simply procedural repetition. They help students compare relationships, recognize underlying structures, and coordinate ideas flexibly across multiple situations. They require retrieval. They do not offer a bank of answers to recognize.

Transfer develops when students encounter the same mathematical structure through multiple representations, contexts, and problem types over time.

Key Idea • Transfer develops when students repeatedly retrieve and apply mathematical relationships across models, contexts, and problem structures.

Conclusion: Building Mathematical Thinkers

Research from mathematics education, cognitive psychology, and neuroscience consistently points toward the same conclusion: students learn mathematics most deeply when they actively construct relationships, organize ideas, draw models, retrieve connected understanding, and refine structures over time.

Horvath (2024) makes this point especially forcefully for younger learners. When students are young, the act of physically drawing, writing down definitions, and constructing representations by hand helps build neural structures that do not form the same way through passive viewing. I’ve watched a sixth grader erase a bar model four times because the scale was wrong — and that struggle taught more than any hint button ever could. The motor act of writing — the deliberate shaping of letters, numbers, and diagrams — engages systems in the developing brain that support attention, memory, and meaning. In other words, when a child writes a definition or draws a bar model, they are helping to build the mental architecture that will hold that idea later. Skipping this step in the name of efficiency comes at a cost that does not show up immediately but accumulates over the years.

As a teacher, you do not need to abandon technology. You do need to stop letting it do the cognitive work your students still need to do themselves.

Students need opportunities to draw and model, compose and decompose units, partition and iterate quantities, compare strategies, justify reasoning, and explain structure. These experiences help students build understanding that is transferable, flexible, and usable in unfamiliar situations.

For example, a student who has built a strong understanding of multiplicative reasoning through bar models and ratio tables in fourth and fifth grade should be able to apply that same structural thinking when they meet proportional reasoning in sixth grade, slope in seventh grade, and linear functions in eighth grade. The surface details change — the context shifts from sharing cookies to mixing paint to comparing speeds — but the underlying multiplicative structure remains the same. Students who recognize that structure transfer their reasoning. Students who only memorize procedures start over each year.

Your workbook, pencil, and student-drawn number lines and bar models are not old-fashioned. They are evidence-based. They are the difference between a student who recognizes an answer and a student who truly understands.

Ultimately, the goal of mathematics instruction is not simply efficient task completion. The goal is to develop students who can think mathematically — students who can recognize structure, transfer understanding, coordinate relationships, and reason flexibly across contexts.

Key Idea • Students learn mathematics by constructing, organizing, retrieving, and refining connected ideas over time. A pencil and a blank page are still two of the most powerful tools in your classroom.

References

Baddeley, A. (1992). Working memory. Science, 255(5044), 556–559.

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A five-year longitudinal study. Developmental Psychology, 47(6), 1539–1552.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Horvath, J. C. (2024). The digital delusion: How modern technology is reshaping the way we think and learn.

Paivio, A. (1990). Mental representations: A dual coding approach. Oxford University Press.

Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.

Students who can compute fluently often stall the moment a problem appears in words. They know how to add, subtract, multiply, or divide, but when those same operations are embedded in language, the task becomes much harder. The issue is rarely the arithmetic itself — it is that students lack a way to organize the quantities and relationships in front of them. In many classrooms, students are taught to rely on keywords rather than analyze how quantities are connected, leading to guesses rather than reasoning.

Bar models also called strip diagrams or tape diagrams address this gap by representing quantities as lengths. Length is spatial, visible, and comparable. Students can see how one quantity is part of another, how two quantities combine, or how one exceeds another. Instead of asking, “What word tells me to add or subtract?” students begin asking, “What is the total? What are the parts? What is known? What is missing?” That is a far more powerful mathematical habit of mind (Ng & Lee, 2009).

Key Idea •Most students who struggle with word problems aren’t failing at computation — they’re failing to see the structure of the situation. By third grade, most students can name the places in a multi-digit number. They can point to the tens place in 347 and the hundreds place in 5,026. Naming positions, however, is not the same as understanding them, and the gap between the two becomes consequential in grades 3–6, where the mathematics begins to demand that students reason about the quantities the digits represent. What matters in these grades is whether the student understands what units the digits represent, how those units are composed, and where the number sits in the broader number system (Fuson, 1990).

Place value is the structure of the base-ten system. It allows any whole number and, later, any decimal to be written with ten digits because the value of each digit depends on its position. Understanding that structure requires coordinating three ideas at once: a digit tells how many units there are, the place tells what kind of unit is being counted, and the number as a whole names a single quantity. In 526, the 5 is not “five” but five units of hundred, the 2 is two units of ten, and 526 is one quantity rather than three separate digits. Coordinating these ideas is a developmental achievement that takes years to build (Ross, 1989).

This overview treats place value as two intertwined understandings — units, or how quantities are composed and decomposed, and magnitude, or how large a number is and its location — and frames them for formative assessment. For each developmental understanding, it describes what the understanding is, why it predicts later achievement, and the kinds of items that reveal whether a student holds it. The aim is to give teachers in grades 3–6 a way to diagnose place-value reasoning, not just confirm that students can label positions.

Key Idea •Place value is the ability to reason about the units a digit represents and the magnitude of the number — two understandings that go well beyond naming positions.

Why Place Value Predicts Later Achievement

Place-value understanding is not a topic completed in the primary grades; it is a way of reasoning that the mathematics of grades 3–8 continues to draw on. Multi-digit addition and subtraction require decomposing and recomposing units — computing 503 − 178, for instance, means partitioning one unit of hundred into ten units of ten. Multiplication and division require working with composite units. Fractions and decimals require nested units and magnitude. Measurement requires iterating units, and algebra requires seeing numbers and expressions as composed quantities. When place value is shallow, students can often succeed on simple cases while failing as numbers grow, as regrouping is required, or as decimals, fractions, and powers of ten enter (National Research Council, 2001).

The predictive evidence is strong and specific. In a longitudinal study following students from first through fifth grade, the accuracy of children’s number-line placements — a direct measure of magnitude understanding — uniquely predicted mathematics achievement and its growth, even after controlling for intelligence, working memory, and processing speed (Geary, 2011). A meta-analysis of 263 effect sizes from more than 10,000 students found that number-line estimation correlates moderately and consistently with broader mathematical competence, with the relationship strengthening as students move into fractions (Schneider et al., 2018). In nationally representative samples from the United States and the United Kingdom, elementary students’ knowledge of fractions and division, both of which rest on place-value and magnitude understanding, uniquely predicted algebra and overall mathematics achievement in high school five to six years later (Siegler et al., 2012).

A student who sees 4,000 as 40 hundreds can divide it by ten at a glance; one who knows only that the 4 is in the thousands place cannot. That is what makes place value a high-value target for formative assessment: the same reasoning a teacher can detect in grade 3 continues to pay off through algebra (Booth & Siegler, 2008).

Key Idea •Because unit structure and magnitude are what predict later achievement, assessing them early while there is still time to act is worth the effort.

Unitizing: The Foundational Predictor

The strongest predictor of place-value understanding is unitizing — the ability to treat a group as a single unit while still knowing what it is made of. A student who sees ten ones only as ten separate objects is reasoning additively; a student who can also see those ten units of one as one unit of ten has begun to unitize. Unitizing is what allows students to work with composite units: a hundred is ten tens or one hundred ones, and a thousand is ten hundreds or one hundred tens. This nesting of units is the structure of the base-ten system (Fuson, 1990; Lamon, 2007).

In grades 3–6, unitizing is the understanding that lets students rename numbers fluidly for computation — seeing 340 as 34 tens when dividing by ten, or 4,000 as 40 hundreds when reasoning about scale. Without it, students may name places correctly while still treating the 4 in 47 as a 4 rather than as 4 tens, which limits their ability to reason about regrouping, multiplication by powers of ten, and decimals (Ross, 1989).

Items that ask students to count in composite units expose whether unitizing is in place: How many tens are in 340? How many hundreds are in 2,500? I have 47 tens what number is that? A student who must build up by ones, or who answers “4 tens” for 340 by reading only the tens digit, is naming positions rather than reasoning about units. The diagnostic value lies in the explanation: ask how the student knows, and listen for whether they coordinate the unit (tens) with the count (34).

Key Idea •Unitizing — treating a group as one unit while knowing its composition is the foundational place-value understanding; assess it by asking how many tens or hundreds are in a number and listening to the reasoning, not just the answer.

Magnitude and Number Sense: Locating Numbers, Not Just Writing Them

Place value also depends on magnitude — understanding how large a number is and where it falls relative to others, beyond how it is written. Students with magnitude understanding know that 80 is much closer to 100 than to 10, that 503 is just over 500, and that 2,000 is ten times 200. A place-value chart shows how a number is composed; a number line shows where it lives. Students need both, and the number line is the representation that most directly externalizes magnitude (Booth & Siegler, 2008).

Magnitude understanding is what keeps students from treating digits independently. A student attending to the quantity recognizes that 402 is greater than 389 even though 389 has larger digits in the tens and ones places. As the longitudinal and meta-analytic evidence shows, the precision of students’ number-line placements is among the most robust predictors of later achievement, which makes magnitude one of the most worthwhile things to assess in grades 3–6 (Geary, 2011; Schneider et al., 2018).

Number-line and comparison items reveal magnitude reasoning directly: Place 472 on a number line from 0 to 1,000. Which is greater, 509 or 590, and how do you know? About how much is 6,000 − 2,950? The most informative version asks for justification. A student who explains the comparison by noting that, after the equal hundreds, 590 has nine tens to 509’s zero tens is reasoning about quantity; one who simply says “590 has a bigger 9” may be comparing digits. Estimation prompts — closer to 3,000 or 4,000? — surface whether students reason about the total before computing.

Key Idea •Magnitude understanding — knowing where a number falls, not just how to write it is among the strongest predictors of later math achievement; number-line placement, comparison-with-justification, and estimation items assess it efficiently.

The Multiplicative Structure of the Places

In grades 4–5, students grasp that the places form a multiplicative structure: each place is worth ten times the place to its right and one-tenth of the place to its left. This is the property that distinguishes a true base-ten understanding from a memorized list of place names (Ross, 1989). It is also what makes multiplication and division by powers of ten meaningful rather than a matter of “adding zeros,” and it is the bridge into decimals, where the same times-ten and one-tenth relationship continues to the right of the ones place.

Because this relationship is often left implicit, it is frequently the missing piece when students struggle with decimals or with scaling. A student who understands that moving a digit one place to the left multiplies its value by ten has a structural account of why 0.1 is ten hundredths and why 30 is ten times 3. Making this relationship an explicit object of instruction — and of assessment — is among the most useful moves a teacher can make in these grades (National Research Council, 2001).

Items that probe the multiplicative relationship are especially revealing: What happens to the value of a digit when it moves one place to the left? How many times greater is the 7 in 7,000 than the 7 in 70? How many tenths are in 5? How many hundredths are in 0.5? A student who answers “ten times” and can connect it to both whole numbers and decimalsholds the structural understanding; one who treats each place as unrelated will often fall back on “adding a zero” and falter when decimals make that rule fail.

Key Idea •Each place is worth ten times the one to its right — the multiplicative structure that turns a list of place names into base-ten understanding; assess it with “how many times greater” and “what happens when a digit moves places”.

Equivalence and Flexible Renaming

A supporting understanding ties units and magnitude together: the recognition that the same quantity can be represented in different but equivalent ways. The number 47 is 4 tens and 7 ones, but it is equally 3 tens and 17 ones, 40 + 7, or 50 − 3. These equivalences are not arithmetic trivia; they are what make flexible computation possible. A student who can rename 304 as 2 hundreds, 10 tens, and 4 ones is prepared to subtract across zeros with understanding rather than by rule (Hiebert & Carpenter, 1992).

This is also where the equals sign should be treated as a statement of balance rather than a signal to compute. Place value gives students a natural context for seeing equality as equivalence: different decompositions of the same number are genuinely equal. Students who hold this relational view of equality are better positioned for the structural reasoning that algebra requires (Skemp, 1976).

Renaming and equivalence items reveal flexibility: Show 304 in two different ways. Explain why 6 tens and 14 ones equals 74. Fill in the blank: 3,000 + ___ + 60 + 2 = 4,762. A student who can produce multiple correct decompositions, and justify why they are equal, demonstrates the flexible unit reasoning that procedural fluency alone does not guarantee.

Key Idea• The same quantity can be named in many equivalent ways, and flexible renaming is what makes regrouping and mental computation make sense; assess it by asking students to represent a number in more than one way and justify the equivalence.

Reading the Errors: What Common Misconceptions Reveal

Place-value errors are rarely careless; they usually point to a specific gap in unit or magnitude understanding, which is what makes them so useful formatively. The most common pattern is treating multi-digit numbers as digits placed side by side reading 36 as “3 and 6” rather than 3 units of ten and 6 units of one. Such a student can often name the tens place while still not understanding that the 3 represents 30 (Ross, 1989).

A second pattern is treating regrouping as a rule rather than a unit relationship. Students who “borrow” or “carry” by procedure often cannot explain that one ten is being partitioned into ten ones. The language matters: describing the action as composing (iterating) and decomposing (partitioning) units, rather than borrowing and returning something, keeps the focus on the mathematics (Skemp, 1976). A third pattern concerns zero as a placeholder — a student who does not understand that the 0 in 406 marks the absence of tens may write four hundred six as 4006 or 46.

A fourth pattern, increasingly important with decimals in grades 5–6, is “longer means larger” reasoning — judging 0.38 to be greater than 0.4 because it shows more digits. This error is a direct signal that magnitude understanding has not yet extended to decimals, and it is precisely what a well-chosen comparison item can surface. In each case the error is diagnostic: it tells the teacher which understanding to rebuild, not merely that an answer was wrong.

Key Idea •Place-value errors are diagnostic, not careless digits-side-by-side, regrouping-as-rule, zero-as-placeholder, and longer-means-larger each point to a specific gap in unit or magnitude understanding that targeted instruction can address.

Conclusion

Place value becomes powerful when students understand that every number is built from units and that those units can be composed, decomposed, compared, iterated, partitioned, and represented in multiple equal ways. For students in grades 3–6, this is not review of a primary-grade topic but the foundation on which multi-digit operations, decimals, fractions, and early algebra are built.

When students struggle, the issue is usually a developmental gap in unitizing or magnitude rather than carelessness, and the research is clear that these are among the understandings that predict long-term mathematical success. Assessment that targets units, magnitude, the multiplicative structure of the places, and flexible equivalence and that listens for reasoning rather than recording answers gives teachers an early, actionable read on where students stand and what they need next.

References

Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79(4), 1016–1031.

Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343–403.

Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age.

National Research Council. (2001). Adding it up: Helping children learn mathematics (J. Kilpatrick, J. Swafford, & B. Findell, Eds.). National Academy Press.

Ross, S. H. (1989). Parts, wholes, and place value: A developmental view. The Arithmetic Teacher, 36(6), 47–51.

Schneider, M., Merz, S., Stricker, J., De Smedt, B., Torbeyns, J., Verschaffel, L., &Luwel, K. (2018). Associations of number line estimation with mathematical competence: A meta-analysis. Child Development, 89(5), 1467–1484.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

The number line has long been a classroom staple, but its deepest power emerges when students construct it themselves choosing endpoints, units, spacing, and labels. When students build the model, they engage essential mathematical actions rather than observe them. Research shows that constructing number lines strengthens proportional and spatial–numeric reasoning (Cohen et al., 2014), helping students map numbers onto meaningful distances. These actions connect number to magnitude, rather than leaving numbers as abstract symbols on a page.

Drawing and scaling lines also activate dual coding, linking verbal and visual systems in ways that support understanding and long-term memory (Paivio, 1986). This is why number-line construction becomes a bridge—from discrete counting to continuous reasoning, and from early arithmetic to algebra and graphing. This Insight distills research across mathematics education and cognitive psychology to show why student-constructed number lines deepen structural understanding and how teachers can embed this practice in instruction and professional learning.

Key Idea • When students construct number lines instead of receiving them, they engage the proportional, spatial, and measurement reasoning that pre-made lines quietly remove.

Cognitive Foundations: From Number to Measurement and Scale

Instead of simply placing numbers on pre-made lines, students engage in measurement-based thinking that drives proportional reasoning —a finding supported by research showing that number-line performance depends heavily on scaling, not just numerical magnitude (Barth & Paladino, 2011; Cohen et al., 2014). When constructing lines, students practice unit iteration and equal partitioning, defining a unit length and repeating it across the line. These are the same skills needed for measurement, fractions, and early algebra.

By mapping numbers to space, students learn that numerals correspond to physical, repeatable lengths rather than just positions on a static line—aligning with cognitive studies showing the role of spatial processing in magnitude understanding (Leibovich et al., 2014). Studies also show that many students hold rigid conceptions of the number line such as the idea that zero must always be centered or that increments must always be one. Constructing lines encourages flexibility and conceptual growth (Ünal et al., 2024), preparing students for later work in algebra and modeling.

Key Idea • Building a line is measurement-based reasoning: students iterate a unit, partition equally, and map number to space, the same actions that underlie magnitude understanding.

Measurement and Fraction Reasoning: Connecting Number to Length

Constructing number lines strengthens understanding of measurement because students connect numeric values directly to physical length. Research shows that coordinating numeric and linear measurement—literally drawing and scaling lines—deepens conceptual understanding (Saxe et al., 2013). As students partition lines into fourths and tenths, they experience fraction values as proportional distances rather than memorized points—an argument supported by work on spatial–numeric integration (Cohen et al., 2014). This work also supports embodied cognition, as drawing and dividing lines engage sensory–motor and spatial networks (Leibovich et al., 2014). This embodied grounding helps students make sense of equivalence, comparison, and scaling across fraction contexts.

Key Idea• Treating a unit as a length lets students experience fractions as proportional distances rather than memorized points, grounding equivalence, comparison, and scaling.

Extending Number-Line Understanding to Data and Graphing

A line plot is essentially a number line with data layered on it, and research shows that students better understand data displays when they construct the axes themselves—selecting the range, tick spacing, and scale (Lehrer & Schauble, 2007). In bar graphs, constructing axes helps students realize that equal spacing represents equal units, reinforcing structural ideas from measurement that do not always transfer when graphs arrive pre-formatted.

Coordinate graphing also becomes more intuitive when students recognize that the x-axis is a scaled number line. Students who have not constructed number lines often struggle with origin placement and scale—a difficulty observed repeatedly in graphing research (Robertson, 2023). These construction experiences develop continuous and proportional thinking, supporting algebraic modeling and early function reasoning.

Key Idea• An axis is a scaled number line; students who build scales themselves carry equal-spacing and origin reasoning directly into line plots, bar graphs, and the coordinate plane.

Instructional Design: Turning Research into Practice

Classroom routines that begin with blank lines help students take ownership of scale and structure, rather than relying on templates. This echoes research showing that student-created tools promote deeper reasoning (Saxe et al., 2013). Tasks with varied endpoints (0–1, 0–50, 2–10) push students to adapt their unit choices and scaling, supporting cognitive flexibility across contexts.

Integrating fraction and measurement contexts ensures that students connect physical measurement to visual and symbolic representations, reinforcing the structural ideas behind units and partitions. Drawing axes for data or coordinate grids builds graphing fluency through scale-making, a key shift emphasized in data-literacy research (Lehrer & Schauble, 2007). Reflection prompts—such as How did you choose your unit? make students’ structural reasoning visible, which is essential for developing conceptual understanding. Student-created number lines provide rich assessment evidence because they reveal how students think about spacing, scale, and labeling, not just whether they placed points correctly.

Key Idea • Routines that start with a blank line, vary the endpoints, and ask students to justify their scale make structural reasoning visible and give teachers rich assessment evidence.

Professional Development: Supporting Teachers as Designers of Structural Learning

When teachers construct number lines during PD, they experience firsthand how scaling, spacing, and unit decisions shape reasoning, building pedagogical content knowledge grounded in students’ cognitive actions. Using consistent structural language—unit, partition, iterate, compose, decompose, equal—helps teachers anchor classroom discourse in mathematical actions that promote sense making (Brendefur & Strother, 2021). Connecting number-line work to measurement, data, and graphing helps teachers see the number line as a unifying model that supports the K–8 trajectory (Robertson, 2023). Analyzing student-created lines enables teachers to identify misconceptions such as uneven spacing or fixed-zero thinking and to design follow-up tasks that target structural understanding.

Key Idea • When teachers build and analyze number lines together, using shared structural language, they develop the pedagogical content knowledge to design tasks that target reasoning, not just placement.

Conclusion: Empowering Mathematical Thinking Through Construction

Research across learning sciences and mathematics education shows that constructing number lines leads to stronger, more transferable understanding than using pre-drawn models (Cohen et al., 2014; Saxe et al., 2013). Through drawing, partitioning, and scaling, students internalize mathematical relationships rather than simply perform them. When construction becomes a routine across grades, students learn to design mathematics—not just record it, transforming the number line into a powerful medium for thinking, modeling, and sensemaking.

Key Idea•Constructing number lines produces more transferable understanding than using pre-drawn ones, because students internalize relationships rather than perform them.

References

Barth, H., & Paladino, A. M. (2011). The development of numerical estimation: Evidence against a representational shift. Developmental Science, 14(1), 125–135. https://doi.org/10.1111/j.1467-7687.2010.00962.x

Brendefur, J., & Strother, S. (2021). Developing mathematical fluency: Helping children make sense of facts and strategies. DMTI Press.

Cohen, D. J., Blanc-Goldhammer, D., Courtney, E. A., Jensen, M. B., & Runeson, B. (2014). The relation between spatial and numerical abilities in children and adults. Frontiers in Psychology, 5, 1060. https://doi.org/10.3389/fpsyg.2014.01060

Lehrer, R., & Schauble, L. (2007). Thinking with data. Lawrence Erlbaum Associates.

Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2014). From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition. Frontiers in Psychology, 5, 962. https://doi.org/10.3389/fpsyg.2014.00962

Paivio, A. (1986). Mental representations: A dual coding approach. Oxford University Press.

Robertson, D. (2023). The power of number line models and scales. Ontario Institute for Studies in Education (OISE) Blog. https://www.oise.utoronto.ca

Saxe, G. B., Shaughnessy, M. M., Shannon, A., & Bowling, D. (2013). Coordinating numeric and linear measurements: Students’ strategies and mathematical understandings. ZDM: The International Journal on Mathematics Education, 45(3), 407–420. https://doi.org/10.1007/s11858-012-0477-4

Ünal, O., Ertekin, E., & Güler, G. (2024). Conceptual stages and student reasoning on the number line. ERIC. https://eric.ed.gov

Students who can compute fluently often stall the moment a problem appears in words. They know how to add, subtract, multiply, or divide, but when those same operations are embedded in language, the task becomes much harder. The issue is rarely the arithmetic itself — it is that students lack a way to organize the quantities and relationships in front of them. In many classrooms, students are taught to rely on keywords rather than analyze how quantities are connected, leading to guesses rather than reasoning.

Bar models also called strip diagrams or tape diagrams address this gap by representing quantities as lengths. Length is spatial, visible, and comparable. Students can see how one quantity is part of another, how two quantities combine, or how one exceeds another. Instead of asking, “What word tells me to add or subtract?” students begin asking, “What is the total? What are the parts? What is known? What is missing?” That is a far more powerful mathematical habit of mind (Ng & Lee, 2009).

Key Idea •Most students who struggle with word problems aren’t failing at computation — they’re failing to see the structure of the situation.

Historical Foundations: The Singapore Model Method

Bar models became widely known through Singapore mathematics in the 1980s, when the Singapore Ministry of Education worked to strengthen problem solving in the national curriculum. Educators found that many students could carry out arithmetic procedures but still struggled with non-routine and contextual problems. The issue wasn’t a lack of practice; students needed a better bridge between the story context and the symbolic mathematics. The Model Method gave students a consistent way to represent quantities before formalizing their thinking with equations (Singapore Ministry of Education, 2012).

This work fits closely with Bruner’s theory of representation, which describes a progression from enactive experience to iconic representation and then to symbolic abstraction (Bruner, 1966). Bar models sit in the critical iconic stage. They are not concrete manipulatives, but they are also not yet abstract symbols. Instead, they function as a bridge: students can move from acting on quantities to drawing them and then to expressing them symbolically. This bridging role explains why bar models extend coherently across addition, subtraction, multiplication, division, fractions, ratios, and early algebra.

Bar models were never intended as a one-time strategy for a specific type of word problem. Their strength comes from coherence the same representational logic can be used across topics and grade levels. A student who uses bars to represent part–whole relationships in Grade 1 can later use the same structure to compare quantities, represent multiplicative relationships, and reason about unknowns in pre-algebra.

Key Idea •Bar models occupy Bruner’s iconic stage — a bridge between concrete experience and symbolic abstraction that extends coherently across topics from arithmetic through early algebra.

What Bar Models Reveal About Mathematics

Bar models are powerful because they reveal underlying structure that is often hidden in symbols alone. In join, separate, and part–whole situations, a single bar represents a total quantity that can be decomposed into parts. Students see that quantities are not isolated values but connected parts of a whole.

Consider 245 = 200 + 40 + 5. The bar below shows how a three-digit number is composed of its place-value parts. Each rectangle is sized in proportion to its value, so the visual itself reinforces what the digits mean: 200 is much larger than 40, which in turn is much larger than 5.

Figure 1. Place-value decomposition of 245. The bracket above the bar represents the whole; the parts are sized proportionally.

In a comparison situation, two bars represent different quantities, and the difference appears as extra length. This allows students to see the difference as a relationship rather than infer it from words. Below, a 10-cm stick is compared with an 8-cm worm. The arrow between the two bars marks the unknown — how much longer is the stick than the worm? Students can see the relationship before they compute it, then reason that the missing length must be 2 cm to complete the comparison.

Figure 2. Comparison model. Two bars are drawn to scale; the arrow marks the unknown difference in length between them.

When units are equal, these same structures extend naturally into multiplication and division. A bar divided into equal units shows that the total is composed through iteration. Across all cases, students begin with structure rather than operations. They reason about how units fit together and come apart, rather than selecting procedures from a list (Carpenter, Fennema, Franke, Levi, & Empson, 1999).

Key Idea •Most arithmetic and early algebra problems come down to two structures part–whole and comparison and bar models make both visible..

Cognitive Foundations: Why Bar Models Work

Bar models are effective in part because they reduce the load on working memory. Word problems require students to process language, track quantities, and coordinate relationships at the same time. Drawing the relationship moves it out of the student’s head and onto the page, where it can be examined rather than held. This frees cognitive resources for reasoning rather than tracking (Sweller, 1988).

Bar models also support dual coding because students represent information both verbally and visually. These dual representations reinforce each other and improve comprehension and recall (Paivio, 1990). Over time, students begin to recognize recurring structures part–whole, comparison, equal groups and these structures become schemas that support transfer to new contexts (Hiebert &Grouws, 2007).

A final mechanism is spatial reasoning. Quantities become lengths, and relationships become spatially visible. Spatial representation helps students perceive abstract relationships more clearly and is strongly associated with long-term success in mathematics. One of the most important reasoning strategies supported by bar models is identifying the value of a single unit and then iterating or partitioning it a strategy that connects multiplication, division, fractions, and ratios into a single coherent way of thinking.

Key Idea •Bar models reduce working memory load, support dual coding, and build transferable schemas three mechanisms that make them effective even for students who struggle with traditional word-problem instruction.

From Bar Models to Algebraic Thinking

Bar models play a critical role in supporting the transition from arithmetic to algebra. Students first experience unknowns as quantities within a structure rather than as abstract symbols. As they grow more sophisticated, the same representation that helped them combine parts and compare quantities now supports reasoning with unknowns.

The bar below represents a problem like 4x + 5 = 53, where x is an unknown quantity that appears four times. The four equal bars labeled x show that the unknown is iterated four times; the smaller bar of 5 completes the whole, and the bracket above marks the total of 53. A student reasoning that the four x bars together must equal 48 and therefore each x must be 12 — is doing exactly the work that solving 4x + 5 = 53 makes formal. The structure of the equation is preserved in the diagram before it is written symbolically.

Figure 3. Algebraic reasoning. Four equal bars of x and a bar of 5 compose a whole of 53; reasoning about the structure leads directly to 4x + 5 = 53.

In this way, bar models support relational thinking, which is foundational to algebra (Carpenter et al., 1999). Students learn to see equations as representations of relationships rather than procedures to execute a shift that pays dividends well beyond elementary school.

Key Idea •Bar models preserve the structure of an equation before it is written symbolically, giving students a meaningful entry point into algebraic reasoning.

Developmental Progression: How Bar Models Grow with Students

The strength of bar models lies in how they evolve with the mathematics, not just repeat across grades. In the early grades, students use bar models to represent joining, separating, and part–whole situations, focusing on identifying the total (or whole) and its parts. Then this expands to comparison situations and equal groups, where students begin coordinating multiple quantities and recognizing relationships such as “more than,” “less than,” and “times as many.”

In upper grades, the representation becomes more precise and more powerful. Students use bar models to reason about fractions, scaling, and multiplicative relationships, working with partitioned units and iterating those units to build new quantities. By middle school, the same structure supports reasoning with unknowns, allowing students to represent relationships that lead directly to equations and algebraic thinking.

Rather than learning a new strategy each year, students refine a single representation that grows alongside the mathematics. This coherence is one of the most important features of the model and is a central reason it is associated with strong long-term outcomes in proportional and algebraic reasoning (Ng & Lee, 2009).

Key Idea •The same representation evolves with the mathematics from K through 6, supporting whole numbers, fractions, ratios, and early algebra students refine one tool, not many.

Using Bar Models Well and Where Instruction Goes Wrong

Bar models are most effective when used as a tool for making sense of relationships, not as a set of drawing steps to follow. When instruction focuses on replicating a diagram “draw this bar, label it here” students may produce correct-looking models without understanding the quantities they represent. The emphasis should be on using the diagram to answer questions such as: What does this part represent? How do these quantities relate? What is the unknown?

A critical instructional focus is helping students attend to units. Every bar represents a quantity, but more importantly, it represents a unit that can be composed, decomposed, partitioned, or iterated. As problems become more complex, students must track not just the numbers, but what those numbers represent. Without this attention to units, students may draw accurate diagrams that do not support correct reasoning.

Three failure modes show up repeatedly in classrooms. The first is treating bar models as a drawing procedure rather than a reasoning tool, which produces neat diagrams that don’t reflect the quantities in the problem. The second is skipping unit reasoning, so the model loses its mathematical meaning and becomes decorative rather than analytical. The third is reserving bar models only for “difficult problems” rather than using them consistently to build structure over time, which prevents students from developing the habit of reasoning about relationships.

Key Idea •The effectiveness of a bar model depends on attention to units and relationships, not on how accurately the diagram is drawn.

Conclusion: A Single Representation, Many Years of Reasoning

Bar models earn their place in classrooms because the same simple representation a length standing for a quantity carries students from kindergarten part–whole problems all the way to early algebra. Along the way it makes structure visible, takes load off working memory, and gives students a stable anchor for reasoning about units.

From both mathematics education and cognitive psychology perspectives, bar models work because they make structure perceptible, support schema development, and promote transfer. When taught with an emphasis on units, relationships, and reasoning rather than drawing rules, they become a high-leverage practice for building durable mathematical understanding from the early grades into algebra.

Key Idea• A single coherent representation used over time builds deeper understanding than many disconnected strategies layered year after year.

References

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

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

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. National Governors Association Center for Best Practices and Council of Chief State School Officers.

Hiebert, J., &Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age.

Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313.

Paivio, A. (1990). Mental representations: A dual coding approach. Oxford University Press.

Singapore Ministry of Education. (2012). Mathematics syllabus: Primary. Curriculum Planning and Development Division.

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.

Social Media Post

Your students can compute. So why do word problems still stop them cold?

Because solving for an answer isn’t the same as understanding a relationship.

Our latest DMT Insight on Bar Models looks at why this single representation a length standingfor a quantity is one of the highest-leverage tools in elementary mathematics:

• Structure over keywords. Bar models help students see part–whole and comparison relationships instead of hunting for trigger words.

• Less load, more reasoning. Externalizing relationships onto paper frees working memory for thinking.

• One model, many years. The same diagram a first grader uses for part–whole becomes the foundation for fractions, ratios, and algebra.

• Units are everything. The model is only as strong as the student’s understanding of what each bar represents.

This isn’t about teaching a new strategy. It’s about giving students a consistent way to see structure from kindergarten through algebra.

Read the full research overview at www.dmtinstitute.com · MathSuccess.io

Where do your students struggle most with word problems seeing the structure, or choosing the operation? Share your thoughts at contact@dmtinstitute.com.

Students who perform well on addition often struggle when mathematics requires them to work in reverse. This is not about effort—it is about how subtraction is taught. When subtraction is taught as a mirror-image procedure rather than a relational operation, students lack the framework to reason flexibly about difference, comparison, and missing quantities. The tools that could build that foundation visual models, part-part-whole reasoning, varied problem structures—are frequently bypassed in favor of procedures that produce answers without understanding (Nunes & Bryant, 1996).

This DMT Insight draws on research from mathematics education and cognitive psychology to answer a practical question: What does subtraction require students to understand—and why do so many struggle to learn it?

Key Idea •Students struggle with subtraction not because it is harder than addition, but because they are not shown the underlying structure.

The Core Problem: Subtraction Without Relational Structure

Subtraction is often taught as a procedure, but it is fundamentally about relationships between quantities. At its core, subtraction asks: What is the total? What quantity do we know? What quantity is missing?

Students must understand that a total is composed of quantities, and that those quantities can be composed or decomposed while the total remains equal. When this structure is not developed, subtraction becomes a set of disconnected steps. Students may produce correct answers in familiar situations, but they struggle when the context changes because they are not reasoning about the relationship between quantities (Fuson, 1992; Nunes & Bryant, 1996).

Mathematics education researchers have observed that subtraction is not a single operation but a family of problem situations. Verschaffel, Greer, and De Corte (2007) identified at least three distinct semantic structures:

· Take-away: removing a quantity from a set (the most commonly taught form)

· Missing addend: determining what must be added to reach a total

· Comparison: finding the difference between two quantities.

Figure 1. The same numbers, 26 and 17, generate three different questions, three different bar model structures, and three different number line strategies.

Students often succeed with take-away problems but struggle with comparison or missing-addend problems—even when the numbers are the same—because they are reasoning about different relationships. When instruction emphasizes procedures instead of relationships, students must rely on memory rather than understanding. They try to follow steps without a clear sense of what the quantities represent or how they relate to the total (Sweller, 1988; Hiebert & Carpenter, 1992).

Key Idea •Subtraction is about reasoning with quantities and totals—students must understand how quantities combine, separate, and relate within a total, not just how to follow steps.

Predictable Errors and Their Instructional Roots

The errors students make in subtraction are not random. They are predictable outcomes of instruction that emphasizes procedures over structure. The most widely documented error is the “smaller-from-larger” pattern (Brown &VanLehn, 1982), in which students subtract the smaller digit from the larger one regardless of place value (e.g., 274 − 138 = 164). These errors occur when students try to follow a procedure they do not fully understand.

From a cognitive perspective, subtraction places high demands on attention and memory. Students must track units, steps, and relationships at the same time. When these demands exceed what students can manage, they resort to shortcuts that seem logical but yield incorrect results. Gathercole and Alloway (2008) showed that children made significantly more errors on subtraction problems requiring regrouping. The procedure demands that students simultaneously hold partial results, track position in the algorithm, and retrieve subtraction facts—overloading a system that was never given a conceptual foundation to lean on.

Research and classroom evidence reveal a consistent pattern of errors when subtraction is taught procedurally (Carpenter, Fennema, & Franke, 1996; Baroody, 2003):

· Smaller-from-larger errors

· Treating subtraction as reversible

· Applying take-away reasoning to all situations

· Failure to transfer to fractions and decimals

Key Idea •Most subtraction errors are not careless mistakes—they are logical responses to instruction that does not make structure visible.

What Cognitive Science Suggests

Working memory plays a central role in subtraction. When students solve subtraction problems—especially with larger numbers—they must track quantities, monitor their steps, and maintain relationships simultaneously. Research shows that when these demands exceed working memory capacity, errors increase, even when students understand the numbers involved (Gathercole & Alloway, 2008; Geary, 2011). This helps explain a common classroom pattern: students may appear to understand subtraction in simple cases but struggle as soon as additional steps or place value are introduced. The difficulty is not just the mathematics—it is the cognitive load placed on the learner (Sweller, 1988).

Subtraction also depends on how students represent quantities. Students draw on language (number words), magnitude (size of quantities), and a mental number line (distance between quantities) when reasoning about subtraction (Dehaene, 2011; Siegler & Ramani, 2009). When these representations are weak or disconnected, students struggle to interpret what subtraction is asking. Finally, affect plays a role. When students experience repeated difficulty, anxiety can reduce attention and working memory, leading to more errors (Beilock& Maloney, 2015). Visual models, clear language, and low-stakes opportunities to explain thinking help strengthen these representations and reduce cognitive load.

Key Idea• Subtraction requires students to coordinate quantities, language, and magnitude under cognitive load—strong representations and supportive instruction make this coordination possible.

A Coherent Instructional Pathway

Research points to a consistent instructional sequence that supports understanding across grade levels:

• Begin with meaning using concrete and visual models

• Introduce all three problem structures

• Develop the addition–subtraction inverse

• Build from structure to symbolic notation

• Delay formal procedures until understanding is established

This progression aligns with Bruner’s enactive–iconic–symbolic (EIS) framework, where students move from action to representation to abstraction (Bruner, 1966). Studies on representational sequences show that students who experience concepts through multiple representations develop a more flexible and durable understanding than those who begin with symbolic procedures alone (Witzel, 2005; Flores, 2010).

Number lines and bar models are especially powerful because they help students see subtraction as distance and relationship, not just removal. Research shows that linear representations support students’ understanding of magnitude and improve their ability to reason about numerical relationships (Siegler & Ramani, 2009).

Multi-digit subtraction should be built on place-value understanding, where students decompose and recompose units while maintaining equality. When procedures are introduced before this understanding is secure, students rely on steps rather than reasoning, which limits transfer to new contexts (Fuson, 1992; Hiebert & Carpenter, 1992).

Key Idea •Effective subtraction instruction moves from meaning to structure to procedure—not the other way around.

Conclusion

Subtraction is challenging, not because students are incapable, but because it requires them to coordinate quantities, units, and relationships while managing cognitive load. When instruction focuses primarily on procedures, students may produce correct answers in familiar situations but struggle to explain their thinking or transfer it to new contexts. In contrast, when subtraction is built on a foundation of meaning—grounded in quantities and totals, developed across multiple representations, and connected to place value and the addition–subtraction inverse—students develop a coherent understanding they can apply across topics. This progression, from enactive to iconic to symbolic, supports students in seeing subtraction as a relationship rather than a rule. When students can reason about how quantities combine, separate, and remain equal, subtraction becomes not just something they can do, but something they understand and can use flexibly across the mathematics they encounter.

Key Idea• Subtraction becomes accessible and transferable when students understand the relationships between

References

Baroody, A. J. (2003). The development of adaptive expertise and flexibility. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 1–33). Lawrence Erlbaum.

Beilock, S. L., & Maloney, E. A. (2015). Math anxiety: A factor in math achievement not to be ignored. Policy Insights from the Behavioral and Brain Sciences, 2(1), 4–12.

Brown, J. S., &VanLehn, K. (1982). Towards a generative theory of “bugs.” In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 117–135). Lawrence Erlbaum.

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3–20.

Dehaene, S. (2011). The number sense: How the mind creates mathematics (Rev. ed.). Oxford University Press.

Flores, M. M. (2010). Using the concrete-representational-abstract sequence to teach subtraction with regrouping to students at risk for failure. Remedial and Special Education, 31(3), 195–207.

Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). Macmillan.

Gathercole, S. E., & Alloway, T. P. (2008). Working memory and learning: A practical guide for teachers. SAGE.

Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental & Behavioral Pediatrics, 32(3), 250–263.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Nunes, T., & Bryant, P. (1996). Children doing mathematics. Blackwell.

Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology, 101(3), 545–560.

Sweller, J. (1988). Cognitive load during problem solving. Cognitive Science, 12(2), 257–285.

Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557–628). Information Age Publishing.

Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities: A Contemporary Journal, 3(2), 49–60.

Social Media

Why do students who can add still struggle when mathematics requires them to subtract?

Because following steps is not the same as understanding relationships.

Our latest DMT Insight shows:

• Three structures, not one. Subtraction is a family of relationships between quantities and totals — take-away, comparison, and missing-quantity — not a single procedure.

• Same numbers, different relationships. Students often master take-away but stall on comparison and missing-quantity problems with the same numbers, because the relationships are different.

• Errors are predictable, not careless. When procedures outpace understanding, working memory overloads — and the errors that follow are systematic, not random.

• One structure, many years. The same part-quantity-total reasoning that anchors whole-number subtraction in Grade 1 carries forward into fractions, decimals, and algebra.

This isn’t about teaching subtraction differently. It’s about giving students a way to see the relationships behind the operation — from early arithmetic through algebra.

Read the full research overview at mathsuccess.io · dmtinstitute.com

Where do your students get stuck — understanding the relationship, or following the steps? Share your thoughts at contact@dmtinstitute.com.

Why do students who can recite their multiplication facts still fall apart when they hit division, fractions, decimals, and proportions—and what does a ratio table have to do with it?

Students who can correctly recall multiplication facts often fall apart when mathematics becomes more complex. This is not a student effort issue—it is an instructional design issue. Students do not understand the relationships behind them. When multiplication is taught as fact retrieval rather than relational thinking, students lack a framework for scaling, rates, and proportions. The tools that could build that foundation—visual models, structured tables, flexible strategies—are frequently skipped in favor of procedures that produce answers without understanding (Lamon, 2007).

Ratio tables are not a worksheet activity or a shortcut for computation. They are a symbolic structure that makes multiplicative relationships visible, flexible, and transferable. Ratio tables are a flexible algorithm students can use to solve multiplicative relationships from grade 3 to algebra—they are a structure students learn to think with. This DMT Insight examines what ratio tables are, why students need them, and what research tells us about teaching them effectively.

The Core Problem—Procedural Multiplication Without Relational Structure

What happens in classrooms where multiplication means memorizing facts rather than understanding relationships?

The central instructional problem is a failure to develop multiplicative reasoning—the ability to reason about relationships among quantities, rather than just to retrieve isolated facts.

• Cannot scale efficiently (e.g., move from 4 × 3 to 40 × 3 or 17 × 3)

• Rely on counting strategies (1, 2, 3, 4…) well beyond primary grades

• Struggle with missing value problems

• Fail to transfer knowledge to division, decimals, fractions, ratios, and algebra

From a cognitive psychology perspective, fact-first instruction imposes high extraneous cognitive load: students juggle memorized answers without a coherent mental model of what those answers represent, making transfer to new contexts extremely difficult (Sweller, 1988; Hiebert & Carpenter, 1992). The result: students can compute, but they cannot reason. These patterns are not accidental. They point directly to why ratio tables—and the way they are taught—matter.

Ratio Tables Are Symbolic Models—Not Visual Models

How do ratio tables fit into a broader learning progression—and where do they ultimately lead?

A bar model shows the size of a quantity. A ratio table shows the relationship between quantities. Ratio tables are symbolic structures that organize relationships numerically.

That is why visual models must come first or alongside them. Bar models and double number lines help students see and feel equal units (groups) and multiplicative relationships. Ratio tables then encode those relationships symbolically, requiring students to coordinate values abstractly (Lesh, Post, & Behr, 1987).

This progression is not a one-time sequence. Visual models should continue alongside ratio tables when introducing new contexts or addressing misconceptions. The goal is integration, not replacement. The endpoint of this progression is algebraic graphing. The coordinate pairs in a ratio table, when plotted, produce a straight line through the origin. The constant in a ratio table is the slope of the line.

This makes ratio tables a direct precursor to y = mx, where m is the unit rate students have been reasoning about. Proportional reasoning, rate, and linear functions are not separate topics—they are one coherent idea expressed in increasingly formal representations. Understanding ratio tables as symbolic structures—rather than visual aids or computation shortcuts—is what allows teachers to use them as a genuine bridge to algebra.

Predictable Misconceptions and Their Instructional Roots

What are the most common errors teachers see—and what does typical instruction do to produce them?

Research and classroom evidence reveal a consistent pattern of errors when multiplicative reasoning is underdeveloped (Carpenter, Fennema, & Franke, 1996; Fosnot & Dolk, 2001):

• Additive Thinking: Students add a constant instead of maintaining a multiplicative relationship (1→4, 2→6, 3→8)

• Losing the Unit: Students lose the “1 column” anchor and produce values that break equivalence

• Sequential Counting: Students rely on counting (1, 2, 3, 4…) even for large numbers like 37 × 6

• Lack of Transfer: Students cannot extend reasoning to fractions, decimals, or rates

These are not random mistakes—they are predictable outcomes of instruction. These errors arise when multiplication is treated as fact memorization before structural understanding is established, and when ratio tables—if they appear at all—are treated as fill-in-the-blank activities rather than flexible reasoning tools.

Misconceptions as Diagnostic Feedback

What are student errors actually telling us—and how should that change what we do next?

Student errors are not just mistakes—they are evidence of how instruction shaped students’ thinking. Additive errors (1→4, 2→6, 3→8) signal that students were never asked to distinguish between adding a constant and maintaining a multiplicative relationship. Over-reliance on sequential counting signals that instruction did not help students see that larger numbers can be composed from known parts.

Failure to transfer signals that ratio tables were taught as a procedure for a specific context, not as a generalizable structure. These patterns collectively indicate that instruction introduced symbolic shortcuts before conceptual understanding was established, and failed to make the unit relationship explicit and central (Gravemeijer, 1999; Lesh et al., 1987). When teachers read errors as diagnostic signals rather than simple wrong answers, instruction becomes more precise, more responsive, and more effective.

The Developmental Trajectory: From Iteration to Flexible Reasoning

How does ratio table thinking develop over time—and what should teachers do at each stage to move students forward?

When students first encounter ratio tables, they naturally begin sequentially—starting at 1 and building to 2, 3, 4. This is developmentally appropriate and should be connected to the construction of bar models. The 1–4 building phase gives students a firm anchor in the unit relationship. This early phase is where students establish what “1” represents—the foundation of all multiplicative reasoning. Rushing past it, or treating the sequential table as a mere warm-up, undermines everything that follows.

What distinguishes skilled multiplicative reasoners is the move toward flexibility: scaling by landmark numbers (×5, ×5, ×10), decomposing non-landmark numbers (13 = 10 + 3), and composing known columns to find unknown ones. Fosnot and Dolk (2001) describe this as the shift from additive to multiplicative thinking—one of the most significant conceptual leaps in elementary mathematics. Lamon (2007) emphasizes that this shift requires explicit instruction in unitizing: students must learn to visualize numbers not just as counts but as composed units that can be strategically manipulated.

This progression does not happen automatically. Without explicit teacher prompts—“How can you jump to 10? How can you decompose 7?”—Students remain stuck in sequential counting well into middle school. The flexibility has to be taught.

A Coherent Instructional Pathway

What does a well-designed instructional sequence actually look like—and how does it hold together across grades?

The goal is not a single good lesson—it is instructional coherence across grades, where each phase builds directly on the last. Research points to five interlocking moves that, taken together, build durable multiplicative understanding:

First, build meaning through iconic models before introducing the ratio table. Bar models and double number lines give students a visual experience of the unit relationship—what one unit (group) is worth—before any symbolic encoding begins.

Second, anchor every table to the unit column. Introduce the table starting from 1, and make that anchor explicit and non-negotiable: every other column must be verifiable against it.

Third, develop flexibility explicitly rather than waiting for it to emerge. Prompt students to iterate through 2, 3, 4, and 5, then look for patterns. Then decompose non-landmark numbers. This shift—from counting up to composing strategically—is an example of multiplicative reasoning.

Fourth, generalize the structure by introducing the 1–2–5–10 landmark table as the full flexible algorithm. Once students can build and use landmark columns fluently, they have a tool that works for whole numbers, decimals, fractions, and rates.

Fifth, delay formal algorithms until understanding is established. Cross multiplication is efficient—but only for students who already understand what a proportional relationship is. Introducing it too early replaces reasoning with rule-following (Skemp, 1976).

Detailed classroom tasks, lesson sequences, and teacher moves for each of these phases are available in the DMTI companion guide. This pathway works because it manages cognitive load deliberately: concepts are sequenced logically, visual tools reduce abstraction at each new stage, and symbolic reasoning is built on a foundation of relational understanding rather than imposed on top of memorized facts (Sweller, 1988).

Conclusion

The persistent difficulties students experience with proportional reasoning are not a mystery, nor are they the result of students not trying hard enough. These difficulties are the predictable outcome of instruction that introduces symbolic shortcuts before conceptual understanding is established. When multiplication is taught through ratio tables—grounded in the unit relationship, supported by visual models, and developed toward flexible strategies—students build a durable understanding that transfers across contexts.

The same structure that helps a third grader reason about equal groups helps a sixth grader reason about unit rates and a seventh grader understand proportional relationships. One structure, taught well, carries students from Grade 3 to algebra. Ratio tables are a bridge from arithmetic to algebra, from equal groups to linear functions. Teaching them well requires an instructional design that makes structure visible, honors developmental progression, and insists that every student understand not just what the answer is, but why the relationship holds. This is how we move from students getting answers to students understanding mathematics.

Download Free Companion Document

References

Brendefur, J. L., & Pitingoro, L. (1998). Dividing fractions using the ratio table. Mathematics Teaching in the Middle School, 4(2), 122–127.

Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction. The Elementary School Journal, 97(1), 3–20.

Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Heinemann.

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age.

Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Erlbaum.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.

Social Media

Why do students who know their times tables still fall apart when they hit fractions, rates, and proportions?

Because memorizing facts is not the same as understanding relationships. Our latest DMT Insight shows:

• Ratio tables are a relational structure—not a worksheet activity

• The unit relationship (“what does 1 represent?”) is the foundation everything else rests

• Sequential counting (1, 2, 3, 4) is a starting point, not a destination

• The same structure that builds multiplication in Grade 3 leads directly to y = mx in Grade 7

This isn’t a student problem—it’s an instruction problem.

When teachers ground ratio tables in the unit relationship, explicitly develop flexibility, and connect tables to graphs and linear functions, students understand, and that understanding transfers. Check out our work at www.dmtinstitute.com and MathSuccess.io.

What’s one place your students get stuck moving from multiplication to proportional reasoning? Share at contact@dmtinstitute.com

Why do so many students insist that a square cannot be a rectangle — and what does that tell us about how we teach quadrilaterals?

These are not random errors. They are predictable outcomes of how quadrilateral instruction is typically designed. In most classrooms, learning about quadrilaterals begins and ends with naming shapes—matching words to pictures. Students learn to recognize what shapes look like, not what must be true about them. Rarely do they investigate what must always be true about a shape, regardless of how it is oriented or drawn.

This matters because quadrilaterals sit at a critical crossroads in mathematics education. They are one of the first sustained opportunities students have to reason about attributes, classification systems, and logical relationships. When instruction develops that reasoning, quadrilaterals become a foundation for algebraic thinking, proportional reasoning, and eventually proof. When instruction reduces them to vocabulary, students build fragile, picture-based categories that collapse the moment a familiar shape is rotated or presented in an unfamiliar form.

Research in mathematics education and cognitive psychology points to the same conclusion: durable geometric understanding begins when students learn to see shapes not as finished pictures, but as structures defined by relationships among lines (Hiebert & Carpenter, 1992; Lehrer & Schauble, 2015). This DMT Insight examines what that shift requires, why students struggle without it, and what a more effective instructional path looks like.

The Problem: Picture-Based Geometry

What happens when students learn shapes as pictures rather than as structures defined by properties?

When geometry instruction centers on memorizing the appearance of shapes, students develop what researchers call prototype-based reasoning. They form mental templates from repeated examples—the upright triangle, the horizontal rectangle, the square resting flat on a side—and judge whether a shape belongs to a category by how closely it resembles that template (Rosch, 1978). The result is a geometry of appearances rather than a geometry of properties.

The consequences are well-documented. Students routinely:

• Reject a rotated square as “not a real square” or label it a diamond

• Insist that a square is not a rectangle because rectangles are “the long ones.”

• Fail to recognize non-prototypical examples—an irregular quadrilateral and a non-isosceles trapezoid

• Accept shapes as belonging to a category based on overall look rather than defining properties

These errors are not signs of inattention. They are the logical result of instruction that emphasizes what shapes look like over what must be true about them. Clements and Battista (1992) found that students frequently believe a square is not a rectangle because they interpret category labels as mutually exclusive. De Villiers (1994) showed that failure to teach hierarchical inclusion—the idea that one category can be a subset of another—undermines students’ capacity for geometric reasoning well into middle school.

The same pattern shows up in everyday classroom materials. Worksheets show parallelograms only leaning to the right. Definitions describe rectangles as “longer than they are tall.” These materials do not merely fail to correct misconceptions—they actively train them (Dagli & Halat, 2016; Verdine et al., 2016).

The Cognitive Foundations: Why This Is Hard

What does cognitive psychology tell us about why quadrilateral learning is particularly challenging?

Cognitive research offers a clear explanation for why these misconceptions are so persistent. Two phenomena are especially relevant: prototype effects and the distinction between concept image and concept definition.

Prototype effects describe the well-established tendency to judge category membership by similarity to a typical example rather than by formal criteria (Rosch, 1978). In geometry classrooms, the typical rectangle is horizontal and longer than it is tall; the typical square sits flat. When students encounter atypical examples—a tall, narrow rectangle, a rotated square—they often reject them as non-members, not because they lack the defining properties, but because they do not match the mental template.

The concept image/concept definition distinction, developed by Vinner and Hershkowitz, helps explain why this problem persists even after formal instruction. Concept image refers to everything a student mentally associates with a concept—all the visual impressions, prior examples, and informal rules accumulated over time. Concept definition is the formal, precise description of what makes something a member of the category. Students may be able to recite a definition correctly while their concept image—built from years of prototype-heavy exposure—continues to drive their actual classification decisions (Vinner, 1991).

Working memory research adds another layer. When a student encounters a new shape, the visual stimulus automatically activates the concept image system through fast, pattern-matching recognition that requires minimal cognitive effort. The formal definition, by contrast, must be deliberately retrieved and applied—a slower, more effortful process. Under typical classroom conditions, the automatic system wins. Students say “that’s a rectangle” before they have a chance to check whether the properties match (Sweller, 1988).

This is why varied exposure matters so much. Variation in orientation, size, and regularity is not enrichment—it is the mechanism by which formal definitions gain genuine meaning. Without it, definitions remain inert, and concept images continue to govern reasoning.

The Mathematical Structure: Lines, Properties, and Hierarchies

What does it mean to understand quadrilaterals as structures defined by line relationships rather than as pictures to name?

A more powerful instructional foundation begins not with shape names, but with lines. Every quadrilateral is, at its core, the bounded region formed when four line segments connect in a closed path. The properties that differentiate quadrilateral types—parallelism, perpendicularity, and congruence of sides and angles—are relationships among those lines.

From this perspective, a parallelogram is the region bounded by two pairs of parallel lines. A rectangle is a parallelogram whose lines intersect at right angles. A rhombus is a parallelogram whose paired lines are equally spaced. A square satisfies both conditions simultaneously. Understood this way, the hierarchical structure of quadrilateral families becomes logical rather than arbitrary: all squares are rectangles because every square satisfies all the conditions for a rectangle and then some. The category is inclusive, not exclusive.

This line-intersection foundation is mathematically important for several reasons. It makes orientation irrelevant—the relationship between two parallel lines does not change when the figure is rotated. It connects shape categories to their defining constraints rather than their typical appearances. And it provides a bridge between a student’s visual experience and formal mathematical reasoning: a student can look at a rotated rectangle and ask, “Are these opposite lines parallel? Do these adjacent lines form right angles?” rather than asking, “Does this look right?”

Research confirms that students who understand shapes through their underlying structural properties are better equipped to reason about class inclusion, to recognize shapes across orientations, and to transfer that thinking to novel problems (Fujita & Jones, 2007; Clements & Sarama, 2009).

What Research Tells Us About Developmental Progressions

How does geometric thinking develop, and what does this mean for instruction?

Research on geometric thinking describes a progression from visual recognition to property-based analysis to relational reasoning about hierarchies and inclusion. At early levels, students identify shapes based on overall resemblance. At intermediate levels, they attend to specific attributes such as the number of sides or the presence of right angles. Only at more advanced levels do they reason about logical relationships—understanding why one category can be a subset of another, or why a single shape can simultaneously belong to multiple categories (Clements & Battista, 1992).

Underlying this progression is an understanding of lines themselves. A line is not a segment drawn on paper—it is an infinite object, extending in both directions without end. What students see in a shape are traces of lines: the sides of a quadrilateral are segments, but they lie on lines that continue beyond the figure. Parallel lines never meet, no matter how far extended; perpendicular lines meet at exactly a right angle. When students grasp lines as objects with these properties in space—not merely as visible edges—they gain the conceptual foundation for understanding why quadrilateral categories exist and how they relate to one another.

This progression does not happen automatically. Research consistently shows that without explicit instruction designed to move students from visual to relational reasoning, students remain at descriptive levels well into middle school. Simply presenting definitions does not produce conceptual reorganization. What moves students forward is a carefully designed instructional experience: exposure to varied examples and non-examples, sorting tasks that require justification, and opportunities to reason explicitly about what properties define a category and why some shapes qualify for more than one.

The cognitive challenge at the relational level is significant. Understanding that all squares are rectangles requires grasping that the set of squares is a proper subset of the set of rectangles—that satisfying more constraints is a stronger condition, not a different one. This logical structure is cognitively demanding, particularly for students who have been taught to treat shape names as mutually exclusive labels. It develops gradually and requires instruction that makes the logic visible, not just the vocabulary.

Moving Forward: What Effective Instruction Looks Like

What are the highest-leverage shifts educators can make?

The research points to several instructional principles that can shift quadrilateral learning from picture-based memorization to property-based reasoning. Effective instruction begins with lines rather than shape names, prioritizes the question “What must be true?” over “What does this look like?”, and uses varied, systematic example spaces—including rotated, stretched, and non-prototypical examples alongside carefully chosen non-examples. Making hierarchical relationships explicit through nested diagrams helps students internalize inclusive logic rather than defaulting to mutually exclusive thinking. Crucially, strong formative assessment reveals student reasoning, not just answers: questions like “What would have to change for this to become a square?” distinguish genuine understanding from pattern-matching. For specific teacher moves and classroom strategies that bring these principles to life, see our Quadrilateral Companion Document. Note: for a classroom-ready visual reference, see the DMTI Quadrilateral Poster, available on Teachers Pay Teachers and Redbubble.

Conclusion

The persistent difficulties students experience with quadrilaterals—the square-rectangle confusion, the orientation errors—are not mysteries. They are the predictable outcomes of instruction that teaches shapes as pictures rather than as structures defined by invariant properties.

When instruction is redesigned around line relationships, property-based classification, and explicit hierarchical reasoning, quadrilaterals become something far more powerful than a vocabulary unit. They become grounds for logical thinking, for the discipline of asking “What must be true?” rather than “What does this look like?”—and for the intellectual habits that will serve students across every domain of mathematics. Properties over Pictures.

References

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). Macmillan.

Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. Routledge.

Dagli, U. Y., & Halat, E. (2016). Young children’s conceptual understanding of triangle. Eurasia Journal of Mathematics, Science & Technology Education, 12(2), 189–202.

de Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. For the Learning of Mathematics, 14(1), 11–18.

Fujita, T., & Jones, K. (2007). Learners’ understanding of the definitions and hierarchical classification of quadrilaterals. Research in Mathematics Education, 9(1–2), 3–20.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Lehrer, R., & Schauble, L. (2015). Learning progressions: The whole world is not a stage. Science Education, 99(3), 432–437.

Rosch, E. (1978). Principles of categorization. In E. Rosch & B. Lloyd (Eds.), Cognition and categorization (pp. 27–48). Erlbaum.

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K., & Newcombe, N. S. (2016). The shape of things: The origin of young children’s knowledge of the names and properties of geometric forms. Journal of Cognition and Development, 17(1), 142–161.

Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Kluwer.

Depth of Knowledge (DOK), developed by Webb (1997, 2002), is often described as a hierarchy of rigor. In practice, this misunderstanding quietly shapes classroom instruction. When rigor is equated with larger numbers, more steps, or longer assignments, lesson design drifts toward procedural intensity rather than conceptual depth. But DOK was never intended to measure how hard a problem feels. It was designed to classify the kind of thinking a task requires. If we misunderstand that distinction, we unintentionally design lessons that train students to equate mathematics with speed and rote memorization rather than with structure and reasoning.

A more precise way to understand DOK is as a taxonomy of mental actions. When students engage with a math task, what must they do cognitively? Must they retrieve a fact, interpret a context, construct a model, justify a claim, or analyze a pattern? DOK describes the complexity of reasoning required, not the surface features of a task. This shift is not theoretical — it directly affects how we design lessons and how we ask questions during instruction. The questions teachers pose determine the depth of thinking students practice.

From a cognitive psychology perspective, each DOK level engages different systems of thinking. Retrieval activates long-term memory; modeling requires integration of visual and symbolic representations; justification engages executive functioning and self-explanation (Sweller, 1988; Paivio, 1990). A task can be effortful — such as multi-digit multiplication with several carrying steps — without being cognitively complex. True rigor comes from the reasoning demanded, not from the size of the numbers.

DOK 1: Retrieval and Reproduction in Multiplication

DOK 1 tasks require recall or execution of well-practiced procedures. In multiplication, this might simply be computing 7 × 8. The defining feature is reproduction without strategic decision-making. DOK 1 is foundational. Fluency reduces cognitive load and frees working memory for higher-order reasoning (Roediger & Karpicke, 2006). Students must move from consciously thinking about facts to thinking with them (Willingham, 2009). However, increasing the size of the numbers does not increase depth. A page filled with larger multiplication problems remains DOK 1 if students are simply executing a procedure. Automaticity supports reasoning, but automaticity alone does not reveal conceptual understanding (Hiebert & Carpenter, 1992). When lesson design emphasizes only DOK 1 tasks, instruction centers on performance rather than structure.

DOK 2: Applying and Representing Mathematical Structure

DOK 2 tasks move beyond reproduction and require application or representation. In elementary mathematics, it is helpful to distinguish between two forms: contextual interpretation and conceptual modeling.

DOK 2A: Contextual Problem Solving and Quantitative Interpretation

A DOK 2A contextual task might state, “There are 7 rows of chairs with 8 chairs in each row. How many chairs are there?” Students must interpret the quantities, determine the relationship, and select multiplication as the operation. The mental action is translating context into mathematical structure. Research shows that difficulty with word problems often arises from the coordination of language and quantitative schemas (Kintsch & Greeno, 1985). Effective lesson design ensures that contextual tasks measure mathematical reasoning rather than reading complexity.

DOK 2B: Conceptual Modeling Through Representation

In DOK 2B, students represent mathematical structure using visual models. A task might ask students to draw an array or area model for 7 × 8 and explain what each dimension represents. Research on dual coding suggests that linking visual and symbolic forms strengthens understanding and supports transfer (Paivio, 1990; Fyfe et al., 2014). When students construct arrays or area models, they are building mental structures that later help them understand fractions, algebra, and proportional reasoning. Representing ideas visually makes mathematical structure visible. In this way, DOK 2B supports relational understanding — knowing not just how to get an answer, but why the mathematics works (Skemp, 1976). Teachers can deepen this level by asking students to compare different models, such as an array and a repeated addition model, to see how they both represent the same operation..

DOK 3: Strategic Thinking and Justification

DOK 3 tasks require strategic thinking and justification. In multiplication, students might be asked to explain why 7 × 8 equals 8 × 7 using a visual model, or to compare two strategies for solving 36 × 25 and determine which is more efficient. The mental action shifts from doing mathematics to reasoning about mathematics. When students justify, they articulate relationships using precise structural language — unit, compose, decompose, iterate, partition, and equal. Explanation strengthens conceptual networks and promotes durable learning (Chi et al., 1989). It transforms procedural knowledge into relational knowledge (Skemp, 1976). Instructionally, this requires teachers to pause and ask, “Why does that work?” or “How do you know the order does not change the product?” If such questions are absent, DOK 3 rarely occurs, regardless of how complex the numbers appear.

DOK 4: Extended Reasoning Across Conditions

DOK 4 tasks require sustained reasoning across multiple representations or conditions and push students beyond single problems into structured investigation. In multiplication, for example, students might generate all possible rectangular arrays with an area of 48 square units and analyze how the perimeter changes as factor pairs vary. This level demands synthesis and pattern analysis across cases; students must investigate systematically, make predictions, test conjectures, and draw conclusions from related data. Such work engages planning, monitoring, and generalization—forms of higher-order thinking identified as central to mathematical proficiency (National Research Council, 2001). Although extended reasoning tasks may not occur daily, they promote deep structural insight by requiring students to decide what to examine, how to organize their findings, and how relationships shift across conditions. In doing so, DOK 4 moves learners from solving isolated problems to recognizing how mathematical ideas connect within a broader system.

Implications for Educators and Instructional Practice

Educators sometimes wonder why children are asked to draw multiplication models instead of jumping straight to the standard algorithm. However, representational competence—the ability to use and connect models—predicts long-term mathematical success (Clements & Sarama, 2014). Drawing arrays for 7 × 8 builds mental structures that later support understanding of area models in algebra. Models are not detours from rigor; they are the foundation of rigor. From a cognitive science perspective, these models give students a mental image to “anchor” the abstract symbol, making it more meaningful and memorable.

Educators can support deeper thinking by asking students to explain their reasoning or show another representation. Questions such as “How do you know?” or “Can you show that another way?” raise the cognitive demand without increasing stress. Simple prompts can meaningfully increase the depth of thinking. For example, after a child computes 6 × 4, an educator might say, “Show this using a bar model or an area model,” or “If you forgot that fact, what is another way you could figure it out?” These questions gently move the child from DOK 1 toward DOK 2B (modeling) and DOK 3 (justification).

Clarifying Misconceptions About DOK

Several misconceptions about DOK persist. Higher DOK does not mean larger numbers or longer problems. Not all word problems are DOK 3; most are actually DOK 2A. DOK is not a staircase that students must climb step by step; it is a way to classify the demands of a task (Webb, 2002). DOK describes the nature of thinking, not the order of instruction. Furthermore, a task is not permanently “at” a certain DOK level. Its level depends on the thinking required of the student. A task that asks for an explanation (DOK 3) becomes a DOK 1 task if the teacher has already provided and practiced the exact explanation. Effective classrooms move flexibly among levels, reinforcing fluency while deepening conceptual reasoning. Cognitive growth happens through this movement across levels, not by abandoning foundational skills.

Conclusion

Depth of Knowledge provides educators with a precise tool for aligning instruction, cognition, and assessment. In multiplication and across all mathematical domains, balanced cognitive demand supports conceptual understanding, procedural fluency, and strategic competence. True mathematical rigor lies in the quality of reasoning students are asked to perform. When instruction and assessments intentionally include retrieval, contextual interpretation, representation, and justification, they reflect the full, multidimensional nature of mathematical proficiency (National Research Council, 2001). Through careful application of DOK, K–5 educators can design learning experiences that cultivate not only correct answers, but well-structured understanding—knowledge that lasts and transfers to new situations.

References

Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182.

Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.). Routledge.

Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92(1), 109–129.

National Research Council. (2001). Adding it up: Helping children learn mathematics. National Academy Press.

Paivio, A. (1990). Mental representations: A dual coding approach. Oxford University Press.

Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). Psychology Press.

Roediger, H. L., & Karpicke, J. D. (2006). The power of testing memory: Basic research and implications for educational practice. Perspectives on Psychological Science, 1(3), 181–210.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.

Webb, N. L. (1997). Research monograph number 6: Criteria for alignment of expectations and assessments in mathematics and science education. Council of Chief State School Officers.

Webb, N. L. (2002). Depth-of-knowledge levels for four content areas. Wisconsin Center for Education Research.

Willingham, D. T. (2009). Why don’t students like school?: A cognitive scientist answers questions about how the mind works and what it means for the classroom. Jossey-Bass.

Why do the same area and perimeter errors show up year after year, even when these topics are “covered” multiple times?

Students’ most common misconceptions about area and perimeter are not random; they are the predictable result of typical instructional sequences that blur the fundamental dimensional nature of these concepts. When instruction treats area and perimeter as a pair of similar formulas attached to the same diagrams, rather than as distinct measurement quantities, students build fragile, formula-driven schemas that fail outside of routine problems (Battista, 2004; Lehrer & Wilson, 2011). The tools students use often deepen the confusion, as the same square tile may be used ambiguously for both perimeter and area, obscuring whether the edge (1D) or the face (2D) constitutes the unit (Lehrer, Jenkins, & Osana, 1998).

The Core Problem—Obscured Dimensionality and Cognitive Load

How do dimension, units, and cognitive load interact to make area and perimeter harder than they appear?

The central instructional problem is a failure to make explicit the dimensional nature of measurement quantities. Perimeter is a one-dimensional (1D) quantity measuring length around a boundary, while area is a two-dimensional (2D) quantity measuring surface coverage (Clements & Sarama, 2014). Teaching these concepts simultaneously, emphasizing formulas prematurely, or using ambiguous tools, obscures this distinction and leads to systematic misconceptions. From a cognitive psychology perspective, this approach imposes high extraneous cognitive load: students must juggle symbols, visual interpretation, and procedural recall without a clear mental model of what is being measured, overwhelming working memory and hindering schema construction (Sweller et al., 2011; Mayer,).

Predictable Misconceptions and Their Instructional Roots

What are the most common errors, and how does current instruction set them up?

Research and classroom evidence reveal a consistent pattern of errors (Battista, 2004):

• Formula Swapping: Using length × width for perimeter or adding sides for area.

• Area-Perimeter Conflation: Believing larger perimeter means larger area, or that shapes with equal area must have equal perimeter.

• Grid-Counting Errors: Counting boundary squares for area, interior squares for perimeter, or grid intersections instead of units.

• Lack of Transfer: Inability to reason about non-rectangular or composite shapes without a procedural cue.

These misconceptions arise predictably from common instructional practices:

1. Teaching Area and Perimeter Together: Presenting them in the same unit frames them as procedural variations of the same task, rather than distinct concepts (Battista, 2004).

2. Starting with Formulas: Introducing A = l × w and P = 2(l + w) before establishing the concepts of unit iteration reduces measurement to abstract symbol manipulation (Lehrer et al., 1998).

3. Using Dimensionally Ambiguous Tools: Employing the same tool (e.g., square tiles) for both perimeter and area without clarifying the shift from a 1D edge unit to a 2D surface unit sends mixed signals (Clements & Sarama, 2014).

Misconceptions as Diagnostic Feedback

What are students’ errors telling us about our teaching?

Student errors are not just mistakes—they are clues about how instruction shaped students’ thinking. Area-perimeter conflation signals a lack of experience with contrast tasks in which one quantity varies while the other is held constant (Battista, 2004). Formula swapping indicates that procedures were memorized without connection to the underlying unit structure. Grid‑counting mistakes reveal that students were not explicitly taught “what counts as one unit” for each dimension. Failure with irregular shapes indicates an overreliance on formula spotting rather than reasoning through decomposition and unit iteration. These patterns collectively indicate that instruction blurred dimensional distinctions, introduced symbolic shortcuts too early, and used tools that did not consistently embody the intended attribute (Sweller et al., 2011).

A Dimensional Framework for Measurement (0D–1D–2D)

How can a dimensional storyline organize student thinking?

A coherent dimensional progression provides a powerful conceptual framework. Learning trajectories research supports building understanding from foundational 1D concepts to more complex 2D and 3D ones (Clements & Sarama, 2014; Lehrer & Wilson, 2011):

• 1D (Length/Perimeter): Iterating linear units along a path.

2D (Area): Structuring space into an array of square units and coordinating them multiplicatively.
This progression represents a qualitative shift in reasoning. Explicitly discussing “Are we measuring theboundary (1D) or the surface (2D)?” helps students categorize quantities and select appropriate strategies.

Aligning Tools and Units with Dimensions

Which tools best highlight 1D and 2D measurement, and how should we use them?

Tools must be chosen and used to make dimensionality perceptually obvious (Battista, 2004; Clements & Sarama, 2014):

Dimension Quantity Ideal Tools & Units Purpose

1D Perimeter, String, ruler, tape; linear units, To embody iteration of length around a boundary.

2D Area , Square tiles, grid paper; square units, To make surface coverage and array structure visible.

The principle is intentional alignment: use tools whose form and function match the dimension of the attribute being measured, and explicitly name that match. Avoid using square tiles to measure perimeter, as this conflates the 2D object with the 1D unit. Digital tools should be chosen for their ability to preserve unit visibility and focus attention on structure, not just dynamic manipulation (Mayer, 2020).

An Improved Instructional Pathway

How can we redesign instruction to build lasting understanding?

Effective instruction inverts the common formula‑first sequence:

1. Separate and Establish 1D: Develop a robust concept of length and perimeter through unit iteration with linear tools.

2. Build 2D Concept from Units: Introduce area as covering with square tiles. Focus on tiling, counting, and structuring into rows/columns long before naming a formula.

3. Contrast to Clarify: Once each concept is stable, use contrast tasks (e.g., fixed perimeter with varying area) to solidify the dimensional distinction (Lehrer & Wilson, 2011).

4. Generalize with Formulas: Introduce formulas only as efficient records of the unit‑iteration processes students already understand.

5. Apply and Transfer: Use decomposition and composition tasks with complex figures to reinforce reasoning from units, not shape recognition.

This pathway manages cognitive load by sequencing concepts logically, using supportive tools, and delaying symbolic abstraction until conceptual schemas are formed (Sweller et al., 2011).

Conclusion: The Evidence is Clear

The persistent difficulties students experience with area and perimeter are not a mystery, nor are they the result of students “not trying hard enough.” From both mathematics education and cognitive psychology perspectives, these errors are the predictable outcome of instruction that blurs dimensional distinctions, minimizes units, and introduces symbolic shortcuts before conceptual understanding is established.

When perimeter is taught as a one-dimensional quantity that measures length around a boundary, and area is taught as a two-dimensional quantity that measures surface coverage through unit iteration, students are far less likely to confuse the two. When tools clearly embody the attribute being measured and instruction progresses from concrete action to visual structure to symbolic representation, cognitive load is reduced, and understanding becomes durable.

Area and perimeter are not isolated topics; they are gateways to proportional reasoning, algebraic thinking, and spatial sense. Teaching them well requires more than better worksheets or clearer explanations—it requires instructional designs that align mathematical structure with how students learn. When that alignment is present, formulas become meaningful, misconceptions become instructional feedback, and students gain understanding that transfers beyond the page.

References

Battista, M. T. (2004). Applying cognition-based assessment to elementary school students’ development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185–204. https://doi.org/10.1207/s15327833mtl0602_4

Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.). Routledge.

Lehrer, R., Jenkins, M., & Osana, H. (1998). The construction of space in measurement. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 67–100). Lawrence Erlbaum Associates.

Lehrer, R., & Wilson, M. (2011). Developing understanding of measurement. In K. J. Leatham & B. R. Peterson (Eds.), Learning progressions in mathematics (pp. 83–101). National Council of Teachers of Mathematics.

Mayer, R. E. (2020). Multimedia learning (3rd ed.). Cambridge University Press.

Sweller, J., Ayres, P., &Kalyuga, S. (2011). Cognitive load theory. Springer..

Summer math programs often focus on skill practice, but true mathematical understanding requires more: critical thinking, precise language, and the ability to explain reasoning. This research explores how effective summer math programs go beyond worksheets to foster deep, lasting mathematical growth. We investigate the relationship between rich early math experiences (incorporating storytelling, problem-solving, and mathematical discourse) and improved student outcomes, including stronger academic performance and enhanced language skills. This overview addresses key questions for educators designing impactful summer programs: How can we cultivate flexible mathematical understanding? What progression of models and language best supports this growth? When should we encourage student-generated strategies versus formal methods? The answers provide practical guidance for creating engaging and effective summer math initiatives.

From Math Stories to Academic Success

How do early math experiences in summer programs shape students’ future academic achievement?

Research consistently shows that early mathematics skills are foundational for later math learning and are among the strongest predictors of later reading and overall academic success, often surpassing the predictive power of early reading skills (Duncan et al., 2007; Nguyen et al., 2021). This strong correlation stems from the development of crucial cognitive and linguistic skills through early math experiences. Executive functions such as working memory and cognitive flexibility are significantly enhanced by engaging with mathematical concepts, as is the acquisition of precise mathematical language. Consequently, these skills are transferable and support both mathematical and literacy development. These findings underscore the importance of early, intentional math experiences in shaping students’ long-term academic trajectories. Effectively designed summer programs significantly strengthen students' cognitive and linguistic skills by incorporating activities focused on math storytelling, rich problem-solving tasks, and language-rich routines, setting them up for greater success in subsequent academic years.

Evidence: Impact on Learning and Engagement

How do summer math programs influence students’ academic growth and engagement?

Recent research demonstrates that summer math programs have a significant, positive impact on student learning and engagement. A comprehensive meta-analysis of 37 experimental and quasi-experimental studies found that students who participated in summer math programs achieved notably higher mathematics outcomes than their peers, with an average effect size of +0.10 standard deviations, which represents a meaningful improvement in math scores (Lynch, An, & Mancenido, 2022). These results highlight that the benefits of summer math programs are not limited to academic achievement but extend to fostering positive attitudes toward mathematics and school participation. These gains are consistent across both higher- and lower-poverty settings, highlighting the equity potential of well-designed summer programs.

This positive impact on engagement is directly linked to program design, as studies show improvements particularly in programs that prioritize hands-on activities, collaborative learning, and rich mathematical discourse (Lynch et al., 2022). These findings strongly support the implementation of active, student-centered summer math programs as an effective strategy for improving both academic achievement and student engagement in mathematics.

Persistent Challenges and Their Solutions.

Why do some students struggle to make gains in traditional summer math programs?

A common flaw in many summer math programs is their emphasis on remediation and rote skill practice, which often limits student engagement and deeper understanding. This emphasis on speed and memorization, often reflected in program design, perpetuates persistent misconceptions such as viewing math as a set of isolated procedures and hinders the development of deeper understanding. Furthermore, when foundational topics like measurement, spatial reasoning, problem-solving, and the use of visual models are neglected, students miss out on essential building blocks for later mathematical achievement. Research shows that early mastery of these areas strongly predicts later success in mathematics and reading (Duncan et al., 2007; Verdine et al., 2017; Mix & Cheng, 2012). Programs that integrate math with literacy, through storytelling, collaborative problem-solving, and culturally relevant contexts, yield greater gains in both mathematics and reading comprehension scores than literacy-only approaches. Embedding mathematics within literacy-rich environments accelerates math learning and enhances literacy. Addressing these persistent challenges requires intentional program design that values depth over speed and provides equitable access to rich mathematical experiences for all students.

An Instructional Models

What happens when students use real-world models and stories before formalizing mathematical procedures?

Effective summer math programs, such as the DMTI Summer Program, utilize a carefully sequenced progression of learning experiences, moving from concrete, hands-on activities (e.g., using story mats and manipulatives to solve word problems involving addition and subtraction) to visual models (e.g., bar models and number lines to represent problem situations) and finally to symbolic representations and equations. Manipulatives foster deep conceptual understanding, while visual models seamlessly bridge the gap between the concrete and the abstract, allowing students to connect their actions to symbolic notation and build lasting understanding. Each week's curriculum is structured around rich problem-solving activities that are framed within culturally relevant contexts, emphasizing language development, strategic formalization, and varied practice, including both independent and collaborative work. This approach ensures students connect mathematical concepts to language, culture, and real-life contexts. This enactive-iconic-symbolic approach, grounded in well-established learning theories like Bruner's three modes of representation, builds a strong foundation for future learning (DMTI, 2025).

Integrating Mathematical Language and Literacy

How can integrating mathematical language and literacy strategies in summer math programs improve both students’ conceptual understanding and their overall academic achievement?

Integrating mathematical language and literacy practices is essential for fostering deep conceptual understanding and academic growth in summer math programs. Research demonstrates that when teachers explicitly emphasize precise mathematical vocabulary (such as unit, compose, decompose, iterate, partition, and equal), facilitate structured mathematical discourse, and incorporate writing activities like math journaling and story creation, students’ academic language proficiency is significantly strengthened a critical foundation for mathematical reasoning and comprehension (Lynch et al., 2022; Reynolds & Yavuz, 2022). Embedding these language-rich strategies within problem-based learning and literacy activities such as story mats, read-alouds, and collaborative discussions not only supports comprehension and mathematical modeling (Lenhoff et al., 2020) but also increases engagement and leads to improved mathematics achievement. By connecting math problems to students’ local contexts and encouraging reflection through both writing and oral explanation, summer programs enhance student learning, promote long-term retention, and bridge the gap between math and literacy skills. This integrated approach supports not only higher mathematical achievement but also broader academic success, particularly for students from diverse backgrounds (Lynch et al., 2022; Reynolds & Yavuz, 2022). This interaction between language and mathematics not only supports achievement in both domains but also prepares students for the increasingly interdisciplinary demands of future learning.

Conclusion: The Evidence is Clear

How can summer programs become launchpads for reasoning and confidence?

Research overwhelmingly demonstrates that effective summer math programs prioritize conceptual understanding, integrated literacy, and foundational skills moving far beyond rote practice and remediation. By centering instruction on mathematical discourse, precise vocabulary, and writing activities, and by connecting learning to meaningful, real-world contexts, school districts can create engaging environments where students build both academic skills and confidence. Incorporating visual and symbolic models, as well as opportunities for reflection through journaling and oral explanations, ensures that learning is both meaningful and enduring. Ongoing professional development and collaborative planning among educators further maximize program impact. Thoughtfully designed, research-driven summer math experiences can truly transform students’ academic trajectories, supporting every child’s growth, confidence, and future success.

References

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Developing Mathematical Thinking Institute (2025). www.dmtinstitute.com Boise, ID.

Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., ... & Sexton, H. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446.

Lenhoff, S. W., Somers, C., Tenelshof, B., & Bender, T. (2020). The potential for multi-site literacy interventions to reduce summer slide among low-performing students. The Urban Review, 52(4), 633–655.

Lynch, K., An, L., & Mancenido, Z. (2022). The impact of summer programs on student mathematics achievement: A meta-analysis. Review of Educational Research. Advance online publication.

Mix, K. S., & Cheng, Y.-L. (2012). The relation between space and math: Developmental and educational implications. In H. J. Ross (Ed.), Advances in child development and behavior (Vol. 42, pp. 197–243). Academic Press.

Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., & Bailey, D. H. (2021). Early mathematics knowledge and later achievement: A longitudinal analysis. Developmental Psychology, 57(9), 1502–1516.

Reynolds, A., & Yavuz, O. (2022). A mechanism to increase literacy and math skills to reduce summer learning loss. Education Leadership Review of Doctoral Research, 10, 48–68.

Verdine, B. N., Irwin, C. M., Golinkoff, R. M., & Hirsh-Pasek, K. (2017). Links between spatial and mathematical skills across the preschool years. Monographs of the Society for Research in Child Development, 82(1), 7–30.

Social Media

Maximize Summer Learning: Boost Math & Literacy Skills Simultaneously

Invest in high-impact summer math programs that deliver a double return: improved math and literacy skills. Research shows these programs significantly enhance students’ academic language proficiency, mathematical reasoning, and overall confidence. By incorporating evidence-based strategies like precise vocabulary instruction, collaborative discussions, and hands-on activities, you can equip your students for success in the upcoming school year. Effective summer programs reduce the summer slide, improve test scores, and create a more engaging learning environment. This represents a highly efficient use of resources, maximizing student outcomes.

Learn more about evidence-based summer learning strategies: www.dmtinstitute.com

A student points to a square drawn on a vertex and says, “That’s a diamond.” The poster on the wall agrees. The worksheet in the packet agrees too. In that moment, the student is not confused—they are being perfectly consistent with the materials and language they have been given. This is one problem of shape instruction: students are often learning a folk geometry—a system of picture-based categories grounded in orientation, symmetry, and cultural labels rather than the mathematics of invariant properties.

Search “triangle definition for kids.” Browse shape posters from major curriculum publishers. Scan popular geometry activities online. Again and again, definitions and images emphasize what shapes look like instead of what must be true. These materials do not merely fail to correct misconceptions; they actively train them (Dağlı & Halat, 2016; Verdine et al., 2016). This overview argues that repairing shape instruction requires reorganizing it around a single, coherent spine: viewing shapes as systems of intersecting straight lines that form closed structures, from which vertices, sides, and properties emerge in a logical order (Lehrer & Schauble, 2015).

Introduction: The Core Cognitive Shift

Why do so many students and adults believe a shape can change simply because it is turned?

Polygons are one of the earliest places in mathematics where learners must move from “it looks like…” to “it must have…” from “pictures” to “properties. This shift from perceptual familiarity to definition-based reasoning—is not cosmetic. It underlies later learning about transformations, area, similarity, congruence, and proof (Lehrer & Schauble, 2015).

Research from mathematics education and cognitive psychology converges on a shared explanation: students construct shape categories from the examples, language, and images they encounter most often, not from formal definitions alone (Duval, 2006; Verdine et al., 2016). When those examples are narrow and prototype-heavy, categories become fragile and orientation-dependent.

Effective shape instruction therefore, requires three integrated forms of knowledge:

• Mathematical structure (definitions as constraints, not pictures),

• Cognitive development (how concept images form and narrow), and

• Language awareness (how everyday terms like diamond create competing taxonomies).

A unifying instructional stance is to adopt the intersecting-lines lens: treating polygons not as finished pictures, but as structures that emerge when straight lines intersect and close in space.

A Foundational Reframe: Shapes as Lines That Intersect in Space

What if students learned shapes as structures that emerge when lines intersect?

Most shape instruction begins with completed figures a triangle drawn upright, a square resting on a side, a “diamond” tilted on a vertex. Cognitively, this privileges the final image and invites classification by appearance. The intersecting-lines lens reverses that order.

In this framing, shapes are not objects, but outcomes of relationships. When straight lines intersect, they create points of intersection. When those intersections are connected by straight segments, they form edges. When those edges connect in a closed chain, a polygon is created. The named shape—triangle, quadrilateral, pentagon—is the result of these constraints, not the starting point.

This view aligns with research showing that geometric understanding deepens when learners attend to relations, constructions, and transformations, rather than static images (Battista, 2007; Duval, 2006). Lehrer and Schauble (2015) emphasize that learning progressions depend on helping students coordinate representations and relationships—seeing figures as systems that can be composed, decomposed, and reorganized.

From this perspective, vertices are not “corners” of a picture, but points where line segments intersect. This distinction matters. When vertices are defined relationally, students are better able to identify them in concave figures, extremely acute or obtuse angles, and non-prototypical shapes. Research shows that vague “corner” language contributes to systematic errors in vertex counting and shape classification (Duval, 2006; Dağlı & Halat, 2016).

Once intersections are established, attention turns to the segments between them, which must be straight, not curved. Emphasizing straightness early helps students distinguish polygons from curved figures such as circles—distinctions often blurred by definitions that describe polygons merely as “flat shapes” (Verdine et al., 2016).

Next comes closure. A polygon exists only when straight segments connect in a closed loop with no gaps. Studies show that when closure is not treated as a necessary condition, students routinely accept “almost closed” figures as legitimate shapes (Dağlı & Halat, 2016).

Only after straightness and closure are secured does it make sense to attend to the number of vertices (and sides). Counting vertices as intersection points naturally leads to classifying polygons by number three for triangles, four for quadrilaterals, five for pentagons, and so on revealing these shapes as members of a single family differentiated by a structural parameter (Verdine et al., 2016).

Finally, properties—equal side lengths, right angles, parallel sides become tools for refinement rather than sources of confusion. Properties no longer decide whether a figure is a polygon; they decide which polygon it is. This ordering supports inclusive hierarchies and reduces reliance on visual prototypes (Fujita & Jones, 2007).

The Mathematical Lens: Constraints, Hierarchies, and Structure

What must be true—regardless of orientation—for a figure to belong to a polygon category?

Mathematically, polygons are simple, closed plane figures composed of straight line segments, classified by number of sides and refined by properties such as angle measures, parallelism, and symmetry (NGA & CCSSO, 2010). The intersecting-lines lens makes these constraints explicit and inspectable.

This lens provides a productive alternative to appearance-based disagreement. When a student claims a rotated triangle is “not a triangle,” the teacher can redirect attention to invariants: number of intersections, straight segments, and closure—rather than debating orientation.

Quadrilaterals exemplify how mathematical categories are built by adding constraints. Parallelograms require parallel sides; rectangles add right angles; rhombi add equal sides; squares satisfy both constraints simultaneously (Fujita & Jones, 2007). This inclusive hierarchy is mathematically elegant but cognitively difficult, because everyday categories are often exclusive. Understanding this tension is central to teachers’ Mathematical Knowledge for Teaching (MKT).

The Cognitive Hurdle: Prototypes and Concept Images

Why does changing a shape’s orientation disrupt classification even when no defining property changes?

Cognitive research shows that children initially organize shape categories around canonical exemplars—upright triangles, squares resting on a side, highly regular polygons (Verdine et al., 2016). These prototypes support quick recognition but narrow the category. Duval (2006) characterizes this as a tension between concept image (what the category “looks like”) and concept definition (what properties determine membership). When orientation changes, the visual match fails—even though the definition still applies. Robust understanding develops only when learners encounter systematic variation in orientation, size, and regularity. Variation is not enrichment; it is the mechanism by which definitions gain meaning (Verdine et al., 2016).

Applying the Intersecting-Lines Lens to Persistent Problem Areas

When students make predictable shape errors, what rule are they using—and how can instruction replace it?

Triangles: The “Skinny Triangle” Rule

Many students accept only upright, isosceles triangles and reject obtuse or rotated examples (Dağlı & Halat, 2016). The implicit rule is appearance-based. Instructional move: Return to invariants—three straight sides, three vertices, closed—and ask what changed and what stayed the same under rotation (Lehrer & Schauble, 2015).

Quadrilaterals: The Hierarchy Conflict

Students resist “a square is a rectangle” because they treat categories as exclusive. Distinct names imply distinct kinds (Fujita & Jones, 2007). Instructional move: Re-center on defining attributes (four right angles) and use nested diagrams or property grids to make inclusion visible.

The “Diamond” Problem: Language as Taxonomy

Diamond is not a mathematical category, yet its use introduces a parallel, orientation-based taxonomy (Duval, 2006; Verdine et al., 2016). Instructional move: Use diamond as a contrast—everyday word vs mathematical classification—and verify invariants explicitly.

The Tyranny of “Nice” Shapes

Irregular pentagons and hexagons are often rejected as “not real,” especially when curricula overrepresent regular examples (Verdine et al., 2016). Instructional move: Design example spaces that force property-checking rather than aesthetic judgment.

The Instructional Path Forward: From Naming to Structural Reasoning

What daily routines make property-based reasoning inevitable rather than optional?

Research describes a progression from visual recognition to property analysis to relational reasoning about classes and hierarchies (van Hiele, 1986). This progression aligns with standards across K–8 (NGA & CCSSO, 2010).

Instruction strengthens when teachers model structural actions—compose, decompose, partition, and reason about equal sides and angles—rather than relying solely on identification (Clements & Sarama, 2011). These actions connect early shape work to later ideas in area, similarity, and proof.

A practical routine across grades is the 3-check method:

1. Straight sides?

2. Closed figure?

3. Count vertices/sides,

4. Apply properties.

Conclusion: Choosing a Geometry of Pictures or Properties

If students learned geometry only from your examples and language, what theory of “what makes a shape a shape” would they construct?

Every classroom teaches either a geometry of pictures or a geometry of properties. That choice is made not in standards documents, but in posters, examples, definitions, and daily language. The same mechanism that creates misconceptions can eliminate them: change the training set. Broaden examples, reorder definitions, make hierarchy visible, and treat everyday labels as contrasts—not categories (Duval, 2006; Fujita & Jones, 2007; Lehrer & Schauble, 2015; Verdine et al., 2016).

References

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

Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333(6045), 968–970.

Dağlı, Ü. Y., & Halat, E. (2016). Young children’s conceptual understanding of triangle. Eurasia Journal of Mathematics, Science & Technology Education, 12(2), 189–202.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.

Fujita, T., & Jones, K. (2007). Learners’ understanding of the definitions and hierarchical classification of quadrilaterals. Research in Mathematics Education, 9(1–2), 3–20.

Lehrer, R., & Schauble, L. (2015). Learning progressions: The whole world is not a stage. Science Education, 99(3), 432–437.

National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Author.

van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Academic Press.

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K., & Newcombe, N. S. (2016). The shape of things: The origin of young children’s knowledge of the names and properties of geometric forms. Journal of Cognition and Development, 17(1), 142–161.

Across the nation, educators are working to improve mathematics achievement amid rising standards, limited instructional time, and increasing expectations for depth of understanding. One of the most powerful but often overlooked levers for improvement is how instructional time is allocated and structured. Minutes are not neutral; they determine whether teachers can engage students in problem-solving, reasoning, and discourse, or whether instruction is reduced to surface-level coverage. Research from cognitive science, developmental psychology, and mathematics education converges on a clear conclusion: students need sufficient, consistent, and intentionally structured time to develop durable mathematical understanding. Importantly, this research does not suggest that simply adding minutes automatically improves learning. Rather, it shows that without sufficient time, high-leverage instructional practices cannot reliably occur at all, regardless of curriculum quality or teacher effort (Hiebert & Grouws, 2007; National Research Council, 2001).

What Does the Developing Brain Require From Mathematics Instructional Time?

From a cognitive psychology perspective, mathematics instruction must align with the developing brain’s realities. Cognitive load theory demonstrates that working memory is highly limited, particularly when students encounter new procedures, representations, or multi-step reasoning (Sweller, 1988). Long stretches of uninterrupted teacher talk overload this system, reducing comprehension and retention. Instead, effective mathematics instruction requires purposeful segmentation—brief instruction followed by exploration, discussion, and reflection. These instructional shifts are cognitive necessities, not pedagogical preferences, because they allow students to process, organize, and integrate new information into long-term memory (Sweller, 1988; Sweller, Ayres, & Kalyuga, 2011).

Equally important is the spacing effect, one of the most robust findings in learning science. Research shows that learning is stronger and more durable when practice is distributed over time rather than massed (Cepeda et al., 2006). Daily engagement with mathematics through retrieval routines, cumulative review, and repeated encounters with core ideas helps maintain and strengthen understanding. Without consistent daily activation, both conceptual and procedural knowledge decay, particularly for students who rely most on school-based learning opportunities. Automaticity, which frees working memory for higher-order reasoning, requires repeated, distributed practice rather than irregular exposure (Baroody, 2006). Thus, the central issue is not whether teachers value rich instruction, but whether there is sufficient instructional time for that instruction to function as intended.

How Much Daily Mathematics Time Do Students Need?

Research integrating cognitive principles with mathematics education points to a clear developmental progression in daily Tier 1 instructional time. In Grades K–2, students are building foundational number concepts through experiences with quantity, comparison, and structure. These ideas require time for manipulation, discussion, representation, and revisiting concepts. Research supports 70–90 minutes daily of core mathematics instruction, not to accelerate pacing, but to allow young learners to engage in developmentally appropriate cycles of exploration and sense-making. A workshop-style structure—brief instruction followed by extended small-group and hands-on learning—supports attention limits while promoting deep understanding.

In Grades 3–5, students transition to more abstract content, including multi-step operations, fractions, and early algebraic reasoning. These concepts demand opportunities for rich problem solving, strategy comparison, and discourse. A daily block of 60–75 minutes allows teachers to include retrieval routines, conceptual investigation, guided practice, and consolidation. When time is compressed, teachers are often forced to choose between depth and coverage, even though research consistently shows that depth supports long-term retention and transfer more effectively than rapid coverage (Hiebert & Carpenter, 1992; Schmidt, Wang, & McKnight, 2005).

In Grades 6–8, departmentalized schedules typically provide 50–70 minutes of mathematics instruction. Adolescents still benefit from lessons divided into phases to manage cognitive load as abstraction increases. Topics such as proportional reasoning, expressions, equations, geometry, and statistics require sustained problem-solving paired with structured discussion and formalization. Across all grade bands, research supports a minimum of 60 minutes of Tier 1 mathematics daily, not because more time guarantees learning, but because less time reliably prevents high-leverage instructional practices from occurring (Banilower et al., 2013; National Research Council, 2001).

What Happens Inside a 50–70 Minute Mathematics Block?

Research suggests that effective mathematics learning unfolds through a complete cognitive cycle that cannot be compressed into short periods. A sustained block allows students to (a) activate prior knowledge, (b) engage in effortful problem solving, (c) discuss and refine strategies, and (d) consolidate learning into durable understanding. While specific instructional approaches vary, studies consistently show that each phase requires time to emerge (Kapur, 2014; Hattie & Yates, 2014).

Typically, the block begins with a brief activation of prior knowledge, supporting retrieval and attentional readiness. This is followed by an extended period of exploration and reasoning, during which students grapple with mathematical ideas, test strategies, and make sense of representations. Whole-group discussion and consolidation then play a critical role in connecting ideas, addressing misconceptions, and formalizing understanding. Finally, purposeful practice or reflection helps stabilize learning and prepare students for future retrieval.

Shortened or fragmented instructional periods interrupt this sequence. When lessons end before discussion and consolidation, students may complete tasks without integrating their understanding. Research on cognitive load and productive struggle indicates that learning is strongest when students are given enough uninterrupted time to struggle, resolve, and reflect within a single session, rather than restarting the cognitive process multiple times across the day.

How Should Intervention Time Be Added Without Weakening Core Instruction?

Within a Multi-Tiered System of Support (MTSS), one principle is non-negotiable: intervention must supplement, not replace, Tier 1 instruction (Fuchs et al., 2008; Gersten et al., 2009). Pulling students from core mathematics for intervention deprives them of exposure to new content and often creates the very gaps that intervention is intended to address. This concern becomes especially salient when schedules are compressed or instructional days are reduced.

Tier 2 intervention typically consists of 20–30 minutes, three to four days per week, delivered in small groups. Effective Tier 2 instruction is diagnostic and targeted, focusing on prerequisite skills and misconceptions using varied representations and feedback. Tier 2 is not a slower re-teaching of the core lesson, but a strategic support designed to strengthen access to upcoming grade-level learning.

Tier 3 intervention requires 30–45 minutes of daily, highly individualized instruction. Research consistently shows that interventions are most effective when Tier 1 instruction remains intact and when instructional priorities are clearly defined at the system level. Many schools address this need through a school-wide intervention or enrichment block, ensuring students receive support without sacrificing access to core mathematics.

How Do Weekly Schedules Influence Mathematics Learning?

Weekly scheduling decisions interact directly with cognitive principles. The spacing effect indicates that frequent, consistent engagement produces stronger retention than longer but less frequent sessions. A five-day instructional week naturally supports this pattern. In contrast, a four-day week introduces recurring three-day gaps that increase forgetting and require additional re-teaching. While longer instructional days may appear to compensate for reduced frequency, research has not identified scheduling redesigns that fully offset the loss of distributed practice in mathematics (Thompson, 2021; Fitzpatrick, Grissmer, & Hastedt, 2011).

Empirical studies show that mathematics achievement is more sensitive than reading to reduced instructional frequency, with small but cumulative adverse effects over time (Thompson, 2021). Teachers are right to ask whether certain students are affected more than others; evidence suggests that students who are already behind or most dependent on school-based learning experience the greatest harm. When four-day weeks are adopted, research points to the need for deliberate mitigation strategies, including structured review cycles, protected intervention time, and close monitoring of student learning outcomes.

What Is the Impact of Extended Breaks, Including Summer?

Just as spacing influences weekly learning, it also shapes yearly patterns. Lengthy interruptions especially the summer break—lead to significant erosion in mathematical understanding. Research indicates that students lose an average of 1 to 3 months of mathematical proficiency over the summer, with the largest losses occurring in the elementary grades (Cooper et al., 1996). Mathematics is particularly vulnerable because of its cumulative structure and reliance on sustained practice.

Importantly, summer learning loss is not evenly distributed. Longitudinal studies show that students from lower socioeconomic backgrounds experience substantially greater summer declines than their peers, while more advantaged students often maintain or increase skills through enrichment (Alexander et al., 2007). Over time, these unequal seasonal patterns contribute significantly to widening achievement gaps, even when school-year instruction is strong. Research identifies effective responses, including high-quality summer learning programs, balanced calendars, and structured home-learning supports (Augustine et al., 2016). Addressing summer learning loss is therefore both an academic and an equity imperative (Alexander et al., 2007; Quinn & Polikoff, 2017).

Conclusion:

Designing effective mathematics instruction requires more than selecting strong materials or improving individual teaching practices; it requires structuring time in ways that align with how students learn. Achievement strengthens when students receive sufficient, consistent, and cognitively aligned instructional minutes across the day, week, and year. Protecting daily mathematics blocks, structuring lessons to support reasoning and consolidation, adding targeted intervention without replacing core instruction, carefully evaluating shortened weeks, and mitigating long instructional gaps together form a coherent system that supports lasting understanding. The research does not argue for “more math at any cost,” but for aligning instructional time with the realities of learning so that teachers’ efforts can have their intended impact.

References

Alexander, K. L., Entwisle, D. R., & Olson, L. S. (2007). Lasting consequences of the summer learning gap. American Sociological Review, 72(2), 167–180. https://doi.org/10.1177/000312240707200202

Augustine, C. H., McCombs, J. S., Pane, J. F., Schwartz, H. L., Schweig, J., McEachin, A., & Siler-Evans, K. (2016). Learning from summer: Effects of voluntary summer learning programs on low-income urban youth. RAND Corporation.

Banilower, E. R., Smith, P. S., Weiss, I. R., Malzahn, K. A., Campbell, K. M., & Weis, A. M. (2013). Report of the 2012 national survey of science and mathematics education. Horizon Research.

Baroody, A. J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics, 13(1), 22–31.

Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354–380. https://doi.org/10.1037/0033-2909.132.3.354

Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2009). Spacing effects in learning: A temporal ridgeline of optimal retention. Psychological Science, 20(9), 1095–1102. https://doi.org/10.1111/j.1467-9280.2009.02418.x

Cooper, H., Nye, B., Charlton, K., Lindsay, J., & Greathouse, S. (1996). The effects of summer vacation on achievement test scores: A narrative and meta-analytic review. Review of Educational Research, 66(3), 227–268. https://doi.org/10.3102/00346543066003227

Fitzpatrick, M. D., Grissmer, D., & Hastedt, S. (2011). What a difference a day makes: Estimating daily learning gains during kindergarten and first grade using a natural experiment. Economics of Education Review, 30(2), 269–279. https://doi.org/10.1016/j.econedurev.2010.08.004

Fuchs, L. S., Fuchs, D., Powell, S. R., Seethaler, P. M., Cirino, P. T., & Fletcher, J. M. (2008). Intensive intervention for students with mathematics disabilities: Seven principles of effective practice. Learning Disability Quarterly, 31(2), 79–92. https://doi.org/10.2307/25474677

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060). National Center for Education Evaluation and Regional Assistance.

Hattie, J., & Yates, G. (2014). Visible learning and the science of how we learn. Routledge. https://doi.org/10.4324/9781315885025

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). Macmillan.

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age.

Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38(5), 1008–1022. https://doi.org/10.1111/cogs.12107

National Research Council. (2001). Adding it up: Helping children learn mathematics. National Academy Press. https://doi.org/10.17226/9822

Quinn, D. M., & Polikoff, M. S. (2017). Summer learning loss: What is it, and what can we do about it? Brookings Institution.

Rohrer, D., & Pashler, H. (2010). Recent research on human learning challenges conventional instructional strategies. Educational Researcher, 39(5), 406–412. https://doi.org/10.3102/0013189X10374770

Schmidt, W. H., Wang, H. C., & McKnight, C. C. (2005). Curriculum coherence: An examination of US mathematics and science content standards from an international perspective. Journal of Curriculum Studies, 37(5), 525–559. https://doi.org/10.1080/0022027042000294682

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285. https://doi.org/10.1207/s15516709cog1202_4

Sweller, J., Ayres, P., & Kalyuga, S. (2011). Cognitive load theory. Springer. https://doi.org/10.1007/978-1-4419-8126-4

Thompson, P. N. (2021). Does a four-day school week improve student achievement? Evidence from Oregon and Colorado. Educational Evaluation and Policy Analysis, 43(2), 307–329. https://doi.org/10.3102/0162373720988479

If districts continue investing in curriculum and PD, why aren’t math outcomes improving?

Classroom instruction in mathematics requires teachers to make hundreds of decisions each day about representations, questions, pacing, and how to respond to student thinking—often within the constraints of new materials and limited instructional support. Research consistently shows that we have overestimated what curriculum alone can accomplish and underestimated the power of teachers’ mathematical knowledge for teaching (MKT) (Ball, Thames, & Phelps, 2008; Hill, Rowan, & Ball, 2005). This overview examines how districts allocate funds, what the research says about the relative impact of curriculum versus professional learning, and why shifting even a small percentage of existing budgets toward content-focused math PD is one of the most cost-effective strategies available.

How Districts Spend: Curriculum vs. Professional Development

Where does the math dollar actually go—and how much reaches teacher learning?

Most districts spend 80-85% of their overall budgets on salaries and benefits, leaving just 15-20% for transportation, utilities, technology, curriculum, assessments, and professional learning (Education Resource Strategies, 2020). Within that discretionary portion, curriculum and PD together account for only 3 to 6% of total spending (Center for American Progress, 2018). When annualized over adoption cycles, math curriculum often represents just $7-$25 per student per year, while math-specific PD averages $20-$60 per student (Learning Policy Institute, 2017).

This means that math teaching—one of the strongest predictors of long-term student success typically receives less than 1% of total district resources, and only a fraction of that PD focuses on building teachers’ deep mathematical understanding. To put this in perspective: For every $100 a district spends, the entire engine of math improvement both the tools and the training to use them amounts to little more than loose change found in the couch cushions. It is no surprise this level of investment fails to produce transformative results.

Figure 1. Typical allocation of a $100 million district budget. After fixed costs, only a small portion remains for curriculum and professional learning combined.

Why Curriculum Alone Rarely Delivers Meaningful Gains

If curriculum is improving, why isn’t student learning improving with it?

High-quality materials matter, but decades of research demonstrate that the impact of curriculum is mediated by teacher knowledge and instructional practice (Cohen & Hill, 2001; National Research Council, 2001). Teachers who lack strong content and pedagogical content knowledge often reduce rich tasks to procedural routines, limiting opportunities for reasoning (Ball et al., 2008). When outcomes fail to improve, districts often initiate another expensive adoption cycle a response that masks the underlying challenge: insufficient investment in teacher knowledge and instructional capacity (TNTP, 2015).

Publisher-led trainings, while helpful for orientation, are rarely designed to build a deep understanding of mathematics or learning trajectories, and short-term workshops rarely produce lasting instructional change (Garet et al., 2001; Yoon et al., 2007). The result is a system that repeatedly changes materials rather than addressing the conditions required for strong implementation.

Figure 2. Breakdown of professional learning spending within a typical district. Only a small fraction supports sustained, content-focused mathematics professional development.

The Multiplier Effect: Why Math-Specific PD Improves Learning More Than Curriculum Purchases

What changes when districts invest in teachers’ mathematical knowledge instead of relying on materials to carry the load?

A large body of research shows that teachers’ expertise is the most influential in-school factor affecting student achievement (Hattie, 2009). In mathematics specifically, teachers with stronger MKT produce significantly larger learning gains, even with the same curriculum (Hill et al., 2005). Unlike curriculum, which expires every 6-8 years, teacher knowledge is a durable, compounding asset.

High-quality PD helps teachers understand why mathematical ideas develop the way they do, how representations support thinking, and how to respond to student misconceptions. This means a teacher can instantly recognize why a student is consistently adding denominators when adding fractions a misconception that the curriculum script may not address and can use a visual model to rebuild understanding in the moment. This is the difference between covering material and teaching children. Sustained, content-focused PD — rather than isolated workshops has been shown to improve instructional practice and student outcomes meaningfully (Desimone, 2009; Garet et al., 2001; Yoon et al., 2007).

Rebalancing the Investment: Spending the Same Dollars More Strategically

How can districts use existing funds more effectively to improve math learning?

Districts do not need new money to improve mathematics outcomes; they need a smarter allocation of the funds they already spend. The shift required is not from low spending to high spending, but from a transactional investment in materials to a transformational investment in teacher expertise. In many districts, annual spending on professional learning alone exceeds $1 million, yet only a small portion of that investment is directed toward deepening teachers’ mathematical knowledge for teaching (Education Resource Strategies, 2020).

A Strategic Reallocation: A $1 Million Thought Experiment

Consider a district that spends approximately $1 million per year on professional learning across all subjects. Even a modest reallocation within this existing budget can have outsized effects. For example, redirecting just $100,000–$200,000—10–20% of the professional learning budget—toward sustained, content-focused math professional learning can double or triple the district’s current investment in teacher mathematical knowledge without increasing overall expenditures.

This shift is not about cutting support, but about targeting resources more effectively. Districts can make this reallocation by selecting high-quality but less expensive instructional materials, reducing reliance on low-impact, one-day workshops, and integrating curriculum implementation with ongoing math-specific professional learning. When curriculum and professional learning are treated as a unified strategy rather than separate line items districts build internal instructional capacity that strengthens the impact of any curriculum, present or future.

Equity and Long-Term Impact

How does investing in teacher knowledge advance equity in mathematics?

Students in historically marginalized communities are more likely to be taught by novice teachers and less likely to receive instruction grounded in strong mathematical understanding (von Hippel et al., 2018). When new materials are distributed without corresponding investment in teacher learning, implementation gaps widen, and students who most need conceptual instruction receive the least.

By prioritizing deep, sustained math PD—especially in high-need schools districts can improve consistency, reduce remediation needs, and create more equitable access to meaningful mathematics. This is how investment in teacher knowledge becomes an act of educational justice.

Conclusion:

If curriculum is the map, teacher knowledge is the driver. Because math curriculum and math-specific PD together account for well under 2% of a typical district’s budget, even small strategic shifts can produce substantial gains in instruction and achievement. Real, lasting improvement will not come from the next adoption cycle; it will come from sustained investment in teachers’ mathematical knowledge for teaching—the most important and enduring asset in the system.

Here’s an additional document to provide more insight on the topic:

Companion Document

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

Center for American Progress. (2018). Lessons from school districts on curriculum spending.

Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.). Routledge.

Cohen, D. K., & Hill, H. (2001). Learning policy: When state education reform works. Yale University Press.

Desimone, L. (2009). Improving impact studies of teachers’ professional development. Educational Researcher, 38(3), 181–199.

Education Resource Strategies. (2020). Resource use in schools: A systems perspective.

Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What makes PD effective? American Educational Research Journal, 38(4), 915–945.

Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. Routledge.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406.

Learning Policy Institute. (2017). Effective teacher professional development.

National Research Council. (2001). Adding it up: Helping children learn mathematics. National Academies Press.

TNTP. (2015). The mirage: Confronting the truth about teacher development.

von Hippel, P. T., Workman, J., & Downey, D. (2018). Inequality in teachers’ access to professional development. AERA Open, 4(3), 1–20.

Yoon, K. S., Duncan, T. S., Lee, S. W.-Y., Scarloss, B., & Shapley, K. (2007). Reviewing the evidence on how teacher PD affects student achievement. Institute of Education Sciences.

Why do we continue to see low math performance despite better standards, assessments, and curricula?

Across the nation, elementary math performance has remained stagnant for decades. Although schools have adopted clearer standards and higher-quality materials, a central issue remains unresolved: teachers are not being adequately prepared or supported to teach mathematics for deep understanding. Research consistently shows that the single most powerful school-based factor in student learning is the quality of instruction (Hattie, 2009). However, most teacher preparation programs and workshops fail to build teachers’ mathematical knowledge, understanding of how children learn, or skill with high-leverage pedagogical practices (Ball et al., 2008; AMTE, 2017). As a result, teachers enter classrooms underprepared, and schools depend on professional development to rebuild the foundation that preparation programs should have provided.

Cognitive Foundations: Why Strong Teacher Knowledge Changes Student Thinking

What becomes possible in classrooms when teachers possess deep mathematical and pedagogical knowledge?

Effective math teaching requires teachers to understand concepts, structures, and representations—not just procedures. Research on Mathematical Knowledge for Teaching (MKT) demonstrates that teachers with strong content and pedagogical content knowledge significantly improve student outcomes (Hill et al., 2005). When teachers understand the “why” behind mathematical ideas, mathematics becomes coherent, and sense making becomes central. In contrast, teachers without deep preparation rely heavily on rules and demonstrations, which results in procedural knowledge that does not transfer. In this way, teacher preparation directly shapes the mathematical experiences available to students.

Structural Knowledge: The Mathematical Actions Teachers Must Understand

What must teachers understand about mathematical structure before they can help students learn it?

Students develop understanding through actions such as composing, decomposing, unitizing, iterating, and partitioning. Teachers must understand these actions deeply because students cannot learn conceptual mathematics through procedures alone. However, many preparation programs treat these foundational ideas superficially (CBMS, 2012). Teachers often graduate without experience using number lines, area models, or manipulatives to build reasoning, and without tools for analyzing student thinking. Without a firm grasp of structure, teachers cannot diagnose misconceptions or connect big mathematical ideas across grades. This leaves students with a fragmented and unstable understanding.

How Students Learn Math: The Missing Psychology in Teacher Preparation

Why must teacher preparation include a deep understanding of the psychology of how children learn mathematics?

Most teacher preparation programs and professional learning workshops offer little to no experience in how children actually learn math and develop mathematical understanding. Learning-sciences research on developmental trajectories, spatial reasoning, dual coding, working memory, and conceptual–procedural relationships is rarely taught to future teachers (Clements & Sarama, 2014; Mix & Cheng, 2012; Paivio, 1986; Rittle-Johnson & Alibali, 1999). Teachers who understand how children learn can better interpret errors, anticipate misconceptions, and design tasks that build deep reasoning. However, because most programs neglect learning psychology, teachers enter classrooms without a clear sense of how children move from informal ideas to formal concepts. This gap limits their ability to support meaningful sense making.

University Preparation: Misalignment Between What Teachers Need and What Programs Provide

Why do universities consistently fall short in preparing teachers for real math instruction?

Most teacher preparation programs underemphasize mathematics and provide little professional development in the psychology of learning or practice-based pedagogy (NRC, 2001; AMTE, 2017). Elementary majors often take only one or two courses that resemble liberal arts math rather than mathematics for teaching. This leaves future teachers without the content and pedagogical foundation required for effective instruction. Compounding this, math and education faculty often work in isolation from each other, leading to programs in which content and methods are disconnected. Candidates rarely practice teaching routines, analyze misconceptions, or build representational fluency. As a result, teachers graduate without the knowledge needed to support their students’ conceptual understanding.

Instructional Consequences: How Preparation Gaps Affect Classrooms

What does instruction look like when teachers lack deep preparation in math, psychology, and pedagogy?

When teachers lack foundational preparation, instruction defaults to rules such as “cross-multiply,” “stack the numbers,” or “move the decimal.” Students learn isolated steps but do not build conceptual understanding. The result is procedural fluency without meaning, which collapses when students encounter more complex ideas such as fractions or algebra. Teachers who lack preparation may use manipulatives or models incorrectly, causing students to see them as add-ons rather than thinking tools. These gaps disproportionately harm students in communities served by uncertified or fast-track teachers (von Hippel et al., 2018). In this way, inadequate preparation becomes an equity issue.

Why Typical Workshops Fail

Why are traditional workshops ineffective at improving math instruction?

One to two-day workshops provide exposure, not transformation. Research shows that such professional development (PD) rarely improves practice because teachers need sustained learning, not isolated strategies, to change instruction (Garet et al., 2001). Workshops do not build conceptual content knowledge, misunderstanding analysis, or pedagogical fluency. They often replicate the weaknesses of teacher preparation short, disconnected, and removed from classroom practice. As a result, schools spend valuable time and money on PD that produces little change in student learning.

High-Quality Professional Development: Rebuilding What Teacher Preparation Did Not Provide

How does sustained, content-rich PD transform teacher practice and student learning?

High-quality professional development builds teacher expertise across mathematics, learning psychology, and pedagogy. Effective PD is long-term, content-focused, and grounded in models, representations, and student thinking (Desimone, 2009). It provides teachers with opportunities to rehearse instructional routines, analyze errors, and apply new practices in their classrooms. The most effective programs also promote a consistent structural language such as unit, partition, iterate, decompose, compose, and equal which anchors reasoning and creates coherence across lessons. When teachers build deep knowledge and pedagogical skill, classrooms shift from memorizing steps to mathematical sense making

Conclusion:

Elementary mathematics success depends on teachers’ knowledge, not just on curriculum materials. Strong preparation programs and sustained PD must build teachers’ conceptual understanding, learning-science knowledge, and pedagogical skills. When teachers have strong foundations, classrooms become places where students reason, model, explain, and connect ideas. Improving teacher knowledge is not just best practice it is an equity imperative. The path forward is clear: meaningful improvement in math requires meaningful investment in the adults who teach mathematics.

Here’s a companion document for school and district leaders who have read WHY TEACHER KNOWLEDGE IS THE KEY TO FIXING ELEMENTARY MATH. Click the link

companion document

References

AMTE. (2017). Standards for preparing teachers of mathematics. Association of Mathematics Teacher Educators.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5), 1860–1863.

CBMS. (2012). The mathematical education of teachers II. Conference Board of the Mathematical Sciences.

Clements, D. H., & Sarama, J. (2014). Learning and teaching early math. Routledge.

Desimone, L. (2009). Improving impact studies of teachers’ professional development. Educational Researcher, 38(3), 181–199.

Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What makes professional development effective? American Educational Research Journal, 38(4), 915–945.

Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Harvard University Press.

Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teacher preparation: Practice-based teacher education. Teachers and Teaching, 15(2), 273–289.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406.

Hattie, J. (2009). Visible learning. Routledge.

Mix, K. S., & Cheng, Y.-L. (2012). The relation between spatial skill and math performance. Developmental Psychology, 48(4), 1227–1244.

National Mathematics Advisory Panel. (2008). Foundations for success. U.S. Department of Education.

National Research Council. (2001). Adding it up: Helping children learn mathematics. National Academy Press.

Paivio, A. (1986). Mental representations: A dual-coding approach. Oxford University Press.

Rittle-Johnson, B., & Alibali, M. (1999). Conceptual and procedural knowledge: Beyond dichotomies. Developmental Review, 19(3), 325–346.

von Hippel, P. T., et al. (2018). Teacher preparation programs and teacher quality. Educational Evaluation and Policy Analysis, 40(1), 21–44.

Yoon, K. S., Duncan, T., Lee, S. W., Scarloss, B., & Shapley, K. (2007). Reviewing the evidence on how teacher professional development affects student achievement. Regional Educational Laboratory Southwest.

How does allowing students to create number lines rather than simply use pre-made ones transform the kind of mathematical thinking we see in our classrooms?

The number line has long been a classroom staple, but its deepest power emerges when students construct it themselves choosing endpoints, units, spacing, and labels. When students build the model, they engage essential mathematical actions rather than simply observe them. Research shows that constructing number lines strengthens proportional and spatial–numeric reasoning (Cohen et al., 2014), helping students map numbers onto meaningful distances. These actions connect number to magnitude, rather than leaving numbers as abstract symbols on a page.

Drawing and scaling lines also activates dual coding, linking verbal and visual systems in ways that support understanding and long-term memory (Paivio, 1986). This is why number-line construction becomes a bridge—from discrete counting to continuous reasoning, and from early arithmetic to algebra and graphing. This Insight distills research across mathematics education and cognitive psychology to show why student-constructed number lines deepen structural understanding and how teachers can embed this practice in instruction and professional learning.

Cognitive Foundations: From Number to Measurement and Scale

What kinds of thinking happen when students draw, scale, and label a number line from scratch?

Instead of simply placing numbers on pre-made lines, students engage in measurement-based thinking that drives proportional reasoning a finding supported by research showing that number-line performance depends heavily on scaling, not just numerical magnitude (Barth & Paladino, 2011; Cohen et al., 2014). When constructing lines, students practice unit iteration and equal partitioning, defining a unit length and repeating it across the line. These are the same skills needed for measurement, fractions, and early algebra.

By mapping numbers to space, students learn that numerals correspond to physical, repeatable lengths, not just positions on a static line aligning with cognitive studies showing the role of spatial processing in magnitude understanding (Leibovich et al., 2014). Studies also show that many students hold rigid conceptions of the number line such as thinking zero must always be centered or increments must always be one. Constructing lines encourages flexibility and conceptual growth (Ünal et al., 2024), preparing students for later work in algebra and modeling.

Measurement and Fraction Reasoning: Connecting Number to Length

What changes when students see a unit not just as “one count” but as a measurable distance?

Constructing number lines strengthens measurement understanding because students connect numeric values directly to physical length. Research shows that coordinating numeric and linear measurement—literally drawing and scaling lines deepens conceptual understanding (Saxe et al., 2013). As students partition lines into fourths, tenths, they experience fraction values as proportional distances rather than memorized points an argument supported by work on spatial–numeric integration (Cohen et al., 2014). This work also supports embodied cognition, as drawing and dividing lines engages sensory–motor and spatial networks (Leibovich et al., 2014). This embodied grounding helps students make sense of equivalence, comparison, and scaling across fraction contexts.

Extending Number-Line Understanding to Data and Graphing.

How does drawing number lines prepare students to reason about scale, spacing, and data in graphs?

A line plot is essentially a number line with data layered onto it, and research shows that students better understand data displays when they construct the axis themselves selecting range, tick spacing, and scale (Lehrer & Schauble, 2007). In bar graphs, constructing axes helps students realize that equal spacing represents equal units, reinforcing structural ideas from measurement that do not always transfer when graphs arrive pre-formatted.

Coordinate graphing also becomes more intuitive when students recognize that the x-axis is a scaled number line. Students who have not constructed number lines often struggle with origin placement and scale a difficulty observed repeatedly in graphing research (Robertson, 2023). These construction experiences develop continuous and proportional thinking, supporting algebraic modeling and early function reasoning.

Instructional Design: Turning Research into Practice

What would it look like if every grade treated number-line construction as a high-leverage routine?

Classroom routines that begin with blank lines help students take ownership of scale and structure, rather than relying on templates. This echoes research showing that student-created tools promote deeper reasoning (Saxe et al., 2013). Tasks with varied endpoints (0–1, 0–50, 2–10) push students to adapt their unit choices and scaling, supporting cognitive flexibility across contexts.

Integrating fraction and measurement contexts ensures that students connect physical measurement to visual and symbolic representations, reinforcing the structural ideas behind units and partitions. Drawing axes for data or coordinate grids builds graphing fluency through scale-making, a key shift emphasized in data-literacy research (Lehrer & Schauble, 2007). Reflection prompts such as How did you choose your unit? make students’ structural reasoning visible, which is essential for developing conceptual understanding.

Student-created number lines provide rich assessment evidence because they reveal how students think about spacing, scale, and labeling, not just whether they placed points correctly.

Professional Development: Supporting Teachers as Designers of Structural Learning

How can teacher learning communities use number-line construction to strengthen both student understanding and instructional design?

When teachers construct number lines during PD, they experience firsthand how scaling, spacing, and unit decisions shape reasoning, building pedagogical content knowledge grounded in students’ cognitive actions. Using consistent structural language—unit, partition, iterate, compose, decompose, equal—helps teachers anchor classroom discourse in mathematical actions that promote sense making (Brendefur & Strother, 2021).Connecting number-line work to measurement, data, and graphing helps teachers see the number line as a unifying model that supports the K–8 trajectory (Robertson, 2023). Analyzing student-created lines enables teachers to identify misconceptions such as uneven spacing or fixed-zero thinking and to design follow-up tasks that target structural understanding.

Conclusion: Empowering Mathematical Thinking Through Construction

What lasting differences emerge when number lines are something students build, not just use?

Research across learning sciences and mathematics education shows that constructing number lines leads to stronger, more transferable understanding than using pre-drawn models (Cohen et al., 2014; Saxe et al., 2013). Through drawing, partitioning, and scaling, students internalize mathematical relationships rather than simply perform them. When construction becomes a routine across grades, students learn to design mathematics not just record it, transforming the number line into a powerful medium for thinking, modeling, and sense making.

References

Barth, H., & Paladino, A. M. (2011). The development of numerical estimation: Evidence against a representational shift. Developmental Science, 14(1), 125–135. https://doi.org/10.1111/j.1467-7687.2010.00962.x

Brendefur, J., & Strother, S. (2021). Developing mathematical fluency: Helping children make sense of facts and strategies. DMTI Press.

Cohen, D. J., Blanc-Goldhammer, D., Courtney, E. A., Jensen, M. B., & Runeson, B. (2014). The relation between spatial and numerical abilities in children and adults. Frontiers in Psychology, 5, 1060. https://doi.org/10.3389/fpsyg.2014.01060

Lehrer, R., & Schauble, L. (2007). Thinking with data. Lawrence Erlbaum Associates.

Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2014). From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition. Frontiers in Psychology, 5, 962. https://doi.org/10.3389/fpsyg.2014.00962

Paivio, A. (1986). Mental representations: A dual coding approach. Oxford University Press.

Robertson, D. (2023). The power of number line models and scales. Ontario Institute for Studies in Education (OISE) Blog. https://www.oise.utoronto.ca

Saxe, G. B., Shaughnessy, M. M., Shannon, A., & Bowling, D. (2013). Coordinating numeric and linear measurements: Students’ strategies and mathematical understandings. ZDM: The International Journal on Mathematics Education, 45(3), 407–420. https://doi.org/10.1007/s11858-012-0477-4

Ünal, O., Ertekin, E., & Güler, G. (2024). Conceptual stages and student reasoning on the number line. ERIC.https://eric.ed.gov

Over the last decade, particularly since the COVID-19 pandemic, digital learning has become increasingly integrated into elementary education. Adaptive software, online games, and video-based lessons now occupy a substantial part of children’s learning time. These tools promise personalized pacing and instant feedback, yet an essential question persists: Can elementary children truly learn deeply by sitting in front of a screen?

While children can gain information, practice skills, and pass digital assessments, the more important question is what kind of learning is occurring and what might be lost in the process. Learning in early and middle childhood (ages 5–11) is shaped through movement, conversation, and shared attention experiences that are hard to replicate digitally. Screens can support learning, but when they dominate, they can displace the cognitive, social, and emotional experiences foundational to development.

Theoretic Foundations

How do children actually learn?

Cognitive development theories from Piaget and Vygotsky emphasize that children construct understanding through physical and social engagement with their world. Piaget described learning as active construction assimilation and accommodation through interaction. Vygotsky emphasized the social mediation of knowledge within the zone of proximal development.

Neuroscience supports these foundations: learning merges perception, movement, and language. When children manipulate objects, gesture during reasoning, or collaborate with peers, the brain integrates sensory, motor, and symbolic systems (Barsalou, 2008; Glenberg, 2010). Spatial reasoning and fine motor activity, for instance, are strongly predictive of later mathematical achievement (Verdine et al., 2017).

Screen-based tasks often simulate engagement clicking or dragging but these actions lack the physical and sensory feedback and negotiative qualities of real interaction. A child stacking five wooden blocks experiences texture, weight, and balance sensations that build neural pathways connecting perception to number and shape. Screens, though efficient, flatten this multidimensional learning experience and risk replacing embodied construction with symbolic mimicry. The result is often knowledge that is performative rather than deeply understood.

What Screens Do Well and Where Do They Fall Short

When is technology a tool, and when does it become a crutch?

Digital platforms offer several benefits, including adaptive feedback, visual representation of abstract concepts, and flexible practice. Virtual manipulatives and interactive animations can meaningfully supplement instruction (Sarama & Clements, 2009; Uttal et al., 2013).

However, research cautions that technology’s promise depends on how it is used. Clark and Feldon (2014) remind educators that media are not methods it is pedagogy, not platform, that determines impact. Many comparative studies have shown modest or mixed effects of technology on teacher-led instruction (Cheung & Slavin, 2013).

Most digital environments privilege individualized interaction over shared meaning-making. The learner’s “partner” becomes the algorithm, not another human. Without collaborative dialogue, reasoning is reduced to a pattern of input and response rather than a process of conceptual refinement. When children learn alone on screens, they often master procedures without the conceptual coherence that comes from conversation and collaboration. They may learn what to think, but not how to think.

The Missing Elements.

What happens when screens replace human interaction?

Embodied Cognition: Thought grows from movement. When children count steps, fold paper, or act out problems, they bind physical and conceptual meaning (Goldin-Meadow, 2014). Virtual manipulatives approximate but do not replicate the full sensory experience of building, turning, and feeling objects. Without kinesthetic grounding, children tend to memorize symbols rather than internalize structure.

Dialogue and Social Reasoning: Learning thrives in dialogue. In lively classrooms, children articulate their ideas, question one another, and refine their meanings (Chapin, O’Connor, & Anderson, 2009). Through this process, they develop metacognition and cognitive flexibility. Most screen-based programs, optimized for efficiency, remove these conversations. They offer correctness feedback, not conceptual negotiation. As a result, fluency may increase while understanding remains shallow and fragile.

Emotion and Motivation: From a self-determination perspective (Ryan & Deci, 2000), motivation depends on autonomy, competence, and relatedness. Teachers provide emotional attunement adjusting pace, offering encouragement, and celebrating effort. Screens can mimic reward structures but typically foster extrinsic motivation through the use of points and badges. Extended solo screen time may also disrupt attention regulation (Christakis, 2019). Children need relational co-regulation found in play, dialogue, and shared discovery to sustain motivation. A teacher’s nod, smile, or tone of voice communicates safety and belonging in a way no program can.

Deep vs. Shallow Learning

Are students learning to compute or to think?

Mathematics education highlights the divide between digital automation and conceptual growth. Many digital systems promote instrumental understanding knowing how to obtain answers rather than relational understanding, which involves grasping why procedures work and how ideas connect (Hiebert & Carpenter, 1992).

On-screen fraction tasks might ask children to match shaded regions to symbols; hands-on explorations such as folding paper or slicing fruit let them act out equivalence. Likewise, spatial reasoning the strongest predictor of later math success strengthens when children rotate solids, build with blocks, or navigate real environments, not just tap polygons on a flat screen.

Equally critical is the feedback loop. In classrooms, errors become teachable moments. Teachers invite students to analyze misconceptions, cultivating resilience and curiosity (Boaler, 2016). Screens often reward speed and accuracy, implicitly discouraging productive struggle and mistake analysis key ingredients of a growth mindset. Without these discussions, learning becomes transactional right or wrong rather than transformational.

Feedback matters. In classrooms, teachers turn mistakes into moments of insight (Boaler, 2016). Screens reward speed and accuracy, not productive struggle. Without dialogue, learning becomes transactional right or wrong rather than transformational.

Reframing Technology: Human-Centered Integration

How Can We Reclaim Technology as a Tool for Thinking?

Technology should amplify, not replace, human pedagogy. The goal is a balanced ecosystem where screens serve, not lead, the learning process. Practical principles include: EIS progression (Enactive → Iconic → Symbolic), Designed Interactivity, Collaborative Digital Spaces, Educator Mediation, and Blended Learning. These principles realign digital education with developmental science, ensuring that technology supports curiosity, communication, and cognitive growth.

Implications for Educators and Policymakers

How Should We Redefine Quality in a Digital Age?

Limit passive screen time following pediatric guidelines—prioritizing quality, context, and social interaction over minutes logged. Embed discourse and gesture prompts within digital lessons. Provide professional learning for teachers to evaluate when and how technology supports conceptual depth. Design curricula that nurture the whole child mind, body, and emotion. Assess depth of reasoning and interpersonal engagement, not just digital accuracy metrics. Ultimately, technology should serve the human agenda of education: helping children think critically, connect relationally, and act creatively.

Conclusion

Elementary children can learn through screens but not because of them. Learning is physical, social, and emotional before it becomes digital or abstract. Overreliance on screens risks narrowing education to information transfer rather than meaning-making. The challenge is not technological rejection but intentional reintegration. When digital tools complement hands-on exploration, conversation, and play, they extend human intelligence rather than replace it. The future of learning will depend less on brighter screens and more on brighter, connected minds those that grow through doing, talking, feeling, and imagining together.

References

American Academy of Pediatrics. (2019). Media and young minds. Pediatrics, 138(5), e20162591.

Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645.

Boaler, J. (2016). Mathematical mindsets. Jossey-Bass.

Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom discussions in math. Math Solutions.

Cheung, A., & Slavin, R. E. (2013). Educational technology and mathematics achievement. Educational Research Review, 9, 88–113.

Christakis, D. A. (2019). Digital addiction in children. JAMA, 321(23), 2277–2278.

Clark, R. E., & Feldon, D. F. (2014). Questionable principles about multimedia learning. In R. Mayer (Ed.), The Cambridge handbook of multimedia learning. Cambridge University Press.

Glenberg, A. M. (2010). Embodiment as a Unifying Perspective for Psychology. Wiley Interdisciplinary Reviews: Cognitive Science, 1(4), 586–596.

Goldin-Meadow, S. (2014). Gesture as a window onto thought. Oxford University Press.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning. Macmillan.

Piaget, J. (1952). The origins of intelligence in children. International Universities Press.

Reich, J., et al. (2021). Digital learning in the time of COVID-19. Educational Researcher, 50(1), 27–37.

Ryan, R. M., & Deci, E. L. (2000). Self-determination theory. American Psychologist, 55(1), 68–78.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research. Routledge.

Verdine, B. N., et al. (2017). Links between spatial and mathematical thinking. Developmental Psychology, 53(2), 260–276.

Vygotsky, L. S. (1978). Mind in Society. Harvard University Press.

Social Media

From Screens to Sensemaking: What Are We Missing?

Screens can capture students’ attention, but can they capture their thinking?Today’s classrooms are filled with adaptive programs, games, and digital lessons. While these tools make learning look efficient, we must ask: What kind of learning is actually happening?

Our new DMT Insight explores what research in cognitive science and math education tells us about screen time and deep understanding.

Here is what you will discover:
• Why hands-on movement, conversation, and shared attention form the foundation for long-term learning.
• How digital tools can support but not replace embodied, social, and emotional experiences.
• What happens to curiosity and motivation when screens become substitutes for human interaction?
• Practical ways to rebalance technology, using it as a tool for sense making rather than simple repetition.

If we want students who can think not just click we must design classrooms that connect body, mind, and meaning.

What’s one way you’ve seen technology enhance or limit real mathematical thinking in your classroom? Share your experiences below!

In K-8 mathematics, fluency with basic addition and multiplication facts is a non-negotiable foundation. However, a significant gap exists between mere memorization and true proficiency. We see this starkly in the data: while many 3rd graders can recall their math facts, by 8th grade, only about 17% of students have maintained this fluency. This dramatic drop-off reveals that short-term recall is not the same as long-term mastery. The ultimate goal is to create flexible thinkers and problem solvers who can compose, decompose, and reason with numbers, not just recall them quickly (Baroody, 2006; Brendefur & Strother, 2015).

Despite this, many instructional programs default to drill-heavy approaches such as timed tests, repetitive flashcards, and massed practice. While these can improve short-term speed, they consistently fail to develop the conceptual understanding and strategic competence necessary for long-term success and application (Boaler, 2014).

This DMT Insight synthesizes research from cognitive psychology and mathematics education to argue that an integrated approach combining explicit strategy instruction rooted in structural language (unit, compose, decompose, iterate, partition, equal) with purposeful retrieval practice is the most effective path to building durable, flexible, and transferable fact fluency.

Redefining Fluency: Automaticity Rooted in Understanding

What if being ‘fast’ at math is actually slowing your students down?

True fluency is more than speed and accuracy; it is the efficient, flexible, and appropriate application of facts in problem-solving contexts (NRC, 2001). This requires a dual perspective:

From Cognitive Psychology: Automatic fact retrieval frees up limited working memory for higher-order tasks (Sweller et al., 2011). However, this automaticity is fragile if it is built solely on rote memorization. Durable recall depends on facts being embedded in rich, interconnected networks of meaning (Bransford et al., 2000).

Mathematics Education: Fluency involves the ability to deconstruct and reconstruct numbers using strategies like making ten, doubling, iterating, and partitioning (e.g., seeing 6 × 7 as (5 × 7) + (1 × 7)). Brendefur & Strother (2015) crucially distinguish between fluency (fast, accurate recall) and flexibility (the ability to derive facts using reasoning), noting that the latter is a prerequisite for robust, long-term mastery of the former.

Thus, we must redefine fluency as automatic retrieval, strategic flexibility, and conceptual understanding.

The Shortfalls of Drill-Only Approaches

Why do students who ace their timed tests in 3^rd grade often fail word problems and forget them over the next few years?

Programs relying solely on massed drills and timed tests exhibit several documented shortcomings:

Limited Transfer and Brittle Knowledge: Students may pass a fact test but be unable to apply those facts in novel problems. Strategy instruction, not drill, leads to improved performance on transfer tasks (Baroody, 2006).

Inhibition of Strategic Flexibility: Drill-centric practice teaches that there is one “right” way to get an answer—quickly. It does not foster the adaptive, multi-strategy reasoning required for complex problem-solving.

Vulnerability to Interference: Similar facts (e.g., 6×7, 7×6, 6×8) compete and cause confusion. Drills do not help students build cognitive networks to suppress this interference, leading to shallow encoding and forgetting.

Induction of Math Anxiety: The focus on speed creates anxiety, which consumes the very working memory capacity that automaticity is meant to free up (Ramirez et al., 2018).

Equity Concerns: Rigid, speed-based approaches disproportionately harm students with learning differences, executive functioning challenges, or math anxiety, widening achievement and confidence gaps (Boaler, 2016).

Evidence in Action: A study by Brendefur et al. (2015) found that after a 5-week intervention, students in a strategy-based program gained an average of 6.08 correct facts per minute, compared to only 0.79 facts for students in a drill-only program—a nearly 8-fold difference in efficacy.

An Evidence-Based Model: Strategy First, Then Retrieval

What’s the secret to making math facts ‘stick’ for the long term?

The most effective model is a phased, integrated approach that builds conceptual understanding before demanding automaticity.

Phase 1: Build Strategic Understanding. Explicitly teach strategies using structural language (unit, compose, decompose, partition, iterate, equal) and progress through representations (visual → abstract) to build deep conceptual schemas. Teachers should model this language aloud (“I decomposed 14 into 10 and 4,” or “I partitioned 12 into three equal groups”) to strengthen students’ structural awareness.

Phase 2: Implement Purposeful Retrieval Practice. Once strategies are understood, use spaced and varied retrieval practice to strengthen recall pathways. This practice is meaningful because it reinforces connected knowledge. Interleaving facts of different operations (addition, subtraction, multiplication, and division) requires students to choose appropriate strategies and enhances long-term retention (Rohrer & Taylor, 2007).

Phase 3: Foster Flexible Transfer. Embed fact use in complex problems and encourage metacognition (“How did you solve it? Solve it using a different strategy.”) to promote adaptive reasoning. For example, if a student knows 6 × 7 = 42, teachers might ask, “How could you use that to find 6 × 8?” This type of question builds relational connections among facts.

This model ensures that speed is built upon a foundation of sense-making, resulting in fluency that is both durable and flexible.

Conclusion:

The evidence is clear: drill-only programs are an inefficient and often counterproductive method for building the mathematical thinkers our students need to become. To cultivate true mathematical proficiency, we must intentionally redesign fluency instruction to focus on structure and strategy.

For educators, coaches, and parents, this means:

• Prioritizing strategy instruction and structural language (unit, decompose, compose, iterate, partition, and equal)

• Replacing high-stakes timed tests with low-stakes retrieval games and rich discussions like Number Talks.

• Celebrating strategic thinking and reasoning as much as, if not more than, speed.

By integrating strategic reasoning with thoughtful practice, we move beyond creating “drill masters” and instead empower all students as confident, capable, and flexible mathematicians.

References

Baroody, A. J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics, 13(1), 22–31.

Bjork, R. A. (1994). Memory and metamemory considerations in the training of human beings. In J. Metcalfe & A. Shimamura (Eds.), Metacognition: Knowing about knowing (pp. 185–205). MIT Press.

Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages, and innovative teaching. Jossey-Bass.

Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (2000). How people learn: Brain, mind, experience, and school. National Academy Press.

Brendefur, J., & Strother, S. (2015). Developing multiplication fact fluency. Advances in Social Sciences Research Journal, 2(10), 166–176. https://doi.org/10.14738/assrj.210.166

Brendefur, J., & Strother, S. (2021). Math facts: Kids need them. Here’s how to teach them. Developing Mathematical Thinking Institute.

Dweck, C. S. (2006). Mindset: The new psychology of success. Random House.

National Research Council. (2001). Adding it up: Helping children learn mathematics. National Academy Press.

Ramirez, G., Chang, H., Maloney, E. A., Levine, S. C., & Beilock, S. L. (2018). On the relationship between math anxiety and math achievement in early elementary school: The role of problem-solving strategies. Journal of Experimental Child Psychology, 167, 404–414.

Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498.

Sweller, J., Ayres, P., & Kalyuga, S. (2011). Cognitive load theory. Springer.

Social Media

So, what is the secret to building fluency that lasts a lifetime, not just until the next test?

Our latest DMT Insight reveals the decisive shift from Drills to Strategies:

· The “Why”: Discover why strategy-based instruction leads to 8x greater gains in fact fluency compared to drill-only practice, creating durable, flexible knowledge.

· The “How”: Learn how using structural language (like decompose and compose) builds the neural networks that prevent forgetting and enable transfer.

· The “What Now”: Get a clear, 3-phase model (Strategy First, Then Purposeful Retrieval) to ensure speed is built on a foundation of sense-making.

Stop the 3rd-to-8th-grade slide. It is time to replace short-term drills with long-term thinkers.

What is the biggest challenge you face in helping students truly retain a solid understanding of math? Share your experience below!

What if practice could build mathematical flexibility instead of rigid routine?

In many classrooms, practice means pages of nearly identical problems 20 addition facts, 15 fraction conversions, or a full sheet of long division. Students learn to repeat, but not to reason. They may master procedures, yet fail to connect them to meaningful contexts or visual representations. This separation between contextual understanding, visual modeling, and symbolic notation weakens transfer and limits flexibility.

Varied Practice offers a different approach. Each task invites students to work across three representations of mathematical ideas the contextual (story or situation), the iconic (visual model), and the symbolic (equation, algorithm, or verbal explanation). In a three-column format, one column is provided while the other two must be constructed. For instance, a student might be given a bar model and must write both the matching story problem and symbolic equation. This intentional translation among forms develops deep understanding and flexible reasoning.

This structure is grounded in decades of research from mathematics education, cognitive psychology, and learning science. It draws from Jerome Bruner’s modes of representation, Allan Paivio’s dual coding theory, and the mathematics education literature on multiple representations and structural reasoning. Together, these frameworks explain why moving among story, model, and symbol is not just pedagogically sound it’s cognitively powerful.

Theoretic Foundations

How do students build mathematical meaning?

As Jerome Bruner (1966) argued, students construct understanding through three interconnected modes of representation: enactive (action-based), iconic (visual or pictorial), and symbolic (abstract or linguistic). In mathematics, these correspond naturally to manipulatives and real-world actions (enactive), visual models such as bar models or number lines (iconic), and equations or algorithms (symbolic). When instruction emphasizes all three modes and the ability to move between them students build representational fluency, the capacity to express a single idea in multiple forms and recognize their underlying equivalence.

Modern mathematics education researchers (Ainsworth, 2006; Lesh, Post, & Behr, 1987) affirm that multiple representations are essential for deep conceptual understanding. Ainsworth describes the “complementary roles” of different representations: visuals reveal relationships that symbols conceal, while symbols allow for generalization and abstraction beyond a specific model. When students can flexibly translate across these modes, they are better prepared to apply mathematics to new situations.

For teachers, this framework reframes practice itself: not as repetition of form, but as variation of representation. A first grader using a bar model showing 14 red apples and 9 green apples to match 14 + 9 = 23, or a fifth grader linking a fraction bar model to the equation 3/4 × 24 = 18, are each engaging in meaning-making across Bruner’s modes..

Dual Coding Theory: Why Seeing and Saying Math Strengthens Memory

Why is the iconic model so powerful?

Cognitive psychologist Allan Paivio’s Dual Coding Theory (1971, 1986) provides the neurological explanation for why varied practice works so effectively. Paivio proposed that the human mind encodes information through two interconnected systems: a verbal channel for language and a non-verbal channel for imagery. When ideas are encoded through both, recall and understanding improve dramatically because learners build two pathways to access the same knowledge.

Engaging both channels simultaneously:

• Activates more regions of the brain, forming stronger neural connections (Clark & Paivio, 1991).

• Provides multiple retrieval cues (verbal and visual), strengthening long-term memory.

• Reduces cognitive load by distributing processing across both channels, aiding comprehension (Sweller, 1994).

In mathematics, story problems primarily activate the verbal system, while visual models engage the non-verbal system. Equations and algorithms though symbolic occupy a middle ground within the verbal channel, functioning as a language of structure (Sfard, 2008). When equations are intentionally paired with visual models, such as linking a bar model to 3 + 2 = 5, both channels operate in tandem. This dual-coded representation allows students to move fluidly between seeing relationships and expressing them symbolically.

Students who engage in dual coding are not merely memorizing equations they are constructing mental images of structure. For example, when solving 24 ÷ 6 = 4, a student who pictures “24 fish divided evenly among 6 trays” is leveraging both cognitive systems. The equation becomes meaningful because it is anchored in imagery and context

Cognitive Benefits: Translating Representations Builds Transferable Understanding.

How does this design change the way students learn?

The act of translating between representations is cognitively demanding and deeply generative. It forces students to reorganize and re contextualize their knowledge, building flexible, interconnected schemas rather than isolated facts. Research across cognitive psychology and mathematics education identifies several key benefits:

Conceptual Understanding Beyond Procedures
Creating or interpreting a representation demonstrates what a student truly understands. For example, when given the equation 9 – 5 = 4, a child who writes “I had 9 apples and gave away 5” shows a separating model of subtraction. A child who writes “Tom has 9 apples and Mary has 5; Tom has 4 more” shows a compare model revealing a deeper grasp of subtraction’s multiple structures.

Metacognition and Self-Monitoring
Translating between forms naturally encourages reflection. When a student’s bar model does not align with their equation, a cognitive conflict arises prompting self-correction and metacognition (Schoenfeld, 2016). Varied practice, by design, builds in this feedback loop.

Mathematical Flexibility and Transfer
Real-world problems rarely appear in symbolic form. To solve them, students must move from situation → iconic model → symbols, and often back again. Practicing these translations develops cognitive flexibility, enabling transfer to novel tasks (Rittle-Johnson & Star, 2007). Over time, students internalize not only procedures, but the relationships among representations.

In short, mathematical meaning lives in the movement the active coordination between story, image, and symbol. When students can move flexibly among them, their knowledge becomes transferable, durable, and richly interconnected.

Designing Varied Practice: Making Representations the Practice

How can teachers bring this idea to life?

Effective varied practice is not accidental it’s intentionally designed to challenge students to connect representations. A well-constructed Three-Column Worksheet(DMTI, 2020) can serve as the vehicle for this practice. Each row addresses a single mathematical idea and includes three columns:
(1) Contextual (Story) | (2) Iconic (Visual Model) | (3) Symbolic / Language (Equation or Explanation)

Only one column is filled in; students must generate the others. For example:

• A teacher gives an equation (8 + 4 = ?; 8 + 4 = 12). Students create a bar model or number line and write a story.

• Another time, students are given a bar model and must create the corresponding story and equation.

Design principles include:

• Consistency: Ensure the same mathematical relationship underlies all three columns.

• Variation: Alternate which column is given, avoiding predictable patterns.

• Progression: Begin with two provided columns (to scaffold) and move toward generating two from one.

• Intentional models: Use bar models, number lines, or area models that naturally represent the concept.

Optional extension: The third column may also focus on language, where students describe the conceptual action (“I partitioned one into 4 equal units”). This encourages precision in mathematical communication and deepens conceptual awareness.

When used regularly, these tasks transform worksheets into cognitive workouts promoting not only skill fluency but conceptual agility. They make visible what students know and how they think.

Conclusion: Building Thinkers, Not Just Solvers

Varied Practice is more than a worksheet strategy it is a cognitive framework for building relational understanding. It integrates the insights of Bruner’s representational theory, Paivio’s dual coding, and modern structural mathematics education into a single, powerful classroom routine. By requiring students to move between story, iconic model, and symbol, we teach them to see mathematics not as disconnected tasks, but as a coherent system of meaning.

For educators, the essential question shifts from “Can my students compute the answer?” to “Can my students show the meaning in multiple ways?” This approach transforms practice from repetition to reasoning, helping students become thinkers who understand the why behind the how.

As the Developing Mathematical Thinking Institute emphasizes, mathematical structure is the bridge between context, iconic model, and symbol. When students compose, decompose, iterate, and partition across representations, they are not just doing math they are developing the habits of thought that make mathematical reasoning a lifelong tool.

References

Ainsworth, S. (2006). DeFT: A conceptual framework for learning with multiple representations. Learning and Instruction, 16(3), 183–198.

Bruner, J. (1966). Toward a theory of instruction. Harvard University Press.

Clark, J. M., &Paivio, A. (1991). Dual coding theory and education. Educational Psychology Review, 3(3), 149–210.

Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Lawrence Erlbaum.

Paivio, A. (1971). Imagery and verbal processes. Holt, Rinehart & Winston.

Paivio, A. (1986). Mental representations: A dual coding approach. Oxford University Press.
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? Journal of Educational Psychology, 99(3), 561–574.

Schoenfeld, A. H. (2016). How we think: A theory of goal-oriented decision making and its educational applications. Routledge.

Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.

Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4(4), 295–312.

& Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498.

Social Media

Are We Teaching Math Procedures or Building Mathematical Thinkers?

It is easy to create worksheets where every problem looks the same. Students learn to mimic a procedure and get the right answer. However, does this build a true, flexible understanding? Research says no. If we want students who can solve real-world problems, we need to change how we practice.

Our new DMT Insights dives into Varied Practice Worksheets a simple but powerful shift:

· Why moving between stories, iconic models, and equations builds deeper neural pathways than repetition alone.

· How a simple 3-column format (Story - Visual - Equation) forces students to think, not just compute.

· The key cognitive science (like Dual Coding Theory) that makes this approach so effective for long-term learning.

· Practical ways to scaffold this in any elementary classroom, for any topic.

Ready to move beyond answer-getting and toward building agile mathematical minds?

What is one topic (like fractions or word problems) where your students struggle to connect the idea to the procedure? Share below!