Why Number Lines Should Be Built
This DMT Insight shows how letting students build number line not just use them strengthens their structural understanding by helping them reason about units, scale & dramatically improve number sense
Introduction: Reframing the Number Line in Classroom Practice
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