Fractions are essential for expressing quantities that fall between whole numbers, such as lengths, weights or volume.
Fractions are written with one number above another and separated by a line. However, it is important to understand that a fraction represents a single number, not two numbers stacked on top of each other.
The differences between fractions and whole numbers can make them challenging to learn.
Measurement is a fundamental aspect of mathematics that bridges geometry and number, forming a core component of children’s mathematical development.
Additional reading
Fractions
Fractions are a fundamental concept in mathematics that help to describe parts of a whole. However, fractions are not just simple representations of parts—they are multidimensional in nature. For example, fractions are essential for expressing quantities that fall between whole numbers, such as lengths, weights or volume. Fractions play a crucial role in everyday life. They help in telling time (e.g. quarter past, half past, quarter to), measuring quantities (e.g. half a kilogram of flour, a quarter teaspoon of salt), and shopping half-price sales.
Comparison with whole numbers can help explain why learning fractions may be challenging (Namkung et al., 2018; Siegler, 2013). For example, fractions do not have unique successors; between any two fractions, there can be infinitely many other fractions. Common misconceptions and errors include viewing numerators and denominators as independent numbers, comparing fraction magnitudes based on whole-number knowledge (e.g. 1/3 > 1/2 because 3 > 2) and ordering fractions based on whole-number knowledge (e.g. 1/2 < 1/5 < 1/10 because 2 < 5 < 10). When multiplying fractions, the result can be smaller than the original fractions, and when dividing fractions, the result can be larger. These properties highlight the differences between fractions and whole numbers and can make them challenging to learn. Understanding whole numbers is necessary for comprehending fractions, but it is crucial to understand that fractions represent a ratio between two integers. The numerator indicates how many parts are being considered, while the denominator shows how many equal parts the whole is divided into.
Children beginning school are typically familiar with the concept of fractions through everyday play activities as they become acquainted with the idea of sharing a number of items fairly between two or more people. Fair sharing is an informal analogy for equal partitioning, serving as the conceptual foundation for formal instruction on various numerical and arithmetic concepts, including measurement. A fraction represents how many parts of a whole exist. For example, when an object is divided into two equal parts, each part is called a half. When divided into four equal parts, each part is called a quarter. Story problems focusing on fractions put the child in a real-life situation to support their conceptual understanding of fractions (e.g. There are two pizzas of the same size. One pizza is cut into two equal-sized parts and the other is cut into four equal-sized parts. What can a child do so that he and three friends can get the same share size from the two pizzas?)
Children need opportunities to explore and experience fractions to develop the necessary skills for success in school and beyond. At an early stage, understanding parts and wholes is a crucial concept in developing fraction skills. For example, if an orange (a whole) is cut in half, there will be two pieces, each representing a part of the original whole.
Visual representations make understanding fractions easier. Diagrams can effectively illustrate fraction concepts. For example, if we take a guava and divide it equally among four people, each person will have one-quarter of the fruit. If one person takes two-quarters, they have half of the guava. Three out of four pieces eaten means that three-quarters of the guava is gone, leaving one-quarter. A whole guava consists of four out of four (or four-quarters).
Fractions are written with one number above another and separated by a line. The numerals indicate how many parts of a whole are being considered. For example, 3/4 represents three out of four equal parts (part-to-whole model).
The numerator (top number) indicates how many parts are being considered.
The denominator (bottom number) indicates how many equal parts make up the whole.
Rectangular diagrams can illustrate parts of a whole. Consider a rectangle divided into ten equal boxes, four of which are shaded. The fraction representing the shaded portion is 4/10.
It is important to emphasize that a fraction represents a single number, not two numbers stacked on top of each other. Children benefit from seeing fractions represented in multiple ways. Diagram representations help them understand fractions as parts of a whole, but diagrams alone do not clarify that fractions are numbers. Number lines complement diagrams by showing fractions as points with precise numerical positions; this supports comparison, understanding of magnitude and the idea that there are infinitely many fractions between any two values. Using both diagrams and number lines strengthens children’s conceptual understanding and helps them develop a flexible and robust sense of fractions.
Measurement
According to the National Research Council of the National Academies (2009), measurement is a fundamental aspect of mathematics that bridges geometry and numbers, forming a core component of children’s mathematical development. It encompasses early understandings of size, length, weight, capacity and spatial relationships, all of which are grounded in the concepts of attribute, unit and scale. The attributes children compare may include area, length, volume and time (Lembrér, 2013). Measurement also supports the development of broader cognitive skills such as counting, sorting, comparing and analytical thinking (Clements & Sarama, 2009).
Young children typically begin exploring measurement through nonstandard units rather than conventional ones, such as centimetres or inches. Research shows that using familiar objects—such as blocks, paper clips or fingers helps children grasp the idea of units and comparison without the added complexity of standard measurement systems (Clements & Sarama, 2014). Storytelling and other child-centred, playful approaches further support this learning, ensuring that measurement remains meaningful and enjoyable (Carlsen, Erford & Hundeland, 2020). Early measurement learning also relies heavily on mathematical language, enabling children to express comparisons of length, weight, width and volume in everyday terms. This emphasis on descriptive vocabulary is reflected in the early childhood curriculum, which encourages children to compare objects using words such as ‘many’/‘few’, ‘tall’/‘short’, ‘thick’/‘thin’ and ‘big’/‘small’, without requiring the use of standard units or formal measurement tools (United Republic of Tanzania (URT), 2016; 2023). Such language-based comparisons align with research showing that children instinctively describe measurements through familiar adjectives—for example, ‘This stick is long!’ or ‘This book might be too tall for the shelf’ (Clements & Sarama, 2014).
As with other mathematical domains, learning trajectories provide a useful framework for understanding how children’s measurement thinking develops over time. Studies on length measurement (Clements & Sarama, 2014; Szilágyi et al., 2013) indicate that children first view length as an inherent property of an object and later as a comparative attribute that allows objects to be ordered or measured. Initially, children can notice differences in length when objects are placed side by side, even if they do not yet understand the importance of alignment. Eventually, they recognize that accurate comparison requires aligning endpoints and can compare lengths indirectly by using a third object or by arranging items in serial order.
This development continues as children enter formal schooling. Measurement learning from Grades 1 to 3 forms a coherent continuum in which children gradually move from informal comparison towards structured, accurate measurement using standard tools. At the beginning of schooling, learners work with hands-on comparisons and nonstandard units, building foundational vocabulary and concepts by directly comparing lengths, masses and volumes. In Grade 2, these foundations expand to include standard units such as centimetres, kilograms and litres. Children begin using simple measuring tools, estimating and recording results and extending their knowledge to time and area in everyday contexts. By Grade 3, students apply a broader range of standard measurement tools with increasing precision. They measure straight and curved lengths, perform simple calculations with measurements, compare areas using square units and tell and set time to the minute. Across these three grades, the learning trajectory reflects a shift from intuitive, experiential exploration to formalized measurement practices using conventional units and tools.
References
Carlsen, M., Erford, I. & Hundeland, P. S. (eds). (2020). Mathematics education in the early years: Results from the POEM4 conference, 2018. Springer.
Casey, B. M., Dearing, E., & Springer, K. (2001). Preschoolers’ spatial reorientation: The importance of a dual representation. Developmental Psychology, 37(5), 699–712.
Clements, D. H., & Sarama, J. (2009). Learning and teaching early maths: The learning trajectories approach. Routledge.
Clements, D. H., & Sarama, J. (2014). Learning and teaching early maths: The learning trajectory approach (2nd ed.). New York, NY: Routledge.
Clements, D. H., & Stephan, M. (2004). Measurement in pre-K to grade 2 mathematics. Teaching Children Mathematics, 10(6), 298–305.
Lamber, D. (2013). Young children’s use of measurement concepts. Working Group 13. Malmö University.
Namkung, J. M., Fuchs, L. S., & Koziol, N. (2018). Does initial learning about the meaning of fractions present similar challenges for students with and without adequate whole-number skill? Learning and Individual Differences, 61, 151–157. https://doi.org/10.1016/j.lindif.2017.11.018
National Research Council of the National Academies. (2009). Mathematics learning in early childhood: Paths towards excellence and equity. The National Academies Press.
Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994–2004. http://dx.doi.org/10.1037/a0031200
Szilágyi, J., Clements, D. H., & Sarama, J. (2013). Young Children’s Understandings of Length Measurement: Evaluating a Learning Trajectory. Journal for Research in Mathematics Education, 44(3), 581-620. Retrieved Nov 2, 2025, from https://doi.org/10.5951/jresematheduc.44.3.0581
United Republic of Tanzania (URT). (2016). Pre-primary Education Curriculum. Ministry of Education, Science and Technology.
United Republic of Tanzania (URT). (2023). Pre-primary Education Curriculum (Revised Edition). Ministry of Education, Science and Technology.
Verdine, B. N., Golinkoff, R. M., Hirsh-Pasek, K., & Newcombe, N. S. (2014). Finding the missing piece: Blocks, puzzles, and shapes fuel school readiness. Trends in Neuroscience and Education, 3(1), 7–13.
