Alberto Guerrero-Velázquez is a philosopher, cognitive scientist, and educator with extensive experience in teaching and educational development. He has taught philosophy, ethics, bioethics, critical writing, and philosophy of artificial intelligence at different educational levels, and has worked in educational leaderships, teacher training, and the design and implementation of educational programmes. His research lies at the intersection of philosophy of mind and cognitive science, with a particular interest in autobiographical memory and its relationship to personal and social identity.
Most teachers have experienced the same situation: a student can read a large number correctly but struggles to explain why each digit has a different value. Constructing numbers is a key skill in the development of logical-mathematical thinking. It involves counting, abstraction, and symbolisation, and helps students use quantities flexibly when problem-solving. However, it is common to find students who can read and write numbers but do not fully understand their meaning or function. This difficulty becomes particularly evident when working with place value. Could it be that we are asking students to think abstractly before they have had enough opportunity to experience what they are trying to understand?
Mathematics in the classroom: Between practice and outcomes
In recent years, there has been a strong emphasis on promoting meaningful learning in mathematics. However, classroom experience and research suggest that students can sometimes perform mathematical procedures without fully understanding the ideas behind them. Understanding place value requires students to connect what numbers represent with the position of each digit, and difficulties with this can affect later mathematical learning (1).
Students may be able to work with ones, tens, hundreds, thousands, and millions while still finding it difficult to explain why a digit changes its value when its position changes. In some cases, students may also struggle with the meaning and function of zero (2).
This becomes particularly relevant in Years 5 and 6, when students are increasingly expected to work confidently with larger numbers and more abstract mathematical ideas. Research with sixth-grade students, for example, has shown that zero can still present conceptual difficulties at this stage of schooling (2). These difficulties can affect later learning, including understanding quantity, estimation, basic operations, and problem-solving.
In mathematics teaching, the development of abstract thinking is often prioritised from the early years, sometimes alongside a gradual reduction of concrete and experiential forms of learning. This is understandable: abstraction is an important part of mathematical thinking, and students are expected to become increasingly independent in working with symbols and concepts. However, the difficulties observed in the classroom invite us to reconsider how the experiences that support abstraction are built, and when students are actually ready to rely mainly on abstract representations.
If students can perform mathematics without understanding it, what exactly are they learning?
The body enters the classroom
This question points to a deeper issue: if students are expected to think abstractly, what kinds of experiences support the development of that capacity?
From the perspective of embodied cognition, learning does not occur solely in the mind, but through the interaction between body, environment, and action (3). Varela, Thompson, and Rosch proposed an approach to cognition in which knowing is closely connected to embodied experience and engagement with the world (3). Other work has also suggested that concepts are not formed in isolation from the world, but are shaped through relationships among perception, context, and environment (4).
This perspective changes the way we might think about learning abstract concepts. Understanding is not only about manipulating symbols in the mind; it can also be supported by experiences that give those symbols meaning.
Applied to teaching, principles of embodied cognition suggest that understanding complex concepts, such as those involved in mathematical thinking, can be supported when they are connected to bodily action. If learning is connected, at least in part, to experiences involving the body and environment, why is this foundation sometimes set aside when abstract thinking is expected to develop?
The transition from the concrete to the abstract does not necessarily have to be abrupt. Research on concreteness fading, for example, suggests that moving gradually from concrete representations toward more abstract ones can help learners transfer what they have learned (5). This research was conducted with university students, so it does not directly tell us how this approach should be used with younger children. However, it supports an important idea: concrete experience and abstract thinking do not have to be treated as separate or opposing ways of learning.
In practice, this means creating opportunities for students to engage with mathematical ideas through movement, interaction, and space. Students might use their bodies to represent numbers, move through space to explore relationships, or physically model ideas that are often taught only through symbols.
An example of the use of the body in processes of representation and abstraction is hand gesture. Hand movements can represent different aspects of an action, ranging from more complete movements to increasingly abstract representations. Novack et al. found that, although both physical action and gesture supported children´s learning of mathematical-equivalence problems, only gesture helped children successfully apply what they had learned to problems that required generalisation (6). This suggests that bodily representation may help students move from a specific action towards a more abstract understanding.
With this perspective in mind, Karla has designed a range of educational interventions to support the understanding of abstract ideas. One example is the Living Board, an approach developed from her teaching experience to foster understanding of place value through movement and spatial interaction (7).
The activity involves constructing a dynamic number system in which students take on different roles: some represent ones, others tens or hundreds, positioning themselves according to their function. As larger numbers are introduced, new “families”, such as thousands and millions, are incorporated and organised spatially.
For example, a student holding the number 5 represents “five” in the ones place, but becomes “fifty” or “five hundred” by moving to a different position. The shift is not only symbolic but physically experienced through movement. Proprioceptive and visual feedback reinforce the relationship between position and value: place value is enacted through action—it is not merely calculated, but experienced.
The important point is not that every mathematics lesson needs to become a physical activity. Rather, teachers can ask whether a concept that students are expected to understand abstractly might first benefit from being represented through movement, space, or gesture.
