Article: Why the Brain Needs to Touch to Understand Mathematics
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Why does the brain need to touch to understand mathematics?
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Working with mathematics through manipulative materials and flexible approaches is not a trend or mere methodological strategy, but an educational necessity if we truly want our students to understand what they learn. Mathematics, by its very nature, is abstract: we speak of numbers, relationships, operations, or structures that cannot be seen or touched. For many children, this poses great difficulty from the beginning. In this sense, Cuisenaire rods and other manipulative materials act as a fundamental bridge between the concrete and the abstract, allowing students to experience, build, compare, and verify with their own hands what they will later represent with symbols.
When a student uses manipulative materials, they are not merely “playing”; they are constructing meaning. This significantly reduces the level of abstraction and facilitates deep understanding. Students stop memorizing procedures and begin to understand why things work the way they do. Furthermore, rods allow for detecting errors, correcting them, and reflecting on them, fostering much more solid and lasting learning.
As José Antonio Fernández Bravo advocates, “mathematics is a mental activity.” This idea is key to understanding how it should be taught. Learning mathematics does not consist of repeating steps or copying procedures but of thinking, reasoning, analyzing, making decisions, and establishing relationships. For this reason, it is essential to propose activities that foster reasoning, investigation, and experimentation, and to move away from activities centered solely on mechanical processes and routine calculations.
Alongside manipulative work, it is imperative to change the traditional approach to mathematics, which has been based for years on the repetition of closed algorithms. For a long time, “knowing mathematics” has been identified with doing long calculations quickly, when in reality true mathematical mastery lies in understanding what you are doing and why. The goal should be to develop number sense: understanding how numbers relate to each other, anticipating results, estimating, checking whether an answer is reasonable, and choosing the best strategy for each situation.
Mental calculation and decomposition play a fundamental role here. Teaching students to decompose numbers, to compensate, to reorganize quantities, or to seek equivalences allows them to face calculations with flexibility and confidence. It is not about everyone solving in the same way, but about each student being able to find strategies they understand and control. In this way, students gain autonomy, confidence, and ability to adapt to different situations.
In this context, flexible algorithms become a key tool. Contrasting with traditional algorithms, which are often applied without understanding, flexible algorithms start from number sense and reasoning. They allow adjusting the procedure to each calculation, facilitate estimation, and strengthen understanding of the decimal system. Moreover, they connect much better with real life, where problems are not solved by following rigid columns but by thinking and making decisions.
Another fundamental aspect of this approach is that it favors attention to diversity. The use of rods and flexible strategies allows each student to progress at their own pace, from their own level, and with their own tools. Everyone can participate, everyone can find a path to understanding, and everyone can feel competent. This reduces math anxiety, improves motivation, and generates a more positive relationship with the subject.
Ultimately, working with mathematics manipulatively and through flexible proposals means committing to meaningful teaching. It means placing students at the center of learning, allowing them to investigate, make mistakes, reflect, and construct their own knowledge. It means prioritizing understanding over memorization, reasoning over repetition, and connection with reality over empty calculation. Only in this way will we succeed in forming capable, confident, critical, and competent students, prepared to use mathematics as a true tool for understanding the world.
Author: Francisco Blas Navarro (@matesmaniacosoaoa)