Rethinking the Curriculum of the Future: Our Brain is Ready to Learn Math

Resumen ejecutivo

• Being able to understand and operate with numbers is crucial for real life: we use them to know if we are running late, to find our bus at a station, to understand a blood test, and even to decide what we should buy at the supermarket.
• Mathematics is so fundamental that there is a number sense that we share with other species, which, although rudimentary, allows us, even as infants, to estimate numerical quantities and to operate with them.
• That number sense is not enough. Establishing concrete connections with symbolic mathematics and practicing these relationships in a supported, gradual, and sustained way will foster initial numerical comprehension and advanced mathematical reasoning.
• Examples are given of simple interventions that can enhance understanding of numerical magnitudes and lead to mastery of mathematical concepts.
• Activities such as estimating, comparing, counting, using appropriate vocabulary, and employing the number line are essential to understanding magnitudes and quantities, facilitating efficient and effective numerical operations, and developing a more natural way to deal with numbers.
• This approach is discussed in the context of current challenges, with a call for reflection on how virtual money management and high-inflation environments in some countries may impact mathematical learning.

Learning reshapes the brain and, in doing so, changes how we perceive, understand, and respond to the world. Educational experiences, whether formal (like schooling) or informal (like playing or shared reading), can cause neural and behavioral changes. However, these changes do not happen arbitrarily; behavior, and the neural architecture supporting it, are constrained by physiological limits shaped by nature and prior experiences[i]. To be effective, education should build on cognitive abilities, of which the rudiments are already present in our brains. This is the case, for example, of mathematics.

Mathematics

Babies can count. Although they may not consciously think “one, two, three,” within days of birth, they can distinguish between significantly different quantities[ii]. We are born with a set of neurobiological abilities independent of language that, among other things, allow certain perception and discrimination of magnitudes. Some authors suggest that this basic capacity is innate and connected to visual and auditory perception[iii], which is why it is often identified as a kind of sense[iv], the acuity of which improves with age—up to a certain point. However, there is no clear evidence of a direct causal relationship between numerical sense and formal mathematical performance[v]. Another branch of research argues that what is relevant is how innate abilities are shaped and modified by experiences and that, therefore, it is essential to provide children with opportunities to develop those skills[vi].

Although these positions are somewhat opposing in the literature[vii], their common points are crucial for education. Because number sense is the behavioral expression of a neural, physiological foundation that, with appropriate challenges, allows us to engage with, and eventually master, symbolic mathematics. While our nervous system is prepared to understand and estimate numerical quantities and to operate with them, it needs to be exercised. Over time, with proper stimuli, we learn that different quantities can be given (verbal) “numerical labels”, and, with sufficient scaffolding, we practice and gradually become able to represent and manipulate them more accurately.

This integrated theory of numerical development[viii] suggests that as a person (whether child, adolescent, or adult) gains more knowledge and understanding of quantities and magnitudes, they will be more effective and efficient in operating with them. Consequently, enhancing explicit and implicit numerical understanding will lead to improvements in learning and overall mathematical performance.

Tapping into Physiological Foundations to Improve Math Performance

Early primary school math learning partially depends on numerical skills acquired in preschool[ix]. Moreover, differences in math proficiency often persist throughout schooling: children who show weaker math skills even at the beginning of primary school generally continue to lag behind their peers, making it less likely that they will pursue math-related carreers later on[x]. While these choices could be seen as matters of free will or biological limitations, we now know that everyone is physiologically prepared to learn math, for which they need appropriate support, strategies, and challenges.

Thus, there are simple interventions to improve understanding of numerical magnitudes, which, in turn, increase the likelihood of mastering mathematical concepts. Below, we examine three potential interventions.

Case 1: Building Connections between Different Representations

Knowledge of symbolic mathematics mediates the transition between informal and formal math learning and is a predictor of academic performance. Although studies have found that individual differences in the accuracy of number sense[1] correlate with symbolic math performance from an early age[xi], across various age groups[xii], and even in particularly talented adolescents[xiii], there is consensus that exercising the number sense requires explicitly incorporating links to symbolic mathematics[xiv].

Along with the non-symbolic understanding, the symbolic and ordinal system also develops. Regardless of whether one prevails over the other or whether they form a sort of virtuous circle, it is necessary to introduce symbols and explicitly link them with symbolic and non-symbolic quantities (e.g., saying “five” while pointing to or drawing a 5 or counting on fingers). In this way, early incorporation of formal concepts like symbols and numbers happens naturally through informal and playful activities[xv] such as counting, playing board games with numeric content, comparing sets of objects, recognizing numbers and shapes, and so on. In general, exposing children early to interacting with quantities in as many ways as possible[xvi] helps build the connections between non-symbolic and symbolic representations, setting the stage for effective math learning.

Case 2: The Number Line and the Importance of Space

Understanding numerical concepts involves learning that quantities can be represented as magnitudes and ordered along a linear (mental) continuum that typically extends horizontally, with greater magnitudes incrementally placed to the right on the line, though this orientation may be culturally determined[xvii]. This spatial-numerical relationship is pre-literate and is found in infancy[xviii], suggesting it may even be innate, as has been observed in babies younger than 3 days of life[xix].

The internal spatial representation of numbers organized along a number line is a dynamic structure that starts with a few small quantities and gradually expands to include larger numbers, then negatives, fractions, and decimals. Representing numbers mentally in this way is associated with formal math skills[xx]. Performance on the number line at age 6, for instance, predicts symbolic calculation skills at age 8[xxi], possibly because early practice with a representation like the number line facilitates understanding the concept of linearity[2], which is foundational to formal math.

Though establishing an organized representation involves spatial skills, it remains unclear what exactly explains the link between spatial and numerical reasoning abilities[xxii]. Nonetheless, the relationship exists[xxiii], and studies suggest that spatial reasoning can be fostered through tasks that demand it, such as those requiring physical or imagined navigation, movement, or orientation in space. Specially designed video games[xxiv] or physical-world activities like building with blocks, solving puzzles, origami[xxv], running, jumping, and hide-and-seek can support children’s spatial reasoning development.

Case 3: Culture and Language

Language shapes, and is shaped by, cognition. Math performance relies on acquiring appropriate language[xxvi], as this influences how children understand and reason with numbers and symbols, ultimately allowing them to solve arithmetic problems. For instance, some cultures with less exact numerical representations than those in the West[xxvii] (because they have not needed nor exercised it) lack most numeric words, which limits their ability to accurately identify anything beyond very small quantities[xxviii]. The importance of language proficiency also becomes evident in the classroom, where instructions often depend on verbal explanations, and solving problems requires specialized vocabulary, which often has different meanings inside and outside the classroom (words like “and,” “by,” “minus,” “volume,” etc.). Increasing early vocabulary, both general and mathematical, can enhance educational equity in math. Integrating language that not only allows counting, but also enables comparison and operations with different quantities and magnitudes, will foster the development of numerical space, laying the foundation for a solid understanding of symbolic mathematics.

The importance of language is also evident in how information is conveyed. The choice of words can impact performance. In a series of studies[xxix], math students of various ages solved problems presented in different formats: as symbols (X + 3 = 7), as straightforward word problems (“If you add 3 to a number, you get 7; what is that number?”), or as part of a story (“Carlos had some candies, and his mom bought him 3 more at a kiosk. When he counted all his candies, he found he had 7. How many did he have before the kiosk?”). Researchers not only assessed the students, but also analyzed how teachers thought their students would perform with each problem format. Most teachers expected that students would do best with the symbolic version, as it directly presented the equation to be solved. In contrast, they assumed that a story-based problem would require students to translate the words into numbers and symbols before solving it, which they believed would lower performance. However, the results showed that students actually found the story format easier than the purely symbolic one, solving more problems correctly when presented in that third format. Teaching effectively involves understanding where students encounter difficulties and what they find easier or harder. When you’re very knowledgeable about a topic, it’s easy to lose that beginner’s perspective, and it takes effort to regain it. As a result, even highly skilled professionals may struggle to teach well if they can’t grasp the difficulties their students face or recognize when students need more context to fully understand a situation and solve it.

The three previous cases are not exhaustive examples of what happens, or can happen, in an educational context, but they help to illustrate how some everyday situations can be modified in a relatively simple way so that learning mathematics does not feel so foreign to a substantial portion of the population.

Mathematics is hard

There are various explanations for why mathematics is challenging[xxx]. Some cognitive theories point to the intrinsic difficulty our minds have in understanding fractions or decimals and the counterintuitive nature of working with them (for example, because ½ + ⅓ is not 2/5, or because 0.5 x 2 is less than 2). Other explanations suggest that much of what is taught in mathematics involves solving a problem using an algorithm without understanding the reasoning behind each step. Often, procedures are repeated mechanically. For instance, we may remember the algorithm for division, yet struggle with—or even find it impossible—to solve a division in an alternative way. The same applies to “moving” numbers from one side of an equation to another: in X + 3 = 7, for example, the “+3” “magically” becomes “-3” when it switches sides. For some students, solving these types of exercises can almost become a game, which is highly desirable as long as the pedagogical objectives are met. Unfortunately, in many cases, without a clear rationale or motivation, simply solving exercises “without thinking” can become repetitive, tedious, or, worse still, unhelpful for “real life.”

Practicing estimation, for example, is often undervalued, yet it can be beneficial for both teachers and students. For teachers, it provides insight into the accuracy of students’ mental representations of quantities, which can help in deciding teaching strategies—whether it’s time to increase the complexity of tasks on the number line and certain operations or if there is a need to reinforce foundational knowledge. For students, estimation practice allows them to recognize if the answer they calculated is reasonable or unlikely, and if so, to rework the problem and/or evaluate alternative methods. For this exercise to be effective, it’s crucial that teachers value the skill of estimating a result and avoid overly penalizing a close-but-incorrect answer. After all, in everyday life, exact results are rarely as important: when budgeting for shopping or calculating travel time, having a general idea of the final figure is often more practical than needing an exact number.

Learning for the Real World

If the goal of education is to prepare students for the “real world” they will face, it is desirable to consider the contextual and cultural environment of each group. In this regard, another potential learning challenge raised in the literature is that examples or exercises provided in class or textbooks[xxxi] are often not varied enough, not fully inclusive, culturally relevant, or reflective of students’ daily lives. For example, while the same rational number can be represented as either a fraction or a decimal, the choice of representation can impact understanding. Research suggests that using decimal numbers is more helpful when division or distribution is required, while teaching fractions might be more effective when approached in a perceptual way, enabling students to develop skills for recognizing structures and patterns[xxxii].

At a cognitive level, solving problems through formal representations (whether in mathematics, physics, or chemistry equations) requires that some knowledge is learned explicitly (such as the meaning of each symbol or the order in which to solve parts of a problem). However, the process of understanding and generalizing these concepts is largely an implicit learning experience—there’s a point when something “clicks” in our minds. To reach this level of mastery with magnitudes, practice is essential. To begin working mentally with quantities and making estimates, manageable magnitudes are necessary. Starting with small numbers, simple operations, and contextually relevant examples can be extremely helpful because “tangibility” aids learning[xxxiii]. For instance, children who accompany their parents shopping or play games like Monopoly or role-play buying and selling items are practicing a contextualized development of numerical understanding. Although more research is needed, it’s reasonable to think that the shift to virtual money, now increasingly common, may hinder this “natural” exercise, since there are no longer bills or coins to count physically. High-amounts currency in many countries, a product of years of accumulated inflation, likely complicates this everyday practice, which used to be part of daily life.

Final Reflections

The evidence indicates that, while we possess a natural, innate foundation that allows numerical understanding as well as the learning and knowledge of formal mathematics to emerge, this foundation needs to be exercised, with adequate, sustained, and supported training. It needs practice. This can only be achieved through education. Education builds upon those natural basis to refine them, making them more precise and useful for the modern world.

Estimating, comparing, measuring, counting, listing, building vocabulary, and understanding the meaning and scope of the operations to be used are some of the practices mentioned in this brief that, with appropriately chosen and progressively challenging (yet not frustrating) demands, help foster a natural proficiency with numbers and magnitudes, generating the fluency needed to understand the underlying concepts that are seek to be addressed. It is crucial to consider the real goals of the education we aim for, the “why” behind what we teach, and to evaluate whether these goals are reflected in lesson plans, teacher training, curricula, bibliography, and materials. To consider whether what we are teaching, and the way in which we are teaching lays the foundation that prepares students to step into a changing and uncertain world. Achieving a balance between content, procedures, and effective mastery so that students can better navigate the real world is part of what will enable today’s students to tackle the novel challenges of the future.

[1]    The tasks that are usually used to perceptually stimulate or evaluate number sense consist of performing exercises that unequivocally demand it. For example, comparing, adding or subtracting two non-symbolic quantities (clouds of dots or sets of objects) that are displayed for such a short period of time that it is not possible to count them (one and a half seconds, or even less). The difficulty of the task can be modulated by increasing the closeness of the two quantities (the more similar they are, the more difficult it is to distinguish between them).

[2]    Linearity implies, for example, that between 1 and 2 there is the same distance as between 1001 and 1002, although proportionally the first difference is very relevant while the second is insignificant.

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