If you are a parent, I am sure you have occasionally taken disciplinary decisions that, despite being difficult, you knew would guide your child towards becoming a pleasant adult and generally respectful of society’s rules (or, at least some of them!). In English, we often say: “best to bend it while it is a twig”. You know that for your child to have a fulfilling future, they need to learn important life lessons from an early age.
Similarly, you know that introducing your child to languages, musical instruments, and sports from a young age will benefit their growth and skills development.
In the academic context, exposing kids to books, reading, and vocabulary-enhancing conversation helps them thrive in literacy, self-expression, and all life opportunities relating to words – which are indeed many!
What about mathematics? What impact does early exposure to solid mathematics learning materials and activities have on your child’s future? Well, being accomplished in mathematics is widely considered a key to financial achievement (Duncan et al., 2007), better socioeconomic positioning (Ritchie & Bates, 2013), and an enhanced perception of health risks and medical decision-making (Reyna & Brainerd, 2007). It follows then that engendering good mathematics knowledge and skills from a young age will set kids on the right path for their futures.
In this article, we will look at early mathematical abilities: What are they? Why are they important? How do they finally lead to our child doing that kind of calculus that you were never able to master?
Math skills taught in early childhood education are designed to pave the way for children to thrive in elementary school and beyond. Research has shown that poor math knowledge from the very beginning can cause gaps that will hinder success and can persist across a child’s entire educational career (e.g., Hornung, Schiltz, Brunner, & Martin, 2014; Jordan et al., 2010; Jordan, Kaplan, Ramineni, & Locuniak, 2009; Krajewski & Schneider, 2009; Lefevre et al., 2010).
Therefore, the preschool years are a fundamental phase. In addition, a child’s math skills at this age can reveal what can be expected of their future maths abilities. Even more than that – early mathematical skills have also been predictive of many other school subjects! As such, it can provide an overall indication of a child’s future academic success.
Why is it that the early years of math development are so crucial to future success? The answer lies in the hierarchical nature of mathematical skills. The foundational numeracy skills learned at a young age form the critical building blocks on which the future exploration of math concepts will be built.
A case in point is problem-solving. To solve the simple calculation of 2+4, a child needs to possess different levels of prior knowledge of the numerical system. For example, a child needs to know what “2” and “4” are and what they represent. Then they need to understand the concept of addition and that it is signified by the symbol “+”.
The major building blocks of early mathematical development were described by Sarama and Clements (2004) as:
1) Visuo-spatial competencies and
2) Numerical knowledge.
Visuo-Spatial Skills (VSA) is a broad term, and there is not much consensus on all its sub-components. Some researchers define VSA as “how individuals deal with materials presented in space, whether in one, two, or three dimensions or with how individuals orient themselves in space” (Carroll, 1993, p. 304). Other researchers describe it is as “skill in representing, transforming, generating, and recalling symbolic, non-linguistic information” (Linn and Petersen (1985)).
Regardless of their distinct definitions, however, both researchers differentiate between three types of VSAs: 1) spatial perception, 2) mental rotation, and 3) spatial visualization.
Spatial perception is considered the ability to discover spatial relationships with respect to the orientation of one’s body, regardless of any distracting information. Spatial perception is essential in daily life, and it is the skill we use to avoid walking into walls or chairs! It makes us aware of where we stand and builds our orientation accordingly. Visual perception allows us to visualize and interpret the visual information around us, and to analyze and make sense of what we are looking at.
Mental rotation refers to the ability to rotate two- or three-dimensional figures in space while the features of the figure remain intact. Such tasks tap into mental representation and transformation. A good example of mental rotation is the 3D board game Ubongo, in which players have to rotate 3 of the Ubongo pieces to fit perfectly in a 2D plane.
Spatial visualization demands more complicated and multistep manipulations of the information introduced. These tasks can integrate aspects of the first two categories (spatial perception and mental rotation). The critical difference between spatial visualization and the other two types of VSA is that it is likely to require multiple solution strategies. Some everyday spatial visualization tasks are embedded figure tasks or block design tests (Linn & Petersen, 1985).
Another critical aspect of VSA is visuo-motor integration (VMI). The significant difference between VMI and VSA rests on the motor component needed to solve the respective tasks (Linn and Petersen (1985)). VMI tasks require the coordination between the processing of visual input (i.e., visuospatial processing) and motor output (i.e., motor activity) (Cameron et al., 2015). VMI is especially essential for learning how to paint, draw, copy, or write what is seen. VSA also taps into working memory, and most VSA-related activities require the child’s working memory capacity.
Spatial skills are regularly used in everyday tasks such as estimating distances, reading maps, and solving math problems. If we consider our 2+4 example, the child should be able to perceive the visual representation of the numbers and the symbol, the relation between the numerical values, and how their positions can be important for solving the calculation. For instance, in the 2 – 4 and 4 – 2 patterns, the results will not be the same when the number placement changes (Fuson, 1988). Other examples of the need for spatial abilities to solve math problems would be the ability to create and rotate geometrical shapes or to find patterns in them (Casey et al., 2015; Hermer & Spelke, 1994).
Numbers are intricately associated with our day-to-day life in activities ranging from trade, shopping, timekeeping, and the communication of statistics, amongst many others. Nevertheless, do we even remember how we learned the numbers? Learning numerical cognition was not a simple task. Many are the steps we have been through to have the knowledge of numbers we have now. To understand this process, researchers have developed many theories.
Several models of numerical cognition have been created, such as the Triple Code Model (TCM) proposed by Dehaene (1992) or the Four-step Developmental Model by von Aster and Shalev (2007), which conceptualize and describe different ways to represent numerical cognition (and the number-specific knowledge that can be derived from it), their interrelations, and their development.
The TCM model of numerical cognition (Dehaene, 1992) is premised on the existence of three primary representational codes (depicted in Figure 1.1). For number compromises, an analogue magnitude representation (e.g. ***), an auditory verbal number representation (e.g., three), and a visual Arabic number representation (e.g., 3). Even though the Triple Code Model has probably been the most influential model of numerical cognition, it still does not shed light on information related to developmental aspects, meaning how children learn numerical cognition. This model shows how we represent numbers, and it presumes that the three “codes” work in parallel or simultaneously. However, the model does not show how we learn such representations. It does not indicate if we acquire such codes simultaneously or one after the other.
Von Aster and Shalev (2007) put forward the four-step developmental model of numerical cognition that partially relies on Dehaene’s triple code model (Dehaene, 1992). It deconstructs the developmental course of different representations of numbers and is therefore well suited to provide a comprehensive description that helps further the understanding of young students’ numerical development. These representations develop quasi-hierarchically, with every step building on the preceding one. For example, Dehaene, von Aster, and Shalev (2007) differentiated between semantic and symbolic (verbal and Arabic) number representation. Notably, von Aster and Shalev further subdivided the semantic number system into two components: an early, implicit core system of magnitude (Step 1 in the model presented in Figure 1.2) and a later, explicit representation of a mental number line (step 4).
The graph shows that the development of numerical cognition that will lead to mathematical knowledge depends on many factors. It depends on the individual’s capacities (their working memory). It is dependent on the brain area that will be activated (for step 1, it is the bi-pariental). Then, you have the ability to learn how it is cognitively represented. Finally, you have the steps. If you look at step 1, you observe that it is the infancy period. There you are at the point of learning comparisons (cardinality). Children create a mental image of comparisons, and once a comparison task appears, like the father asking the child to pick up the bag with more apples, the children activate this knowledge to perform the task.
Only after passing through step 2 in which the child learns the verbal number system (that we use numbers to count and they have names – one, two, three) and step 3 (that we represent this number words with symbols – 1, 2 and 3), will the child understand the order. That 1 will always be before 2 and that 3 will always be after 2 (step 4 – ordinality). The mental number line (ordinality), therefore, will develop successively, relying on previous ways of representing numerical magnitude with verbal and Arabic symbols and on the growing capacities of the person’s general abilities domain (e.g., working memory).
To sum up, we discussed the significance of the preschool years in laying down the foundation on which mathematical skills will continue to build and thus the importance of developing early competencies in mathematics. We also pinpointed the precursors of mathematical knowledge. We provided a framework for the different types of knowledge children need to be equipped with and are therefore required to be trained in during the preschool years and ahead of formal schooling.
Does this mean that my child is now capable of solving those complex calculus problems we mentioned at the beginning of this blog? Go easy! We learned that many are steps for learning basic numeracy – cardinality, verbal number system, Arabic system, and then ordinality. For the basics of math, one must put in a lot of effort to learn, and it takes some years of working hard to master numerical cognition… You can imagine that is the same for calculus. After learning that math mainly consists of numbers, for calculus children will “un-learn” a bit of math in the sense that they will see this:
x + y = 14
and ask themselves: Who put letters on this equation if we are learning math?
Let’s talk about calculus after children master basic numeracy! One step after the other and between – many, many exercises and activities to consolidate knowledge!