Mathematics is the subject that studies geometric and numeric entities. It’s essential for all sciences, both hard and soft ones. Because it is such a significant domain of our knowledge, understanding how its learning works is crucial. More in depth, a survey about whether or not mathematical cognition has a collocation in the brain and if that can be changed through neuroplasticity is the starting point for studying the mechanisms of mathematical learning in general. Acknowledging how the mathematical foundations are grasped could prevent mathematical learning disabilities (MLD), as shown later on in this article, but before discussing it in neuroscientific terms, it will be clarified what 'mathematical foundations' are, more specifically those of numerical nature.
Figure 1: A blackboard with mathematical exercises. Photo by Dan Cristian Pădureț on Unsplash
Although mathematics became a science in the 5th century B.C. in Ancient Greece, the meta-mathematical study of its foundations started just in the 19th century. The philosopher and mathematician Gottlob Frege (1848–1925), considered the father of the contemporary mathematical logic, began his foundational research in the 1870s. His point of view is ascribable to logicism, a school of thought that considers logic as the base for any kind of knowledge, mathematics included. Frege gave a solid and rigorous basis to the arithmetic of natural numbers through his Begriffsschrift ('concept-writing'), an axiomatic systematization in which he utilizes the predicate calculus, a formal language where the truth value of a complex predicate relies on the truth of its single, atomic components. Frege’s aim was the reduction of all mathematical concepts under logical terms. From this perspective, the predicate calculus leads to the demonstration of whole theorems. More in depth, a number is the extension of a logical concept. A brief recall on what 'extension' means in semantics: the set of objects united by a property or idea that applies (or extends) to those objects. The logicist perspective from which numbers are just extensions of logical concepts has had a historical impact on developmental psychology, starting from Jean Piaget, as it will be explained in the next paragraphs.
Logicism illustrates the foundations of mathematics as analytical concepts, and that means they are implicitly contained in logical concepts by definition. This is in contrast with the Kantian systematization of geometry and arithmetic as a priori sciences. In Kant’s view, an arithmetic statement such as 5+7=12 is an a priori synthetic judgement, whereas from Frege’s standpoint, it’s an a priori analytic judgement. From Kant's outlook, the arithmetic statements are synthetic because they predicate something new that is not implicit in the subjects of the sentence. Frege's point of view is not only opposed to the Kantian one; it's also in contrast with intuitionism and with the above mentioned Piaget.
Figure 2: Bronze bust of Gottlob Frege (1848–1925) by Karl-Heinz Appelt (1940–2013). Image downloaded from Wikimedia Commons and edited by cantorsparadise.com
The meta-mathematical debate surely didn’t end with Frege; he was rather just the beginner of foundational research. During the period of the intuitionist school, which arose in the first half of the 20th century, ideas that were opposed to the logicist ones emerged. In fact, from the intuitionist perspective, the mathematical concepts can’t be reduced to logical ones. More specifically, the basilar intuition is an unlimited source for the construction of mathematical buildings. The internal experience is irreducible and the formal language is not the base for the staple notions of mathematics, it’s rather a technique for the communication of these concepts.
Whether we agree with logicists or intuitionists, what’s clear is that the first component of the mathematical structure resides in arithmetic. In order to understand how mathematical learning functions, the pursuit should start with how arithmetic innate abilities work in the human mind.
One of the very first psychologists who studied the functioning of mathematical learning was Jean Piaget (1896–1890). Against the logicist school, Piaget argued that the nature of a number can not be reduced to logical concepts, since it needs the notions of both "class" and "relation". In mathematics, a class is a collection of objects while the relation is a law that associates some elements of an A set to one or more elements in the B sets. From Piaget's outlook, these concepts are examples of the irreducibility of mathematics, which he later on also proved with experiments (Piaget J., Szeminska A., 1967). In the psychologist’s view, logic and arithmetic are co-dependent and irreducible. This standpoint is mirrored in two phases that Piaget traced in the development of a child: a pre-logic stage and a pre-numeric one. The two periods were indissoluble and formed one single "module".
Figure 3: Elementary arithmetic operations on paper. Photo by Chris Liverani on Unsplash
In the contemporary study of mathematical learning, a major contribution has been given by the neuroscientists Stanislav Dehaene and Brian Butterworth. If Piaget looked at the arithmetic comprehension as inextricably linked to the logic one, the two scientists put an effort to isolate the numerical cognition in order to understand its mechanisms. The two main innate abilities of the human brain that approach numbers are 'subitizing' and 'approximation'. Subitizing is the immediate recognition of a small quantity of elements without counting them. The term originates from Latin (subitus) which means 'immediate'. The time employed by the brain to recognize a number of elements under three is not linear and this suggests that those units are not counted but just instantaneously recognized. Approximation, on the other hand, is the innate ability to make a rough estimate of a number of elements that is out of the subitizing range, so that is bigger than three.
Figure 4: Response times for the recognition of increasing quantities. Note: above number 3, the growth of the response time is linear. (Dehaene, “The number sense”, 1997, redrawn from Mandler and Shebo 1982)
Because both subitizing and approximation are innate abilities that can’t be learnt, it makes sense to investigate where their biological collocation is in the brain, although the information about this mapping is still incomplete. Subitizing depends on neuronal circuits in the parietal inferior cortex, in particular in the Brodmann Area 39. Injuries in the left parietal region cause acalculia, while the manipulation of numbers and arithmetic abilities depend on the activation of the intraparietal sulcus (T. Klingberg and M. Schel 2017; Dehaene 1997; Butterworth 1999). While acalculia is a disorder that is secondary to a brain injury, dyscalculia is a disability in which mathematics can still be learnt, but in a much more difficult and longer way. Dyscalculia is considered a genetic condition that concerns the 2.5% of the population, according to the data of the International Academy for Research in Learning Disabilities. For both teachers and scientists then, it should be interesting to understand why the mathematical learning disabilities are experienced by 20% of the children. In an experiment (2015, T. Iuculano et al.) that followed thirty children aged between seven and nine with MLD and without any other neurological or psychiatric conditions, the children showed a hyper-activation in many areas of the brain, particularly in the prefrontal and parietal cortex. The children were subjected to mathematical games with a 1:1 tutoring, meaning one tutor for every child. The experiment lasted eight weeks, and neuroscientists observed a relevant change in the neuroplastic configuration of children’s brains, that hyper-activation had been normalized. This adjustment made the students able to process the information in a gerarchic and organized way, which is essential for arithmetic problem solving. This is an example of how MLD could be challenged through education.
Figure 5: Children at school. Photo by Austrian National Library on Unsplash
The meta-mathematical foundational investigation started almost 150 years ago and breathed new life into the developmental psychology for the studying of the numerical cognition, which was then channeled into the contemporary neurosciences. The debate upon the basic concepts of mathematics defined and restricted the domain of research, which turned out to a better understanding of the collocation of the numerical cognition in the brain. The comprehension of the biological mechanisms of this kind of learning improves the teaching approach and gives the opportunity to understand how mathematical illiteracy can be challenged. This emphasis on education is an opportunity to consider the mathematical learning as an issue that can be improved through the environment and discourages the extremisms of a totally innatist hypothesis.
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Cover: Johns, Jaspers (n.d), Retrieved from http://scrogginsart.weebly.com/pre-k-and-k.html
Figure 1: Photo by Dan Cristian Pădureț on Unsplash
Figure 2: Image downloaded from Wikimedia Commons and edited by cantorsparadise.com
Figure 3: Photo by Chris Liverani on Unsplash
Figure 4: Picture taken by Dehaene, “The number sense”, 1997, p. 67, redrawn from Mandler and Shebo 1982
Figure 5: Children at school. Photo by Austrian National Library on Unsplash