Handbook of Mathematical Cognition

Journal of experimental child psychology, 2009

Despite the importance of mathematics in our educational systems, little is known about how abstract mathematical thinking emerges. Most research on mathematical cognition has been dedicated to understanding its more simple forms such as seriation and counting. Although these forms constitute the foundational plinth upon which all other maths skills develop, the gap between basic skills and the processing of complex mathematical concepts is poorly understood. What has come to be sufficiently well understood, however, is how numeracy is acquired. The 90s marked a change in our approach to human cognition in general and to mathematical cognition in particular. Neuroimaging technologies have enabled localization of neural activity, revealing that mathematical cognition, like other forms of cognition and skills, depends upon a network of activation. The key finding from neuroimaging and single cell recording is that numerical information is held in the intraparietal sulcus. Now that the core of mathematical cognition has been identified it is time to understand how basic skills are used to support the acquisition and use of abstract mathematical concepts. Chassy and Grodd (2012) opened the door for abstract mathematical cognition by examining for the first time the neural correlates of negative numbers, an abstract mathematical concept that emerges early on in mathematical curricula. The present issue reports crucial advances in our understanding of the neural underpinnings of abstract mathematical cognition. For a general introduction to the topic the reader is referred to the article signed by Moeller et al. The article offers an excellent overview of the networks that are involved to some degree in processing quantities, the very basis of mathematical cognition. The authors' conclusion strengthens the view that a frontal parietal network constitutes the essence of our abilities in mathematics. The fronto-parietal network has been highlighted by a number of studies and is thought to underpin the learning of mathematical concepts. By increasing the complexity of the concepts stored in our memory, we improve the quality of our understanding of the physical world in the first stages of mathematical cognition. Abstract concepts are then able to emerge from concrete, physical quantities. On the path of mathematical development, the first step toward an abstract representation of concepts is the shift from concrete, object-based cognition to the use of symbols. The symbols, though arbitrary, represent concrete quantities that help children quantify and thus understand the world around them. Roesch and Moeller support this view by suggesting that an internal representation of fingers contributes to the actual ability to represent quantities. In a similar vein, a cross cultural study authored by Bender and Beller compares the Western counting system to a Polynesian language of the Tonga island, offering a unique view of how concrete counting of different objects leads to an abstract representation of numbers; thus demonstrating that the roots of abstract mathematical cognition emerge from basic, sensory abilities (a long standing view that finds a new echo here). By highlighting the concrete roots of mathematical cognition, the authors of these studies open the debate on the inheritance of mathematical skill by pointing toward very concrete sensory performance. The symbols in a later stage of mathematical development are used to represent concepts of an abstract nature. That is, once the notion of natural number is acquired, the next step toward expertise is to formalize operations as abstract entities. For example, the operation 5 + 4 = 9 is concrete and can be taught by using objects. Dowker demonstrates that pupils tend to use the same problem-solving strategies to solve problems in subtraction and addition problems. Since the properties of the two operations differ the application of the same strategy leads the pupil to commit errors. Pupils have to learn a new set of properties to be able to solve subtraction. Similarly, Huber et al. argue that mental representations of fractions do not differ from natural numbers; what do differ are the strategies used to encode information. Dowker's and Hubet et al.'s views are in line with the study of Mihulowicz et al. who, by comparing left and right lesioned patients, showed that arithmetic operations are underpinned by different networks. The view of some educators, that subtraction and addition are mirror operations, is mistaken. It is interesting to note that teaching might be adapted so that different approaches could be used to teach different operations. The studies highlight the fact that learning arithmetic includes knowledge that is not purely numerical. This is our first hint indicating that educational strategies might have a huge influence on the ability of students to learn abstract concepts. The next stage in mathematical learning is the step consisting in moving from concrete (arithmetic) to abstract (algebraic) relationships. A study by Susac et al. looked at this move and showed that it requires about 4 years of training to master this new step toward abstract thinking in mathematics. It is crucial to note that these 4 years are in addition to the many years required for correctly mastering the basics. Mathematical learning is a long road. It calls for pedagogical approaches that are specific to each level. Two main variables might modulate the acquisition of mathematical expertise: Educational system and inherited factors. The idea that teaching practices impact heavily on the ability of students to develop their skills in abstract mathematical cognition is demonstrated by Prado et al. The authors ran a cross cultural study comparing Chinese and American students on problem-size effects, and show that educational practices, which differ in the 2 countries, impact on the wiring of the network in charge of symbolic arithmetic. In line with this result, McLean and Rusconi attempt to bridge the gap between the findings of academic science and the practical problems faced by teaching institutions when dealing with students with mathematical difficulties. After revealing the cognitive factors underpinning the acquisition of mathematical knowledge, McLean and Rusconi discuss the types of interventions that may help students with mathematical difficulties. With respect to inherited factors, Zhang et al. have shown that gifted adolescents display a highly integrated fronto-parietal network, hence displaying a more efficient link between the representation of numbers in the parietal cortex and working memory in the prefrontal cortex. The many findings of the articles in this special topic call for further research to see how specific neural networks serve various abstract mathematical concepts.

Cortex

From our very early school years we start to realize that numbers govern much of our life. A glance at the headlines will tell us a crucial parliamentary bill was defeated by 149 votes, that inflation is steady at .9%, that the GNP has declined by 1% and so on. A flick of our telephone gives us the time (in digits) and date, the telephone numbers of our friends, with apps to furnish our bank balance, and how many steps we have made today. However, these symbolic representations of quantity, usually by Arabic numerals, capture only a small fragment of our daily experience with numerable quantities, and how these quantities guide our behaviour, and the ways we exploit our inner ability to "sense" the numerosity of these quantities. By showing that birds can perform both simultaneous visuo-spatial and temporal-sequential coding of the numerosity of simple visual items (clouds of dots), the German zoologist Otto Koehler (1941; 1950) was among the first to suggest that the symbolic mathematical competence that characterises much human activity might be grounded in phylogenetically older systems that allow approximate, but behaviourally adaptive, estimates of numerosity. During biological evolution these rudimentary mathematical abilities might have been crucial for survival and adaptation by allowing, for example, the recognition and memorization of environments with more or fewer food items, or by favouring

Trends in Neurosciences, 1998

). The number domain is a prime example where strong evidence points to an evolutionary endowment of abstract domain-specific knowledge in the brain because there are parallels between number processing in animals and humans.The numerical distance effect, which refers to the finding that the ability to discriminate between two numbers improves as the numerical distance between them increases, has been demonstrated in humans and animals, as has the number size effect, which refers to the finding that for equal numerical distance,discrimination of two numbers worsens as their numerical size increases.

Annual Review of Neuroscience, 2009

Number symbols have allowed humans to develop superior mathematical skills that are a hallmark of technologically advanced cultures. Findings in animal cognition, developmental psychology, and anthropology indicate that these numerical skills are rooted in nonlinguistic biological primitives. Recent studies in human and nonhuman primates using a broad range of methodologies provide evidence that numerical information is represented and processed by regions of the prefrontal and posterior parietal lobes, with the intraparietal sulcus as a key node for the representation of the semantic aspect of numerical quantity.

Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities that are subject to mathematical ruminations—they are represented, used, embodied, and manipulated in order to achieve many different goals, e.g., to count or denote the size of a collection of ob jects, to trade goods, to balance bank accounts, or to play the lottery. Consequently, numbers are both abstract and intimately connected to language and to our interactions with the world. In the present paper we provide an overview of research that has addressed the question of how animals and humans learn, represent, and process numbers.

2009

During the last three decades, a large body of research was devoted to analyzing number representation in humans. The findings in neuroscience and their adequate interpretation in relation with cognition may reshape the traditional ways of teaching and learning. The paper synthesizes the following achievements of research on human cognition: there is a complex relationship mind-andbrain; mathematical tasks activate some specific cortex zones; human beings possess inborn numerical predispositions independent of language; language plays a scaffolding role in developing the computational capacities; there are cultural tools developed by humans based on innate predispositions and environmental interactions. Educational consequences derived from here might be: giving more importance to some capacities that traditionally are neglected in teaching, such as estimations and approximations, which facilitate access to inborn predispositions and connect to real life; taking into account, especially in young ages, the natural predispositions and building up knowledge starting from these; developing a dynamic training that stimulates connections, relations, semantic associations.

Trends in Cognitive Sciences, 2000

2 , F e b r u a r y 2 0 0 0 monkeys (Macaca mulatta). J. Comp. Psychol. 111, 286-293 44 Itakura, S. and Tanaka, M. (1998) Use of experimenter-given cues during object-choice tasks by chimpanzees (Pan troglodytes), an orangutan (Pongo pygmaeus), and human infants (Homo sapiens).

Mende MA, Shaki S and Fischer MH (2018) Commentary: The mental representation of integers: An abstract-to-concrete shift in the understanding of mathematical concepts. Front. Psychol. 9:209. doi: 10.3389/fpsyg.2018.00209, 2018

Decision times during processing of positive number symbols (1, 2, 3 etc.) inform our understanding of mental representations of integers (Holyoak, 1978; Dehaene et al., 1993; Fischer and Shaki, 2014). Effects of number magnitude on cognition include distance effects (faster discrimination for larger numerical differences in a number pair), size effects (faster processing of smaller numbers), Spatial-Numerical Association of Response Codes (SNARC; faster left/right responses to small/large numbers), linguistic markedness (MARC; faster left/right responses to odd/even numbers) and semantic congruity effects (faster smaller/larger decisions over smaller/larger number pairs). Results converge on the notion of a spatially oriented mental number line (MNL) where numerically smaller number concepts exist to the left of larger number concepts. How do these performance signatures help us to understand the cognitive representation of negative number symbols (−1, −2, −3 etc.)? Unlike natural number symbols, negative number symbols lack corresponding real entities that support sensory-motor learning. We discuss a recent proposal by Varma and Schwartz (2011) with implications for developmental research.

Frontiers in psychology, 2017