# Bibliography

Alvarez, G. A., & Cavanagh, P. (2004). The capacity of visual short-term memory is set both by visual information load and by number of objectsPsychological science15(2), 106-111.

ABSTRACT: Previous research has suggested that visual short term memory has a fixed capacity of about four objects. However, we found that capacity varied substantially across the five stimulus classes we examined, ranging from 1.6 for shaded cubes to 4.4 for colors (estimated using a change detection task). We also estimated the information load per item in each class, using visual search rate. The changes we measured in memory capacity across classes were almost exactly mirrored by changes in the opposite direction in visual search rate ($r^2 = .992$ between search rate and the reciprocal of memory capacity). The greater the information load of each item in a stimulus class (as indicated by a slower search rate), the fewer items from that class one can hold in memory. Extrapolating this linear relationship reveals that there is also an upper bound on capacity of approximately four or five objects. Thus, both the visual information load and number of objects impose capacity limits on visual short-term memory.

Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematiciansProceedings of the National Academy of Sciences113(18), 4909-4917.

ABSTRACT: The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.

Amalric, M., & Dehaene, S. (2018). Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain’s semantic networks. Philosophical Transactions of the Royal Society B: Biological Sciences, 373(1740), 20160515.

ABSTRACT: Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that ‘the sine function is periodical’ or that ‘London buses are red’, activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders. This article is part of a discussion meeting issue ‘The origins of numerical abilities’.

Amalric, M., & Dehaene, S. (2019). A distinct cortical network for mathematical knowledge in the human brainNeuroImage189, 19-31.

ABSTRACT: How does the brain represent and manipulate abstract mathematical concepts? Recent evidence suggests that mathematical processing relies on specific brain areas and dissociates from language. Here, we investigate this dissociation in two fMRI experiments in which professional mathematicians had to judge the truth value of mathematical and nonmathematical spoken statements. Sentences with mathematical content systematically activated bilateral intraparietal sulci and inferior temporal regions, regardless of math domain, problem difficulty, and strategy for judging truth value (memory retrieval, calculation or mental imagery). Second, classical language areas were only involved in the parsing of both nonmathematical and mathematical statements, and their activation correlated with syntactic complexity, not mathematical content. Third, the mere presence, within a sentence, of elementary logical operators such as quantifiers or negation did not suffice to activate math-responsive areas. Instead, quantifiers and negation impacted on activity in right angular gyrus and left inferior frontal gyrus, respectively. Overall, these results support the existence of a distinct, non-linguistic cortical network for mathematical knowledge in the human brain.

Amalric, M., Denghien, I., & Dehaene, S. (2018). On the role of visual experience in mathematical development: Evidence from blind mathematiciansDevelopmental cognitive neuroscience30, 314-323.

ABSTRACT: Advanced mathematical reasoning, regardless of domain or difficulty, activates a reproducible set of bilateral brain areas including intraparietal, inferior temporal and dorsal prefrontal cortex. The respective roles of ge- netics, experience and education in the development of this math-responsive network, however, remain un- resolved. Here, we investigate the role of visual experience by studying the exceptional case of three professional mathematicians who were blind from birth (n = 1) or became blind during childhood (n = 2). Subjects were scanned with fMRI while they judged the truth value of spoken mathematical and nonmathematical statements. Blind mathematicians activated the classical network of math-related areas during mathematical reflection, si-milar to that found in a group of sighted professional mathematicians. Thus, brain networks for advanced mathematical reasoning can develop in the absence of visual experience. Additional activations were found in occipital cortex, even in individuals who became blind during childhood, suggesting that either mental imagery or a more radical repurposing of visual cortex may occur in blind mathematicians.

Arsalidou, M., Pawliw-Levac, M., Sadeghi, M., & Pascual-Leone, J. (2018). Brain areas associated with numbers and calculations in children: Meta-analyses of fMRI studiesDevelopmental cognitive neuroscience30, 239-250.

ABSTRACT: Children use numbers every day and typically receive formal mathematical training from an early age, as it is a main subject in school curricula. Despite an increase in children neuroimaging studies, a comprehensive neuropsychological model of mathematical functions in children is lacking. Using quantitative meta-analyses of functional magnetic resonance imaging (fMRI) studies, we identify concordant brain areas across articles that adhere to a set of selection criteria (e.g., whole-brain analysis, coordinate reports) and report brain activity to tasks that involve processing symbolic and non-symbolic numbers with and without formal mathematical operations, which we called respectively number tasks and calculation tasks. We present data on children 14 years and younger, who solved these tasks. Results show activity in parietal (e.g., inferior parietal lobule and precuneus) and frontal (e.g., superior and medial frontal gyri) cortices, core areas related to mental-arithmetic, as well as brain regions such as the insula and claustrum, which are not typically discussed as part of mathematical problem solving models. We propose a topographical atlas of mathematical processes in children, discuss findings within a developmental constructivist theoretical model, and suggest practical methodological considerations for future studies.

Artemenko, C., Soltanlou, M., Ehlis, A. C., Nuerk, H. C., & Dresler, T. (2018). The neural correlates of mental arithmetic in adolescents: a longitudinal fNIRS studyBehavioral and Brain Functions14(1), 5.

ABSTRACT: Arithmetic processing in adults is known to rely on a frontal-parietal network. However, neurocognitive research focusing on the neural and behavioral correlates of arithmetic development has been scarce, even though the acquisition of arithmetic skills is accompanied by changes within the fronto-parietal network of the developing brain. Furthermore, experimental procedures are typically adjusted to constraints of functional magnetic resonance imaging, which may not reflect natural settings in which children and adolescents actually perform arithmetic. Therefore, we investigated the longitudinal neurocognitive development of processes involved in performing the four basic arithmetic operations in 19 adolescents. By using functional near-infrared spectroscopy, we were able to use an ecologically valid task, i.e., a written production paradigm. A common pattern of activation in the bilateral fronto-parietal network for arithmetic processing was found for all basic arithmetic operations. Moreover, evidence was obtained for decreasing activation during subtraction over the course of 1 year in middle and inferior frontal gyri, and increased activation during addition and multiplication in angular and middle temporal gyri. In the self-paced block design, parietal activation in multiplication and left angular and temporal activation in addition were observed to be higher for simple than for complex blocks, reflecting an inverse effect of arithmetic complexity. In general, the findings suggest that the brain network for arithmetic processing is already established in 12–14 year-old adolescents, but still undergoes developmental changes.

Badre, D., & Nee, D. E. (2018). Frontal cortex and the hierarchical control of behavior. Trends in cognitive sciences22(2), 170-188.

ABSTRACT: The frontal lobes are important for cognitive control, yet their functional organization remains controversial. An influential class of theory proposes that the frontal lobes are organized along their rostrocaudal axis to support hierarchical cognitive control. Here, we take an updated look at the literature on hierarchical control, with particular focus on the functional organization of lateral frontal cortex. Our review of the evidence supports neither a unitary model of lateral frontal function nor a unidimensional abstraction gradient. Rather, separate frontal networks interact via local and global hierarchical structure to support diverse task demands.

Bonnefond, M., Kastner, S., & Jensen, O. (2017). Communication between brain areas based on nested oscillations. eneuro4(2).

SIGNIFICANCE: To understand how the brain operates as a network, it is essential to identify the mechanisms supporting communication between brain regions. Based on recent empirical findings, we propose a mechanism for
selective routing based on cross-frequency coupling between slow oscillations in the alpha and gamma bands.

ABSTRACT: Unraveling how brain regions communicate is crucial for understanding how the brain processes external and internal information. Neuronal oscillations within and across brain regions have been proposed to play a crucial role in this process. Two main hypotheses have been suggested for routing of information based on oscillations, namely communication through coherence and gating by inhibition. Here, we propose a framework unifying these two hypotheses that is based on recent empirical findings. We discuss a theory in which communication between two regions is established by phase synchronization of oscillations at lower frequencies (25 Hz), which serve as temporal reference frame for information carried by high-frequency activity (40 Hz). Our framework, consistent with numerous recent empirical findings, posits that cross-frequency interactions are essential for understanding how large-scale cognitive and perceptual networks operate.

Bull, R., Espy, K. A., & Wiebe, S. A. (2008). Short-term memory, working memory, and executive functioning in preschoolers: Longitudinal predictors of mathematical achievement at age 7 yearsDevelopmental neuropsychology33(3), 205-228.

ABSTRACT: This study examined whether measures of short-term memory, working memory, and executive functioning in preschool children predict later proficiency in academic achievement at 7 years of age (third year of primary school). Children were tested in preschool (M age = 4 years, 6 months) on a battery of cognitive measures, and mathematics and reading outcomes (from standardized, norm- referenced school-based assessments) were taken on entry to primary school, and at the end of the first and third year of primary school. Growth curve analyses examined predictors of math and reading achievement across the duration of the study and revealed that better digit span and executive function skills provided children with an immediate head start in math and reading that they maintained throughout the first three years of primary school. Visual-spatial short-term memory span was found to be a predictor specifically of math ability. Correlational and regression analyses revealed that visual short-term and working memory were found to specifically predict math achievement at each time point, while executive function skills predicted learning in general rather than learning in one specific domain. The implications of the findings are discussed in relation to further understanding the role of cognitive skills in different mathematical tasks, and in relation to the impact of limited cognitive skills in the classroom environment.

Butterworth, B. (1999). What counts: How every brain is hardwired for math. The Free Press.

KIRKUS REVIEW: A neuropsychologist (University College, London) argues that the ability to do math is inborn, not learned. Butterworth proposes a “number module” in the brain, containing the ability to count and to understand numbers. The evidence for this is drawn from history, animal studies, infant learning, and an impressive range of other disciplines. While few of us are professional mathematicians, numbers are an inescapable feature of everyone’s life: grocery prices, phone numbers, children’s ages, sports scores, speed limits, interest rates, and many other examples. The ability to use these numbers on some basic level appears to be as widespread as the ability to use language; yet the two appear not to be directly related. Number systems were developed independently in several parts of the world, and there are marked differences between them; the Babylonians used a base of 60, the Mayans one of 20, as counterexamples to the 10-based math Western cultures use. This argues against some single prehistoric genius having come up with an idea that then diffused to other cultures. In fact, the ability to distinguish between quantities and to perform primitive calculation seems inherent in infants and even in those animals and birds that have been tested. Studies of stroke patients who have lost their math ability indicates that key mathematical functions reside in the left parietal lobe of the brain. A fascinating chapter on the history of various methods of counting on fingers (or other body parts) shows a similar relationship between another specific brain region and math ability. Other chapters explore the question of why some of us are particularly good or bad at math and the ways that children learn math at home, on the streets, and in school. Butterworth writes clearly and entertainingly, with plenty of examples drawn from everyday life and flashes of humor that belie the notion that math is a dry subject. A pioneering study of a fascinating area of the human mind.

Davis, G. E., & McGowen, M. A. (2004). A Rush of Connections and Insights, a Glorious Moment of ClarityNetworks: An Online Journal for Teacher Research7(2), 116-116.

INTRODUCTION: We examine in some detail a deep example of productive long-term memory for mathematics. This example occurred during our regular teaching sessions and was noticed by us because of our focus on how memory formation occurs in classroom settings. It was something we noticed explicitly because of our theoretical orientation to memory. The student, Sandra, who exhibited this productive memory was enrolled in a mathematics methods course for pre-service elementary teachers. Her experiences and written reflections during the semester indicate that she imagined herself to have a particularly bad memory for mathematics. This focus, on what pre-service elementary teachers imagine mathematics is about and what needs to be remembered is typical in our experience … Prior to our focus on memory, our teaching was concerned with presenting mathematical material in a logically coherent manner. For example, in teaching college level mathematics we would, when teaching elementary descriptive statistics, first introduce the idea of a function as an organizing idea  and then introduce the average as a particular type of function. This sort of teaching is powerful because it introduces concepts, such as that of function, as a general organizing principal for a lot of other mathematics. With our focus on memorable episodes, however, we wanted students to begin to look more deeply and to see connections where previously they saw none. So the sequence of tasks we present in the next section was designed to explicitly provoke, without our telling, students’ understanding of the connectedness of many parts of mathematics. Their ability to express these connections, we argued, would be evidence for long-term powerful memory for mathematics in a way that students had not encountered before.

Dehaene, S. (1992). Varieties of numerical abilitiesCognition44(1-2), 1-42.

ABSTRACT: This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.

Dehaene, S. (2010). The calculating brain. Mind, brain and education: Neuroscience implications for the classroom, 179-200.

Dehaene, S. (2011). The number sense: How the mind creates mathematics. OUP USA.

GOODREADS REVIEW: The Number Sense is an enlightening exploration of the mathematical mind. Describing experiments that show that human infants have a rudimentary number sense, Stanislas Dehaene suggests that this sense is as basic as our perception of color, and that it is wired into the brain. Dehaene shows that it was the invention of symbolic systems of numerals that started us on the climb to higher mathematics. A fascinating look at the crossroads where numbers and neurons intersect, The Number Sense offers an intriguing tour of how the structure of the brain shapes our mathematical abilities, and how our mathematics opens up a window on the human mind

Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processingMathematical cognition1(1), 83-120.

ABSTRACT: A model is prosed for the mental processes and neuroanatomical circuits involved in number processing and mental arithmetic. The model elaborates on Dehaene’s (1992) triple-code model and assumes that arabic and magnitude representations of numbers are available to both hemispheres, but that the verbal representation that underlies arithmetic fact retrieval is available only to the left hemisphere. Speculations as to the anatomical substrates and connections of these representations are proposed. We review a large number of single-case studies of acalculia and show that our model predicts in some detail the nature of the cognitive impairment in relation to the site of the lesion. The compatibility of our views with recent brain imaging studies of number processing in normal subjects is also examined.

Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science284(5416), 970-974.

ABSTRACT: Does the human capacity for mathematical intuition depend on linguistic competence or on visuo-spatial representations? A series of behavioral and brain-imaging experiments provides evidence for both sources. Exact arithmetic is acquired in a language-specific format, transfers poorly to a different language or to novel facts, and recruits networks involved in word-association processes. In contrast, approximate arithmetic shows language independence, relies on a sense of numerical magnitudes, and recruits bilateral areas of the parietal lobes involved in visuo-spatial processing. Mathematical intuition may emerge from the interplay of these brain systems.

Devlin, K. (2010). The mathematical brain. Mind, brain and education: Neuroscience implications for the classroom, 163-178.

Eger, E. (2016). Neuronal foundations of human numerical representations. In Progress in brain research (Vol. 227, pp. 1-27). Elsevier.

ABSTRACT: The human species has developed complex mathematical skills which likely emerge from a combination of multiple foundational abilities. One of them seems to be a preverbal capacity to extract and manipulate the numerosity of sets of objects which is shared with other species and in humans is thought to be integrated with symbolic knowledge to result in a more abstract representation of numerical concepts. For what concerns the functional neuroanatomy of this capacity, neuropsychology and functional imaging have localized key substrates of numerical processing in parietal and frontal cortex. However, traditional fMRI mapping relying on a simple subtraction approach to compare numerical and nonnumerical conditions is limited to tackle with sufficient precision and detail the issue of the underlying code for number, a question which more easily lends itself to investigation by methods with higher spatial resolution, such as neurophysiology. In recent years, progress has been made through the introduction of approaches sensitive to within-category discrimination in combination with fMRI (adaptation and multivariate pattern recognition), and the present review summarizes what these have revealed so far about the neural coding of individual numbers in the human brain, the format of these representations and parallels between human and monkey neurophysiology findings.

Helfrich, R. F., & Knight, R. T. (2016). Oscillatory dynamics of prefrontal cognitive control. Trends in cognitive sciences, 20(12), 916-930.

ABSTRACT: The prefrontal cortex (PFC) provides the structural basis for numerous higher cognitive functions. However, it is still largely unknown which mechanisms provide the functional basis for flexible cognitive control of goal-directed behavior. Here, we review recent findings, which suggest that the functional architecture of cognition is profoundly rhythmic and propose that the PFC serves as a conductor to orchestrate task-relevant large-scale networks. We highlight several studies that demonstrated that oscillatory dynamics, such as phase resetting, cross-frequency coupling and entrainment, support PFC-dependent recruitment of task-relevant regions into coherent functional networks. Importantly, these findings support the notion that distinct spectral signatures reflect different cortical computations supporting effective multiplexing on different temporal channels along the same anatomical pathways.

Jacob, S. N., Vallentin, D., & Nieder, A. (2012). Relating magnitudes: the brain’s code for proportionsTrends in cognitive sciences16(3), 157-166.

ABSTRACT: Whereas much is known about how we categorize and reason based on absolute quantity, data exploring ratios of quantities, as in proportions and fractions, are com- paratively sparse. Until recently, it remained elusive whether these two representations of number are connected, how proportions are implemented by neurons and how language shapes this code. New data derived with complementary methods and from different model systems now shed light on the mechanisms of magnitude ratio representations. A coding scheme for proportions has emerged that is remarkably reminiscent of the representation of absolute number. These novel findings suggest a sense for ratios that grants the brain automat- ic access to proportions independently of language and the format of presentation.

Liang, Y., Liu, X., Qiu, L., & Zhang, S. (2018). An EEG study of a confusing state induced by information insufficiency during mathematical problem-solving and reasoningComputational intelligence and neuroscience2018.

Nieder, A. (2016). The neuronal code for numberNature Reviews Neuroscience17(6), 366.

ABSTRACT: Humans and non-human primates share an elemental quantification system that resides in a dedicated neural network in the parietal and frontal lobes. In this cortical network, ‘number neurons’ encode the number of elements in a set, its cardinality or numerosity, irrespective of stimulus appearance across sensory motor systems, and from both spatial and temporal presentation arrays. After numbers have been extracted from sensory input, they need to be processed to support goal-directed behaviour. Studying number neurons provides insights into how information is maintained in working memory and transformed in tasks that require rule-based decisions. Beyond an understanding of how cardinal numbers are encoded, number processing provides a window into the neuronal mechanisms of high-level brain functions.

Peters, L., & De Smedt, B. (2018). Arithmetic in the developing brain: A review of brain imaging studiesDevelopmental Cognitive Neuroscience30, 265-279.

ABSTRACT: Brain imaging studies on academic achievement offer an exciting window on experience-dependent cortical plasticity, as they allow us to understand how developing brains change when children acquire culturally transmitted skills. This contribution focuses on the learning of arithmetic, which is quintessential to mathematical development. The nascent body of brain imaging studies reveals that arithmetic recruits a large set of interconnected areas, including prefrontal, posterior parietal, occipito-temporal and hippocampal areas. This network undergoes developmental changes in its function, connectivity and structure, which are not yet fully understood. This network only partially overlaps with what has been found in adults, and clear differences are observed in the recruitment of the hippocampus, which are related to the development of arithmetic fact retrieval. Despite these emerging trends, the literature remains scattered, particularly in the context of atypical development. Acknowledging the distributed nature of the arithmetic network, future studies should focus on connectivity and analytic approaches that investigate patterns of brain activity, coupled with a careful design of the arithmetic tasks and assessments of arithmetic strategies. Such studies will produce a more comprehensive understanding of how the arithmetical brain unfolds, how it changes over time, and how it is impaired in atypical development.

Pinheiro-Chagas, P., Daitch, A., Parvizi, J., & Dehaene, S. (2018). Brain mechanisms of arithmetic: A crucial role for ventral temporal cortexJournal of cognitive neuroscience30(12), 1757-1772.

ABSTRACT: Elementary arithmetic requires a complex interplay between several brain regions. The classical view, arising from fMRI, is that the intraparietal sulcus (IPS) and the superior parietal lobe (SPL) are the main hubs for arithmetic calculations. However, recent studies using intracranial electroencephalography have discovered a specific site, within the posterior inferior temporal cortex (pITG), that activates during visual perception of numerals, with widespread adjacent responses when numerals are used in calculation. Here, we reexamined the contribution of the IPS, SPL, and pITG to arithmetic by recording intracranial electroencephalography signals while participants solved addition problems. Behavioral results showed a classical problem size effect: RTs increased with the size of the operands. We then examined how high-frequency broadband (HFB) activity is modulated by problem size. As expected from previous fMRI findings, we showed that the total HFB activity in IPS and SPL sites increased with problem size. More surprisingly, pITG sites showed an initial burst of HFB activity that decreased as the operands got larger, yet with a constant integral over the whole trial, thus making these signals invisible to slow fMRI. Although parietal sites appear to have a more sustained function in arithmetic computations, the pITG may have a role of early identification of the problem difficulty, beyond merely digit recognition. Our results ask for a reevaluation of the current models of numerical cognition and reveal that the ventral temporal cortex contains regions specifically engaged in mathematical processing.

Polspoel, B., Peters, L., Vandermosten, M., & De Smedt, B. (2017). Strategy over operation: Neural activation in subtraction and multiplication during fact retrieval and procedural strategy use in children. Human brain mapping38(9), 4657-4670.

ABSTRACT: Arithmetic development is characterized by strategy shifts between procedural strategy use and fact retrieval. This study is the first to explicitly investigate children’s neural activation associated with the use of these different strategies. Participants were 26 typically developing 4th graders (9- to 10-year-olds), who, in a behavioral session, were asked to verbally report on a trial-by-trial basis how they had solved 100 subtraction and multiplication items. These items were subsequently presented during functional magnetic resonance imaging. An event-related design allowed us to analyze the brain responses during retrieval and procedural trials, based on the children’s verbal reports. During procedural strategy use, and more specifically for the decomposition of operands strategy, activation increases were observed in the inferior and superior parietal lobes (intraparietal sulci), inferior to superior frontal gyri, bilateral areas in the occipital lobe, and insular cortex. For retrieval, in comparison to procedural strategy use, we observed increased activity in the bilateral angular and supramarginal gyri, left middle to inferior temporal gyrus, right superior temporal gyrus, and superior medial frontal gyrus. No neural differences were found between the two operations under study. These results are the first in children to provide direct evidence for alternate neural activation when different arithmetic strategies are used and further unravel that previously found effects of operation on brain activity reflect differences in arithmetic strategy use.

Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010). Working memory and mathematics: A review of developmental, individual difference, and cognitive approachesLearning and individual differences20(2), 110-122.

ABSTRACT: Working memory refers to a mental workspace, involved in controlling, regulating, and actively maintaining relevant information to accomplish complex cognitive tasks (e.g. mathematical processing). Despite the potential relevance of a relation between working memory and math for understanding developmental and individual differences in mathematical skills, the nature of this relationship is not well-understood. This paper reviews four approaches that address the relation of working memory and math: 1) dual task studies establishing the role of working memory during on-line math performance; 2) individual difference studies examining working memory in children with math difficulties; 3) studies of working memory as a predictor of mathematical outcomes; and 4) longitudinal studies of working memory and math. The goal of this review is to evaluate current information on the nature of the relationship between working memory and math provided by these four approaches, and to present some of the outstanding questions for future research.

Sousa, D. A. (2014). How the brain learns mathematics. Corwin Press.

GOODREADS REVIEW: Text examines: Children’s innate number sense and how the brain develops an understanding of number relationships; Rationales for modifying lessons to meet the developmental learning stages of young children, preadolescents, and adolescents; How to plan lessons in PreK-12 mathematics; Implications of current research for planning mathematics lessons, including discoveries about memory systems and lesson timing; Methods to help elementary and secondary school teachers detect mathematics difficulties; Clear connections to the NCTM standards and curriculum focal points.

Taylor, S., & Egeto-Szabo, K. (2017). EXPLORING AWAKENING EXPERIENCES: A STUDY OF AWAKENING EXPERIENCES IN TERMS OF THEIR TRIGGERS, CHARACTERISTICS, DURATION AND AFTER EFFECTS. Journal of Transpersonal Psychology49(1).

ABSTRACT: Awakening experiences are temporary experiences of an intensification and expansion of awareness, with characteristics such as intensified perception, a sense of connection and well-being. Ninety awakening experiences were collected and thematically analysed to identify their triggers and characteristics, and also their duration and after-effects. Four main triggers of awakening experiences were found: psychological turmoil, contact with nature, spiritual practice and engagement with spiritual literature (or audio or video materials). Characteristics were found to be positive affective states, intensified perception, love and compassion, a transcendence of separateness, a sense of revelation and inner quietness. The duration of the majority of experiences was from a few minutes to a few hours. The most prevalent after-effects were a desire to recapture the experience and a shift in perspectives and values. The study confirms the importance of psychological turmoil in generating awakening experiences, and that most awakening experiences occur spontaneously, outside the context of spiritual practices and traditions.

Thurston, W. P. (1998). On proof and progress in mathematics. New directions in the philosophy of mathematics, 337-355.

INTRODUCTION: This essay on the nature of proof and progress in mathematics was stimulated by the article of Jaffe and Quinn, “Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics”. Their article raises interesting issues that mathematicians should pay more attention to, but it also perpetuates some widely held beliefs and attitudes that need to be questioned and examined. The article had one paragraph portraying some of my work in a way that diverges from my experience, and it also diverges from the observations of people in the field whom I’ve discussed it with as a reality check. After some reflection, it seemed to me that what Jaffe and Quinn wrote was an example of the phenomenon that people see what they are tuned to see. Their portrayal of my work resulted from projecting the sociology of mathematics onto a one-dimensional scale (speculation versus rigor) that ignores many basic phenomena. Responses to the Jaffe-Quinn article have been invited from a number of mathematicians, and I expect it to receive plenty of specific analysis and criticism from others. Therefore, I will concentrate in this essay on the positive rather than on the contranegative. I will describe my view of the process of mathematics, referring only occasionally to Jaffe and Quinn by way of comparison.

Voytek, B., Kayser, A. S., Badre, D., Fegen, D., Chang, E. F., Crone, N. E., Parvizi, J., Knight, R.T. & D’esposito, M. (2015). Oscillatory dynamics coordinating human frontal networks in support of goal maintenanceNature neuroscience, 18(9), 1318.

ABSTRACT: Humans have a capacity for hierarchical cognitive control—the ability to simultaneously control immediate actions while holding more abstract goals in mind. Neuropsychological and neuroimaging evidence suggests that hierarchical cognitive control emerges from a frontal architecture whereby prefrontal cortex coordinates neural activity in the motor cortices when abstract rules are needed to govern motor outcomes. We utilized the improved temporal resolution of human intracranial electrocorticography to investigate the mechanisms by which frontal cortical oscillatory networks communicate in support of hierarchical cognitive control. Responding according to progressively more abstract rules resulted in greater frontal network theta phase encoding (4–8 Hz) and increased prefrontal local neuronal population activity (high gamma amplitude, 80–150 Hz), which predicts trialby-trial response times. Theta phase encoding coupled with high gamma amplitude during inter-regional information encoding, suggesting that inter-regional phase encoding is a mechanism for the dynamic instantiation of complex cognitive functions by frontal cortical subnetworks.