Many species can discriminate nonsymbolic quantities, but only humans transform such perceptual magnitudes into symbolic numbers and mathematics. The process by which this transformation occurs remains debated. Here, we propose a computational model of human quantity estimation which synthesizes four general concepts established over the history of cognitive science: (1) nonsymbolic quantity representations; (2) a general capacity for comparative judgment; (3) associative memory; and (4) an anchoring-and-adjustment mechanism. We tested model predictions in preregistered experiments where participants estimated briefly displayed dot arrays. The model reproduced key psychophysical signatures of human quantity estimation across stimulus durations (100, 500, and 1000 ms) and replicated human calibration effects induced by exposure to veridical reference quantities. The results suggest that the psychophysics of symbolic estimation observed in humans can be explained by an extension of the evolutionarily ancient capacity for quantity discrimination.

Perceiving the number of objects in a scene is fundamental to human behaviour and mathematics. Standard numerosity experiments assume that brief stimuli block higher cognitive processes, implying that the signature characteristics of human numerosity estimation—such as exact enumeration of small numbers, underestimation of large numbers, and scalar variability (Weber’s law)—arise from specialised sensory representations. Here we challenge this century-old assumption and find that it does not hold. Combining qualitative, correlational, and experimental methods, we provide evidence that several estimation signatures, previously attributed to specialised sensory representations, likely arise from general cognitive processes. We conducted standard numerosity estimation experiments with brief (100 ms) dot patterns and visual masking. In interviews, participants reported using common cognitive processes, including comparing to previous examples and counting from visual memory. Quantitative surveys confirmed that the reported processes correlated with their expected estimation signatures. Preregistered experiments further validated several predictions for how the processes affect their corresponding signatures. These results call for a re-evaluation of current models of numerical cognition and the development of new experimental methods to better understand how numerosity is represented and processed in sensory circuits.

How to influence and assess whether students engage in conceptual thinking are longstanding methodological problems in mathematics education. Recently, eye-tracking technology has fueled a discussion on whether eye movement analysis can support valid inferences about mathematical thinking. This study investigates whether eye movement analysis can distinguish between conceptual and non-conceptual thinking in a geometric classification task where both modes of thinking lead to identical responses. Participants were asked to classify geometric shapes while we tracked their eye movements and to report their thinking verbally. Our findings indicate that self-reported conceptual thinking is characterised by fewer eye movements between task shapes and response shapes, and that self-reported non-conceptual thinking involves comparing the shapes’ similarity directly. A logistic regression model correctly classified the self-reported ways of thinking in 80.3

In this paper, we provide a conceptual framework of the central aspects of mathematical definitions discussed in the mathematics education literature. Based on a systematic literature review, we found that characterizations of definitions in the mathematics education literature can be classified into five main themes: requirements, preferred features, role and function, nature, and types of definition. Within each theme, a refined set of aspects was found by means of inductive coding. We discuss each aspect and point out areas where we believe further research is needed. The framework can be used both as a research tool and a didactical tool, as it highlights many aspects of mathematical definitions that call for attention by teachers and researchers.

I denne studien undersøkte vi tallforståelsen hos 77 femåringer fra fem ulike barnehager med digitale oppgaver. Kvantitative analyser av datamaterialet viste at det allerede før skolestart var stor spredning i barnas tallforståelse. Sammenlignet med det som er kjent fra før om norske barns utvikling av tallforståelse i barnehagealder, viste gruppen vi undersøkte overraskende godt utviklet tallforståelse. Vi diskuterer hva resultatene kan bety for den pedagogiske aktiviteten i barnehagen med tanke på å tilpasse aktiviteter til barns tallforståelse. ENGLISH ABSTRACT An insight into five-year-old children’s number sense in Early Childhood Education using digital tasks In this study we investigated the number sense of 77 five-year-old children from five early childhood education institutions using digital assessment tasks. From quantitative analyses of the data, we found a large variability in the children’s number sense already before the start of formal education. Compared to what is known about Norwegian children’s development of number sense in Early Childhood Education, many children showed a surprisingly well- developed number sense. We discuss what these results could mean for learning activities in early childhood education.

Symbolic numbers are a remarkable product of human cultural development. The developmental process involved the creation and progressive refinement of material representational tools, such as notched tallies, knotted strings, and counting boards. In this paper, we introduce a computational framework that allows the investigation of how material representations might support number processing in a deep reinforcement learning scenario. In this framework, agents can use an external, discrete state to communicate information to solve a simple numerical estimation task. We find that different perceptual and processing constraints result in different emergent representations, whose specific characteristics can facilitate the learning and communication of numbers.

A system for approximate number discrimination has been shown to arise in at least two types of hierarchical neural network models—a generative Deep Belief Network (DBN) and a Hierarchical Convolutional Neural Network (HCNN) trained to classify natural objects. Here, we investigate whether the same two network architectures can learn to recognise exact numerosity. A clear difference in performance could be traced to the specificity of the unit responses that emerged in the last hidden layer of each network. In the DBN, the emergence of a layer of monotonic 'summation units' was sufficient to produce classification behaviour consistent with the behavioural signature of the approximate number system. In the HCNN, a layer of units uniquely tuned to the transition between particular numerosities effectively encoded a thermometer-like 'numerosity code' that ensured near-perfect classification accuracy. The results support the notion that parallel pattern-recognition mechanisms may give rise to exact and approximate number concepts, both of which may contribute to the learning of symbolic numbers and arithmetic.

When learning to count, children actively engage with a variety of counting tasks and observe demonstrations by more knowl-edgeable others. We investigate how a single neural network-based agent, situated in a multimodal learning environment, can learn from observing such demonstrations to perform multiple number tasks such as counting temporally and spatially distributed objects, and a variant of the give-N task. We find that i. the agent can learn different tasks that require counting, ii. learning progresses in similar stages for different tasks, iii. sequential learning of subtasks aids learning of the full task of counting spatially distributed objects, and iv. a mechanism for updating memory when each object is counted emerges from learning the task. The work relies on generic deep learning processes in widely used neural network modules rather than mechanisms specialized for mathematics learning, and provides an architecture in which aspects of a sense of number emerge from learning several different number related tasks.

In this study we investigate whether transformations between different representations of mathematical objects constitute a suitable framework for the assessment of students' comprehension of fraction addition. Participants (N = 164) solved a set of 20 fraction addition problems constructed on the basis of Duval's (2017) theory of the role of representational transformations in mathematical comprehension. Using Rasch measurement theory and principal component analysis, we found that the items could be separated into three levels of difficulty based on the transformation involved. This large-scale structure was consistent across gender and across subgroups of preservice teachers and middle-grade students. On a finer scale, the production of diagrammatic representations, and the type of diagrammatic representation involved, constitute potential subdimensions of the instrument. We conclude that transformations between representations can be productive for the assessment of fraction addition comprehension as long as care is taken to curtail the potential effects of multidimensionality.

Learning the procedure of counting represents a major step in children's development of the concept of the natural numbers. How children acquire generalized concepts of number and counting skills i...

The aim of this study was to examine when children learn to read and how learning to read depends on a foundation of alphabetic knowledge. 356 children aged 5-6 years completed assessments of letter-sound knowledge, i.e. the names and sounds of uppercase and lowercase letters of the Norwegian alphabet. Each child was tested at the start, the middle and the end of the school year. The time that each child broke the reading code was also recorded. The results indicated that 11% of the children knew how to read before starting school and 27% of the children did not learn to read by the end of the first year. The remaining children typically knew 21 uppercase letter sounds before they were first able to read, and only a few (<5%) knew less than 11 uppercase letter sounds when they broke the reading code. The average of all four letter-scores at the time they broke the reading code was 19 +/- 5 letters (mean +/- standard deviation). Although letter sound knowledge was associated with the ability to read, it was not sufficient for breaking the reading code. 40% of children who knew 23 letter sounds or more, enough to read more than 80% of the most common Norwegian words, and 15% of children who knew all 29 letter sounds still could not read. Based on these data, it seems reasonable to advocate learning letter-sound correspondences early in the first year of school to form the best possible basis for breaking the reading code.

Episodic-like memory is thought to be supported by attractor dynamics in the hippocampus. A possible neural substrate for this memory mechanism is rate remapping, in which the spatial map of place cells encodes contextual information through firing rate variability. To test whether memories are stored as multimodal attractors in populations of place cells, recent experiments morphed one familiar context into another while observing the responses of CA3 cell ensembles. Average population activity in CA3 was reported to transition gradually rather than abruptly from one familiar context to the next, suggesting a lack of attractive forces associated with the two stored representations. On the other hand, individual CA3 cells showed a mix of gradual and abrupt transitions at different points along the morph sequence, and some displayed hysteresis which is a signature of attractor dynamics. To understand whether these seemingly conflicting results are commensurate with attractor network theory, we developed a neural network model of the CA3 with attractors for both position and discrete contexts. We found that for memories stored in overlapping neural ensembles within a single spatial map, position-dependent context attractors made transitions at different points along the morph sequence. Smooth transition curves arose from averaging across the population, while a heterogeneous set of responses was observed on the single unit level. In contrast, orthogonal memories led to abrupt and coherent transitions on both population and single unit levels as experimentally observed when remapping between two independent spatial maps. Strong recurrent feedback entailed a hysteretic effect on the network which diminished with the amount of overlap in the stored memories. These results suggest that context-dependent memory can be supported by overlapping local attractors within a spatial map of CA3 place cells. Similar mechanisms for context-dependent memory may also be found in other regions of the cerebral cortex.

, 136 (2008); 321 Science et al. Christopher D. Harvey Dendritic Spine The Spread of Ras Activity Triggered by Activation of a Single This copy is for your personal, non-commercial use only. clicking here. colleagues, clients, or customers by , you can order high-quality copies for your If you wish to distribute this article to others here. following the guidelines can be obtained by Permission to republish or repurpose articles or portions of articles ): November 18, 2013 www.sciencemag.org (this information is current as of The following resources related to this article are available online at http://www.sciencemag.org/content/321/5885/136.full.html version of this article at: including high-resolution figures, can be found in the online Updated information and services, http://www.sciencemag.org/content/suppl/2008/06/12/1159675.DC1.html can be found at: Supporting Online Material http://www.sciencemag.org/content/321/5885/136.full.html#related found at: can be related to this article A list of selected additional articles on the Science Web sites http://www.sciencemag.org/content/321/5885/136.full.html#ref-list-1 , 8 of which can be accessed free: cites 28 articles This article 33 article(s) on the ISI Web of Science cited by This article has been http://www.sciencemag.org/content/321/5885/136.full.html#related-urls 30 articles hosted by HighWire Press; see: cited by This article has been http://www.sciencemag.org/cgi/collection/neuroscience Neuroscience subject collections: This article appears in the following

The medial entorhinal cortex (MEC) is part of the brain's circuit for dynamic representation of self-location. The metric of this representation is provided by grid cells, cells with spatial firing fields that tile environments in a periodic hexagonal pattern. Limited anatomical sampling has obscured whether the grid system operates as a unified system or a conglomerate of independent modules. Here we show with recordings from up to 186 grid cells in individual rats that grid cells cluster into a small number of layer-spanning anatomically overlapping modules with distinct scale, orientation, asymmetry and theta-frequency modulation. These modules can respond independently to changes in the geometry of the environment. The discrete topography of the grid-map, and the apparent autonomy of the modules, differ from the graded topography of maps for continuous variables in several sensory systems, raising the possibility that the modularity of the grid map is a product of local self-organizing network dynamics.

Allocentric space is mapped by a widespread brain circuit of functionally specialized cell types located in interconnected subregions of the hippocampal-parahippocampal cortices. Little is known about the neural architectures required to express this variety of firing patterns. In rats, we found that one of the cell types, the grid cell, was abundant not only in medial entorhinal cortex (MEC), where it was first reported, but also in pre- and parasubiculum. The proportion of grid cells in pre- and parasubiculum was comparable to deep layers of MEC. The symmetry of the grid pattern and its relationship to the theta rhythm were weaker, especially in presubiculum. Pre- and parasubicular grid cells intermingled with head-direction cells and border cells, as in deep MEC layers. The characterization of a common pool of space-responsive cells in architecturally diverse subdivisions of parahippocampal cortex constrains the range of mechanisms that might give rise to their unique functional discharge phenotypes.

We report the existence of an entorhinal cell type that fires when an animal is close to the borders of the proximal environment. The orientation- specific edge- apposing activity of these " border cells" is maintained when the environment is stretched and during testing in enclosures of different size and shape in different rooms. Border cells are relatively sparse, making up less than 10% of the local cell population, but can be found in all layers of the medial entorhinal cortex as well as the adjacent parasubiculum, often intermingled with head- direction cells and grid cells. Border cells may be instrumental in planning trajectories and anchoring grid fields and place fields to a geometric reference frame.

Grid cells are topographically organized in the sense that, within the dorsal part of the medial entorhinal cortex, the scale of the grid increases systematically with anatomical distance from the dorsal border of this brain area. The ventral limit of the spatial map is currently not known. To determine if the grid map extends into the intermediate and ventral parts of the medial entorhinal cortex, we recorded activity from entorhinal principal cells at multiple dorsoventral levels while rats shuttled back and forth on an 18 m long linear track. The recordings spanned a range of more than 3 mm, covering approximately three quarters of the dorsoventral extent of the medial entorhinal cortex. Distinct periodic firing fields were observed at all recording levels. The average interpeak distance between the fields increased from ∼50 cm in the most dorsal part to ∼3 m at the most ventral recording positions. The increase in grid scale was accompanied by a decrease in the frequency of theta modulation and the rate of phase precession. The increase in average spacing and field size was approximately linear but this relationship coincided with a substantial increase in the variability of each measure. Taken together, the observations suggest that the spatial scale of the grid representation increases progressively along most of the dorsoventral axis of the medial entorhinal cortex, mirroring the topographical scale expansion observed in place cells in the hippocampus. © 2008 Wiley‐Liss, Inc.