An assessment of non-standardized tests of mathematical competence for Norwegian secondary school using Rasch analysis

International audienceDo non-standardized, publisher-provided tests for lower secondary school provide valid and reliable measures of mathematical competence? We analysed a sample of items pooled from tests accompanying three different Norwegian textbooks using Rasch analysis. The pooled sample of items was found to be sufficiently unidimensional for measuring function competence, with four strands of sub-competencies in accordance with theory. The competence associated with an increasing difficulty of items could be qualitatively characterised by shifts from a) identifying through constructing to reasoning about representations, b) using visual to using algebraic representations, and c) local to global interpretations of functions. While the individual tests differed substantially in the distribution of items across strands of mathematical competence, minor adjustments to the combined instrument were sufficient for providing a valid and reliable measure of mathematical competence

Neural representations of Euclidean space

Moderne nevrovitenskap bygger på antagelsen om at kognitive fenomener som bevissthet, hukommelse, og stedsans oppstår fra den samlede aktiviteten av individuelle nerveceller. Hjerneområdene hippocampus og entorhinal cortex (EC) er kritiske for hukommelse og stedsans hos både mennesker og dyr. Hos rotter utgjør ’stedceller’ i hippocampus egne kart for hvert miljø rotta utforsker mens ’gitterceller’ i EC utgjør et koordinatsystem som passer til alle miljøer. Både kart og koordinatsystem finnes i ulike skalaer i den øverste (dorsale) delen av hjerneområdene mens det hittil har vært uklart om dypere (ventrale) deler er involvert i stedsansen. Vi angrep spørsmålet ved å måle aktiviteten i enkeltceller langs hele lengden av begge hjerneområder hos rotter som løp frem og tilbake på en 18m spesialkonstruert løpebane, og fant stedceller og gitterceller langs hele lengden av begge hjerneområdene (Artikkel II og III). Skalaen til både stedcellene og gittercellene økte fra å representere steder på mindre enn 1m i den dorsale enden til opp mot 10 meter i den ventrale enden av hjerneområdene. Videre utviklet vi en matematisk modell for hvordan minnekart i hippocampus kan være knyttet til stedskoordinater i EC (Artikkel I), og kom fram til et koblingsskjema mellom gitterceller og stedceller som utnytter den systematiske økningen i skala fra dorsal til ventral. Modellen stemmer godt med de anatomiske koblingene mellom EC og hippocampus, og får støtte av nye eksperimentelle data. Til slutt undersøkte vi hvordan stedskartene i hippocampus er knyttet til miljøets utforming. Tidligere matematiske beregninger har vist at kartenes geometriske forankring kan forklares hvis det finnes en tredje celletype som signaliserer rottas avstand og retning til ulike grenser i miljøet. Vi målte aktivitet i enkeltceller i EC mens rotter utforsket miljøer med ulike geometriske former(Artikkel IV). Blant de andre celletypene fant vi en liten gruppe ”grenseceller” som bare er aktive når rotta løper i nærheten av en vegg eller bordkant. Avhandlingen fremlegger ny evidens for at beregning av hierarkisk organisert stedsinformasjon er et grunnprinsipp for hvordan hippocampus og EC fungerer (f.eks. ved behandling av minner), påviser en hittil ukjent enhet for representasjon av geometri i EC, og antyder hvordan nevrale enheter kan samhandle for å støtte opp under kognitive funksjoner som stedsans og hukommelse.  As cognitive phenomena are believed to arise from neural activity, uncovering how neurons represent Euclidean one- and two-dimensional space provides a foundation for understanding how the brain organizes and processes information about terrestrial objects and events. Neurons in the hippocampus and medial entorhinal cortex (MEC) of rats exhibit discrete spatial receptive fields at a scale that increases with the neuron’s distance from the dorsal pole of both structures. To find out whether spatial processing is a cardinal function of these structures, we recorded neural activity along the dorsal-most 85% of the CA3 area of the hippocampus (Paper II), and dorsal-most 75% of the MEC (Paper III) while rats explored an 18m linear track. Neurons at all dorsoventral levels of both structures displayed spatial receptive fields, implying functional homogeneity within the hippocampus and MEC. Spatial scale increased from dorsal to ventral in both CA3 and MEC. In hippocampus, field length ranged from less than 1m to more than 10m. In the MEC field length ranged from less than 50cm to approximately 3m, and inter-peak distance ranged from less than 1m to at least 8m. The parallel increase in spatial scale suggests a simple transformation from the repetitive spatial metric of grid cells to the unary place-cell representation of space. Developing a mathematical firing-rate model of place-cell activity to exploit this fact, we showed that place fields can be formed from converging grid-cell inputs that cover a range of spatial scales and orientations but have an overlapping firing peak in the placefield center (Paper I). Inferring metric relationships between entities in hippocampal association maps may therefore rely on interaction with the MEC coordinate system. Because metric information is in turn contingent on the geometric layout of the external environment, we initiated a search for neural representations of geometric features in the parahippocampus. A small proportion (< 10%) of cells that discharged close to environmental borders was found in all cellular layers of MEC as well as in pre- and parasubiculum (Paper IV). ‘Border cells’ typically had a firing field apposing one or more walls of the recording enclosure regardless of enclosure shape, size, or which room the rat was exploring, and responded to any wall, drop, or partition that impeded the rat’s exploration. Taken together, this thesis demonstrates that hierarchically organized spatial processing is an integral property of the hippocampus and MEC, extends the evidence for a modular organization of spatial cognition, and suggests how such modules may interact to support behaviorally relevant functions like spatial memory and navigation

Linear figural patterns as a teaching tool for preservice elementary teachers – the role of symbolic expressions

International audienceFigural patterns connect several aspects of mathematical activity central to the work of teaching mathematics. In this pilot study, we investigated the solutions of 16 preservice elementary teachers to linear figural patterns of different levels of complexity after the completion of a series of six teaching sessions of a course in mathematics education. We found that a) most students were able to generalize and find the figural number of an arbitrary figure in the sequence; b) about half of the students produced mathematically imprecise formulas when translating from an arbitrary number into a general algebraic expression; c) the formulas students produced frequently lacked structural correspondence with the figural patterns and d) students had difficulties in interpreting figural patterns that are more complex. These results indicate that although the course successfully trains students to generalize with linear figural patterns, more attention to precisely formulating mathematical ideas and to interpretation of more difficult patterns can further improve the training of preservice elementary teachers

Linear figural patterns as a teaching tool for preservice elementary teachers – the role of symbolic expressions

International audienceFigural patterns connect several aspects of mathematical activity central to the work of teaching mathematics. In this pilot study, we investigated the solutions of 16 preservice elementary teachers to linear figural patterns of different levels of complexity after the completion of a series of six teaching sessions of a course in mathematics education. We found that a) most students were able to generalize and find the figural number of an arbitrary figure in the sequence; b) about half of the students produced mathematically imprecise formulas when translating from an arbitrary number into a general algebraic expression; c) the formulas students produced frequently lacked structural correspondence with the figural patterns and d) students had difficulties in interpreting figural patterns that are more complex. These results indicate that although the course successfully trains students to generalize with linear figural patterns, more attention to precisely formulating mathematical ideas and to interpretation of more difficult patterns can further improve the training of preservice elementary teachers

Learning exact enumeration and approximate estimation in deep neural network models

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

Place Cell Rate Remapping by CA3 Recurrent Collaterals

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

Emerging Representations for Counting in a Neural Network Agent Interacting with a Multimodal Environment

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 is still under debate. Here we investigate how a neural network agent develops representations for key concepts of counting while learning to perform several different counting tasks in a multimodal, interactive environment. We identify neural activity and connection patterns that realize a) a representation of the entity to count that was invariant to the task, b) a mapping from entity to number-word, and c) a representation of the number of entities that have been counted that was shared between tasks. The results support the notion that abstract representations of number can arise from integrating experiences across a range of number-related tasks