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The Challenge of Modeling the Acquisition of Mathematical Concepts.

ABSTRACT: As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer simulation should have a primary role in filling this gap because it allows identifying the finer-grained computational mechanisms underlying complex behavior and cognition. Modeling efforts will be most effective if carried out at cross-disciplinary intersections, as attested by the recent success in simulating human cognition using techniques developed in the fields of artificial intelligence and machine learning. In this respect, deep learning models have provided valuable insights into our most basic quantification abilities, showing how numerosity perception could emerge in multi-layered neural networks that learn the statistical structure of their visual environment. Nevertheless, this modeling approach has not yet scaled to more sophisticated cognitive skills that are foundational to higher-level mathematical thinking, such as those involving the use of symbolic numbers and arithmetic principles. I will discuss promising directions to push deep learning into this uncharted territory. If successful, such endeavor would allow simulating the acquisition of numerical concepts in its full complexity, guiding empirical investigation on the richest soil and possibly offering far-reaching implications for educational practice.

SUBMITTER: Testolin A

PROVIDER: S-EPMC7099599 | biostudies-literature | 2020

REPOSITORIES: biostudies-literature

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The Challenge of Modeling the Acquisition of Mathematical Concepts.

Testolin Alberto A

Frontiers in human neuroscience 20200320

As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer sim ...[more]

PMID: 32265678

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Project description:Using laser capture microdissection to isolate over 100 specific cell populations, we report the profiling of prostate cancer progression from benign epithelium to metastatic disease. By analyzing these expression signatures in the context of over 15,000 â??molecular conceptsâ??, or sets of biologically connected genes, we generated an integrative model of prostate cancer progression. Keywords: disease state analysis Experiment Design: â?¢ Type of experiment: This study used microarray experiments to profile prostate cancer progression through the isolation of pure cell populations using laser capture microdissection (LCM) and OmniPlex Whole Transcriptome (WTA) Amplification. â?¢ Experimental factors: This study profiled various pure cell populations isolated by LCM from prostate tissue. â?¢ The number of hybridizations performed in the experiment: A total of 104 hybridizations were performed. â?¢ The type of reference used for the hybridizations: A single commercially available pool of benign prostate tissue (CPP, Clontech) was used as the reference in all hybridizations. For the reference sample, 10 ng of CPP total RNA was converted into a WTA cDNA library in a volume of 25 Âµl and five 5 Âµl aliquots were amplified by WTA PCR and purified. Five ng aliquots of the pooled products were subjected to a second WTA PCR amplification in the presence of amino-allyl dUTP and all products were pooled before labeling to ensure equal representation across all of the hybridizations. â?¢ Hybridization design: For all hybridizations, the experimental prostate sample isolated by LCM was labeled with Cy5 and the common reference of benign pooled prostate as described above was labeled with Cy3. â?¢ Quality control steps taken: In order to assess biological and technical reproducibility, we captured identical regions from two specimens (PCA_11 and MET_HR_3) and subjected one sample, PCA_30, to two hybridizations. For PCA_11, adjacent regions of the same section were captured and treated as separate samples. For MET_HR_3, two separate foci of the liver metastasis (from separate frozen blocks) were captured and treated as separate samples. For PCA_30, two aliquots of the primary WTA PCR products were separately subjected to a second round of WTA PCR amplification. Replicates from PCA_30 and MET_HR_3 were also hybridized to arrays from each of the print runs utilized in this study. Samples used, extract preparation and labeling: â?¢ The origin of the biological sample (for instance, name of the organism, the provider of the sample) and its characteristics: All samples used were of human origin. The table in the MIAME Checklist (Supplmentary Methods) summarizes the samples used. Hybridizations are named according to the class of sample profiled and the background color is the same as shown in Figure S3. Two print runs were used during the course of the study and the indicated run for each hybridization is given. For prostate cancer samples, the Gleason patterns present on the sample section used are indicated, as well as the Gleason patterns of the actual cells isolated by LCM. A unique case number for all profiled samples is given. Descriptions of samples are given where applicable, including the location of metastatic foci. â?¢ Manipulation of biological samples and protocols used: Laser Capture Microdissection (LCM) was performed from frozen tissue sections with the SL Microtest device using ÂµCUT software (MMI). Approximately 10,000 cells were captured for each sample. Serial sections were used if cells could not be obtained from a single section. Total RNA was isolated from captured cells with the RNAqueous Micro kit (Ambion) and treated with DNAse I according to the manufacturer's instructions. RNA quantification was perfomed using Ribogreen (Molecular Probes). â?¢ Protocol for preparing the hybridization extract: For each sample, total RNA (~10ng) in a volume of 25 Âµl was converted into an OmniPlex WTA cDNA library and amplified by WTA PCR using reagents and protocols according to the beta-commercial TransPlex WTA kit (Rubicon Genomics, Ann Arbor, MI). For each sample, a single 10 Âµl aliquot of the WTA cDNA library was amplified by WTA PCR and products were purified. Four or five 5 ng aliquots of product were subjected to a second WTA PCR amplification in the presence of amino-allyl dUTP for post amplification labeling and products were pooled before proceeding with the hybridization. Yields after all WTA PCR amplifications were between 2 to 5 Î¼g per reaction, and reaction progress was monitored in real time using SYBR green I (Molecular Probes) on the iCycler IQ (BioRad) or ABI 7300 Real Time PCR system (Applied Biosystems). Real-time PCR amplifications were terminated at or before plateau phase as measured by fluorescence incorporation to preserve maximum representation. â?¢ Labeling protocol(s): For all WTA amplified cDNA samples with amino-allyl dUTP incorporated (10-20 ï?g), nuclease free water was added to 500 ï?l, the sample was transferred to a Microcon YM-30 filter, and centrifuged at 13,000 rcf at RT for 12 minutes. The spin column was inverted in a new collection tube and centrifuged at 13,000 rcf for 2 min at RT. The amount of sample was measured and the volume was adjusted to 5 ï?l with nuclease free water. For labeling, 5 ï?l of 1 M sodium bicarbonate (pH 9.0) was added and allowed to incubate at RT for 15 min. Nine ï?l of anhydrous DMSO were added to Mono Reactive Cy3 and Cy5 dye packs (Amersham, Buckinghamshire, England), mixed thoroughly, and 1 ï?l of the proper dye was added to each sample. The labeling mixture was incubated in the dark at RT for 1 hour. The reaction was stopped by the addition of 9 ï?l of 4 M Hydroxylamine for 15 min. For each labeling mixture, RNase free water was added to 100 ï?l, 500 ï?l of PB buffer was added and Cy3 and Cy5 mixtures were added to separate Qiaquick (Qiagen, Valencia, CA) spin columns and centrifuged at 13,000 rcf. Columns were washed twice with 700 ï?l of buffer PE, and centrifuged dry at 13,000 rcf for 2 min. To elute, 60 ï?l of EB buffer was placed on the column, incubated for 5 min and centrifuged at 13,000 rcf for 1 min. The Cy3 labeled and Cy5 labeled cDNA for each hybridization were mixed and 1.5 ï?l were used for analysis on a ND-1000 spectrophotometer. â?¢ External controls (spikes): Not utilized Hybridization procedures and parameters: â?¢ The protocol and conditions used during hybridization, blocking and washing: For hybridization, 40 ï?g of human Cot-1 DNA (Invitrogen, Carlsbad, CA) was added to 120 ï?l of labeled cDNA, the probe was transferred to a Microcon YM-30 filter and centrifuged at 13,000 rcf for 6 min at RT. The spin column was inverted in a new collection tube and centrifuged at 13,000 rcf for 2 min at RT. The eluted probe volume was adjusted to 18.6 ï?l with nuclease free water and the following were added: 4 ï?l of yeast tRNA (10 ï?g/ï?l, Invitrogen, Carlsbad, CA), 4.9 ï?l of 20X SSC and 0.84 ï?l of 10% SDS. The probe was denatured at 100 Co for 3 min and centrifuged at 13,000 rcf for 45 sec. The probe was added directly to the microarray and a cover slip was added. Slides were hybridized overnight in DIE-TECH 1 or 5 slide hybridization chambers in a 65 Co water bath. After hybridization, cover slips were removed by incubation in 2x SSC / 0.05% SDS. Slides were then washed in 2x SSC / 0.05% SDS for 5 min and 0.2x SSC / 0.05% SDS for five minutes. Slides were then washed in 0.2x SSC for 10 sec and centrifuged dry at 500 rpm for 5 min. Measurement data and specifications: â?¢ Type of scanning hardware and software used: Microarrays were scanned using a GenePix 4000B scanner (Axon Instruments, Union City, CA). â?¢ Type of image analysis software used: Images were gridded and spots were quantified using GenePix Pro 4.0 software (Axon Instruments, Union City, CA). â?¢ A description of the measurements produced by the image-analysis software and a description of which measurements were used in the analysis: The measurements made by GenePix Pro 4.0 are described in the sample files. For all hybridizations, the log2 normalized Median of Ratios (as described below) was used. â?¢ Data selection and transformation procedures: Arrays were auto-gridded by GenePix 4.0. Features flagged by GenePix as not found during grid

Project description:A research link between conditional reasoning and mathematics has been reported only for late adolescents and adults, despite claims about the pivotal importance of conditional reasoning, i.e., reasoning with if-then statements, in mathematics. Secondary students' problems with deductive reasoning in mathematics have been documented for a long time. However, evidence from developmental psychology shows that even elementary students possess some early conditional reasoning skills in familiar contexts. It is still an open question to what extent conditional reasoning with mathematical concepts differs from conditional reasoning in familiar everyday contexts. Based on Mental Model Theory (MMT) of conditional reasoning, we assume that (mathematical) content knowledge will influence the generation of models, when conditionals concern mathematical concepts. In a cross-sectional study, 102 students in Cyprus from grades 2, 4, and 6 solved four conditional reasoning tasks on each type of content (everyday and mathematical). All four logical forms, modus ponens (MP), modus tollens (MT), denial of the antecedent (DA), and affirmation of the consequent (AC), were included in each task. Consistent with previous findings, even second graders were able to make correct inferences on some logical forms. Controlling for Working Memory (WM), there were significant effects of grade and logical form, with stronger growth on MP and AC than on MT and DA. The main effect of context was not significant, but context interacted significantly with logical form and grade level. The pattern of results was not consistent with the predictions of MMT. Based on analyses of students' chosen responses, we propose an alternative mechanism explaining the specific pattern of results. The study indicates that deductive reasoning skills arise from a combination of knowledge of domain-general principles and domain-specific knowledge. It extends results concerning the gradual development of primary students' conditional reasoning with everyday concepts to reasoning with mathematical concepts adding to our understanding of the link between mathematics and conditional reasoning in primary school. The results inspire the development of educational interventions, while further implications and limitations of the study are discussed.

Project description:Background and objectivesLiposuction continues to be one of the most popular procedures performed in cosmetic surgery. As the public's demand for body contouring continues, laser lipolysis has been proposed to improve results, minimize risk, optimize patient comfort, and reduce the recovery period. Mathematical modeling of laser lipolysis could provide a better understanding of the laser lipolysis process and could determine the optimal dosage as a function of fat volume to be removed.Study design/materials and methodsAn Optical-Thermal-Damage Model was formulated using finite-element modeling software (Femlab 3.1, Comsol Inc). The general model simulated light distribution using the diffusion approximation of the transport theory, temperature rise using the bioheat equation and laser-induced injury using the Arrhenius damage model. Biological tissue was represented by two homogenous regions (dermis and fat layer) with a nonlinear air-tissue boundary condition including free convection. Video recordings were used to gain a better understanding of the back and forth movement of the cannula during laser lipolysis in order to consider them in our mathematical model. Infrared video recordings were also performed in order to compare the actual surface temperatures to our calculations. The reduction in fat volume was determined as a function of the total applied energy and subsequently compared to clinical data reported in the literature.ResultsIn patients, when using cooled tumescent anesthesia, 1064 nm Nd:YAG laser or 980 nm diode laser: (6 W, back and forth motion: 100 mm/s) give similar skin surface temperature (max: 41 degrees C). These measurements are in accordance with those obtained by mathematical modeling performed with a 1 mm cannula inserted inside the hypodermis layer at 0.8 cm below the surface. Similarly, the fat volume reduction observed in patients at 6-month follow up can be determined by mathematical modeling. This fat reduction depends on the applied energy, typically 5 cm3 for 3000 J. At last, skin retraction was observed in patients at 6-month follow up. This observation can be easily explained by mathematical modeling showing that the temperature increase inside the lower dermis is sufficient (48-50 degrees C) to induce skin tighteningDiscussion and conclusionLaser lipolysis can be described by a theoretical model. Fat volume reduction observed in patients is in accordance with model calculations. Due to heat diffusion, temperature elevation is also produced inside the lower reticular dermis. This interesting observation can explain remodeling of the collagenous tissue, with clinically evident skin tightening. In conclusion, while the heat generated by interstitial laser irradiation provides stimulate lipolysis of the fat cells, the collagen and elastin are also stimulated resulting in a tightening in the skin. This mathematical model should serve as a useful tool to simulate and better understand the mechanism of action of the laser lipolysis.

Dataset Information

The Challenge of Modeling the Acquisition of Mathematical Concepts.

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The Challenge of Modeling the Acquisition of Mathematical Concepts.

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