Computer Vision Modeling of the Development of Geometric and Numerical Concepts in Humans

AI summary

Overview

Research area: Computational cognitive science at the intersection of computer vision and developmental psychology — specifically, whether the training trajectories of CV models mirror the developmental trajectories of mathematical concepts in children.

Technical level: Intermediate. The paper is readable without deep mathematics, but it assumes familiarity with CNNs, representation vectors, and psychophysical effects (distance, size, ratio).

Scope: A two-experiment case study of a single model, ResNet-50, trained on ImageNet-1k and evaluated at every training epoch on (1) a 43-concept geometric/topological odd-one-out task and (2) numerosity comparisons across six stimulus sets, with each trajectory compared against human lifespan data.

What This Paper Is About

Prior work has shown that computer vision models trained only to classify images nonetheless develop latent representations of geometric and numerical concepts that resemble those of human adults — a result called cognitive alignment. What has not been tested is developmental alignment: whether a model's improving performance across training follows the same progression that children show as they grow up. This paper asks whether ResNet-50, trained on ImageNet, becomes sensitive to geometric and topological concepts and develops a human-like "mental number line" along the same developmental arc observed in people from childhood through adulthood.

Key Contributions

1. First developmental-alignment test of a standard CV pipeline. Prior developmental modeling (Testolin et al. 2020) used a custom network, a non-standard layer-wise training procedure, and abstracted stimuli. This study uses a standard ResNet-50 architecture, standard ImageNet training, and naturalistic images, checking model behavior after each of 90 epochs.
2. A concept-class-level comparison for geometry and topology. The study tracks the model's sensitivity to 43 geometric/topological (GT) concepts grouped into 7 classes and correlates each class's training curve with human lifespan curves from Izard and Spelke (2009).
3. A checkpoint-by-checkpoint analysis of numerical representation. It measures the distance, size, and ratio effects at every epoch across six numerosity stimulus sets, and reconstructs the model's latent number line at intermediate checkpoints using multidimensional scaling (MDS).
4. A re-framing of the core-knowledge debate. The results are used to argue which GT concepts may come "for free" from learning to perceive the visual world, and which may instead require core knowledge or explicit mathematics instruction.

Main Findings

• Overall developmental alignment holds, but the model lags in absolute performance. Mapping 2 epochs of training to 1 year of human development (90 epochs against the 45-year span of ages 6–51), the Pearson correlation between human and model accuracy curves is r = 0.50 (p < 0.01). A power function fits the human data at R² = 0.40 and the model data at R² = 0.66. The paper states humans "decisively outperform the model," though per-epoch absolute accuracy values are not reported numerically.

• Alignment is present for 4 of the 7 GT classes, absent for 3. The model's accuracy improves with training for Euclidean Geometry, Geometric Figures, Metric Properties, and Topology, but is low and barely improves for Chiral Figures, Geometric Transformations, and Symmetrical Figures. Humans, by contrast, improve across all 7 classes. For context, Izard and Spelke (2009) found 3–6-year-old children above chance on 27 of the 43 concepts (63%), including all 8 Euclidean Geometry concepts and none of the 8 Geometric Transformations concepts.

• Class-level human–model correlations are strongest for Euclidean Geometry and Geometric Figures. Table 2 reports Pearson r (p) between human and model curves as 0.5049 (p = 0.0044) for Euclidean Geometry and 0.5434 (p = 0.0019) for Geometrical Figures. The remaining classes show weak or negative correlations: Metric Properties 0.2255 (p = 0.2308), Topology −0.0482 (p = 0.8003), Chiral Figures −0.0124 (p = 0.9481), Geometric Transformations −0.1936 (p = 0.3053), and Symmetrical Figures −0.0285 (p = 0.8812). Corresponding model power-function fits range from R² = 0.6168 (Euclidean Geometry) to R² = 0.0008 (Symmetrical Figures).

• A mental number line emerges over training. At epochs 1, 2, 10, and 90, the model shows no distance, size, or ratio effects early in training; these effects manifest as training proceeds, indicating that a human-like number representation is learned as a side effect of image classification.

• The distance and ratio effects are robust; the size effect is not. The distance effect appears early in training with the canonical negative correlation, and the paper reports it holds for all but stimulus sets 1 and 6. The ratio effect follows the canonical form with high R² for all but stimulus sets 5 and 6. The size effect has positive but near-zero correlations and is weakest in the "easiest" sets — set 1 (equal-area circles) and set 2 (equal-circumference circles).

• Growth curves are more developmentally plausible for distance and ratio than for size. In the power-function fits (Table 3, R² for distance / size / ratio): set 1 = 0.49 / 0.04 / 0.35; set 2 = 0.43 / 0.00 / 0.39; set 3 = 0.39 / 0.13 / 0.36; set 4 = 0.38 / 0.25 / 0.48; set 5 = 0.00 / 0.26 / 0.15; set 6 = 0.12 / 0.15 / 0.01. The paper reports developmentally plausible growth for distance and ratio in all but the most varied set (6), with the size effect fitting less well.

• The reconstructed number line becomes increasingly canonical. MDS on the 9×9 cosine-similarity matrix for stimulus set 1, projected to one dimension, shows the model's latent number line sharpening over training.

Methodology in Plain English

Model and training. The researchers trained a standard ResNet-50 (~25.6M parameters) from scratch on ImageNet-1k (ILSVRC-2012), using the official 1.28M training / 50k validation split. Images were RandomResizedCrop'd to 224×224 (scale [0.08, 1.0], aspect ratio [3/4, 4/3]), randomly horizontally flipped (p = 0.5), and normalized. They trained for 90 epochs with SGD, global batch size 256, initial learning rate 0.1, step-decayed by a factor of 0.1 every 30 epochs, ending at 1×10⁻³, on a single A40 (48 GB) GPU. A full checkpoint was saved after every epoch. Final validation accuracy matched the standard ResNet-50 reference (top-1 ≈ 76%, top-5 ≈ 93%).

Experiment 1 (geometry and topology). The stimuli were the 43-concept odd-one-out task from Dehaene et al. (2006). Each stimulus shows 6 images; 5 embody a target GT concept and 1 does not, and the task is to select the non-embodying image, so chance is 1/6. After each epoch, each of the 6 images was resized/cropped to 224×224, passed through the model, and the 2048-dimensional representation before the final prediction layer was collected. The model's odd-one-out choice was the image with the lowest average cosine similarity to the other five. Performance was aggregated overall and by the 7 classes. Human comparison data came from Experiment 2 of Izard and Spelke (2009): 400 Western participants aged 6–51, who completed 2 practice trials and 43 experimental trials by clicking their choice.

Experiment 2 (number). The stimuli were 720×720-pixel images of numerosities 1–9 from Upadhyay and Varma (2023), organized into 6 sets of increasing difficulty. Set 1 controls total black-pixel area across five levels of 103–518 pixels; set 2 controls total circumference across five levels of 100–300 pixels; sets 3 and 4 randomize shape and area; set 5 ("anything goes") randomizes individual items as mixtures of circles, squares, and triangles; set 6 uses naturally occurring, mostly clip-art-like numerosities found via Google Search and manually verified. For each of the 36 unordered pairs of unequal numerosities, the researchers sampled stimuli, extracted the pre-classifier vector for each, and computed cosine similarity. Their linking hypothesis: less similar vectors mean more discriminable numerosities, hence a faster predicted human judgment. This yielded three measures — the correlation of similarity with distance |n₁ − n₂| (negative = human-like distance effect), the correlation of similarity with average size (n₁ + n₂)/2 (positive = human-like size effect), and the R² of a negative exponential fit predicting similarity from the ratio max(n₁, n₂)/min(n₁, n₂) (closer to 1 = human-like ratio effect).

Alignment metric. Two epochs were mapped to one year of development because the model ran 90 epochs and the human age range spanned 45 years.

Why This Matters

The paper offers a concrete method for using machine learning models as developmental models rather than only as adult-cognition models, and it turns a modeling result into evidence about a long-standing debate: whether geometric and numerical concepts are part of innate "core knowledge" or are acquired from experience with the visual world. The finding that four classes of GT concepts and a mental number line emerge "for free" from image classification favors the learning account for those cases, while the failure to improve on chiral, transformational, and symmetrical concepts suggests those may rely on core knowledge or on supervised mathematics instruction.

Real-world applications:

• Educational technology and curriculum design: Understanding which concepts appear readily from visual experience versus which require instruction could inform the ordering and emphasis of early

Authors’ abstract

Mathematical thinking is a fundamental aspect of human cognition. Cognitive scientists have investigated the mechanisms that underlie our ability to thinking geometrically and numerically, to take two prominent examples, and developmental scientists have documented the trajectories of these abilities over the lifespan. Prior research has shown that computer vision (CV) models trained on the unrelated task of image classification nevertheless learn latent representations of geometric and numerical concepts similar to those of adults. Building on this demonstrated cognitive alignment, the current study investigates whether CV models also show developmental alignment: whether their performance improvements across training to match the developmental progressions observed in children. In a detailed case study of the ResNet-50 model, we show that this is the case. For the case of geometry and topology, we find developmental alignment for some classes of concepts (Euclidean Geometry, Geometrical Figures, Metric Properties, Topology) but not others (Chiral Figures, Geometric Transformations, Symmetrical Figures). For the case of number, we find developmental alignment in the emergence of a human-like ``mental number line'' representation with experience. These findings show the promise of computer vision models for understanding the development of mathematical understanding in humans. They point the way to future research exploring additional model architectures and building larger benchmarks.