Minimal Embodiment Enables Efficient Learning of Number Concepts in Robot

Zhegong Shangguan; Alessandro Di Nuovo; Angelo Cangelosi

cs.ROApr 13, 2026

Minimal Embodiment Enables Efficient Learning of Number Concepts in Robot

Authors: Zhegong Shangguan, Alessandro Di Nuovo, Angelo Cangelosi

Organizations: Cognitive Robotics Lab, Department of Computer Science, University of Manchester · Smart Interactive Technologies (SIT) Research Laboratory, Department of Computing, Sheffield Hallam University

Abstract

Robots are increasingly entering human-interactive scenarios that require understanding of quantity. How intelligent systems acquire abstract numerical concepts from sensorimotor experience remains a fundamental challenge in cognitive science and artificial intelligence. Here we investigate embodied numerical learning using a neural network model trained to perform sequential counting through naturalistic robotic interaction with a Franka Panda manipulator. We demonstrate that embodied models achieve 96.8% counting accuracy with only 10% of training data, compared to 60.6% for vision-only baselines. This advantage persists when visual-motor correspondences are randomized, indicating that embodiment functions as a structural prior that regularizes learning rather than as an information source. The model spontaneously develops biologically plausible representations: number-selective units with logarithmic tuning, mental number line organization, Weber-law scaling, and rotational dynamics encoding numerical magnitude ( $r = 0.97$ , slope $= 30.6°$ /count). The learning trajectory parallels children's developmental progression from subset-knowers to cardinal-principle knowers. These findings demonstrate that minimal embodiment can ground abstract concepts, improve data efficiency, and yield interpretable representations aligned with biological cognition, which may contribute to embodied mathematics tutoring and safety-critical industrial applications.

Figures & tables

Figure 1: Embodied numerical cognition: from human to robot. (A) Humans use pointing gestures to coordinate counting with objects. (B) Neural basis: visual (V1/V2/V3) and proprioceptive (S1/S2) signals converge in the intraparietal sulcus (IPS), supporting abstract numerical knowledge. (C) Robotic implementation: a Franka Panda manipulator performs sequential counting with a wrist-mounted camera. (D) Model architecture: visual and motor inputs are integrated by an LSTM, which outputs number classification and next-step motor prediction.

Figure 2: Embodied learning yields superior data efficiency, distinctive learning dynamics, and structured numerical representations. a, Validation accuracy across training data fractions (10%, 50%, 100%) for embodied (blue) and vision-only (red) models, with and without ImageNet pretraining. Embodied models consistently outperform vision-only baselines across all conditions. Notably, pretraining enhances embodied model performance while impairing vision-only models, a negative transfer effect most pronounced in low-data regimes. Error bars indicate standard deviation across 3 random seeds. b, Learning trajectories over training epochs (10% data condition). The vision-only model (red) exhibits rapid early improvement followed by plateau, whereas embodied models (blue) show an initial latent period (20–50 epochs) before sharp performance gains that ultimately surpass vision-only baselines. Shaded regions indicate standard deviation. c, Grad-CAM visualizations of visual attention across convolutional layers (Conv1, Conv3, Conv5) for numerosities 1, 2, 4, and 8. Embodied models maintain focused attention on target objects across all conditions, while vision-only models, particularly with pretraining, allocate attention broadly to background regions, especially for small numerosities. d, PCA visualization of visual encoder (AlexNet) representations, with colors indicating numerosity (1–10, purple to yellow). Pretrained embodied models develop linearly organized representations resembling a mental number line, while vision-only models exhibit distorted U-shaped geometry. e, Per-number learning trajectories across training epochs. Embodied models (blue shades) acquire numbers sequentially (1 $\rightarrow$ 2 $\rightarrow$ 3 $\rightarrow$ …), mirroring children’s developmental progression from subset-knowers to cardinal-principle knowers. Vision-only models (red shades) show approximate discrimination across all numbers from early training, resembling the Approximate Number System (ANS). All panels show results from models trained on 10% of the full dataset. Vision-only results are from the single-image baseline (V1 and V3); the sequential-image baseline (V2 and V4) yielded nearly identical performance (see Table 1 ).

Model	Conditions			10% Data		50% Data		100% Data
	Pretrain	Joint Shuffle	Curriculum	Acc (%) $\uparrow$	MSE $\downarrow$	Acc (%) $\uparrow$	MSE $\downarrow$	Acc (%) $\uparrow$	MSE $\downarrow$
E1	$\times$	$\times$	Easy $\rightarrow$ Hard	85.7(2.7)	1.73(0.04)	97.2(0.2)	1.54(0.04)	95.8(0.7)	1.77(0.09)
E2	$\times$	$\times$	Hard $\rightarrow$ Easy	84.0(3.3)	1.74(0.04)	96.3(0.2)	1.67(0.08)	94.2(0.4)	1.87(0.21)
E3	$\times$	$\times$	Random	90.1(2.7)	1.70(0.03)	99.9(0.1)	1.68(0.08)	100.0(0.0)	1.05(0.15)
E4	$\times$	✓	Easy $\rightarrow$ Hard	85.5(3.3)	2.05(0.05)	97.2(0.4)	1.97(0.07)	97.2(1.7)	2.19(0.07)
E5	$\times$	✓	Hard $\rightarrow$ Easy	84.0(3.7)	2.10(0.03)	85.2(18.1)	2.11(0.14)	82.1(27.1)	2.31(0.10)

Table 1: Validation performance at best checkpoint. Metrics shown as mean (std) over 3 seeds. Accuracy is final_val_count_accuracy (%), MSE is joint MSE (

\times 10^{2}

). Bold indicates best per data fraction and metric.

Figure 3: Motor signals function as structural priors rather than information sources. a, Schematic of the shuffle experiment. Within each training batch, joint position labels are randomly permuted across samples while visual inputs and count labels remain unchanged. This manipulation preserves motor prediction as an auxiliary training objective but eliminates veridical visuomotor correspondence. b, Counting accuracy for intact (solid bars) versus shuffled (striped bars) visuomotor correspondence across data fractions. Shuffling has negligible effect on counting performance: models with disrupted correspondence achieve accuracy comparable to intact models across all conditions. c, Motor prediction MSE for intact versus shuffled conditions. Unlike counting accuracy, motor prediction error increases substantially when visuomotor correspondence is disrupted, confirming that the shuffle manipulation successfully eliminated meaningful visual-motor relationships. d, Learning trajectories for intact (solid lines) versus shuffled (dashed lines) conditions across data fractions (10%, 50%, 100%). The characteristic two-phase learning profile, an initial latent period followed by rapid improvement, is preserved regardless of visuomotor correspondence. Together, panels b–d demonstrate a double dissociation: shuffling impairs motor prediction while leaving counting accuracy and learning dynamics intact, indicating that motor signals function as structural priors rather than information sources.

Figure 4: Curriculum learning reveals intrinsic developmental constraints in numerical acquisition. a, Schematic of three curriculum strategies. In easy-to-hard, training samples are ordered by numerosity from 1 to 10. In hard-to-easy, the order is reversed from 10 to 1. In random, samples are shuffled without systematic ordering. b, Final counting accuracy across curriculum strategies and data fractions for pretrained (left) and non-pretrained (right) models. Random curriculum yields the best performance, followed by easy-to-hard, with hard-to-easy performing worst, particularly in low-data regimes. Error bars indicate standard deviation across 3 random seeds. c, Heatmaps of per-number accuracy across training epochs for all curriculum and pretraining conditions (10% data). Y-axis represents numerosity (1–10), X-axis represents training epochs, and color indicates accuracy (purple: low, yellow: high). Across all conditions, high accuracy (yellow) emerges first for small numbers (top rows) and progressively extends to larger numbers (bottom rows), regardless of curriculum strategy. This curriculum-invariant developmental pattern demonstrates that the small-to-large acquisition order reflects intrinsic computational constraints of the counting task rather than training order artifacts.

Figure 5: Neural population dynamics reveal number-line organization and rotational counting mechanism. (a) Tuning curves of representative number-selective units preferring numerosities 2, 5, and 8. (b) Population activity of positive (red) and negative (blue) numerosity detectors as a function of log(numerosity), showing logarithmic encoding ( $R^{2}=0.90$ and $0.89$ , respectively). (c) Distribution of preferred numerosities across units (left) and proportion of positive, negative, and non-selective units (right). (d) Neural distance between adjacent numbers decreases with magnitude, consistent with compressed number representations (logarithmic fit $R^{2}=0.45$ ). (e) Representational dissimilarity matrix (RDM) showing graded similarity structure along the number line. (f) Neural distance scales with numerical ratio (min/max), with both linear ( $R^{2}=0.85$ ) and Weber-law ( $R^{2}=0.84$ ) models providing good fits. (g) RSA correlation (left) and rotation quality (right) remain stable across training conditions (E3, E6, E9, E12) and data fractions. (h) PCA trajectories in Layer1 showing rotational dynamics at 100%, 50%, and 10% training data. (i) jPCA projections reveal clear rotational structure across data conditions. (j) Terminal positions in jPCA space show orderly arrangement by numerical magnitude. (k) Phase encoding: terminal jPCA phase correlates linearly with final count (100%: $r=0.974$ , slope $=30.6^{\circ}$ /count; 50%: $r=0.975$ ; 10%: $r=0.873$ ), demonstrating that counting corresponds to traversing a single rotational cycle.

Model	Layer	Conditions			10% Data			50% Data			100% Data
		Pretrain	Shuffle	Curriculum	RSA $\rho$	$R^{2}$	Quality	RSA $\rho$	$R^{2}$	Quality	RSA $\rho$	$R^{2}$	Quality
E3	L1	×	×	Random	0.66	0.87	0.77	0.72	0.84	0.58	0.74	0.86	0.58
E3	L2	×	×	Random	0.72	0.77	0.55	0.74	0.79	0.68	0.74	0.64	0.84
E6	L1	×	✓	Random	0.67	0.87	0.73	0.72	0.84	0.56	0.73	0.86	0.52
E6	L2	×	✓	Random	0.73	0.74	0.62	0.74	0.78	0.67	0.76	0.68	0.82
E9	L1	✓	×	Random	0.70	0.83	0.67	0.72	0.82	0.63	0.70	0.85	0.64

Table 2: Neural dynamics and representational structure across training conditions for both LSTM layers. RSA

\rho

= Spearman correlation between neural and numerical distance (all

p<0.001

);

R^{2}

= linear dynamics model fit quality; Quality = rotation quality score (0–1 scale). Rotational component explained 100% of dynamics variance in all conditions. Random curriculum, Epoch 250.

Figure 6: Overview of the experimental robotic platform. A Franka Emika Panda manipulator is equipped with an Intel RealSense D435i RGB-D camera for visual perception. The system performs object detection on colored balls (bottom-left inset) and controls the robot joints to sequentially reposition each object into the center of the camera’s field of view, enabling active perception and counting

Figure 7: Overview of three modeling paradigms: (a) an embodied model that integrates visual observations and motor signals over time using an LSTM to predict sequential states and actions; (b) single-image classification using a CNN; and (c) sequential-image classification, where multiple frames are processed by CNNs and aggregated (e.g., via pooling) to produce a final prediction.

Figure 8: Dataset distribution. Number of counting sequences for each numerosity (1–10), following a Zipf-like distribution.

Figure 9: Learning trajectory of all training configurations, presented as validation-accuracy curves over epochs in a unified multi-panel layout (six panels per row). Panel titles indicate experimental settings, while color and legend denote model families.

Figure 10: Heatmaps of per-number accuracy across training epochs for all experimental settings. Each panel corresponds to one result file, with epoch on the x-axis, number category (0–10) on the y-axis, and color indicating accuracy (0–1). Embodied Counting (EC) models (viridis colorbar, teal border) are labeled by training data proportion (10%/50%/100%), pre-training condition (pre0/pre1), shuffle joint setting (sj0/sj1), and curriculum order (easy-to-hard, hard-to-easy, or random). Single Image (SI) models (magma colorbar, purple border) are labeled by training data proportion, pre-training condition, and random seed, and serve as a non-embodied baseline for comparison.

Figure 11: Per-file jPCA trajectories for the 10% data condition, layer1. Each subplot corresponds to one CSV result file (single model setting at a specific epoch), arranged in a 6-column grid. Within each subplot, colored curves denote count-specific trajectories in the jPC1-jPC2 plane. This panel enables fine-grained comparison of rotational structure under low-data training.

Figure 12: Per-file jPCA trajectories for the 10% data condition, layer2. Each subplot corresponds to one CSV result file, arranged in a 6-column grid. The panel highlights how rotational dynamics in layer2 vary across curriculum orders, pre-training conditions, shuffle-joint settings, and epochs when training data is limited.

Figure 13: Per-file jPCA trajectories for the 50% data condition, layer1. Each subplot corresponds to one CSV result file, arranged in a 6-column grid. Compared with 10% training, this panel allows inspection of increased trajectory stability and organization under intermediate data availability.

Figure 14: Per-file jPCA trajectories for the 50% data condition, layer2. Each subplot corresponds to one CSV result file, arranged in a 6-column grid. This panel provides a condition-wise view of layer2 rotational geometry at intermediate data scale across all curricula and training setups.

Figure 15: Per-file jPCA trajectories for the 100% data condition, layer1. Each subplot corresponds to one CSV result file, arranged in a 6-column grid. This panel captures full-data rotational dynamics and supports direct comparison with the 10% and 50% conditions.

Figure 16: Per-file jPCA trajectories for the 100% data condition, layer2. Each subplot corresponds to one CSV result file, arranged in a 6-column grid. Together with the other five split panels and the global overlay figure, this panel completes the full comparison of jPCA rotational dynamics across data regimes, layers, and training configurations.

Embodied Cognition

Embodied cognition studies how thought, meaning, and learning arise through bodily perception and action in interaction with the environment.

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Abstract

Embodied Cognition

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