Jerome Bruner argued in 1964 that concepts must be traversed in three stages: enactive (bodily action), iconic (image), symbolic (language and notation). The order is not a preference — it is a developmental logic. Symbols that arrive before their sensorimotor grounding are thin; they may produce correct test performance while leaving the concept unrooted.

Maria Montessori, working fifty years before anyone had the vocabulary of embodied cognition, designed learning materials that implement Bruner’s sequence with unusual precision. The Golden Bead cube for “one thousand” is about the size of a large fist and weighs roughly one kilogram. You cannot represent “one thousand” on a tablet screen in a way that competes with carrying that weight across a room ten times.

This post is about what embodied cognition research tells us, why Montessori implements it correctly, and what we are giving up when we substitute glass surfaces for physical materials.

Bruner’s Three Modes

Jerome Bruner proposed in a 1964 paper and the subsequent book Toward a Theory of Instruction (Bruner, 1964; 1966) that knowledge is represented in three distinct, developmentally ordered modes:

Enactive: Knowledge encoded in action patterns. You know how to ride a bicycle; you cannot fully describe it in words; the knowledge is in your body. An infant knows what “cup” means because she has grasped cups hundreds of times — before she has the word.

Iconic: Knowledge encoded in images or perceptual representations. You can visualise the route without navigating it. You recognize a melody without playing it.

Symbolic: Knowledge encoded in language or other arbitrary symbol systems. The numeral “7” has no visual resemblance to seven objects. Its meaning is purely conventional and rule-governed.

The developmental sequence matters. A child who acquires a symbol before the underlying enactive and iconic representations are established has a label without a referent. She can produce the word or numeral correctly — and her understanding of it is correspondingly brittle. Transfer to novel contexts is poor; the concept does not generalise.

This is not a fringe view. It is the core claim of embodied cognition research, which has spent thirty years producing experimental evidence for it.

What Embodied Cognition Actually Shows

Lawrence Barsalou’s 2008 review in Annual Review of Psychology is the canonical synthesis (Barsalou, 2008). The central claim: cognition is not implemented in an abstract, modality-free computational system separate from the body. Perception, action, and interoception are constitutive of — not merely scaffolding for — conceptual thought. When you think about “lifting,” the motor cortex activates. When you think about “rough texture,” the somatosensory cortex activates. Concepts are grounded in the sensorimotor systems through which they were originally experienced.

This has a direct pedagogical implication. If mathematical concepts are represented using perceptual-motor simulation systems, then the quality of that simulation depends on the richness of the founding sensorimotor experience. A child who has handled physical objects of different weights has richer representational resources for arithmetic and measurement than one whose entire numerical experience has occurred on a flat, weightless, textureless glass surface.

Arthur Glenberg and colleagues tested this experimentally. In a 2004 study, first- and second-graders read short texts describing farm scenes (Glenberg et al., 2004). Children who physically moved toy objects (horse, barn, fence) to enact the described events showed dramatically better comprehension and inference performance than children who merely read and re-read the passages. The effect size approached two standard deviations in some conditions. Children who imagined moving the objects also improved, but less than those who actually moved them. The physical action was not decorative. It was causally relevant to understanding.

Glenberg extended this logic to arithmetic word problems (Glenberg, 2008). Children who physically manipulated objects while working through problems were better at identifying what was relevant and computing correct answers. The enactive engagement was improving not just memory of the text but mathematical reasoning.

Montessori Got There First

Maria Montessori opened the Casa dei Bambini on 6 January 1907 in a San Lorenzo tenement in Rome, enrolling approximately fifty children aged two to seven. She had no Barsalou. She had no Glenberg. She had children, materials, and the patience to watch what happened when children were allowed to choose their own work.

What she built was a pedagogical system that implements the Bruner sequence without exception.

The Golden Bead Material is the canonical example. Units: single glass beads. Tens: ten beads wired into a bar. Hundreds: ten bars wired into a flat square. Thousands: ten squares wired into a cube. The child can hold a unit bead between two fingers. She needs two hands to lift the thousand cube. The physical weight scales with place value. She experiences — proprioceptively — that “one thousand” is categorically heavier and larger than “one hundred” before she has seen the numeral or heard the word “thousands place.”

The Knobbed Cylinder Blocks illustrate a different principle. Four wooden blocks, each containing ten cylinders varying in height, diameter, or both. The child removes all cylinders and replaces them. If any cylinder goes into the wrong socket, the remaining cylinders will not all fit. The task cannot be completed incorrectly and left that way. Error control is mechanical, built into the material. The teacher need not intervene. The child corrects herself, alone, through the physical feedback of the materials.

Montessori called this controllo dell’errore — control of error. It is one of her most important insights: if the feedback is physical, the child internalises the standard rather than depending on external evaluation. The authority is in the material, not in the adult’s judgment.

The evidence that this works has accumulated across more than a century. Angeline Lillard and Nicole Else-Quest published a landmark study in Science in 2006, using a lottery-based design: children who had won a lottery to attend public Montessori schools compared with those who had not (Lillard & Else-Quest, 2006). Montessori five-year-olds showed significantly higher letter-word identification, phonological decoding, and applied mathematical problem-solving. The lottery controlled for family self-selection.

A 2025 national randomised controlled trial — 588 children across 24 public Montessori schools, with lottery-based assignment — found significant advantages in reading, short-term memory, executive function, and social understanding at the end of kindergarten, with effect sizes exceeding 0.2 SD (Lillard et al., 2025). These are not small effects for field-based school research. And the costs per child were lower than conventional programmes.

Korczak and the Right to Make Mistakes

Janusz Korczak ran an orphanage in Warsaw and wrote How to Love a Child in 1919 (Korczak, 1919) and The Child’s Right to Respect in 1929 (Korczak, 1929). His central argument was that children are not pre-adults — they are persons with full moral status and a right to their own experience, including the experience of making mistakes.

In August 1942 German soldiers came to his orphanage. Korczak was offered false papers, safe houses, multiple escape routes arranged by friends and admirers. He refused each time. He led approximately 192 children and staff to the Umschlagplatz and did not return.

I mention Korczak not as an appeal to emotion but because his argument is structurally connected to Montessori’s. If a child has moral status, she has the right to encounter the actual consequences of her choices — including physical ones. A material that makes incorrect placement physically impossible before the child has had the experience of trying and correcting is a different kind of education from a screen that prevents error altogether through invisible software constraints, or one that simply supplies the correct answer.

Error is information. Physical error is particularly rich information. Taking it away is not protection — it is impoverishment.

Buber: What a Screen Cannot Offer

Martin Buber’s essay “Education,” delivered as an address in 1925 and published in Between Man and Man (Buber, 1947), argues that genuine education requires what he calls an I-Thou relation: an encounter in which the other is met as a whole, irreducible subject, not an object to be managed.

A touchscreen is the paradigmatic I-It relation. It is smooth, frictionless, optimised for engagement, responsive to exactly the touch it was designed to respond to. There is no otherness, no resistance, no genuine encounter. The screen does not push back. The Knobbed Cylinder Block does — literally. If you try to force a cylinder into the wrong socket, the material resists. That resistance is not a flaw in the pedagogical design; it is the pedagogical design.

Buber also introduced the concept of Umfassung — inclusion — by which a teacher must simultaneously stand at their own pole of the educational encounter and imaginatively experience the pupil’s side. A screen cannot do this. It has no pole. Its responsiveness is a simulation of attention, not attention itself. Turkle’s later phrase — “simulated empathy is not empathy” — is the same argument in a different register.

The Tablet Problem

The educational technology industry has produced an enormous quantity of “educational apps” for young children. The research is beginning to catch up.

Kathy Hirsh-Pasek and colleagues identified four pillars that distinguish educational from merely entertaining digital content: active engagement, depth of engagement, meaningful learning, and social interactivity (Hirsh-Pasek et al., 2015). Reviewing commercially available apps, they found that most fail on three or four of these criteria. They produce interactions in the shallow sense — tapping, swiping — without the kind of self-directed, goal-oriented, socially-embedded activity that drives genuine cognitive development.

A 2021 meta-analysis of 36 intervention studies found that educational apps produced meaningful gains when measured by researcher-developed instruments targeting constrained skills (letter naming, counting), but small to negligible effects on standardised achievement tests (Kim et al., 2021). The apps teach what they teach. Transfer is limited.

By contrast, a 2023 scoping review of 102 studies found that physical manipulatives — block building, shape sorting, paper folding, figurine play — showed consistent benefits across mathematics, literacy, and science that transferred to standardised measures (Byrne et al., 2023).

The fundamental problem is haptic. A 2024 review of haptic technology in learning found that force feedback and texture information substantially improve spatial reasoning, interest, and analytical ability (Hatira & Sarac, 2024). Standard capacitive touchscreens — every tablet your child has encountered — provide no force feedback and no texture differentiation. Every object, regardless of its symbolic “weight” or “size,” feels identical under the fingertip.

The Golden Bead thousand cube weighs approximately one kilogram. You cannot represent that experience on a tablet. The symbol arrives without the sensation, and Bruner’s sequence is violated from the first tap.

What We Should Ask

The question is not whether tablets have educational uses — they clearly do, particularly for older children working at the iconic and symbolic levels, and for content where direct physical manipulation is impossible or dangerous. The question is whether we are using them in developmental contexts where the enactive stage has not yet been established.

A child who has carried the thousand cube across a room, stacked the hundreds into the square, and felt the weight difference in her hands has a different representation of place value from one who has tapped numerals on a flat screen. Both may perform identically on a constrained test tomorrow. Ask them a transfer question in six months and the difference will appear.

We are teaching children to operate symbols before giving them the physical experiences that make those symbols mean anything. The result is not ignorance — the children can tap the correct numeral — but brittleness. The concept is a label, not a root.

Montessori knew this. Bruner formalised it. The haptics literature is now confirming it experimentally. The difficult question is why we are still buying flat glass rectangles for classrooms when a box of wooden cylinders costs less and works better.

References

• Bruner, J. S. (1964). The course of cognitive growth. American Psychologist, 19(1), 1–15.
• Bruner, J. S. (1966). Toward a Theory of Instruction. Harvard University Press (Belknap Press).
• Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645. DOI: 10.1146/annurev.psych.59.103006.093639
• Glenberg, A. M., Gutierrez, T., Levin, J. R., Japuntich, S., & Kaschak, M. P. (2004). Activity and imagined activity can enhance young children’s reading comprehension. Journal of Educational Psychology, 96(3), 424–436. DOI: 10.1037/0022-0663.96.3.424
• Glenberg, A. M. (2008). Embodiment for education. In P. Calvo & A. Gomila (Eds.), Handbook of Cognitive Science: An Embodied Approach (pp. 355–371). Elsevier.
• Lillard, A. S., & Else-Quest, N. (2006). The early years: Evaluating Montessori education. Science, 313(5795), 1893–1894. DOI: 10.1126/science.1132362
• Lillard, A. S., Loeb, D., Berg, J., Escueta, M., Manship, K., Hauser, A., & Daggett, E. D. (2025). A national randomized controlled trial of the impact of public Montessori preschool at the end of kindergarten. Proceedings of the National Academy of Sciences, 122(43). DOI: 10.1073/pnas.2506130122
• Korczak, J. (1919). Jak kochać dziecko [How to Love a Child]. Warsaw.
• Korczak, J. (1929). Prawo dziecka do szacunku [The Child’s Right to Respect]. Warsaw.
• Buber, M. (1947). Between Man and Man (trans. R. G. Smith). Kegan Paul. (Original German publication 1947; contains “Education,” address delivered 1925, and “The Education of Character,” address delivered 1939.)
• Hirsh-Pasek, K., Zosh, J. M., Golinkoff, R. M., Gray, J. H., Robb, M. B., & Kaufman, J. (2015). Putting education in “educational” apps: Lessons from the science of learning. Psychological Science in the Public Interest, 16(1), 3–34. DOI: 10.1177/1529100615569721
• Kim, J. S., Gilbert, J., Yu, Q., & Gale, C. (2021). Measures matter: A meta-analysis of the effects of educational apps on preschool to grade 3 children’s literacy and math skills. AERA Open, 7. DOI: 10.1177/23328584211004183
• Byrne, E. M., Jensen, H., Thomsen, B. S., & Ramchandani, P. G. (2023). Educational interventions involving physical manipulatives for improving children’s learning and development: A scoping review. Review of Education, 11(2), e3400. DOI: 10.1002/rev3.3400
• Hatira, A., & Sarac, M. (2024). Touch to learn: A review of haptic technology’s impact on skill development and enhancing learning abilities for children. Advanced Intelligent Systems, 6. DOI: 10.1002/aisy.202300731

Changelog

• 2026-02-03: Changed “lottery-based quasi-experimental design” to “lottery-based design” for Lillard & Else-Quest (2006). A lottery provides genuine random assignment; “quasi-experimental” implies the absence of randomisation, which is the opposite of what the lottery design achieved.

A three-year-old cannot put her shoes on before her socks. Not because she lacks motor skills — because the operations do not commute.

The same structural constraint, dressed in the language of operators on a Hilbert space, is why Heisenberg’s uncertainty principle holds. This post is about that connection: the accidental algebra lesson built into getting dressed, and why the physicists of 1925 had to abandon one of arithmetic’s most taken-for-granted assumptions.

Getting Dressed Is a Non-Abelian Problem

Start with the mundane. Your morning routine imposes a strict partial order on operations: underwear before trousers, socks before shoes, cap before chin-strap if you cycle. Try reversing any pair and the sequence fails — physically, not just socially. You cannot pull a sock over a shoe.

The operation “put on socks” followed by “put on shoes” produces a wearable human; the reverse produces neither, and no amount of wishing commutativity into existence will help.

In the language of abstract algebra, two operations \(A\) and \(B\) commute if \(AB = BA\) — if doing them in either order yields the same result. Everyday life is full of operations that do not commute: rotate a book 90° around its vertical axis then 90° around its horizontal axis; now reverse the order. The final orientations differ. Turn right then turn left while driving; left then right. Different positions.

The intuition is not hard to build. What is surprising is how rarely we note it, and what it costs us when we finally hit a domain — quantum mechanics — where non-commutativity is not an inconvenient edge case but the central fact.

Piaget Said Seven; Toddlers Disagreed

Jean Piaget argued that children do not acquire operational thinking — the ability to mentally perform and reverse sequences of actions — until the concrete operational stage, roughly ages seven to eleven (Inhelder & Piaget, 1958). Before that, he claimed, children lack the understanding that an operation can be undone or reordered.

Post-Piagetian research pushed back hard. Patricia Bauer and Jean Mandler tested infants aged sixteen and twenty months on novel, multi-step action sequences (Bauer & Mandler, 1989). For causally structured sequences — where step A physically enables step B — infants reproduced the correct order after a two-week delay. They were not told the order was important. They had no language to encode it. They just knew, implicitly, that the operations had a necessary direction.

A 2020 study by Klemfuss and colleagues tested 100 children aged roughly two-and-a-half to five on temporal ordering questions (Klemfuss et al., 2020). Children answered “what happened first?” questions correctly 82% of the time. The errors that did appear followed an encoding-order bias — children defaulted to reporting the next event in the sequence as originally experienced, regardless of what was asked. The ordering knowledge was intact. What children lack, for Piaget’s full seven years, is the formal recursive conception of reversibility. The procedural knowledge — that some sequences must be done in the right order and cannot be freely rearranged — is there from the second year of life.

Which means: learning that \(AB \neq BA\) is not learning something exotic. It is articulating something the nervous system already knows.

The Mathematician’s Commutator

Abstract algebra formalized this intuition in the nineteenth century. A group is abelian (commutative) if every pair of elements satisfies \(ab = ba\). Integers under addition: abelian. Rotations in three dimensions: not.

Arthur Cayley’s 1858 memoir established matrix algebra as a formal theory (Cayley, 1858). Multiply two \(2 \times 2\) matrices:

\(AB \neq BA\). Non-commutativity is not a curiosity; it is the generic condition for matrix products. Commutativity is the special case — and requiring justification.

William Rowan Hamilton had already gone further. On 16 October 1843, walking along the Royal Canal in Dublin, he discovered the quaternions and carved their multiplication rule into the stone of Broom Bridge:

From this it follows immediately that \(ij = k\) but \(ji = -k\). Hamilton’s four-dimensional number system — the first algebraic structure beyond the complex numbers — was non-commutative by construction. He did not apologize for it. He celebrated it.

The Lie algebra structure underlying these commutator relations is the same skeleton that governs Messiaen’s modes of limited transposition, which I traced in a previous post on group theory and music — a very different physical domain, but identical algebraic machinery.

Born, Jordan, and the Physicist’s Shock

Classical mechanics treats position \(x\) and momentum \(p\) as ordinary real numbers. Real numbers commute: \(xp = px\). The Poisson bracket \(\{x, p\} = 1\) encodes a classical relationship, but the underlying quantities are scalars, and scalars commute.

In July 1925, Werner Heisenberg published a paper that could not quite bring itself to say what it was doing (Heisenberg, 1925). He replaced classical dynamical variables with arrays of numbers — what we would now call matrices — and found, uncomfortably, that the resulting quantum condition required order to matter.

While Heisenberg was on vacation, Max Born and Pascual Jordan finished the translation into matrix language (Born & Jordan, 1925). They wrote the commutation relation explicitly, recognized it as the fundamental law, and showed that it reproduced the known quantum results:

Non-commutativity of position and momentum was not a mathematical accident. It was the theory.

The uncertainty principle followed four years later as a theorem, not an additional postulate. Howard Robertson proved in 1929 that for any two observables \(\hat{A}\) and \(\hat{B}\), the Cauchy–Schwarz inequality on Hilbert space yields (Robertson, 1929):

Substituting \(\hat{A} = \hat{x}\), \(\hat{B} = \hat{p}\), \([\hat{x}, \hat{p}] = i\hbar\):

This is the uncertainty principle. It does not say nature is fuzzy or that measurement disturbs systems in some vague intuitive sense. It says: position and momentum are operators that do not commute, and the Robertson inequality then constrains their joint variance. Non-commutativity is the uncertainty principle. Put the shoes on before the socks and the state is not defined.

The same logic applies to angular momentum. The three components satisfy:

This is the Lie algebra \(\mathfrak{su}(2)\). You cannot simultaneously determine two components of angular momentum to arbitrary precision — not because the measurement apparatus is noisy, but because the operations of measuring them do not commute.

The fiber bundle language that underlies these rotation groups also appears, in different physical dress, in the problem of the falling cat and geometric phases — another case where the order of rotations has non-trivial physical consequences (see that post).

Connes and Non-Commutative Space

Alain Connes asked what happens if we allow the coordinates of space itself to be non-commutative. In ordinary geometry, the algebra of coordinate functions on a manifold is commutative: \(f(x) \cdot g(x) = g(x) \cdot f(x)\). Connes’ non-commutative geometry replaces this with a spectral triple \((\mathcal{A}, \mathcal{H}, D)\): an algebra \(\mathcal{A}\) of operators (possibly non-commutative) acting on a Hilbert space \(\mathcal{H}\), with a generalized Dirac operator \(D\) encoding the geometry (Connes, 1994).

The payoff was remarkable. With Ali Chamseddine, Connes showed that if \(\mathcal{A}\) is chosen as a specific non-commutative product of the real numbers, complex numbers, quaternions, and matrix algebras, the spectral action principle reproduces the full Lagrangian of the Standard Model coupled to general relativity from a single geometric principle (Chamseddine & Connes, 1996). The Higgs field, the gauge bosons, the graviton: all from the geometry of a non-commutative space.

Classical geometry is the special case where the coordinate algebra is commutative. Drop that assumption and you open up a vastly richer landscape. Quantum mechanics lives in that landscape. Possibly, so does the structure of spacetime at the Planck scale.

The Lesson Pre-Schoolers Already Know

There is an irony here that I cannot quite leave alone. Students learning linear algebra for the first time consistently make the same mistake. Anna Sierpinska documented it carefully: they assume \(AB = BA\) for matrices because they have spent years in arithmetic and scalar algebra where multiplication commutes (Sierpinska, 2000). The commutativity of ordinary multiplication is so deeply internalized that abandoning it feels like breaking a rule.

But the pre-schooler in the sock-and-shoe scenario never had that problem. Her procedural memory, documented in infants as young as sixteen months by Bauer and Mandler, encoded the correct asymmetry directly. The order of operations is the first thing a developing mind learns about actions in the world, before the arithmetic of school teaches it the convenient fiction that order is irrelevant.

Arithmetic is the outlier. \(3 + 5 = 5 + 3\) because counting does not depend on where you start. But putting on clothes, multiplying matrices, rotating rigid bodies, measuring quantum observables: these operations carry memory of order, and they repay the attention a child already brings to them before she can name a number.

The universe is non-abelian. We are born knowing it. School briefly convinces us otherwise. Physics eventually agrees with the pre-schooler.

References

• Inhelder, B., & Piaget, J. (1958). The Growth of Logical Thinking from Childhood to Adolescence. Basic Books.
• Bauer, P. J., & Mandler, J. M. (1989). One thing follows another: Effects of temporal structure on 1- to 2-year-olds’ recall of events. Developmental Psychology, 25, 197–206.
• Klemfuss, J. Z., McWilliams, K., Henderson, H. M., Olaguez, A. P., & Lyon, T. D. (2020). Order of encoding predicts young children’s responses to sequencing questions. Cognitive Development, 55, 100927. DOI: 10.1016/j.cogdev.2020.100927
• Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17–37. DOI: 10.1098/rstl.1858.0002
• Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik, 33, 879–893.
• Born, M., & Jordan, P. (1925). Zur Quantenmechanik. Zeitschrift für Physik, 34, 858–888.
• Robertson, H. P. (1929). The uncertainty principle. Physical Review, 34, 163–164. DOI: 10.1103/PhysRev.34.163
• Connes, A. (1994). Noncommutative Geometry. Academic Press. ISBN 0-12-185860-X.
• Chamseddine, A. H., & Connes, A. (1996). Universal formula for noncommutative geometry actions: Unification of gravity and the standard model. Physical Review Letters, 77, 4868–4871. DOI: 10.1103/PhysRevLett.77.4868
• Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the Teaching of Linear Algebra (pp. 209–246). Kluwer Academic Publishers. DOI: 10.1007/0-306-47224-4_8

Changelog

• 2026-02-03: Corrected the age range for the Klemfuss et al. (2020) study from “two to four” to “roughly two-and-a-half to five” — the actual participants were aged 30–61 months.
• 2026-02-03: Updated the characterisation of Klemfuss et al. (2020) findings to reflect the paper’s central result: errors follow an encoding-order bias (children default to the next event in encoding sequence). The paper’s title — “Order of encoding predicts young children’s responses” — names the mechanism.