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.

This story is not supported by the evidence.

The relationship between socioeconomic background and educational outcomes is one of the most replicated findings in social science. PISA data from Germany consistently show one of the steepest socioeconomic gradients in the OECD — the correlation between parental education and student performance is higher here than in most comparable countries. This is not a recent finding. It has been stable for decades. The system produces it reliably, which means the system is, in some meaningful sense, designed to produce it — even if no individual actor intended that design.

Understanding why requires a different vocabulary than the one most educational institutions use about themselves.

Bourdieu’s Three Capitals

Pierre Bourdieu spent much of his career developing an account of how social inequality reproduces itself through culture and education. The core concept is capital — but not only the economic kind.

Bourdieu (1986) distinguishes three forms:

Economic capital is material resources: money, assets, time purchased through money. This is the most visible form of advantage. Wealthier families can pay for tutoring, for better-resourced schools, for the unpaid internships that build CVs, for the years of postgraduate study that increasingly function as the entrance requirement for professional careers.

Cultural capital is more subtle. It includes dispositions, skills, and knowledge that are valued by educational institutions and professional fields — but valued in a way that tends to favour those who acquired them at home, in childhood, before formal education began. The ease with which a student navigates a seminar. The familiarity with the tacit conventions of academic writing. The sense that the university is, broadly, a place made for people like you. These are not things that are explicitly taught; they are things that are transmitted, Bourdieu argues, through families whose own cultural capital aligns with what the institution expects.

Social capital is networks: the web of relationships that provide information, referrals, opportunities, and vouching. Who you know, in the flattest possible terms.

All three reinforce each other. Economic capital can be converted into cultural capital through education and into social capital through exclusive networks. Cultural capital eases access to prestigious institutions, which build social capital. The system is not static, but it has a strong gravitational pull toward reproduction.

Bourdieu and Passeron (1977) developed this into a theory of education as reproduction: the function of educational institutions is not primarily to transmit knowledge but to legitimate the transmission of social position from one generation to the next. The process is misrecognised — by students, teachers, and institutions — as meritocracy. This misrecognition is essential to the function. If it were transparent, it would lose its legitimising power.

The Hidden Curriculum

Philip Jackson (1968) coined the term hidden curriculum for everything that schools teach that is not in the official syllabus. How to sit still. How to wait your turn. How to speak to authority. How to navigate institutions, read implicit expectations, manage bureaucracies. How to understand that your job is to demonstrate competence within a form that someone else has set.

For students whose home culture aligns with the institutional culture, the hidden curriculum is invisible. They already know it; it requires no effort; it is simply how things are. For students whose home culture diverges, it is a second curriculum that must be decoded while simultaneously managing the official one.

Lareau (2003) documented this in careful ethnographic detail. Middle-class families engage in what she calls concerted cultivation — a mode of child-rearing that practises precisely the dispositions valued by educational institutions: articulate self-advocacy with adults, a sense of entitlement to ask questions and seek explanations, activities structured around developing discrete skills. Working-class and poor families, in her study, more often practised accomplishment of natural growth — providing security, affection, and freedom without the institutional structuring. Neither is better parenting. But one of them is what the school expects.

The child who arrives at school already knowing how to talk to teachers, how to present themselves, how to advocate for their own needs, has a significant advantage that is invisible in the transcript. It does not appear as “privilege”; it appears as “ability” or “maturity”. The institutional category does the misrecognising work.

Privilege as Invisible to Those Who Have It

Peggy McIntosh (1989) wrote what became one of the most cited — and most contested — essays in education: “White Privilege: Unpacking the Invisible Knapsack”. Her core observation is structural: privilege is the absence of disadvantage, and absences are invisible to those who live inside them. You do not notice the ease with which you move through a system that was designed for people like you, any more than you notice breathing.

This is not an accusation. It is a description of a structural feature with consequences for self-understanding.

My background is in physics; I now work in universities, having grown up in a household with books and educated parents and the background assumption that higher education was something that people like us did. I was not aware of most of this as an advantage while it was happening, because it did not feel like an advantage — it felt like normal. The awareness came later, with effort, and it remains incomplete.

The invisible entrance fee is what you have already paid, in cultural capital, before you walk through the door. The institution does not ask about it explicitly. It simply rewards those who have it and attributes the reward to merit.

What This Means for Accessibility

The previous post in this series argued that full accessibility — Barrierefreiheit — is structurally impossible in a society organised as ours is; that the honest goal is Barrierearmut, the ongoing reduction of barriers. The connection to privilege is direct.

Barriers to education are not only physical. They include the cultural distance between the home environment and the institutional culture. They include not knowing that office hours exist and are meant for you, not just for students with problems. They include the inability to identify as “the kind of person who does a PhD” because you have never met anyone who did one. They include the exhaustion of navigating a system that requires you to translate yourself at every step, while your better-resourced peers spend that cognitive energy on the actual work.

None of these barriers appear on an accessibility audit. They are not visible from inside the institution looking out. They require actively listening to people whose experience differs from the institutional default, and then being willing to revise the default rather than add an exception.

The PISA gradient in Germany is a measurement of accumulated, unreduced barriers. It is not a measurement of the distribution of talent or effort. The system is producing the outcome; the students are receiving the label.

The Meritocracy Problem

Meritocracy is an appealing concept and a damaging ideology when taken seriously. The appeal: rewards should go to those who earn them, and earning should depend on effort and ability rather than inherited position. This is genuinely better than aristocracy.

The problem: in a society with steep inequality in the distribution of cultural, economic, and social capital, “merit” is not a neutral measurement. It is a measurement of the match between a person’s accumulated resources and the demands of the institution. Calling that match “merit” names the outcome without naming the process that produced it.

Michael Young, who invented the word “meritocracy” in 1958, intended it as a satire. His book The Rise of the Meritocracy depicted a dystopia in which the illusion of fairness made inequality more entrenched, not less, because it stripped the legitimacy from those who were left behind. If outcomes are fair, then failure is your fault. The ideology provides the institutional absolution; the individuals bear the moral weight of structural disadvantage.

This is precisely the dynamic that Bourdieu’s theory of misrecognition describes. The student from a poorly resourced background who does not reach the outcomes of their better-resourced peer is seen — by themselves, by teachers, by the institution — as less talented or less motivated, rather than as navigating a steeper gradient with fewer tools.

What Institutions Can Actually Do

The structural critique is not an argument for fatalism. Institutions can do things that matter.

They can make the hidden curriculum visible — explicitly teaching what is usually assumed. That means orientation programmes that actually explain institutional culture, not just procedures. It means academic writing support that is not remedial but normative. It means mentoring that connects first-generation students with people who understand the landscape.

They can audit their practices for whose default they assume. The timed closed-book exam was designed for a particular set of conditions; asking what it actually measures, and whether there are better instruments, is not lowering standards — it is interrogating what the standard is measuring.

They can diversify their faculty and staff, not as a cosmetic gesture but as a structural change in whose tacit knowledge is embedded in the institution. If the people who design the curriculum all navigated it from the same starting position, the curriculum will encode that starting position as normal.

They can name the entrance fee. Acknowledging that outcomes correlate with background, that this is a systemic feature and not a distribution of merit, is the first step toward taking institutional responsibility for the gradient rather than attributing it to the students.

None of this resolves the structural problem. The structural problem requires political change at scales well beyond any individual institution. But institutions are not passive. They can reduce the barriers they control, while being honest about the ones they do not.

A Personal Note

I sit in institutional positions that this analysis would identify as advantaged. I teach in a university. I benefited from the gradient in ways I cannot fully account for. The point of naming this is not guilt; it is responsibility. Being advantaged by a system you did not design does not make you complicit in its worst outcomes — but it does make you responsible for using whatever institutional leverage you have to make the system less exclusive.

The connection to accessibility is this: both inaccessibility and privilege are about whose defaults are built into the system and who is required to adapt to defaults they did not set. Reducing barriers and interrogating privilege are the same project, approached from different angles.

Neither is completable. Both are necessary.

References

• Bourdieu, P. (1986). The forms of capital. In J.G. Richardson (Ed.), Handbook of Theory and Research for the Sociology of Education (pp. 241–258). Greenwood Press.
• Bourdieu, P. & Passeron, J.C. (1977). Reproduction in Education, Society and Culture. Sage. (Original French edition 1970.)
• Jackson, P.W. (1968). Life in Classrooms. Holt, Rinehart and Winston.
• Lareau, A. (2003). Unequal Childhoods: Class, Race, and Family Life. University of California Press.
• McIntosh, P. (1989). White privilege: Unpacking the invisible knapsack. Peace and Freedom, July/August, 10–12.
• OECD (2023). PISA 2022 Results (Volume I): The State of Learning and Equity in Education. OECD Publishing.
• Young, M. (1958). The Rise of the Meritocracy. Thames and Hudson.

I want to argue that this is not only unachievable in practice — which most people in the field will readily concede — but structurally impossible in a society organised the way ours is. The honest term is Barrierearmut: poverty of barriers, reduction of barriers, a direction rather than a destination. The difference is not just linguistic. It shapes what we promise, what we measure, and what we allow ourselves to stop doing.

Two Models of Disability

The medical model of disability, which dominated institutional thinking for most of the twentieth century, locates the problem in the individual. A person is disabled by their impairment — by the deafness, the mobility limitation, the cognitive difference. The solution, in this frame, is treatment, cure, rehabilitation: changing the person to fit the world.

The social model, developed in the 1970s by disability activists — particularly through the work of the Union of the Physically Impaired Against Segregation in the UK — inverts this (UPIAS, 1976). The distinction is between impairment (a physical or cognitive difference) and disability (the disadvantage created by a society that does not account for that difference). A wheelchair user is not disabled by their legs; they are disabled by a building with no ramp. A deaf student is not disabled by their hearing; they are disabled by a lecture delivered without captioning.

Oliver (1990) developed this into a full political framework. Disability is not a medical category but a social relation — a product of how societies organise space, communication, labour, and meaning. The implication is radical: to address disability, you do not fix the person; you change the society.

This model has transformed disability law, architecture, and educational policy. The UN Convention on the Rights of Persons with Disabilities (2006) is explicitly built on it. WCAG — the Web Content Accessibility Guidelines — embodies it for digital environments. The Behindertengleichstellungsgesetz in Germany draws on it.

And yet.

The Limit of the Social Model

The social model is politically necessary and descriptively powerful. It is also incomplete.

Shakespeare and Watson (2002) offer a careful critique: the strict social model, in its effort to relocate disability from body to society, ends up treating impairment as irrelevant — as a neutral fact that only becomes disabling through social organisation. But impairment is not neutral. Pain is real. Fatigue is real. Cognitive load is real. Some impairments impose limits that no architectural or digital intervention fully removes, because the limits are not externally imposed but intrinsic to how a particular nervous system processes the world.

The WHO’s International Classification of Functioning, Disability and Health (ICF, 2001) offers a biopsychosocial synthesis: disability as an interaction between health condition, body function and structure, activity, participation, and contextual factors (both environmental and personal). This is less politically clean than the social model — it does not attribute all disablement to society — but it is more honest about the complexity.

The point is not to retreat from the social model’s insights but to acknowledge that “removing all barriers” is an incomplete goal even in its own terms. Impairment is real; context is transformable; and the interaction between them is irreducibly particular. There is no single intervention that produces accessibility for everyone.

Why Barrierefreiheit Is a False Promise

Consider what full accessibility would require. It would require physical spaces that accommodate every mobility profile, every sensory profile, every energy and endurance pattern. It would require information architectures that are simultaneously navigable by users with very different cognitive and perceptual systems. It would require communication norms, cultural contexts, and institutional practices that do not privilege any particular neurotype, any particular communication style, any particular relationship to time and deadlines and social convention.

None of that is achievable in a society with the historical sediment ours has. Our cities were built for able-bodied adults with average sensory capacity and without requirement for cognitive accessibility. Our universities were built — institutionally, not just physically — for a particular kind of learner with a particular kind of background, deploying a particular kind of intelligence. Retrofitting accessibility onto these structures is possible, valuable, and necessary. But it is not the same as having built for full human variation from the start. The ramp bolted onto the side of the neoclassical building solves the wheelchair problem and leaves everything else intact.

Kafer (2013) makes a more radical version of this argument. The concept of “normal” function — the standard against which accessibility is measured — is not neutral. It encodes a history of who was considered the default human, and who was considered an exception requiring accommodation. Achieving “accessibility” within a framework that still treats certain bodies and minds as exceptions to be accommodated does not escape that framework; it manages it.

This is why a building can pass every accessibility audit and still function as an excluding institution. The audit measures physical features. It does not measure whether disabled students are welcomed into the culture of the institution, whether their modes of participation are genuinely valued, whether the hidden curriculum of “how to be a student” is legible to someone whose processing differs from the assumed default.

What Barrierearmut Means

If Barrierefreiheit is the impossible promise, Barrierearmut — barrier reduction — is the honest goal. It is not lesser. It is more accurate.

Barrier reduction as a framework asks: which barriers, for which people, with which effects, can be reduced through which interventions, at what cost, with what trade-offs? It treats accessibility as an ongoing practice rather than a checkable state. It acknowledges that every design decision — physical, digital, institutional — makes some things easier for some people and harder for others, and that the question is always whose needs are centred and whose are treated as exceptions.

Universal Design (Mace, 1997) moves in this direction: designing from the start for the broadest range of users, rather than designing for the norm and retrofitting for exceptions. A kerb cut is the standard example — designed for wheelchair users, also useful for people with pushchairs, luggage, bicycles, temporary injuries. But Universal Design, honestly applied, acknowledges that no design is truly universal. Every design embeds assumptions. The honest goal is to minimise the distance between those assumptions and the actual diversity of users.

For digital environments this is particularly visible. WCAG 2.2 defines four principles — Perceivable, Operable, Understandable, Robust — and success criteria that can be tested against. Meeting WCAG AA is a meaningful achievement. It is not the same as being accessible to all users. Screen reader users with different software behave differently with the same page. Cognitive accessibility — making content understandable, not just perceivable — is addressed by WCAG 3.0 drafts but is notoriously difficult to operationalise. The standards improve; the gap remains.

Institutional Honesty

I work in a university. Universities have accessibility offices, procedures, documentation requirements. A student with a disability can request accommodations: extended exam time, written materials in accessible formats, individual arrangements. These accommodations are real and valuable. They are also, structurally, a system for managing exceptions to a norm that the institution has no intention of revising.

The student who needs extended time is asking the institution to adjust its standard procedure for their case. The institution does so, often generously. But the standard procedure — the timed exam, the lecture format, the office-hours model — remains the standard. The exception is granted; the norm persists. This is barrier management, not barrier reduction.

Barrier reduction would mean asking, as a matter of institutional practice: what is the actual pedagogical purpose of the timed exam, and are there better ways to assess that competency that do not exclude students whose processing differs? It would mean asking what the lecture format assumes about the listener, and whether those assumptions are necessary. These questions are uncomfortable because they challenge practices that are also convenient, and because the people who benefit from the current norms are the ones with the institutional power to change them.

This is not a problem unique to universities. It is the general structure of the problem.

A Direction, Not a Destination

I am not arguing for giving up on accessibility work. The opposite. I am arguing that naming the goal honestly — barrier reduction, not barrier freedom — produces better practice than the false promise of an achievable endpoint.

Barrierefreiheit as a legal standard can be met by a compliant building that is still a hostile institution. Barrierearmut as a practice requires continuous attention to who is being excluded and by what, and ongoing effort to reduce that exclusion knowing that it will never be complete.

That is harder. It does not allow the institution to certify itself as done. It requires asking the uncomfortable questions about whose default is encoded in the design — a question that leads, quickly, to the question of privilege.

That is the next post: The Invisible Entrance Fee: On Privilege, Education, and the Institutions That Reproduce Both.

References

• Kafer, A. (2013). Feminist, Queer, Crip. Indiana University Press.
• Mace, R.L. (1985). Universal Design: Barrier Free Environments for Everyone. Designers West, 33(1), 147–152.
• Oliver, M. (1990). The Politics of Disablement. Macmillan.
• Shakespeare, T. & Watson, N. (2002). The social model of disability: an outdated ideology? Research in Social Science and Disability, 2, 9–28.
• UPIAS (1976). Fundamental Principles of Disability. Union of the Physically Impaired Against Segregation.
• WHO (2001). International Classification of Functioning, Disability and Health (ICF). World Health Organization.
• UN General Assembly (2006). Convention on the Rights of Persons with Disabilities (A/RES/61/106).

Changelog

• 2025-11-05: Corrected the Mace reference from (1997) Designers West 44(1) to (1985) Designers West 33(1), 147–152. The year 1997 relates to the separate “Principles of Universal Design” publication by Connell, Jones, Mace et al. at NC State, not the Designers West article.

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.

The short answer is that YouTube does not work fine, for reasons that matter specifically to universities. The longer answer is what this post is about.

What Is Wrong with YouTube

YouTube is the world’s dominant video platform. It is free to use, globally available, handles any file size, transcodes automatically, and comes with an audience of two billion logged-in users. For individual creators who want reach, it is genuinely hard to beat.

For universities it fails in at least three important ways.

Data protection. When you upload a lecture or a concert recording to YouTube, the content, the metadata, and the viewing behaviour of your students go to Google’s servers — which are predominantly in the United States. Under the GDPR, transferring personal data to third countries requires either an adequacy decision, standard contractual clauses with additional safeguards, or explicit informed consent. After the Schrems II ruling (Court of Justice of the EU, 2020), the adequacy of US-based data transfers became legally contested in a way that makes institutional YouTube use genuinely difficult for European universities. Using it for anything with identifiable students — which includes most teaching content — is a compliance problem.

Platform logic. YouTube is designed to maximise watch time. Its recommendation algorithm is not neutral. It will recommend whatever comes after your lecture, and what comes after your lecture is not under your control. For educational content — especially sensitive material, or content that should remain in a defined pedagogical context — this is a real problem. The platform is not indifferent to what is hosted on it; it shapes how it is consumed.

Institutional fragility. YouTube is free until it isn’t. Platform terms change; monetisation policies change; content is demonetised or removed based on automated systems with imperfect appeal mechanisms. Building institutional infrastructure on a free commercial service is a bet that the commercial incentives of that service will remain aligned with your needs. That bet has a poor historical record.

None of this means that individual instructors should never use YouTube. It means that universities should not make YouTube their default institutional solution.

educast.nrw: A Cooperative Model

educast.nrw is a project of the Digitale Hochschule NRW — a cooperative of North Rhine-Westphalian universities building shared digital infrastructure. The concept is a state-wide video service, run by universities for universities, for the recording, processing, management, and distribution of video content in teaching and research. The platforms calls itself “Hochschul-YouTube”, which is both accurate and slightly underselling what makes it different.

The HfMT Köln participates as a user institution, which is where I come in as the IT contact for setting it up on our end.

The technical foundation is Opencast, an open-source video management system developed by a consortium of universities. This matters: the software is auditable, the development direction is set by the institutions that use it rather than by advertising revenue, and the infrastructure runs on German servers that are explicitly GDPR-compliant. Licenses on uploaded content are freely choosable — CC-BY-SA is an option, which means the university’s teaching materials can be open access if that is what the instructor wants.

What It Can Do

The feature set covers the actual use cases of a university, not the use cases of a content creator trying to build a following.

Recording. The Opencast Studio browser app records in three modes: screen only, camera only, or screen and camera simultaneously as a synchronised multi-stream. That last option — Presentation and Presenter as separate streams, played back with the viewer switching focus between them, or as picture-in-picture — is the format that works for a lecture. You get the slides and the speaker in the same video, but the viewer can choose which to focus on. That flexibility is not something you get from a simple screen recording uploaded to YouTube.

Multi-perspective video. For a music university this is the feature that changes things. A concert or a masterclass is not well-served by a single camera angle. The platform supports simultaneous recording from two camera perspectives — a wide shot and a detail shot, say, or a front view and a hands view for a piano performance. The viewer can switch between them in playback, or the institution can set a default presentation. This is infrastructure that makes the teaching use of concert recordings actually feasible, not just technically possible.

Formats. Video up to 4K with adaptive bitrate streaming (the player adjusts automatically to the viewer’s connection), audio up to broadcast quality (48kHz/16-bit, exceeding CD’s 44.1kHz), with FLAC at up to 96kHz/24-bit in development. No file size limit. These specifications matter for music. A piano recording compressed to whatever YouTube decides to do with it is not the same as an uncompromised audio stream. The difference is audible.

ILIAS integration. This is the practical hinge. Video that lives in a separate platform is video that students may or may not find. Video embedded directly in the ILIAS course page, in the learning module, at the point in the curriculum where it is relevant — that is video that is part of the course rather than adjacent to it. The integration between educast.nrw and ILIAS is direct: upload to the video platform, embed in ILIAS on pages, in learning modules, or as standalone video objects, all from within ILIAS.

Access rights. The granularity here is what distinguishes it from any public platform. Each video can be set to: public (anyone on the internet), institution-wide (anyone logged in at the university), course-wide (only enrolled students in a specific course), individual (specific named people), or private (only the uploader). A graduation concert might be public. A practice session for student feedback might be course-only. A recording made for an individual student’s reflection might be shared only with that student and their teacher. These are all normal use cases in a music university; they all require different settings; the platform handles all of them.

Use Cases in a Music University

The general university use case — lecture recording, video tutorial, self-study module — applies at HfMT as much as anywhere. But a music and dance university has some specific ones.

Concert recordings. HfMT Köln runs performances at both its Cologne and Wuppertal sites. Recording these and making them available to students, faculty, and selectively to the public used to mean someone had to manage files, find hosting, deal with YouTube’s automated copyright detection flagging student performances of copyrighted repertoire, and explain to students why their graduation concert had been muted by an algorithm. The controlled platform makes all of this manageable.

Stage presence as a reflective tool. Watching yourself perform is a standard part of performance training. It is uncomfortable, useful, and until recently required either dedicated recording equipment or the ad-hoc use of a phone propped against something. A proper recording infrastructure with controlled access — the student sees the video, their teacher sees the video, nobody else does — changes the pedagogical viability of this approach. The barrier to actually using video feedback in practice teaching drops substantially.

Theory and practice. This is the institutional argument I made in the presentation and stand by: video infrastructure that works for a lecture also works for a concert. The same system that stores the introduction to music theory also stores the masterclass by a visiting artist. This is not incidental — it is the point of a shared infrastructure. You do not need to choose a platform for academic content and a different one for performance content. The platform works for both.

The Argument Behind the Argument

There is a broader principle at work here that extends beyond video platforms. Public universities are funded by public money. The infrastructure they build with that money — software, platforms, data, content — should be under their control, governed by their values, and ideally available to other public institutions. The commercial platform model inverts this: you get free hosting in exchange for your data, your students’ attention, and your institutional dependence.

educast.nrw is an example of what the alternative looks like in practice: a cooperative of public institutions building shared infrastructure on open-source software, governed collectively, with data on European servers under European law. It is not perfect — the setup overhead is real, the user experience does not match YouTube’s, and the feature roadmap (automatic subtitling, H5P support, livestreaming, annotation tools) is still catching up to what commercial platforms have had for years. But the model is right.

The question of who owns the video infrastructure of a university is the same question as who owns its email, its learning management system, its student data. The answer should be: the university, operating under law, answerable to its students and to the public that funds it.

The slides from the Tag der Lehre 2022 presentation are available on request. For educast.nrw setup at HfMT Köln, contact the IT department.

Links

• educast.nrw
• Opencast
• Opencast Studio
• Tobira Videoportal
• opencast-ilias plugin (GitHub)

Changelog

• 2025-08-18: Corrected the capitalisation of “OpenCast” to “Opencast” throughout (matching the project’s official spelling on opencast.org).