Systematic phonics instruction transformed literacy outcomes by identifying the precise semiotic mechanism through which alphabetic writing encodes speech — the grapheme-phoneme correspondence at the lower level of a two-level architecture — and making that mechanism the explicit object of instruction. This paper argues that early numeracy faces the same structural problem. Positional numeral notation encodes arithmetic structure through a two-level architecture whose lower level — the indexed ordinal foundation that gives each digit’s position its meaning — is concealed by the written symbol. We call this positional opacity. No existing manipulative addresses it directly. Multi-base Arithmetic Blocks (MAB, Dienes 1960) operate at the upper level: they represent the iconic magnitude of multi-digit quantities richly and usefully, but on the prior assumption that the lower level is already secure — an assumption that cardinal-first instruction cannot itself warrant.
Drawing on Peircean semiotics, we develop a theory of signs for arithmetic that distinguishes reference semiotics — in which signs point to an independently existing world — from model semiotics — in which signs constitute the very reality they describe. Counting is a model semiotic: the counting gesture simultaneously constitutes ordinal position and cardinal quantity in a single act, the ordinal-cardinal weld. The arithmetic numeral “4” is not the name of a collection but the invariant produced by the convergence of all 4! counting paths across any four objects — a hypostatic abstraction in Peirce’s sense (CP 4.235). What developmental psychology has described as the sequential stages through which children come to know number are, on this account, the cognitive imprints of the semiotic practices through which they are taught: the psychology is consequent on the semiotics, not the reverse.
The linear abacus — a string of one hundred beads with an excess of string that enables manipulations — is introduced as a theoretically grounded resource that addresses positional opacity directly. Each bead’s numerical identity is constituted by the counting gesture that placed it (genuine Peircean index, CP 2.248); zero is represented as indexed absence; the fold gesture enacts place value as a constitutive act rather than a designation; and arithmetic reversibility is structurally immediate. A concrete instructional technique — spelling the number — makes the operator/operand structure of positional notation perceptually explicit through colour-coded notation paired with the fold gesture. The Talk–Do–Write framework operationalises Peirce’s pragmatic maxim (CP 5.402) as structured bidirectional practice across gesture, speech, and symbol: mathematical understanding is constituted by the full set of these practices, not by any one of them. Classroom observations illustrate the framework’s explanatory power: students confined to additive descriptions of number by prior cardinal-based tools acquired multiplicative structural understanding within a single session once the constitutive gesture was available to them.
We propose a sequenced curriculum in which the linear abacus establishes the ordinal-indexical foundation first, after which MAB becomes interpretable as a compressed record of processes already understood: the linear abacus makes sense of the MAB in the same way that phonics instruction makes sense of whole-language reading practice, by securing the lower level that the upper-level tool presupposes. This paper is the theoretical introduction to the Linear Abacus Research Programme — a sustained theoretical and empirical project establishing the semiotic foundations for a unified approach to primary arithmetic instruction.
Children are not born knowing how to read. But they are not born unable to read either. What they typically lack is explicit instruction in the system connecting letters to sounds. Once that system is taught directly — grapheme by grapheme, in structured sequence — most children learn to decode written language with surprising speed. This is the insight behind systematic phonics instruction, confirmed by research synthesis across multiple countries (Rose 2006; National Reading Panel 2000) and now the basis of most evidence-aligned literacy curricula.
The mechanism is straightforward. Alphabetic writing encodes speech through symbol-to-sound correspondences. Those correspondences are not transparent — the letter b does not look like the sound /b/ — so they must be taught. Phonics removes what we call the phonological opacity of alphabetic spelling: the gap between the written symbol and its spoken basis. This paper asks whether an analogous opacity afflicts early number learning, and whether an analogous remedy is available. Positional numeral notation — the system in which 87 means eight tens and seven ones — is one of the most powerful and compact representational inventions in human history. It is also deeply opaque. Nothing in the written symbol 87 announces that the 8 is doing a different job to the 7, that one is a count of groups and the other a count of leftovers, or that both are connected to a specific position in the ordered sequence of natural numbers. Children must somehow infer this structure from symbols that conceal it. We call this positional opacity.
The linear abacus — a string of one hundred beads that can be slid, partitioned, and folded along an extended cord — is a direct manifestation of the heuristic definition of number developed in this paper: a number is a named place in an order that answers the question “how many”. Each bead occupies a position constituted by the counting gesture that placed it; the string as a whole functions as a material numeral — not a representation of a number but a physical instantiation of the ordinal structure that number notation encodes. As a material numeral the linear abacus is agnostic with respect to naming conventions: it does not presuppose any particular base, and it accommodates both additive systems (such as Roman numerals) and the multiplicative positional system of decimal notation. It removes positional opacity in the same way phonics removes phonological opacity: by making the structural composition of the numeral system explicit, physical, and teachable. A concrete instructional technique — spelling the number — operationalises this for classroom use and has been successfully implemented with Grade 1 students. The theoretical grounding for this claim requires a precise account of what kind of sign the linear abacus is, what kind of sign MAB are, and why that difference matters. We develop that account in what follows.
Contemporary semiotic approaches to mathematics education — notably the Theory of Objectification (Radford 2014) and related work on semiotic mediation (Sáenz-Ludlow 2020) — have established that mathematical learning is irreducibly semiotic: objects are not discovered through tools but constituted in and through sign activity. This paper works within that broad commitment while departing from it on a foundational point. Where objectification theory grounds meaning in embodied, cultural-historical activity, the present account grounds it in the internal grammar of the sign system itself — in the role-binding structure that makes a numeral mean what it means, independently of the cultural context in which it is wielded (Waywood 2025). The departure is consequential: it shifts the locus of pedagogical failure from insufficient social participation to structural mismatch between tool and mathematics.
The paper advances the semiotics of mathematics in three respects. First, ontological diagnosis: it provides a principled account of why certain manipulatives fail — not because students lack engagement or teachers lack skill, but because the tool’s semiotic solidarity (the alignment of what is expressed with how it is expressed — (Hjelmslev 1961)) is ontologically incompatible with the mathematics of positional notation (ordinal structure constituted by counting). The failure is structural and therefore predictable. Second, constitutive rather than representative semiotics: it moves beyond the claim that tools mediate access to pre-existing quantities, arguing instead that the right tool constitutes its objects — that arithmetic number does not exist prior to the sign system that produces it, and that choosing the wrong sign system therefore does not merely obscure number but produces a different and deficient object in its place. Third, transcoding as grammar morphism: it treats the pedagogical transitions between gesture, speech, and symbol not as simple translation between equivalent representations, but as a structure-preserving map between sign systems with different grammars of role-binding — a morphism that can succeed or fail, and whose failure mode is diagnosable in semiotic terms.
A sign, in the tradition established by Charles Sanders Peirce, is anything that stands for something else to someone in some respect (CP 2.228)1 . Three things are always in play: the sign itself (what Peirce calls the representamen), the object it stands for, and the understanding it produces in a mind (the interpretant). What matters for our purposes is how the sign is connected to its object — and Peirce identifies three fundamentally different kinds of connection (CP 2.247–2.249):
| Sign type | Connection to object | Simple example |
| Icon | Resemblance — the sign shares a quality with its object | A portrait resembles its subject; a scale model shares proportions with a building |
| Index | Existential connection — the sign is really produced by or causally linked to its object | A footprint is produced by a foot; smoke indexes fire; a thermometer reading is produced by temperature |
| Symbol | Convention — the connection is established by rule or habit | The word dog means dog by social agreement, not because it looks or sounds like one |
Most signs are mixed in practice — a photograph is both iconic (it resembles the subject) and indexical (light from the subject caused the image). What matters is which connection is doing the primary work.
Before applying Peirce’s categories to arithmetic, we need a prior distinction that runs through all that follows. Consider the difference between the sentence There are seven apples in the bowl and the act of counting the apples. The sentence refers to a state of affairs that exists independently of the sentence. The apples are there whether or not anyone describes them. Language is a reference semiotic: its signs point toward a world that precedes and exists independently of the signs used to describe it.
Counting is different. When a child counts the apples — touching each one, saying one, two, three… — the number is not discovered pre-existing in the bowl. It is constituted by the counting act itself. Before counting, there are apples; after counting, there are seven apples. The number seven is the product of the process, not a property that was there waiting to be found. Counting is a model semiotic: its signs construct the very reality they describe (Waywood 2026).
This distinction has implications for the founding level of each system:
This is why phonics and linear abacus instruction are genuine analogues, not merely convenient metaphors. Both are cases of explicit instruction at the founding level of a semiotic system — the level at which the abstract sign is welded to the bodily act that gives it meaning.
When a child counts correctly — one gesture, one item, one number word — something precise happens at each step. Consider the third counting act that constitutes counting to three:
These are not two separate pieces of information discovered by two separate processes. They are simultaneously constituted by a single gesture. The third counting act cannot produce one without the other. We call this joint constitution the ordinal-cardinal weld: the counting gesture enacts a position, and that position simultaneously is a quantity. This weld is the foundational semiotic act of arithmetic. It is the reason a number is both “where” in a sequence (ordinal) and “how many” in a collection (cardinal) at once, and it is the reason that counting — not collection-viewing — is the correct foundation for number instruction: the weld is enacted in the gesture and cannot be read off from a static display of objects. The research literature on early number development provides empirical grounding for the importance of counting as a developmental process (Gelman and Gallistel 1978; Fuson 1988)2 , though where cognitive accounts describe counting as a skill to be acquired, the ordinal-cardinal weld is here proposed as the foundational semiotic act—a move that treats the coordination of position and quantity as a matter of sign-constitution rather than just mental representation. Where English references ’four’ and ’fourth’ Arithmetic has 4 as constituted by counting.
The ordinal-cardinal weld explains what happens in a single counting act. A deeper question is what kind of object arithmetic number is — what “4” refers to when it appears not in the sentence four apples but in the equation 2 + 2 = 4. The question begins with something that seems obvious but is not: where does the ordering come from?
Consider two arrangements of the same nine countable items (Figure 1). The linear arrangement suggests an already-imposed ordering — a sequence where each item has a predetermined position. This arrangement subtly misleads us about the nature of counting by presenting the ordering as inherent to the objects rather than imposed by the counting process itself. The scattered arrangement exposes the problem: while each symbol can be seen as “one,” nothing in the symbols themselves tells us which “one” is first, which second, which third. The ordering is not a property of the things; it is a contribution of the counter.
To see this precisely, consider four card suits scattered in a region (Figure 2). To count them, one of them must become “first” in my attention — but which one? Any of the four can be first; the objects themselves are silent on the question. This is an embodied choice: counting is not a passive observation but an act in which the counter imposes a sequence on what has no sequence of its own. What governs this act is a single constraint: don’t point at the same thing twice.This constraint (Thirdness in Peirce’s sense3 ) elevates the counting sequence from a series of brute contacts into a genuine sign with internal structure — the structure that makes it possible for the sequence to have a determinate result at all.
The mark “1st” can fall on any of the four suits, generating different counting sequences. There are 4! = 24 possible orderings, each a valid count sequence, all yielding the same result:
| ♠♡♢♣ | ♠♡♣♢ | ♠♢♡♣ | ♠♢♣♡ | ♠♣♡♢ | ♠♣♢♡ |
| ♡♠♢♣ | ♡♠♣♢ | ♡♢♠♣ | ♡♢♣♠ | ♡♣♠♢ | ♡♣♢♠ |
| ♢♠♡♣ | ♢♠♣♡ | ♢♡♠♣ | ♢♡♣♠ | ♢♣♠♡ | ♢♣♡♠ |
| ♣♠♡♢ | ♣♠♢♡ | ♣♡♠♢ | ♣♡♢♠ | ♣♢♠♡ | ♣♢♡♠ |
The arithmetic numeral “4” is not the record of any particular sequence in this table. It is the invariant across all of them. Figure 4 makes this visible by a different route. In the sequential arrangement (left), each column represents a completed count, the number of elements in the column matching its position in the sequence. The structure makes explicit what is implicit in counting practice: each successive count incorporates all previous counts while adding one more, so that the count to twelve contains within it the count to eleven, which contains the count to ten, and so on. This recursive containment is not an incidental feature of counting but the source of the ordered structure that defines arithmetic number. The randomised arrangement (Figure 3b) shows the same numerals with the ordinal-cardinal coordination destroyed. Were this structure to define our number system, expressions such as 2 + 3 = 5 and 6 + 9 = 8 would be equally valid, since nothing in the structure would privilege one relation over another. Arithmetic becomes impossible not because the numerals are missing, but because their positions are no longer governed by an invariant order.
This invariance across all 24 orderings is the precise meaning of Peirce’s concept of hypostatic abstraction (CP 4.235): the logical operation that takes a predicate — is the result of counting four things — and promotes it to a subject — 4 — that can itself be operated on, placed in relations, and used as an object of further reasoning. The arithmetic “4” does not describe any particular collection in the world. It is what the rule-governed counting process constructs when applied to any collection of that size, considered as an object in its own right, independent of any particular instance or sequence. This is why arithmetic number belongs to a model semiotic and not a reference semiotic. “Four suits” uses four as a predicate in a reference semiotic: it describes a property of a particular collection that exists independently of the description. The arithmetic “4” is the hypostatic abstraction of the process of counting-to-four: a model-semiotic object constituted by the convergence of all 24 counting paths, not a reference to anything that exists prior to those paths being taken (Waywood 2026).
The relation between the arithmetic “4” and the spoken word four is therefore not one of synonymy but of connotation in Hjelmslev’s (1961) sense: a semiotic whose expression plane is itself a complete semiotic system.4 The arithmetic position constituted by counting becomes the expression plane for the spoken form: four takes the arithmetic sign as its signifier and adds a culturally variable spoken content. This is why “eighty-seven” and quatre-vingt-sept both name the same arithmetic object — the spoken forms are contingent on language; the position is not. There is no arithmetic for the linguistic form: f-o-u-r + t-h-r-e-e has no grammar, no rewriting rules, no operational semantics. The spoken name names what the counting process has already made; it does not make it. Less obviously nor can arithmetic be done with arrangements:
The distinction matters for instruction. It means that arithmetic number cannot be shown to a child by displaying a collection of objects, however carefully arranged5 . Four blocks do not contain “4.” They contain the occasion for 24 counting paths that will produce “4” if followed. The numeral is the invariant product of the process; it is not resident in any arrangement of things. A child who has only ever been shown collections — never led through the constituting process — has been shown occasions for number without being shown number itself.
The argument to this point is ontological — it concerns what arithmetic number is. It has a methodological consequence that should be stated explicitly. The dominant framework in mathematics education research treats the psychology of number development as foundational: children possess cognitive structures — schemas, mental models, conceptual stages — that precede and constrain what semiotic tools can accomplish. On this view, manipulatives and representations are vehicles for accessing what is already, in principle, cognitively available at the appropriate developmental moment.
We invert this direction. What appear as cognitive structures are, on our account, the imprints of semiotic practices — the inner precipitate of outward participation. There are not developmental stages that children traverse and then express in their counting behaviour; there are semiotic practices that constitute particular psychologies in the children inducted into them. The parallel is Wittgenstein’s: what we treat as inner cognitive structures are constituted by participation in outward practices, not the reverse — the inner stands in need of outward criteria (Wittgenstein 1953, PI §580). This is not an external import from a different philosophical tradition: it follows from Peirce’s own later claim that every thought is a sign [CP 5.265], which entails that there are no pre-semiotic cognitive givens for semiotic practice to act upon — the psychology and the semiotics are not two things in a causal relation but one thing at two levels of description. A child inducted into cardinal-first, collection-designating counting practices acquires a psychology oriented toward collection-size. That psychology is real; it is not, however, explanatorily foundational. It is a consequence.
The practical implication is direct: difficulties that the research literature has theorised as developmental — the additive-to-multiplicative transition, place value confusion, procedural fluency without structural understanding (Kamii 1986; Ross 1989; Mix et al. 2014) — may be consequences of the semiotic practices our culture deploys rather than facts about children’s cognitive architecture. If so, the remedy is not to wait for cognitive readiness but to change the practice. The classroom observations in §6.5 are the first evidence; the Linear Abacus Research Programme is designed to test this systematically.
The model semiotic account of counting provides, for the first time, a natural and mathematically honest representation of zero. Zero is not a collection of nothing — that concept is strained and cognitively late-developing. Zero is the state in which no counting gesture has yet occurred: the starting condition before the first act of counting, not the result of a failed attempt to count something absent. On the linear abacus, zero is represented as the absence of slid beads — the gold end of the string before any bead has been moved into the counted position. This is an indexed absence: the absence of the index (no bead in counted position) stands for the absence of the process (no counting gesture performed), and the representation is structurally honest.
MAB has no natural zero. There is no “zero block,” and zero must therefore be introduced as a separate convention — an arbitrary symbol for “no blocks here.” This is symptomatic of the deeper problem: MAB represent number as a thing (a collection of blocks of certain sizes), and things cannot represent their own absence without conceptual contortion. A model semiotic grounded in process represents its own starting condition naturally: before the process begins, the index is absent.
Language, Martinet (1960) observed, has a remarkable architecture. It operates on two levels simultaneously:
A small inventory of phonemes — typically 40–44 in English — combines to produce an indefinitely large vocabulary. Phonics instruction teaches Level 1 explicitly: what are the units, what symbols represent them, and how do they combine?
Positional numeral notation has exactly the same two-level architecture:
The numeral 87 records two gesture-counts: 8 encodes eight applications of the fold gesture at the ten-level; 7 encodes seven applications of the counting gesture at the one-level. The written symbol compresses all of this into two marks that look identical in kind, differing only in position — and nothing in the marks themselves announces which level of gesture each is recording. This is positional opacity: the written symbol conceals its own gestural foundation in exactly the way that alphabetic spelling conceals its phonemic foundation from the reader who has not been taught the code.
As in spelling, the underlying structure must be made explicit through instruction. The linear abacus makes the Level 1 / Level 2 connection visible, just as phonics does for alphabetic writing. The precise semiotic mechanism by which each bead acquires its identity through the counting gesture is established in Section 5; the fold as the constitutive act of the ten-level is developed in Section 5.3. The fold-boundary enacts the moment at which the gesture of the current level is complete and the next level of recursive indexing begins (Section 5.3), and the colour-coded spelling notation records that gesture directly (Section 6) — blue for the count of gestures, red for the unit each gesture constitutes (Section 6.2).
The parallel is structural, but the founding levels differ in the way already described:
| Level 1 (founding, bodily) | Level 2 (written encoding) | Instruction welds |
|
| Literacy | Phoneme — unit of sound | Grapheme — letter encoding the sound | Phonics: letter ↔ sound |
| Numeracy | Indexing gesture — counting act / fold | Digit-in-position — coefficient encoding the gesture | Spelling: digit ↔ gesture |
This difference explains why a direct transplant of phonics methods cannot work for numeracy — the pedagogy must be adapted to the gestural, rather than acoustic, founding level. But the structural goal is the same: make the two-level architecture explicit, and teach the connection between the abstract symbol and the founding act that gives it meaning.
Multi-base Arithmetic Blocks (MAB, also called Dienes blocks, Dienes 1960) are the dominant concrete manipulative for early place value instruction. Their value is real and we are not arguing for their removal. But they carry a set of semiotic problems that cannot be resolved by improved design, because they arise from a categorical misfit: MAB is a semiotic of space deployed in a domain that requires a semiotic of number. The two systems have incompatible grammars, and the consequences of that incompatibility run through every aspect of how MAB represents quantity.
In a geometric semiotic, objects are spatial magnitudes and operations are type-changing: length × length = area, area × length = volume. Each application of × produces an object of a new ontological kind. This is why Euclid keeps arithmetic and geometry in entirely separate books — the operations are incommensurable in kind. In an arithmetic semiotic, objects are positions in an ordered sequence constituted by gesture, and operations are type-preserving: the fold gesture applied to ten-units produces hundred-units of the same kind, one level up in the same recursive indexing structure. The × never changes ontological type; it iterates the same act on the same kind of object.
Each MAB piece is an icon in Peirce’s sense (CP 2.277): it represents its object by sharing a sensory quality with it. The unit cube represents one by its point-like volume; the ten-rod represents ten by its linear extension; the hundred-flat represents one hundred by its planar area; the thousand-cube represents one thousand by its three-dimensional volume. In each case the sign and its object share a geometric quality — size, shape, spatial extension — and that resemblance is the sole ground of the representation. The iconic sign represents a number assumed to exist independently of the sign. Where the iconic sign represents, the indexical sign constitutes: the number is made in the gesture, not pointed to from outside — a distinction whose full semiotic consequence is developed in Section 5.
The misfit between the two semiotics is not contingent on the particular geometric vehicles Dienes chose. It is entailed by the sign relation itself: an iconic sign, connecting to its object by shared quality, cannot but import the ontological type of its material into the object domain. Any iconic spatial sign for arithmetic carries geometric ontology with it. MAB imports the geometric semiotic wholesale. The unit cube, ten-rod, hundred-flat, and thousand-cube are not objects of different sizes within a single system — they are objects from different geometric categories: point-like, linear, planar, volumetric. The iconic chain is a Euclidean dimensional chain, not an arithmetic iterative chain. A child reading MAB is being asked to read arithmetic structure off geometric structure whose grammar actively contradicts it: where arithmetic requires × to be type-preserving iteration, the geometric vehicle presents × as type-changing dimensional extension. The problems diagnosed in what follows — the cardinal fallacy, the heterogeneous iconicity, the exhaustion of the iconic chain at 103, the absence of zero — are consequences of this single foundational misfit.
The deepest problem with MAB is not practical but ontological. MAB present number as a thing: a collection of blocks of certain sizes, with certain permitted operations. The ten-rod is ten, not because any counting process produced it, but because it was manufactured to proportions that resemble numerical magnitude iconically, and then underwritten by teacher declaration. The iconic resemblance cannot do the work alone — a rod ten times as long as the unit cube does not announce its own arithmetic meaning — so the symbol must underwrite the icon. The child using MAB is learning the rules of what we might call block-world: this object means ten, ten of these can be exchanged for one of those. Block-world is a reference semiotic — the blocks refer to numbers by convention, the way the word ten refers to ten.
Number-world is different in kind. As the phenomenology of Section 2.4 establishes, the arithmetic “4” is the hypostatic abstraction of the convergence of all 4! counting paths — the invariant of a process, not the designation of an object. The linear abacus instantiates number-world directly: the tenth bead is in position ten because ten counting gestures placed it there, and “10” names the invariant of that process. No designation is needed; the constitution is immediate.
This is what we call the cardinal fallacy: the implicit claim, built into MAB’s design, that the size of a collection is the foundation of number. It is not wrong that four blocks are four — cardinality is genuinely a property of number. But it inverts the correct ontological order. Cardinality is a result of the counting process; it is not its foundation. Presenting cardinality first, as MAB does, installs the consequence before the cause and leaves the child with a conception of number that is ontologically mislocated.
The mechanism of this mislocation is precise. The ordinal-cardinal weld, established in Section 2.3, holds that position and quantity are simultaneously constituted in a single gesture: the n-th counting act enacts ordinality and cardinality at once, neither preceding the other. The cardinal fallacy severs this weld. When a child encounters number first as a collection of blocks, cardinality is presented without its ordinal constitution — the blocks have sizes but no positions generated by a counting sequence. This is precisely the condition shown in Figure 3b: the same numerals, but their coordination destroyed. In that condition, arithmetic relations become arbitrary. 2 + 3 = 5 is not privileged over 2 + 3 = 8 because nothing in a cardinality-only conception of number constrains which results follow from which operations. Arithmetic operations require ordered structure; they cannot be performed on collection sizes alone. The cardinal fallacy does not merely mislabel number — it actively blocks access to the arithmetic structure that written notation encodes.
The transfer problem from MAB to written arithmetic is therefore not primarily perceptual — not a matter of blocks looking different from symbols. It is ontological: the child who has learned block-world must make a conceptual leap into number-world, and nothing in MAB instruction provides the bridge. The blocks never show her the 4! paths converging on “4.” They show her four unit cubes already assembled into a collection, designated as meaning four — the cardinality presented as the thing itself, prior to and independent of any counting process that might have produced it. This is why children who handle MAB competently often cannot correctly read or write multi-digit numerals (Kamii 1986; Ross 1989): they are fluent in block-world, and the notation belongs to number-world.
There is a further consequence that reaches beyond early arithmetic. The invariant ordinal structure constituted by counting is the foundation from which algebraic generality grows — the reason that the relation a + b = b + a holds not as a fact about particular collections but as a law about positions in an ordered structure. A number concept grounded exclusively in cardinal magnitude has no such invariants to generalise. MAB can model the written algorithms for multi-digit computation — and in this function they have genuine, if circumscribed, pedagogical value — but the algorithms are a surface implementation that works by virtue of arithmetic structure without revealing it.6
The following table sets out six distinct semiotic problems with MAB, compared to the linear abacus. Each is a separate issue, though they reinforce one another.
|
Table 3: Six semiotic problems with MAB, compared to the linear abacus
|
||
|
Problem |
MAB |
Linear abacus |
|
1. Sign type |
Icon (CP 2.277): each piece shares a geometric quality with its object — point-like volume, linear extension, planar area, volumetric extension. Four geometrically distinct iconic vehicles across the place values (see Section 4.3) |
Genuine index (CP 2.248): each bead’s position is constituted by the counting gesture that placed it — not by any quality the bead possesses |
|
2. Foundation |
Cardinal — represents how many in a collection. The ordinal sequence is absent |
Ordinal — represents position in a sequence generated by counting. Cardinality follows as a consequence |
|
3. Zero |
No natural representation. Zero must be introduced as a separate arbitrary symbol |
Natural indexed absence: zero is the state in which no counting gesture has occurred — no beads in the counted position |
|
4. Grouping |
Pre-packaged. The ten-rod arrives already assembled. The child observes the group but never constructs it |
Performed by fold gesture. The child constructs each group of ten by folding the string. The group is made, not given |
|
5. Transition |
Static. The crossing of a place value threshold is invisible. There is no representation of what happens at ten |
Dynamic and reversible. The fold enacts the transition: ten counted beads become one fold. The fold can be undone and repeated |
|
6. Exchange |
Requires trading different physical objects (ten unit cubes for one ten-rod). The identity 10 ×1 = 1 ×10 is asserted by teacher authority, not demonstrated |
No exchange. The same beads fold and unfold. The identity is visible in the unbroken string: the fold is the grouping |
Because each MAB piece is an icon whose vehicle is geometric extension, the system is forced to change vehicle at every level of magnitude. Peirce distinguishes a subtype of icon he calls an image: a sign that shares a sensory quality with its object (CP 2.277). MAB use geometric extension as that quality:
| Place value | MAB object | Iconic vehicle | Perceptual operation required |
| 100 = 1 | Unit cube | Volume (point-like object) | Direct comparison |
| 101 = 10 | Ten-rod | Length (linear extension) | Length judgement |
| 102 = 100 | Hundred-flat | Area (planar extension) | Area judgement |
| 103 = 1,000 | Thousand-cube | Volume (three-dimensional) | Volume judgement |
| 104 = 10,000 | (no distinct object available) | (geometric space exhausted) | (no iconic representation possible) |
The ratio between successive place values is constant (always 10:1) but the perceptual operations required to appreciate it are not. Length judgements are direct and reliable. Area judgements are less so — psychophysical research establishes that perceived area grows as a power function of actual area with an exponent less than 1 (Stevens 1961), meaning a ten-fold increase in area looks like considerably less. Volume judgements are less reliable still. MAB systematically deploys less perceptually reliable vehicles at precisely the higher place values where perceptual clarity matters most. The iconic chain is also structurally bounded: physical space provides exactly three extensions beyond the point — length, area, volume — so MAB exhausts its representational logic at 103 by necessity, at precisely the moment the curriculum most needs the representation to continue.
The child has no perceptual basis for seeing that the relation between a unit cube and a ten-rod is the same kind of relation as the relation between a ten-rod and a hundred-flat — because in a geometric semiotic, it is not.
Section 4 established that MAB’s semiotic problems are not correctable by redesign: they are entailed by the choice of an iconic sign relation, which necessarily imports geometric ontology into a domain whose grammar is arithmetic. The linear abacus answers this diagnosis directly. Where the iconic sign represents a number assumed to exist independently of the sign, the indexical sign constitutes the number in the gesture itself. The bead does not resemble its number; it is that number, because the counting gesture made it so.
The linear abacus uses a single indexical vehicle throughout: discrete position in a sequence constituted by gesture. The same beads, the same fold gesture, the same act of counting applies at 1, 10, 100, 1,000, 10,000, and beyond. These are the fold-boundaries of base ten — the positions at which the string is re-indexed: the beads counted to that point are gathered into one group of, and that group becomes the new unit whose instances the next sequence of gestures will count. For base eight the fold-boundaries would be 1, 8, 64, 512, … — the same re-indexing logic, a different grouping threshold, the same bead string. At each fold-boundary, digits are revealed as coefficients rather than cardinals: they are counts of grouping gestures, not names of quantities.7 This is why the representation scales indefinitely: because the same gesture — count to the fold-boundary, re-index, count again — can always be applied one more time to the same kind of object.
Figure 5 shows both representations of 287 side by side, making the contrast between a geometric iconic chain and a single recursive indexical vehicle directly visible.
A genuine index, for Peirce, is a sign “really affected by its Object” — connected to its object through an existential, causal, or sequential process, not through resemblance or convention (CP 2.248). The weathervane is moved by the wind it indicates. The photograph is causally produced by the light from its subject. The pointing finger directs attention through spatial contiguity.
The n-th bead on the linear abacus is in position n because n counting gestures have been performed — one for each bead slid from uncounted to counted position. The bead does not look like n. It was not designed to mean n by convention. It is at position n because a real sequential process placed it there, and for no other reason. This is an existential connection in exactly Peirce’s sense: the bead’s numerical identity is constituted by the history of gestures that produced its position.
Critically, this indexicality is non-comparative. The 87th bead is an index of the 87th counting gesture regardless of whether the 88th has occurred. Its meaning is self-standing, constituted by its own history, not by its structural relation to adjacent beads. This is why the 100th bead is an index of 100 in exactly the same sense as the 57th bead is an index of 57, even if the string stops there.
The block is a thing whose meaning is assigned; the bead is a trace whose meaning is constituted.
The construction of the linear abacus makes the indexical logic physical. The string has three sections:
Counting proceeds by sliding beads one at a time from the bead bank to the counted position. Each slide is a counting gesture, and the bead in its new position is the resulting index. The name of each bead — “one,” “two,” “three” — is attached to the bead after it is placed, not before. The name follows the act. This processual priority — Dewey’s (1938) principle that process is prior to thing — means the number is never a pre-existing object that the bead merely represents. The number is the bead in position, as constituted by the gesture. The manipulative and the mathematics are not two things in a representational relationship; they are one event with two descriptions.
When ten beads have been counted, they can be gathered into a group by folding the string concertina-style. This fold gesture introduces a new semiotic element. The individual beads remain genuinely indexical — each is still the trace of its own counting gesture — but the fold creates a spatial grouping whose relational structure represents the multiplicative relationship between place values.
Peirce distinguishes a type of icon he calls a diagram: a sign that represents its object not by resembling its surface qualities but by sharing its relational structure (CP 2.277). He notes that diagrams are the primary vehicle of mathematical reasoning: “all necessary reasoning is diagrammatic” (CP 2.279). A diagram does not look like its object; it is structured like its object.
The folded bead string is a diagram of the place value structure of positional notation. Eight folds of ten beads each, plus seven loose beads, produces a spatial structure whose relationships mirror the arithmetic relationships encoded in the numeral 87: eight groups of ten, plus seven ones. The fold takes the bead string from purely indexical (individual counting acts) to diagrammatic (relational structure of the number system), and this transition — enacted physically by the child’s own hands — is the moment at which counting becomes arithmetic. This progression from index to diagram corresponds precisely to the curriculum progression from counting to place value. The two are not separate topics requiring separate manipulatives; they are continuous stages in the development of a single semiotic structure.
A consequence of using a single indexical vehicle throughout is that the linear abacus is honest about scale in a way no iconic manipulative can be. Each bead is one cubic centimetre. This means the decimal place value system and the metric measurement system — the quantitative scaffolding of the physical world children already inhabit — are calibrated to the same standard.
| Place value | Linear abacus | Metric equivalent | Embodied experience |
| 100 = 1 | 1 bead | 1 cm | Held between finger and thumb |
| 101 = 10 | 10 beads | 10 cm | A hand span |
| 102 = 100 | 100 beads (1 string) | 1 m | Roughly the child’s own height |
| 103 = 1,000 | 10 strings | 10 m | A classroom length |
| 104 = 10,000 | 100 strings | 100 m | A sprint distance |
| 105 = 100,000 | 1,000 strings | 1 km | A walk of several minutes |
The significance of this calibration is that the embodiment is not metaphorical. Most manipulatives exploit metaphorical embodiment: the physical action stands for the mathematical operation, with a representational gap between the two. The linear abacus closes that gap at the level of scale. The child is not experiencing a representation of exponential growth — she is experiencing exponential growth in the metric system itself, which is structurally identical to the decimal place value system. Large numbers are not abstractly large; they are as far as the football oval, as tall as the school building. The bigness is remembered in the body, not merely stated in a symbol.
This also provides an honest corrective to a well-documented human tendency to underestimate exponential growth (Wagenaar and Sagaria 1975). MAB conceals the true scale of exponential growth by compressing each place value into a compact object; the thousand-cube fits in a child’s hand, which actively misrepresents what one thousand means at human scale. The linear abacus, precisely because it uses the same linear vehicle throughout, makes the scale honest: 1,000 beads stretched across a school corridor is genuinely surprising. That surprise is the beginning of understanding.
The digit 8 in 87 and the digit 7 in 87 look the same: both are black marks of similar size and shape. But they are doing structurally different things. In the expression (8 ×10) + (7 ×1):
Standard positional notation uses position alone to distinguish these roles, so that a child sees two digits that look alike with no perceptual basis for understanding that they are doing structurally different jobs. This is positional opacity at the level of grammatical role.
The instructional technique we call spelling the number addresses this by using colour to make the operator/operand distinction perceptually explicit:
The number 87 is spelled:
The student produces this expression by performing the fold gesture: 8 folds of 10 beads each, then 7 single beads. The blue coefficients are counted in the gestures; the red units are counted in the beads. The colour-coding maps directly onto the physical distinction between what the hands are doing (folding — blue) and what the beads are being (grouped units — red). A three-digit example, 954, is spelled:
This notation makes the operator/operand structure visible, and simultaneously reveals that place value is multiplication: having 9 hundreds is taking the hundred-unit nine times, a fact that is gestured before it is named.
The fold gesture enacts multiplication before children have been formally introduced to the concept. Eight folds of ten beads each is 8 ×10 — not a representation of it, not a preparation for it, but the thing itself performed by the child’s hands. The fold is discrete (each fold creates a distinct countable chunk), reversible (the string unfolds back to the linear sequence), and transparent (the child made it and therefore understands exactly what it is). This has a curriculum consequence of some significance: place value and multiplication are not separate topics requiring separate introduction. They are two readings of the same gesture. The child who spells 87 has already enacted multiplication; the formal introduction of the × symbol later is the naming of something already understood.
As in phonics, spelling the number is explicitly bidirectional:
| Direction | Phonics analogue | Numeracy task |
| Encoding | Spoken word → written spelling | Written numeral → fold-and-bead configuration |
| Decoding | Written spelling → spoken sound | Fold-and-bead configuration → written numeral |
Both directions must be practised. A child who can produce the bead configuration from 87 but cannot recover 87 from the bead configuration has not internalised the structure — she has learned a procedure. Fluency requires symmetry8 . If the mapping only works from symbol to gesture (Write → Do), the sign remains an index of a command rather than a diagram of a structure. True transcoding requires the child to recognize the invariant across all six directions of the Talk-Do-Write triangle . Rosenshine’s (2012) principles of effective explicit instruction emphasize cumulative rehearsal and systematic review to move students toward automaticity; the spelling technique operationalizes this by requiring the symmetrical practice necessary for structural internalisation . Without this bidirectional symmetry, the child risks learning a one-way procedural mapping rather than owning the full pragmatic concept of the number.
The parallel with systematic phonics instruction is precise here, not merely analogical. The research case for phonics rests on a structural observation: alphabetic writing is sufficiently opaque that incidental exposure to print does not reliably produce decoding competence in a significant proportion of children. The remedy is explicit instruction in the code itself — grapheme by grapheme, in structured sequence, practised in both directions until the correspondence is automatic (NationalReadingPanel2000; Rose 2006). The argument for the spelling technique rests on exactly the same structural observation about positional notation: the operator/operand distinction is sufficiently concealed by the written symbol that incidental exposure to numerals does not reliably produce structural understanding. The remedy is the same in kind — explicit instruction in the code, practised in both directions. What Rosenshine’s principles describe for instruction in general, systematic phonics demonstrates for alphabetic writing specifically, and the spelling technique operationalises for positional notation: the structure must be made the explicit object of instruction, and fluency requires symmetrical practice in both directions of the mapping.
The two classroom encounters reported below were not designed as formal studies. They are observational accounts from instructional settings in which the linear abacus and the spelling technique were introduced to children who had been working with other manipulatives. We report them as qualitative evidence of the kind of understanding that becomes available when children are given a tool that models the structure of number rather than illustrating a quantity — evidence that the theoretical distinctions drawn in earlier sections are not merely analytical but have observable pedagogical consequences. The encounters are reported in the spirit of Grouios and Waywood (2026), which documents the instructional approach in detail; systematic empirical investigation is a priority for subsequent work.
Two classroom encounters stay with us as particularly vivid illustrations of what becomes possible when children are given the right tools and the right invitation to think. The first took place in a remediation setting. Students in this group had been working primarily with ten-frames and counters — familiar resources, but ones that had quietly limited how these children understood number. A numeral like 34 meant three things and four things. There was no grouping, no structure, no sense that the digits were in conversation with one another. Place value, in any meaningful sense, had not yet arrived. In a single session introducing the linear abacus alongside colour-coded numerals and sentence frames, something shifted. By the end of the lesson, these same students could model a number on the bead string, describe what they saw — “I can see 4 groups of 10 and 5 groups with 1” — and record it in writing as a numeral with real meaning behind it. These children had not been struggling because they lacked ability. They had been struggling because nothing had yet shown them the structure that was always there.
The second context was a Grade 1/2 class beginning a unit on place value with MAB blocks. Many students were finding it difficult to represent numbers beyond 100, and most described numbers additively — 76 is 70 and 6 — without reaching the deeper, multiplicative relationships that positional notation actually encodes. Then, in a single lesson with the linear abacus and the spelling technique, students folded the bead string themselves, constructed the base-ten groupings with their own hands, and began to understand not just what each digit stood for, but why the whole system worked the way it did. They moved from additive descriptions to multiplicative ones, expressing numbers using the full spelling notation: 954 = (9 ×100) + (5 ×10) + (4 ×1). This was not imitation. The structure made sense because they had built it. Several students, without being prompted, began wondering aloud about where 1000 would fall on the string — reaching, spontaneously, toward the infinite generative pattern that underlies the entire number system.
These two encounters carry three quiet but important truths. High-level structural understanding is not beyond young children — it is beyond the tools they are usually given. When a tool genuinely models the structure of number, rather than simply illustrating a quantity, children do not just learn a procedure; they grasp a concept. And the pedagogy is everything: it was the act of constructing the groupings, not watching them arrive pre-assembled, that made understanding possible. The child who folds the bead string knows what a ten is — because she just made one.
Peirce’s pragmatic maxim holds that the meaning of a concept is constituted entirely by its practical consequences — by every way it can be put to use in the world (CP 5.402). Applied to mathematical understanding: to understand a number is not to hold a mental image of it, nor to have memorised a definition. It is to be able to act with it fluently across every semiotic register in which it can appear — to gesture with the beads, to say the number name, to write the expression, and to move between all three without friction. Understanding, on this view, is not a state that precedes and enables fluent action; it is that fluency. The pragmatic maxim thus makes a direct pedagogical claim: a concept has been understood only when all its uses are available. Partial use — being able to fold the beads but not to write the expression, or to write the expression but not to build the fold — is not partial understanding of a fully understood thing. It is a different, thinner concept, one whose meaning is genuinely more limited. This implies that learning occurs not within any single semiotic register but in the act of moving between registers. We call this movement transcoding9 : the translation of meaning across sign systems that differ in modality and structure. Each act of translation tests the learner’s grasp of the invariant structure that all representations share, and deepens it when the translation is successful.
The Talk–Do–Write framework operationalises the pragmatic principle as a structured pedagogical cycle. The three modes are not equivalent — they engage fundamentally different semiotic registers, and their differences are theoretically load-bearing:
| Mode | Semiotic role | What it does | Example: 87 |
| Do | Model semiotic — the founding act | Enacts the ordinal-cardinal weld. Constitutes the material numeral through gesture. The number is made here, not named. | Folding 8 groups of ten, then counting 7 remainder beads |
| Talk | Reference semiotic — the cultural name | Attaches a linguistically contingent label to the position already constituted in Do. Names vary across languages; the position does not. | “Eighty-seven” (English); “Quatre-vingt-sept” (French) |
| Write | Symbolic record — the persistent notation | Records the place-value structure established in Do in a form that is language-independent and computation-ready. | 87; or explicitly: 8 × 10 + 7 ×1 |
Do is listed first because it is theoretically primary. Talk and Write are derivative: they name and record what Do has constituted. This ordering matters for instruction — it is why the fold gesture precedes the written expression, not merely accompanies it. The French example in Talk is not decorative: it demonstrates that two entirely different spoken names (“eighty-seven” and “four-twenty-seven”) attach to the same material numeral. The name is culturally contingent; the position is not.
Three modes generate six directed connections. Peirce’s pragmatic maxim requires that a fully understood concept must be usable in every direction — not just in the directions that happen to be practised in class. Each of the six directions corresponds to a distinct instructional question:
| Direction | Instructional question |
| Do → Write | “Here is the bead configuration — write the numeral and its spelling” |
| Write → Do | “Here is the numeral — build it on the string” |
| Do → Talk | “Here is the configuration — say the number name” |
| Talk → Do | “I will say a number — build it on the string” |
| Write → Talk | “Here is the written expression — read it aloud and describe what each part means” |
| Talk → Write | “I will say a number — write its spelling” |
A child who has practised all six directions owns the concept of that number in Peirce’s pragmatic sense: its meaning is constituted by the full set of its uses, and she has those uses. A child who has practised only two or three directions has a concept that is genuinely more restricted — not a full understanding partially expressed, but a thinner understanding that will constrain what she can do with the number later. This also explains why mismatches between modes are pedagogically productive rather than merely signs of error. When a child’s bead configuration does not match her written expression, she faces a concrete semiotic problem: two representations of the same number are contradicting each other. The invariant structure that both should express is not yet stable in her understanding. Resolving the mismatch — not by being told the answer, but by working across the modes until they agree — is the process by which that structure is consolidated. The Talk–Do–Write framework creates structured opportunities for exactly this kind of productive tension. It is not a teaching strategy bolted onto a mathematical curriculum; it is a curriculum built around the semiotic architecture of mathematical understanding itself.
One of the most significant features of the linear abacus is that the four arithmetic operations do not require four different representational systems. They are four interpretations of a single family of gestures performed on the same physical object.
| Operation | Gesture | Symbolic form |
| Addition | Merge beads forward across a zero pinch from a start position | a + c =? (start known, change known, end unknown) |
| Subtraction | Separate beads backward from an end position | ? + c = b (start unknown, change known, end known) |
| Multiplication | N-fold: gather beads into repeated groups of fixed size, count the folds | n ×c =? |
| Division (quotitive) | Count how many complete folds of a given size fit in a total | ? ÷c = n |
The underlying structure in all four cases is the difference triple (additive) or ratio triple (multiplicative): three quantities standing in a determined relationship, with one unknown. The gesture changes direction (forward/backward) and granularity (individual beads/groups), but the same string, the same beads, and the same fundamental act of counting are present throughout.
This unity has consequences beyond curriculum efficiency. Standard instruction introduces operations sequentially, each with its own manipulative conventions and representations, implicitly suggesting they are four separate kinds of mathematical activity. On the linear abacus, their unity is perceptually present from the beginning. The child who has practised addition and subtraction on the bead string can see, when multiplication is introduced, that folding is a structured version of repeated addition — not as an abstract principle but as a visible fact about the same object she has been using all along.
The unity extends to fractions. When a fold does not divide evenly into a total — as
when 42 beads are folded into groups of 5, yielding 8 complete folds and 2 remainder beads
— the remainder is naturally expressed as a fraction of the group size: 8
. Fractions are
not a new representational system introduced with new rules; they are the answer to a
question that arises from the same fold gesture that grounds multiplication. The linear
abacus provides a single representational substrate for the full structure of primary
arithmetic.
The argument above is not a case for discarding MAB. Their iconic representation of cardinal magnitude has genuine pedagogical value, particularly for modelling written algorithms for multi-digit computation, and for making the proportional relationships between place values concretely visible once those relationships are understood. The case is for their proper positioning within a sequenced curriculum.
The problem with introducing MAB first — as current practice almost universally does — is that it installs the cardinal fallacy as the foundation of the child’s number concept. A student who meets number first as a collection of blocks acquires an iconic-cardinal model of what number is, and subsequently faces the positional notation system whose logic is indexical-ordinal, with no instructional bridge between them. The research literature on place value difficulties is substantially a literature about the consequences of this unmanaged transition (Kamii 1986; Ross 1989; Mix et al. 2014). The persistent difficulties in place value acquisition documented by (Mix et al. 2014) are not merely developmental hurdles; they are structural symptoms of positional opacity. As the child moves from the additive logic of spoken names to the multiplicative logic of notation, the linear abacus provides the missing ‘gestural foundation’ that bridges these two disparate semiotic systems.
The sequencing we propose follows three stages:
Linear abacus and spelling: establish the ordinal-cardinal weld through counting gesture. Teach numeral encoding and decoding using colour-coded spelling notation. Develop fluent bidirectional practice across all Talk–Do–Write transcoding directions with two- and three-digit numbers. Ground large-number intuition in metric embodied experience.
MAB: introduce iconic-cardinal representation once the ordinal-indexical foundation is secure. The ten-rod is now readable as an abbreviation of ten folds already understood — a compressed record of a process the student has herself performed. The student acquires two complementary representations, indexical and iconic, and can move between them.
Number line: bridge from discrete indexical model to continuous linear representation, supporting measurement, negative numbers, and the later development of rational arithmetic.
This sequence parallels the well-established Simple View of Reading (Gough and Tunmer 1986): reading comprehension is the product of decoding skill and language comprehension, and instruction must develop both, in the right order. A Simple View of Numeracy would analogously hold that numerical understanding requires both ordinal-indexical structure (linear abacus) and cardinal-iconic magnitude (MAB), taught explicitly and in sequence.
The phonics parallel clarifies what is at stake in the sequencing proposal. The argument for systematic phonics is not merely that it produces better outcomes than the alternatives — though the evidence that it does is substantial (NationalReadingPanel2000; Rose 2006). The deeper argument is that alphabetic writing encodes spoken language through a two-level architecture whose lower level — the grapheme-phoneme correspondence — is opaque in the written symbol. Instruction that does not make this level explicit leaves the learner to infer the code from exposure to instances of its use, which a significant proportion of children cannot do reliably. The structure of the problem determines the structure of the remedy.
The argument for the proposed curriculum sequence rests on the same logic. Positional notation encodes arithmetic structure through a two-level architecture whose lower level — the operator/operand distinction, the indexed ordinal foundation of each digit’s meaning — is concealed by the written symbol. Instruction that does not make this level explicit leaves the learner to infer it from exposure to numerals and procedures, which the research literature on place value difficulties shows a substantial proportion of children cannot do reliably (Kamii 1986; Ross 1989; Mix et al. 2014). The linear abacus addresses the lower level directly, as systematic phonics addresses grapheme-phoneme correspondence directly. The MAB addresses the upper level — the iconic magnitude of multi-digit quantities — as whole-language approaches address the upper level of written language: richly and usefully, but on the prior assumption that the lower level is already secure. The sequencing proposal is therefore not merely a pragmatic recommendation about which manipulative to use first. It is the numeracy equivalent of the phonics finding: that explicit instruction at the founding semiotic level is not optional for a significant proportion of learners, and that no amount of upper-level richness compensates for its absence.
Systematic phonics instruction transformed literacy outcomes not because it introduced a new and better reading programme, but because it identified the precise semiotic mechanism through which alphabetic writing encodes speech — the grapheme-phoneme correspondence at the lower level of the writing system’s double articulation — and made that mechanism the explicit object of instruction. The result was a shift from approaches that assumed structural understanding would develop incidentally from exposure, to approaches that taught the structure directly and produced measurably more reliable outcomes for the full range of learners. The key insight was structural: alphabetic writing has a two-level architecture; the lower level is opaque; opacity requires explicit instruction to overcome.
This paper has argued that early numeracy faces the same structural problem and admits the same structural solution. Positional notation has a two-level architecture. Its lower level — the indexed ordinal structure that gives each digit’s position its meaning, the operator/operand distinction that the written symbol conceals — is opaque in exactly the sense that grapheme-phoneme correspondences are opaque: not hidden by complexity, but by the very economy of the notation that makes it powerful. The linear abacus removes this opacity in the same way phonics removes phonological opacity: by making the founding semiotic level physical, explicit, and teachable. The spelling technique is the numeracy equivalent of systematic phonics instruction. The Talk–Do–Write framework is the numeracy equivalent of the bidirectional, cumulative, explicit practice that phonics research identifies as the mechanism of reliable decoding acquisition.
What the MAB/LA comparison establishes is not that one manipulative is better than another, but that the two tools operate at different semiotic levels — and that the level the linear abacus addresses is foundational in the same sense that grapheme-phoneme correspondence is foundational to reading: it is what the upper-level tool presupposes and cannot itself supply. This is why the linear abacus makes sense of the MAB rather than replacing it, just as phonics instruction makes sense of whole-language reading practice rather than replacing it: the two operate at different levels of the same two-level system, and the lower level must be secure before the upper level can do its work.
The argument developed in this paper can be compressed into five claims, presented here in order of their direct bearing on primary teaching and learning. The theoretical order of the argument in the body of the paper runs in the opposite direction — from ontological foundation to pedagogical application — for reasons of philosophical rigour; the reversal here is deliberate.
So we conclude: place value difficulties are not developmental — they are semiotic. The child has been taught to play the wrong language game (Block world), and no amount of further practice in that game produces the understanding that belongs to a different one (Number world). The linear abacus, used with the spelling technique and Talk–Do–Write framework described here, corrects the grammar of number at its source — and in doing so provides early numeracy with an explicit, unified, semiotic foundation whose structure matches the mathematics it is meant to teach.
Dewey, John (1938). Logic: The Theory of Inquiry. New York: Henry Holt.
Dienes, Zoltán P. (1960). Building Up Mathematics. London: Hutchinson.
Fuson, Karen C. (1988). Children’s Counting and Concepts of Number. New York: Springer.
Gelman, Rochel and Charles R. Gallistel (1978). The Child’s Understanding of Number. Cambridge, MA: Harvard University Press.
Gough, Philip B. and William E. Tunmer (1986). “Decoding, Reading, and Reading Disability”. In: Remedial and Special Education 7.1, pp. 6–10.
Grouios, Genovieve and Andrew Waywood (2026). A Concise Instructional Manual for the Linear Abacus®: For Teachers, Parents, and Children. First published January 7, 2026. Australia: Linear Abacus®.
Hjelmslev, Louis (1961). Prolegomena to a Theory of Language. Trans. by Francis J. Whitfield. Trans. F. J. Whitfield. Original Danish 1943. Madison: University of Wisconsin Press.
Kamii, Constance (1986). “Place Value: An Explanation of Its Difficulty and Educational Implications for the Primary Grades”. In: Journal of Research in Childhood Education 1.2, pp. 75–86.
Kress, Gunther (2010). Multimodality: A Social Semiotic Approach to Contemporary Communication. London: Routledge.
Martinet, André (1960). Elements of General Linguistics. Trans. by Elizabeth Palmer. Trans. E. Palmer. London: Faber & Faber.
Mix, Kelly S., Richard W. Prather, Linda B. Smith, and Jerri D. Stockton (2014). “Young Children’s Interpretation of Multidigit Number Names: From Emerging Competence to Mastery”. In: Child Development 85.3, pp. 1306–1319.
National Reading Panel (2000). Teaching Children to Read: An Evidence-Based Assessment of the Scientific Research Literature on Reading and Its Implications for Reading Instruction. Washington, DC: National Institute of Child Health and Human Development.
Peirce, Charles Sanders (1931–1958). Collected Papers of Charles Sanders Peirce. Ed. by Charles Hartshorne, Paul Weiss, and Arthur W. Burks. 8 vols. Cited as CP [volume].[paragraph]. Cambridge, MA: Harvard University Press.
Radford, Luis (2014). “On the Role of Representations and Artefacts in Knowing and Learning”. In: Educational Studies in Mathematics 85.3, pp. 405–422.
Rose, Jim (2006). Independent Review of the Teaching of Early Reading. London: Department for Education and Skills.
Rosenshine, Barak (2012). “Principles of Instruction: Research-Based Strategies that All Teachers Should Know”. In: American Educator 36.1, pp. 12–39.
Ross, Sharon H. (1989). “Parts, Wholes, and Place Value: A Developmental View”. In: Arithmetic Teacher 36.6, pp. 47–51.
Sáenz-Ludlow, Adalira (2020). “Classroom Collective Argumentation as Semiotic Mediation”. In: ZDM Mathematics Education 52.3, pp. 411–423.
Stevens, Stanley S. (1961). “To Honor Fechner and Repeal His Law”. In: Science 133.3446, pp. 80–86.
Wagenaar, Willem A. and Sabato D. Sagaria (1975). “Misperception of Exponential Growth”. In: Perception & Psychophysics 18.6, pp. 416–422.
Waywood, Andrew (2025). “Role-Binding Hypostasis: A Semiotic Mechanism for Meaning Constitution and the Genesis of Solidarity”. Preprint (Part I of II), December 13, 2025. PhilPapers.
— (2026). “Operational Arithmetic as Numerogrammar: A Semiotic Foundation for Arithmetic Learning”. Preprint (Part II of II), January 27, 2026. PhilPapers.
Wittgenstein, Ludwig (1953). Philosophical Investigations. Trans. by G. E. M. Anscombe. Blackwell.
