Sfard, Anna. "When the Rules of Discourse Change, but Nobody Tells You: Making Sense of Mathematics Learning From a Commognitive Standpoint." The Journal of the Learning Sciences, vol. 16, no. 4, Lawrence Erlbaum Associates, 2007, pp. 565–613. https://doi.org/10.1080/10508400701525253
About the Original Article's Tone
This is a theoretical framework paper published in the Journal of the Learning Sciences — a peer-reviewed academic journal aimed at researchers in cognitive science, education, and the learning sciences. Sfard is writing for an audience that already knows Vygotsky, Wittgenstein, and Cobb. She is not explaining herself to a general audience; she is staking a theoretical position in an ongoing debate.
It uses:
Dense philosophical argument rooted in Wittgenstein's language game framework and Vygotskian participationist theory
Original coined terminology (commognitive, commognitive conflict) that she defines carefully but then uses freely throughout
Empirical case studies as illustrations of the theory — not as primary evidence for it (the framework was developed independently)
Long, subordinate-clause-heavy sentences that reward slow reading and punish skimming
A rhetorical strategy of defining her position against prior frameworks (acquisitionism vs. participationism, cognitive conflict vs. commognitive conflict) before asserting her own
Occasional moments of real intellectual humility — the studies don't produce clean results, and she says so
The vibe: A philosopher-researcher who has spent years thinking about one problem and is now formalizing the vocabulary to describe it. Think tight theoretical manifesto with empirical windows rather than classroom-teacher-friendly practitioner guide. You will need to read slowly and probably re-read the theoretical section before the case studies make full sense.
What it glosses over: Sfard is almost entirely uninterested in practical implementation at scale. She shows two case studies — one seventh-grade class, two first graders — and draws sweeping theoretical conclusions. The cross-cultural generalizability question (both studies are from Israeli classrooms) goes unaddressed. The paper also doesn't engage with what a classroom of 30 heterogeneous students looks like when the teacher does understand commognitive conflict — she identifies the problem more sharply than the solution.
Visual Metaphor
You've seen footage of slime mold solving a maze. Physarum polycephalum — a single-celled organism with no brain, no neurons, no central processor — spreads exploratory tendrils through a maze, finds the food at both ends, and within hours retracts every dead-end route, leaving a single efficient path. Researchers have replicated the Tokyo rail network with it. They built a model of the Roman road system.
The stunning part isn't the solution. It's that the organism doesn't think about the maze and then act on it. There is no separate computation happening somewhere that gets transmitted to the tendrils. The chemical signaling through the cytoplasm is the intelligence. The network is not the carrier of thought — the network is the thinking, full stop.
Sfard is making the same claim about mathematical learning. When a student says "x squared" instead of "x two," they are not translating a thought from inner language into outer words. The saying is the thinking. Pull the discourse out and there is no mathematical cognition underneath. The rules of the discourse are the rules of the mind.
What This Is Really About
You know that moment in math class when a student can work the problem — they follow the steps, they get the answer — but when you ask them to explain it, something breaks down? Traditional cognitive psychology would say: the concept isn't fully internalized yet. The thinking is there; the language is lagging behind. Sfard says that framing is exactly backwards, and getting it backwards is why teaching at certain mathematical junctures keeps failing.
Commognition is the word she coins — from cognitive and communicational — to assert that thinking and communicating are not two separate processes where one precedes and causes the other. Thinking mathematically is a form of mathematical communication. Mathematical communication that has been internalized enough to happen without another person present is what we call mathematical thinking. They are the same activity, running in different modes.
The Core Idea: Mathematical Discourse as the Object of Learning
Most researchers treat discourse as a means for learning mathematics. "Classroom discussion helps students learn math." Sfard argues this is backwards. Discourse is the object of learning. When you learn mathematics, what you're learning is a set of discursive practices — specific word uses, specific visual mediators, specific endorsed narratives, specific routines. You are not acquiring concepts that then get expressed in mathematical language. You are learning to participate in mathematical discourse, and that participation is the concept.
Mathematical discourse has four defining characteristics that Sfard identifies and analyzes throughout the paper:
Word use: Mathematical words have specific disciplinary uses that differ from everyday use. "Negative" as a unary prefix (negative 5) vs. "minus" as a binary operation (7 minus 5) is an example — the discourse has rules about these that are almost never made explicit.
Visual mediators: Mathematical discourse uses specially created symbolic artifacts — number lines, algebraic formulae, geometric diagrams — not just images of physical things. How you scan a number line with your eyes while adding is part of the discourse; it is not separate from the thinking.
Endorsed narratives: Mathematical discourse has specific criteria for what counts as true. What can be endorsed and what can't is determined by the rules of the discourse — not by empirical correspondence to an external reality.
Routines: Well-defined repetitive patterns in interlocutors' actions. Some routines are explicit (object-level rules, like "multiply before you add"). Others are tacit meta-rules — rules about the discourse that almost no one can articulate but everyone is expected to follow.
Two Types of Learning — and Why Metalevel Learning Is So Hard
Here is where the framework gets genuinely useful for MASL. Sfard distinguishes two types of mathematical learning:
Object-level learning: Expanding the existing discourse — new vocabulary, new routines, new endorsed narratives. This is what most math instruction targets. "Add this formula to your toolkit."
Metalevel learning: Changing the meta-rules of the discourse — the tacit governing rules about how the game is played. This is dramatically harder, because the meta-rules are almost always invisible to learners, and often to teachers.
The reason metalevel learning is so hard: the old meta-rules don't seem wrong to students. A student who thinks "x two" is the right way to say x² isn't confused about the symbol — they're following a perfectly coherent meta-rule (say the base, say the exponent number). The rule seems to work everywhere else. They have no incentive to change it because, from inside their current discourse, nothing is broken.
Commognitive Conflict: The Mechanism of Metalevel Learning
If the old meta-rules don't seem wrong, what forces the change? Enter commognitive conflict — the situation that arises when two interlocutors are operating according to different meta-rules. They may use the same words but mean different things by them. They may endorse contradictory narratives both of which seem completely valid from within their respective discourses. The key insight: this is not a factual disagreement where one person is right and one is wrong. It is an encounter between incommensurable discourses.
The paper's comparison of commognitive conflict and cognitive conflict is worth spelling out:
Dimension
Cognitive Conflict (Acquisitionism)
Commognitive Conflict (Sfard)
Locus
Between the learner's belief and the world
Between incommensurable discourses of two interlocutors
Role in learning
Optional — useful for removing misconceptions
Practically indispensable for metalevel learning
How resolved
Student's rational effort against external evidence
Student's acceptance and individualization of expert's discursive ways
The resolution of commognitive conflict is not logical argument. It's gradual enculturation — "thoughtful imitation" in Sfard's words. You learn to participate in the new discourse before you fully understand why its rules exist. The understanding comes through the participation, not before it. This is uncomfortable for progressive pedagogy that prizes student agency, but Sfard is blunt about it.
What the Two Classroom Studies Show
Sfard applies the framework in two case studies:
Study 1 (Negative Numbers, Grade 7): Students learning negative numbers hit a hard wall at "minus times minus is plus" — not because they lack computational skill, but because the justification for that rule requires a metalevel shift in what counts as mathematical proof. The old meta-rule: a claim is valid if a concrete model confirms it. The new meta-rule: a claim is valid if it is consistent with axioms. No concrete model exists for minus times minus equals plus. Students who haven't made the metalevel shift have no basis to accept the claim except teacher authority — and teacher authority alone is insufficient when the rules of the discourse haven't been negotiated.
Study 2 (Triangles and Quadrilaterals, Grade 1): Two first graders (Eynat and Shira) can sort triangles correctly using direct perception — but can't articulate why, and resist applying the definition to elongated triangles that "don't look right." The metalevel shift required: from direct identification ("this IS a triangle by nature") to discursively mediated identification ("this CAN BE CALLED a triangle because the definition says so"). This sounds simple. It took weeks of scaffolded repetition to partially achieve.
The Learning–Teaching Agreement
Both studies end imperfectly, and Sfard uses this to make her strongest practical argument: the conditions for learning, not just its mechanisms, matter enormously. She proposes that successful navigation of commognitive conflict requires a learning–teaching agreement — an unwritten set of understandings about three things:
The leading discourse: Both parties must agree on which discourse (the expert's) will provide the standard. This is inherently a power question.
Interlocutors' roles: The teacher must actually teach (not withhold her discursive expertise to honor student discovery); the student must genuinely commit to following the expert's lead (not just tolerating it).
The nature of the change: Both parties must accept that the new discourse will initially feel foreign and arbitrary — that the student will practice it as discourse for others (for the teacher's benefit) before it becomes discourse for oneself (spontaneous mathematical thinking).
The seventh-grade teacher violated the learning–teaching agreement by refusing to demonstrate her discursive ways — hoping students would discover the meta-rules themselves. They couldn't. The first-grade teacher honored it by simply repeating the correct discursive sequence patiently, dozens of times. Her results were better, though still incomplete by the study's end.
The Big Picture
The most important claim in this paper — the one that makes it a foundational text for MASL — is this: you cannot fix a discourse problem with more content exposure. Students who say "x two" for x² are not going to self-correct by doing more algebra. They are operating in a fully coherent internal discourse. Only a direct encounter with the conventional discourse, scaffolded by an expert who makes the conflict visible and demonstrates the new form patiently, will produce the metalevel change. The title says it plainly: when the rules of discourse change, but nobody tells you, you keep playing by the old rules. MASL is, at its core, a system for telling students — explicitly and repeatedly — that the rules have changed.
Key Vocabulary
Core terms in alphabetical order. Academic definition first, then a plain-language "Simply:" translation.
Acquisitionism
Simply: The old-school view that learning means picking up mental objects — concepts, ideas, knowledge — that live in your head and grow over time.
The theoretical tradition that conceptualizes learning as the acquisition of entities such as concepts or ideas. Acquisitionists focus on cross-situational invariants and view individual cognition as the primary unit of analysis. Sfard argues this framework glosses over the interpersonal texture of learning and is blind to metalevel change.
Commognition
Simply: A portmanteau that erases the wall between thinking and talking — they are not two things but one activity viewed from different angles.
A portmanteau of cognitive and communicational, coined by Sfard to describe the view that thinking and interpersonal communication are the same category of activity. Thinking is individualized (intrapersonal) communication; mathematical discourse is interpersonal commognition. The term signals that whatever is said about one applies to the other.
Commognitive Conflict
Simply: The productive disorientation that happens when your mathematical conversation partner is playing by different game rules — and neither of you knows it.
The situation that arises when interlocutors communicate according to different meta-rules, producing seemingly contradictory narratives that cannot be adjudicated by shared criteria. Unlike cognitive conflict (where one belief is simply wrong), commognitive conflict arises from incommensurable discourses — neither party is wrong within their own system. It is the primary driver of metalevel mathematical learning.
Discourse
Simply: A flavor of communication — not just vocabulary, but the whole package of rules, objects, gestures, and habits that define how a group talks about something.
A type of communication practice that brings certain people together and excludes others, characterized by particular word uses, visual mediators, endorsed narratives, and routines. Mathematical discourse is one such type — distinct from colloquial discourse in all four features. Learning mathematics means joining the mathematical discourse community, not extracting mathematical objects from an independent reality.
Discourse for Others / Discourse for Oneself
Simply: The difference between speaking a second language to please a teacher and dreaming in it. One is performance; the other is internalization.
Sfard's developmental distinction within learning. A discourse for others is one the learner practices because expert interlocutors use and value it — it feels foreign, and the learner enacts it for communicative purposes rather than spontaneous reasoning. Discourse for oneself is the same form of communication after individualization — it becomes the learner's spontaneous tool for thinking. The transition from the former to the latter is the goal of mathematical education, and it cannot be rushed.
Endorsed Narrative
Simply: A statement the discourse community agrees to accept as true — and the criteria for what counts as "true" are built into the discourse itself.
Any text framed as a description of objects or relations that is subject to acceptance (labeled as true) or rejection (labeled as false) within a discourse. Mathematical endorsed narratives include theorems, definitions, and proofs. Crucially, what counts as endorsable depends on the discourse's own meta-rules — not on mind-independent empirical reality. The shift from concrete-model-required to axiom-consistency-required is a change in what kinds of narratives can be endorsed.
Learning–Teaching Agreement
Simply: An unwritten contract between teacher and student about whose discourse leads, what roles each plays, and the understanding that the new language will feel weird before it feels natural.
Sfard's term for the set of unspoken but essential understandings that enable commognitive conflict to catalyze learning rather than stall communication. Requires consensus on: (1) which discourse sets the standard (the leading discourse), (2) each party's role as teacher or learner, and (3) the expectation that new discourse will initially be practiced as a discourse for others. Without this agreement, commognitive conflict becomes an obstacle rather than an opportunity.
Mediated Identification
Simply: When you count a triangle's sides before you call it a triangle — instead of just knowing it's a triangle because it looks like one.
An identification procedure in which an object's name is assigned through an explicit, communicable discursive sequence — as opposed to direct identification (spontaneous, perception-based, non-articulable). The shift to mediated identification in geometry requires recognizing that naming is a human decision about discourse, not a natural fact about the world. It splits the formerly merged act of recognition-and-naming into two distinct steps.
Meta-rules (Metadiscursive Rules)
Simply: The silent rulebook of the discourse — the rules about how to play the game that nobody explicitly teaches you but you're still expected to follow.
Rules that govern discourse practices rather than the objects of discourse — rules about how mathematical communication is conducted, how words may be used, what counts as valid proof, and how definitions function. Meta-rules are rarely explicit; they are learned through participation, not instruction. Their tacit nature is precisely what makes metalevel learning so difficult: you can't teach someone to follow rules you've never stated.
Metalevel Learning
Simply: The deeper and harder kind of math learning — where you have to change your assumptions about how the game is played, not just learn new moves in the game.
Learning that involves transformation of meta-rules — the governing tacit rules of mathematical discourse. As opposed to object-level learning (expanding vocabulary, adding routines, producing new narratives within existing meta-rules), metalevel learning requires changing the rules themselves. It is almost never self-initiated, cannot be produced by logical argument alone, and requires sustained encounter with the new discourse through a knowledgeable interlocutor.
Objectification
Simply: When a symbol or word stops feeling like a label and starts feeling like a real thing with properties you can manipulate — the sign gets "filled in."
The discursive process through which mathematical nouns (e.g., negative five) cease to be mere labels and become objects in their own right with independent properties and relations. An objectified use of negative five treats it as an integrated numerical entity; a non-objectified use treats the minus sign as an annotation appended to the "real" number five. Students' degree of objectification is directly observable in their word use and routine performance.
Participationism
Simply: The newer view that humans develop by participating in collective practices — learning is a change in what and how you do things with others, not a private accumulation of mental stuff.
The theoretical tradition that views developmental transformations as changes in patterned human practices rather than in individual mental acquisitions. Participationists trace learning to Vygotsky and activity theory; they view collective activity as developmentally prior to individual cognition. Sfard's commognitive framework is participationist but goes further by equating thinking with self-communication rather than just describing thought as socially shaped.
Routine
Simply: A habitual, recognizable pattern in how participants of a discourse act — the scripts and moves that define "doing mathematics" in a given community.
A well-defined, repetitive pattern of action characteristic of a given discourse. Routines partially overlap with word use, mediator use, and narrative endorsement but are broader. Object-level routines are explicit and grounded in mathematical properties (calculation procedures). Meta-level routines are tacit patterns governing how the discourse is conducted (how one identifies a geometric figure, how one justifies a definition). Routine change is the core observable indicator of mathematical learning in the commognitive framework.
Visual Mediator
Simply: The symbolic tools mathematicians use to point at and coordinate their talk about mathematical objects — not illustrations, but the actual medium of mathematical thought.
Artifacts used to identify the object of mathematical talk and coordinate communication. Unlike colloquial discourse mediators (images of physical objects), mathematical visual mediators are specially created symbolic artifacts — formulae, graphs, number lines, diagrams. Within commognition, mediators are not auxiliary aids for expressing pre-existing thought; they are constitutive of the thinking itself. How a person uses a mediator is an observable component of their mathematical discourse.
🎯 MASL Connection
This Study Supports:
All Four MASL Interventions — at the level of mechanism, not method. Sfard's commognitive framework is the theoretical engine that makes MASL's design choices defensible. Each intervention maps cleanly:
Baseline Language Assessment: The assessment's core design premise — that verbal production of algebraic notation is a measurable, diagnosable dimension of mathematical competency — is grounded in commognition. Sfard establishes that word use is a primary observable feature of mathematical discourse. Students who say "x two" for x² are not making a vocabulary error; they are participating in a non-conventional mathematical discourse. The assessment diagnoses which students have not been inducted into conventional algebraic discourse — from a commognitive lens, this is a cognitive diagnosis, not a vocabulary check.
Partner Card Sort: The "We Say" card in the MASL Trio is not decorative. Sfard is explicit: the spoken name IS part of the mathematical object students are learning to think with. The card sort that forces students to repeatedly match notation (Math ★) to spoken form (We Say ▲) to meaning (Meaning ●) is commognitive practice — it trains the three-way fusion that constitutes objectified discourse participation. The sentence frames ("I say this is a match because...") scaffold the mediated identification routine that Sfard shows is the key developmental shift.
Worked Example: Language Frames: Sfard's learning–teaching agreement requires that the expert interlocutor model the correct discourse before students can be expected to participate in it. Worked examples with sentence frames are precisely this — a scaffolded demonstration of the conventional discourse, structured to allow students to participate (complete the frames) before they have fully individualized the form. Sfard is explicit that this "discourse for others" stage is not a sign of failure; it is the mechanism of learning.
Suggest Improvements: This activity is a designed commognitive conflict. Students receive a mathematical statement using informal/invented notation (the student's current discourse), identify the imprecision, and replace it with the conventional MASL form (the expert discourse). Sfard's framework predicts this conflict is where metalevel learning happens — but only if the conflict is visible and the leading discourse is clear. The Suggest Improvements activity makes the conflict explicit and provides the conventional form as the resolution standard. This is exactly what Sfard says the negative-numbers teacher failed to do when she withheld her discursive leadership.
Design Implications:
Suggest Improvements must come AFTER students have used the "wrong" form, not before. Sfard is clear: commognitive conflict requires a genuine encounter between two active discourses. If students have never produced "x two" spontaneously, having them correct it in a worksheet exercise is object-level vocabulary practice, not metalevel learning. The baseline assessment should precede Suggest Improvements by at least one instructional cycle — the student needs to have operated in their current discourse long enough for the conflict to be real.
The teacher's role in the Partner Card Sort is not neutral facilitation — it is active discourse modeling. Sfard's negative-numbers case study shows what happens when the teacher withholds her expert discourse to honor student discovery: the students default to whichever peer voice is most socially authoritative, not the most mathematically correct. The teacher must be willing to demonstrate the conventional spoken form and to assert it as the leading discourse, especially during the early card sorts.
The "We Say" card's spoken form needs to be heard aloud repeatedly before it becomes discourse for oneself. Sfard's framework implies there is no shortcut through the "discourse for others" stage. Students who see "x squared" written but never say it aloud, never hear the teacher say it emphatically, and never practice it in sentence frames have not had the conditions Sfard identifies for individualization. The verbal production requirement is not pedagogical enthusiasm — it is mechanistically necessary.
The Math Irregulars framework targets meta-rule changes, not object-level additions. Students who say "x to the two" are not missing a vocabulary item — they are applying a perfectly coherent meta-rule (say the operation, say the exponent value) that works everywhere except for x² and x³. Correcting this requires metalevel learning: changing the meta-rule itself. Sfard predicts this will be slow, require sustained scaffolded encounter, and will not be produced by explanation or logical argument alone.
Evidence Strength for MASL:
Theoretical Foundation
Sfard (2007) provides the strongest possible theoretical warrant for MASL's central claim — that verbal production is a cognitive mechanism, not a reporting mechanism. The commognitive framework is internally rigorous, extensively cited (4700+ article views; foundational in the math education and learning sciences literature), and directly applicable to algebraic notation discourse. However, this paper provides no effect sizes, no intervention data, and no direct evidence about algebraic notation specifically. It supports MASL at the level of mechanism: it explains why verbal production should matter, but does not demonstrate how much it matters at the instructional level. The two case studies are Israeli middle and elementary school — secondary Algebra 1/2 is not represented. For evidence that the mechanism produces measurable outcomes in secondary math, you need Ke & Newton (2024), Booth et al., and the Barbieri meta-analysis.
Connections to MASL Framework (click to expand)
MASL Trio (Math / We Say / Meaning cards): The three-card structure maps directly onto Sfard's four features of mathematical discourse: the Math ★ card activates visual mediator attention; the We Say ▲ card targets word use (the most direct point of entry into conventional discourse); the Meaning ● card targets endorsed narratives. The trio is not arbitrary — it covers the discourse dimensions Sfard identifies as most observable and teachable.
Sentence frames: Sfard's "thoughtful imitation" is the mechanism that sentence frames operationalize. The frames provide the conventional discursive form before the student has the metalevel understanding to generate it independently — exactly the learning–teaching agreement condition she prescribes. The Barko-Alva & Chang-Bacon critique (that frames reduce production to fill-in-the-blank) must be answered by ensuring frames target reasoning, not just conclusions.
Irregular forms instruction (Mathematical Language Irregulars): Every MASL irregular is a meta-rule exception. "x squared" instead of "x to the 2" is not a lexical anomaly — it is a metadiscursive rule inherited from Greek geometry that overrides the general exponent naming rule. Sfard's framework predicts students will actively resist this kind of metalevel correction unless the conflict is made explicit and the leading discourse is clearly demonstrated.
Compatible frameworks: Arcavi's symbol sense (1994) identifies the linguistic register of symbol reading as integral to competency — compatible and complementary. Moschkovich's Academic Literacy in Mathematics framework similarly treats discourse as inseparable from content. Long's interaction hypothesis in SLA (corrective feedback forcing form-meaning mapping) maps onto the commognitive conflict mechanism in second-language acquisition terms. All four frameworks converge on the same core claim through different disciplinary lenses.
💬 Key Quotes
Copy-paste ready for papers, discussions, and capstone arguments.
"The focus of this article is on one particular type of discourse (thus thinking), called mathematical."
p. 571Thesis
Why this quote: The parenthetical "(thus thinking)" is the entire argument compressed into four words. It collapses the thinking/discourse distinction in one stroke — use this as the opening salvo for any commognition argument.
"Thought is not an incorporeal process which lends life and sense to speaking, and which it would be possible to detach from speaking." (Wittgenstein, 1953, p. 108, quoted in Sfard)
p. 570Foundational
Why this quote: Sfard grounds her entire framework in Wittgenstein rather than inventing it from scratch — this quote is the philosophical anchor. Use it when you need to show that commognition isn't Sfard's idiosyncrasy; it's a specific application of a 70-year philosophical tradition.
"Learning mathematics may be defined as individualizing mathematical discourse, that is, as the process of becoming able to have mathematical communication not only with others, but also with oneself."
p. 573Definition
Why this quote: The operational definition of math learning in commognitive terms. "Individualizing discourse" is the conceptual move that makes MASL's verbal production activities learning activities, not just output activities. Students practicing "x squared" out loud are individualizing the discourse.
"Without other people's example, children may have no incentive for changing their discursive ways. From the children's point of view, the discourse in which they are fluent does not seem to have any particular weaknesses as a tool for making sense of the world around them."
p. 574Challenge
Why this quote: This is the MASL irregulars problem stated in the sharpest theoretical terms possible. A student who says "x two" has no reason — from inside their current discourse — to change. The old form works. Only a direct encounter with the conventional discourse, manufactured by an activity like Suggest Improvements, creates the incentive.
"Discourses are not mastered by overt instruction (even less so than languages, and hardly anyone ever fluently acquired a second language sitting in the classroom), but by enculturation ('apprenticeship') into social practices through scaffolded and supported interaction with people who have already mastered the Discourse." (Gee, 1989, p. 7, quoted in Sfard)
p. 605Provocative Claim
Why this quote: The uncomfortable truth that MASL has to reckon with honestly. "Overt instruction" alone doesn't produce fluency. MASL's system is designed around scaffolded social practices — the card sort, the partner frames, the suggest-improvements activity — not lecture-and-drill. This quote is the theoretical justification for why the interactive component of MASL is not optional.
"At this initial stage, children's participation is possible only if heavily scaffolded by expert participants. For some time to come, the child cannot be expected to be a proactive user of the new discourse: In his or her eyes, this form of talk is but a discourse for others, that is, a discourse that is used for the sake of communication with those to whom it makes sense."
p. 607Practical Application
Why this quote: Direct theoretical warrant for why the "We Say" card and sentence frames need heavy scaffolding at introduction. The goal is not immediate independent production; it is scaffolded participation in the conventional discourse. Internalizing comes later, through repetition.
"Introducing students to mathematical discourse is 'like teaching someone to dance': It necessarily involves 'some telling, some showing, and some doing it [by the teacher] with [the students], along with regular rehearsals.'" (Lampert, 1990, p. 58, quoted in Sfard)
p. 609Example
Why this quote: The most quotable and accessible summary of what teaching mathematical discourse requires. "Regular rehearsals" is exactly what the MASL card sort and partner talk activities provide. Use this in the capstone paper's methods section to justify the repeated-practice structure.
📚 References & Further Reading
Expand references with read/skip recommendations
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.
Must Read
What it is: The book-length expansion of the commognitive framework — this 2007 article is essentially a preview chapter. Tone: Dense but more developed than the article. Why it matters: The full theoretical apparatus, including the treatment of mathematical objects as discursive constructs, is here. Verdict: If you cite Sfard (2007) in the capstone, you should have this on your shelf; the book is the more comprehensive citation for theoretical sections.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.
Foundational
What it is: The Vygotsky collection that contains zone of proximal development and the arguments about the social origins of higher cognitive functions. Tone: Short chapters, accessible relative to primary Vygotsky. Why it matters: Sfard's framework is explicitly Vygotskian — the individualization/communalization distinction and the role of scaffolding both derive from here. Verdict: You don't need to re-read it for MASL, but knowing the framework is assumed.
Wittgenstein, L. (1953). Philosophical investigations. Blackwell.
Advanced
What it is: Wittgenstein's second major work, containing the language-game framework and the argument that meaning is use. Tone: Aphoristic, deliberately unsystematic — reads like numbered notes, not an argument. Why it matters: Sfard's entire ontological move (discourse produces its own objects; thinking is communication) is rooted here. Verdict: Skip the primary text; the secondary literature on Wittgenstein's implications for education is more efficient for your purposes.
Gee, J. P. (1989). Literacy, discourse, and linguistics: Introduction. Journal of Education, 171(1), 6–17.
Worth Reading
What it is: Gee's introduction of "big D Discourse" — the full social/cultural/political package that comes with entering a discourse community. Tone: Short and readable. Why it matters: The enculturation argument Sfard quotes on p. 605 is the heart of Gee's position. Useful for the MASL argument that symbol-speaking involves identity, not just vocabulary. Verdict: 15-minute read; adds the social-identity dimension that Sfard mentions but doesn't develop.
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. Journal of the Learning Sciences, 10, 113–164.
Worth Reading
What it is: Cobb and colleagues' participationist framework for classroom norms and mathematical practices — Sfard's closest theoretical relative and occasional sparring partner. Tone: Dense empirical-theoretical hybrid. Why it matters: Sfard explicitly locates commognition relative to Cobb's work; understanding the difference (sociomathematical norms vs. metadiscursive rules) sharpens your use of both. Verdict: Worth reading if you're writing about classroom discourse practices at length; skip if you're focused on the individual acquisition mechanism.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–64.
Worth Reading
What it is: Lampert's case study of her own mathematics teaching, developing the argument that doing mathematics is a social and discursive practice. Tone: Accessible; blends theory with classroom narrative. Why it matters: The "teaching as dance" quote Sfard borrows from here is the most practice-friendly framing of the learning–teaching agreement. Verdict: Worth an hour of your time — one of the most readable entry points to the "discourse as mathematical practice" tradition.
Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35.
Must Read
What it is: The foundational MASL paper that establishes symbol sense — the ability to read, manipulate, and speak about algebraic symbols with understanding. Tone: Accessible and example-rich. Why it matters: Arcavi and Sfard are compatible frameworks; Arcavi names what students need (symbol sense), Sfard explains why verbal production is part of that competency (because discourse is cognition). Together they provide the MASL theoretical spine. Verdict: Essential — read this alongside Sfard (2007).
🧠 Quiz
Six conceptual questions about the ideas — not definitions.
1. A student correctly solves every problem involving negative numbers but consistently argues that "minus times minus should be minus because you can't get positive from two negatives." According to Sfard, this student is experiencing:
Correct. The student's argument is internally coherent within a discourse where mathematical claims must be grounded in concrete models. Their meta-rule — "valid mathematical claims must have real-world instantiations" — is the exact rule that signed number arithmetic requires abandoning. This is a metalevel learning challenge, not a factual error. Showing them more examples of minus-times-minus won't help, because the problem isn't the example — it's the standard of endorsement. The wrong answers mistake the nature of the conflict: it's not a logic error (a) or a memory gap (b), and it's not the acquisitionist-style cognitive conflict between a belief and external reality (d). It's a discursive incommensurability — two endorsement systems producing different verdicts.
Not quite. Think about what Sfard means by meta-rules versus object-level rules. The student isn't making a computational mistake and isn't simply missing a fact. They're applying a perfectly coherent framework — that math claims require concrete-model grounding — that the new discourse has replaced with an axiom-based framework. This is a change in the governing rules of the discourse, not a gap within the existing rules. That's what makes it metalevel.
2. Sfard argues that mathematical discourse is fundamentally different from most prior research perspectives because it treats discourse as:
Correct. This is the core ontological claim. Most research says "classroom discourse helps students learn mathematics" — discourse as a means, with mathematics as a separate thing being learned. Sfard inverts this: learning mathematics IS changing one's mathematical discourse. The discourse doesn't help you get to the concept; the discourse IS the concept, in its individualized communicative form. Options (a), (c), and (d) all treat discourse as instrumental — a tool in service of something else. Sfard is arguing there is no something else. Mathematical cognition is mathematical discourse that has been individualized.
The key distinction is whether discourse is a tool for something else (a means) or whether it IS the thing being learned (the object). Most of the wrong answers treat discourse as instrumental — helpful scaffolding, social lubricant, or assessment window. Sfard's radical claim is that none of those framings go far enough. Learning mathematics means becoming a participant in mathematical discourse. There is no separate cognitive event that the discourse expresses or supports.
3. The seventh-grade teacher in Sfard's negative numbers study allowed students to debate "minus times minus" for two full class periods before intervening with the correct answer. According to commognitive analysis, the main problem with this approach was:
Correct. Sfard's analysis is explicit: by withholding her expert discourse, the teacher created a void. Students can't resolve a commognitive conflict by themselves — they don't know which discourse should set the standard, because the meta-rules of the new discourse are exactly what they haven't learned yet. When social authority (who's popular, who sounds confident) fills the leadership vacuum, the mathematically correct student (Sophie) loses the debate to the mathematically incorrect but socially dominant framing. The wrong answers miss the mechanism: it's not about time efficiency (a), not about student authority per se (b), and commognitive conflict can't be replaced by cognitive conflict (d) — the problem type is wrong for that resolution.
The key issue is the learning–teaching agreement, specifically the requirement that the teacher actively establish the leading discourse. When the teacher deliberately steps back, students face a commognitive conflict with no anchor point — they don't know which discourse rules to follow because that's exactly what they're supposed to be learning. Sfard is clear: you can't resolve meta-rule conflicts through student-led inquiry, because the meta-rules are the very thing students don't yet have. The teacher's well-intentioned restraint inadvertently prevented the learning.
4. Commognitive conflict is described as "incommensurable" rather than simply "incompatible." The difference matters because:
Correct. This distinction is philosophically precise and pedagogically consequential. If two discourses were merely incompatible, you could find shared criteria and demonstrate which one is correct — like resolving a factual dispute. But incommensurable discourses operate by different internal standards entirely. There's no view from nowhere that both parties share. This is why showing students a concrete model for "minus times minus" doesn't work: the model is compelling only to the person who already endorses concrete-model validation. The new discourse has replaced that standard with axiom-coherence — a criterion the student's current discourse doesn't recognize. Resolution requires adopting the new discourse, not being shown who won the argument within the old one.
Incommensurability is Rorty's term (cited by Sfard) and it means something specific: the two discourses don't share a meta-standard for resolving their conflicts. It's not that one is harder than the other (a), not a mathematical-vs-philosophical split (c), and definitely not about empirical evidence (d) — evidence presupposes shared criteria for what counts as valid evidence. The whole problem is that the two discourses have different ideas about what valid mathematical justification even looks like.
5. In the triangles study, Shira and Eynat initially use "direct identification" rather than "mediated identification." The key difference is:
Correct. This is Sfard's most philosophically interesting observation in the geometry study. For Shira and Eynat, calling something "a triangle" is like noticing it is red — a fact about the world, not a decision about language. When the teacher says the elongated figure is a triangle, the girls resist not because they disagree about the definition but because their ontological framework doesn't allow for names to be definitions. A triangle IS what a triangle looks like. The metalevel shift required: moving from "the world determines the word" to "the word is a human convention governed by a definition." This has nothing to do with accuracy (a), measurement tools (c), or who leads the activity (d).
The distinction is ontological, not procedural. It's not about speed, accuracy, tools, or who's in charge. Direct identification treats the world as imposing word use on us — shapes have "natural kinds" and names just label what we find there. Mediated identification treats naming as a human discursive act, governed by definitions that the discourse community has agreed upon. An elongated triangle "doesn't look like a triangle" to Shira because for her, triangularity is a natural property of shapes, not a name we apply based on counting sides. That's a metalevel belief, and changing it requires changing the meta-rule, not just learning the definition.
6. Sfard argues that requiring students to invent mathematical rules through discovery pedagogy, before the teacher demonstrates the conventional discourse, is problematic at metalevel learning junctures. Her strongest reason is:
Correct. This is the subtlest and most important argument in the paper. Meta-rules are contingent — they are useful customs, not logical necessities. "Minus times minus is plus" is not the only possible rule; mathematicians chose it because it preserves certain desirable algebraic properties. There's no way to discover this rule by examining the world or by logical deduction from prior rules. You can only arrive at it by being shown that the mathematical community adopted it, and why. Students asked to invent the rule will produce their own coherent alternatives (as Roi does) that follow logically from their existing meta-rules. There's nothing wrong with those alternatives within their own discourses. Discovery only works when there's something to discover in the world; metalevel changes involve human decisions about discourse conventions.
The answer isn't about student ability (a) or about discovery vs. direct instruction as a general matter (b). Sfard isn't anti-discovery for object-level learning. And her argument is not about authority structures (d) — she is quite careful to say that accepting the teacher's leading discourse doesn't require mindless obedience. The key is the contingency of meta-rules: they are not facts about the world that students can discover, nor logical necessities that students can derive. They are conventions that must be demonstrated and then adopted. Without a demonstration to adopt, students invent their own perfectly coherent alternatives.
🃏 Card Sort
Match each concept on the left to its correct characterization on the right. Drag and drop.
Concepts & Terms
Commognition
Metalevel Learning
Commognitive Conflict
Learning–Teaching Agreement
Object-Level Learning
Objectification
Mediated Identification
Meta-rule
Discourse for Others → Oneself
Incommensurability
What It Means / Does
The fusion of cognitive and communicational — thinking IS a form of communication, not a process that precedes or feeds into it
Changing the tacit governing rules of a discourse — not learning new moves, but changing the rules of the game itself
When two interlocutors operate by different meta-rules, producing contradictory narratives neither party recognizes as a rule difference
The unwritten understanding about leading discourse, roles, and the expectation that new discourse feels foreign before it becomes one's own
Expanding an existing discourse — new vocabulary, new routines, new endorsed narratives — without changing the governing meta-rules
When a mathematical symbol or noun stops being a label and becomes a manipulable entity with its own properties in a student's discourse
Assigning a geometric name through an explicit, communicable discursive sequence — counting sides before calling it a triangle
A tacit rule about how discourse is conducted, rarely made explicit, learned through participation — the "how you play the game" rather than moves within it
The developmental trajectory from using a discourse to please an expert, to using it as your own spontaneous cognitive tool
When two discourses share no meta-standard for adjudicating their conflicts — neither is wrong on its own terms; they play by incompatible rulebooks
Reflect
You opened with an image of slime mold solving a maze — no brain, just chemical signaling that IS the intelligence. Sfard's claim is that mathematical thinking works the same way: there's no thought underneath the discourse that the discourse expresses. But the slime mold analog also reveals a tension Sfard doesn't fully resolve: slime mold changes its network when conditions change. What conditions need to change in a classroom before a student's mathematical discourse changes? Does commognition explain the conditions, or just the mechanism?
Sfard describes a student — Sophie — who has objectified negative numbers, and a student — Roi — who hasn't. Sophie says "minus 12." Roi says "plus 12, because 6 is bigger." Both are articulate. Both are reasoning. According to commognition, Sophie is mathematically thinking; Roi is not — or is operating in a discourse that doesn't yet count as the mathematical one. Does this make you comfortable? What does it mean to say someone is not yet "mathematically thinking" when they're clearly thinking hard about mathematics?
Commognition was developed from small case studies — one class, two first-graders. Sfard draws sweeping theoretical conclusions from these episodes. The framework is compelling, but here's the scaling problem: in a heterogeneous class of 30 students, there may be students at five different metalevel stages simultaneously, running five different discourse systems, some never having been told that the rules changed at all. What does commognitive conflict look like when you can't give it three months of 1-hour discussions? Does the theory survive contact with the actual conditions of secondary algebra instruction?
Sfard's most politically charged claim: entering a new discourse requires accepting the teacher's discursive ways as the leading discourse — even before you understand why. She is careful to distinguish this from mindless obedience, but the line is thin. Who decides which discourse is "expert"? For students from communities where Standard Academic Mathematics is a site of historical exclusion — where "real math" has been used to track, gate, and exclude — is "accept the leading discourse first, understand it later" an epistemically neutral instruction? Or is there something in the learning–teaching agreement that needs to name the power dynamics more directly?
The MASL title phrase "Mathematical Language Irregulars" maps directly onto Sfard's insight that metalevel changes are triggered by encounters with the new discourse — but nobody tells you. A student who says "x to the 2" is not wrong; they're coherent. The irregular is "x squared" — a Greek geometry inheritance that breaks the general rule. Now flip the frame: if MASL tells students explicitly that "x squared" is an exception, does that make the learning easier (the conflict is now visible) or does it bypass the productive disorientation of commognitive conflict? Is there a version of Sfard's argument in which making the irregulars explicit reduces the depth of learning?
Sfard ends the paper with a frank admission: the commognitive framework has a sign reading "under construction." She's particularly uncertain about how learning–teaching agreements can be cultivated given cultural factors external to the classroom. Think about a specific student you know or have taught — or imagine the most reticent student in your future classroom. What would it take for that student to accept mathematical discourse as a "leading discourse" worth adopting? What does the learning–teaching agreement look like when the student's relationship to school mathematics has already been shaped by years of deficit messaging?