Symbol Sense: Informal Sense-Making in Formal Mathematics

About the Original Article's Tone

This is a theoretical framework paper published in For the Learning of Mathematics — a journal explicitly devoted to the intellectual side of math education rather than the experimental side. It is written for researchers and curriculum theorists, but it arrives with an unusual warmth: Arcavi is not laying out a formal model, he is sharing something he noticed and cannot stop thinking about.

It uses:

• Philosophical framing — questions asked in the spirit of "what might this mean?" rather than "what does the data show?"
• Anecdote as primary evidence — classroom episodes, student moments, teacher behaviors, rendered in close observational detail
• Admissions of incompleteness — Arcavi repeatedly notes that the characterization is partial, provisional, not settled
• Literature borrowed across disciplines — number sense, Gestalt psychology, Freudenthal's phenomenology, linguistic register
• Normative language — what symbol sense should include, what instruction ought to do

The vibe: This feels like a particularly good faculty colloquium talk written down. It moves like conversation — here is something curious, let me show you an example, now here is what I think it means, but honestly I'm still working it out. Think "brilliant colleague sketching an idea on a whiteboard" rather than "peer-reviewed results section."

What it glosses over: Arcavi does not operationalize symbol sense in a way that could be measured or systematically assessed. The definition is a list of components, not a rubric. There is no discussion of how symbol sense develops in students with different mathematical backgrounds, language backgrounds, or disability profiles. The implicit subject of the article is a fairly confident algebra learner; the framework's applicability to students who struggle with symbol manipulation at a basic level remains unaddressed.

Visual Metaphor

What This Is Really About

You know that feeling when you look at an algebraic expression and something feels off before you've done any calculation — like the equation can't have a solution, or the formula is probably simpler than it looks, or this setup is about to go somewhere interesting? That's symbol sense. Arcavi's 1994 paper is the first serious attempt to name it, describe it, and argue that it should be taught — not just hoped for.

Academic researchers gave this phenomenon a name in the early 1990s, building on parallel work in number sense. But the instinct they were describing — the fluent, meaningful relationship with mathematical notation — is as old as mathematics itself. What Arcavi did do is catalog it, argue it's multi-dimensional, and insist that it's teachable. That last part is the fight. Because a lot of math education implicitly treats symbol fluency as something you either develop or you don't.

The Core Idea

Symbol sense is a multi-dimensional competency — not a single skill, not a personality trait. It is the capacity to read mathematical notation with meaning, not just move it around according to rules. Arcavi distinguishes it sharply from symbol manipulation: you can be excellent at manipulation and have almost no symbol sense. The student who runs a procedure correctly but can't pause mid-computation and ask "does this answer even make sense given what I know about the problem?" has manipulation without sense.

Crucially, symbol sense includes knowing when not to use symbols. Part of the competency is recognizing when a graph, a diagram, or a verbal description serves better than formal notation. This is not a failure of algebraic fluency; it is a higher form of it.

The Components: What Symbol Sense Actually Includes

Arcavi proposes a provisional list of components. This is not a final taxonomy — he is explicit that the list is a "working tool for further reflection" — but it is the most precise characterization available. The components are:

1. Friendliness with symbols: An understanding of and aesthetic feel for the power of symbols — when they should be used to display relationships, generalizations, and proofs that would otherwise be hidden. This includes the complementary sense: knowing when symbols obscure rather than reveal, or when the work required is too costly for the gain.
2. The ability to manipulate AND to "read through" symbolic expressions as complementary aspects: Efficient manipulation requires detaching from meaning — a global "gestalt" view that moves quickly. Reading through symbols toward meaning requires the opposite: slowing down, inspecting the expression, extracting implications. Competent algebraists move between these modes fluidly. Most students only develop one.
3. Awareness that one can engineer symbolic relationships: The recognition that symbols are tools you can construct, not just formulas you apply. Knowing that you can write a symbolic expression to represent a desired verbal or graphical situation — and having the skill to actually do it.
4. The ability to select — and revise — symbolic representations: Choosing a variable assignment for a problem (e.g., whether to use n, n+1, n+2 or n−1, n, n+1 for consecutive integers) and having the courage to abandon a choice that isn't working and search for a better one.
5. Checking symbol meanings during a procedure: The habit of pausing mid-computation to ask whether the symbols are still tracking the original problem. Not just checking arithmetic — checking whether the symbolic treatment is still faithful to the situation it's supposed to represent.
6. Awareness of different roles symbols play in different contexts: Recognizing that x can be an unknown to solve for, a variable to graph, a parameter to manipulate, or a placeholder in an identity — and developing an intuitive feel for which role is active in a given context.

The Classroom Culture Problem

Here is what makes this paper more than a taxonomy: Arcavi argues that whether symbol sense develops is not primarily a cognitive question. It's a cultural one. He presents a case study of IA, a mathematically able Israeli high school student who demonstrated clear symbol sense — and explicitly suppressed it during problem-solving. Why? Because his classroom had rewarded symbolic procedure and penalized sense-making. Students who spent time making sense of a problem "usually have neither time nor energy to do the symbols... and fail the exams."

This is not a story about what students can't do. It's a story about what classrooms teach students not to do. Symbol sense is dormant in most students — not absent. The question is whether the instructional environment calls it into action or starves it of oxygen.

What Actually Happens When Symbol Sense Is Instructionally Supported

Arcavi describes a classroom episode from an 8th-grade pilot study on introducing inequalities. A student, asked to express "all heights below 2.90 meters" mathematically, offers x < 2.90. Another student suggests y < 2.90 as a different representation — because for students without deep algebraic acculturation, different letters genuinely feel like different representations. Rather than correcting this with a mini-lecture, the teacher asks both students what each representation "says."

A girl then produces an unprompted insight that becomes the paper's most quoted moment: "x < 2.90 shows something, one something less than 2.90, but the line shows all the numbers at once." This is explicit verbalization of symbol sense — her perception of the difference between a symbolic and a graphical representation of the same relationship, voiced because the classroom allowed it. The paper argues that this kind of meta-mathematical talk, if repeated and normalized, is the instructional mechanism through which symbol sense develops.

The Patience Argument

One of the more underappreciated parts of the paper is Arcavi's claim about intellectual patience. Drawing on Sfard's circularity argument (you need to use a concept to understand it, but you need to understand it to use it) and Tobias's research on humanities scholars who won't become science scholars, he argues that a core component of algebraic competence is the ability to tolerate partial understanding — to keep working without the comfort of full clarity, trusting that meaning will emerge.

This is developable. But it requires a classroom culture that stops treating "I don't fully understand this yet" as failure and starts treating it as the normal state of learning in progress. Teachers who compulsively resolve every open question — who cannot leave "unresolved issues hanging in the air" because it feels irresponsible — are inadvertently teaching students that partial understanding is a crisis, not a stage.

What This Challenges

The paper challenges two entrenched assumptions: (1) that symbol manipulation is the foundation and symbol sense grows on top of it once the foundation is solid; and (2) that symbol fluency is an inborn mathematical ability that some students simply have. Arcavi explicitly argues for nurture over nature and for sense-making as concurrent with and interwoven through symbol manipulation, not sequentially after it.

The Big Picture

Arcavi (1994) is ground zero for a thirty-year conversation about what it means to be fluent in mathematical language. The paper does not solve the problem it names — Arcavi is admirably honest about how incomplete the characterization is. But it establishes that there is a problem worth naming, that it's multi-dimensional, that it's teachable, and that the most important variable is not student ability but classroom culture. Every subsequent effort to design instruction that targets the spoken and written dimension of algebraic notation — including MASL — is standing on this paper.

Key Vocabulary

🎯 MASL Connection

This Study Supports:

• The theoretical rationale for MASL as a whole: Arcavi's central argument — that symbol sense is a multi-dimensional competency that includes the capacity to read through symbols toward meaning and to speak about what one perceives in notation — is the theoretical foundation on which MASL rests. MASL exists because Arcavi identified a gap between procedural symbol manipulation and genuine symbol fluency; MASL's four activity types are each designed to develop a different dimension of that gap.
• The "We Say" card in the MASL Trio: Arcavi's Component 2 (reading through symbolic expressions) and Component 5 (checking symbol meanings during a procedure) both require the learner to verbalize what they perceive in an expression. The "We Say" card is not a label added for EL access — it is the production mechanism Arcavi's framework implies. The student who says "x squared" rather than "x to the 2" is performing symbol sense, not reporting that they already have it.
• The Baseline Language Assessment: Arcavi's framework establishes that the verbal-production dimension of symbol sense is a genuine component of algebraic competency, not a secondary reporting layer. This provides the theoretical warrant for measuring it: if verbal production is part of the competency, then an instrument that assesses verbal production is measuring something real. The Baseline Language Assessment fills a gap Arcavi's framework implies must exist.
• The Mathematical Language Irregulars framework: Arcavi's Component 6 (awareness of different roles symbols play in different contexts) directly anticipates the Irregulars problem. When the symbol "−" means "minus" in one context and "negative" in another, or when "f(x)" means "f of x" but looks like multiplication, students cannot build symbol sense through exposure and inference alone — the notation has no consistent rule to sense. The Irregulars framework is a direct extension of Arcavi: it identifies the specific cases where Component 6 requires explicit instruction rather than cultivated intuition.
• Suggest Improvements activity: Arcavi's student girl — "x < 2.90 shows something, one something less than 2.90, but the line shows all the numbers at once" — is a student whose symbol sense about what she perceives is genuine but incomplete. The Suggest Improvements activity creates the instructional conditions Arcavi calls for: students are given a statement using informal spoken notation, asked to sense what's imprecise about it, and required to replace it. This targets exactly the verbalization-of-symbol-perception that Arcavi identifies as the developmental mechanism.
• The meta-mathematical talk principle underlying all four activities: Arcavi argues that meta-mathematical talk — discourse about notation itself, not just through notation — is the key instructional mechanism for developing symbol sense. Every MASL activity is structured around generating this kind of talk: naming symbols, discussing what expressions "say," correcting informal language, sorting representations.

Design Implications:

• The "We Say" card should require verbal production (spoken aloud), not just written identification — because Arcavi's Components 2 and 5 are real-time, in-the-moment capacities, not retrospective labels. A card sort where students only read the cards silently would not develop the dimension of symbol sense Arcavi identifies.
• The Baseline Language Assessment's two-prompt format ("I say this as:" / "I would describe it as:") directly corresponds to the distinction Arcavi draws between conventional reading (Component 1: "I know what this symbol is") and genuine sense-making (Component 2: "I can say what this expression is doing"). Both prompts are needed; one captures mastery, the other captures the sense-making capacity.
• MASL activities should normalize intellectual patience — Arcavi's argument that tolerating partial understanding is a component of algebraic competence means that activities which rush toward closure (correcting students immediately, confirming right answers quickly) actively undermine the disposition symbol sense requires. Suggest Improvements should leave the "why this is wrong" discovery with the student for as long as productive struggle is occurring.
• The classroom culture argument suggests that MASL activities need explicit teacher moves that signal symbol sense is valued — not just correct procedures. The card sort's role-rotation structure (placer/challenger) is one such move: it makes the act of questioning a placement a valued and required classroom behavior.
• The Mathematical Language Irregulars framework should be framed not as "exceptions to memorize" but as cases where symbol sense cannot be inferred — borrowing Arcavi's framework to explain why these forms are difficult (no rule to sense) rather than just listing them as hard vocabulary.

Evidence Strength for MASL:

This is theoretical foundation, not empirical warrant. Arcavi (1994) establishes the conceptual landscape and names the multi-dimensional competency that MASL addresses — but it provides no effect sizes, no controlled comparison, and no assessment instrument. Its strength for MASL is maximal for the "what is the problem we're solving" argument and zero for the "our solution produces measurable gains" argument. It should be cited to establish what symbol sense is and why the verbal-production dimension is a legitimate target; it cannot be cited as evidence that any particular MASL activity works. For that, MASL requires the empirical studies in its bibliography (Ke & Newton, Barbieri et al., Mercer & Sams, etc.).

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📚 References

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