Enacting genericity: Coming to see the general in the particular while proving with manipulatives, diagrams, and symbols

Enacting genericity: Coming to see the general in the particular while proving with manipulatives, diagrams, and symbols | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Enacting genericity: Coming to see the general in the particular while proving with manipulatives, diagrams, and symbols Julien Putz, Muhammed Fatih Doğan, Dor Abrahamson This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8409836/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract A pervasive form of mathematical reasoning involves using a specific instance to draw conclusions about a general class. In mathematics education, researchers have conceptualized this practice as reasoning with “generic examples” and proposed it as a scaffold supporting students’ transition from informal exploration to the construction of proofs. Whereas existing research has largely investigated generic examples in terms of their epistemic status or didactic use within proving, it has paid less attention to the cognitive processes by which learners come to perceive a particular instance as standing for the general. Adopting an enactivist perspective, we conceptualize genericity not as an intrinsic property of an inscription but as a subjective cognitive achievement. We report on findings from task-based clinical interviews with five undergraduates in a STEM education course, interleaved with brief micro-phenomenological probes of lived experience. Participants worked through a sequence of proof tasks across registers (manipulatives, diagrams, symbols) designed to elicit generic reasoning. Through micro-genetic, multimodal analysis, we derived a three-phase model of enacting genericity: reaching, in which one case is transformed into neighboring cases; foregrounding, in which case-invariant structures become salient; and contracting, in which these structures are condensed into portable semiotic forms. Illustrative case analyses show how these micro-actions enable learners to treat an example-based argument as general. On this basis, we propose design principles for selecting and sequencing cognitively ergonomic examples that afford reaching, foregrounding, and contracting, and we argue that proofs retain explanatory power for learners when they emerge as contractions of prior example-based exploration. generic examples mathematical proof diagrammatic reasoning enactivism phenomenology Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8409836","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":571754764,"identity":"345ce4df-de23-4fde-9bbf-5a2725b65ed0","order_by":0,"name":"Julien 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