{"schema_version":1,"research_id":"eoa_inverse_innovation_exp06_external_evaluation_20260803","source_assessment_id":"ritualized_meaning_and_commitment_enactment__mathematics:P2:v0","cell_id":"ritualized_meaning_and_commitment_enactment__mathematics","search_queries":["site:maa.org CUPM geometry axiomatic systems recommendations undergraduate mathematics curriculum proof","students difficulties axioms non-Euclidean geometry education study","embodied learning geometry physical manipulation axioms non-Euclidean education research","concept map dependency graph axioms theorems geometry teaching","\"Student interpretations of axioms in planar geometry\" full text","\"axioms\" \"non-Euclidean geometry\" students conceptions mathematics education empirical","site:maa.org \"Geometry Study Analysis\" axioms","site:ams.org undergraduate geometry course axiomatic non-Euclidean teaching","\"Analysing signs emerging\" \"designed dependency task\"","\"designed dependency task\" geometry students","\"Assumption Digging\" geometry dissertation full text","mathematics education manipulatives theorem dependency cards axioms classroom","10.1080/0020739X.2023.2246979","Højsted Mariotti designed dependency task dynamic geometry 2024 abstract"],"sources":[{"source_id":"S1","title":"An investigation into senior high school students’ performance and difficulties in proving Euclidean geometry in mathematics curriculum: Evidence from Ghana","publisher":"Contemporary Mathematics and Science Education","url":"https://www.conmaths.com/article/an-investigation-into-senior-high-school-students-performance-and-difficulties-in-proving-euclidean-17990","source_class":"PRIMARY_RESEARCH","publication_date":"2026-02-27","accessed_at":"2026-08-03","claims_supported":["In a purposive sample of 62 students, learners struggled with correct geometric proofs, logical reasoning, and connecting geometric concepts.","Geometry-proof difficulty is consequential enough for researchers to call for targeted pedagogy, although this study does not specifically measure illicit transfer between axiom systems."]},{"source_id":"S2","title":"Student interpretations of axioms in planar geometry","publisher":"Taylor & Francis, author-uploaded full text hosted by ResearchGate","url":"https://www.researchgate.net/publication/322688661_Student_interpretations_of_axioms_in_planar_geometry","source_class":"PRIMARY_RESEARCH","publication_date":"2018-01-24","accessed_at":"2026-08-03","claims_supported":["Four semesters of neutral-geometry teaching experiments found recurrent student interpretations of axioms that conflicted with formal mathematical norms.","Formal attention to dependence and independence among axioms rarely appeared initially and often emerged only after reflective discussion.","Guided reinvention and reflection on creating and using axioms are established instructional approaches adjacent to the proposal."]},{"source_id":"S3","title":"Teaching Non-Euclidean Geometries in High-School: an experimental study","publisher":"University of Camerino","url":"https://pubblicazioni.unicam.it/handle/11581/483528","source_class":"PRIMARY_RESEARCH","publication_date":"2023-02-16","accessed_at":"2026-08-03","claims_supported":["The dissertation reports that axiomatic geometry was problematic for students and that changing a few axioms can be used to expose plural geometric systems.","Short seminars and workshops involving 154 students and 57 teachers establish feasibility of bounded instruction on non-Euclidean geometries.","The work identifies limitations in earlier qualitative evidence and the need for better-documented, larger-scale experiments."]},{"source_id":"S4","title":"Assumption Digging in Euclidean Geometry","publisher":"arXiv","url":"https://arxiv.org/abs/2106.01419","source_class":"PRIMARY_RESEARCH","publication_date":"2021-06-02","accessed_at":"2026-08-03","claims_supported":["A teaching experiment in two Indian schools recursively traced claims to premises, definitions, and axioms using argument trees.","Students mostly justified instructor-questioned claims; only some questioned instructor arguments, showing both feasibility and the importance of facilitation.","Tracing dependencies from conclusions to assumptions is close prior art for the proposal’s mathematical core."]},{"source_id":"S5","title":"Analysing signs emerging from students’ work on a designed dependency task in dynamic geometry","publisher":"Taylor & Francis; record hosted by PURE Faroe Islands","url":"https://www.pure.fo/en/publications/analysing-signs-emerging-from-students-work-on-a-designed-depende/","source_class":"PRIMARY_RESEARCH","publication_date":"2023-09-28","accessed_at":"2026-08-03","claims_supported":["A design-based study implemented geometry dependency tasks with four Danish grade-eight students using dynamic geometry.","Students did not immediately grasp hierarchical dependencies; specific prompts and explanations of unexpected observations appeared necessary.","Interactive dependency representation is established adjacent prior art, but the study did not use a recurring governed changeover ritual."]},{"source_id":"S6","title":"Geometry: CUPM Curriculum Guide Study Group Report","publisher":"Mathematical Association of America","url":"https://maa.org/wp-content/uploads/2024/06/Geometry.pdf","source_class":"OFFICIAL_GUIDANCE","publication_date":"2015","accessed_at":"2026-08-03","claims_supported":["The MAA Geometry Study Group recommends substantial geometry content and at least one geometry-focused undergraduate course.","The report says understanding the axiomatic method should be part of every mathematics major’s education and recommends exposure to non-Euclidean geometries.","Mathematics departments and geometry instructors are identifiable adopters or authorizers for a course-level pilot."]},{"source_id":"S7","title":"Guidance on Web Accessibility and the ADA","publisher":"U.S. Department of Justice, Civil Rights Division","url":"https://www.ada.gov/resources/web-guidance/","source_class":"OFFICIAL_GUIDANCE","publication_date":"2022-03-18","accessed_at":"2026-08-03","claims_supported":["Inaccessible digital content can deny equal access, and public entities must provide effective communication.","Color-only cues, missing text alternatives, mouse-only navigation, and inaccessible forms are recognized barriers.","The proposal’s digital and visual alternatives require institution-specific accessibility review; nominal modality choice alone does not establish accessibility."]},{"source_id":"S8","title":"Occupational Employment and Wages, May 2023: Mathematical Science Teachers, Postsecondary","publisher":"U.S. Bureau of Labor Statistics","url":"https://www.bls.gov/oes/2023/may/oes251022.htm","source_class":"GOVERNMENT_OR_REGULATOR","publication_date":"2024-04-03","accessed_at":"2026-08-03","claims_supported":["BLS reported a 2023 mean annual wage of $95,320 and median of $81,020 for postsecondary mathematical-science teachers.","Instructor, teaching-assistant, accessibility-review, and evaluation labor dominate resource-equivalent pilot costs; 2026 estimates require inflation and overhead assumptions."]}],"problem_evidence":{"support":"MODERATE","rationale":"Multiple studies visibly establish difficulties with geometric proof, metamathematical interpretations of axioms, and hierarchical dependency reasoning. Dawkins is particularly relevant to alternative axiomatic systems. None directly measures the proposal’s exact observable state—students importing theorems immediately after a course changes systems—within the intended comparative-geometry course, so prevalence and local importance remain unverified.","source_ids":["S1","S2","S3","S5"]},"stakeholder_evidence":{"support":"MODERATE","rationale":"The MAA explicitly identifies mathematics departments and geometry instructors as the audience responsible for teaching axiomatic systems and non-Euclidean geometry, and Cardinali reports teacher participation and instructional concern. This establishes a credible adopter/authorizer class and expressed curricular need, but no named institution or instructor has expressed demand for the ritualized assembly itself or committed a course section, staff time, or funds.","source_ids":["S3","S6"]},"prior_art":{"proximity":"ADJACENT_PRIOR_ART","closest_analogues":[{"name":"Guided reinvention of neutral axiomatic geometry","similarity":"Students create, compare, use, and reflect on axioms and provability relations; reflective discussion develops formal interpretations.","remaining_difference":"No recurring symbolically marked system-change ceremony, witnessed closure, opt-out governance, reciprocal commitments, or retirement process was reported.","source_ids":["S2"]},{"name":"Assumption Digging","similarity":"Recursively maps a conclusion to premises, definitions, and axioms through an argument tree and facilitated student questioning.","remaining_difference":"It investigates assumptions within Euclidean geometry rather than ceremonially replacing an axiom and reversibly suspending dependent conclusions at repeated system transitions.","source_ids":["S4"]},{"name":"Designed dependency tasks in dynamic geometry","similarity":"Interactive manipulation is deliberately used to focus attention on hierarchical logical dependencies among geometric properties.","remaining_difference":"It uses dynamic-software tasks, not removable axiom/conclusion cards or a governed collective changeover lifecycle; evidence involved only four grade-eight students.","source_ids":["S5"]},{"name":"Short non-Euclidean geometry seminars and workshops","similarity":"Students encounter plural geometric worlds created by modifying axioms in bounded classroom interventions.","remaining_difference":"The reported intervention is an instructional course/workshop, without evidence for the proposed ritual sequence or its incremental benefit over an ordinary dependency-map lesson.","source_ids":["S3"]}],"distinctive_claim_remaining":"Holding mathematical content, dependency display, distributed turns, follow-up task, and instructor support constant, adding the proposal’s marked changeover sequence—reversible premise replacement, witnessed classification, nonretaliatory participation, reciprocal commitments, and governed debrief/retirement—will reduce illicit axiom-system imports or improve diagnosis of them without increasing confusion between underivability and falsity, accessibility burden, or perceived participation pressure.","confidence":"MODERATE"},"implementation_evidence":{"support":"MODERATE","rationale":"The component techniques—guided axiomatizing, short non-Euclidean workshops, argument trees, dependency tasks, reflection, and simple physical/digital representations—have been implemented in educational settings, and the proposed pilot needs no novel technology. Feasibility still depends on an instructor-verified dependency map, equivalent nonvisual and asynchronous materials, a local accessibility check, protection from informal grading, and an institutional determination about classroom-evaluation data. No source validates the integrated ritual workflow or its safety in an undergraduate comparative-geometry course.","source_ids":["S2","S3","S4","S5","S7"]},"scores":{"meaningful_impact":{"score":3,"rationale":"Axiom-relative reasoning is a core curricular objective, and invalid premise transfer can undermine proof. The exact local error prevalence and downstream effect remain unmeasured.","source_ids":["S1","S2","S6"]},"stakeholder_pull":{"score":3,"rationale":"Professional guidance expresses a clear need to teach axiomatic systems and non-Euclidean geometry, but there is no committed adopter or demand for this specific assembly.","source_ids":["S3","S6"]},"incremental_advantage":{"score":2,"rationale":"The proposal isolates a testable addition to dependency-map instruction, but no comparative evidence shows that ritual marking improves learning beyond guided inquiry, dependency tasks, or ordinary manipulatives.","source_ids":["S2","S4","S5"]},"distinctiveness_plausibility":{"score":3,"rationale":"The mathematical core substantially overlaps established guided axiomatizing and dependency tasks; the governed recurring changeover bundle remains contrastively distinctive in the searched sources.","source_ids":["S2","S3","S4","S5"]},"technical_implementability":{"score":4,"rationale":"Cards, dependency paths, scripts, digital alternatives, and ungraded classification tasks are technically simple. Mathematical validation and accessible equivalent representations are the principal implementation burdens.","source_ids":["S3","S4","S5","S7"]},"adoption_authority_feasibility":{"score":3,"rationale":"An instructor and department can plausibly authorize a bounded activity inside an existing geometry course, but no actual course owner has agreed and local curriculum, accessibility, and research-review authority is unknown.","source_ids":["S6","S7"]},"evidence_readiness":{"score":4,"rationale":"The proposal states observable errors, a close comparator, immediate experience measures, and strong falsifiers. A credible design still needs comparable sections or transitions and predeclared scoring rules.","source_ids":["S2","S3","S5"]},"safety_net_benefit":{"score":3,"rationale":"Pass rights, modality alternatives, withdrawal of erroneous displays, debriefing, and retirement reduce coercion and misconception risks. Their effectiveness cannot be inferred from the written script, and accessibility requires local verification.","source_ids":["S7"]},"scalability":{"score":3,"rationale":"Materials are inexpensive and reusable, but each axiom transition requires expert dependency checking, accessible alternatives, facilitation, and monitoring for ritual drift; scaling across systems and sections is therefore labor-bound.","source_ids":["S3","S7","S8"]}},"score_confidence":"MODERATE","costs":{"first_evidence":{"band_2026_usd":"10K_TO_50K","scope":"Design and evaluate one comparator-controlled transition across two comparable course sections or discussion sections, including instructor/TA preparation, mathematical review, accessibility review, anonymous instruments, scoring, and analysis.","confidence":"LOW","assumptions":["Approximately 100-250 total professional hours across instructor, TA, accessibility specialist, and evaluator.","2023 BLS teacher compensation is escalated to 2026 and loaded for benefits/overhead.","Existing classrooms and learning-management systems are available without new licensing.","This is resource-equivalent cost, not necessarily incremental cash spending."],"source_ids":["S7","S8"]},"initial_deployment_startup":{"band_2026_usd":"10K_TO_50K","scope":"Create and independently check reusable physical, verbal, and digital dependency displays; write scripts, scoring rubrics, opt-out procedures, correction/rollback protocol, and staff training for one course.","confidence":"LOW","assumptions":["Roughly 120-300 professional hours depending on the number of axiom systems and accessibility accommodations.","No custom software is built.","Institutional legal, disability-services, or research-review work is absorbed as staff time."],"source_ids":["S7","S8"]},"operational_launch":{"band_2026_usd":"UNDER_10K","scope":"Run one 20-minute assembly, administer immediate anonymous feedback and one fresh ungraded task, correct artifacts, and conduct a short review.","confidence":"MODERATE","assumptions":["One instructor, one TA or access monitor, and one evaluator use previously prepared materials.","Total launch and analysis effort remains below roughly 100 loaded labor hours.","No student incentives or external venue costs are required."],"source_ids":["S8"]},"annual_recurring":{"band_2026_usd":"10K_TO_50K","scope":"Maintain and run approximately three transitions per year across two to four sections, refresh accessible artifacts, train rotating facilitators, review work samples, and perform an annual drift/harm audit.","confidence":"LOW","assumptions":["Approximately 150-400 loaded professional hours annually.","Mathematical displays are rechecked whenever the course’s axiom systems or worked theorems change.","The upper end reflects multiple sections and independent accessibility/evaluation support."],"source_ids":["S7","S8"]}},"verified_pipeline_gates":{"externally_supported_problem":{"status":"YES","reason":"Primary research supports difficulties with geometric proof, formal interpretations of axioms, and dependency reasoning, although the exact course-specific prevalence is not known.","source_ids":["S1","S2","S3","S5"]},"externally_credible_adopter_or_authorizer":{"status":"YES","reason":"The MAA identifies mathematics departments and geometry instructors as responsible for including axiomatic systems and non-Euclidean geometry; these roles can authorize a bounded course activity. No specific institution has committed to adopt it.","source_ids":["S6"]},"distinct_testable_incremental_claim":{"status":"YES","reason":"The added ritual package can be compared against an otherwise identical dependency-map lesson using axiom identification, unavailable-premise detection, false-versus-underivable classification, perceived choice, and accessibility outcomes.","source_ids":["S2","S4","S5"]},"bounded_next_evidence_step":{"status":"YES","reason":"A single-transition, two-section comparison is bounded in duration, population, materials, outcomes, and stopping rules.","source_ids":["S3","S5"]},"no_unresolved_safety_or_authority_stop":{"status":"UNCERTAIN","reason":"The safeguards are thoughtful, but no institution-specific accessibility review, course-owner agreement, data-governance determination, or research/quality-improvement review has been documented. Visible nonparticipation and informal grading pressure require live verification.","source_ids":["S7"]},"credible_cost_scope_and_range":{"status":"YES","reason":"The intervention uses ordinary classroom materials and its dominant resource is professional labor. BLS compensation data support the order of magnitude, while explicit staffing, overhead, and reuse assumptions bound the estimates.","source_ids":["S8"]}},"next_evidence_step":"Secure one comparative-geometry instructor, department approval, an accessibility review, and any required institutional determination for classroom-evaluation data. At one axiom transition, assign two comparable course or discussion sections to (A) the full Axiom Changeover Assembly and (B) the same checked dependency display, distributed response opportunities, instructional time, and follow-up support without threshold marking, synchronized turning, witnessed ritual closure, or commitment renewal. Before seeing outcomes, publish scoring rules for a fresh ungraded task requiring students to name the active system, identify one unavailable premise, and classify a conclusion as false, unproved, or undecided. Also collect anonymous perceived-choice, accessibility, pressure, and enactment-versus-proof confusion items. Stop rather than scale if the baseline error is absent; if arm A does not outperform B on the reasoning task within a predeclared educationally meaningful margin; if A increases false/refutation confusion or perceived pressure; if pass or nonvisual paths are conspicuous or inequivalent; or if the dependency display is materially disputed. Treat the result only as a pilot-specific estimate, not realized impact.","blocking_evidence":["No direct prevalence estimate for illicit theorem transfer at axiom-system transitions in the intended course population.","No comparative evidence that ritual marking adds benefit beyond identical dependency content, active inquiry, and follow-up support.","No named instructor, department, or curriculum coordinator has committed a course section.","No institution-specific accessibility verification for the physical, verbal, and digital dependency representations.","No documented local determination concerning collection and reuse of student work and anonymous experience data.","No measured evidence that passing is socially costless or that visible participation remains independent of grading.","No validated scoring threshold or sample-size rationale for the proposed fresh proof-classification task."],"research_disposition":"PARTNERED_RESEARCH_PROGRAM","world_novelty_boundary":"The search evaluated only publicly accessible educational research, professional guidance, and government sources returned by the preserved queries. It found close component practices but no exact integrated governed changeover assembly. That absence is not evidence of world novelty. Patentability, freedom to operate, comprehensive product/practice coverage, market size, and realized impact were not measured.","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_version":0,"controller_recommendation":{"action":"STOP_EMPIRICAL_RESEARCH_NEEDED","repairable":false,"material_progress_observed":true,"progress_targets":["Obtain a named instructor/department partner and document authority for the bounded pilot.","Complete institution-specific accessibility and data/research-review determinations before collecting outcomes.","Run the matched-content comparator so the ritual package, rather than the dependency lesson, is the only material contrast.","Predeclare task scoring, educationally meaningful advantage, pressure/accessibility thresholds, and all stop rules.","Report baseline prevalence, arm-specific reasoning outcomes, false-versus-underivable confusion, opt-out experience, missing data, and fidelity.","Retire the assembly if it provides no incremental advantage or produces coercion, inequivalent access, or mathematical misconception."],"reason":"Bounded web research verifies the general instructional problem, credible adopter roles, adjacent prior art, and low-complexity implementation, but it cannot establish local prevalence, incremental learning advantage, noncoercive participation, or equivalent accessibility. Those decisive claims require a partnered live classroom comparison. Under the controller rule, evidence requiring fieldwork or live testing must stop for empirical research, and every STOP is non-repairable in this evaluation cycle."},"proposal_index":2}