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Teaching Geometric Proof with Tech: Pitfalls and Possibilities

Hwei-Shin Harriman, Wode Ni, Yuchen Jin, Dominik Moritz, Joshua Sunshine

arXiv:2608.25117v1cs.HC

TL;DR

Geometric proof is difficult to learn, and existing technology offers limited support for it. The paper interviews geometry teachers and reviews proof tools, finding that annotation-centered, visual workflows remain fragmented across digital tools.

  • Problem

    Geometric proof is difficult because students must integrate spatial and logical skills, while existing tools provide limited support for this work.

  • Method

    The paper interviews geometry teachers to identify educational proof-tool requirements and uses them to review existing tools.

  • Results

    Teachers’ proof-solving workflow is inherently visual, but reviewed tools separate diagram annotation from other proof activities.

  • Takeaways & Limitations

    Educational proof tools should connect live diagrammatic annotation with proof construction rather than forcing teachers to choose between separate supports.

  • Takeaways & Limitations

    Supporting multiple solution paths and proof representations is technically challenging, while the common two-column format is deceptive.

Abstract

from arXiv · show

Geometric proof is a foundational yet challenging topic in mathematics, requiring students to integrate visual, logical, and notational skills. While technology has enhanced learning in other mathematical domains, its impact on geometric proof remains limited. To investigate this gap, we interviewed 18 geometry teachers to establish the technical requirements of educational proof tools. These requirements inform our review of 33 commercial and research tools. Our findings reveal a critical mismatch: while teachers value certain digital tools for initial planning and exploration activities, they revert to pen-and-paper for formal proof because it supports diagram annotation and provides space for multiple approaches to proof-solving. Annotating the diagram is a key component of the proof-solving workflow that existing tools do not support. We propose four technical and human-centered design guidelines for educational proof tools to meet teacher needs at scale: integrating diagram and proof, generating problems and feedback automatically, supporting multiple proof formats, and reducing accidental complexity in the user experience.

1 Introduction · 2 An Example Geometric Proof · 3 Background and Related Work

Geometric proof is difficult because students must integrate spatial, logical, and symbolic skills, while existing digital tools do not comprehensively support proof-solving. The paper examines teacher needs, defines geometric proof structure, reviews learning challenges and technologies, and identifies design directions for better tools.

  • 1 Introduction: Geometric proof requires simultaneous integration of spatial, logical, and symbolic skills, making it difficult for students to master.Only 30% of geometry students were reported to produce acceptable solutions to proof-based problems.
  • 1 Introduction: 18 geometry-teacher interviews revealed that teachers use digital tools for exploration and hypothesis formation but rely on pen-and-paper for formal proof.Teachers value pen-and-paper because students can add marks and annotate the proof while working.
  • 1 Introduction: A review of 33 tools found that none provide comprehensive support, with diagram annotation and visual reasoning remaining major omissions.The reviewed tools included intelligent tutoring systems, dynamic geometry environments, large language models, and automatic theorem provers.
  • 1 Introduction: The paper proposes four improvements: integrating diagram and proof, generating proof content automatically, supporting multiple proof formats, and reducing user-experience complexity.These improvements target teacher and student needs while reducing the manual labor required to create proof problems.
  • 2 An Example Geometric Proof: A geometric proof problem specifies a construction, givens, and a goal, while its two-column solution pairs each statement with a given or known geometric reason.The example uses triangle congruence through SAS and CPCTC to establish an angle relationship.
  • 3.1 Challenges of learning proof: Students struggle to shift from procedural mathematics to deductive reasoning, including identifying proof starting points and selecting applicable definitions or theorems.Geometric proof also requires students to distinguish diagram-based observations from properties that require deduction.
  • 3.1 Challenges of learning proof: Students may incorrectly treat diagrams as scale representations or assume proofs apply only to the specific paired diagram.The two-column format can also make proof-solving appear sequential even though proofs have graph-shaped logical structures and multiple possible paths.
  • 3.2 Teacher as a mediator of curriculum and technology: Teachers mediate curriculum and technology by adapting materials, framing tasks, choosing software, and guiding students toward open-ended deductive reasoning.Textbooks’ emphasis on memorization and low-level tasks can conflict with the reasoning required for proof.

4 Interview Study

The study used semi-structured remote interviews with 18 geometry teachers, organized into three phases progressing from pedagogical goals to technology practices and a concrete proof problem. Contextual inquiry and artifact elicitation captured teachers’ existing workflows and instructional strategies while allowing conversational variation.

  • Study design: 18 geometry teachers participated in semi-structured interviews conducted remotely via Zoom, with sessions lasting 50–80 minutes.One researcher led each interview while a second took notes; audio, video, and screen shares were recorded with consent.
  • Interview protocol: The interview protocol used three chronological phases: pedagogical goals, situated technological practices, and a concrete problem-solving scenario.This structure addressed RQ1 through discussions of proof-teaching philosophy and RQ2 through technology and interactivity questions.
  • Interview protocol: Teaching strategies were discussed before technology to reduce priming effects and capture pedagogical goals independently of existing tools’ constraints or affordances.The ordering was intended to separate participants’ teaching philosophies from specific digital-tool influences.
  • Situated practices: Contextual inquiry asked participants to screen-share existing curricular materials and demonstrate classroom workflows without creating new content.Shared materials included digital handouts, interactive modules, problem sets, and online resources; teaching materials were used to avoid revealing student data.
  • Problem-solving scenario: Artifact elicitation used a common depiction of the Figure 1 proof to elicit concrete instructional strategies, scaffolding tactics, and limitations of existing tools.The semi-structured format allowed participants to introduce related topics, deviate from the procedure, or skip questions as conversations developed.

Background

The background section identifies the interview topics used to characterize geometry teachers: teaching experience, school type, and geometry level taught.

  • The interviews asked teachers how long they had been teaching geometry.
  • The interviews asked what kind of school each teacher taught at.
  • The interviews asked what level of geometry each teacher taught.

Teaching geometric proof (targeting RQ1)

The section frames RQ1 through teacher interviews about classroom emphasis, the value and teaching process of geometric proof, instructional resources, and students’ areas of difficulty.

  • The interviews asked teachers how much they focus on teaching geometric proof in their classrooms.
  • Teachers were asked about the value of learning proof for students and their procedure for teaching geometric proof.
  • The interview covered teachers’ use of textbook problems, online resources, handouts, and other materials.
  • Teachers identified which parts of geometric proof students understand most easily and which are hardest to understand.

Use of technology and interactivity (targeting RQ2)

This section targets RQ2 by examining teachers’ use of interactive tools and technology for teaching proofs. The interview questions address tool types, classroom activities and frequency, perceived benefits and limitations, nonuse, proof-specific technology, and a recent classroom-use episode.

  • The interview asked teachers which interactive tools they use and what classroom activities they support.
  • Questions examined how often teachers use interactive tools and what they value about them, alongside challenges or limitations.
  • The interview also asked why teachers might use no interactive tools, what technology they use specifically for proofs, and details of their latest interactive-tool use.

Artifact elicitation: proof walkthrough

The artifact walkthrough asked teachers to compare a proof with expected student work, explain solution strategies, identify likely difficulties, and describe instructional guidance. Interviews involved screened U.S. high-school geometry teachers and were analyzed through inductive thematic analysis.

  • Artifact walkthrough: The final protocol asked teachers to assess student-work similarity, explain expected solution paths, identify challenging proof components, and propose guiding strategies.These questions structured the artifact walkthrough around expectations, problem-solving, difficulties, and instructional support.
  • Participant screening: Participants had to be U.S.-based teachers with at least 2 years of high-school geometry experience, including geometric proof, plus curriculum-development experience.Participants also had to be willing to share their teaching experience with researchers.
  • Participants: 18 participants averaged 19 years of geometry-teaching experience, ranging from 3 to 35 years, across public and private schools.The participant backgrounds are summarized in Table 1.
  • Data collection: Interviews were recorded, machine-transcribed, manually corrected, and supplemented with high-level notes and affinity diagrams.The authors periodically reviewed emerging insights and adjusted the interview protocol to probe identified blind spots.
  • Analysis: The authors used inductive thematic analysis, beginning with independent open coding before collaboratively refining themes against the dataset until theoretical saturation.Coding discrepancies were resolved by revisiting codebook definitions and merging or splitting ambiguous entries as needed.

5 Interview results targeting RQ1

Teachers described proof as central to mathematics but exceptionally difficult to teach, because students must shift from procedural problem-solving to unfamiliar logical argumentation. Their reported practices emphasize diagram annotation, strategies for getting unstuck, and flexibility across proof structures, while existing tools inadequately support formal proof.

  • 5 Interview results targeting RQ1: Teachers widely use dynamic geometry environments for general geometry but reported no adequate software for teaching formal argumentation.GeoGebra and Desmos were commonly used, yet participants described available proof tools as a poor fit and rigid platforms as frustrating.
  • 5.1 Challenges of teaching proof: Teachers called proof both central to mathematics and the most difficult topic to teach.They described proof as necessary for mathematical facts to count as established and as especially challenging instructionally.
  • 5.1 Challenges of teaching proof: Proof introduces logical argumentation that differs from students’ prior procedural algebra experience and often requires creative, non-linear reasoning.Teachers reported that students have not previously been asked to think this way, while proofs may have multiple valid structures and avenues of thought.
  • 5.2 Skills teachers build: Teachers use checklists, decomposition, rule banks, and other tactics to help students proceed when they are unsure how to continue.One participant described a checklist or algorithm as helpful for getting unstuck, while other reported strategies manage complex constructions and support rule selection.
  • 5.2 Skills teachers build: All participants reported teaching students to mark up diagrams so they can see relationships and track information established during the proof.Teachers specifically emphasized adding tick marks to constructions, a practice the authors translate into a tool requirement.
  • 5.2 Skills teachers build: Teachers want students to recognize that proofs can have multiple valid structures and to access formats beyond linear two-column presentations.They described interchangeable proof steps, tree-like dependencies flattened by textbook formats, and differing student preferences for flow or other formats.

6 Interview results targeting RQ2

Teachers use digital tools mainly for content curation, exploration, and selected feedback activities, but proof-specific software remains limited and often poorly aligned with classroom proof practices. Key shortcomings include guessable or single-path proof tasks, accidental interface complexity, and weak integration between diagram annotation and logical reasoning.

  • Resource constraints: N = 10 teachers reported using no proof-specific software, reflecting limited availability of tools compared with general-purpose classroom resources.Teachers’ time constraints lead them to adapt external materials and rely on general-purpose tools.
  • Dynamic Geometry Environments: All participants used Dynamic Geometry Environments for induction, discovery, intuition-building, and hypothesis testing, but regarded them as proof-adjacent rather than formally rigorous.Teachers used DGEs to manipulate objects and explore diagrams before transitioning to full proofs.
  • Intelligent Tutoring Systems: Proof activities in four Intelligent Tutoring Systems fell short of expectations despite promises of automatic feedback, personalization, and interactive materials.Teachers valued immediate, scaffolded, and difficulty-scaled feedback, but feedback quality varied widely.
  • Proof activity design: Partial-proof exercises were often guessable, encouraging elimination or button mashing instead of teaching students to reason from the diagram through the proof.Multiple attempts and limited answer choices let students infer answers without reading or understanding the proof process.
  • Proof activity design: Teachers wanted proof tools to accept multiple valid approaches and support complete proofs from the given statement to the goal.Only DeltaMath and ALEKS offered full-proof activities, using libraries of pre-programmed valid solutions.
  • Diagram–proof integration: Existing software poorly supports the workflow of annotating diagrams while reasoning, while usability inconsistencies add accidental complexity unrelated to geometric argumentation.Teachers requested clickable diagram marks, correctness feedback, visual hints, and integration of these annotations into proof-solving.

7 Existing Tool Review

The review finds substantial gaps between teachers’ geometric-proof requirements and existing tools: no category or individual tool supports all proof activities. Digital tools help with exploration and feedback, but pen-and-paper remains important for freeform annotation and multiple proof paths.

  • Coverage of Existing Tools: 33 tools across 8 categories were reviewed, yet no category or individual tool supports all activities in the proof lifecycle.The requirements span exploration, annotation, formal deduction, feedback, and teacher-led curation.
  • Exploration and Annotation: DGE-supported compass-ruler constructions effectively support exploration, with three tools fulfilling all four core exploration requirements EX01-EX04.These tools let students test whether specific observations are valid, but ATP feedback is binary: the property is provable or “not provable.”
  • Exploration and Annotation: Teachers’ annotation practices remain underserved because few tools provide marking features, and those features are generally optional rather than integrated with feedback.Participants use diagram marking to visualize the proof state and make sense of the proof.
  • Flexibility and Proof Formats: Tools must support multiple proof solutions and formats because two-column proofs flatten tree-like dependencies and leave no space for alternate paths.Transitioning among formats can help students understand proof structure rather than viewing it as a linear sequence.
  • Feedback and Authoring: Immediate feedback is software’s primary advantage, but existing feedback is inconsistent and usually text-only, with few tools offering visual feedback or hints.The review identifies auto-generated feedback as a scaling challenge and calls for integrated graphical interventions that connect feedback to the diagram.
  • Feedback and Authoring: The cost of authoring a single flexible proof problem with feedback must be dramatically reduced for teacher authoring to become realistic.Teachers often adapt or select problems because proof instruction spans only a few weeks, while creating flexible feedback imposes an unrealistic workload.

8 Discussion

The discussion identifies diagram–proof integration, scalable automated generation and feedback, and combined tool capabilities as key directions for next-generation proof education tools. It also emphasizes that technical challenges and accidental complexity must be addressed while preserving visual and logical aspects of geometric proof.

  • Diagram–proof integration: Geometric proof is visual as well as logical, but existing tools do not integrate diagrams, student annotations, and written proofs.Teachers emphasized referencing, marking, and redrawing diagrams during proof solving; existing scribble features cannot verify or require correct markings.
  • Scalable generation and feedback: Automated theorem proving tailored to high-school geometry could reduce the one-to-many content bottleneck by generating solution spaces and corrective feedback.The prover should accept only high-school-level properties and produce understandable proofs while supporting multiple approaches.
  • Scalable generation and feedback: Combining the rigor of an ATP with the flexibility of an LLM could support personalized assistance, additional questions, worked examples, and unexpected questions.This combination addresses the limits of pre-authored feedback and can provide solutions with different levels of detail without constant teacher intervention.
  • Multiple representations and usability: Multiple views and representations of the same proof can support different ways of working, while reducing switching costs and other sources of accidental complexity remains necessary.The discussion proposes toggling between representations as students work and identifies menu interactions as a source of accidental complexity.

9 Conclusion

Geometric proof instruction depends on a visual, scaffolded deductive cycle, but existing tools separate diagram annotation from logical proof and feedback. The paper calls for interdisciplinary, user-centered tools with semantically linked live diagrams, automated problem support, multiple visual proof formats, and reduced accidental complexity.

  • Teacher needs: 18 geometry teachers identified a scaffolded deductive cycle of Fact Collection, Visual Search, and Translation for proof instruction.The cycle is inherently visual and relies on diagrammatic annotation and pattern-matching.
  • Technology landscape: 33 existing tools expose a segmented workflow in which annotation and immediate feedback are supported by separate tools.The largest oversight is the lack of integration between the diagram’s visual state and the proof’s logical state.
  • Design implications: The teaching challenge is both a content problem and an interaction design problem requiring formal precision combined with user-centered design.The paper positions HCI as able to reduce accidental complexity and support the non-linear process of student learning.
  • Design implications: “Live” diagrams should be interactive and semantically linked to the formal proof, while reducing the effort needed to build proof problems.These capabilities would help implement, evaluate, and explore different visual proof formats.
  • Design implications: Tool designers must avoid accidental complexity because additional complexity can overwhelm students already facing a complex subject.Aligning algorithmic capabilities with classroom realities could make geometric proof an accessible gateway for logical reasoning.

10 Generative Artificial Intelligence Use Disclosure

The authors used Gemini 2.5 Pro for initial drafting and phrasing, then reviewed and refined its output. Scite supported parts of the literature review, with all supplied references manually validated before inclusion.

  • 10 Generative Artificial Intelligence Use Disclosure: Gemini 2.5 Pro converted outlines into rough text and brainstormed alternative phrasing for selected sections.
  • 10 Generative Artificial Intelligence Use Disclosure: The authors thoroughly reviewed, refined, and replaced Gemini-generated text as necessary to ensure accuracy and alignment with the paper’s intent.
  • 10 Generative Artificial Intelligence Use Disclosure: Scite assisted the literature review, specifically in developing Sections 1 and 3, while authors manually validated all provided references before final inclusion.

11 Funding Disclosure

The work was supported by multiple National Science Foundation awards and a Graduate Research Fellowship grant, with additional funding from the ARCS Foundation’s Pittsburgh Chapter.

  • National Science Foundation support included Award Nos. 2447499, 2346174, and 2119007.
  • The project also received National Science Foundation Graduate Research Fellowship Program Grant No. DGE2140739.
  • The authors state that the expressed opinions, findings, conclusions, and recommendations do not necessarily reflect the views of the National Science Foundation.
  • Additional funding came from the ARCS (Achievement Rewards for College Scientists) Foundation, Pittsburgh Chapter.

13 Author Contributions

The authors divided responsibilities across study conception and design, data analysis and interpretation, and manuscript drafting and critical revision. All authors approved the final manuscript and accepted accountability for the work.

  • Hwei-Shin Harriman contributed to study conception and design, data analysis and interpretation, manuscript drafting, and critical revision.
  • Wode Ni contributed to data interpretation, manuscript drafting, and critical revision for intellectual content.
  • Yuchen Jin designed the study and analyzed the data, while Dominik Moritz and Joshua Sunshine contributed to study conception, data interpretation, and critical manuscript revision.
  • All authors provided final approval of the published version and agreed to accountability for all aspects of the work.
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