Abstract

Introduction: A central challenge in the teaching and learning of geometry in secondary school is ‎supporting students’ transition from practical geometry to theoretical geometry (Mariotti, 2007). In ‎practical geometry, the validity of a proposition is often inferred from visual or experiential cues. ‎In contrast, theoretical geometry requires students to rely on logical connections among accepted ‎propositions, definitions, and theorems (Balacheff, 2010). One of the earliest points at which this ‎transition becomes visible in the geometry curriculum is triangle congruence proofs, when students ‎are expected to construct formal geometric proofs using precise language, explicit premises, and ‎recognized criteria (Wang et al., 2018).‎

The purpose of this study was to investigate the learning obstacles (Brousseau, 2002) that ‎eighth grade students encounter when constructing triangle congruence proofs. While prior research ‎has documented common errors (Wang et al., 2018), misconceptions (Cirillo & Hummer, 2019), ‎and the competencies of high attaining students (Cirillo & Hummer, 2021), these studies have not ‎examined how such difficulties emerge dynamically within the social context of classroom and ‎during learning.‎

Theoretical Framework: In Brousseau’s (2002) theory of didactical situations, an obstacle is a ‎conception that is valid and efficient within a familiar context but becomes a source of error when ‎applied outside that context. Obstacles are thus rooted in prior successful experiences and are ‎resistant to change precisely because they have been productive in the past. This framework allows ‎us to interpret students’ difficulties not as deficits but as natural consequences of their existing ‎knowledge structures.‎

Method: To gain an initial understanding of students’ conceptions of congruence and proof, the ‎authors conducted exploratory interviews with 13 eighth garde students. Drawing on insights from ‎these interviews, the first author and three collaborating teachers designed a sequence of lessons for ‎teaching triangle congruence proofs in eighth grade in four one-hour sessions. These lessons began ‎with a “Triangle Congruence Story,” a scaffold intended to support students’ early reasoning and ‎gradually fade as students developed more formal proof practices.‎

The lessons were implemented in three eighth grade classrooms. Data consisted of audio and video ‎recordings from 12 classroom sessions, all of which were transcribed for analysis. Students’ proofs ‎and the classroom discussions surrounding them were examined to identify errors, misconceptions, ‎and reasoning patterns. These were coded using descriptors aligned with Brousseau’s notion of ‎obstacle. ‎

Findings: Three broad categories of learning challenges emerged: 1) Challenges related to symbolic ‎representations, 2) Challenges related to identifying the premises of the proof, and 3) Challenges ‎related to selecting and using triangle congruence criteria.‎

Challenges Related to Symbolic Representations:‎

At the outset, students struggled with the conventions of geometric notation. Two recurring issues ‎were observed:‎

• Confusing segment and angle notation, such as writing “A = C” instead of “AC = …,” ‎which obscured the intended meaning.‎

• Misusing Farsi derived abbreviations for congruence criteria, particularly mixing symbols ‎that resembled one another.‎

Challenges Related to Identifying Premises:‎

Seven challenges emerged in students’ attempts to identify and articulate the premises needed for a ‎proof. These challenges reflect a developmental progression from perceptual reasoning toward ‎theoretical justification:‎

• Perceptual claims: asserting equality based solely on appearance (“they look equal”). ‎

• Perceptual justifications: explaining claims through visual impressions (“as much as that ‎went up…”). ‎

• Ritualistic reasoning: invoking geometric terms without conceptual grounding (“because it’s ‎a hypotenuse”). ‎

• Inadequate arguments: misapplying general statements (“opposite sides are equal”). ‎

• Ignoring conceptual objects: overlooking key givens or relationships stated in the problem. ‎

• Conceptual confusions: mixing up median, altitude, and angle bisector. ‎

• Misidentifying triangle elements: selecting equal parts that do not belong to the target ‎triangles.‎

These challenges illustrate the tension between students’ prior perceptual practices and the ‎epistemic demands of formal proof. They also reveal how students gradually shift from intuitive ‎reasoning to more structured and conceptually grounded argumentation.‎

Challenges Related to Selecting and Using Congruence Criteria

Students also encountered obstacles in applying triangle congruence criteria appropriately:‎

• Treating AAA as a valid congruence criterion, despite its insufficiency. ‎

• Using SAS or ASA without verifying betweenness, leading to incorrect applications. ‎

• Assuming that right triangles always require HA or HS, overlooking the applicability of SAS ‎and ASA. ‎

• Applying HA without distinguishing between an acute angle and the right angle, resulting in ‎invalid conclusions.‎

These obstacles reflect students’ attempts to generalize from familiar patterns without yet ‎appreciating the structural constraints of each criterion.‎

Discussion and Conclusion: The findings indicate that challenges related to applying congruence ‎criteria were less varied but more complex than those related to identifying premises. This suggests ‎that the heart of students’ struggle lies not merely in recalling or selecting the correct criterion but in ‎constructing a coherent chain of reasoning grounded in appropriate premises.‎

A learning trajectory informed by these findings would begin by acknowledging students’ initial ‎reliance on perceptual claims. As students recognize the limitations of such claims, they naturally ‎shift toward perceptual justifications. When these also prove insufficient, students begin to invoke ‎theoretical terms—initially in ritualistic ways, and later with increasing conceptual depth. Along this ‎trajectory, students learn to attend to the conceptual objects embedded in the problem statement, ‎though superficial understanding may still lead to errors. Finally, as diagrams become more ‎complex, students must learn to distinguish between relevant and irrelevant equalities.‎

In learning to apply congruence criteria, students overcame obstacles sequentially. They first ‎confronted the misconception that AAA guarantees congruence, a belief dispelled through ‎counterexamples. They then learned to attend to betweenness in SAS and ASA, recognizing its ‎importance through tasks where neglecting it led to incorrect proofs. Students next broadened their ‎understanding of right triangle proofs, realizing that HA and HS are not the only applicable criteria. ‎The final obstacle involved refining their use of HA by distinguishing between acute and right ‎angles.‎

Overall, this study highlights how students’ prior knowledge can become an obstacle in the domain ‎of geometric proof. By identifying these obstacles and understanding their epistemic roots, ‎educators can design instructional trajectories that better support students’ transition from ‎perceptual to theoretical geometry, ultimately strengthening their capacity to construct valid and ‎meaningful geometric proofs.‎