Research report: Two-tier teaching experiment and third-order modeling

A new mathematics-education methodology proposes a two-tier teaching experiment for examining how teachers’ interpretations of students’ thinking develop through sustained interaction, alongside a system of third-order modeling that separates students’ reasoning, teachers’ interpretations and researchers’ accounts of those interpretations. The paper develops the concept and procedures rather than reporting outcomes from a classroom trial, so its contribution is a testable framework for future empirical studies. Its central aim is to make visible the fine-grained process through which mathematical ideas attributed to students can perturb, extend and reorganize a teacher’s broader mathematical meanings for teaching.

The design responds to a limitation in constructivist teaching experiments. In the traditional format, a teacher-researcher interacts instructionally with an individual learner, a pair or a small group while generating and testing hypotheses about how the learners interpret mathematical tasks. The teacher-researcher’s interpretations become progressively more analytical as tasks are designed to prompt learners to reconsider their ideas.

Liang and colleagues therefore identify two focal levels of learning: students’ learning at Tier 1 and teachers’ learning from students at Tier 2. The proposed design aims to trace not merely whether teachers recognize particular student strategies or misconceptions, but how their interpretations change across repeated encounters with authentic student activity. To specify that process, the framework combines radical constructivism with a modeling account of mathematical meanings.

Radical constructivism holds that people actively construct knowledge through interaction and that one person has no direct access to another person’s knowledge. Students’ observable actions therefore do not reveal their mathematical meanings by themselves; an observer must infer meanings that make those actions intelligible. A student’s first-order model comprises the personal mathematical meanings that organize the student’s activity.

A teacher’s second-order model is the teacher’s interpretation of those meanings: an attempt to determine what the student may be thinking and why an answer, strategy or representation makes sense from the student’s perspective. The framework treats such a model not as a copy of the student’s mind but as a viable, revisable explanation built from the teacher’s own mathematical experience. Constructivist theory supplies a mechanism for describing how these interpretations can change.

A teacher normally assimilates new student activity into existing mathematical schemes. When a student produces a result that conflicts with the teacher’s expectations, that conflict can create perturbation or disequilibrium. If the teacher modifies or reorganizes the relevant meanings to accommodate the unexpected activity, accommodation has occurred.

The proposed analysis therefore examines both what teachers say about students and how their mathematical schemes are activated, challenged and revised. Teachers’ mathematical meanings of students are also treated as part of, rather than separate from, their broader first-order meanings for teaching: existing mathematical ideas constrain what a teacher can infer, while interpretations of students may eventually become resources that reshape the teacher’s own mathematical system. Third-order modeling adds a further distinction that prevents the perspectives from being collapsed.

Researchers observing a teacher cannot directly inspect the teacher’s second-order model of a student. They must construct a third-order model from evidence in the teacher’s language, actions, written work and reflections. A third-order model is thus the researcher’s second-order model of a teacher’s second-order model of a student’s mathematical meanings.

It may differ from the teacher’s actual interpretation and from the researcher’s separate model of the student, and the equivalence of these models cannot be directly verified. This layered approach also creates an anti-deficit safeguard: the researcher’s view is not automatically treated as the standard against which a teacher should be judged. Disagreement instead becomes an opportunity to examine the mathematical substance and experiential grounding of each interpretation.

Operationally, the method unfolds through repeated cycles of planning, teaching and reflection. During planning, the teacher initially helps select a topical area and proposes task ideas or materials, preserving substantial ownership of the investigation. The researcher and teacher then choose or adapt tasks that create opportunities for students to express mathematical thinking.

In later cycles, task design has two purposes: eliciting rich student activity and testing existing hypotheses about students’ meanings. The researcher may suggest constructivist principles, including designing for perturbation, while treating the teacher’s existing pedagogical commitments as both a resource and an object of inquiry. Tasks are selected to be open-ended and responsive to students’ likely meanings, and contexts, representations or numerical values may be varied to test how students assimilate or accommodate changes.

The teacher is also invited to articulate a personal solution, allowing the researcher to construct an initial model of the teacher’s mathematics as a baseline for later analysis. At Tier 1, the classroom teacher assumes the role of teacher-researcher during task-based interviews with one student, a pair or a small group. The teacher works directly with students to build, test and refine second-order models of their mathematical meanings, while the researcher acts as witness-researcher, observing the interaction and recording artifacts.

The authors regard authentic interaction with the teacher’s own students as indispensable rather than optional. Videos of unfamiliar classrooms or anonymous student work can support discussion, but they lack the relational history and practical constraints that shape a teacher’s situated reasoning. The method therefore places model construction in consequential interactions that teachers themselves experience as teaching, allowing the researcher to examine how teachers respond to students rather than merely asking them to interpret prepared cases.

Reflection converts those interactions into material for further learning. Researchers can replay selected video clips and ask what the teacher was thinking at particular moments, what mathematical ideas the students appeared to use, and why those ideas might be reasonable from the students’ perspectives. Other prompts ask how student thinking resembles or differs from the teacher’s own, how it relates to earlier ideas and how it compares across students.

These questions seek to move analysis beyond the correctness of solutions toward the underlying meanings that make an action intelligible, while also moving teachers away from deficit explanations such as simply stating that students do not know something. The researcher can then test whether apparently responsive behavior reflected attention to student thinking, established assumptions or intentions that were not visible in the original interaction, and can compare the teacher’s retrospective reconstruction with the in-the-moment behavior recorded on video. The data collection is deliberately intensive.

It includes audio and video recordings of all research sessions, selected video clips, student and teacher written work, task materials, researcher field notes and interview protocols. The authors describe a typical program involving one focal teacher, four to five cycles, three sessions per cycle and weekly participation across a semester. Planning and reflection sessions generally last 90 to 120 minutes, while teaching sessions run for 45 to 60 minutes; additional planning or reflection time may be scheduled when discussion exceeds the allocated period.

The sessions occur outside regular instructional time, such as during planning periods or after school. Clear links must be maintained between reflection clips and their corresponding teaching segments, with timestamps and durations recorded so the original episodes can be located reliably during analysis. Analysis constructs three evolving models from the researcher’s perspective: a second-order model of the student’s mathematical meanings, a second-order model of the teacher’s broader mathematical meanings and a third-order model of the teacher’s interpretations of students.

Through conceptual analysis, the researcher develops hypothetical mental actions and operations that explain observable activity at particular moments, then revises or confirms them against evidence in videos, transcripts and written artifacts. Planning-session accounts establish a baseline for the teacher’s mathematics. Teaching-session evidence supports models of students and tentative third-order models based on the teacher’s responses.

Reflection sessions then permit the researcher to refine the third-order model using the teacher’s retrospective accounts, in-the-moment reasoning and reconstructed model of students, while also examining how the teacher coordinates second-order models with broader first-order meanings. The researcher’s own model of the student serves as a reference rather than a normative standard, and discrepancies can prompt the researcher to revise it. Important limits accompany the proposal.

The framework is intended for intensive experimental settings in which teachers can attend closely to students, conditions that relax the pacing, management and curricular pressures of ordinary classrooms. Its claims are therefore situated and require empirical testing before extension to broader educational practice. The researcher also shapes the experience by choosing tasks, selecting clips, directing reflection and deciding which developments merit analysis, making the resulting descriptions interpretations situated within theoretical commitments and power relations rather than neutral observations.

The design deliberately supports teacher learning as part of investigating it, so the paper does not claim that teachers would spontaneously produce the modeled processes without facilitation. The method has not demonstrated that it improves classroom outcomes or that teachers learn in one prescribed way. It instead supplies a conceptual and empirical framework for future studies of teacher cognition, with possible applications to research on perspective-taking, mentoring, professional collaboration and other settings where people learn through understanding others’ thinking.

Subject of Research: Methodology for studying teachers’ mathematical learning from students

Article Title: Two-tier teaching experiment and third-order modeling

Article References: Liang, B., Wu, Y., & Liu, R. (2026). Two-tier teaching experiment and third-order modeling. Frontiers in Education, 11, Article 1783442. https://doi.org/10.3389/feduc.2026.1783442

Image Credits: AI Generated

DOI: 10.3389/feduc.2026.1783442

Keywords: two-tier teaching experiment, third-order modeling, teacher learning, student thinking, mathematical meanings, constructivism, mathematics education, teacher knowledge, research methodology, classroom research, Two-tier, teaching