Andrew Egbert · Applied Arts

Design & Development Research

Developing an AI‑Based Mathematics Tutor

A Type 1 design and development research study for a Scaffolded LLM Tutoring Design Framework


The study

Formative validation of a framework for LLM tutoring

Intelligent tutoring systems deliver measurable gains, but building one demands infrastructure and technical depth out of reach for most individual designers. Tutoring built on large language models is far more accessible, yet it has lacked a structured framework connecting prompt design to learning theory. This study is a formative validation of the Scaffolded LLM Tutoring Design Framework (SLTDF): a design model that translates established learning science into auditable tutoring behaviors that can be prompt‑engineered into an LLM.

The framework's first instantiation is the Math Tutor GPT (v4.1), the latest of eight development iterations, built for community college students preparing for math placement testing. Eight expert reviewers, selected through purposive maximum‑variation sampling across AI/LLM prompting, mathematics education, and instructional design, evaluated the Tutor through seven scenario‑based interactions.

RQ1 · Design fidelityTo what extent does the current Math Tutor GPT design exhibit the principles of evidence‑informed tutoring, as reported by expert reviewers?
RQ2 · RefinementWhat design refinements do expert reviewers recommend for future versions of the Tutor?

Three ways in

Watch it, read it, or go deep

The project is presented in three formats: a recorded presentation, a 16‑page interactive graphic novel that retells the study, and the full written report.

Video · Presentation

Project presentation

A guided walkthrough of the study: the problem, the framework, the expert review, and where the Tutor goes next.

Watch on YouTube →

Report · PDF · 149 pages

Full project report

The complete study: theoretical framework, methodology, instrument, per‑behavior results, discussion, and the five design principles, with full reviewer response data in the appendices.

Download the report (PDF, 2.2 MB)

What the experts found

Findings at a glance

Expert reviewers

8

Maximum-variation sample; math education, instructional design, AI/LLM design

Behaviors reliably present

6 of 7

Judged fully or partially exhibited by ≥75% of reviewers

Grand mean appropriateness

3.02

On a 1–4 scale, across all seven behaviors

Moderate-or-high fidelity

87.5%

Of reviewers, in overall design fidelity ratings

Behavioral appropriateness by tutoring behavior

Mean expert rating on a 1–4 appropriateness scale (n = 8; n = 7 for spaced retrieval)

Motivational framing B7 3.50
Worked examples with fading B3 3.25
Mastery check gating B4 3.25
Diagnostic questioning B1 3.13
Scaffolded hint ladder B2 2.88
Error classification & remediation B6 2.88
Spaced retrieval prompts B5 2.29

The scaffolded hint ladder was the most reliably present behavior (100% of reviewers), yet its appropriateness rating shows presence and quality are not the same thing: hints sometimes revealed answers instead of guiding toward them. Spaced retrieval is the one clear fidelity gap.

View the data as a table
Per-behavior appropriateness statistics
BehaviornMeanSDMedian% fully exhibited
B7 · Motivational framing83.500.764.087.5%
B3 · Worked examples with fading83.250.893.575.0%
B4 · Mastery check gating83.251.044.062.5%
B1 · Diagnostic questioning83.130.643.050.0%
B2 · Scaffolded hint ladder82.880.833.0100%
B6 · Error classification & remediation82.881.133.062.5%
B5 · Spaced retrieval prompts72.290.952.012.5%

Reviewers converged on clear refinement priorities: hint scaffolding that occasionally gave answers away (flagged by five of eight reviewers), spaced retrieval prompts that rarely surfaced, and error feedback that treated conceptual and procedural mistakes the same. Each is logged for the next Tutor revision, with spaced retrieval first in the modification triage.

What the field can take away

Five design principles for prompt-engineered tutors

Stated in the Richey and Klein format for design and development research, each principle pairs a design move with its theoretical rationale and the evaluation evidence behind it.

Hold the answer back

Scaffolded hint ladders should withhold answer‑revealing structure until the learner has attempted the problem: Vygotskian scaffolding requires productive struggle within the zone of proximal development.

Evidence: five of eight reviewers flagged hints that handed over the answer (hint‑ladder M = 2.88).

Engineer session state

Spaced retrieval cannot rely on the model remembering prior content. The system should engineer explicit session state, because the spacing effect depends on deliberate reintroduction of material over time.

Evidence: spaced retrieval rated fully exhibited by only 12.5% of reviewers and absent by 62.5% (M = 2.29).

Classify before correcting

Tutor feedback should classify each error as conceptual, procedural, or careless before responding, and route to type‑specific remediation: process‑level feedback yields larger gains than uniform correction.

Evidence: three of eight reviewers flagged undifferentiated error feedback (M = 2.88).

Require the learner's move

Self‑explanation prompts should require learner input before the tutor proceeds: the self‑explanation effect depends on active generation by the learner, not a rhetorical pause.

Evidence: three of eight reviewers reported strategy checks continuing without waiting for a response.

Specify before you prompt

Designers should produce a written behavior specification before prompt engineering begins: Type 1 fidelity assessment requires a design contract mapping each principle to an auditable behavior.

Evidence: overall fidelity of M = 3.00, moderate but not full, with several flagged issues tracing to implicit specification.

The framework

The Scaffolded LLM Tutoring Design Framework

The SLTDF is a three‑tier model. It synthesizes ten learning science constructs into seven operationalizable tutoring behaviors, then acts as a session manager deciding which behaviors surface, and which recede, at any moment in a chat based on the learner's actions and session state.

Ten learning science constructs

The theoretical base

Cognitive load theoryDirect instructionMastery learningDeliberate practiceError-based learningScaffold fadingRetrieval & spacingSkill fluencyTransfer-oriented designMetacognitive structuring

SLTDF session manager

Synthesizes the constructs and arbitrates which behaviors are evidenced or reduced, turn by turn

Seven operationalizable tutoring behaviors

The auditable design contract evaluated in this study

Diagnostic questioningScaffolded hint ladderWorked examples with fadingMastery check gatingSpaced retrieval promptsSelf-explanation promptingError diagnosis & differential response
Instantiated in the Math Tutor GPT · v4.1

Type 1 research asks whether the design is adequate, not yet whether learners gain from it. This study contributes the first piece of formative evidence that the framework's behaviors are recognizable to experts as theoretically grounded, and it maps the road to the learner‑facing research that follows.