LLMs Are Already Good Tutors: Training-Free Prompt Optimization for Pedagogical Math Tutoring

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LLMs Are Already Good Tutors: Training-Free Prompt Optimization for Pedagogical Math Tutoring

Unknown · 2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

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machine learning, deep learning, neural networks

LLMs Are Already Good Tutors: Training-Free Prompt Optimization for Pedagogical Math Tutoring Unggi Lee1,† Minchul Shin2,† Jeongsu Moon5 Kyungtae Joo5 1

Yeil Jeong3,† Eunjoo Lee2

Sookbun Lee4 Hoilym Kwon6

Korea University Sejong Campus 2 Gyeonggi Institute of Education 3 Indiana University Bloomington 4 Opentutorials 5 Chosun University 6 Korea University Korean Studies Center †

Abstract

0.15

Aligning LLMs for math tutoring typically requires RL-based training with multi-GPU infrastructure. We investigate whether trainingfree prompt optimization-evolving only the system prompt via API calls-can serve as a practical alternative. We adapt 7 published methods and propose 5 education-specialized methods, evaluating these 12 methods under 5 conditions on 2 OOD benchmark suites. All 12 best-per-method configurations surpass the strongest RL-trained baseline (Rtotal = 0.633), and our ParetoGrad achieves the best Pareto balance across post-test solve rate, leak control, and helpfulness, rather than dominating any single component. Behavioral analysis with an 82-code educational codebook reveals that training-free methods rely on teachingknowledge patterns at 2-3× the rate of RLtrained models, with a compensating ∼10 percentage-point reduction in intent-level scaffolding. We also find a task-dependent reasoning mode effect consistent across training-free and RL-based paradigms. Our approach enables efficient development of pedagogically aligned LLM tutors with prompts alone and minimal compute.

0.20

Leak Rate ( better)

arXiv:2605.27088v1 [cs.CL] 26 May 2026

Corresponding authors [email protected] [email protected]

[email protected]

Published Proposed ParetoGrad (Rtot best) Size Helpfulness

0.25

ParetoGrad

LeakShield

0.30

EvoPrompt MIPROv2

GEPA

0.35

TF-GRPO

CondBridge MetaBlend TextGrad

0.40

Frame OPRO

0.45 ACE

0.50 RL Ped. Think R : Rsol = 0.29, leak = 0.17

0.55 0.48

0.50

0.52

0.54

Rsol (post-test,

0.56

better)

0.58

0.60

Figure 1: Post-test solve rate (Rsol , K=8) vs. leak rate trade-off across the 12 listed training-free methods; bubble size reflects helpfulness. Our proposed ParetoGrad (⋆) achieves the best Pareto balance across the three objectives without updating model weights.

training problems, creating a significant barrier for educators and researchers. Meanwhile, automatic prompt optimization methods such as OPRO (Yang et al., 2024), TextGrad (Yuksekgonul et al., 2024), and GEPA (Agrawal et al., 2026) have demonstrated that LLM behavior can be steered by evolving the system prompt without weight updates. Yet these methods have never been applied to multi-turn educational dialog, where optimization must jointly balance scaffolding quality, answer leak prevention, and student learning outcomes. We investigate whether training-free prompt optimization can serve as a practical alternative to RL-based tutor alignment (Figures 1 and 2). We first adapt 7 published prompt optimization methods to multi-turn educational dialog. Through extensive experimentation with diverse optimization

Introduction

Large language models (LLMs) hold significant promise as personalized math tutors (Tack and Piech, 2022; LearnLM Team, Google, 2025), but aligning them with sound pedagogy-guiding students toward solutions without simply revealing answers-remains challenging (Chi and Wylie, 2014). Recent RL-based alignment approaches (Dinucu-Jianu et al., 2025; Lee et al., 2026a) train with multi-objective rewards (solve rate, leak prevention, helpfulness) using GRPO (Shao et al., 2024), producing effective 7B tutors competitive with frontier models. However, these approaches require multi-GPU infrastructure and thousands of 1

RL-Based Tutor Alignment

Training-Free Prompt Optimization Tutor LLM (Frozen)

A. Prompt Population 𝑷

Let’s solve this together!

ParetoGrad Highlights

Student Model

Beats RL 0.633

Think about The absolute value.

Maintain Diverse Trade-Offs

Pareto Front

Weights 𝜽 Fixed

Keeps prompts with

···

Tutor LLM (trainable)

Best Overall Reward 𝑹𝒕𝒐𝒕𝒂𝒍 = 0.719

Hint 1: |𝒙 + 𝟑| = 𝟓 . What is 𝒙?

different

B. Multi-Turn Tutoring Dialog Simulation

Student Model

Solve/ Non-Leak / Let’s think step by step. Hint 1: |𝒙 + 𝟑| = 𝟓

Reward Evaluation(Per Turn/Dialog)

𝒙 = 𝟐 or 𝒙 = −𝟖

Turn 1 → Turn 𝑻 (𝑻 ≤ 5)

Helpful Trade-Offs

Weakness-Targeted Update Improves the weakest

C. Objective Evaluation(Per Dialog) Solve

No-Leak

Helpful

Multi-Objective Evaluation

Thinking Solve Rate Maximize

GRPO Updates Weights 𝜽

Non-Leak Rate Maximize (1 − Leak)

Helpfulness Maximize

Preserve Trade-Offs (Do Not Collapse)

metric without losing other strengths

Population Search

𝜽𝒕

Uses crossover +

𝜽𝒕+1 D. ParetoGrad Update (Best One of Our Proposed Methods) Find Weakest Objective

Expensive Training

Variation Operators

Weakness-Targeted Reflection

Non-Dominated Sorting

Next Population

Crossover

Mutation for diverse prompt exploration

Frozen Weights Training-Free prompt evolution

Mutation

Multi-GPU

10K+

Many

Problems

Training Steps

Identify the lowest performing metric

Pareto Front + Crowding Distance

Text gradient on weakest metric

Keep diverse high-quality prompts

100 x less data, lightweight inference

Figure 2: Left shows RL-based tutor alignment and center shows our training-free prompt optimization, illustrated with ParetoGrad as a representative instance; right lists method highlights. RL updates tutor weights θ via GRPO over reward components (solve, non-leak, helpfulness, thinking), requiring multi-GPU training over 10K+ problems. In contrast, our approach keeps the tutor and student models frozen and evolves a population of system prompts P through four steps - (A) maintaining diverse prompt trade-offs, (B) simulating multi-turn tutoring dialog, (C) evaluating each prompt on three objectives (Rsol , Rleak , Rhelp ), and (D) updating the population via weaknesstargeted reflection, crossover and mutation, and non-dominated Pareto sorting with crowding distance. ParetoGrad achieves the best Pareto balance across solve rate, leak control, and helpfulness with 100× less data and single-GPU inference, and all 12 listed training-free methods surpass the strongest RL-trained baseline.

strategies, we identify and propose 5 educationspecialized methods that embed pedagogical priors - scaffolding, leak prevention, meta-optimization - directly into the prompt evolution process. Using the same reward framework as PedagogicalRLThinking (Lee et al., 2026a), we evaluate all 12 methods under 5 conditions on 2 OOD benchmark suites (Lee et al., 2026b; Macina et al., 2025), yielding 792 evaluation runs. Our key contributions are: • We show that training-free prompt optimization can match or exceed RL-trained baselines on the same multi-objective reward using only inference-time compute.1 • We propose five education-specialized prompt optimization methods (ParetoGrad, CondBridge, LeakShield, Frame, MetaBlend) inspired by empirical failure analysis of existing approaches. • We provide the first adaptation of published prompt optimization methods to multi-turn edu-

cational dialog, together with a systematic behavioral comparison against RL-trained approaches using the same educational codebook.

Method

2.1

Problem Formulation

Given a frozen LLM M with fixed weights θ, we seek a system prompt P ∗ that maximizes tutoring quality: P ∗ = arg max Ex∼X [Rtotal (Dialog(M, P, x))] P

(1) Unlike RL-based approaches (Dinucu-Jianu et al., 2025; Lee et al., 2026a) that update θ via GRPO, we keep θ frozen and evolve only P , requiring only black-box API access. Each candidate prompt P is evaluated by simulating multi-turn tutoring dialogs between the tutor M and a student model S for up to T = 5 turns on problem x.

https://anonymous.4open.science/r/ tf-openlearnlm-anon-C780/README.md

2.2

2.4.1

Reward Design

Frame extends TextGrad (Yuksekgonul et al., 2024) with a post-gradient reframing pass:

We adopt the reward framework from Lee et al. (2026a). Given a dialog D = Dialog(M, P, x), we compute four reward components: (1) Rsol (D): post-dialog student solve rate over K = 8 attempts; (2) Rleak (D) ∈ {0, 1}: 1 if the tutor avoided leaking the answer, 0 otherwise (we report the leak rate 1 − Rleak in tables for interpretability; ρ = −0.94 between leak rate and Rtotal across 137 method×condition runs); (3) Rhelp (D) ∈ [0, 1]: pedagogical helpfulness; (4) Rthink (D) ∈ [0, 1]: thinking quality (reasoning models only). The total reward is: ( Rsol +Rleak +Rhelp Rtotal (D) =

2.3

3 Rsol +Rleak +Rhelp +Rthink 4

(no think)

Pt+1 = Refr(Pt − ∇text (Pt , D))

(3)

where Refr(·) applies three linguistic transformations: (1) converting prohibitive language into behavioral guidance; (2) replacing brevity directives with substance-focused instructions; (3) injecting concrete tutoring exemplars. In pilot analysis, reframing alone reduced leak rate from 0.891 to 0.761. 2.4.2

(2)

Leak Prevention

CondBridge evaluates each candidate P under both NoThink (MNT ) and Think (MTH ) conditions:

(think)

Adapted Published Methods NT TH P ∗ = arg max αRtotal (P ) + (1−α)Rtotal (P )

We adapted 7 published prompt optimization methods to the tutoring domain, replacing their evaluation functions with our dialog simulation and reward Rtotal . GEPA (Agrawal et al., 2026) uses reflective prompt evolution with a Pareto frontier over Rsol , Rleak , and Rhelp . ACE (Zhang et al., 2026) extracts reusable skills, which we redefine as pedagogical strategies such as scaffolding and questioning. OPRO (Yang et al., 2024) uses an LLM-as-optimizer approach, where we include tutoring failure cases in the meta-prompt. EvoPrompt (Guo et al., 2024) applies genetic algorithms over a population of tutor system prompts {P1 , . . . , PN }. TextGrad (Yuksekgonul et al., 2024) computes textbased gradients ∇text from dialog quality feedback and updates Pt+1 = Pt −∇text . MIPROv2 (OpsahlOng et al., 2024) performs Bayesian selection over pedagogical instruction variants. TF-GRPO applies the group relative scoring from GRPO (Shao et al., 2024) to rank prompt candidates by Rtotal without updating θ. 2.4

Pedagogical Scaffolding

(4)

where α = 0.5. This dual-condition objective drives CondBridge to the lowest leak rate (0.204) among all training-free methods at NoThink, while at Think Reward it reaches Rtotal = 0.691. 2.4.3

Dual-Objective

LeakShield uses two-stage optimization: Stage 1 (first 40%) minimizes leakage; Stage 2 maximizes Rsol and Rhelp under a leak ceiling τ : P ∗ = arg max Rsol (P )+Rhelp (P ) s.t. 1−Rleak (P ) ≤ τ (5) P

If leakage exceeds τ during Stage 2, the optimizer reverts to anti-leak updates. 2.4.4

Meta-Optimization

MetaBlend collects the best prompts ∗ } from m completed runs, ex{P1∗ , . . . , Pm tracts common structural patterns, and synthesizes an initial prompt P0 . Phase 2 refines P0 via TextGrad for 300 iterations, preserving patterns as constraints.

Proposed Education-Specialized Methods

Error analysis of 19,500 dialogs from the adapted methods revealed three tutoring-specific failure patterns: answer leakage (82.8% of failures), rigid instruction following, and lack of pedagogical structure despite high Rtotal (Chi and Wylie, 2014). Inspired by these empirical observations, we designed 5 education-specialized methods targeting these failure patterns. We present these methods below; additional exploratory variants are documented in Appendix D.

2.4.5

Population-Gradient Hybrid

ParetoGrad maintains a population {P1 , . . . , PN } with N = 5 and applies NSGA-II non-dominated sorting across (Rsol , Rleak , Rhelp ), yielding the best Pareto balance across the three objectives (Rtotal = 0.719 at Think NoReward). Each generation produces offspring via weakness-targeted TextGrad, crossover, and mutation over 100 generations. 3

Table 1: In-domain BigMath and OOD benchmark results. Each training-free row shows its best of 5 conditions, encoded by the Th./Th.R/Prompt indicators. Rsol is the K=8 post-dialog solve rate; Rtotal = (Rsol + Rleak + Rhelp )/3. ParetoGrad achieves the best Pareto balance across the three components, and all 12 listed training-free methods surpass the strongest RL-trained baseline. BigMath (ID) Method

OpenLearnLM (OOD)

MTBench

Th. Th.R Prompt Rsol ↑ Leak↓ Help↑ Rtotal ↑ CK↑ PK↑ SK↑ Att↑ Avg↑ Sc↑ Pd↑ Avg↑

Frontier Models (Zero-shot)† GPT-5.2 (Ped.) ✓ Claude-4-Opus (Ped.) ✓ DeepSeek-V3.2 (Ped.) ✓ -

Ped. Ped. Ped.

.340 .350 .390

.000 .090 .110

.440 .760 .820

.593 .673 .700

8.08 8.46 8.63 8.68 8.46 6.63 8.61 8.82 8.45 8.13 7.46 7.32 8.63 8.77 8.05

RL-Trained Models† NoThink (RL) Think NR (RL) Think R (RL) Ped. Think NR (RL) Ped. Think R (RL)

✗ ✓ ✓ ✓ ✓

✗ ✗ ✓ ✗ ✓

Gen. Gen. Gen. Ped. Ped.

.120 .281 .284 .275 .294

.300 .180 .182 .214 .172

.180 .730 .764 .766 .776

.333 .604 .621 .607 .633

7.95 7.95 7.95 7.95 7.99

Baseline No optimization

✗

Gen.

7.43 7.43 7.43 7.43 7.43

7.77 7.76 7.76 7.76 7.77

7.57 7.79 7.71 7.79 7.86

7.68 7.73 7.71 7.73 7.76

.120

.300

.180

.333

7.56 6.87 8.53 7.93 7.72 7.17 5.25 6.21

Training-Free: Published Methods (Adapted) EvoPrompt ✓ ✓ Gen. .566 MIPROv2 ✓ ✓ Gen. .563 TF-GRPO ✓ ✓ Gen. .561 GEPA ✓ ✗ Gen. .542 OPRO ✓ ✓ Ped. .574 TextGrad ✓ ✗ Ped. .527 ACE ✓ ✗ Ped. .551

.286 .298 .312 .324 .382 .374 .460

.847 .848 .845 .841 .846 .845 .833

.711 .707 .700 .686 .684 .666 .642

7.62 6.97 6.78 6.41 6.64 6.97 7.11

7.54 7.42 6.71 6.87 6.67 6.67 6.63

8.69 8.56 7.95 7.27 7.70 7.45 8.62

8.50 8.21 8.50 8.43 8.21 8.36 8.57

8.09 7.79 7.49 7.24 7.31 7.36 7.73

8.16 8.09 8.29 8.03 7.95 7.70 8.23 8.00 7.84 7.65 8.01 7.72 8.21 8.06

8.13 8.16 7.82 8.12 7.75 7.86 8.13

Training-Free: Proposed Methods (Ours) ParetoGrad ✓ ✗ Gen. LeakShield ✓ ✗ Gen. CondBridge ✓ ✓ Gen. Frame ✓ ✗ Ped. MetaBlend ✓ ✓ Gen.

.252 .264 .336 .354 .392

.845 .847 .843 .845 .843

.719 .693 .691 .683 .676

6.92 6.97 6.88 6.97 6.88

6.55 6.87 6.71 6.83 6.79

7.99 7.12 7.91 7.62 7.68

8.07 8.00 8.14 7.93 8.29

7.38 7.24 7.41 7.34 7.41

7.77 7.79 7.37 7.87 7.82 7.84 7.89 7.72 7.88 7.63

7.78 7.62 7.83 7.81 7.76

.563 .496 .552 .558 .562

†

Frontier/RL/Baseline rows show ∆Solve in the Rsol column as reported by Lee et al. (2026a,b); training-free rows use the K=8 post-test Rsol defined above. Bold = best per column (for Rsol and Rtotal , restricted to the 12 listed training-free methods due to the Rsol metric difference). Underline = best per column among non-Frontier rows (RL + 12 training-free) when distinct from the bolded entry, else 2nd best in that pool. Full appendix results in §A.

Experiments

3.1

Setup

2 OOD suites: OpenLearnLM (Lee et al., 2026b) (4 sub-benchmarks) and MathTutorBench (Macina et al., 2025) (2 sub-benchmarks), totaling 1,334 items. Our approach requires only a single RTX 3090 for vLLM inference, compared to 4×H100 GPUs for RL training (Lee et al., 2026a), with a small amount of API costs for GPT-4o-mini and LLaMA-3.1 via OpenRouter. We conduct 792 evaluation runs in total. We focus on the 5 proposed methods listed in Table 1; additional exploratory variants are in Appendix A.

We match the setup of Dinucu-Jianu et al. (2025); Lee et al. (2026a). The tutor model is Qwen2.57B-Instruct for the NoThink condition with a maximum of 256 output tokens, and Qwen3-8B for all thinking conditions with 384 output tokens and a thinking budget of 1,024 tokens. The student model is LLaMA-3.1-8B-Instruct with a maximum of 512 tokens. Reward judgments are made by GPT-4o-mini (Zheng et al., 2023), and prompt improvements are proposed by GPT-4o as the reflection model. Each dialog runs up to 5 turns under 5 conditions identical to Lee et al. (2026a): NoThink (Qwen2.5-7B), Think NoReward and Think Reward (Qwen3-8B, with/without Rthink ), and their pedagogical-seed variants. Whether each condition enables thinking, applies Rthink , or seeds with a pedagogical prompt is encoded in Table 1 as the Th./Th.R/Prompt indicator triplet. Optimization uses 100 BigMath (Albalak et al., 2025) problems filtered to medium-to-hard difficulty (student solve rate 1-60%). We evaluate on

3.2

Main Results

Table 1 presents performance on both in-domain and OOD benchmarks; for compactness each training-free method is shown with its best of 5 conditions ranked by Rtotal , and the corresponding configuration is encoded by the indicator triplet. RQ1: Is training-free prompt optimization effective for math tutoring? Yes - and more strongly than initial estimates suggested. All training-free methods improve over the unoptimized baseline (Rtotal ≈ 0.33); all 12 listed methods even surpass the strongest RL4

In-domain (BigMath)

0.82 0.81

0.8 0.6

0.55

0.2

7.86

7.42

7 0.54 0.53

0.43

0.4

NoThink Think (avg of 4)

0.34 0.24

OOD score

1.0

trained model (Ped. Think R, Rtotal = 0.633). Our proposed ParetoGrad (Think, no reward) reaches Rtotal = 0.719, the most balanced result across the three objectives. ParetoGrad does not dominate any single component but is uniformly strong, with Rsol = 0.563 (3rd, tied with MIPROv2), Leak= 0.252 (best), and Help= 0.845 (top tier). Specialists exist for individual components - OPRO leads Rsol (0.574) and CondBridge attains the lowest leak (0.204) at NoThink - but neither balances all three components as well as ParetoGrad, illustrating the persistent solve-leak tension (Figure 1).

NoThink Think7.79 (avg of 4) 6.92

6 5 4 3 2 1

0.0

R_sol

1-Leak

Help

R_tot

OL-Avg

MTB-Avg

Figure 3: Left compares NoThink and Think (avg of 4 think conditions) on in-domain metrics (0-1 scale). Thinking modestly degrades leak control (1−Leak drops from 0.55 to 0.43). Right shows OOD benchmark averages. Think improves MathTutorBench (+0.87) but hurts OpenLearnLM (−0.44), revealing a taskdependent reasoning mode effect.

RQ2: Do education-specialized methods outperform general-purpose ones?

ure 3 right), thinking improves MathTutorBench but degrades OpenLearnLM. Figure 3 left shows that in-domain leak control degrades sharply under thinking. This task-dependent reasoning mode effect is consistent across both training-free and RL paradigms (Lee et al., 2026a), suggesting it reflects a fundamental property of reasoning-enabled models rather than an artifact of the optimization method. After the full sweep, the best condition per method is a Think variant for all 12 listed methods, with the choice between Rthink on/off and pedagogical/general seed differing across methods - encoded in Table 1 by the indicator triplet rather than collapsed into a single representative condition.

The answer is nuanced under the completed evaluation. On in-domain BigMath Rtotal , the top12 split into 7 published-adapted and 5 proposed methods; the most balanced is our proposed ParetoGrad (0.719), with published EvoPrompt (0.711) and MIPROv2 (0.707) close behind on aggregate reward. On OOD, published methods now lead OpenLearnLM-Avg (EvoPrompt 8.09) and MathTutorBench-Avg (MIPROv2 8.16), with proposed methods narrowly behind (CondBridge OL-Avg 7.41; CondBridge/Frame MTB-Avg 7.83/7.81). Pedagogicalseed prompts help OOD specifically for ACE, TextGrad, and Frame, while non-pedagogical seeds win on Rtotal for ParetoGrad/LeakShield/GEPA. Education-specialization therefore yields gains in leak control and in selected OOD subscores, rather than a uniform advantage on Rtotal (see §4.1).

Analysis

4.1

In-Domain vs. OOD Generalization

Figure 4 reveals essentially no correlation between in-domain Rtotal and OOD MathTutorBench-Avg across the 12 listed methods (ρ = 0.01, p = 0.96), and only a weak correlation with OpenLearnLMAvg (ρ = 0.25, p = 0.41). The most striking dissociations are: ACE ranks 12th in-domain but 2nd on MTB (and 3rd on OL), LeakShield ranks 5th indomain but last (12th) on both MTB and OL, and ParetoGrad ranks 1st in-domain but 9th on MTB and 7th on OL. Codebook analysis suggests the mechanism: methods that minimize leak via tight scaffolding (ParetoGrad, LeakShield) optimize the in-domain reward rubric, while content-deliveryrich methods (ACE, MIPROv2, EvoPrompt) transfer their structured tutoring patterns more readily to held-out benchmarks. This indicates that indomain reward maximization and OOD generalization rely on fundamentally different pedagogical strategies.

RQ3: How do training-free methods compare to RL-trained approaches? Training-free optimization can match or exceed RL on aggregate reward: ParetoGrad reaches Rtotal = 0.719 vs. RL Ped. Think R at 0.633, while using 100× less data (100 vs. 10K problems) and only inference-time compute on a single consumer GPU rather than the multi-GPU training stack required by RL. RL still attains slightly tighter leak control under thinking (RL Ped. Think R leak 0.172 vs. ParetoGrad 0.252), and conversely training-free can exceed RL on ∆Sol (0.394 vs. 0.294). The paradigms are complementary: RL provides tight behavioral control through weight updates, while training-free offers accessibility, interpretability, and immediate deployability with a much smaller compute footprint. Comparing Think and NoThink (Table 1, Fig5

8.0

Praise

Step-by-step

Proportion (%)

OOD MTBench

MIPROv2

+0.01 s =ACE 8.1

Exploratory Q

7.9

Info provision 7.8

ParetoGrad

Strategy sel. PedRL NoThink PedRL-T Think CondBridge Frame

7.7 7.6

Proposed Published

0.64

0.65

Hint provision

LeakShield 0.66

0.67

0.68

0.69

Rtotal (in-domain)

0.70

0.71

0.72

Frequency (%)

50 40 30 20 10 0

T2 Scaffold (High) Scaffold (Low)

Turn Execute (High) Execute (Low)

Understand (High) Understand (Low)

Figure 4: Left plots in-domain Rtotal vs. OOD MTB-Avg across the 12 listed methods. Reward maximization is essentially uncorrelated with OOD MathTutorBench performance (ρ = 0.01, p = 0.96) and only weakly correlated with OpenLearnLM (ρ = 0.25, p = 0.41). Center compares sentence-multilabel code frequency (%) between RL-trained models (Dinucu-Jianu et al., 2025; Lee et al., 2026a) and training-free methods (CondBridge, Frame). Right reports Polya phase progression for top-6 vs. bottom-6 methods by Rtotal (NoThink); high-performance methods sustain Scaffold codes throughout the dialog.

4.2

Trained vs. Training-Free Behavioral Comparison

(49-73%), and high-performance methods maintain consistently higher Scaffold than low-performance methods (+9.8pp at T1, +8.8pp at T5). Conversely, low-performance methods substitute direct Execution codes (calculation, step-by-step procedure) by 3-5pp across mid-late turns. Sustained scaffolding-continuing to guide rather than execute on the student’s behalf-is the key behavioral differentiator between high- and low-performing optimization methods.

We labeled 415,775 tutor sentences across 60 method-condition configurations (12 methods × 5 conditions) with GPT-4o-mini using the 82-code educational codebook from Lee et al. (2026a). Figure 4 center compares two training-free methods (CondBridge, Frame) with RL baselines on six core codes (sentence-multilabel %). Training-free methods strongly suppress Praise (CondBridge 5.94%, Frame 3.20%; comparable to RL Ped. Think Reward 5.78% and far below the unoptimized baseline 11.76%), with MIPROv2 as a notable exception whose Rthink -activated runs increase rather than reduce Praise (Appendix C.12). However, they redistribute behavior toward content delivery rather than explicit scaffolding: Information provision is 23-31% under training-free versus 8-9% under RL, while explicit Hint provision drops to 0.5-1.4% versus 9-11% under RL. At the category level, this manifests as a 2-3× elevation of Mathematical Knowledge for Teaching (training-free 19-23% vs. RL 7-8%) and a ∼10pp deficit in Pedagogical Intent Utterance (training-free 46-55% vs. RL 6065%), suggesting that prompt-level optimization recruits teaching-knowledge patterns where RL recruits intent-level scaffolding moves. 4.3

4.4

Optimization Convergence

Figure 5 left shows convergence patterns across methods. All methods share a budget of 500 reward evaluations, but iterations differ because population methods (ParetoGrad, N = 5) consume ∼50 evaluations per iteration while single-candidate methods (TextGrad) use ∼10. Gradient-based methods converge rapidly within 10 iterations, while dualobjective methods (CondBridge, LeakShield) show slower but steadier improvement as they balance competing objectives. 4.5

Transition Patterns

Transition analysis (Figure 5 center) reveals a code-type dependent pattern. On question codes, low-performance methods exhibit higher selfrepetition: Exploratory question self-loop probability is 0.408 for low-performance vs. 0.357 for highperformance methods. By contrast, on contentdelivery codes, high-performance methods show stronger self-chaining (Information provision 0.314 vs. 0.172; Step-by-step instruction 0.347 vs. 0.212), reflecting sustained explanation rather than fragmented question repetition. The pedagogical sig-

Polya Phase Progression

We mapped the 82 codes to Polya’s problemsolving phases and tracked progression across dialog turns (Figure 4 right; top-6 vs. bottom-6 methods by Rtotal , NoThink). Scaffold codes (hints, exploratory questions, prompts) dominate every turn 6

0.15

TextGrad ParetoGrad CondBridge Frame MIPROv2 OPRO

Step-by-step

+0.12

Explor. Q

+0.07

-0.11

+0.06

Info prov.

+0.13

-0.06

+0.13

+0.09

Hint

-0.12

+0.20

Strategy

+0.07

-0.09

Restate. +0.07

1 -0.17 -0.09

-0.05

+0.07

0.10

te.

ep t

sta

4 T1

Co nc

nt Hi

r. Q

op rov .

plo Ex

1 2

-0.10

Inf

tep

0.05

0.15

ais

y-s

Iteration

Ste

0.10

0.05

+0.11

Concept

0.0 10

-0.13

0.00

0.2

-0.05

ate gy

0.4

-0.07

Str

0.6

+0.06

p-b

Normalized Best Score

0.8

+0.10

Praise

Freq. (pp)

1.0

Praise

Step-by-step

Explor. Q

Turn

Figure 5: Left shows optimization convergence for 6 methods under a shared 500-evaluation budget; gradient methods converge within 10 iterations while dual-objective methods improve gradually. Center displays codeto-code transition probability differences (high − low). High-performance methods chain content delivery codes (Step-by-step, Information provision), while low-performance methods chain question codes (Exploratory question). Right reports per-turn frequency differences between successful (top 25%) and failed (bottom 25%) dialogs in the high-performance group. Successful dialogs deploy fewer Step-by-step instructions and substitute more Exploratory questions and Praise toward the dialog’s conclusion.

nature of strong methods is thus a shift of selfrepetition from question loops toward explanation chains.

spect and modify.

4.6

While the preceding aggregate analyses characterize behavioral shifts in distribution, the excerpts here clarify what those shifts look like in dialog. First, the leak-rate differences in Table 1 and the Polya phase progression in §4.3 point to a shared mechanism: higher-performing methods avoid solution takeover while keeping the student responsible for the next mathematical move. ParetoGrad illustrates this delegation: “Let’s start by defining the number of red marbles as R. The problem gives us three relationships involving R. Can you write these relationships as equations?” The tutor supplies variable naming and structural framing, but leaves equation formulation to the student (Wood et al., 1976). In pedagogical terms, this preserves the problem-solving cycle: the tutor supports entry into the task without collapsing Understand, Plan, and Execute into tutor-performed computation (Pólya, 1945). The same pattern appears across both proposed and adapted methods, so the qualitative evidence is best read as a behavioral signature of successful prompt optimization rather than a proposed-versus-published contrast. Second, the MKT/PIU redistribution discussed earlier appears qualitatively as a shift toward concept-rich explanation (Shulman, 1986; Ball et al., 2008). CondBridge, for example, frames inequalities and symmetry as a “key insight” for exploiting mathematical structure, explaining why a representation is useful rather than merely which

4.8

Success vs. Failure Dialogs

Within the same high-performance methods, successful dialogs (top 25% Rtotal ) differ from failures across turns (Figure 5 right). Successful dialogs deploy fewer Step-by-step instructions early (−4.0pp at T1, −2.2pp at T2) and at closure (−4.0pp at T5), substituting more Exploratory questions (+1.2pp at T1, +2.6pp at T5) and more Praise (+2.3pp at T5). The pattern suggests that successful tutoring elicits the student’s understanding through questions rather than dictating procedure, and reserves praise for confirmed progress near the dialog’s conclusion. Decomposing this effect by reasoning mode and turn position (Appendix C.11) shows that the lateturn Praise contribution to Rtotal is concentrated in NoThink dialogs at T4+, providing the behavioral substrate for the closing-turn pedagogical wrap-up. 4.7

Cost Comparison

PedagogicalRL-Thinking (Lee et al., 2026a) requires substantial multi-GPU compute for hundreds of hours of GRPO training, while our approach uses only inference-time serving on a single consumer GPU during prompt optimization. Training-free can match or exceed RL on aggregate reward (ParetoGrad Rtotal = 0.719 vs. RL Ped. Think R 0.633) with substantially lower GPU requirements, while producing interpretable, humanreadable prompts that educators can directly in7

Qualitative Analysis

operation to perform. RL-trained tutors more often foreground interactional pacing through praise, questions, and hints. Thus, training-free optimization and RL appear to emphasize different but compatible pedagogical resources: explanatory depth and scaffolding density. Finally, some higher-performing dialogs elicit student-model turns with features of mathematical communication, including variable definition, function transformation, and explicit reasoning connectives. After a structured ParetoGrad exchange, the simulated student writes, “let a = log10 2 . . . Therefore f (a) + f (−a) = . . .” These are features of student-model generated discourse, not evidence of human learning. Still, they suggest that future rewards could explicitly value transferable scaffolds, such as Polya-phase maintenance (Pólya, 1945), to better connect in-domain optimization with OOD teaching competencies.

als than fine-tuned models.

Related Work

6.1

Prompt Optimization

Prompt optimization refines black-box LLM prompts without weight updates via metaprompting (Zhou et al., 2023; Yang et al., 2024), textual gradients (Pryzant et al., 2023; Yuksekgonul et al., 2024), modular Bayesian programs (Khattab et al., 2024; Opsahl-Ong et al., 2024), evolutionary populations (Guo et al., 2024), or reflective evolution (Agrawal et al., 2026; Zhang et al., 2026). These methods have been benchmarked primarily on single-turn NLP tasks under a single scalar reward, with reflective evolution recently matching or exceeding reinforcement learning on agentic tasks. 6.2

LLM-Based Math Tutoring

LLM-based math tutors guide students through multi-turn dialog rather than supplying answers (Tack and Piech, 2022; LearnLM Team, Google, 2025), with scaffolding and answer-leak prevention as the defining design constraints (Chi and Wylie, 2014); recent work operationalizes this trade-off through dialog corpora (Macina et al., 2023), multiobjective RL alignment (Dinucu-Jianu et al., 2025; Lee et al., 2026a), and OOD benchmarks (Macina et al., 2025; Lee et al., 2026b). Our training-free alternative matches or exceeds the strongest RLtrained baseline using only inference-time prompts that educators can directly inspect, and enables systematic behavioral comparison over the same 82-code educational codebook (Lee et al., 2026a).

Discussion

Training-free prompt optimization works because the prompt itself is an explicit pedagogical prior over the LLM’s frozen distribution: short instructions can directly invoke teaching-knowledge patterns (Shulman, 1986; Ball et al., 2008) that RL approximates only through many gradient updates over scalar rewards (Dinucu-Jianu et al., 2025; Lee et al., 2026a). The observed MKT/PIU redistribution is consistent with this account: prompts recruit declarative resources, while RL shapes behavioral pacing (Chi and Wylie, 2014), making the two paradigms complements rather than substitutes. The dissociation between in-domain reward and OOD generalization qualifies what “best method” means: tight-scaffolding methods win the indomain rubric, while content-delivery-rich methods transfer better to held-out benchmarks (Macina et al., 2025; Lee et al., 2026b). Since aggregate scores cluster tightly, ParetoGrad’s value lies less in topping the table than in lacking weakness across the three reward components - Pareto balance is a more honest summary than scalar dominance for multi-objective evaluation. For educators, training-free tutoring’s most useful property is its artifact: a plain-text prompt that can be read, edited, and shared, rather than opaque weights. Combined with the sustained scaffolding (Wood et al., 1976) and Polya-phase structure (Pólya, 1945) visible in the dialogs, these prompts function more like inspectable instructional materi-

Conclusion

We investigated whether training-free prompt optimization can serve as a practical alternative to RL-based tutor alignment for math tutoring. Adapting published prompt optimization methods to multi-turn educational dialog and proposing five education-specialized variants inspired by empirical failure analysis, we found that training-free methods match or exceed the strongest RL-trained baseline using only inference-time compute, while recruiting more teaching-knowledge patterns and less intent-level scaffolding than RL. These results position training-free prompt optimization not as a cheaper substitute for RL but as a complementary paradigm that offers interpretability, accessibility, and the ability to explore diverse pedagogical strategies through method design alone. 8

Limitations

David Dinucu-Jianu, Jakub Macina, Nico Daheim, Ido Hakimi, Iryna Gurevych, and Mrinmaya Sachan. 2025. From problem-solving to teaching problemsolving: Aligning LLMs with pedagogy using reinforcement learning. In Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing (EMNLP).

All tutoring dialogs use a simulated student (LLaMA-3.1-8B) rather than real learners; while standard in the field (Dinucu-Jianu et al., 2025; Lee et al., 2026a), student behavior may differ from authentic interactions. Experiments are restricted to mathematics; generalization to other subjects is untested. Evaluation relies on LLM-as-judge (Zheng et al., 2023) without human evaluation. We report single-run results without seed variance, though the breadth of comparison (792 runs) partially mitigates this. Experiments use 7B/8B models; larger models may narrow the gap between methods. Cross-paradigm comparison with RL uses different OOD benchmarks, limiting direct numerical comparison. Codebook analysis uses GPT-4o-mini for automatic labeling (vs. GPT-5mini in Lee et al. 2026a); a labeler-induced gap appears in absolute Praise rate, though the directional finding that training-free methods suppress Praise below the unoptimized baseline holds.

Qingyan Guo, Rui Wang, Junliang Guo, Bei Li, Kaitao Song, Xu Tan, Guoqing Liu, Jiang Bian, and Yujiu Yang. 2024. Connecting large language models with evolutionary algorithms yields powerful prompt optimizers. In Proceedings of the Twelfth International Conference on Learning Representations (ICLR). Omar Khattab, Arnav Singhvi, Paridhi Maheshwari, Zhiyuan Zhang, Keshav Santhanam, Sri Vardhamanan, Saiful Haq, Ashutosh Sharma, Thomas T. Joshi, Hanna Moazam, Heather Miller, Matei Zaharia, and Christopher Potts. 2024. Dspy: Compiling declarative language model calls into self-improving pipelines. In International Conference on Learning Representations. LearnLM Team, Google. 2025. proving Gemini for learning. arXiv:2412.16429.

Use of Generative AI

LearnLM: ImarXiv preprint

Unggi Lee, Jiyeong Bae, Jaehyeon Park, Haeun Park, Taejun Park, Younghoon Jeon, Sungmin Cho, Junbo Koh, Yeil Jeong, and Gyeonggeon Lee. 2026a. Rewarding how models think pedagogically: Integrating pedagogical reasoning and thinking rewards for LLMs in education. arXiv preprint arXiv:2601.14560.

We used Claude (Anthropic) to assist with drafting, editing, and code generation during the preparation of this manuscript. All scientific claims, experimental design, and data analysis were conducted and verified by the authors.

Unggi Lee, Sookbun Lee, Heungsoo Choi, Jinseo Lee, Haeun Park, Younghoon Jeon, Sungmin Cho, Minju Kang, Junbo Koh, Jiyeong Bae, Minwoo Nam, Juyeon Eun, Yeonji Jung, and Yeil Jeong. 2026b. OpenLearnLM benchmark: A unified framework for evaluating knowledge, skill, and attitude in educational large language models. arXiv preprint arXiv:2601.13882.

References Lakshya A Agrawal, Shangyin Tan, Dilara Soylu, Noah Ziems, Rishi Khare, Krista Opsahl-Ong, Arnav Singhvi, Herumb Shandilya, Michael J Ryan, Meng Jiang, Christopher Potts, Koushik Sen, Alexandros G. Dimakis, Ion Stoica, Dan Klein, Matei Zaharia, and Omar Khattab. 2026. GEPA: Reflective prompt evolution can outperform reinforcement learning. In Proceedings of the Fourteenth International Conference on Learning Representations (ICLR). Oral.

Jakub Macina, Nico Daheim, Sankalan Pal Chowdhury, Tanmay Sinha, Manu Kapur, Iryna Gurevych, and Mrinmaya Sachan. 2023. MathDial: A dialogue tutoring dataset with rich pedagogical properties grounded in math reasoning problems. In Findings of the Association for Computational Linguistics: EMNLP 2023.

Alon Albalak, Daman Agarwal, Pratyush Maini, Jon Saad-Falcon, and Tatsunori Hashimoto. 2025. BigMath: A large-scale, high-quality math dataset for reinforcement learning in language models. arXiv preprint arXiv:2502.17387.

Jakub Macina, Nico Daheim, Ido Hakimi, Manu Kapur, Iryna Gurevych, and Mrinmaya Sachan. 2025. MathTutorBench: A benchmark for measuring openended pedagogical capabilities of LLM tutors. In Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing (EMNLP).

Deborah Loewenberg Ball, Mark Hoover Thames, and Geoffrey Phelps. 2008. Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5):389–407.

Krista Opsahl-Ong, Michael J Ryan, Josh Purtell, David Broman, Christopher Potts, Matei Zaharia, and Omar Khattab. 2024. Optimizing instructions and demonstrations for multi-stage language model programs. arXiv preprint arXiv:2406.11695.

Michelene T.H. Chi and Ruth Wylie. 2014. The ICAP framework: Linking cognitive engagement to active learning outcomes. Educational Psychologist, 49(4):219–243.

George Pólya. 1945. How to Solve It: A New Aspect of Mathematical Method. Princeton University Press. Reid Pryzant, Dan Iter, Jerry Li, Yin Tat Lee, Chenguang Zhu, and Michael Zeng. 2023. Automatic prompt optimization with “gradient descent” and beam search. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing. Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y.K. Li, Y. Wu, and Daya Guo. 2024. DeepSeekMath: Pushing the limits of mathematical reasoning in open language models. arXiv preprint arXiv:2402.03300. Lee S. Shulman. 1986. Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2):4–14. Anaïs Tack and Chris Piech. 2022. The AI teacher test: Measuring the pedagogical ability of blender and GPT-3 in educational dialogues. In Proceedings of the International Conference on Artificial Intelligence in Education. David Wood, Jerome S. Bruner, and Gail Ross. 1976. The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2):89–100. Chengrun Yang, Xuezhi Wang, Yifeng Lu, Hanxiao Liu, Quoc V. Le, Denny Zhou, and Xinyun Chen. 2024. Large language models as optimizers. arXiv preprint arXiv:2309.03409. Mert Yuksekgonul, Federico Bianchi, Joseph Boen, Sheng Liu, Zhi Huang, Carlos Guestrin, and James Zou. 2024. TextGrad: Automatic “differentiation” via text. arXiv preprint arXiv:2406.07496. Qizheng Zhang, Changran Hu, Shubhangi Upasani, Boyuan Ma, Fenglu Hong, Vamsidhar Kamanuru, Jay Rainton, Chen Wu, Mengmeng Ji, Hanchen Li, Urmish Thakker, James Zou, and Kunle Olukotun. 2026. Agentic context engineering: Evolving contexts for self-improving language models. In Proceedings of the Fourteenth International Conference on Learning Representations (ICLR). Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, and 1 others. 2023. Judging LLM-as-a-judge with MT-bench and chatbot arena. Advances in Neural Information Processing Systems, 36. Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. 2023. Large language models are human-level prompt engineers. In International Conference on Learning Representations.

Table 3: OOD average across 4 Think conditions. ⋆ = best condition per method. T = Think, PT = Ped. Think, NR = NoReward, R = Reward.

Full Results

Across the full experiment we designed 19 education-specialized prompt optimization methods targeting the three failure patterns identified in §3.2 (answer leakage, rigid instruction following, lack of pedagogical structure). The 5 representative methods (ParetoGrad, CondBridge, LeakShield, Frame, MetaBlend) are described in §3.2 and listed in Table 1; the remaining 14 exploratory variants are described in §D. All 19 methods were evaluated under the NoThink condition with the same multiturn dialog simulation and multi-objective reward framework. Table 2 presents in-domain BigMath results for the full set, sorted by Rtotal . Rtotal uses the K=8 post-test Rsol (consistent with Table 1); the ∆Sol column is reported separately for reference. Two published methods (TextGrad, GEPA) are included with an asterisk for direct comparison under the same NoThink condition.

∆Sol Leak Help Rtotal

CondBridge TextGrad* ParetoGrad GEPA* HintGrad CurriculumOpt DisCo HintChain PrincipleHint DecompReward SokRat PopGrad Frame ThinkGuard LeakShield MetaBlend PromptDistill Adversarial ContrastOpt AnchorBoost DualLoop

.167 .173 .394 .237 .263 .232 .180 .304 .292 .167 .284 .165 .195 .206 .167 .117 .121 .162 .177 .105 .091

.204 .208 .492 .340 .370 .346 .294 .448 .454 .320 .446 .350 .392 .424 .508 .492 .500 .700 .720 .668 .786

.848 .840 .849 .842 .846 .846 .843 .847 .845 .840 .845 .840 .840 .784 .782 .801 .785 .821 .799 .794 .705

C.1

.614 .612 .594 .590 .589 .587 .586 .577 .572 .571 .570 .562 .558 .531 .489 .486 .479 .437 .428 .421 .346

Think Condition Comparison

Table 3 compares OOD performance across the four Think conditions for all 12 training-free methods. Think Reward achieves the highest average (7.66), though differences across conditions are small (spread 0.08). The best condition varies by method (marked with ⋆), suggesting no single Think configuration is universally optimal.

T-R

PT-NR PT-R

MIPROv2 EvoPrompt ACE GEPA ParetoGrad OPRO TF-GRPO CondBridge MetaBlend Frame TextGrad LeakShield

7.76 8.00 7.98 7.54 7.52 7.46 7.54 7.54 7.44 7.44 7.34 7.37

⋆7.92 7.90 7.91 ⋆8.10 7.78 7.78 7.82 7.87 ⋆8.00 ⋆7.76 7.59 7.61 ⋆7.62 7.49 7.37 ⋆7.60 7.44 7.45 ⋆7.60 7.51 7.54 7.55 ⋆7.59 7.36 7.52 ⋆7.58 7.37 ⋆7.55 7.49 7.53 7.48 ⋆7.53 7.50 7.39 7.47 ⋆7.52

Average

7.58

7.66

7.60

7.58

Labeling Methodology Details

We use the 82-code educational codebook from Lee et al. (2026a), organized into seven toplevel categories: Mathematical Problem Solving (MPS), Mathematical Knowledge for Teaching (MKT), Cognition, Metacognition, Pedagogical Intent Utterance (PIU), Student Intent Utterance (SIU), and Affect/Discourse (A/D). Tutor responses are labeled with OpenAI GPT-4omini (openai/gpt-4o-mini via OpenRouter) at temperature 0. For thinking-mode trajectories, <think>...</think> blocks are stripped before sentence-level labeling, and sentences shorter than 15 characters are excluded. PedRL-Thinking originally reports labels obtained with GPT-5-mini; absolute rates therefore should not be directly compared across the two studies, but the within-study rank-based comparisons (Spearman) used in Appendix C.11 are robust to labeler-induced offsets.

*Published method shown for reference.

T-NR

§4.2 and §4.3. All training-free rows are 12-method averages under each condition; RL reference rows are from Lee et al. (2026a).

Table 2: All 19 proposed methods on BigMath (NoThink). Method

Method

Two rate definitions appear in this work. The single-label rate (used by aggregate category percentages following Lee et al. 2026a) is the fraction of sentences whose label set is exactly {c}; the multi-label rate (used for dialog-level aggregation and for the core-code tables in this appendix) is the fraction of sentences whose label set contains c. Spearman correlation tests in Appendix C.11 apply Benjamini-Hochberg correction across the 5 conditions × 4 reward components grid (20 tests, q < 0.05).

Behavioral Analysis Details

This section provides condition-level aggregates and additional cuts of the behavioral analysis in 11

Table 4: Labeled corpus statistics. Quantity

Count

Labeled tutor sentences (60 method-condition cells) Labeled dialogs with Rtotal Schoenfeld paragraphs (thinking only)

415,775 22,126 30,954

C.2

stable across conditions (17.5-18.7% under Think variants). Table 7: Core code frequencies (sentence-multilabel %, 12-method average per condition). Each entry is the fraction of sentences containing at least one instance of the code.

Category Distributions

Table 5 reports the 4-category code-instance % per condition (the basis for the body’s MKT/PIU finding), and Table 6 expands the Interaction category into Cognition, Metacognition, Student Intent Utterance, and Affect/Discourse. The residual ∼5−8pp is dominated by Affect/Discourse, with small Cognition and Metacognition contributions that grow modestly under Think conditions.

MPS MKT

PIU

Interaction

NT T-NR T-R PT-NR PT-R

20.29 20.37 19.68 24.03 23.73

54.81 49.37 49.69 46.22 47.27

5.59 7.68 7.51 7.17 6.23

RL reference (Lee et al., 2026a) RL NoThink 16.73 7.95 65.14 RL Ped. Think 25.06 7.15 59.54

9.11 6.97

19.31 22.57 23.12 22.57 22.77

SIU A/D 0.27 0.31 0.35 0.36 0.20

20.29 20.37 19.68 24.03 23.73

19.31 22.57 23.12 22.57 22.77

0.69 1.91 2.06 1.59 1.51

0.80 1.82 1.80 1.91 1.82

54.81 49.37 49.69 46.22 47.27

Strat

1.30 1.45 1.44 1.32 1.11

0.04 0.10 0.11 0.07 0.08

8.91 9.42 8.12 10.89

3.45 4.89

18.73 18.31 17.46 13.57 14.04

37.10 22.28 22.22 23.20 27.28

Code

T-NR

T-R

PT-NR PT-R

Information provision Step-by-step instruction Concept explanation Exploratory question Application of rules/formulas Providing justification Performing calculations Open-ended question Praise Encouraging participation Request for clarification Context check/Solution comp. Clarification Emotional Support Use of examples

37.10 29.95 21.10 18.73 11.61 11.45 6.93 6.36 5.48 4.70 3.72 3.19 3.04 2.94 2.71

22.28 14.47 17.09 18.31 8.08 13.28 4.24 4.60 5.99 3.79 3.77 4.02 5.26 2.39 2.69

22.22 14.03 16.80 17.46 7.15 12.35 3.84 3.74 5.89 3.38 4.54 3.52 4.58 2.37 2.64

23.20 16.66 16.44 13.57 7.42 12.25 4.07 3.83 4.02 2.39 2.99 3.50 5.84 1.61 2.67

C.4

27.28 18.61 17.74 14.04 6.94 11.36 3.75 3.78 3.35 1.89 2.73 3.51 5.74 1.13 2.12

Polya Phase Distribution

Table 9 summarises the Polya phase distribution over the full dialog, complementing the per-turn view in Figure 4 right. Scaffold dominates every condition (58-63%), while the pedagogical seed shifts mass from Interaction toward Understand (NT 8.2% → PT-R 13.3%).

3.83 3.65 3.31 3.31 2.69

Table 9: Polya phase distribution over the full dialog (code-instance %, 12-method average per condition).

Cog = Cognition, Met = Metacognition, SIU = Student Intent Utterance, A/D = Affect/Discourse.

Condition

C.3

29.95 14.47 14.03 16.66 18.61

Table 8: Top-15 codes by overall sentence-multilabel frequency, 12-method average per condition. Rows are sorted by NoThink frequency.

Table 6: Extended 7-category distribution (codeinstance %, 12-method average per condition). Expands Table 5 by splitting Interaction into Cognition, Metacognition, Student Intent Utterance, and Affect/Discourse.

NT T-NR T-R PT-NR PT-R

Hint

5.48 5.99 5.89 4.02 3.35

Step = Step-by-step instruction, ExplQ = Exploratory question, Info = Information provision, Hint = Hint provision, Strat = Strategy selection.

MPS = Math Problem Solving, MKT = Math Knowledge for Teaching, PIU = Pedagogical Intent Utterance, Interaction = remaining categories.

Condition MPS MKT Cog Met PIU

Praise Step ExplQ Info

NT T-NR T-R PT-NR PT-R

RL reference (Lee et al., 2026a) RL NoThink 11.76 15.18 12.85 RL Ped. Think 5.78 18.22 13.27

Table 5: Major category distribution (code-instance %, 12-method average per condition). Rows sum to 100. NT = NoThink, T-NR = Think NoReward, T-R = Think Reward, PT-NR = Ped. Think NR, PT-R = Ped. Think R. Condition

Condition

Code Frequencies

NT T-NR T-R PT-NR PT-R

Table 7 compares six core codes from the body’s discussion against RL references, and Table 8 expands the view to the 15 most frequent codes (by NoThink prevalence). Information provision, Stepby-step instruction, and Concept explanation dominate under NoThink (37.1%, 30.0%, 21.1% of sentences), while Exploratory question is the most

C.5

Understand

Execute

Scaffold

Review

Interaction

8.20 9.91 10.01 13.22 13.29

9.75 8.22 7.71 8.32 7.77

63.39 58.48 59.03 58.57 61.64

3.82 5.97 5.81 5.99 6.01

14.84 17.42 17.44 13.89 11.29

Turn-Level Behavioral Diversity

Table 10 reports two diversity metrics per tutor turn: Shannon entropy of the code distribution 12

and the mean number of distinct codes assigned. Think conditions raise entropy slightly above NoThink (1.51 nats → 1.59-1.62), consistent with the broader code mass shift visible in Tables 7 and 8.

Table 12: Response length (mean words per turn) and math content (LaTeX math character fraction, %) by condition (12-method average). Condition

Vis WC

Thk WC

Total WC

Vis Math%

Thk Math%

Table 10: Turn-level behavioral diversity (12-method average per condition). Entropy is the Shannon entropy (nats) of the code distribution within a tutor turn; unique codes per turn is the mean number of distinct codes assigned to a tutor turn.

NT T-NR T-R PT-NR PT-R

110.1 146.9 141.1 150.2 145.9

0.0 201.3 200.4 204.0 201.7

110.1 348.2 341.5 354.3 347.6

17.97 11.17 10.38 10.30 11.27

0.93 0.95 1.02 1.02

NT T-NR T-R PT-NR PT-R

C.6

Entropy (nats)

Unique codes / turn

1.505 1.621 1.586 1.609 1.570

7.82 8.11 7.45 8.21 7.72

method identity or condition (NoThink vs. Think vs. Pedagogical seed) drives the dominant behavioral pattern. MIPROv2 (NT) CondBridge (NT) ParetoGrad (NT) ParetoGrad (T-NR) MIPROv2 (T-R) MIPROv2 (PT-R) LeakShield (T-R) LeakShield (T-NR) Frame (T-NR) LeakShield (PT-R) TextGrad (T-R) TF-GRPO (T-R) MetaBlend (PT-NR) ACE (T-NR) LeakShield (NT) MetaBlend (T-NR) OPRO (PT-NR) EvoPrompt (PT-R) ACE (T-R) OPRO (PT-R) Frame (PT-R) LeakShield (PT-NR) ACE (PT-NR) MetaBlend (PT-R) ACE (PT-R) OPRO (NT) ACE (NT) TF-GRPO (NT) Frame (NT) GEPA (NT) EvoPrompt (NT) EvoPrompt (PT-NR) MetaBlend (T-R) MIPROv2 (T-NR) ParetoGrad (PT-R) OPRO (T-R) TF-GRPO (T-NR) TextGrad (PT-R) CondBridge (PT-R) TextGrad (PT-NR) ParetoGrad (PT-NR) CondBridge (T-NR) Frame (PT-NR) TextGrad (T-NR) MIPROv2 (PT-NR) ParetoGrad (T-R) OPRO (T-NR) GEPA (PT-NR) TextGrad (NT) MetaBlend (NT) GEPA (PT-R) CondBridge (PT-NR) TF-GRPO (PT-R) TF-GRPO (PT-NR) EvoPrompt (T-NR) GEPA (T-NR) EvoPrompt (T-R) CondBridge (T-R) GEPA (T-R) Frame (T-R)

Schoenfeld Sanity Check

Table 11 reports the Schoenfeld 3-phase distribution over <think> content. This is included only as a sanity check against Lee et al. (2026a) Table 10; the body’s behavioral analysis does not rely on it. Our Explore rate of 20-29% diverges substantially from the 0.75% reported under RL Ped. Think Reward, likely because our <think> blocks are short procedural traces rather than extended exploratory reasoning. Table 11: Schoenfeld 3-phase distribution (%) of <think> content (paragraph-level, 12-method average per thinking condition). Condition

Explore

General

Verify

T-NR T-R PT-NR PT-R

26.73 28.52 20.60 20.82

25.20 24.20 26.52 25.60

48.08 47.28 52.87 53.58

81.02

18.23

Reference (Lee et al., 2026a) RL Ped. Think R 0.75

C.7

70 60 50

Code-instance %

Condition

Vis WC = visible response words, Thk WC = thinking-trace words.

40 30 20 10

MPS

MKT

Cog

Met

PIU

SIU

A/D

Figure 6: Hierarchical clustering (Ward linkage on Euclidean distance) of 60 method-condition cells by their 7-category behavioral signature.

Response Length and Math Content

C.9

Condition Effect Analyses

Table 12 reports visible response length, thinking trace length, and the LaTeX math character fraction. NoThink responses are roughly 25% shorter than Think visible responses (110.1 vs. ∼145 words) but contain a higher math character fraction (18.0% vs. ∼11%), reflecting the tendency of NoThink to deliver direct computational content.

Figures 7 and 8 isolate the per-method effect of the two thinking-condition axes. The Rthink reward (Think Reward − Think NoReward) produces small shifts (|∆| ≤ 2pp in most cells), while the pedagogical seed produces larger and methodspecific shifts, most prominently increased MPS and reduced PIU for several methods.

C.8

C.10

Method Behavioral Clustering

Figure 6 applies hierarchical clustering to the 60 method-condition cells using their 7-category behavioral signature. Cells with similar code-instance distributions group together, surfacing whether

Published vs. Proposed Group Comparison

Figure 9 aggregates the 7-category distribution across all 5 conditions and compares the 7 adapted Published methods against our 5 Proposed meth13

ACE

+1.4

CondBridge

-2.8

-4.3

+4.8

EvoPrompt

-0.9

+1.7

-0.8

Frame

+8.9

+9.3

-9.5

+2.6

-2.0

-3.5

-3.3

+6.3

MIPROv2

-17.1

-1.2

+19.5

+2.3

15 -8.7

10 5

-1.3

MetaBlend

-2.0

+3.1

-2.8

+1.8

OPRO

+5.3

+5.4

-8.6

-2.1

ParetoGrad

+10.8

+2.0

-14.3

+1.5

TF-GRPO

-1.9

-4.5

+4.1

+2.2

TextGrad

-6.2

-2.9

+7.5

+1.6

MPS

MKT

PIU

Interaction

0 5

Code-instance %

LeakShield

+5.1

-3.1

-2.2

+4.5

-2.5

-1.3

-0.6

EvoPrompt

+4.6

-2.5

-1.3

-0.8

Frame

+7.1

+3.6

-7.3

-1.2

-3.4 +1.9

LeakShield

+6.8

-4.7

-1.5

-0.6

MIPROv2

-2.8

-1.5

+1.9

+2.3

MetaBlend

+1.5

-2.5

-1.3

+2.3

OPRO

+2.9

-1.2

ParetoGrad

+10.1

+8.6

0 MPS

0 5

-4.5

TF-GRPO

-1.2

-3.5

+5.4

-0.8

TextGrad

+8.0

+8.4

-11.0

-5.4

MPS

MKT

PIU

Interaction

PIU

SIU

A/D

Table 13: Per-condition Spearman correlation between dialog-level Praise rate and reward components. Significance after Benjamini-Hochberg correction over 20 tests. ∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 10−15 . Condition NoThink T-NR T-R PT-NR PT-R

ρ(Rsol ) ∗∗

6,000 +0.126 4,131 +0.048 4,206 +0.056∗ 3,900 −0.019 3,889 −0.003

ρ(Rleak )

ρ(Rhelp )

+0.016 −0.007 −0.010 +0.005 −0.010

∗∗

+0.077 +0.011 +0.023 +0.029 +0.004

ρ(Rtotal ) +0.106∗∗∗ +0.013 +0.026 +0.003 −0.014

Table 14: Turn-position decomposition of Spearman correlation between dialog Praise rate and reward. The T4+ effect drives the bulk of the Rsol contribution.

Figure 8: Per-method effect of the pedagogical seed prompt on the 4-category distribution, averaged over the Rthink axis (Pedagogical seed − no seed, percentage points). Annotated cells have |∆| ≥ 0.5pp.

ods. The two groups overlap substantially across all categories, indicating that the behavioral signature of training-free tutoring is largely driven by the prompt-optimization paradigm itself rather than by specific method design. C.11

Met

dialog’s closing turns (T4+), consistent with a lateturn motivational signal that carries measurable post-test contribution.

10 5

Cog

MKT

Figure 9: Distribution of 7-category code-instance % across method-condition cells, grouped by Published (7 methods × 5 conditions) vs. Proposed (5 methods × 5 conditions). Boxes show interquartile range with median; whiskers extend to 1.5×IQR.

-1.4 -14.1

% (Ped seed no seed)

ACE CondBridge

Figure 7: Per-method effect of Rthink reward on the 4-category distribution (Think Reward − Think NoReward, percentage points). Annotated cells have |∆| ≥ 0.5pp.

GEPA

Published (7 methods × 5 cond) Proposed (5 methods × 5 cond)

% (Think R Think NR)

GEPA

-1.5

C.12

Turn position

ρ(Rsol )

ρ(Rtotal )

Significance

T1-T3 (mid) T4+ (close)

+0.046 +0.093

+0.062 +0.110

q < 0.001 q < 10−28

MIPROv2 Reward-Driven Behavioral Shift

The Praise-suppression generalization in §4.2 is a method-level aggregate. MIPROv2 is a notable exception: when Rthink is activated, its Praise rate increases by +8.49pp, its Pedagogical Intent Utterance share by +19.5pp, and its turn-level codedistribution entropy collapses by −0.91 bits - each the most extreme value among the 12 listed methods (ParetoGrad moves in the opposite direction with ∆Praise = −5.03pp). Table 15 lists four behavioral category cells where MIPROv2 exceeds the 12-method mean by ≥ 2σ, all sharing a “high

Praise Mode-Specific Analysis

Across n = 22,126 labeled dialogs, we decompose the dialog-level Spearman correlation between Praise rate and reward components by reasoning condition and turn position. Table 13 shows that the positive correlation between Praise rate and Rtotal is concentrated in the NoThink condition, while Think variants show null or slightly negative effects. Table 14 further localizes the effect to the 14

reward effect (reward on − reward off, Think condition) on three behavioral indicators: Praise rate (sentence-multilabel %), PIU share (code-instance %), and turn-level code-distribution entropy (bits). The reward signal acts differently across methods in both magnitude and direction: MIPROv2 sharpens to a single high-Praise high-PIU pattern (entropy collapse), while ParetoGrad moves in the opposite direction, suppressing Praise and PIU while broadening the code distribution. Aggregate Rtotal does not distinguish these qualitatively different policies.

PIU, low MPS” signature consistent with a rewarddriven tilt toward direct telling at the expense of Socratic questioning. These cells contribute to MIPROv2’s competitive Rtotal = 0.707 despite the behavioral divergence, illustrating that scalar reward alone can mask qualitatively different policies. Table 15: Behavioral category cells where MIPROv2 exceeds the 12-method mean by ≥ 2σ. PIU = Pedagogical Intent Utterance, MPS = Mathematical Problem Solving. Condition