The Tutoring Effectiveness Index: Predicting LLM Math Tutor Quality from Four Conversation Signals

Aligning large language models (LLMs) as math tutors typically demands costly reinforcement-learning (RL) training and external LLM judges. We ask whether a frozen model's internal reasoning signals can replace both. We propose the Tutoring Effectiveness Index (TEI), a training-free, judge-free four-signal index that combines a Schoenfeld-Verify keyword ratio, a math-step density, an ends-question rate, and a deep-reasoning gate from the Deep-Thinking Ratio (DTR) probe. Selecting from N candidates with TEI (the TEI@N rule) raises the improvement rate on pre-incorrect scenarios from 59.0% to 81.9% at N=8 on a frozen DeepSeek-R1-8B base, with no training and no external judge. We also measure the alignment tax of pedagogical GRPO. Thinking length drops from 1,764 to 119 words per turn (-93%), Content-Knowledge and Pedagogical-Knowledge accuracy fall by -71% and -80% relative, and the student's Delta Solve Rate crosses from +0.180 to -0.012. To anchor the behavioural reading, we reproduce an 82-code educational codebook on 119,009 tutor sentences with a one-shot structural classifier. Together, these results offer a cost-effective recipe for building math-tutoring LLMs without RL training or external judges.