What algorithms do LLMs actually learn and use to solve problems? Studies addressing this question are sparse, as research priorities are focused on improving performance through scale, leaving a theoretical and empirical gap in understanding emergent algorithms. This position paper proposes AlgEval: a framework for systematic research into the algorithms that LLMs learn and use. AlgEval aims to uncover algorithmic primitives, reflected in latent representations, attention, and inference-time compute, and their algorithmic composition to solve task-specific problems. We highlight potential methodological paths and a case study toward this goal, focusing on emergent search algorithms. Our case study illustrates both the formation of top-down hypotheses about candidate algorithms, and bottom-up tests of these hypotheses via circuit-level analysis of attention patterns and hidden states. The rigorous, systematic evaluation of how LLMs actually solve tasks provides an alternative to resource-intensive scaling, reorienting the field toward a principled understanding of underlying computations. Such algorithmic explanations offer a pathway to human-understandable interpretability, enabling comprehension of the model's internal reasoning performance measures. This can in turn lead to more sample-efficient methods for training and improving performance, as well as novel architectures for end-to-end and multi-agent systems.