Large Language Models (LLMs) have shown strong potential in educational question answering, yet they often exhibit cognitive misalignment by generating responses that emphasize linguistic fluency rather than the cognitive depth required by different levels of Bloom's Taxonomy. This limitation arises from three structural gaps: Bloom classifiers are typically disconnected from the generation process, Chain-of-Thought (CoT) reasoning lacks explicit cognitive scaffolding, and Retrieval-Augmented Generation (RAG) verifies factual consistency only at the final output. To address these limitations, this paper proposes Cognitive Chain-of-Thought (Cog-CoT), a four-module framework that integrates Bloom's Taxonomy into the reasoning process of LLMs. The framework consists of (1) a LinearSVC-based Cognitive Classifier (weighted F1 = 0.9915) for Bloom-level prediction, (2) a Hierarchical Cognitive Decomposer that constructs a bottom-up sequence of Bloom-aligned sub-questions, (3) a Cog-CoT Reasoning module employing six level-specific prompt templates with step-wise cosine similarity verification against retrieved context ( = 0.65, MAX_RETRY = 3), and (4) a Bottom-Up Aggregator that synthesizes verified intermediate responses into a coherent final answer. Experiments using three open-weight LLMs (Gemma-3-4B-IT, LLaMA-3.1-8B-Instruct, and Qwen2.5-7B-Instruct) on an Indonesian Social Studies dataset and external English benchmarks (SQuAD 2.0 and ASQA) demonstrate that Cog-CoT consistently outperforms Zero-Shot prompting, achieving improvements of 4.16%–7.36% in GT Cosine Similarity while also producing superior performance across BLEU-4, ROUGE-L, METEOR, and BERTScore. These results demonstrate the effectiveness of integrating cognitive scaffolding with structured reasoning and retrieval verification for educational question answering.