Normal fuzzy sets and Pythagorean cubic fuzzy sets are the best means to deal with fuzziness. Combining both of these structures in our current work, we establish the idea of Pythagorean cubic normal fuzzy set. We define some basic operational laws for Pythagorean cubic normal fuzzy set and introduce a number of aggregation operators, including Pythagorean cubic normal fuzzy weighted averaging operator, Pythagorean cubic normal fuzzy weighted geometric operator, Pythagorean cubic normal fuzzy order weighted averaging operator and Pythagorean cubic normal fuzzy order weighted geometric operator. We examine several favorable properties, including monotonicity, boundedness, and idempotency for the proposed operators. We develop an algorithm for the solution of multicriteria decision-making problems. Moreover, we propose an extended form of the TODIM (Portuguese acronym for Interactive Multi-Criteria Decision Making) method. We present a multicriteria decision-making example related to assessing the educational needs of students with disabilities. The techniques and operators defined in the current work provide greater generality and accuracy and give precise results. Ultimately, a detailed illustration is provided to show the closure process of these specified procedures and functions, demonstrating their credibility and efficacy.