There is little empirical research related to how elementary students develop computational thinking (CT) and how students apply CT in problem-solving. To address this gap in knowledge, this study made use of learning trajectories (LTs; hypothesized learning goals, progressions, and activities) in CT concept areas such as sequence, repetition, conditionals, and decomposition to better understand students' understanding of CT. This basic interpretive, qualitative study aimed at gaining a deeper understanding of elementary students' CT by having students express and articulate their CT in assessment-based problem-solving scenarios. This study implemented eight math-CT integrated lessons aligned to both the Common Core State Standards for Math (CCSS-M) and the LTs throughout the school year 2019-2020 with third- and fourth-grade students. The current study specifically examined the students' CT in the areas of sequence, repetition, conditionals, and decomposition. By using the cognitive interview technique with a think-aloud protocol, this study revealed the CT thought processes of 22 elementary students while they worked on CT assessment items. Participants' CT articulation was examined using a priori codes translated verbatim from the learning goals in the LTs and mapped to the learning goals and progressions in the LTs. Results suggested that students demonstrated a more consistent articulation of CT in the area of sequence and varied understanding in repetition, conditionals, and decomposition, with some participants demonstrating more complex CT than others. As such, participants' CT articulation was generally mapped to the intermediate-level in the Sequence LT and varied levels in the other LTs. By collecting empirical data on how students expressed and articulated their CT, this study made theoretical contributions by generating initial empirical evidence to support the hypothesized learning goals and progressions in the LTs. This study also has methodological implications by employing cognitive interviews to study students' CT and practical implications by identifying learning gaps and suggesting targeted instructional activities to support the development of elementary students' CT. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]