Previous research suggests that in classes that take an integrated approach to science, technology, engineering, and math (STEM) education, students tend to engage in fulfilling goals of their engineering design challenges, but only inconsistently engage with the related math and science content. The present research examines these inconsistencies by focusing on student engagement, or effort, towards math and science concepts while working on an engineering challenge, through the lens of expectancy-value theory. Specifically, we examine how students' perceptions of the "value" of math and science and "expectancy" for success with the math and science relate to the "efforts" they put towards using math and science while working on engineering challenges. Our results suggest that subjective task value significantly predicts efforts towards both math and science, whereas neither expectancy, nor the interaction between expectancy and value predicted effort. We argue that integrated learning environments need to help students understand how the domains of math, science, and engineering support their work in fulfilling the engineering project design goals. In other words, we argue that we, as educators, must help students to recognise the value of each of the domains addressed within STEM integrated learning environments. This paper discusses strategies for accomplishing this goal.