As the world grapples with the ongoing COVID-19 pandemic, a particularly thorny set of questions surrounds the reopening of primary and secondary (K-12) schools. The benefits of in-person learning are numerous, in terms of education quality, mental health, emotional well-being, equity and access to food and shelter. Early reports suggested that children might have reduced susceptibility to COVID-19, and children have been shown to experience fewer complications than older adults. Over the past few months, our understanding of COVID-19 has been further shaped by emerging data, and it is now understood that children are as susceptible to infection as adults and have a similar viral load during infection, even if asymptomatic. Based on this updated understanding of the disease, we have used epidemiological modeling to explore the feasibility and consequences of school reopening in the face of differing rates of COVID-19 prevalence and transmission. We focused our analysis on the United States, but the results are applicable to other countries as well. We demonstrate the potential for a large discrepancy between detected cases and true infections in schools due to the combination of high asymptomatic rates in children coupled with delays in seeking testing and receiving results from diagnostic tests. Our findings indicate that, regardless of the initial prevalence of the disease, and in the absence of robust surveillance testing and contact-tracing, most schools in the United States can expect to remain open for 20-60 days without the emergence of sizeable disease clusters. At this point, even if schools choose to close after outbreaks occur, COVID-19 cases will be seeded from these school clusters and amplified into the community. Thus, our findings suggest that the debate between the risks to student safety and benefits of in-person learning frames a false dual choice. Reopening schools without surveillance testing and contact tracing measures in place will lead to spread within the schools and within the communities that eventually forces a return to remote learning and leaves a trail of infection in its wake.