This paper introduces the concept of infrastructure into discussions on climate change and education. We focus on the links between the increased use of digital data and the central role of data infrastructures in education, and the energy infrastructure needed to support their growing use in schools and school systems. We elaborate a need for a greater accounting of the climate and related social costs of these interwoven digital and energy infrastructures of schooling. We suggest this is part of the 'disposition' of the infrastructures of schooling that should be weighed into decisions on whether and how to continue with digital technologies in schools. By acknowledging the climate and environmental incommensurability of digital infrastructures, education leaders and young people can more fully understand their dispositions and their costs. We propose three implications for education governance that entail greater consideration of the limits of current school climate change infrastructures such as 'eco school' programs and EdTech 'AI for good' initiatives, pushes for 'computing within limits' without substantial changes, and current school governance practices which unnecessarily rely on digital infrastructures. Instead, what is needed may be a reversal of the extensive use of digital infrastructures by schools and education governance bodies.