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Connections across curricular topics, within and beyond mathematics, are important. Setting out from this premise, and deploying a commognitive lens, I see the incorporation of said connections in students' learning as fostering "fuel" as opposed to "fossil" narratives about mathematics – where "fuel" is meant as meaningful engagement with mathematical, and other, "realizations" of a mathematical object and "fossil" is meant as externally imposed reverence for a mathematical object, yet alienation from its meaning, utility and purpose. In this vision of connectedness, there is a dialectic, dynamic and productively blurred boundary between intra- and extra-mathematical connections. I make the case for this vision of connectedness through drawing on studies that explore students' narratives about mathematics at large and about the significance of intra- and extra- mathematical connections (the latter in the case of one mathematical object, exponential growth). The analyses I sample from indicate how connectedness-informed pedagogies can facilitate students' navigating across "literate" (mathematical) and "colloquial" (everyday) discourses and moving towards "fuel" narratives about mathematics.