© The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. A continuing challenge in mathematics education at the university level is how to create learning environments that give students the opportunity to progress from their informal ways of reasoning to more formal ways of reasoning. This article shows how the emergent models heuristic (Gravemeijer, 1999, Math. Think. Learn, 1, 155-177) provides this opportunity to first-year university students, specifically for them to construct the Linear Algebra concepts of spanning set and span. For this, a design-based research methodology was used. A hypothetical learning trajectory (HLT) was applied, created within the framework of this heuristic. The results provide evidence that the emergent models heuristic support the construction of these concepts of Linear Algebra. Moreover, useful tips are given for using this heuristic to construct mathematical concepts.