An application in mathematics is any context, within science or broader, which involves or requires some kind of quantitative thinking. For instance, arguments involving risk, chance, or uncertainty use probabilistic concepts. Every time we interpolate or extrapolate from a given set of data we employ functional relationships. In discussing dynamics of drug absorption, we use exponential or more complex mathematical models. Describing viruses infecting bacteria or studying interactions between species in an ecosystem requires that we use mathematics tools.

In this paper I study certain teaching and learning situations, named tensions, which arise when the students, practitioners, or instructors engage with applications in mathematics that require modifications to our cognitive models. Tensions can be identified in the ways we formulate the problem of our inquiry, in defining the objects we study, in the implicit and explicit assumptions we make, in the interpretations of results of experiments and mathematical calculations, in visual interpretations, and in other situations.

Using specific examples, I illustrate that these tensions could be viewed as living in a specific zone of proximal development. This concept provides a framework within which we contrast what a single discipline can achieve, compared to fresh new visions and insights generated when the diverse views of mathematics and other science disciplines are brought together.