This is a question of basic maths. And I am sorry if this is a stupid question, but I am teaching it and there is something I remember seeing once and ignoring it when I was a child. I understand it is a consequence of a bad procedure in the use of units, but would like to confirm with your answers and thoughts.
The case is that, for example, when I have radians and degrees, if I express the equality $\pi radians=180 degrees$ I am able to construct the neutral fraction $\frac{\pi radians}{180 degrees}=1$ with which perform cancellations and change units. The problem is that this student had the interpretation of units as variables, so he proceeded to state from the first equation $radians=\frac{180}{\pi } degrees$ and so concluding the exact inverse ratio. Though it is a correct equation, he used it as a functional relationship. The proportion that does hold is $\frac{180}{\pi }=\frac{degrees}{radians }$ taking now degrees and radians as variable quantities. How can one justify as clearly as possible this false duality?
EDIT: Thanks for the answers. I was going through the same mental process of clarifying the difference of units (using the example one of you gave: $s$ is different if it is used as the unit or the quantity one second), but he made me acknowledge something: if it is not possible to handle it as such, why can we cancel them (the units)?
Your student is interpreting the statement, "$\pi$ radians equals $180$ degrees" as if it meant, "$pi$ times the number of radians equals $180$ times the number of degrees," which of course, it doesn't.
It's as if we took the statement "$1$ dollar equal $100$ cents" to mean, "The number of dollars equals $100$ times the number of cents."