Biominerals are typically composites of hard matter such as calcite, and soft matter such as proteins. There is currently considerable interest in how the soft matter component is incorporated into the hard matter component. This would typically be a protein that does not fold up into a single rigid domain but is closer to a simple polymer, being incorporated into a growing inorganic crystal in aqueous solution. Here I use computer simulation to study a very simple (2D lattice gas) model of a growing phase and a polymer. This allows me to study the microscopic dynamics of incorporation or rejection of a single polymer by the growing phase. It also allows me to look at how high concentrations of absorbing polymer can both arrest crystal growth, and change the shape of crystals. I find that the incorporation of a single polymer into the growing phase is due to slow dynamics of the polymer at the growth front. These slow dynamics are then unable to keep up with the advancing growth front. This is an intrinsically far-from-equilibrium process and so occurs even when incorporation is thermodynamically highly unfavourable. During the incorporation process, large polymers create large and deep, but transient, pits in the growth front.