A finely divided, starch-containing polymer dispersion which is obtainable by free radical emulsion polymerization of ethylenically unsaturated monomers in the presence of at least one redox initiator and starch, wherein (a) from 0 to less than 40% by weight of at least one optionally substituted styrene, (b) from greater than 60 to 100% by weight of at least one C1-C12-alkyl acrylate and/or C1-C12-alkyl methacrylate, (c) from 0 to 10% by weight of at least one other ethylenically unsaturated copolymerizable monomer, are used as the ethylenically unsaturated monomers, the sum (a)−(b)+(c) being 100% and being based on the active monomer content and (d) from 15 to 40% by weight of at least one degraded starch which has a molar mass Mw of from 1000 to 65 000 g/mol, based on the total weight of solids content of components (a)+(b)+(c)+(d), are used as the starch, and the polymerization being carried out in the presence of at least 0.01% by weight, based on the monomers used, of at least one terpene containing chain-transfer agent, and in which the polymer dispersion optionally comprises an aluminium compound. The use of the finely divided, starch-containing polymer dispersions as sizes for paper, board and cardboard is also claimed.