Aqueous polymer dispersions which are obtainable by free radical copolymerization of (a) from 0.1 to 99.9% by weight of styrene and/or methylstyrene, (b) 0.1-99.9% by weight of 1,3-butadiene and/or isoprene and (c) from 0 to 40% by weight of other ethylenically unsaturated copolymerizable monomers, the sum of the monomers (a), (b) and (c) always being 100, in the presence of from 10 to 40% by weight, based on the monomers used, of at least one degraded starch having a molecular weight Mn of from 500 to 40 000 and of water-soluble redox catalysts are prepared by free radical copolymerization of the monomers (a), (b) and, if required, (c) in an aqueous medium in the presence of a degraded starch having a molecular weight Mn of from 500 to 10 000 and redox initiators and are used as engine sizes and surface sizes for paper.