This perspective paper draws on the interest in ensuring that students who enter primary teacher training programs have a solid background knowledge of mathematics. We describe the access criteria and requirements for admission to the primary education degree programs at the Universidad de Alicante and Universitat Autònoma de Barcelona, in Spain, and the University of Helsinki, in Finland. We present the results of an evaluation of the mathematical knowledge that students bring to their education as teachers at these three institutions. The results show that in each program, the subgroup of students who had followed the longer track of mathematics courses scored significantly higher on the mathematical test, although this was no longer as clear when we compared across universities. We also found that the students who had taken the mathematics section of the entrance examination or the matriculation examination scored higher on the test than those from the same program who had not, but this tendency broke down when cross-university comparisons were made. We also explored how the cap set on the number of students admitted to the three programs -- this being the most striking difference in the admission policies -- could be an explanatory variable for these discrepancies. The comparison between universities leads us to hypothesize that expecting applicants to have met certain requirements in their academic trajectories prior to university entrance and adjusting the cap set on the number of places could ensure a better mastery of mathematical knowledge among those students admitted to the Spanish programs.