Recent educational studies in mathematics seek to justify a thesis that there is a conflict between students' intuitions regarding infinity and the standard theory of infinite numbers. On the contrary, we argue that students' intuitions do not match but to Cantor's theory, not to any theory of infinity. To this end, we sketch ways of measuring infinity developed at the turn of the 20th and 21st centuries that provide alternatives to Cantor's theory of cardinal and ordinal numbers. Some of them introduce new kinds of infinite numbers, others simply define new arithmetic for Cantor's infinite numbers. We also sketch a way how to introduce these new theories in students' courses. To do this the crucial is the concept of an ordered field, since we define the opposition finite vs infinite in terms of Archimedean and non-Archimedean fields.