There is an extraordinary free book about Mathematics Education in Early Childhood (NRC, 2009). In chapter two (p. 21), there is a detailed description of mathematics in this first years. You can take this document as a reference.

But I think that your question is more about the mathematics that are really teached in schools (early childhood settings), and that is a different question. In Spain, mathematics curriculum is minimalist, as in other countries, and mathematical contents of teaching follow traditions (in absence of clear curricular guide) like using Cuisenaire rods, Dienes logic blocks, and so on.  

This is an important question, well-worth considering, especially for parents and teachers.

There is a wealth of material on elementary school mathematics on RG. See, e.g.:

https://www.researchgate.net/profile/Dorit_Patkin/publication/269410487_The_contribution_of_in-service_training_programs_to_the_professional_development_of_mathematics_teachers/links/5502005e0cf2d60c0e6297ce.pdf

https://www.researchgate.net/profile/Anna_Sierpinska/publication/237304806_Research_in_Mathematics_Education_Through_a_Keyhole/links/0c96052c49dfb15031000000.pdf

https://www.researchgate.net/profile/Michael_Mhlolo/publication/281585954_Is_Rote_Learning_of_Number_Concepts_'Inherently_Rotten'_or_Is_It_Just_a_Blame_and_Shame_Game_that_Vitiates_Principles_of_Natural_Progression/links/55eebbbe08ae0af8ee1a2da6.pdf

https://www.researchgate.net/profile/Danielle_Harlow2/publication/230256856_Structures_and_improvisation_for_inquirybased_science_instruction_A_teacher's_adaptation_of_a_model_of_magnetism_activity/links/547f3cd40cf22ca9d34f616e.pdf

Article The contribution of in-service training programs to the prof...

Article Research in Mathematics Education Through a Keyhole

Article Is Rote Learning of Number Concepts ‘Inherently Rotten’ or I...

Article Structures and Improvisation for Inquiry-Based Science Instr...

Dear Carlos,

thank you for sharing the information about the book. I found some interesting data in it which made me thinking about the question even more. By trying to simplify mathematics we actually do contradictions. For example, telling the children that a piece of paper has a rectangular 2D shape we are forgetting its thickness which makes it a 3D shape in reality. And it is a question of how much is it rectangular as well. Are the sides exactly equal in length or are the angles exactly right? No, they are not. Am I right when I think that we (or the teachers) are sometimes forgetting to tell children that mathematical objects are ideal, unexisting?

Dear James,

thank you very much for investing your time in sharing all these relevant recourses. I will now take my time in reading all of them :)

Best, Ana

In the United States under the Common Core State Standards for Mathematics, there is quite substantial mathematics content in the primary grades.

One novel aspect is the inclusion of a variety of types of additive word problems, together with the expectation that students will learn to use different strategies for solving such problems. This shift in standards reflect the influence of research such as that of the CGI group.

http://www.corestandards.org/Math/

http://www.heinemann.com/ChildrensMath/

"Stop teaching kids to add up — maths is more important" (Conrad Wolfram). Sometimes I also have an impression that the time invested in geometry, in contrast to arithmetic, is really vague.

Dear Ian,

thank you very much for sharing your opinion.

Despite all the standards, principles, curricula, lesson plans and other sorts of 'rules' existing in the USA and Germany and elsewhere, the question of how much do (should) we 'rule' (or shape) the learning of mathematics in the early childhood remains open.

Which differences do you see between contemporary mathematics, mathematics in school and research in mathematics education?