Research on how to teach math to children - what schools of thought are there about teaching math? Firstly, I'm not sure if this is the best place to post this question because there isn't quite one answer, but in my opinion what proves to be the "best" answer would be a well supported one with lots of information on different approaches. 
I've been tasked with helping to redesign a math curriculum for an enrichment program and I have very few restrictions on what the curriculum should include or how it should flow. The curriculum is meant to be an enrichment curriculum (supplementary to the existing one here in Ontario). 
I wanted to find some resources on proven and effective ways to design and teach math so that students have a deeper, more rigorous and more intuitive understanding of math - an approach that prepares them to tackle math abstractly and ultimately prepares them for higher level math. The curriculum starts at grade 4 and continues on to grade 11.
 A: I know few facts about this, thus I talk from the ignorance with the intention to provide you information, and references (all in spanish) that I know that have a high quality.
I know that the prestigious analyst Miguel de Guzmán wrote literature about it, and believe about two main subjects (but as I've said, I am not sure about my knowledges and maybe there are more):


*

*strategies for problem solving

*visualization in mathematics
Also he was author of books for high school (I've studied books written by him, and other co-authors). Miguel de Guzmán, see thisone Wikipedia and thatone, founded the ESTALMAT, and there is an university chair in the Universidad Complutense de Madrid named after him. I say the Cátedra UCM Miguel de Guzmán. Here for example in the left link Publicaciones,  there is bibliography  that you can to find and read if you know a spanish friend in your city, and have access to such journals. I know that these publications must be of interest. On the other hand you've also the link Educación from the web page of La GACETA de la RSME. Are six pages (Páginas in spanish)  containing bibliography about articles on education. 
About what is, from an academic point of view, this school of thought I can not explain it, since I am out of this project that began with Miguel de Guzmán and all spanish professors that now work about it. My advice is that if you are working for an official organism, is that maybe you can contact with the people working in Spain. Good luck.

RSME means Real Sociedad Matemática Española
A: There are two main competing approaches, which can be termed respectively internal and external.
The internal approach holds that mathematics has inner coherence and compelling force that need to be taught to the students to give them an idea of the true nature of mathematics in its unadulterated form.  Since set theory is commonly taken to be the foundation of mathematics, in this approach students are likely to see some concepts of set-theoretic type rather early on.  This was the philosophy born out of a conversation between Piaget and Dieudonne and eventually gave us the New Math (sometimes called Modern Mathematics).
The external approach holds that mathematical concepts can only be taught effectively through games of association of real-life phenomena that students are already familiar with, and encourages the use of applications to motivate mathematical concepts. Proponents of this approach agree that mathematics has internal coherence but argue that such coherence (as well as advanced foundational issues) can only be appreciated by mature mathematicians after a period of training involving more hands-on techniques as above.  A major proponent of this approach is Hans Freudenthal.
A fine study of this issue is this article by Christopher J. Philips:

Phillips, Christopher J. In accordance with a "more majestic order": the new math and the nature of mathematics at midcentury. Isis  105  (2014),  no. 3, 540–563.

See also the questions under this tag.
A: I would design a curriculum where you teach students to love math and realize its potential. Show them early on where it hides in nature and how it describes the world around us. Don't make it seem like the chore and burden most schools do
For some useful techniques, there is an article on how to improve teaching methods so that kids learn better and it sticks with them. Here
