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Abstract: "Middle school students lack efficient problem solving skills due to several factors. The writing process has been linked to the development of students' problem solving skills. The present study investigated the impact of the writing process on the mathema tical problem solving skills for middle grades students. N = 96 students participated in a six weeks, after school STEM program, and they were randomly assigned into two groups: one focused on the writing process with mathematical problem solving and the o ther on homework/high stakes test preparation with mathematical problem solving. In this quantitative study, the results provide evidence that the students from the writing process group were more likely to generate and apply better problem solving skills as compared to the control group. This study further contributes to the support and importance of integrating different subjects across the generalized learning realm"

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Abstract: "An innovative mathematics game shown to be effective for low-achieving mainstream students is tested in special education for learners with intellectual disabilities. The game relies on a graphical, intuitive representation for numbers and arithmetic operations to foster conceptual understanding and numbers sense, and provides a set of 2-player games to develop strategic thinking and reasoning skills. The game runs on computers and interactive white boards, and as an augmented reality application at a science centre. We compare its use in special education and mainstream education with respect to usage, performance levels and learning gain. The game has been used by teachers in special educations, with gains in mathematical understanding, strategic thinking and communication skills as effects."

This list of "myths" written with a strong political bias. Take it in with a grain of salt. I like that they include references and links for why they believe what they do. This article would be good fodder for discussion.

I keep writing, in bits and pieces, about the middle way between the two approaches. For example, there should be enough discovery for every kid to firmly believe they could derive anything, if needed. This "just enough" will vary from student to student, too. In "The Glass Bead Game" Joseph took years to trace one short (highly abstract) game transcript to its very roots. He wanted to see for himself if the game was real. He did not do it for every game, though. The idea of "flow channels" between two opposite bad states comes to mind: http://www.secondlanguagewriting.com/explorations/elements/flow.jpg

Experiments by University of Chicago psychologists with deaf people in Nicaragua who learned "homesigns" rather than formal sign language, could not count beyond three. In contrast, people learning sign language learn number words and number concept and understand large numbers. The findings ""suggest that language is an important player in number acquisition." The research is reported in PNAS: http://www.pnas.org/content/early/2011/01/31/1015975108.abstract