Here are some areas of interest and questions that arise in mathematics teaching in the elementary grades:
Curriculum -- What content should be taught? When should it be taught? What are the objectives to teaching a specific idea or group of ideas? How should the content be arranged within a particular year and across the elementary grades? When should algebraic ideas be introduced?
Learning -- How important is discovery learning? How can discovery learning best be utilized in the classroom? What is the most effective use of manipulatives? Which fosters better retention - teaching for understanding or practical applications?
General -- How effective is homework? How much time should be devoted to practicing skills? How should classes be organized? How can we implement differentiated instruction especially as related to gifted or remedial students? Should we insist that elementary teachers be math specialists? Should we integrate computer assisted instruction, and if so what are the long term ramifications and limitations? How should gender inequality be addressed at the elementary level?