Thoughts on splitting topics
Many skills have two parts: mathematical problems and word problems. Word problems should always follow mathematical problems, because word problems are usually mathematical problems with extra steps. Not all skills will have mathematical problems and word problems. For example, age problems are by definition word problems. And on the other side, finding the prime factorization of an integer is strictly a mathematical problem.
A topic may also be split according to how the problem is solved. I consider concrete and visual to be a single topic, as I cannot see a useful distinction between using base ten blocks and drawing figures to represent them. I consider abstract method to be any algorithm where primarily numbers are being manipulated. This distinction is useful, because some students, especially those with learning disabilities, will be able to understand the concrete and visual methods, but not the abstract. Should we teach concrete and visual methods before abstract methods? Or is it better to challenge the student with the abstract method first, then return to concrete and visual methods only if necessary? I feel that much research could be done in this area. What topics should go concrete and visual, then abstract, and which should go vice-versa?