As a student in a fairly good university(Umass), Math tends to be taught exactly how you describe it. "Heres the FT. This is where you use it. This is how you d

As a student in a fairly good university(Umass), Math tends to be taught exactly how you describe it. "Heres the FT. This is where you use it. This is how you do it". Most students will be able to do the FT, and most students can actually find the correct answer. However, because teachers and books tend to formulate very artificial situations where the a specific tool students often are used to looking for very specific patterns inside problems to decide what tool they will use for the job. When a problem falls outside the usual patterns, students will have a hard time identifying the tool they should use. I see this all the time when I help someone with a programming assignment. They may understand the problem, they may understand each individual solution if you explain it to them. However, they usually do not know where they should start. They may understand what a hash table is, and they may even know how to implement one. However if a problem does not fall into one of the patterns they are used to for a "hash table problem" they may not realize right away that they could use such a solution. Kids need to be taught how to break down a problem into its elements, and then realize which elements can be solved most effectively by what tool. This can also be learned through experience. Program enough and you will eventually start to break down problems yourself. But some kids don't seem to take initiative and work on themselves outside school. Which also is the reason why some kids from Umass CS are working at Facebook, Google, Microsoft, and others serve me coffee.