Please, never mind my English it&#x27;s awkward sometimes.<p>Math could be taught qualitatively.<p>First, one can give their students ten or so hairy problems where the goal is to write down what the problem is asking in the most explicit and gory details.<p>Then one can present another batch of hairy looking problems and ask the students what definitions&#x2F;lemmas&#x2F;theorems&#x2F;corollaries look appropriate to apply to the problems and why?<p>Another batch of convoluted problems can be offered where the goal is to simply formulate one&#x27;s questions regarding the problems in the most precise way one can. Asking the right, penetrating questions solves more than half the problem.<p>Ask the students to rewrite all the definitions&#x2F;lemmas&#x2F;theorems&#x2F;corollaries presented in class in their own words.<p>Present students with a text full of unproven statements (most grad level math texts). The exercise here is to identify as many of these statements as possible. The bravest students are encouraged to supply proofs.<p>Give students a bunch of required problems peppered with generalizations and extensions of required concepts(typically reserved for big boy students) and ask them why they are required to solve these problems. What exactly to be gained out of solving every specific problem?<p>After all that students can tackle the required set of problems written up for them by the Department Of Serious Business. At this point math will come naked and explicit in front of one&#x27;s eyes.<p>None of this is going to happen in classrooms, though. Students will have to learn these things on their own as per the traditions.