I think that the author makes a very interesting point in debating the pros and cons from both views of understanding, more precisely on understanding mathematics. I still think that relational understanding is a more valid goal for a mathematics teacher but apparently relational is not independent from instrumental understanding, as one is helped by the other. Relational understanding seems to promote the independence of students when it comes to showing their mathematical knowledge. On the other hand, instrumental understanding leads to too many rules, that once broken, e.g. making a mistake, it is hard to find the right way.
The idea of seeing math as a town is a new, yet sounded somehow familiar to me. Understanding mathematics relationally is like knowing the city so that one can go from one point to the other, and understanding mathematics instrumentally is like knowing a number of routes between a number of points.
An idea I got after reading this was to maybe ask my students to draw their own "mathematics" maps, to see how they can connect their knowledge. And why limit it to mathematics only? I would like to see how math connects within itself and also how all those concepts connect with other areas like art, science, sports, business...
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