Nix the Tricks

Several weeks ago  a colleague asked if I had ever heard of the "butterfly method" for adding fractions with unlike denominators.  I said I hadn't, and she drew this:

She was pretty excited.  She explained how this made adding  fractions with unlike denominators much easier for her kids.   I mumbled something about trying to make sure that the kids had a good conceptual understanding of what was happening, and later looked it up on Nix the Tricks.  Sure enough, it was in there.
  If you haven't seen Nix the Tricks, you should take a look.  It's is a downloadable book that explains how these shortcuts circumvent conceptual understanding, and offers alternative, more meaningful ways to approach the concept.
Yes I am sympathetic, but I'm also sympathetic towards teachers using these shortcuts.  When teachers feel pressured to make sure their kids have "mastered" a skill (adding fractions with unlike denominators, for example) and have a limited amount of time in which to accomplish the task, this is what happens. Unfortunately there is not always time to let some of these skills, and the concepts that underpin them, unfold in meaningful, organic ways.
     Here's something my daughter (grade 8, pre-algebra) drew for me one night several months ago while she was doing a homework assignment that had to do with multiplying integers.  She explained that her teacher had shown her this in class.

It helped her complete the assignment, but she pretty much had no idea what was really going on.

   Nix the Tricks is a work in progress.  You are encouraged to comment on submissions and add your own.  Take a look and let them know what you think.