Maths notation is needlessly complex [video]
This presentation misses a more fundamental point.  It's not about the <i>notation</i> at all, it's about understanding that EXPT, SQRT and LOG are <i>functions</i> with a particular relationship to each other.  That relationship can be expressed using a two-dimensional spatial notation, but that doesn't really help you understand the concept at all because there are a lot of <i>different</i> relationships that are naturally described by putting three things in a triangle.<p>What you <i>really</i> want students to understand is that expt, log, and nth-root are <i>functions</i> that are related in the following way:<p><pre><code>  expt(b, n) = x
  log-x-base-b(x, b) = n
  nth-root(x, n) = b
</code></pre>
It's really that simple.  No fancy notation needed.  In fact, fancy notation <i>always</i> gets in the way of understanding because people naturally think in words, not in spatial relationships.  Mathematical notation was invented not because it aids understanding, but because when you're writing math with pen and ink it's faster and uses less paper to use Greek letters and spatial relationships than full words.  But when you're on a computer, it's easier to write out the names of functions, and that is actually a better impedance match to people's natural mental processes, which involve <i>language</i>.<p>[UPDATE] I would like to revise this: not everyone thinks in language.  But everyone <i>communicates</i> in language.  For <i>communicating</i> mathematical concepts, language is the best tool we have.  There's a reason that the symbology in math papers is invariably wrapped in natural language.  It's the same reason that the video has a narration.  It wouldn't make any sense otherwise.