Last Saturday I went to the Museum of Mathematics, across from Madison Square Park with Casey. The museum of math appears to be primarily a playground where parents bring their children to osmose math from pretty geometric shapes. It was a bit expensive, small, and broken. But it was also pretty fun.
Exhibits were generally interactive activities you could enjoy without thinking about much of anything besides “this is neat,” with explanations and history available at touchscreen kiosks. This idea is great – the museumgoer can be entertained for an hour while the seeds of neat ideas are planted, or can actively seek out an explanation if they’re curious.
The museum seemed very focused on geometric and topological topics in math  but it certainly did a terrific job of divorcing math from arithmetic, algebra and calculus, an important and worthwhile task.
I couldn’t help thinking of how much more I liked the MathWorkssponsored math exhibit at the Boston Science museum  it presented neat ideas with more context, but was less interactive. But I think I learned less from it  I don’t remember any specific new ideas I left that exhibit with, beside feeling like I should do more math.
It also occurred to me that the annual ITP show might be neater too, for its interdisciplinary “everything is awesome” approach, but the average educational quality for me of each exhibit was much higher at the Math Museum.
Unfortunately there was a lot of hamfisted teaching going on: I observed and directly experienced museum staff (probably mostly volunteers, all wellmeaning) spouting lectures as though they were reading an interpretive sign. The advantage of a real person over a sign that you can truly engage and teach instead of lecture, and let the museumgoer drive the experience  but perhaps there’s no time for us to develop the sorts of relationship required for that kind of learning. The element of “bring your kids here so they’ll grow up to be smart,” which I’m sure permeates most museums, seemed particularly strong at MoMath. This made it particularly disappointing that while the museum essentially has free math tutors, I felt they were ineffective.
Some neat ideas from the museum:

There was a fun visualization of the relations of the relative pitches in chords: if we take a chord to be three pitches, we can change each pitch by one half step in either direction to create 6 chords to which this chord is connected. If we take a major chord like C E G, it’s one half step perturbation away from C minor (C E♭ G), C augmented (C E G♯), and 4 others, thereby forming a graph. If we allow relative transposition, we now have a smaller graph that looks like this:
where the numbers represent the intervals. A major chord is in the middle; 4 half steps, then three more half steps. It was a new idea to me, so I found it cool, though the sound design of the interactive exhibit didn’t get the idea across particularly well.

Catenary curves: they’re different from parabolas, but look similar. The seem more natural geometrically  if you don’t go straight to mechanics, your kid should learn these first. (But who am I kidding, of course if I were teaching small children I would teach mechanics first, regardless of what developmental psychology has to say about some ideas being more natural than others.) As an undergraduate physics major, I’m embarrassed to admit this I didn’t remember what the word meant at all from when I last learned about it.

Fractal Tree  a really intuitive description of a fractal: take a picture of yourself, overlay it on your arms. Overlay each of those overlays’ arms with yourself again, etc.