I've spent a lot of time thinking about improving student engagement.  It seems that many of the solutions are simple.  Here's one that worked well last week.

A student asks for help on a homework problem.  You all know the drill.  You do the problem on the board.  Five people pay attention.  You figure that if one person asked, probably the whole class should be following along.  But most aren't.  

I simply decided to take their lack of interest off the table.  I asked the whole class to read the problem.  Then I randomly picked a student to start the problem.

[sidebar:  I have been highlighting my seating chart.  Every time someone participates, whether voluntarily or by my choice, I highlight their name on my seating chart.  When everyone has a yellow highlight, I switch to green.  It gets a bit messy and in time it looks like it's filled with psychedelic asterisks.  It allows me to call choose carefully who I call on, while ensuring that no one can be ignored.]

We scaffold through the problem with about 8 different students being called on to contribute to the solution.  The entire class is engaged and everyone has seen a correct solution.

Now as to why most of class doesn't care about a solution to a challenging homework problem.  Well, if I had all the answers, the comments would be closed.