What does = mean?

There's a lot of work out there on how american kids don't know what = means.  Nicole McNeil, Martha Alibali, and Bethany Rittle-Johnson have done some really remarkable work on this phenomenon recently.

It boils down to all of the arithmetic problems that kids see in an american math class have the form:

• 3 + 4 = ____
• 4 + 29 = ____
• 54*31 = ____
• 81/ 9 = ____
• 1/4 * 2/3 = ____
  

Are you starting to see a pattern?  There's a computation to be performed then an equals sign, then a blank to be filled in.  If this is the only type of math problem that you see then clearly = means 'do the calculation'

From this point of view, the statement 4+5 = 3+6 makes no sense.  There's no computation to be performed.  So it's just confusing.  If you believe that equals means 'write the answer' and there's no answer to write, then that's going to create some cognitive dissonance.  And when students see a problem like 4+5 = 2+ ___, they can't even understand the problem. 

The thing is, this misconception about what = means doesn't go away by high school, or college.  There are even some studies that show that practice like the items in the bulleted list actually make students even worse at mathematics, because they reinforce that misconception about equality.  Students don't understand that equals means 'the two sides are the same' because all of their experience says that equals means 'put the answer'

This is a well studied problem, but it's closely related to something I observed yesterday in a student that came to my office hours.  The class is a 200-level stats class for science and engineering majors.  This guy came in for help studying for today's exam.  He kept asking, so What do I do with a Binomial distribution? what do I do with a Poisson distribution?    I'm not sure I can explain the level to which these questions don't make sense.  It's kinda like asking, What do I do with an elephant?  You could feed the elephant, you could ride the elephant, you could train the elephant to do tricks, you could weight the elephant.  There's no computation that you do on an elephant, it just is.  You can use the elephant to accomplish different tasks, but there's no one thing that an elephant is for.