Building understanding of the equals sign

I recently read a paper called the Importance of Equal Sign Understanding in Middle Grades, where it is argued that most middle-schoolers view the equals sign as a kind of signifier for "That's where the answer goes" (3+4=?), rather than understanding the deeper, relational meaning (3+?=3+29+10 or y=2x+1), and that this significantly hinders their understanding of simple algebra problems.


I still see this misunderstanding whilst currently teaching college-age students. This is worrying.

Ways that understanding can be taught:

• Dave Orritt has created a lesson with resources looking at questions of the form 3 + __ = 2 + 9, etc. of varying difficulty.     I would have liked a little progression such as:
    5 + 21 = 13 + __
    5 + 21 + 154 = 13 + __ + 154
  
  
• Nrich has a nice problem that involves changing equivalences, so that even before you get on to generalising, you can look at how the equals sign still applies as you move around the grid.
  
  
• Here are two (first and second) similar web-apps dealing with balancing equations. They are made to be used to help learn solving equations, but you could also just let students try things out to see what happens to the equation.
  
  
  Explore linear graphs starting with finding values of x and y that make the equation x + y = 5 (or any other) true, then plotting them all as points on a graph at the front of the class. (What happens to x as y increases? etc)
  
  
• Building up equations. Can't remember where I found this, but its the best version of its like that I've seen:
  
  
  
  Which could then be followed up with students building their own similar webs.
  
  
  
• Practicing simple, worded and otherwise, formulae; including substitution of different variables and rearranging. You could also ask if different forms (e.g. F=ma, F/a=m, Fm=a) are always, sometimes or never the same and to justify their answers. 


• Equation chains - e.g. 3 + 2 = 9/? + 2 = (2+?)/3 + 2 = 25/?


• Hands-on equations or something similar

 

How would you do it?

These are some initial suggestions but I'm wondering if anyone has any other ideas.