In the video above, the student describes adding 4 to both sides as “taking the 4 away from one side and putting it on the other.” This is difference then “balance reasoning” or using the subtraction property of equations.
The problem is that this “Give and Take” model for what we can do with equations runs into problems about 50% of the time. It only works when the signs work out, and when it doesn’t it leads to errors:
In the above, the student (I think) took the 3x away from one side of the equation and put it on the left side of the equation. This is perfectly in line with how the student in the video describes his move.
Where does this “Give and Take” model come from? The video above gives a clue, since the student offers the explanation to make sense of adding 4 to both sides. The story that this student developed for himself was that we’re getting rid of the 4 and putting it on the other side. This is naturally appealing, since the right side of the equation ends up with a 0 and the left side ends up with 4 more.
The “give and take” model is a way of seeing equation-moves. It helps make sense of why we would add quantities to both sides of the equation, but it makes wrong predictions about what would work for situations that don’t call for adding something to both sides, like 2x + 3 = 3x – 4.
What can we do about the give/take model? Can it be avoided?
Maybe not. Students need some way to make sense of what we’re doing in 2x + 6 = x – 4. Balance reasoning is no help with the adding 4 to both sides part of this. (Why? Because we almost never have to add stuff to both sides in balance problems, and because negative quantities don’t make a ton of sense in the balance context.) This equation isn’t primed for backtracking, since there are variables on both sides and backtracking doesn’t handle these situations well. Perhaps we want to tell kids to first deal with the variables, using balance reasoning to get x + 6 = -4 and then using backtracking, covering up or guess and check to handle this equation. Fair enough, but this is fragile. Students should be able to make sense of the entirely appropriate move of adding 4 to both sides for 2x + 6 = x – 4 somehow. For a lot of kids, the best way to think of this case is taking stuff away from one side and slapping it on to the other.
It seems unlikely that there’s some other powerful context and type of thinking that we’ve missed. At this point, the devil is in the details. I don’t know if there’s another way to make sense of why we would add 4 to both sides in this case other than the properties of equality, or as an abstraction from balance thinking (i.e. “the mobile will always stay balanced if we add 4 to both sides.”)
I guess we just need to be careful about giving kids too many of these types of equations without justifying the move in terms of balance or equality properties.