I see the “subtract 5” in students who don’t understand the difference between “negative” five and “minus” five. I don’t quite understand the notation (what does “pick 8” mean?) so I can’t really go further with it.

Student understands going backwards but student does not know inverse relationships of operations. Been seeing this often this week with my freshman I’m teaching the same concept almost the same way.

louise

The student understands opposites, but does not understand working from the bottom up. When we use the “undo” method we write the opposite in column two, but then work upwards. We don’t try to flip the list.

I agree with Sadie Estrella, but note that the student stroke out the correct answers, especially the last one for which “mul 4” is strange. What I find interesting is that we never learn to solve equations in this manner in France. I think it’s a good idea, and will try it this year. It should help understand the concept of f^{-1}.

I agree with Louise. The student switched the add to subtract, mult to div. and div. to mult., but did the “opposite operation” in the same order as the original. They DID, however, use the numbers in the reverse order.

They don’t really understand the concept of “undoing” in order to solve, just that “addition changes to subtraction, etc.”

Remediation: Definitely back up and work through some one step then two step solving situations with a variety of operational combinations in a variety of “orders.” With two steps use “cover-up” method to help them see what to “undo” first. Once the student has truly mastered these, he/she should be ready to move on to more complex equations.

Mike Mathews

I’d like to know more about this method. I’ve never seen it taught this way before. I think I really like it but wonder how a problem like “60 – a = -8” would work. Can this method work if there is a variable on both sides?

I’d say “no” and “no” to using backtracking with those sorts of equations. Of course, it’s possible, but at that point we’re stretching the model beyond its usefulness.

But that’s alright. Math is the continuous activity of creating and breaking metaphors and models.