Clearly the kid doesn’t have a deep conceptual understanding of how to solve equations or simplify expressions. True, the kid probably learned some stuff proceduraly as opposed to conceptually. (Though, I can confirm, that in this classroom nobody ever said anything, like, “When you have an equation you need to add something to each side to isolate the x.” The balanced-scale model was used at first.)
There’s still two interesting, deeper questions, to consider. (Possibly more: bring it up in the comments.)
a) Would this kiddo always make this mistake, when presented with an expression to simplify?
b) If not, then what exactly is it about this problem that prompts the kid to employ a basic move from equations?
4 replies on “Solving the Expression”
I think the learner is leaning hard on something that might make sense to them: do the same thing to both sides. So they are finding sides even if they have to look hard! I’d be curious to ask them 4x – 4(x-2)= 6 (though that presents some other issues) or 4x – 3(x-2) = 5. I’d also like to see some situations where students make up expressions to describe relationships, though hard to distinguish that from functions.
John, I have a colleague that refuses (on principle) to distinguish expressions from functions. She would read 4x – 4(x+2) as a relation between a quantity x and another quantity, 4x – 4(x+2). Just because we don’t name that quantity doesn’t mean it isn’t there. Funny that in this case, the other quantity is just 8…
I wonder if the 0 is part of what caused this student to want the answer to be 8x + 8. It may look a lot less fishy to students to have an algebraic expression simplify to ax + b form than just 8. I can imagine a thought process that goes something like “whoah, the 4x and -4x will zero out. I don’t think I can zero out x’s in an expression. Wait, maybe I need to add the same thing to both terms?” One way to test that would be to give John’s expression: 4x – 3(x – 2) without the equality, first. We could also test without the subtraction, to see if the student would simplify 4x + 4(x+2) to 8x + 8 or (highly unlikely) just plain 8.
This mistake could fit the Pershan Model of Math Mistakes: when presented with an unusual or somehow harder feeling situation, student mathematicians revert to rules and procedures that don’t actually make sense.
I noticed that the student learned equations with a balance model. I wonder what models they have to fall back on when things get stressful in the non-equation simplification — maybe comparing a graph or a table of values? Testing some values for x in the simplified and non-simplified versions? How can we help students recognize “something’s fishy” and test their hypothesis?
Does the student know the similarities and differences between expressions & equations?
The thing that I’m not sure about is how much of this is a visual cue (i.e. it looks like it does when you’re solving an equation) and how much of this is a conceptual connection between manipulating expressions and solving equations.