How Do You ‘Fix’ Incorrect Procedures?
Something I’ve been thinking about recently: what is the best way to ‘fix’ a procedural mistake that a student has memorized? In my own musings, the question tends to focus on one of two possible actions: either drill & skill the correct procedure into the student then go back and tie it to the conceptual (if possible), or start with the conceptual and rebuild their knowledge from scratch. What’s the best way to do it?
Here’s the very concrete place where I find this issue: adding and subtracting integers. What I’m finding is that my students have some scrambled intuition of the number line and the knowledge that there exists a rule/procedure, but they aren’t sure on either one of them, so they mess both up. What follows are real conversations I’ve had with students:
“So -5 + -3 = 8”
“Why positive 8? Why not negative 8?”
“Because a negative and a negative is a positive”
(Student Mistake: Confusing the rules for addition/subtraction of integers with multiplication/division of integers)
“So -3 – (-4) = -7”
“Why negative 7?”
“Because you start at -3 and you have a negative 4, so you move 4 in the negative direction”
(Student Mistake: Not understanding how subtraction changes the rules for moving along the number line)
“Subtraction means I switch the signs. So -4 – 5 is the same as 4 + -5, which is -1. So -4 – 5 = -1”
(Student Mistake: Memorized a rule but is applying it incorrectly)
The best way I can describe my observations and conversations is that my students have memorized a bunch of rules and procedures without knowing when to apply them or why they’re applying them in the first place. Their brain is a mess of rules and half-formed ideas that they guess at applying because they’ve been trained to blindly apply procedures without really knowing what’s going on. So… what’s the best way to fix this?
What I Tried: Working one on one with some of these students, I started by going back to the conceptual – revisiting the number line. There is a way to explain addition and subtraction of integers on the number line that I like – the first number describes where you start, the operation tells you which direction you face, and the second number tells you how far to walk in that direction. So “-4 – 5” means you start at -4, subtraction means you face the negative numbers, then the positive 5 means you take 5 steps in that direction – you end up at -9. “-3 – (-2)” means you start at -3, subtraction means you face the negative numbers, then the negative 2 means you walk backwards two spaces – you end up at -1. I like it – it works for me. However, when I tried to explain it to my students… it totally confused them. To use some Piaget vocabulary to describe what I think was happening, they had an already-formed schema labeled ‘adding and subtracting integers’ and I was creating a new ‘adding and subtracting integers’, and the two schemas refused to assimilate in their heads. They kept trying to unify the rules they had memorized with the number line procedure I was giving them and it wasn’t working. It doesn’t help that muddled up in the ‘adding and subtracting integers’ schema are the rules for multiplying and dividing integers, which really should be their own schema.
So, what next? I tried to just obliterate the schema altogether: here are the rules – subtraction means adding the opposite (SMATO!), then there are 3 cases: both positive, both negative, one positive and one negative. Memorize these and drill & kill. Keep doing it until you’re only getting right answers. I have mixed feelings about reteaching this way because I’m committing the same crime that their previous teachers committed: memorize these rules that you don’t fully understand and move on. However, in thinking about this particular concept – adding and subtracting integers – and discussing it with other people (especially non math-savvy folk), I started to realize that the way most of us add/subtract integers is with these rules alone – no ties to the conceptual. Even I, when I see an integer operation problem, will default to my ‘subtract the two, keep the sign of the bigger number’ rule – is this because I understand the conceptual and therefore feel comfortable taking the shortcut? Or did I memorize that rule first, then later made the connection to the conceptual? Is it okay that some people have this rule memorized and no real connection to the number line but can still function well in the real world? And if that is okay, does that mean it’s okay for me to reteach it this way to my students? I’ve had something of an ethical battle going on in my head as I’ve been reteaching this. If there are any middle-school math or freshman algebra teachers out there, I’m curious (1) how much of teaching integer operations is conceptual versus procedural, and (2) how long it takes you to teach this topic.
Let’s Complicate Things Even More: I’m teaching Geometry this year, which means integer operations don’t exactly show up on my pacing calendar. So if I need to spend an extra day reteaching these subjects to all of my classes, how much is a reasonable amount of time to spend reteaching? And how does teaching full-group versus 1-on-1 effect how I can reteach it? If I’m cramped for time, then the schema-obliteration method saves me the most time, whereas the conceptual reteaching means I’m spending a whole day – maybe two – on a topic that isn’t technically a geometry concept. What I ended up doing, for the record, is I started by reteaching at the conceptual level to a few students who came in for help (to mixed results). But when I realized that the majority of my students didn’t know how to do integer operations, I spent half a day giving them the rules for integer operations and doing drill & skill – I was afraid that if I tried to rebuild from a conceptual level, it would (1) take more time than I wanted to take out of full-group instruction, and (2) it would confuse some students even more as they tried to assimilate their previous knowledge with the number line and I wouldn’t be by their side to help.
I’m getting to the point with some students where I really need to sit down and reteach solving linear equations and I’m again debating the best way to do it. They have rules memorized in their head but don’t know how or when to apply them (“I have -7x + 3 – 4x = 12. The opposite of -7x is +7x, so I add 7x and I get 3 + 3x = 12”). Is it worth it to start from scratch with linear equations, or will that confuse them even more as they try and assimilate everything they already know with the new things I’ll be telling them? Or should I commit a teaching sin and teach purely procedural in the hopes that it obliterates all the rules and procedures they already have memorized, then find time to go back and connect it to something procedural?
If Jason Dyer is out there and happens upon this post, I’d love to know (1) if all this makes sense, and (2) your opinion
EDIT: This seems to be a good week to talk about ways of finding student’s misconceptions. Jason Dyer did indeed manage to find this post and share his insights in the comments, which is both fascinating and helpful. If you haven’t already, you should find your way to his blog and see his ongoing investigation into the cognitive science behind student’s wrong answers. Also this week, Sam Shah talks about a question on his most recent test that works wonders at getting right to the heart of student’s misconceptions – I may steal this technique and using it as a 2-part exit ticket and see what kind of formative assessment I get. He’s got some great questions at the end of the post about trying to get to the heart of what’s going on in our student’s heads.