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.

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.

1. Great post. I couldn’t have explained the problem better. I think this is something we all deal with, and have the same inner battle with the drill and kill method. The added pressure of the urgency to correct the misunderstanding as soon as possible because they are in an upper level math class (like Algebra II or Pre-Calculus) where this type of mistake only makes them feel more self-conscious about math in front of their peers, also doesn’t help. Ultimately, I wonder if doing some drill and kill and showing them their procedural mistakes will pick away at their preexisting schema. What I’ve found is that it’s important to first make them understand they’ve developed a misunderstanding. Without doing that, it’s hard to convince them they need to re-learn or learn the technique again. The light bulb moment of “wow, I’ve been doing it wrong all these years” is huge, especially when they realize all of the higher level work they’ve perhaps made mistakes on is only a result of this one problem, which they admit is pretty simple to fix. In Pre-Calculus, that does a lot to convince them they’re not “dumb at math;” they are actually good with the higher order thinking with advanced math, they simply just have held on to a misunderstanding of the basics for a bit too long.

• Thanks for the comment! I think what you’ve added is right on as well – having that clear conversation where the student realizes they have a misconception is key. I fully agree that the light-bulb moment where everything starts to fall into place is huge and really uplifting for the student.

Thinking about the way things played out in my classroom, I had this conversation with a few students where I made them realize they had this integer operation misconception. But I think the more general conversation I had with most of my classes wasn’t so much “Look – you need to realize that you have a misconception and I’m going to help you fix it”, but instead “Look – you all know you have a misconception and I’m not going to let it be okay anymore. I’m going to make you fix it”. I think the conversations I had were more about a shift in expectations rather than a shift in student knowledge. If that makes sense…

Thanks again for the comment – making me reflect even more on how these things play out

2. Everything is so context-sensitive that I am unsure I can give one “right” way of fixing bugs, other than if it is better to explain what’s wrong exactly with what they think is true rather than re-explaining a whole procedure from scratch.

Integers are a perfect case study for this.

The most common error my students still have at this point with my Algebra I (we’ve been doing integers since the beginning of the semester) is adding two negatives and getting a positive. The raw rules (which I did show, given some students do ok with them) don’t seem to help with this, they’re just too easy to get mixed up with negative times negative.

The number line hasn’t done as much good on this as I’d like, either, but I think I’ve traced this to simply: the students don’t stop to visualize. While there exist people who are incapable of visualization (this personal essay is good) they’re fairly rare. However, the students will still plow by without even trying to stop and visualize. So I might recommend getting students to close their eyes, imagine they are standing on the number line, looking down, taking steps matching what operation they’re doing, etc. This resembles mnemotechnic methods of “memory palaces” (used commonly by folks in memorization compeitions) where items being remembered are placed in literal places in the mind.

• Jason,

I’m glad you managed to find this post and share your comments. As this last week has panned out and I’ve continued to tutor students on integer operations, I’ve taken your advice and encouraged them to visualize themselves at the number line. I started to do that, I found myself explaining the whole concept of a number line to some students, which has been an interesting conversation in itself. I think I’ve always been assuming that students at least know what the number line looks like but have been confused on how to translate the operations into movement along the number line – what I’m discovering now, as I’ve tried to get them to visualize, is that many don’t even know what the number line really means or what it represents in terms of numbers and their relationships. I’ve tried to remember if there was ever a time when the concept of a ‘number line’ didn’t make sense to me, but I haven’t been able to.

Anyway – the emphasis on visualization has helped a lot. I think if I had it to do over again in the full-group context, I would’ve spent the whole day just reteaching the number line from a conceptual perspective – their are some really interesting conversations you can have if you start there. My favorite is asking students to consider the concept of a number in the first place – “What is the number 3? Can you point at something in the real world and say ‘that – right there – is the number 3’? Or are you pointing at 3 objects as opposed to the actual number 3? And is there a difference?”. So yah… that’s a conversation I enjoy having.

I guess I’ll keep you updated if I make any more interesting breakthroughs in the ‘reteach integer operations’ field.