In my early years of teaching, a bunch of us used to joke about how the way we knew we were improving as teachers was that, even if we weren't totally sure students had learned what we intended, we at least managed not to un-teach them things. For example, the first time I taught a unit on integers, students felt just fine about basic subtraction in the beginning; after I "taught" them integers, however, if I asked them what 8 - 3 was, they might just look at me and cry. (Only half joking.)
I'm sure I'm not alone in that experience. I've met many other teachers who have taught the integers unit and done all the "right" things, focused on conceptual understanding and number lines and zero pairs and elevators and temperature and manipulatives until they were blue in the face and still, at the end, half the students are not quite sure what to do with -3 - (-8). (Negative 24? Let's go with negative 24.)
One of the classic explanations for why kids need to understand operations with negative numbers is the "real life" explanation--How will they be able to pay their bills, figure out the temperature outside, etc. etc. if they can't work with integers?? But the truth is that the vast majority of kids who fail the integer unit can still correctly answer questions like, "If you owe six people $20, how far in debt are you?" or "If it's -12 outside and you don't want to go out until it's -5, how many degrees does it have to warm up?" or "If you're $100 in debt and you get a windfall of $500, how much money do you have?", even if they can't represent the situation with an equation.
As a math teacher, for the kids that fail the integer unit, I am much, much more concerned about what will happen to them when they start doing formal algebra. For example, if you've taught Algebra 1 or pre-Algebra, you probably lost count in the first week of the number of times you saw errors like this:
Solve for x:
4 - x = 5
x = 5 - 4
x = 1
Or had conversations like this one:
Solve for x:
-x = 7
Student: Well, it can't be 7, because then you get -7 = 7 which is wrong. So it must be -7. So let me put in -7 for x.... But now I still get -7 = 7. It's still wrong!
Teacher: Are you sure you substituted correctly? If we're going to say that x equals -7, shouldn't you leave the first minus sign?
Student: No, the minus sign is already there.
Teacher: Since we are replacing x with -7, shouldn’t we write –(-7)?
Student: You can’t, though, because then it's negative negative 7, and there's no such number.
And, in a way, you're not surprised; they did, after all, fail the integers unit. Working with negative numbers are just plain hard, and now we've invited variables to the party to boot. What did we expect, really?
It's tempting to just go over the rules for algebraic manipulations again, slower and louder, or teach them some kind of mnemonic or other memory trick. Let's be honest, though; if that was going to work, it would have worked already.
Instead, what if we focused on helping students make sense of what equations like this really mean? Not just, "How can I manipulate the terms correctly?", but what does the structure of the the terms tell me? There is evidence that this approach can be significantly more effective than just re-teaching the rules. Helping students understand the meaning behind the symbols, though, sometimes requires that we as teachers do a bit of deeper thinking, ourselves, about what they mean.
As a student teacher, one of the first Algebra A (first semester Algebra 1) lessons I watched my Cooperating Teacher teach was called "The Three Meanings of Minus." Now, I had been a fairly enthusiastic math major and my algebra skills were pretty darn solid, but this was the first I'd heard of "minus" having multiple meanings, so I was intrigued.
To grossly summarize, she had students look at three different expressions and discuss what they thought the minus sign was really telling us in each one.
-7 10 - x -(x + 3)
These were just ordinary East Bay kids in an ordinary 9th grade math class at an ordinary school, so I half-expected they would be too confused to respond and my CT would then explain whatever she had in mind. But after a few minutes of table talk, it turned out that the students had some interesting thoughts to share. They immediately shared ideas like, "In the first one it's a negative, but in the second it's subtraction/take away." The third was a little harder for them to describe, but after some conversation my CT clarified that because this minus applied to everything in the parentheses, it was telling us to take the opposite of what was inside.
At that point in the year, the students had been working with Algebra Lab Gear (similar to Algebra Tiles) for several weeks, so together the class discussed how we might show each expression and each type of minus using the Lab Gear. The students then worked in groups to look at a variety of expressions involving the minus sign, characterize them as to whether it was showing negative, subtraction, or taking the opposite, and how to represent each one using Lab Gear.
In subsequent years, I did similar activities with my Algebra 1 students using Algebra Tiles, and added an extension lesson where we discussed how we could build equivalent expressions that used different types of minus.
"Negative 7" "The opposite of 7"
-(x + 3) -x + (-3)
"The opposite of the sum of x and three." "The opposite of x, added to negative 3."
10 - x (x sitting on top of 10) 10 + -(x)
"Subtract x from 10." "The sum of ten and the opposite of x"
Not only did this help my students translate symbols into meaningful ideas, but paradoxically, distinguishing between "subtract x from 10" and "add the opposite of x to 10" helped them more clearly articulate why these two expressions--and these two types of minus--were mathematically equivalent.
Now, some people will argue that this is all philosophizing, that a negative sign is a negative sign is a negative sign and in the final analysis the rules for symbolic manipulation are all the same, so why make it complicated?
Well, it turns out that the usefulness of distinguishing between these different types of minus is supported by research. When math education researchers study how kids understand integers, they distinguish between these three meanings as well (though instead of negative, subtract, and opposite, they refer to them as the "unary, binary, and symmetric functions" of the minus sign). Research supports the idea that kids make minus sign errors in algebra not because "negatives are just hard," but because they usually do not have a well-developed sense of the unary and symmetric functions of the symbol. (Ie., they understand the idea of subtraction, but not "negative" or "take the opposite.")
The fact is that remembering rules for symbolic manipulation without any meaning to hang them on is already complicated for students, as evidenced by the piles upon piles of negative sign errors we see amassing in our classrooms on a daily basis. Understanding the meaning of the minus sign in different contexts provided my students with deeper conceptual understanding; it gave them a way to think about an expression in terms of what it meant ('x - 3' means 'subtract three from x', while 'x + (-3)' means 'add negative three to whatever x is') rather than just trying to remember the rules for moving things around. Yes, symbolically, the expressions are equivalent, but having a way to express the meaning of a particular expression improved my students' intuition about those rules and why they made sense. It also improved their ability to think both structurally ( (x - 3) is a term in and of itself that I can move around without simplifying) as well as flexibly (if it's more convenient to rewrite (x - 3) as x + (-3), I can do that and it's fine).
Can you identify the different types of minus in play in each of these expressions?
4 - x 4 - (x + 2)
-x + 42 - (x + 2)
-13y - (81y^2 - 2)