In summer school, I’m teaching what used to be known as pre-algebra and happily, my colleagues had a whole bunch of worksheets that I got on a data stick. Very nice, and the curriculum was very good, leaving me time to tweak but not spend all my time inventing.
It’s not like the curriculum was a surprise: integer operations and fractions played a big part.
Of course, when we math teachers say “integer operations”, we mean “operations with negative integers” because while we don’t really care all that much if they’ve memorized their plus nines and times sevens (sorry, Tom!), kids that don’t fundamentally understand the process of addition are usually un-included by high school.
But negative numbers are one of those “Christ, they’ll never get it” topics. I don’t reliably have an entire class of kids who answer 9-11 with -2 until pre-calculus. I’m not kidding. They say 2, of course. But not negative 2. And if you give them -3-9, they will decide it’s 12 or -6 or, god forbid, 6. But not -12. They’re actually not terrible at subtracting negatives, provided that it’s subtracted from a positive. So they know 9-(-12) is 21, but have no idea what -9-(-12) is, and wildly guess -21.
I’ve suddenly realized that negative numbers aren’t really the problem. Subtraction causes the disconnect, as a result of the tremendous bait and switch we pull when moving from basic math to the abstractions needed for advanced math.
In elementary school, kids learn addition and subtraction. They are not told that they are learning addition and subtraction of positive integers. Nor are they told that they are only learning subtraction when the subtrahend is less than the minuend and, by the way, we need new terms. Those are horrible. In fact, kids are told that they can’t subtract in these cases.
At no point are kids told that everything they’ve been taught is temporary, and that much of it will become irrelevant if they move into advanced math. Consider the big fuss over Common Core subtraction, which is all about an operation that has next to no meaning in advanced math other than grab your calculator. (No, this isn’t an argument pro or con calculators, put your hackles down.) Or consider the ongoing drama over the aforementioned “math facts memorization” which, frankly, gets turned ass over tincups with negatives and subtraction.
Common Core requires that sixth grade math introduce negatives. Along with ratios, rates, fraction operations, and statistical analysis, all tremendously complicated concepts. In seventh grade, things get serious:
Never mind that most non-mathies would clutch their pearls at the very thought of parsing these demands, or that these comprise one of nearly twenty standards that have to be covered in seventh grade. Leave that aside.
Focus solely on NSA1B and NSA1C which, stripped of the verbiage, define the way we math teachers reveal the bait and switch.
So first, you teach the kids about these negative numbers and how they work. Then you show them that okay, we kind of lied before when we taught you that addition always increases. Actually, the direction depends on whether the added value is positive or negative.
But that’s it! That’s all you have to know! Just this one little thing. So negative numbers allow us to move in both directions on the number line.
And subtraction? Piffle. Because it turns out that (all together now!) Subtraction is addition of the opposite. Repeat it. Embrace it. Know it. Then everything makes sense.
So we teach them these two things. Yeah, we lied about adding because we had to wait to introduce negative numbers. But there’s this one little change. That’s all you have to know! because subtraction is a non-issue. Just turn subtraction into addition and funnel it all through the same eye of the same needle. Dust your hands. Done, baby.
Well, not done. As I said, we all know that negative numbers are brutal. We build worksheets. We support the confusion. We do what we can to strengthen the understanding.
But over the years, as I started teaching more advanced math, I realized that subtraction doesn’t go away. Subtraction is essential. It’s the foundation of distance, for starters.
And what the standards don’t mention is that introducing negative numbers changes subtraction beyond all recognition. The people who “get” it are those who reorder the integer universe spatially. Everyone else just stumbles along.
Until this summer, I never addressed this issue. I’m pretty sure most math teachers don’t, but I welcome feedback.
How do we change subtraction?
For starters, we violate the rule they’ve been taught since kindergarten. Turns out you can subtract a bigger number from a smaller number. (And, when a kid asks, “Well, in that case, how come we have to borrow in subtraction?” we teachers say…..what, exactly?)
But that’s just for starters. Take a look at the integer operations, broken down by sum and difference. (Much time is spent on teaching students “sum” and “difference”. More on that in a minute.)
So first, a row of numbers like this brings home an important fact: the Commutative Property ain’t just for mathbooks. This provides a great opportunity to show students the relevance of seemingly abstract theory to the real world of math.
But notice how much simpler the addition side is. I color-coded the results to show how discombobulated the subtraction pairs are:
Middle school math teachers spend much time on words like sum and difference, but I’m not entirely sure it helps.
For example, consider the “difference” between -9 and -5, which is -4. First, -5 is greater than -9, a complicated concept to begin with–and -4 is greater than both. And–even more confusing to kids taught to limit subtraction–none of those relationships matter to the result.
So -9 – (-5) = -4. Which is the same as adding a positive 5 to -9. So the difference of -9 and -5 is the same as the sum of -9 and 5.
Meanwhile, -9 – (-5) is subtraction of a negative, which we have hardwired kids to think of as “adding”–which it is, of course, but adding in negative-land is subtracting. So what we have to do is first get kids to change it to addition, then realize that in this case, the addition is a difference.
It’s not illogical, if you follow the rules and don’t think too much. But “follow the rules and don’t think too much” works for little kid math. As we move into algebra, not so much–we discourage zombies. Math teachers are always asking students, “Does your answer make sense?” and how can a student answer if subtraction makes no sense?
One of the things I’m wondering about is the end result of the operation. Any two numbers have a difference and a sum, all expressed in absolute values. 9 and 5 have a difference of 4 and a sum of 14, and no matter what combination of sign and operation used, the answer is the positive or negative of one of these two. So I ordered them by the end result.
Notice that P+P, N+N, P-N, N-P are ultimately collective sums. No matter the relative size, P+P and P-N move to the right, N+N, N-P move to the left, and result in a positive or negative sum of the two terms.
That looks promising, but I’m not sure how to work with it yet, particularly given the confusion of the actual meanings of sum and difference.
Here’s what I’ve got so far, and how I’m teaching it:
- What students think of as “normal” subtraction is actually “subtraction of a positive number”, where the subtracted number is smaller. Subtraction of a positive number always involves a move to the left on the numberline.
- In subtraction, the starting value does not change the direction of the operation–that is, -9 – 5 and 9 – 5 will both go to the left.
- The starting value must not change. This is a big deal. Kids see -9-5 and think oh, this is subtracting a negative so they change the -9 to 9. No. It’s subtracting a positive.
- Please, please PLEASE sketch it out on a numberline. Please? Pretty please?
Hey, it’s a start. I also use a handout I built six years ago, during my All Algebra All The Time year (pause for flashback) and has proved surprisingly useful, particularly this part:
I am constantly reminding kids that subtraction is complicated, that the rules changed dramatically. Confusion is normal and expected. Take your time. I am seeing “success”, with “success” defined as more right answers, less random guessing, more consistent mistakes in conception that can be addressed one by one.
I don’t know enough about elementary and middle school math to argue for change, except to observe that much more time is needed than is given. I once took a professional development class in which a math professor covered an abstruse explanation of negatives and finished up by saying “See? Explain it logically and beautifully. They’ll never forget it again.” We laughed! Such a kneeslapper, that guy.
But I’m excited to get a better sense of why kids struggle with this. It’s not the negatives. It’s subtraction.