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.

June 30th, 2016 at 1:56 pm

I have my thoughts on this and I would like to do a post. The clue is meaning, and electronics is half the answer. Stay tuned.

June 30th, 2016 at 2:24 pm

So would they learn better if they were taught negatives before subtration?

Would it work to let them subtract 5 from two in 92-35, but then insist that they then add -3 to 60 after they have finished subtracting 30 from 90 to even everything out? This seems very similar to what I remember of long division, so shouldn’t be too difficult to teach to kids at this time (long division coming only a year after double digit subtraction, if things are still as I remember).

Would it help them to use a zero as a place holder to add the remaining negative to in P+(-N) when N is larger? Or to use zero as a placeholder in N+N or N-(-N) or even N+(-N) or similar, turning them into 0+N+N, 0+N-(-N), 0+N+(-N). Which would basically create the number line without forcing them to draw it out.

June 30th, 2016 at 3:14 pm

I actually think that the common core method of subtracting, where you work along the number line, would be helpful.

July 7th, 2016 at 11:10 pm

Common Core was right to utilize number lines, for sure.

The visualization is helpful in making students more sure of themselves, whatever the calculator might say. It actually works better as subtraction than adding a negative, judging by how the students’ eyes develop a look of skepticism.

The concept of subtraction and the need for precision in which number is first has been forgotten by some of the same students who can write a fraction correctly, then proceed to put the numbers in the calculator in reverse order, plus some others. Luckily, distance can be dealt with and used even with bad sense of direction.

June 30th, 2016 at 3:03 pm

Give them two colors of blocks, say red and blue. Teach them that each red block stands for a positive number and each blue block stands for a negative number. Teach them that they can pair any number of red blocks with the same number of blue blocks to equal zero. Show them they can add or take away these pairs without affecting the resulting “number”. Now you let them experiment with things they know: adding and subtracting positives. Then show them it is possible to subtract a larger number from a smaller one by adding pairs of red and blue blocks until they have enough red blocks to take away. Show them that the resulting number is negative, as they only have blue blocks remaining. Show them that they can subtract negative numbers the same way, starting with the blue blocks this time and removing the correct number of them to get the answer, adding zeros – pairs of red and blue blocks – as needed. This is a great hands-on method that can really help some kids understand what is really happening when they add and subtracting intigers.

June 30th, 2016 at 3:14 pm

I’ve tried that, and it works with addition of negatives. Doesn’t work with subtraction. Also–and this is a kid thing–they refuse to stop to think. So even the best method only works if they use it.

July 1st, 2016 at 4:10 am

You mean they just don’t understand it when they subtract negative blocks? Of course not every method works for every kid, this method is just the one that clicked best for me when I was learning. Of course, you are right that the kid is only going to learn if they use it, and they are only going to use it if they can see that it has value for them. That’s the real trick, but here are a couple other methods I know.

You could try a number line. Tell them the first number is their starting point, add means move right and subtract means move left. As for the second number, positives are obedient to the sign and negatives are disobedient to the sign (negative numbers move the wrong direction on the line). If you have the time, space, etc. to do so, I would make a number line on the floor and have them walk out problems on it.

Or you could relate it to money. Positive numbers are how many dollars they have, negative numbers are how much debt. Adding is gaining either money (good) or debt (bad) depending on if they are adding a positive or a negative number. Subtracting is losing either money (bad) or debt (good) depending on if they are subtracting a positive or a negative. You could try an activity with play money and i.o.u. slips. Anything to make it hands-on and engaging.

Those are a few of the better methods I know. I know not every kid will respond, but maybe one of these will help.

June 30th, 2016 at 4:54 pm

[…] Source: Education Realist […]

July 1st, 2016 at 7:44 pm

Thank-you, this post really got me thinking.

I’ve always been an “add the opposite” rule follower because it was drilled into my head in high school in the 80’s. And also because of the huge volume of problems we were made to do, which didn’t improve my understanding…

The number line is key and I see that now.

9-5: begin at 5 move right to 9 is 4.

9-(-5): begin at -5 move right all the way to 9 is 14.

-9-5: begin at 5 move left all the way to -9 is -14.

5-9: begin at 9 move left to 5 is -4.

July 1st, 2016 at 10:06 pm

I’d do it the other way round.

9-5: begin at 9, move five to the left. It’s subtracting a positive.

9-(-5): begin at 9, move five to the right. It’s subtracting a negative.

Check that out.

There’s nothing wrong with “add the opposite”, but the problem is that kids don’t realize a) that subtraction is usually subtracting a positive, so adding the opposite is adding a negative and b) they don’t know how to handle it when the starting number is a negative.

July 2nd, 2016 at 9:13 pm

When I taught algebra, I learned to get around this difficulty by treating every potential subtraction of integers problem as an addition example. To get there, I first had to get to mastery of the idea that formulations like – (–5) mean “the opposite of negative 5,” not “subtract negative 5.” Then I had students do every example by circling and adding the integers within.

For 9 – 5, circle the 9 and the –5. 9 plus – 5 is 4.

For 5 – 9, circle the 5 and the –9. 5 plus –9 is –4.

For 9 – (– 5), circle the 9 and the “opposite of negative 5.” 9 plus 5 is 14.

For – 9 – (–5), circle the –9 and the “opposite of negative 5.” –9 plus 5 is –4.

And so on.

Of course, for this to work, students have to be able to add the resulting positive and negative integers, but I found that to be a much easier matter than trying to wrestle with “subtracting a positive” vs.”subtracting a negative.”

No method is perfect, I suppose, but I had a lot of success with this one. I got it from one of John Saxon’s old Algebra I books from the mid-80s. Don’t know if his series still treats the topic this way.

July 3rd, 2016 at 1:25 pm

Yeah, but you’re turning everything into addition. I used to believe people when they told me that subtraction is always addition, but in fact, we need subtraction. So I think it’s better to keep subtraction. Also, regardless of what I want, subtraction is imbued in the kids.

July 4th, 2016 at 5:38 pm

Yes, we need subtraction, I agree. Slope (rate of change) and length of line segment calculations are two which immediately come to mind. Students get these wrong all the time.

July 3rd, 2016 at 3:10 pm

I believe -3-15=-18, not -12. Or was this your intended point?

July 3rd, 2016 at 5:29 pm

Actually, I think I meant 3-15, but that would have been just like the previous problem. So I changed it to -3-9 and threw in some details. Thanks.

July 4th, 2016 at 12:02 am

Good luck! This is one of those things that if you figure out something that works you’d better copyright it, because you will be rich. Seriously, I’ve pounded my head against the wall many years with this. And I’ll sometimes think the kids get it, only to have my hopes dashed a week or two later when it’s as though they’ve never seen it before. Not to be too cynical, but I’m coming to the conclusion there’s only one way to do this – with a calculator.

July 4th, 2016 at 2:57 am

Yes. My hope is to just get them to stop and think.

July 4th, 2016 at 2:09 pm

What are negative numbers ?

The question really is “What are the positive and negative numbers ?”

First of all we have “he numbers”. 1, 2, 3, … and zero for completeness sake, and also that “none” is a number, and “nothing” is not a number.

“The numbers” are actually very different from one another. The counting numbers, 0, 1, 2, … , are very different from the measuring numbers. Measuring numbers have “quantity”, counting numbers have “counts”.

Measuring numbers are cakes, pizzas, watts, feet, mass, volume, area, and so on, where the quantity is “some” or “none”.

Examples of measuring numbers are “half of a foot”, “2/3 of a pizza”, “0.05 square feet”, and they are just “numbers”, attached to units of measurement.

Now there is a problem.

“Apply and extend previous understandings of numbers to the system of rational numbers.” (quote from CCSS)

Just like that !!!!!!

There is NO easy extension, “half of a foot” is NOT extendable to “three feet below sea level”.

The meaning of “positive and negative numbers”, or the “signed numbers”, is not a “some or none” situation at all.

The “signed numbers” are abstractions of “relative position” and “change of position”, and the position of “zero” is often, if not always, arbitrary.

A temperature scale has a zero, and temperatures above zero are “positive”, temperatures below zero are “negative”.

A different temperature scale has a different zero, and, worse still, the scale factors (scales) are different as well.

“Feet above sea level” and “meters above Mount St Helens” are similarly “different”.

An electrical circuit can have chosen a voltage value of zero at any point in the circuit.

In these and all similar situations the zero is chosen by a human, and not as the “none or nothing” value.

The value 9, or the value -5, is marked on the scale as a position relative to the zero on that scale.

Marks on the left, or the “down side”, are conventionally the “negative” marks, and marks on the right, or the “up side” are the “positive” marks.

The positive marks. “+”, are conventionally ignored, but at the start one should put them in.

… to be continued, when the confusion between “negative” and “subtraction” is resolved.

And this CCSS bit is so stunningly superficial.

Grade 6

Apply and extend previous understandings of numbers to the system

of rational numbers.

5. Understand that positive and negative numbers are used together

to describe quantities having opposite directions or values (e.g.,

temperature above/below zero, elevation above/below sea level,

credits/debits, positive/negative electric charge); use positive and

negative numbers to represent quantities in real-world contexts,

explaining the meaning of 0 in each situation.

6. Understand a rational number as a point on the number line. Extend

number line diagrams and coordinate axes familiar from previous

grades to represent points on the line and in the plane with negative

number coordinates.

a. Recognize opposite signs of numbers as indicating locations

on opposite sides of 0 on the number line; recognize that the

opposite of the opposite of a number is the number itself, e.g.,

–(–3) = 3, and that 0 is its own opposite.

July 4th, 2016 at 6:58 pm

“Well, in that case, how come we have to borrow in subtraction?”

I mean, you don’t *have* to. If you’d rather deal with a mix of positive and negative numbers than carry a 1, then yeah, 21-19 = (20-10) + (1-9) = 10 + -8 = 2. Indeed, if we wanted to use negative numbers as digits and had a readable notation for doing so, |1|-8| (representing 1*10^1 + -8*10^0) would just be the answer, and would be equal to |2| (representing 2*10^0).

I don’t mean to suggest that this would be pedagogically helpful, but it might make one feel less deceitful?

July 4th, 2016 at 9:59 pm

Not deceitful, but just understanding we’re changing the rules.

July 5th, 2016 at 12:58 am

Have you seen Project Z?

http://www.sci.sdsu.edu/CRMSE/projectz/presentations.html

I think it supports the big idea that you identified in this post, that thinking of opposites is a way to help kids make sense of this stuff.

I’d add that there are two ways to make sense of addition/subtraction involving opposites:

subtraction is addition of an opposite

subtraction is the opposite of addition

Sometimes in class I help kids decide what subtraction of a negative does by establishing what adding a negative does. 4 + (-3) makes 4 lower, so 4 – (-3) should do the opposite, and make it higher.

Really, though, there is no one idea that is going to be helpful in every situation. Part of this, I think, is that middle school teachers are no used to the way that slight differences in number values can make a problem very different for kids. (So 4 + (-3) is quite different for a kid than 4 + (-5), I think.)

Love these thoughts here, I hope you keep writing about negatives!

July 5th, 2016 at 1:21 am

I do the same thing–tell kids if they’re confused by 9-21, then try 9-(-21), or even 9+21.

And I agree. There’s no trick. That’s the problem. We teach it as a trick, when in fact it’s insanely complicated.

Thanks for the good words. Teaching pre-algebra has been a good kick.

July 5th, 2016 at 9:08 pm

An alternative look at this stuff is to view the positive and negative as attributes, and create a (temporary) sign called “and”.

This will cause the “and” to be the “action” verb, as in

+9 and +5 = +14

+9 and -5 = +4

-9 and +5 = -4

-9 and -5 = -14

The “and” can be dispensed with eventually as it is redundant..

July 5th, 2016 at 10:56 pm

But “and” is just “add”.

July 6th, 2016 at 1:21 am

3 add 4 = 7 ….

3 add -4 = -1 ….

-4 add 6 = 2 ….

do not give the flavor of “and”, and also the difficulties with the two things, “add” as a binary operator and “add” as in “add to”, cause confusion later.

July 6th, 2016 at 2:53 am

Yeah, but you’re still not teaching subtraction, and my position is you need to.

December 29th, 2016 at 12:17 am

This is kind of how I teach it, but math is not my subject area, science is. But I sub math sometimes and help students who show up after school with their math.

Here’s how I approach it, admittedly not as introductory concepts.

You have to start thinking of these problems on a higher level than “add and subtract.” You are combining 2 (or more) numbers to find a value. Don’t look at the + and – signs as telling you to add or subtract, attach them to the number after them as positive x or negative x.

So -5+3= is -5 with +3. If both numbers are the same sign: add. If the numbers are different signs: subtract (remember “find the difference?”). The sign of your answer will be the same as the bigger number (higher absolute value).

I actually show them on a number line (moving right 2x, moving left 2x, or moving 2 different ways). But that is the basics of it.

December 29th, 2016 at 5:25 am

Good ideas.

July 6th, 2016 at 3:12 pm

So subtraction is the process of finding the difference between a and b, where a – b indicates that if b is positive then the result is less positive than a, and if b is negative the result is more positive than a.

(It does look similar to the “rub out the minus” approach that some students like so much!).

My approach is to separate the “and” from the signs, and write it like this:

a – b is …. +a “and” -b

and then do the combination

So we get as well

+a “and” -(-b)

as

+a “and” +b

which is conventionally written a + b

Have fun with this !

July 6th, 2016 at 9:16 pm

for subtraction it is the distance between 2 points. write a number line. put the two numbers on the line. to the right positive left negative. minus 9 minus 5. put minus 9 down. plus 5 down going right to left minus 14. i am sorry this probably just works for me.

July 6th, 2016 at 9:28 pm

No, that’s how we teach it.

August 3rd, 2016 at 3:25 pm

[…] Not Negatives, Subtraction […]

January 4th, 2017 at 7:27 pm

[…] Negatives, Not Subtraction–important concept that I made great strides on this year […]

December 1st, 2017 at 9:23 am

[…] that all the things we told you before ain’t necessarily so. Turns out, for example, you can subtract a big number from a smaller one. Fractions might be “improper” if the numerator is larger than the denominator, but […]

October 22nd, 2018 at 12:58 am

Recently I did a post on Subtraction and Negative Numbers, from apples to algebra. Do read the last bit, on the 19th century view.

https://howardat58.wordpress.com/2018/05/25/subtraction-what-exactly-is-it-bonus-is-8722/