**Editor’s note:** This is the eighth piece in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” by Barry Garelick, a second-career math teacher in California. He has written articles on math education that have appeared in *The Atlantic*, *Education Next*, *Education News* and *AMS Notices*. He is also the author of three books on math education. Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd.” The previous chapters can be found here:

Chapter 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5 and Chapter 6 , Chapter 7

**Chapter 8: The General This ‘n That Shop, Negative Numbers, and Faith**

I went to school at

“Are you going out of business?” I asked the cashier.

Her response: “I hope not.”

It is this type of optimism that I try to instill in my students. And among my seventh grade students the first such test has been negative numbers.

I do not like to prolong the topic. I once observed a teacher taking three weeks to teach it. After two weeks of adding and subtracting negative numbers and the students had it down fairly well, the teacher introduced a new explanation using colored circles. While the approach can be effective, its use at this point only caused confusion. One girl asked “Why are we doing this?”

The teacher answered “I know you know how to add and subtract with negative numbers. Now I want you to understand why it works.”

The girl’s response: “I don’t want to understand!”

This incident was not unique. I’ve found that a lot of the confusion with the addition and subtraction of negative integers is that students are given more techniques and pictorials than are really needed. They are left with the impression that it is a complex process and that there are many different ways to do it. This is ironic considering that subtraction is an extension of addition. In mathematical terms, a – b = a + (-b) where a and b can be positive or negative.

I keep it relatively straightforward with the first day spent using number lines with arrows to compute addition of negative numbers. The next day, the addition is done without pictures.

I then introduce subtraction. I tried this my first year teaching at my previous school with an accelerated seventh grade class. I asked them to compute 6 + (-4) which they knew how to do from the previous lesson. Two was their answer. I then asked them to compute 6 – 4. They saw the connection almost immediately, leading to the general rule of “adding the opposite”.

While it worked well with my accelerated class, I thought maybe I wouldn’t have the same success the next year with the seventh grade class which had large deficits in their math knowledge. But the technique worked just fine, with Kyle shouting out: “Adding a negative number is the same thing as subtraction”. This became a quote that I posted on the quote wall.

The only rub in all this is the subtraction of a negative number. I used to introduce this by first asking if anyone could solve 10 – (-5), and then linking the question to football for those who liked or played It.: “After a loss of 5 yards, how many yards do you need to get a first down?” As I mentioned in an earlier chapter, Jimmy had answered “Can’t you just punt it?” I have since changed my tactics. I specify that the ball has to be run and then use any number of non-football examples such as: “It was -10 degrees yesterday and 20 degrees today. By how much did it increase?”

My goal in teaching adding and subtracting of negative numbers is to achieve a level of automaticity, so that students can ultimately solve a problem like 3 – 7 without using pictures or writing it as 3 + (-7). At the same time, I try to get them to develop a number sense as to whether their answer is going to be negative or positive. I give them models to use such as: “If I gain 3 yards and lose 7 am I ahead or behind and by how much? If I earn $3 but owe $7, am I ahead or “in the hole” and by how much? They do get it, though they need reminders through the year.

And as far as multiplying negative numbers I provide an illustration of why things work as they do. I use an example of making a video of someone riding a bike backwards, and running the video backwards.

“What if they were skateboarding?” Jimmy asked.

“Whatever you want,” I said.

If the backwards rate is represented as -3 mph and we run the video backwards at 2 times the normal speed—represented as -2—the person appears to be riding a bike or skateboarding forward at 6 mph. A similar model can be used to show why a negative number times a positive is a negative number. (For the more curious students, and certainly in accelerated classes and in eighth grade algebra, I show the proof using the distributive rule.)

While my backwards video example generally does the job, it didn’t with Jimmy even when I used skateboarding rather than a bicycle rider. He tended to be quite literal. “That’s in a video; does it work in real life?” he asked thus opening up the question of whether mathematics can apply to images. He finally accepted the example used in JUMP Math where someone on a mountain is descending at a rate of 30 ft per minute or -30 ft/min. The example asks how one would represent where the mountain climber had been relative to his present position 3 minutes earlier, or -3 min. Jimmy agreed that the person would be higher 3 minutes previous and further accepted that the situation is represented by -3 x -30, or +90.

While Jimmy understood it, a girl said she did not understand either example.

“For now, just work with the rule,” I said. “You’ll get it the more you work with these kind of problems.” The girl did understand the examples a few days later. “Sometimes you just have to have faith in the math,” I told her. She evinced no expression so I said nothing more. Which is a good thing because I doubt she really cared about how we’re all sometimes like the cashier at the General This n’ That Shop.

I learned the concept of subtraction in the early 60s as the distance between two numbers on the number line .. but it’s really the directional distance. How do you understand the final sign for (-3) – (-6) in the grade you are supposed to be able to do this math? Even if you could write a proof, it doesn’t mean that you have the skills to do all variations correctly. My son had to derive Maxwell’s Equations in a college math class, but that never meant that he could solve all problem variations and applications. There is a world of difference between conceptual understanding or even proofs and full application problem solving understanding. This goes for even the simplest math. Full understanding ONLY comes with lots of problem set practice as clearly given in traditional textbooks. All proper textbooks introduce conceptual understanding and then use problem set variations to build deeper connections and understanding.

There are a lot of subtleties with signs in addition, subtraction, multiplication, and division that get worse with complex rational expressions. I distinctly remember trying to understand all of the possible variations of having a plus or minus sign in the numerator, denominator, or just sitting out to the left right at the dividing line. It ties in with so many other skills and mathematical understandings like factors. Any silly colored circles (or whatever) explanation is only an attempt to provide conceptual understanding and not anything more. That’s nothing new in the history of teaching math – only different, and as Barry says, it’s redundant and annoying if done in several ways. In all cases, teachers have to move on to mechanical techniques to handle all variations correctly as in -((x^2-4)/(-2-x)). How do colored circle understandings evolve to handle that? I just learned this for addition:

+ (+) is +

+ (-) is –

– (+) is –

– (-) is +

And yippee, its the same for multiplication and division. It’s not quite that simple, and you have to understand that a sign can also be viewed as a factor of (+1) or (-1), but that rule is not directly derived from any one conceptual understanding. Solving is not always derived from understanding. It’s often the other way around – you develop understanding AFTER and from the mechanical skill techniques that work in all situations.

“Sometimes you just have to have faith in the math,” I told her.

Yes, as I liked to say to my students, let the math do the thinking for you.