Several years ago on the first day of school, in order to get a glimpse inside their heads, I asked each of my incoming 7th grade math students to draw a number line starting at 10 and counting down by ones until they couldn’t go any further, or ran out of paper. The number lines I got back were, shall we say, interesting. Some samples:
Hoo boy. Each of these kids had just spent a whole year studying opposites, absolute values, and positive and negative (x, y) coordinates (all 6th grade Common Core standards), and clearly very little of it had stuck. Why? Most likely because they hadn’t spent enough time a.) drawing number lines, and b.) experimenting with them. (We only remember things we’ve actually done - and done a lot.)
Let’s discuss the drawing part first. As an experiment, I asked the same students that day to place 0 at the center of a new number line, and to think of it as a mirror, with positives to the right of it, and negatives to the left. This time the number lines came back looking like this:
Why did this strategy work? Because they already knew how mirrors work (7th grade vanity). It was instantly connectible to their experience.
Why did the previous strategies fail? Probably because their teachers hadn’t given them enough practice drawing. As psychologist Jerome Bruner noted as far back as 1967, children progress through a hierarchy of three modes as they learn: the enactive (or action-based) mode (where they do things like combine three apples and four more apples to assemble a group of seven apples); the iconic (or representational) mode (where they draw three apples and four more apples and count to a total of seven apples); and the symbolic (or abstract) mode (where they forget about apples entirely and perform the generalized operation 3 + 4 = 7).
Too often in schools, and particularly in the upper grades, we bypass the drawing mode and jump straight to the symbolic. Why? The reasons, I think, are twofold: we assume that drawings are somewhat juvenile, and we worry that some of our students “can’t draw.”
As for the first point, I’m reminded of a student teacher years ago, who actually stopped students from drawing number lines during a lesson on negatives because “you’re not little kids.” He subsequently relented when they also stopped getting the answers right. Why shouldn’t students make drawings when they need to? Mathematicians do!
And as for the second point, math drawings don’t have to be perfect; they just have to be good enough. As long as everything’s in the right place and the numbers are correct, the drawing’s good to go. (This is a good thing too, because “imperfect but good enough” drawings can be super-helpful when you’re learning to add and subtract negatives, as we shall see.)
Before we start actually adding and subtracting negatives, let’s start with what students already know. Adding two positive numbers, like 43 and 216, always looks like this on the number line, with two jumps next to each other, starting from zero:
The diagram clearly demonstrates that the two jumps (of 43 and 216) are equivalent to a combined jump of 259, and thus 43 + 216 = 259. Notice that the final destination is on the positive side of zero, so the sum is positive. Notice also that the scale of the drawing does not matter, and that the diagram is so simple that anyone can draw it.
Note that the sizes of the jumps in the diagram above and the sizes of the actual jumps on the number line are not related to each other. This point about scale is crucial; it enables students to create drawings representing any two numbers. This differs substantially from common methods of teaching adding and subtracting negatives - like the current method wikiHow uses - where students are asked to count their way to the answers on the number line; needless to say, counting ceases to be a practical strategy once the numbers get too big.
Adding two negative numbers looks the same, except that the jumps are on the negative side of the number line. Consider -8 + (-591) and its related drawing:
Again the diagram is clear: a jump of -8 plus another jump of -591 is equivalent to a combined jump of -599, thus the sum of -8 and -591 is -599. If a student understands how to add two positives, they should now understand how to add two negatives.
But what if we start with a positive number and subtract a negative number? Subtracting a negative is the same thing as adding; going backwards backwards is the same thing as going forward. We can simply turn the - - into a + and add like usual; 43 - (-216) is equal to 43 + 216, and thus the diagram is exactly the same as the positive diagram above.
What if we start with a negative number and subtract a positive number? If we start with a negative and subtract a positive, we’re getting more negative: -8 - 591 is equal to -599, which is more negative than either of the numbers we started with; the diagram is no different from the one above. (This is due of course to the fact that adding a negative is the same thing as subtracting; going forward backwards is the same thing as just going backwards.)
The key thing to notice here is that whenever we have two jumps next to each other, we add the lengths of the jumps and take the sign of the final destination.
When we subtract a positive number from a larger positive number (as in 175 minus 23), it always looks like this on the number line, with a jump inside of a jump:
Notice that the final destination here is positive, and that the larger jump (175) minus the smaller jump (23), combined with the sign of the final destination, yields the answer of positive 152.
Whenever we end up with a jump inside of a jump, we subtract the smaller jump from the larger jump and take the sign of the final destination. Some examples:
or
52 + (-86) = -34
or
-17 - (-79) = 62
-368 + 46 = -322
or
-368 - (-46) = -322
“Wait,” you may be thinking. “You want kids to do these kinds of drawings every time they have to add or subtract negatives?! Can’t they just memorize the rules instead?"
First of all, the rules are the problem, not the solution. Kids (as you may have noticed) are terrible at 1) memorizing rules and 2) following them. And the rules for adding and subtracting negatives are worse than most, because they rely on prior knowledge (regarding absolute values) that the kids may not have (and generally don’t).
Secondly, humans pass through the iconic (or drawing) mode to get the symbolic mode, so if there’s any “rule” to be followed here it’s pictures first, then rules. In fact, the students should get to the point where they understand the pictures (i.e., situations) so well that they can infer the rules for themselves. (When they do get to this point, they can begin to wean themselves off of the pictures, of course; they’ve successfully made it to the symbolic mode.)
Case in point: by drawing and examining the following, students should be able to not only infer the rule that a - b is the opposite of b - a, but also to see why this is so (bearing in mind that zero is its own opposite).
-15 - 12 = -27
For a “next level” lesson, students can draw and examine the following to infer the rule that a - b = -b + a (keeping in mind that the negative of zero is itself).
The drawing approach is vastly superior to teaching students rules that they can’t “see,” of course, partly because should they forget the rules, they can easily re-construct them from memory, but also because the ability to see math in your head is a large part of what constitutes the all-important “math sense.” This approach also disavows students of the faulty notion that math rules are arbitrary - like the rules of baseball, say - and instead helps them frame them more accurately as logical deductions.
“Okay,” you may be thinking. “This all sounds logical, but how the heck does a teacher teach it?” The quickest, surest way is to have the students solve some problems by drawing along with you (by starting at zero and drawing their way to each solution), and then by having them work their way through “see it, do it, check it” activities containing worked-out examples (with drawings), problems related to the examples (requiring them to draw), and worked-out solutions to the problems (containing correct completed drawings). Adding and subtracting negatives is one of the trickiest of all things to teach (mirrors and counter-intuitive concepts are naturally confusing, even for adults), thus students need large amounts of trial-and-error practice with it before it “hardens.” (Contact me using the Twitter link below if you’d like some sample “see it, do it, check it” activities on this topic to use with your kids.)
We are now past the rapids; once students have mastered adding and subtracting negatives, the rest of the topic is relatively easy to navigate. How can we teach multiplying and dividing negatives so that students master that as well?
Stay tuned for Part 2!