Adding and Subtracting Integers

Any mathematics educator will agree that for many students, the arithmetic of positive and negative integers is a challenge.  Often students get confused when they have to subtract numbers, especially once dealing with negative numbers, since their initial instinct to count it out on their fingers no longer seems viable.

It is also true that this frustration that many students have ultimately leads to their disenfranchisement in the math classroom, and is largely responsible for many people deciding they don't like math at all.  Thus, in comes the current approach to reconcile this problem:

-2 + -3 = ?

3 - +2 = ?

5 - -3 = ?

Bad Idea: Keep Change Change

Students initially learned what addition was in a very natural way.  Its simple to say that if one started with 2 of something, and added 3 more, then clearly they would have 5 things.  This is where math plays on preexisting human intuition (interestingly enough, this is where mathematics begins in the first place).

But as soon as the problem is subtraction, it gets a little bit harder.  This can be remedied by thinking of subtraction as a reverse kind of addition.  In other words, the problem:

5 - 3 = ?

can simply be seen as asking "What number do you add 3 to, in order for the total to be 5?"  This very natural approach to subtraction does have its foundations in underlying mathematical rigor, and is thus very conceptually acceptable.  Of course, things get much messier when the numbers involved can include positive or negative numbers, mainly because it is much harder to visualize the difference between the two.

Thus, the current pedagogy proceeds as follows:

5 - -3 = ?

• Keep the first number the same
• Change the operation from subtraction to addition
• Change the sign of the second number
• Then perform the operation as though it were addition

So the student is expected to rewrite the problem as 5 + +3, which is clearly 5 + 3, which he/she can easily and intuitively deduce is 8.  Problem solved.

Since the students are now equipped with a procedural tool to make their lives easier, it is sometimes easy to ignore situations where this method utterly and totally fails.  Consider the following:

-5 - -3 = ?

The only thing I changed is I made the initial number negative to begin with.  Now the student has to do the same KCC procedure, which yields:

-5 + +3 = ?

Is this any better?  Is it any clearer what the answer is?  Now we need to come up with procedural rules to help student add numbers possibly of opposite signs.  So then we have something to the effect of:

When adding numbers with signs:

• If the signs are different, subtract them, if the same, add them
• Take the sign of the larger of the two numbers

I hardly know where to begin here.  We've run into a procedural nightmare here, and all we've been trying to do is add and subtract integers.  We've given them so much procedure to consider and look at that we've divorced it entirely from the context in which we began.  And its not actually any easier for students, I think. Instead, math has become a list of steps to memorize.

But the proceduralist has a solution to this problem as well:  When in doubt, have the students use their calculators.

This final step invalidates any and all previous steps, because the calculator is never wrong (as long as you use it correctly, which is then another whole giant thing sometimes).

I hope I have adequately illustrated the giant mess that is this current pedagogical practice.  Now, onto the solution.

Good Idea: The Number Line

First, a note on mathematical foundations.  In geometry, a very well-known axiom of continuity was decided upon by Richard Dedekind, named Dedekind's axiom in his honor.  What it basically states is that lines (from geometry) and the set of real numbers (of algebraic and analytic fame) are actually the same thing.  So numbers correspond to distances or locations on lines, in a very tangible and practical way.  This axiom is noteworthy in higher mathematics because it bridges the gap between many sophisticated mathematical fields.

But why is it noteworthy for middle-school and high-school-aged students?  While its not necessary to present it to them specifically and by name, it is important for math educators to realize that strong, foundational mathematics gives us the answer to helping students with a very simple problem.  The real numbers are points on a line.  Points on a line are real numbers.  This fact is inescapable, and has a tremendously powerful application to help solve our addition/subtraction problem.

Here is a number line:

Its very simple, really.  You start with a line.  You pick some point in the middle of it, and call it "0".  As you move to the right, the numbers get bigger, and to the left, they get smaller.  Clear so far?  Good.

As can be seen and verified quite easily, any number can be found on this number line.  Its always possible to draw a quick line, and label some point on it "-2", or "56", or "-24,563" if necessary.  So here is how I propose the addition/subtraction problem be solved:

-5 - -3 = ?

-5 is a real number, its a real location on this line.  Draw the line on paper if you need to (or visualize it in your mind if you can do that easily enough, which of course comes with practice).  Find "-5", wherever you want it.

Now, ask yourself, "what am I doing, adding or subtracting?"  If adding, then you wanna think "go to the right, that's where numbers are bigger", and to the left if you're subtracting.  Finally ask "is the number I'm adding/subtracting positive or negative?"  If its positive, your initial "direction" is correct.  If negative, reverse it, because again, numbers go down as you go to the left, up as you go to the right.

I'll admit, it can be argued that this method is at least as procedural as KCC.  But inherent in that procedure is an underlying concept.  KCC just feels like a "set of rules to follow, arbitrarily set down".  Here, I think the student understands what's really happening, and is in touch with the actual mathematics of the situation.

The advantages of this method can be summed up as follows:

• If a student is struggling with the idea of numbers themselves, thinking of them as distances and points on a line gives them a visual interpretation.  This visualization adds a dimension to their ability to comprehend, and thus can help many students whose visualizing skills are stronger than their algebraic abstraction skills.
• This is real math going on here, not a long-winded, arbitrary list of rules.
• When students see the coordinate plane for the first time, being familiar with the number line will make their lives much easier.  After all, the coordinate plane is just 2 number lines.