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To be completed: notion of opposite, other images (i.e., other than number line) of negative and explanation of each image...
Children's alternative conceptions of 'what happens below zero'
Preliminary list: Hit the wall (zero forever, or more zeros); cloned (but not reflected) pattern (i.e., 3, 2, 1, 0, -10, -9, -8, ..., with all the bugs that has); temporary debt, but no real name for it (I'll have to pay it back, but it's not a "number" that can otherwise be manipulated); ...
Arithmetic with negative numbers
This section is preliminary notes, without adequate illustration or references, intended only as a starter. It is also written with personal opinion as well as fact. Those opinions can be kept, but need to be marked clearly to distinguish them from fact. Please edit to improve.
Arithmetic by the rules
Making sense of the rules: Images and strategies that make sense to young students
The Madison Project (late 60s early 70s?)
The Madison Project (late 60s early 70s?) had an imagery, the set of "Postman Stories," that are mathematically sound, and generally compelling to adults. There is no clear research documenting how well they worked with children. In outline, a postman delivers bills or checks (i.e., adds negative or positive quantities of money), and sometimes realizes that he has made mistakes and returns to take some of these back (i.e., subtracting the same quantities). If he brings a bill, you are expecting to be poorer; if he takes a bill away (subtracts a negative quantity), you're richer. It generalizes to multiplication nicely enough. (Division gets a bit contrived.) One appeal of this set of stories is that it makes the arithmetic with negative numbers feel like common sense to adults who will be teaching, and gets them out of the "it's all rules" frame of mind. It also does work for some children (the older the better), but the complexity of the language and imagery (different kinds of things coming and going) is hard for many children, and worse as the age goes down.
The number line image
The number line image has a different set of advantages and disadvantages. Oddly, it (by itself) makes subtraction clearer than addition. Picture subtraction as answering the question "how different are these two numbers?" or "how far is this number from that one?" or "what is the distance from this one to that one?" That is not quite complete, but close enough for the moment. (The complete truth is that distance is never negative, and that subtraction tells not only the distance, but also which direction we're traversing the distance. But that's a fine point to add later.) So, 302 minus 87 is asking "how far do we have to go from 87 to 302?" Clerks used to make change (before the machine told them what to do) by literally traversing the distance. 87 to 90, 90 to 100, 100 to 300, and 300 to 302, giving a total of 215 steps. So 302–87=302. One beauty of that is that mental arithmetic is easier (no crossing out things and rewriting them as one might on paper with borrowing). The distance idea generalizes perfectly to negative numbers. The if 302–87 means "the distance from 87 to 302" then 53–(-18) means "the distance from -18 to 53." Picturing that on the number line makes clear that the distance is -18 to 0 and then 0 to 53, which shows why we're "adding" 18 to 53 --- nothing about "change the sign and add" (which people then goof up because there's no logic and it's easy to mess up an arbitrary rule by, for example, forgetting the order and adding before changing the sign). See number line for a partial example of this image in play.
To complete the image of subtraction, though, we have to make sure that 10–3 and 3–10 are different. The distance, by itself, in both cases is 7, but we've ignored direction. So to write the answer we must say how far from the second number in the expression to the first number and in what direction (down or up, meaning left or right). If 10–3 means “how far from 3 to 10,” then the answer is 7 up. Then 3–10 is 7 down. Everything stays consistent when we do it to 53–(-18) and (-18)–53, or even -10–(-24).
But, as with any single-image approach, there’s a problem. Kids don’t seem to find this hard at all--the logic is clear even to young ones--but there is still something arbitrary (and therefore forgettable). Why does 10–3 mean “from 3 to 10” not “from 10 to 3”? No reason, just convention. And the other common number-line imagery treats expressions like 7+3=10 as meaning “if you start on 7 and move 3, then you arrive at 10.” It is consistent with the notion of inverse to say, then, that 10–7=3 means “if you start on 7 and arrive at 10, then you have moved 3,” but just as language and logic were in conflict with the Postman Stories they can be here, too. Because the form (structure, grammar) of 7+3=10 and 10–7=3 feel the same and only the vocabulary ('minus' instead of 'plus' as one of the words) changes, people tend to expect the meaning (“if you start on 7 and move 3, then you arrive at 10,” in the case of plus) to transfer (becoming “if you start on 10 and move -7, then you arrive at 3”). And, because mathematics is so consistent, that meaning does transfer, but now we have a confusion about what the start and arrival spots are, and what number represents the move as distinguished from the destination... Either image works, but if we mix the two, it gets messy.
Blue chips and Red chips
Adding means "putting more chips on the table" and subtracting means "taking some chips off the table." If there are 7 red chips and we "add 4 red chips" we can count to find the result. Likewise, if we subtract 5 red chips, we can again count to find the result. Ditto for blue chips on the table. The "obvious" rule is that we can't take away things we don't have so, for example, if there are 7 red chips on the table (and that's all), we can't take away any blue chips.There is also a special rule: at the end of adding or taking away chips, if there are pairs of chips of opposite colors on the table, the two chips in each pair "neutralize" (or "annihilate" or whatever imagery one likes) each other and go away. (You could make it friendlier and have them marry each other and go on honeymoon, but that creates problems later on.)
So, if there are 7 red chips and we "add 10 blue chips," seven of the blues will pick up the seven reds and go away, leaving 3 blue chips on the table. This rule even makes it possible to take away things one doesn't seem to have. If we have seven blue chips on the table and want to take away three reds, we must first get those reds somewhow. So, we add three blue-red pairs (because, together, blue and red "neutralize each other" and count for nothing). This new display has ten blue chips and three red chips. If we just left it alone, the three pairs would go away (showing that it really "is still the same as seven blue chips). But the new display allows us to take away those three reds that we had wanted to take away, resulting in ten blue chips alone on the table. (This, by the way, is the problem with the "marriage" imagery. We have just had three couples break up!)
At first, in this play (which works perfectly even with first and second graders), one doesn't assign positive or negative to either color. The arithmetic (and chip behavior) is totally symmetric, so one is merely getting a sense of the behavior which will later be applied to numbers. (And, whenever you like, pick either color and assign it to either sign, and it all works out.)
There are many other images and techniques as well. Early experiences in games in which one can go temporarily “in debt” and then recover contribute to the necessary intuitions. Tug-of-war activities on a number line (allowing both sides of 0) do, too.
My experience with children suggests that all of these contribute, but none of these is, by itself, perfect. The number line appears to be very effective—and, in fact, both imageries (what number represents the move and what numbers represent the endpoints) are important—but one needs also to have some intuitions that shore up the tendency to get mixed up about directions and order (which, in all domains of cognition and not just in math, tend to be pretty fragile). The chips model, because of its symmetry, is good for that, but the symmetry also makes it incomplete by itself. And the common-sense story line of the Postman Stories also helps, but mostly for older children, and mostly after some other experiences that have already established a lot of intuition both about addition and subtraction and about negative numbers. And, at some point, abstraction must take over. Multiplication of both positives and negatives by a positive are pretty easily derived from any of the above images, but “multiplication by a negative” is pretty hard to make concrete. One must already be ready to generalize/extend. And division is positively arcane in most of these models. It can, of course, be done, but it’s no prettier in the models than if one just said “well, division is just undoing multiplication, so figure it out.”