20160325 MINUS, MINUS, MINUS-MINUS
1 What Color Math?
2 How to Make a Number Line
3 Space Walks
4 Welcome to Debt Hole
5 Poor Physics
6 Through the Looking Glass
7 Do We Live in Burrows?
8 Stick and Stake
9 The minus of a minus
10 Why the Horrors
11 Conclusions
1 What Color Math?
Three years ago, I had to explain subtraction of negative numbers to my child, then 8 years old. As usual, I asked Google what American education was teaching. The page
http://www.purplemath.com/modules/negative2.htm
came out first, as it did today. I looked through it, then through several other Google's favorites, and taught subtraction my way. Use this link to skip this preamble and learn how.
Several days ago, I ran the same query, this time for my youngest American child. I call her TLG or The Little Girl. She was 6 years and 6 months back then.
Now please excuse me, I have to quote this celebrity content before it went away. It is important to understand that the author was a teacher and that she looked for the answer. American math web is huge, and she certainly knew it better than I.
"Whoa! Wait a minute!" you say. "How do you go from ' – (–16)' to ' + 16' in your first step?" This is actually a fairly important concept, and, if you're asking, I'm assuming that your teacher's explanation didn't make much sense to you. So I won't give you a "proper" mathematical explanation of this "the minus of a minus is a plus" rule. Instead, here's a mental picture that I ran across in an algebra newsgroup:
Imagine that you're cooking some kind of stew, but not on a stove. You control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.
If you add a hot cube (add a positive number), the temperature goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.
Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, –2(–3) = + 6.
There's another analogy that I've been seeing recently. Letting "good" be "positive" and "bad" be "negative", you could say:
good things happening to good people: a good thing good things happening to bad people: a bad thing bad things happening to good people: a bad thing bad things happening to bad people: a good thing
The above isn't a technical explanation or proof, but I hope it makes the "minus of a minus is a plus" and "minus times minus is plus" rules seem a bit more reasonable.
After this page, Google offered a generously illustrated one, comparing negative and positive values to balloons and weights. The third link was to YouTube video, on which somebody was actually adding and taking away red and green cubes, This business was dubbed Conceptual Math, and it was probably inspired by the latest fad in education called manipulatives. In short, no "proper" mathematical or technical explanation (let alone a proof) was in plain view.
The expressions like "the minus of a minus" and "- (-16)" looked interesting. I am gathering such details and dwelling on them because I have never been admitted to American school. I can only try to imagine what do American teachers say and how do they think. By the way, if you are familiar with this project, you know that my goal is not to criticize school or try to improve it in any way. I only want my last child to grow American and stay mathematically enabled.
I believe I know how to subtract a negative number and can explain it in many ways. So I decided to share this knowledge before I die, even though my attempt will probably die even sooner because my math is not purple.
I will start from the most essential information, but please read on to the dessert. If I happen to live long enough, I will work on this article couple more times. I don't have anybody to edit it yet. There can be any number of discrepancies, not to mention my obviously bad English.
2 How to Make a Number Line
If you opened the link, you've seen that the issue was raised after discussing addition and subtraction on a number line. That's where I am going to start.
For the reasons beyond my observations, my bigger American schoolchildren grew to believe that the numbers on a number line belonged to the points, so some of those points were negative. Teaching them back was not easy. With TLG, I started talking about number line before she turned 5. You'll see why. On a properly understood number line the "minus of a minus minus" question simply does not exist.
Suppose you drew a fresh line and marked a point on it. What number is here? You may think it's 0. I may think it's 213. Or 3.07. Nothing in this point has anything to do with any number._______________.________________________Let's now look at the number line. Do we need the numbers on it? If we leave only 0 and 1, will we be able to tell where is 5 or any other point?
0___1___________________________________
We absolutely will. Using the space from 0 to 1, we will walk along the line counting. The count of the steps from 0 to another point will tell the number at this point.
To make a line a number line, we need two points. They can be anywhere, but once we made a choice, we are bound to stick with it, or it will be another number line. The points are designated as 0 and 1.
A point has no width, length or height. There is nothing to count or measure in it. No point is bigger or smaller than another point. Several points do make a quantity, but they don't make a line. We could as well line up some counting tokens.
The quantity is enclosed between any two points. It can be compared to the likes of it, and what can be compared can be measured. It's called distance. Look, di-stance. To build a number line, we don't even need the 1 point. We need the distance and the direction to it.
Only two points matter. A distance is a property of a line segment, and any segment is between its beginning and its end. A point labeled 5 has no quantity. The label indicates the number of the steps from 0 to this point.
Distance is always positive because our space is isotropic, which means, it's the same in any direction. Coincidentally, I just finished a new project, in which I tried to explain and demonstrate how important it is to teach isotropic thinking.
The negative numbers to the left from zero are not distances. This little secret of the number line causes all the problems. To find the point for a negative number, we count from 0 to the left. The counts and the distances, however, can only be positive.
The model of space with two infinities and zero between them seems so natural that we may not even notice how passing through zero we change the direction of counting. Everywhere on the number line we may count the distances in any direction. To find them from the numbers, we have to subtract the left one (which is supposed to be smaller) from the right one. Labeling the points on the right we add the distances to 0, while on the left we subtract them from 0.
Let me read this one more time. Did I get it right? Are you sure?
Space comes without arrows and numbers. The arrows and numbers were introduced to numeralize space. It's not clear who and when decided to use a number line to teach math, but it must have happened much later than the first numbered Roman milestone was erected.
At some point, some Ancient Roman must have realized that the milestones are not only telling the distance to and from Rome. On the same road, they allow to figure out the distance between any two of them without counting the calculi again, simply by subtracting the smaller number from the bigger one.
The "as the crow flies" distance between Boston and New York is 190 miles. From New York to Boston it's 190 miles too. Without wind and other encumbrances, the fly time and the energy expenditure will be the same. Subtracting a bigger number from a smaller number today, we get a negative number, which must be smaller than a positive number, but represents the same flight or the stretch of the road!
How to subtract a negative number anyway?
Consider the distance to Rome. To find the distance between 0 and 5, we subtract 0 from 5. OK then, what's the distance between -5 and 0? Of course it's 5, but how to find it from the numbers? I mean, how else can we do it?
3 Space Walks
To obliterate any doubt, we can move 0 and 1. After all, 0 is just our observation point, and 1 is the end of our measuring stick. Moving 0 and 1 while keeping the distance between them the same, changes every number by the distance of the move. The distances between all other points remain unchanged.
| |
-2__-1___0__+1__+2__+3__+4__+5__+6__+7__+8__+9__
| |
-4__-3__-2__-1___0__+1__+2__+3__+4__+5__+6__+7__
| |
-6__-5__-4__-3__-2__-1___0__+1__+2__+3__+4__+5__
| |
-9__-8__-7__-6__-5__-4__-3__-2__-1___0__+1__+2__
+6 - +2 = +4 - 0 = +2 - -2 = -1 - -5 = 4
I set up two pins. They were at +2 and +6. I moved 0 and 1 two 1s to the right, and the pins appeared to be at 0 and +4. Then at -2 and +2. Finally, at -5 and -1. The distance between the pins has always been 4, and nothing else. A child of any teachable age understands this.
The static example with +5, 0 and -5 can be easier to remember, but a sliding number line allows to enter the weirdness, go through it and exit from the other side. I offered both to TLG and she understood them quite easily.
The moving 0 and 1 could have been studied with a ruler, or, better yet, a measuring tape with 0 in the middle. Swapping 0 and 1 and changing the distances between them is even funnier. The tricks like these are essential for understanding number line.
Early arithmetic is about counting and whole numbers. A number line looks like a collection of the numbered points. The kids ignore the line, they count the points to find the numbers. Without the line and the distances though, we would not even find the points to count. It's just a bijection? OK, but why this one?
4 Welcome to Debt Hole
The original meaning of "negative" quantity most likely was: it's a void, but we know how big it is. Debt comes to mind first, and debt is essential for early education. The notion that getting rid of something you owe makes you richer is very obvious.
And it's a rich subject, conceptually. Mom gave you $20 to pay to school cafeteria. It turned out, US Government just realized that if it thinks it has to teach every child, it has to feed them first, and now your family do not owe anything. Somebody surely became $20 richer, but was it you or your mom?
In practical calculations, we avoid negative quantities and numbers by separating negative and positive spaces. We say we have debt. A shopping list is the negative inventory, but do we care? We just use different names for them. If we need to bridge the negative and the positive areas, we count from a negative number to zero using positive 1s.
In this post, I demonstrated how it is possible to represent negative values as holes and build shnumbers out of them. A hole is a good metaphor for debt because holes are inversions of things. An abstract thing can be it's hole turned inside out.
Adapting the hole model for a whole-number line is easy. It can also cover real numbers (see X-bag).
Suppose there is a stack extending from minus infinity to some point on the right from zero. It's neither negative, nor positive. It's just a stack, and it's the same everywhere. For some reasons, we consume this stack. Maybe it's edible (and tasty) or valuable. Can it be a stack of coins? Or cookies?
Turning the following pictures 90° counterclockwise would not be better. I tried.
Every once in a while, we subtract a piece of the stack - a cookie or two - and spent it. At some moment, we came to zero and learned that the left part of the cookie stack belongs to somebody else. To infinitely many owners, probably, because it's infinitely long. Can it belong to school cafeteria? Good thing, the owners agreed to lend us some of their cookies. By the way, we can turn the table and imagine it's us are lending cookies to someone else.
This way or another, somebody keeps chewing on the stack digging the hole in the wall, so the hole - the distance between the wall and the end of the stack - is growing. Every time a piece of the stack is subtracted, the same quantity adds to the hole.
To the left from 0, we can't count the cookies anymore. At any moment there is an infinite quantity of them left. The debt hole brings about a new king of quantity because we can measure the depth of it. If we decide to pay it off, we will be adding to the stack and subtracting from the hole. To return from NCW (net cookie worth) of -23 to +19, one got to pave the way with real cookies, as positive as they get.
Why cookie line is so understandable, and number line was not? Consider the distance from NCW 5 to NCW 10, It's equal to 5, right? However, when we go from 10 to 5 we take 5 cookies, and to go from 5 to 10 we have to bring 5 cookies. Dealing with things in space we see or imagine what we do.
Another reason cookie line if friendly: the negative space is meaningfully different from the positive space. Coloring number lines is nothing compared to debt hole.
5 Poor Physics
Of the purrpular math wisdoms, I would single out the idea that negative is bad and positive is good. Just think of electricity. Technically, however, the hot and cold cubes are worse, not to mention balloons and weights. That was, of course, my incompetent opinion.
The cold and hot cubes seem to be wrong in so many aspects, I don't even know where to start. Besides, this mathematical physics is too hilarious. OK, thermometer scale is not a number line - there is absolute zero - and positive degrees are no different from negative degrees. A hot cube can only affect the temperature of the "stew" becoming cold. Once it's heat passed to the stew, there is no way to take it back, unless we employ one of those creatures called Maxwell's demons. If cubes were heaters, they would need energy (wires?), and if they were coolers... OK, OK. The idea is perfectly fine. Didn't the teacher mention that the cubes are magic? If magic is that they teach, this explains everything.
Without magic, school math to me is the best taught subject. Instead of using physics, let alone teaching it, schools screw it up in every possible way. But - please excuse me - what positive knowledge can be built on fundamentally wrong analogies?
For centuries, physicists and engineers had been converting every quantity to the scales of line segments so we could see them. The 0 on thermometer scale simply appears closer to our operating range. The situation with time is similar, but harder to understand. Time could have begun too long time ago, and we don't know exactly when. Besides, our BC/AD system is lacking 0.
If we can avoid negative numbers counting money and things, and even thermometer scales belongs to the positive half of a number line, where in the world number lines and coordinate systems based on them are being used?
In math, of course. Math is a part of this world, and the biggest one. Otherwise, Euclidean space (generalized as a metric space, or space with distances) is a model of physical space, period. The answer to "the minus of a minus" must be found between the two. If it was not understood, we got to teach better.
Magical cooking and gravity are distracting at best. Another analogy - the one with balloons and weights - no matter how oversimplified, is about two forces, and forces are spatial.
The mathematical model fits only if physical space is isotropic (or, at least, isotropic enough). In isotropic space, there is no zero point. The distance to the first point and direction to it is not marked. Where did all this stuff come from?
From our imagination. Coordinate system is our body in space. Call it an avatar if you like. The point marked 0 is our point of view. It can be anywhere because we assume that space has no beginning. The point marked 1 is the end of the body part, which we use for measuring (like, for example, cubit or foot). The rest of our spatial intuition is derived from experimenting with solid objects, but it only makes sense if we imagine ourselves among them.
Unfortunately, our spatial experience is crippled with gravity and limited by Earth. Another complication is our body plan. We are most capable in one direction, which we identify as right and left. That's barely enough to operate on a two-dimensional plane. The common Cartesian 3D coordinate system is for the monsters with three arms.
A number line, which is a 1D coordinate system, is not so demanding. To play with distances, I use a compass or tile a number line with material objects. Cubes (unless one wants to promote them) are just not right for this. I have matchsticks.
6 Through the Looking Glass
A number line would look like it has a mirror in the middle, if not for those annoying minuses. Let's use colors instead, or stick less obnoxious labels like left and right.
Mirroring provides another look at the "minus of a minus" problem. Here is what we have:
1 = -1 -1 = 1
Everything looks perfectly normal, is it?
But then
1 = --1 --1 = 1
Indeed, if one minus makes blue red and red blue, two minuses circle the colors around.
What makes students miserable is just a notation, a sign of subtraction before a number. When I learned about negative numbers, I thought that the minus signs in front of them meant unsatisfied subtractions. The hungry monsters were ready to take a bite from a positive number or swallow it completely. It did not take long to understand that positive numbers were treating negative numbers just as nicely.
This left a strong sense of misgiving. If we bring together 1 and 1, they will be 11. While -1 and 1 turn into -11, 1 and -1 become 0. Finally, -1 and -1 make -2. Something is profoundly wrong here, don't you think? I do.
7 Do We Live in Burrows?
I'm afraid, the content of this chapter must be X-rated. Nothing from it may be used or mentioned at school. It will not end up well.
Using the red-blue notation, how could we perform the operations? Is a blue number just a negative number without the minus sign?
It can be. Or it can be positive. Or both red and blue can be positive. Or negative. Only the rules of operations would be reworded. Like, for example, in case of all-positive we would add blue and red to find the distance, but we can keep calling this addition subtraction like teachers told us to do.
Seriously, we have three points in a line: A, B, C (in that order). We know the distances from A to B and from B to C. What's the distance from A to C? How school messes up our sense or reality to make this problem not obvious?
Remember the stack of cookies? In this model, the negative and the positive values were different. Debt was a hole, no matter what color I might paint it.
In space, the reality on the left from zero is no different from what we see on the right. Moving, mirroring, rotating the coordinates do nothing. We certainly can not turn open space into a burrow or change it in any way.
Familiar number line is very weird. The minus signs on it do not just tell that their bearers are located to the left from zero. The negative values must be smaller than their positive counterparts. Please excuse me, a space like this is my home. It's not what I see.
Remember, the distances related to the negative numbers are all positive. Why negative numbers are smaller then?
Because teachers told us so. Yet it only makes sense while we are counting (measuring) things. Things can be arranged in space. They are not space. Space itself is different. It has no zero, to begin with. We can move zero along, making the negative places positive and the positive places negative.
Dealing with coins and cookies, sticks and ropes, paper and plywood, water and sand, we know what is zero. Zero means we have no coins or rope. As long as we exist, we can't have no space.
Suppose an Ancient Rome road was there before Rome was build. Then, the builders put the golden milestone in the middle of it. Could it make one half of it negative? Or can it just have two positive halves?
On the ancient road, we could use North and South miles. Adding North to North and South to South, we would move away from the golden milestone. Subtracting North from North or South from South would brings us closer to the golden milestone first, then to the other side from it. Adding North to South or South to North would be like subtracting the same direction miles. Subtracting South from North or North from South would be like adding the same direction miles.
A bigger number would mean the bigger distance from the golden mile, either way. The North and South numbers could be negative or positive. A negative North would be South and the negative South would be North. Now, would the distance between a South milestone and a North milestone be South or North? Or something else?
In space, there would be no reason to prefer North or South. Not so with numbers. It is usually assumed that the natural numbers are positive. Even if they are not, the positive whole and real numbers are bigger than the negative numbers.
Negative numbers make sense for counting/measuring things. Applying them to space was an insanity, which schools forced us to accept. Well, the unified model works. It covers not only space, but credit too. It just challenges our intuition, particularly so because the number line itself is a one-dimensional space.
But then, how many people look at those minuses every day? How many of them cry foul before the teacher asked to subtract? And are you still sure that the students who have troubles understanding number line are the dumbest?
8 Stick and Stake
Since a number line is a matter of distances, can we number them? We can, but some teachers may find the following instructions disturbing. Don't try to do it at school.
I took a liberty to introduce a new unit of measurement: one shoe. And it's not just a name. I will actually use two shoes as the measuring sticks.
The following would be three right shoes (or call it +3 if you like)
Then, here is 3 - 5
Shoes are vectors in disguise, and they are countable. I can use shoes to introduce displacement. And again, I don't see the "minus of a minus" trouble with them.
That's how I make a negative shoe positive and a positive shoe negative. It's so obvious that I could safely label the left shoe with "-" and the right shoe with "+".
Sure enough
Walking through the looking glass is not different from counting away, but it may seem weird. The 1st and the 2nd shoes have the 1st shoe between their heels and the 2nd shoe between their toes. Either way, it's 1 shoe. Both the left 1st and the right 1st shoes are between their toes and they have 0 shoes between their heels. Well, we wanted two infinities.
Speaking of 0, there is no shoe for it. A shoe is a shoe, and 0 is nothing. Zero is simply no shoes.
Not only shoes can be negative and positive - they can walk negatively and positively. This will be useful for multiplication.
We can walk through 0 in four different ways and represent our paths accordingly. It can be counted in the positive (forward walking) shoes, left or right, or in the negative, backward walking. The count will be positive and distance will be the same. We just measure it differently. To me this proves that, unlike the numbers, the shoes cover the space adequately.
In one-dimensional space of a number line, a right shoe can be a plus one, and a left show a minus one, so we can start incrementing and decrementing. We can also play with real numbers. Important difference is that a right shoe is no smaller (let alone better) than a left shoe.
Remember the lack of 0 in BC/AD? I believe, this "the measuring stick" or just the stick model of spatial quantity emerges early in life even if we don't teach it. It works while we operate with small distances using our body parts and seeing the whole scene at once. Transition to the "measuring rope with two stakes" or just the stake model is clear. As we begin measuring and think big, we become small. Our body is a point compared to a mile. That's when we get milestones, number lines, an extra point for every stretch of space and zero as a number. A point can stand for 0 because a point is nothing too.
Two models coexist, and the differences between them causes troubles in learning. The stick is better for the scale of the school number lines. A countless number of times, I had to tell to kids: Never mind the points, count your steps, draw the arches.
An extra point or stake, which messes up counting and disappears if we join the ends of a segment to make a contour, is particularly troubling. I can easily think of toys to address and eliminate this issue long before school - once I searched for them and could not find anything - but this would be too much of a digression.
I believe we have to teach the stick model explicitly and compare it to the stake model at every opportunity. To me, the stick model is just a shnumber line, doubled up and enhanced with vectors. Vector is another "advanced" concept, which very young kids can learn and use in two dimensions.
9 The Minus of a Minus
Trying to read the teacher's mind (because I have no choice), I can only assume that she understood the red minus in
a-(-b))
as a function. It could have been framed as a function, but it was not. To my uneducated eye, we are dealing with three different roles of this humble glyph.
The outer (red) minus must be a sign of operation of subtraction, which in our numeral system is distinctly different from addition, and no algebra can change this fact. I mean, subtraction can be be merely an addition of a negative number, but dealing with numbers we have to subtract.
To my taste, the common method of subtraction is too different from addition. In STEREO LEARNING I described a fully reversible addition, but it needs to be reversed and applied anyway. In the end of this article I will demonstrate a numeral system, in which subtraction can be diced altogether, but this system is for another planet.
Subtraction is a binary operation. It takes two operands, and the minus sign stands between them.
Next meaning: a minus can indicate a negative number. If it stands before the number, this number must be subtracted from 0.
It looks like nothing else can be found in the Books of Education. A sole minus what does not indicate a negative value like in -(-5) appears to throws students and teachers alike into cognitive dissonance.
The painless way to understand a single minus before a number or a variable, is to consider it to be a sign of a unary operation, which makes a "negative" value "positive" and a "positive" value "negative". I prefer to call it inversion. From the point of view of arithmetic, inversion can be understood as a subtraction from 0 or as a multiplication by a negative 1.
I told more about inversion in this post. Let me also mention that unary operations do not require parentheses, affectionately known as "banana peels". You may just go through the signs in -----------7 and discard every pair of them because they cancel each other.
If the number of the minus signs was odd, the last of them would survive, and we have to subtract. If the number of the minuses was even, we have to add, but there is no + signs in expressions like
a - -b
I can be wrong, but, as far as I remember, the bulletproof method of pre-reform math was to replace every two consecutive minuses with a plus, non regarding to the arity of them. Banana peels must prevent students from doing this.
10 Why the Horrors?
We can explain, and a reasonably smart modern children can understand this explanation by their 7 or 8 years, provided that we teach them instead of dumbing into millennia-old wizardry.
Recently, I taught TLG, who was 6 years and 3 months old, the method of subtraction described in this chapter. It's not even a method, it's a standard algorithm applied without fear, favor or prejudice. It allows to inverse a "positive" number subtracting it from 0, and then inverse the resulted "negative" number the same way to get the original number back.
TLG liked this trick so much that it took me several lessons to return her to the government-sponsored subtraction. As usual, I said that teachers are not going to allow this, but there were other reasons. For example, it's not easy to spot even that …981 = -19.
The big reason behind numerological nightmares is that the numbers below zero are not represented in the ancient Hindu-Arabic numeral system, which we still use.
The concept is a piece of cake. The problem is how to deliver it. Keeping minus in front of every negative numeral wouldn't be very unhelpful. Alternative methods include using different colors, underlining/overlining the numerals or turning them upside down. I decided to use three colors. This solution looks ugly, but it works.
I am going to use a handy quinary notation with the numerals 2, 1, 0, 1, 2. One negative numeral would be enough. I just wanted to ensure equal (balanced) representation.
I provide the chart of addition (the left one below) in diagonal form, as introduced in SHNUMBERS. The regular positive-only chart is on the right. The stricken-through cells are the coordinates. The numbers on them are the addends. The sum is found as usual. Going through the counting below you will understand how the system works.
Let's count forwward from 0. In parentheses, I explain every number using our trusted Hindu-Arabic system.
1 (1)
2 (2)
12 (5 - 2 = 3)
11 (5 - 1 = 4)
10 (5)
Allow me to remind you that in this quinary or base-5 system the weight of the positions are 1, 5, 25, 125 and so on. Hence, 10 is equal to Hindu-Arabic 5. The numbers between Hindu-Arabic 2 and 5 are made with blue numerals: 11 means 5 - 1 and makes Hindu-Arabic 4.
I didn't have much trouble counting because I imagined a number wheel counter, in which the wheels were changed to accommodate 5 numerals and relabeled.
Let's count forward a bit more.
11 (5 + 1 = 6) 12 (5 + 2 = 7) 22 (10 - 2 = 8) 21 (10 - 1 = 9) 20 (10) 21 (10 + 1 = 11) 22 (10 + 2 = 12) 122 (25 - 10 - 2 = 13)
The numbers look longer because they are quinary. Subtraction is possible but optional. Subtracting, you don't have to put a bigger number first and change the sign. To invert a number, invert every numeral in it. For example, 122 (that's 13) becomes 122 (-25 + 10 + 2 = -13). Then you can just add it.
I hope it was enough to understand that balanced notations stop discrimination of the blue or left-hand numbers (until, of course, we learn multiplication). Likening subtraction of a negative number to multiplication of two negative numbers (in the language of Purple Math, "minus of a minus is a plus" and "minus times minus is plus") was just another instance of educational magic.
"The minus of a minus" problem is very closely related to division by a fractional number (so-called "invert and multiply" rule), but I've never heard of teachers associating them.
I am sure I could teach TLG the balanced notation, but I am not going to do it yet. We have different priorities. I want long multiplication (with distributivity, exponentiation and other bells and whistles) to settle down first. Looks like TLG will be there before she'll turn 7. Then we will have plenty of time for this and many other mathematical treats.
The balanced notation can replace the positive-only one. It leaves up to us how to use it. We can still assume that the red numbers are bigger or count the shoes.
11 Conclusions
Negative quantities and numbers are being mistaught, and, upon closer examination, I found that it's much worse that I thought. Number lines, widely used as a helping device, are particularly bad. They complicate understanding of operations and they are downright confusing for marking up the space.
As you could see in the debt hole chapter, in the model using things in space, in which a negative number is an empty space (hole) extending all the way to positive infinity, and a positive number is a filled space starting in negative infinity, the operations are very natural and easy to understand.
This model is not at all exotic. We still see it in liquid-filled thermometers, in bar indicators (usually simulated on computer displays) and in bar diagrams. The modern designers often spoil it, drawing the negative values as the bars extending from 0 to the left.
The nature of the hole is different, but it's an interval. I taught TLG that the negative numbers are holes without using the metaphor of a debt hole represented as a burrow. It was just a filled and unfilled space, always counted or measured from zero, which is the only possible way to do it. Adding a positive number to a negative number, we fill the emptiness first. Adding a negative number to a positive number, we clean some space.
TLG quickly learned to use regular number lines. Every time I saw her confused (and I am good at it), I could say: "Remember, a negative number is a hole". In her experience, it wasn't a debt hole though. Handling shnumbers on a pegboard, she did have the holes.
Using number lines in coordinate systems, we parents have to understand, remember and teach that a number line is a special case and a common implementation of a general model with two infinities and zero between them. The negative numbers have nothing to do with space. We have and use some other notations like, for example, the relative or cardinal directions, if not explicitly vectors.
