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I am trying to teach a teenage person math, but he doesn't seem to be able to grasp the concept of negative numbers and $0$.

Again and again he finds $-4$ greater than $-3$ because he has spent several years seeing $4$ greater than $3$. Similarly he has never experienced adding $0$ to stuff and stuff to $0$ and I am unable to make him understand.

Please tell me a few methods which I can try with him to make him understand the concepts

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    $\begingroup$ My first idea would be to draw the numberline, where greater = to the right. $\endgroup$ – Jyrki Lahtonen May 30 '14 at 9:56
  • $\begingroup$ I agree with Jyrki, the problem doesn't seem to be the negative numbers, but rather the partial order. $\endgroup$ – Git Gud May 30 '14 at 10:00
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    $\begingroup$ I agree with previous comments. May be, introducing first the $0$ (which is a difficult concept) could help before moving gradually to the left of the number line. $\endgroup$ – Claude Leibovici May 30 '14 at 10:08
  • $\begingroup$ @Claude : How do you suggest to introduce the concept of 0? $\endgroup$ – rijul gupta May 30 '14 at 10:35
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    $\begingroup$ If the student has experienced freezing temperatures (sub-zero on the Celsius scale), show him a thermometer and talk about his experience at various temperatures, especially at negative values (below zero). You could put the thermometer in a freezer to get a negative reading. If you have remote sensor, put it in the freezer, and watch the temperature drop. $\endgroup$ – Dan Christensen May 30 '14 at 13:40

18 Answers 18

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Draw a number line and label all the integers.

Tell him that adding $x>0$ is moving $x$ units to the right and subtracting $x>0$ is moving $x$ units to the left.

Tell him that adding $0$ is not moving at all.

Tell him that adding $x<0$ is moving $-x$ units to the left and subtracting $x<0$ is moving $-x$ units to the right.

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  • $\begingroup$ A reasonable approach. But be warned that number lines can introduce significant 'fences and fenceposts' confusions for someone who really has only a basic understanding of number. $\endgroup$ – jwg May 30 '14 at 11:23
  • $\begingroup$ @jwg - There are fences and fenceposts for all methods. But the number line is indeed a good concept to introduce positive vs. negative numbers. It might be more tricky to teach the concept of irrational numbers, but then there are other methods for that. Or do you mean that the idea of a number line will CAUSE problems later on? $\endgroup$ – Alec May 30 '14 at 18:53
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    $\begingroup$ +1 for "labelling all the integers" ;-) $\endgroup$ – vonbrand May 30 '14 at 20:41
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    $\begingroup$ @Aleksander, I mean that when you work with a number line there is a subtle distinction between numbers as points (fence posts) and numbers as displacements (fences). For people who are familiar with vectors, the Cartesian plane, etc. it is not at all troubling. For beginners it can lead to huge confusions. $\endgroup$ – jwg Jun 2 '14 at 7:50
  • $\begingroup$ "adding 𝑥>0 is moving 𝑥 units to the right" — actually, it is moving x units in the direction in which the numbers become bigger, which can be to the left or to the top or wherever the arrowhead is directed. For this reason, drawing the number line with two arrowheads makes no sense. $\endgroup$ – Rusty Core Dec 5 at 20:53
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Again and again he finds $-4$ greater than $-3$.

Ask him who is richer, he who has a smaller debt $($like $3$ rupees$)$, or he who has a bigger debt $($like $4$ rupees$)$, assuming both persons have no money, just debts.

He has spent several years seeing $4$ greater than $3$.

A debt of $4$ rupees is indeed bigger than one of $3$ rupees. But the one that owes less is richer than the one who owes more. There's a difference between owing and owning. :-)

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    $\begingroup$ Then his problem lies with basic or elementary logic, not math. Perhaps his strength lies in other kinds of disciplines. $\endgroup$ – Lucian May 30 '14 at 10:56
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    $\begingroup$ IMO identifying negative numbers with debt is a fake 'origin story' and something which is only 'intuitive' to those who already understand negative numbers. Figuring out who owes/is owed more or less introduces several complications into the question What is 5 - 8? or What would you add to 3 to get 0? $\endgroup$ – jwg May 30 '14 at 11:22
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    $\begingroup$ I think you can see the problems in this if you try and use it to justify subtracting a negative being the same as adding a positive. What does it mean to owe someone an amount which is itself a (negative) debt? $\endgroup$ – jwg May 30 '14 at 11:26
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    $\begingroup$ I would downvote this answer if I could. I think somebody who doesn't graps negative numbers would really not understand this example, as it is counter-intuitive at best. $\endgroup$ – Arlaud Pierre May 30 '14 at 12:46
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    $\begingroup$ I disagree that one needs to first understand negative numbers to first understand debt. Indeed, somewhere I read the story of a foreigner who was teaching math in Brazil, and complained about how hopeless his students were at arithmetic. He later learned that his one of his "hopeless" students was running an informal but sophisticated bank on the side, and had no difficulty tracking debts or figuring out interest. $\endgroup$ – Frank Thorne May 30 '14 at 16:05
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Apologies, this should be a comment on the answer provided by @Jasper Loy but I don't have enough rep on this site.

I just wanted to add that in my experience, struggling students have an easier time grasping negative numbers when the number line is oriented vertically rather than horizontally. I think we as humans naturally make the 'up=greater, down=less' association; while the 'right=greater, left=less' association may seem natural to those of us with a stronger math background, it usually isn't natural for students like yours.

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    $\begingroup$ This seems immensely helpful I'll try it and tell how it went. +1 $\endgroup$ – rijul gupta May 30 '14 at 15:03
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    $\begingroup$ So a useful example could be distance above sea level. Also try distinguishing the concepts of magnitude and order: 3m above the surface is a smaller distance than 10m below it, but it's still higher up. $\endgroup$ – deltab May 30 '14 at 15:37
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    $\begingroup$ Likewise, consider the floors in a building with one or more sub-basements. (Especially a good example if you live in a country with the convention that the phrase "1st floor" refers to the floor above ground level, unlike here in the US, where "1st floor" and "ground level" are used interchangably." $\endgroup$ – mweiss May 30 '14 at 16:25
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I want to post this answer just to give show any future prospectors what did work for me in this particular case! and also because it's tooo long for a comment (pardon my over emphasis)

I mixed the concepts of number line first horizontal, but then as he found it difficult I showed him a vertical number line as suggested by @JVL, along with pictures of water level where I marked a portion in middle as 0L.

I also demonstrated the fundamentals of richness in debt as suggested by @Lucian, since the child understood the concept of debt before hand, as he has had such experiences (probably because of being from an economically backward background, or maybe just like that) It was quite easy for him to grasp that he was richer when he was in a debt of Rs 3 rather than when he was in a debt of Rs 4 (I live in India, that's why the rupees)

I have chosen the answer by @JasperLoy because number line was my first approach, and I had given him quite a few problems to work on and he had already grasped quite a bit just by thinking which numbers lied at which side. I had also taught him addition on number line before posting this question and he seemed to be doing better now, even though there were numerous conceptual mistakes.

I would also love to declare that I believe I have finally taken him across this initial strata of basic number line and can soon take him on a fast track (he's quite intelligent, just lacks formal education).

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Perhaps teaching about other negative numbers would be easier, like $-1 > -2$ or $-100 > -200$.

Some possible approaches:

  • Suppose you have $5$ apples. If I took from you $3$ or $4$, in which case would you have more?

    Now suppose you have no apples and again I take $3$ or $4$ from you (that means, should you get a hold of some apple, you have to give it to me, until I have $3$ or $4$ respectively). In which case would you have more?

  • Some elevators have levels numbered via negative numbers (often basement or car parks). You can ask then, which level is on top of the other $-1$ or $-2$. If there is no such lift around, you could talk about stair steps or ladder rungs that leads to basement.

  • Sometimes the temperature in the Arctic drops below $0^\circ$. When it would be warmer, at $-30^\circ$ or $-40^\circ$? You can also turn it around, i.e. when it is colder, at $30^\circ$ or $40^\circ$.

  • If the kid is interested in geography, you could talk about depressions (i.e. places with elevations below sea level). Which one is higher, Lake Asal in Djibouti or the Dead Sea? You could also talk about some other forms like the Mariana Trench, such curiosities may spark some interest.

I hope this helps $\ddot\smile$

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I work with students with disabilities and often have to teach them about negative numbers. I tend to use an elevator analogy that I borrowed from a math methods class I took years ago; I create a fictional building that has 5 levels above ground, a ground floor (0), and then an underground parking structure with a P1, P2, P3, P4. Then I tell them we're going to call the floors below ground, -1, -2, -3, and -4. Using a diagram of the elevator shaft, I can have them do both addition (moving up a certain number of floors) and subtraction (moving down a certain number of floors). When they add and subtract negative numbers, you end up moving in the opposite direction - so adding -5 means to move down 5, and subtracting negative 3 means to move up 3. Using something like this, it becomes clear that -4 is less than -3, since you have the floors in the order they would appear in the building...

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    $\begingroup$ Reminds me of the classic version of SimCalc MathWorlds elevators. intec.concord.org//t3/sc/sceldown.gif $\endgroup$ – JPBurke Jun 22 '14 at 0:14
  • $\begingroup$ Neat! Where do I find the SimCalc software? How far back does their use of the analogy date? $\endgroup$ – James S. Jun 22 '14 at 23:55
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    $\begingroup$ We built SimCalc Elevators in 1994. UMass Dartmouth owns the software. I don't know if any elevators activities still exist in the curriculum. kaputcenter.umassd.edu/products/software $\endgroup$ – JPBurke Jun 22 '14 at 23:59
  • $\begingroup$ Here's a paper on some of the early SimCalc work in which you can see the elevator world. Mostly, the software was designed to research learning about the mathematics of change and variation. sriinternational.com/sites/default/files/publications/imports/… $\endgroup$ – JPBurke Jun 23 '14 at 0:02
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Try to introduce this with his own age. Let's see he's 8 years old, ask him what was his age 4 years ago, then he will say 4 and after you ask him what was his age 3 years ago and he will say 5 so then it will maybe be easier for him to understand that -4 is smaller that -3 because 3 years ago he was older than 4 years ago. It's a simple solution that can work, but all child are differents. Hope this help. Concepts are way more easier to learn than facts.

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Just with respect to the concept of negative numbers being greater than or equal to each other - maybe try talking about it in terms of things on earth, in the air, and under the ground. Things underground are negative, things on the surface of earth are zero, and things in the air are positive. The higher up you are the greater you are. In this system, 1 foot under ground is higher up (more positive, greater, etc.) than 2 feet under ground. Three feet under ground is even lower, and so on.

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I would begin with the definition of the opposite of a number as each of the numbers in a pair:

such as $1$ and $-1$ or $-\frac{1}{2}$ and $\frac{1}{2}$, also called additive inverse.

Then introduce the property of opposites: for every real number $a$ there is a unique real number $-a$ such that $$a+(-a)=0.$$

In the above equation you can ask him to substitute the familiar positive numbers in for $a$. This will then provide the motivation for the natural conclusion that a negative number is what you add to a positive number to obtain zero, as the property of opposites states.

If you feel ambitious, you could use the property of opposites as a special case to motivate the definition of subtraction.

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  • $\begingroup$ This sounds overly theoretical but has the advantage of being easily understandable and testable. I wouldn't say real number though. $\endgroup$ – jwg May 30 '14 at 11:25
  • $\begingroup$ The real numbers can also be defined at an elementary level as any number that is either positive, negative, or zero. $\endgroup$ – skullpatrol May 30 '14 at 11:44
  • $\begingroup$ You are missing the point. If you define the real numbers to be all numbers, then why not just say all numbers? $\endgroup$ – jwg May 30 '14 at 11:46
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    $\begingroup$ I am referring to the imaginary numbers, which they will learn about and find it harder to understand if the teacher lies to them in the beginning by saying that the real numbers are ALL numbers. $\endgroup$ – skullpatrol May 30 '14 at 13:03
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    $\begingroup$ @skullpatrol - All teacher lie at all levels of education. $\endgroup$ – Davor May 30 '14 at 13:28
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Negative numbers are strange beasts, since they're completely artificial constructs. Perhaps we have forgotten how difficult they are to comprehend, since we learned them so long ago?

Negatives are useful for modelling phenomena like opposite directions, debts, etc. but those phenomena themselves contain no negatives; speed is an absolute quantity regardless of direction, debts are sets of rules for moving around absolute quantities, etc.

They're also useful for scales, like temperature, but we can only ever measure absolute quantities (eg. the length of mercury in a thermometer).

There is an excellent analogy to be made with Complex numbers, since they are also artificial constructions which cannot be measured; when they're used to model real phenomena, the solutions always turn out to be Real.

I'm not recommending you teach negative numbers by analogy with complex numbers, but that you see which methods of teaching complex numbers work, and adapt them to help you teach negatives. Complex numbers are usually taught to teenagers, eg. advanced high school or introductory University courses, so I imagine that the difficulties experienced may be similar (eg. at a young age, we may just accept the rules at face value, rather than questioning them like a teenager would).

As for the concept of zero, one way might be to represent numbers as sets of things in containers. "Here is 1 apple, here are 2 apples, together they make 3 apples" works very well, but there's nothing to point at for "0 apples". By putting the apples in boxes, you can use an empty box to represent "0 apples".

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    $\begingroup$ Perhaps even natural numbers are artificial constructs? Since every apple is different, can't we only ever have this apple and that apple, and never two apples? $\endgroup$ – jwg May 30 '14 at 13:49
  • $\begingroup$ Speed does certainly have a direction, it is a vector after all... $\endgroup$ – vonbrand May 30 '14 at 20:43
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    $\begingroup$ @vonbrand I think speed is the magnitude of the velocity. In the one-dimensional case the sign of the velocity indicates direction whereas the absolute value is the magnitude. $\endgroup$ – James S. Cook May 30 '14 at 20:57
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Maybe you should introduce him to the concept of distance away from zero. The negative is only a direction (on the previously suggested scale/timeline). Maybe try a few analogies to help him make the learning relevant, or even suggest that he already understands the concepts in practise.

It sounds like he is struggling with conceptual maths, but the actuality is that he will have used these principles at some stage in his life. Your task here is to help him recognise this.

I mentored a young chap when I was at university that had the same challenges, and we eventually got around most of his problems by making the discipline of maths a less threatening beast, with the added benefit of him accepting that maths was not a bunch of irrelevant theories or contrived questions, but something he was already using and would benefit from understanding better. (For the record, he went on to exceed everybody's expectations in his exams by 3 grades)

I hope this helps, and best of luck.

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This:

http://www.tcm.com/mediaroom/video/241887/Stand-and-Deliver-Movie-Clip-Fill-the-Hole-.html

Seriously. Maybe don't phrase it quite the same way, but the "fill in the hole" analogy always resonated with me.

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The explanation to be given to the student depends upon whether you are teaching him Pure Mathematics or Applied Mathematics. I'd like to write my answer from the Pure Mathematical point of view.

In Mathematics a Number is a Mathematical object, in a less strict language, a symbol. E.g. the natural numbers $2$ and $3$, are symbols which represent the concept of being two and being three. The most common use of the natural numbers is to count things, e.g. $5$ eggs -- $7$ books. Mathematics doesn't bother where these objects(natural numbers) are used in real life or not. Mathematics studies the properties of these objects. Natural numbers $a,b$ and $c$ always follow the law:

$$\begin{cases} \text{If}\ \ a+c=b+c \\ \ \ \ \ \ \ \ \ \ \ \ \ a=b \tag{Law - I} \end{cases}$$

The above mentioned law can be understood intuitively by recalling the concept of numerical equality. Let $a,b$ and $c$ represent the number of apples in three different groups. Now, if it is given that $a+c=b+c$, then the only way this can be true is that a and b represent the same number of apples. Law - I can never be proved, so we take it as an axiom.

Another law that these numbers follow is, $$\begin{cases} \text{If}\ \ a+c> or <b+c \\ \ \ \ \ \ \ \ \ \ \ \ \ a> or <b \tag{Law - II} \end{cases}$$

Then, I'd explain the concept of subtraction. The symbol $a-b$ is defined by the equation, $$(a-b)+b=a$$

In case $a=b$ or $a<b$, we don't have any natural number for $a-b$. Again Mathematics does not care where the concept of subtraction is used in real life. It does not care if the $a-a$ or $a-b$( when $a<b$) have any use in real life or not.

We had the law $1$, if $a+c=b+c$ then $a=b$. By the principle of permanence we define the law to be true for, if $(r-r)+c=(p-q)+c$, then $(r-r)=(p-q)$--whatever the meaning of this symbolic equation is. This definition eventualy implies that any symbol $x-x$ doesn't depend upon the value of $x$. So we choose the symbol $0$ for that expression. Also $a-a=b-b=x-x=0.$ By the definition of subtraction we have, $(b-b)+b=b$. This is equivalent to $(a-a)+b=b$

Having defined $0$, I'd explain the negative numbers. A negative number $-b$ is a brief symbol for $0-b$. By the principle of permanence we define all the laws of natural numbers to be true for the symbol $-b$, e.g. commutative, associative etc; whatever the meaning in real life may be assigned to this symbol, it will obey all those laws, even the law-1 and law-2.

$a+(-b)=a+(0-b)=a+0-b=(a+0)-b=a-b$. So $a-a=a+(-a)$.

Consider two numbers $r$ and $s$, such that $b>a$. We have,
$$\ \ \ \ \ \ \ \ \ \ \ b>a$$ $$ \implies \ \ 0+b>0+a$$ $$ \implies (a-a)+b>(b-b)+a$$ $$ \implies (-a)+(a+b)>(-b)+(a+b)$$ $$ \implies -a>-b$$ Briefly, if $b>a$ then $-a>-b$. In particular for the objects $3,4,-3$ and $-4$ we have $4>3$ so $-3>-4$.

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Show the student a globe. Perth, Australia is -32 degrees latitude (aka 32 degrees south where north is defined as "positive"); the capital city of Canberra is -35 degrees. Perth has a higher (more northern) latitude, while the magnitude of Canberra's latitude is larger.

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If I add 10 to -3, I get 7. If I add 10 to -4 I get 6. 6 is less than 7, so -4 must have been less than -3.

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At the beginig of teaching the negative numbers, I prefere to use the temperture, becaues children need a non - abstract example. It will be easy for them to understand that temperture at $0$ is more than any negative temperture. Consequently $-3 > -4$ will be clear.

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Possible Geometrical method for this particular teenager:

Assuming that he has no difficulty with the whole numbers and the non-negative number line, imagine a reflection about 0. Let us call this operation *. Then each positive whole number has a reflected image on the new extended number line. For example, the reflected image of $3$ is $3^*$ and the reflected image of $4$ is $4^*$. If he believes that $a>b$ if $a$ lies to the right of $b$ on the number line, then he should also believe that $3^*>4^*$. As a bonus you get $(3^*)^*=3$.

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There is an intuitive way to do this.

The solution is the common ground between addition, subtraction, multiplication and division. Mathematics has to be thought in a certain order, all "lower" concepts have to be accepted by the learner first before you advance.

Here is an example:

-4 + 3 does not make sense unless you introduce commutativity. -4 + 3 is equivalent to 3 - 4 which is easy to do equals -1. But wait how does he know that we can go below zero?

(-2) * 6 does not make sense unless you introduce distributivity. (0 - 2) * 6 is equivalent to (0 * 6) - (2 * 6) = 0 - 12 = -12. But wait, how does he know 0 multiplied by anything is 0

-2 + 4 + 1 - 3 does not really make any sense as negative numbers are weird, how can you have -4 apples. Teach him to put a zero before the -2 so it becomes 0 - 2 + 4 + 1 - 3. But wait, how can you subtract from zero?

So to cut a long story short, here is the way to teach basic math. It must be precisely taught in this order:

  1. Teach addition of positive numbers big and small (1 to 99999999). Make sure he can add two and triple digits. Behind his back choose only addition of numbers that when added at any level do not go over 10 AND don't have any zeros in them. For example 123 + 451 = 574, 1+3 or 5+4 *Do not choose any exercises that go over 10 for example 5+6 to 399 + 3. If he asks, tell him you'll show him later something extraordinary :D or give him candy.

wait a while and have him do about 500 additions this way and write them down.

  1. Teach the number 0 as when added to another number the result is the same number. 0+1, 0+7 and so on and all the other numbers at step 1. NOW add 0 to all the other numbers you left out at step 1. 499+0 567+0 and so on.

wait a while and have him do about 100 additions this way and write them down.

  1. NOW add numbers like 489+344 and teach him to "carry" the 1 when 8+4 is 12 and he writes a 2 and adds the one to the higher order and so on. He knows how to add zero to things so you can put 560 or 870 to the mix.

wait a while and have him do about 500 additions this way (it might seem like a lot but he can do 50 per day and it will take him 10 days of practice)

That's it you are done with addition!

Notice how I used the fact he doesn't know math to my advantage. The "trick" to teaching anything is "not to fail", only teach cases that enforce the rules he already knows, avoid the exceptions. Only teach special cases when all the necessary building blocks are there for HIM to draw the conclusion on his own. WAIT for him to do that, sometimes he catches on really fast, sometimes it will take a little bump in the right direction.

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    $\begingroup$ You never got to teaching negative numbers, which was the question. $\endgroup$ – rijul gupta May 30 '14 at 13:07
  • $\begingroup$ Given that you claim that this is the one way to teach basic math, I guess I never was taught it. Thanks for this precious info. $\endgroup$ – Benoît Kloeckner May 31 '14 at 20:00

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