The Catchy Nonsense of “Two Negatives Make a Positive”

My 6th- and 7th-grade students are pretty effective at calculating with negative numbers. They all know, for example, that 5 – (-2) = 7. Ask them why, and you’ll hear this:

“Because two negatives make a positive!”

Then, if you listen carefully, you will hear something else: the low rumble of my teeth grinding together with tectonic force.

“Two negatives make a positive” is one of those math slogans that drives me crazy, because it is so pithy, so memorable, so easy to apply… while also being so vague and non-mathematical that I’m amazed students find it useful at all.

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We can all think of many, many cases where two negatives don’t make a positive. Rain on your wedding day plus grand larceny on your wedding day does not make for a winning combination, despite what “two negatives make a positive” would suggest.

It’s not even true with negative numbers, where -10 + -30 does NOT equal +40 (although I have seen students claim that it does, citing “two negatives make a positive” as their justification).

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In fact, that’s one of my major complaints with “two negatives make a positive”: it is such a swift, over-arching generalization that students wind up applying it in places where it doesn’t make much sense.

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In fact, “two negatives make a positive” doesn’t really make much sense anywhere.

What does make sense is a slight variant, less catchy but far more true: “The opposite of the opposite is just the thing itself.”

What’s the opposite of “the opposite of happy”?

Well, “the opposite of happy” is sad.

So the opposite of that is “happy” again.

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For adding and subtracting with negatives, I tend to favor a debt model.

For multiplying and dividing with negatives, I think a slightly more abstract approach is necessary – it’s all about the properties of multiplication.

Good mental models are more effective than mantras like “two negatives make a positive,” I believe. But even if they weren’t – even if the use of mantras led to error-free computation with negatives – I’d still favor the “mental model” approach. Learning new models engenders the kind of rich thinking that math class is supposed to be about; learning new mantras engenders the uncritical thinking of the cult-follower.

 

 

 

23 thoughts on “The Catchy Nonsense of “Two Negatives Make a Positive”

  1. Haha! Fantastic. I never liked that phrase either, for much the same reasons. I do like the visual of the number line and that the second negative simply changes the direction of the outcome. Minus a positive ends you toward the negative direction, but minus a negative switches the direction — toward the positive end. Even my autistic child gets this.

    Love your posts and cartoons. ๐Ÿ˜€

    • Me too!

      That multiplying by negative one changes the direction you’re pointing ( i.e. rotates you by 180), can be used beautifully to introduce the complex numbers, as multiplying by iota changes your direction by half! (i.e. 90 degrees)
      You’re neither pointing forward nor back, you’re pointing somewhere completely different-the imaginary axis!

      I first came across this in W.W.Sawyer’s wonderful book Mathematician’s Delight.

  2. “Rain on your wedding day plus grand larceny on your wedding day ..” I’m not sure rain + grand larceny is a good thing on any day, but I applaud your specificity. Is there a really interesting story behind that, or do you just have a really vivid imagination?

    • “Rain on your wedding day” from Alanis Morissette… and grand larceny because hey, why not?

      (My own wedding day was mostly a matter of dust-storms, only some of them emanating from the dance floor.)

  3. If we want a phrase in English that implies the correct rule without abandoning “negative,” we can say “the negative OF a negative is positive” where the word “of” in many contexts conveys multiplication, not addition.

    Of course, any attempt at a mnemonic or the like is not guaranteed to give deep understanding, but just a few pegs upon which to hang something that is fairly arbitrary. All sorts of mnemonic devices are great for recalling historical dates, lists (e.g., the presidents of the United States by number, the periodic table of elements), phone numbers, etc., where there is no completely logical reason that things should be as they are. For mathematics, that would be things that are conventions (e.g., order of operations) rather than fundamental to mathematics from a logical perspective. We could have had a different order of operations or a different base or a different way of interpreting composition of functions, etc., and things would not fall apart. But having the product of two negative numbers be positive has underlying logic that we can get at in various conceptual ways. as does having subtraction of a negative number being the same as adding the inverse of that number “make sense.” There are underlying reasons to have x^0 equal 1 for all non-zero x that can be explained. And whenever there is more of an explanation than “that’s the convention,” we should help students grasp the underlying logic rather than simply dismiss their need to know by saying, “That’s the rule. Remember it!”

    Perhaps the worst instance of counter-productive use of mnemonics is the teaching of the execrable FOIL, a device that supplants teaching the mathematics (learning how to apply the distributive property) and hamstrings students from actually understanding the general case that they will need for multiplying any polynomials that aren’t both binomials. What a foolish and harmful misuse of mnemonics.

    The situation of “two negatives makes a positive” has the underlying danger of not only the specific error (do we mean addition or multiplication, or doesn’t make a difference?) but spreading the pernicious notion that math is memorization, not understanding.

    • Well said!
      Even I was going to write something like yours similar, I have now written a note to remove confusions related to what exactly ‘is’ a negative number.Many people think they know what is a negative number,which actually is incorrect.

  4. Before everyone had digital cameras, people used film, and there was a negative of the film. If one took a negative of the film negative, one would obtain something that looked like the original photo. This is an example of “A negative of the negative is (just like) the original.”

  5. Subtraction is just the addition of negative numbers.

    I say we introduce negative numbers in the second grade. Then we can get rid of this subtraction nonsense, and reduce the number of basic operations that need to be taught.

  6. During a lecture the Oxford linguistic philosopher J. L. Austin made the claim that although a double negative in English implies a positive meaning and in French a negative one, there is no language in which a double positive implies a negative. To which Morgenbesser responded in a dismissive tone, “Yeah, yeah.”

  7. Actually,Humans have adopted the convention,that whenever they encounter with a pair of elements,having opposite nature(behaviour), it would be nice to name one of them as positive and the other as negative.The words ‘Positive’ and ‘Negative’ are used in mathematics JUST for the sake of showing oppositeness of the two quantities,ideas or elements.

    Consider in Physics.Why have we named electronic charge as negative and protonic charge as positive? Have scientists figured out some ‘tattoo’ on electron showing its negative charge(and same for proton?)? No.
    We could have worked with Electricity by naming the electron charge as positive and the proton charge as negative! Nothing will change! Except the terminology.

    So,in Mathematics too, we find a natural number quantity and a negative integer quantity as opposite.Think that you just now have 100 pennies.So you have 100,with no complex confusion.
    You can have whatever you can by your owned 100 pennies.

    Again,consider, a different situation.This time you ‘owe’ 100 pennies to one of your friends.In this situation,it is far away of thinking what to buy,instead you are to think how to pay him back 100 pennies.

    Quite different situations, more precisely, the situations are ‘opposite’ to each other.
    So,here, we have a pair of opposites,both regarding 100 pennies.So,if in the first case,you have 100 pennies,then how much do you have in the second case? Yes,the opposite of what you had in the first situation.So what is the opposite of the number 100?

    We can nicely smoothen this problem by using the conventional use of ‘SYMBOLS FOR OPPOSITES’,that is, the words ‘positive’ and ‘negative’. (And in the written sense, by using + and -)

    So let the amount you had in the first case be positive(which is more likely to name ‘positive’),so, the amount you had in the second case is Negative 100!

    Strictly speaking,the pair of Positive and Negative of any elements or ideas are ONLY used to denote a sign of ‘oppositeness’.

    Also one more thing I would like to include here,that the sign minus( – ), is a symbol, that we PUT between 2 numbers.The subtraction operation is supposed to be performed BETWEEN 2 numbers.
    Then what does it mean by, for instance, -5?
    From what is 5 subtracted from? Does that literally make sense?
    Here’s the answer:
    -5, is just a short form of the result of the operation 0 – 5,no matter what the result is!
    Obviously, if you have something nothing(that is 0), and still need to give someone 5 things of what you have,then obviously you are forced to do the operation 0 – 5 ! It would be difficult for us to employ special symbols for the ‘opposite’ numbers (for the natural ones) ,so we used the idea of using the symbols of positives to show magnitude of the negative ,and then simple putting a minus sign ahead.
    In short, without loss of generality, -5 is just a compact form for writing 0 – 5.

  8. John Allen Paulos makes the case in his classic book Innumeracy for using a debt model to understand not only addition with negatives but also multiplication with negatives, too.

  9. Great point – when learning about fractions, I remember being told that ‘whatever you do to the top, you do to the bottom’, which is clearly only true for multiplication and division, but not addition and subtraction! Maths teachers need to make sure what they’re teaching is true, before worrying about catchy statements.

  10. Pingback: The Catchy Nonsense of โ€œTwo Negatives Make a Positiveโ€ โ€” Math with Bad Drawings | Math latest scoop

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