My friend Dr24hours sent me a link via twitter to a new piece of mathematical crackpottery. It’s the sort of thing that’s so trivial that I might lust ignore it – but it’s also a good example of something that someone commented on in my previous post.
This comes from, of all places, Rolling Stone magazine, in a puff-piece about an actor named Terrence Howard. When he’s not acting, Mr. Howard believes that he’s a mathematical genius who’s caught on to the greatest mathematical error of all time. According to Mr. Howard, the product of one times one is not one, it’s two.
After high school, he attended Pratt Institute in Brooklyn, studying chemical engineering, until he got into an argument with a professor about what one times one equals. “How can it equal one?” he said. “If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what’s the square root of two? Should be one, but we’re told it’s two, and that cannot be.” This did not go over well, he says, and he soon left school. “I mean, you can’t conform when you know innately that something is wrong.”
I don’t want to harp on Mr. Howard too much. He’s clueless, but sadly, he’s a not too atypical student of american schools. I’ll take a couple of minutes to talk about what’s wrong with his stuff, but in context of a discussion of where I think this kind of stuff comes from.
In American schools, math is taught largely by rote. When I was a kid, set theory came into vogue, but by and large math teachers didn’t understand it – so they’d draw a few meaningless Venn diagrams, and then switch into pure procedure.
An example of this from my own life involves my older brother. My brother is not a dummy – he’s a very smart guy. He’s at least as smart as I am, but he’s interested in very different things, and math was never one of his interests.
I barely ever learned math in school. My father noticed pretty early on that I really enjoyed math, and so he did math with me for fun. He taught me stuff – not as any kind of “they’re not going to teach it right in school”, but just purely as something fun to do with a kid who was interested. So I learned a lot of math – almost everything up through calculus – from him, not from school. My brother didn’t – because he didn’t enjoy math, and so my dad did other things with him.
When we were in high school, my brother got a job at a local fast food joint. At the end of the year, he had to do his taxes, and my dad insisted that he do it himself. When he needed to figure out how much tax he owed on his income, he needed to compute a percentage. I don’t know the numbers, but for the sake of the discussion, let’s say that he made $5482 that summer, and the tax rate on that was 18%. He wrote down a pair of ratios:
And then he cross-multiplied, getting:
and so 
My dad was shocked by this – it’s such a laborious way of doing it. So he started pressing at my brother. He asked him, if you went to a store, and they told you there was a 20% off sale on a pair of jeans that cost $18, how much of a discount would you get? He didn’t know. The only way he knew to figure it out was to do the whole ratios, cross-multiply, and solve. If you told him that 20% off of $18 was $5, he would have believed you. Because percentages just didn’t mean anything to him.
Now, as I said: my brother isn’t a dummy. But none of his math teachers had every taught him what percentages meant. He had no concept of their meaning: he knew a procedure for getting the value, but it was a completely blind procedure, devoid of meaning. And that’s what everything he’d learned about math was like: meaningless procedures performed by rote, without any comprehension.
That’s where nonsense like Terence Howard’s stuff comes from: math education that never bothered to teach students what anything means. If anyone had attempted to teach any form of meaning for arithmetic, the ridiculous of Mr. Howard’s supposed mathematics would be obvious.
For understanding basic arithmetic, I like to look at a geometric model of numbers.
Put a dot on a piece of paper. Label it “0”. Draw a line starting at zero, and put tick-marks on the line separated by equal distances. Starting at the first mark after 0, label the tick-marks 1, 2, 3, 4, 5, ….
In this model, the number one is the distance from 0 (the start of the line) to 1. The number two is the distance from 0 to 2. And so on.
What does addition mean?
Addition is just stacking lines, one after the other. Suppose you wanted to add 3 + 2. You draw a line that’s 3 tick-marks long. Then, starting from the end of that line, you draw a second line that’s 2 tick-marks long. 3 + 2 is the length of the resulting line: by putting it next to the original number-line, we can see that it’s five tick-marks long, so 3 + 2 = 5.
Multiplication is a different process. In multiplication, you’re not putting lines tip-to-tail: you’re building rectangles. If you want to multiply 3 * 2, what you do is draw a rectangle who’s width is 3 tick-marks long, and whose height is 2 tick-marks long. Now divide that into squares that are 1 tick-mark by one tick-mark. How many squares can you fit into that rectangle? 6. So 3*2 = 6.
Why does 1 times 1 equal 1? Because if you draw a rectangle that’s one hash-mark wide, and one hash-mark high, it forms exactly one 1×1 square. 1 times 1 can’t be two: it forms one square, not two.
If you think about the repercussions of the idea that 1*1=2, as long as you’re clear about meanings, it’s pretty obvious that 1*1=2 has a disastrously dramatic impact on math: it turns all of math into a pile of gibberish.
What’s 1*2? 2. 1*1=2 and 1*2=2, therefore 1=2. If 1=2, then 2=3, 3=4, 4=5: all integers are equal. If that’s true, then… well, numbers are, quite literally, meaningless. Which is quite a serious problem, unless you already believe that numbers are meaningless anyway.
In my last post, someone asked why I was so upset about the error in a math textbook. This is a good example of why. The new common core math curriculum, for all its flaws, does a better job of teaching understanding of math. But when the book teaches “facts” that are wrong, what they’re doing becomes the opposite. It doesn’t make sense – if you actually try to understand it, you just get more confused.
That teaches you one of two things. Either it teaches you that understanding this stuff is futile: that all you can do is just learn to blindly reproduce the procedures that you were taught, without understanding why. Or it teaches you that no one really understands any of it, and that therefore nothing that anyone tells you can possibly be trusted.
So the way I was taught to understand multiplication, a x b is a added to itself b times, so 3 x 4 = 3 + 3 + 3 + 3 = 12; a real world example would be I have 3 cartons of a dozen eggs, so I have 12 + 12 + 12 eggs, or 3 x 12 eggs, aka, 36 eggs. Not sure if that was the Oklahoma school system or the Electric Company on PBS that taught me that, but I like that line of thought much more than turning lines into boxes, even if its a better way to model it when doing higher forms of math, I think it builds on something they already know and doesn’t make it something new and scary (though I am not an educator, so I could be wrong)
I think you’re right on being taught wrong “facts”, especially when it comes to hard science like math can be devastating, I didn’t understand where he was possibly coming from with his theory 1×1=2, but did notice he said ” so what’s the square root of two? Should be one, but we’re told it’s two, and that cannot be.” He’s right, though clearly he didn’t get far enough to realize the correct answer is somewhere between 1 and 2.This broken answer stuck in his head and now he’s rejecting all math. I learned early on sometimes teachers were wrong, rather than trying to ensure books are never wrong I think it makes sense to teach kids that sometimes people and books are wrong; fortunately the Discovery Channel these days is chock full of examples (Ancient Aliens, Real Mermaids, etc)
I’ve encountered similar problems from people (of all ages) who insist that the product of two negative numbers has to be negative. I haven’t figured out a way to represent negative numbers with areas, so I make an attempt with an analogy with velocity.
Distance = velocity x time
Positive velocity and distance is to the right, negative to the left. Positive time is the future, negative is the past. So, I argue, if a car is going to the left, it was to the right in the past, so a negative velocity (left) times negative time (past) gives a positive distance (right). (I know this should be displacement, but I’m avoiding nit-picky vocabulary.) The best response I get from this explanation is a pause while staring at my diagrams skeptically. I don’t think anyone quite believes it if they didn’t already.
Do you have any other ways of teaching this?
One analogy I always liked is filming a car reversing, and then playing the footage backwards. Two reverses make a forward.
There’s a sense that comes into play with certain forms of geometry that adds “sign” to areas (and other higher-dimensional analogues).
You think of a chunk of area as a rectangle spanned by two ordered sides. That is, if you stand at one corner, facing into the rectangle, you can extend your right arm along the first side, and your left arm along the second side. More complicated areas (like circles) are approximated by filling them up with small rectangles.
So, what’s a negative area? simple! it’s just a rectangle with the two sides swapped! That is, your LEFT arm goes along the first of the two sides, and your RIGHT arm goes along the second. We consider the rectangle spanned by two edges in this order to have a negative area, rather than a positive area.
And if you’re thinking this is sounding a lot like the cross product in three dimensions, you’re starting to get a better idea of what exactly was going on with multivariable calculus!
To some extent it’s a choice. You can totally have your own system where – * – = – but only if you’re willing to give up a whole load of nice properties that we intuitively expect from numbers (and that we see in physics)
E.g. in the standard system, the equation
-1 * x = -5
has exactly 1 solution but in their system it has 2 and
-1 * x = 5
has exactly 1 in the standard system but one in their system. Start throwing squares and cubes in there and the standard system gets more complex but still doable whereas theirs just becomes unusable.
This is a super-useful property of numbers and is the reason we can do science.
So they’re free to make their cake out of poop but they shouldn’t be surprised when the people who know how to make real cake are not interested in switching to their recipe.
Also, if they can handle a bit of math, ask them what’s
-1 * 0?
and is that equal to
-1 * (1 – 1) ?
and is that equal to
(-1) * 1 + (-1) * (-1) ?
So now, what’s (-1) * (-1)?
@MZH: When I was tutoring math, I too would use a vector-based approach, although I wouldn’t call it out explicitly as such. I would just work with a number line, with positive and negative integers on it. How much is 2×3? Well, we take 2 copies of a segment of length 3, stacking them up in sequence along the positive half of the number line starting from 0. Now how much is 2 x -3? We start with a segment going from 0 to -3, and stack two of them in the same direction to get -6.
Okay, now how much is -3 x 2? I point out that we would like the commutative law to still hold for negative numbers, so it should be the same as 2 x -3, which we have just seen was -6. I point out that we can get there if we interpret multiplication by a negative number to mean “take a segment representing the second number and flip it to the other side of 0, then stack up how ever many copies of that segment.” So we take the positive segment of length 2, flip it to the other side of 0 (by rotating 180 degrees around the origin), and stack up 3 copies of that to get to -6.
And now, we can ask how much is -3 x -2? Using the insight from that last step, we take the segment from 0 to -2, flip it to the other side of 0, and stack up 3 copies of that to get to +6. A negative times a negative gives us a positive.
Interpreting “multiplying a negative number of times” as a flip plus a positive multiplication is consistent with what a student may later learn about multiplication in the polar complex plane, where a multiplication by any number on the unit circle is interpreted as a rotation about the origin. Multiplying by -1 is a rotation by 180 degrees, multiplying by i is a rotation by 90 degrees, etc. We just keep it simple by restricting attention to the two cases that leave us on the real number line.
Using the “we’d like the same laws that apply to positive integers to also apply in this extended domain” is a strategy I’ve used in a lot of contexts, such as teaching the interpretation of fractional exponents. It seems to be helpful in getting students to understand why these things are defined the way they are, instead of just being a bunch of arbitrary formulas to memorize.
Wow. That’s wackier than the guy I saw who was convinced that division was broken because you can’t divide by 0. Or something. He might have been trying to explain how to divide by 0; I couldn’t tell.
@Fergal Daly
I like that -1*0 demonstration. I’ll try that next time the subject comes up.
I also like the -1*0 way, but I think I like it because I am a mathematician. If I have to give a meaning to it for somebody who “does not like” maths, I would rather go for a physics example, like the movie of a car going backwards.
He said “One times one equals two”, not “One plus one equals two”.
Yeah, I know.
The title was a failed attempt at a riff on him. If he’s right, then the value of 1+1 would be uncertain, because the whole number system would collapse into meaninglessness.
I thought I’d be clever with a title that played with that. I was wrong – a bunch of people have complained about it.
Not maths, engineering. 1 * 1 = 2 for sufficiently large values of 1.
In the english language, one plus one equals not just two but two ones, incomplete answers are not allowed .
Eg test : A. 2
B. Apples
C. 2 ones
D. Zero
C is the correct answer because it is complete