Tuesday, 23 June 2015

Multiplication is Ridiculous

Interpret the following: 6X-3

Do you see “6 times negative three”? Do you see “6 times x minus 3”? After all, it could be either. Because multiplication is ridiculous. Yes, I said it. Multiplying: the operation that can be expressed using nothing, or practically any other math symbol already in use elsewhere. Why do we do this to students? Why do we do this to ourselves?

X marks the spot... of no return.



QUICK CONTEXT


The “X” symbol was first used for multiplication by William Oughtred, which was published in his book “Clavis Mathematicae” (1631). However, around that same time, Thomas Harriot was using the elevated dot “ ” symbol. Gottfried Leibniz preferred that notation... supposedly with the complaint that the “X” resembled an unknown “x” too much. (Even THEN they saw this problem coming!) Incidentally, Leibniz also used the inverted U (“cap”) symbol for multiplication, which now denotes intersection.

Owing to general disagreement, it wasn’t until the 1800s that “X” became popular in arithmetic for denoting multiplication. But the elevated dot is still used too - except where it denotes a decimal point in British textbooks (as Ben Orlin found out). But wait, in senior math for North America, the elevated dot is also used to represent the DOT PRODUCT (or scalar product) of two vectors, not to be confused with the CROSS PRODUCT (or vector product) - that’s the one which uses the X. Though we can write dot product as |a|x|b|x(cosC), where those x’s are (of course) multiplications, not cross products.

Confused yet? No? Okay, then let’s make it worse.


PARENTHESES HELL


Oh dear...
In elementary school, the convention is to teach X means multiplication (why??). Later on, with order of operations, we can put math into parentheses/brackets. What’s inside them gets done first, as in (3-1)X(2+4). Yet once you get to high school, we don’t use that X *AT ALL* (provisionally, until cross product), we simply use the parentheses for multiplying. Except now you must do these special "multiply" parentheses AFTER you do any exponents, despite PEMDAS/BEDMAS! It only makes sense - exponents are a form of repeated multiplication, and you need to multiply before you can multiply. Right?

Take a step back now to evaluate: (5)(4)2

If you’re like a LOT of students, you’d write (20)2, not realizing that the expression is really (5)(4)(4). Of course, by the distributive law of multiplication, it still means (20)(20), because the 5 distributes to both 4’s, yeah? (NO! WRONG! Distribute on addition only.) What also doesn’t help is the HORRIBLY inconsistent way brackets are then used in conjunction with exponents in textbooks: (5)4 is actually the same as (54), and used interchangeably, but they’re NOT the same as (-54) because now the (-) is NOT on the base. Granted, I admit I’m guilty of being inconsistent with parentheses that way myself, unless it’s pertinent to the question at hand.

Which brings me to the huge problem: Whether something involving brackets is multiplied or not tends to be inferred from CONTEXT. Which is terrible when we come to f(x), the notation for a function. There it’s NOT multiplication, those parentheses aren’t the same parentheses as before! Perhaps rightly, every year this BAFFLES some students, who invariably see f(x)=5x as some weird multiplying ritual. To find out when it equals 20, they will write 20(x)=5x. Then divide by 20. (Or worse, subtract 20. x = -15, right?) Similarly, ask the student to find “f(3)” and their last step is invariably a division by that 3, unless they learned the context clues.

Okay, so X is confusing, but parentheses are JUST HORRIBLE by comparison. What’s left? Well, context for multiplying can’t be THAT hard, right? How about we express multiplying without any symbol at all! What could go wrong?


MISSING LINK


Perhaps this “no symbol” idea could work - if we weren’t in a place value system!! Combine that issue with early use of “x” as an unknown, and we’re screwed. After all, why can’t 4x mean 4 in the tens place, and an unknown ones place? (Heck, I suppose it can, if we let x=10, so that the ones place is zero.) Then there’s a typical high school expression like “5-2x”. Is it any wonder students combine unlike terms? There’s a subtraction RIGHT THERE, and no other operations! So 3x, right? (Wrong again!)


That's a deadly Sin...
Except now there’s also the problem of OTHER times math uses no symbol, when we DON’T mean multiplication! Exponents being the more obvious one, what with " 32 " not meaning (3)(2). (Except didn’t exponents mean multiplication after all?) As another example, if I write “sin x”, that doesn’t mean “sin” multiplied by x. “sin” by itself is meaningless! Yet despite that, there’s always that one student who will try to solve “sin 3x=90” by dividing out “sin 3”. (And I cry a little bit. Though here’s a follow-up question: Should I cry more or less if they instead divide the 3 before doing inverse sine?)

Just to round things out, multiplication can also mean division. Because if you’re dividing by 2, that’s the same thing as multiplying by 0.5 - and we generally want to do this if we end up with a rational or trigonometric expression that has fractions on top of divisions. (Such as [(x+1)/(x+2)]/2 ) Oh, and what do we use to show multiplication of two rational expressions? Often the X symbol.

I mean, students don’t know cross product with vector notation yet, so they can’t get confused, right? And once we reach matrices, they’ll know 3x4 is a 3 by 4 matrix, not a calculation, yeah?

What. The. Hell.


THE SOLUTION


The thing that bugs me the most about all confusion this is that there’s an obvious solution to it. Computer scientists have been using it for decades. It’s the asterisk/star (*) symbol! Which you have probably used yourself online, perhaps even in writing up a mathematical blog post!

And before you argue that * is a recent addition, the choice wasn’t completely arbitrary. The asterisk was (supposedly) used in Germany back in the 1600s. Possibly even by Johann Rahn, the same guy who popularized the obelus symbol for division. (Though for division, Leibniz preferred using the colon. Bringing this article full circle.)


Enough! We thought WE were your X girlfriends!

In conclusion, I claim the X isn’t working. So why haven’t we switched to * in this day and age??

Well, aside from the fact that traditions are really hard to break (hey, I’ll admit I’m not using it yet), there’s the problem of needing to reprint all the textbooks. Textbooks that schools can’t afford to buy. And online, we might also lose out on Khan Academy videos like “Why aren’t we using the multiplication sign?”. So I guess we’re stuck. Until the computer uprising.

For more on Mathematical Symbol Origins go to this link! Alternatively, read my prior rants about y=a(x-h)2+k or cross multiplication. Or merely comment below.

16 comments:

  1. I vote for the asterisk. It doesn't have any conflicting uses provided one avoids footnotes. Textbooks have the option of changing fonts to make the various uses of x obvious, but it's difficult to do with handwriting. The asterisk also has the advantage of being like the dot you mentioned but being visibly unique.

    Another option is to stop using x, y, and z as unknowns and variables. Those letters get a workout in math, being used for variables and for dimensions (though I will admit the uses are related). The use of c for a constant is natural, seeing the first letter of the word. I suppose this is what happens when a system developed in one language with its own characters is transformed over to a different language.

    ReplyDelete
    Replies
    1. Footnotes or *emphasis*? ;) Honestly though, some people don't pay really close attention to fonts, so I don't consider a font change argument to be very valid. As to the "x", "y", "z", if those were around back in the 1600s, I think they would be worse to change than the X multiply symbol, which is only about 200 years old. And "c" gets something of a workout too (I think the only time it's consistently a constant is when doing integration).

      Ah well - fair point, and math really is kind of it's own language, borrowing bits and pieces from everywhere. And (like some coding languages) at times one wonders as to the syntax.

      Delete
  2. I used to teach fourth grade and we used the Everyday Math textbook. The textbook used the asterisk for multiplication. Many, many people thought it was very confusing. I thought it made a lot of sense - but I was seriously outnumbered. Our district doesn't use that text any more and now I am back at the middle school, but I still use the asterisk sometimes out of habit. Sometimes, even when something makes sense, one has to decide if going against the masses is worth the struggle :-)

    ReplyDelete
    Replies
    1. Fascinating! Hm, I wonder why it was seen as "confusing" - because it was honestly being mistaken for something else, or simply because it wasn't "the way we're used to doing it". Suspect the latter. Of course, I'm not one to talk... I haven't actually tried the * myself, it merely seemed sensible after my rant.

      Good on you for making the attempt! So true about going against the masses; one has one's own sanity to consider. Small steps though, perhaps.

      Delete
  3. Sigh. I just entered something very thoughtful. And I hit Preview and logged in, and it was gone. I'll try again later ... after I retype it in Notepad or something and cut and paste.

    ReplyDelete
    Replies
    1. Looking forward to it. That's one thing I hate about blogger, first click always logs you in and wipes out your content.

      Delete
    2. Wish I'd known that sooner. :(

      Delete
  4. This came up recently in our programming class where someone was using x as a variable and there was multiplication involved.

    While I would agree that * is a useful symbol and could be our go-to choice, I think the broader point is that math has a lot of these cases where context is important for understanding what is happening. As teachers, I think it helps for us to recognize this and have pro active strategies about helping students both establish context and shift context. Also, it opens us up to the possibility that, when we disagree, it isn't because either side is *wrong* but that we are framing the situation from different perspectives . . . even in math class.

    ReplyDelete
    Replies
    1. That's a really good broad point - I wonder if it's a bit like choosing their/there/they're in English, based on the stuff around the word. Because like that issue, constantly reiterating "f(x) is notation" doesn't necessarily fix the problem. I do like that idea of framing allowing both perspectives to be correct!

      I'm also reminded of a comment Glenn Waddell Jr made to me on Twitter: "The fact that we know it is a problem and feed it is our fault. We must be hyper aware of the issue. (same with equal signs)"

      Delete
  5. Second try: Two things

    First, problems like (5)(4)^2 happen because of a little bit of laziness when it comes to teaching the Distributive Property: who uses its full name? The Distributive Property of Multiplication Over Addition. I don't. I probably assume that they've already heard of it in middle school, so why waste the extra breath and board space. (Likewise, I rarely mention, say, the Associative Property of Addition and the Associative Property of Multiplication when I'm reviewing.)

    Both addition and multiplication need to be in the problem for it to be distributive property problem. And yet, I occasionally see area of a triangle answered as
    (1/2)(6)(8) = (3)(4) = 12.

    Second, a problem that I have going back to when I started teaching. Why is -3^2 = -9? No, I'm serious. Why does it need to be written as (-9)^2. Yes, I know what 0 - 3^2 is, but that isn't the same thing. There's an operation there. But -3 is a single thing, a concept, a position on the number line. The calculators even have separate buttons for subtraction and negative to acknowledge this ... And yet they don't.

    I did an informal survey years ago asking colleagues "What is negative three squared?" They all said nine. I didn't leave a pause after "three". On the other hand, when I left a little pause after "negative", I got -9.

    Basically, they verbally heard -3^2 and said 9, and -(3)^2 and said -9. No one would have a problem with the latter, but if I wrote down the former, they would said that that wasn't what they said or what I said.

    However, I come "gotten with the programmed" calculators and textbooks. I don't need to confuse the kids any more. Multiplication will do that enough.

    ReplyDelete
    Replies
    1. First point is a really good one - I've actually told my classes that mathematicians tend to adopt "lazy notation" for things... such as rewriting an expression (like -4+5) with the negative second to avoid a + symbol, or assuming a root is '2' unless written otherwise. (And why not? So long as everyone understands...) It hadn't occurred how it's affecting definitions too.

      I think the second issue goes back to the fact that subtraction and negative often ARE seen as interchangeable. What you say makes some sense, but then we get an expression like the discriminant, b^2-4ac. I have seen a substitution for this be: -5^2-4(1)(3) = (25)(-12) = -300. The subtraction somehow vanishes, and it's all multiplying. Would putting brackets around the (-5) stop that? Maybe not, but for me, it at least makes it clear WHICH of the - is an operation, and WHICH is a value on the number line. Of course, it could be that I'm "programmed" too.

      Delete
  6. This comment has been removed by a blog administrator.

    ReplyDelete
  7. Thank you for the interesting post Gregory.

    6X-3 might also be read as "six X minus three". The Times Roman font capital X is a letter of the alphabet. To be a little bit more mathematical, I recommend using the multiplication symbol, × which you get when you enter 'Alt 0215' on a Windows PC.

    Then insert spaces and unpack, so 6X-3 becomes 6 × (0 - 3). This can now better be read (IMHO) as six multiplied by 'zero minus three'.

    The reason is simple. Multiplication distributes over BOTH addition and subtraction. Unpacking the concepts (usually skipped) means 6 × +3 = 6 × (0 + 3) = 6 × (0 + 1 + 1 + 1) = 0 + 6 + 6 + 6 = 0 + 18 = "positive 18".

    Similarly, 6 × -3 = 6 × (0 - 3) = 6 × (0 - 1 - 1 - 1) = 0 - 6 - 6 - 6 = 0 - 18 = "negative 18".

    So with multiplication on the integers, multiplication involves either repeated addition from zero, OR, repeated subtraction from zero, according to the symbol in front of the multiplier.

    After 1631, William Oughtred must have received a lot of 'negative' feedback about the + and - symbols each having two meanings. In the introduction to his first edition he said the + sign meant either plus or positive and the - sign meant either minus or negative. In later introductions, he dropped the signs explanation and just explained the + and - symbols as operations.

    To make the use of plus vs positive clearer, I follow the old sensible practice of using superscript + and - for signs and normal case + and - for operations.

    BTW, the algorithmic 'repeated addition' definition of multiplication (cited since 1570) is also ridiculous, because it literally produces wrong answers. If you are curious, please goto
    http://www.jonathancrabtree.com/euclid/elements_book_VII_definitions.html

    ALT CODES http://usefulshortcuts.com/alt-codes/maths-alt-codes.php

    ReplyDelete
    Replies
    1. Thanks for the comment! I admit that I didn't give much consideration to font choice (or spacing), mainly because when we write by hand, it's basically up to interpretation. Perhaps I should be a bit more rigorous online, or scan to illustrate. (Some posts I even default to ^ rather than the exponentiation I used here.)

      Thanks for the codes link, I'll try to remember it (often I cut and paste instead). I do find it interesting that we both consider the addition and subtraction going from zero to clarify - and I agree with the "off by one" problem of "repeated addition". (I feel like there's a similar problem when contrasting sequences to line equations.) Hadn't considered superscript, but that makes sense.

      Incidentally, when you first said "Times Roman" I thought we were headed into "Roman Numerals", since X is ten. :)

      Delete
  8. Hello! This post was recommended for MTBoS 2015: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to tina.cardone1@gmail.com whether or not you grant us permission to include your post. Thank you, Tina and Lani.

    ReplyDelete
    Replies
    1. For the public record, I did respond to this comment via email. (It would appear I self nominated by pointing out my most popular math post of 2015. This post has over 550 views, +11 on G+, and the most comments.) Held off on saying anything so that I could point out where you could get a copy of the book. Which is now a reality! Enjoy.

      More information/preview here:
      http://drawingonmath.blogspot.ca/2016/03/the-best-of-math-teacher-blogs-2015.html
      Direct purchase here:
      https://www.createspace.com/6027355

      Delete