8 ÷ 2(2 + 2) = ?
This is really a PoSWW. Except, there are a lot of words.
Above is one of those stupid BODMAS things, which appear in the media about once a month. Except, this one has just been sorted by a couple of Canadian Maths Ed professors, in a Conversation article titled The Simple Reason a Viral Math Question Stumped the Internet. Regular readers will be aware of our method of resolving such questions, but we think there are aspects of the Conversation article that warrant specific whacks.
Have fun.
I think the biggest issue with this article is that it was written at all. It’s like debating about the correct spelling of “colour”.
* “we will explain the definitive answer to this question — and why the manner in which the equation is written should be banned.”
I think it should be banned because there isn’t a definitive answer.
* “Key to the debate, we contend, is that the multiplication symbol before the parentheses is omitted.”
I don’t think so. With brackets, you can omit multiplication symbols in arithmetic, no problem. I think the issue is missing brackets. USBB!
* “had the expression been correctly “spelled out” that is, presented as “8 ÷ 2 × (2 + 2) = ? ”, there would be no going viral, no duality, no broken internet, no heated debates”
I highly doubt that. Pretty sure I’ve seen similar things with a multiplication symbol that sparked just as much debate, if not a Conversation piece.
And then there’s the comments. I should learn, whenever I see one of these sorts of things, scroll on by.
Yes the weird part is they explicitly mention different conventions in the US and Canada and only in a comment reply bring up a third convention. None of which are concerned with how multiplication is implied.
Both authors are from the faculty of education. WitCH solved.
I suppose this is more evidence of the limitations of those authors. They just don’t seem to understand simple logic. I’m stunned at how bizarre this article is. I suppose they were trying to evoke a kind of “mathematical” Eats Shoots and Leaves.
“The real reason, then, that 8÷2(2+2) broke the internet stems from the practice of omitting the multiplication symbol, which was inappropriately brought to arithmetic from algebra. ”
I think this speaks for itself.
That is indeed a killer-stupid quote.
I spy three candidates for Crap in this question, even without the other words:
1. The obelus sign (division sign in case some of my colleagues are reading this) is awkward even at the best of times and really should not be used in conjunction with any other operations. In the first known use of the sign I can find (in German, which took some translating) the sign was used to mean “everything to the left of this symbol divided by everything to the right of this symbol”. In other words
is meant to be read as 
. I understand many teachers would disagree with this and say that order of operations has moved on… Well, so has my argument, better luck next time.
2. There is some debate amongst programmers as to whether implied multiplication takes priority over other operations. In other words writing
should be read as 
whereas 
is read as 
. The debate erupts if you then ask whether 
is therefore 
or 
(actually, debate doesn’t erupt, the dumb silence remains.
3. The
is pretty stupid.
Thanks, RF. The history thing is interesting. I’ll check it out.
If I find the scan I took from the book, I will email it through.
Not saying this is the first ever use of the obelus sign, just the earliest I am aware of.
I looked it up. The ÷ sign was first used to mean division by a Swiss guy in the 17th century, and then caught on in England. Before that it had been used on occasion to indicate subtraction. I couldn’t see any mention of teh implied brackets, but it might have just been understood.
I’m a bit worried that they don’t describe the rule that I teach. This is that multiplication and division are given equal priority and we work left to right. Instead they describe it as a question of which comes first out of multiplication and division. (I see now they mention it in response to a comment.)
Either acronym is terrible if the rule is the one I think is conventional. I honestly can’t remember learning these acronyms at school. I only learned of their existence during my teaching course, but I see they are now widely taught. I don’t understand their appeal at all. They cause confusion and don’t resolve any.
Order of operations is a quite a difficult mine-field to tread when speaking with teachers. BODMAS is one of the most horrific acronyms I know and the PEDMAS, BIDMAS, PEMDAS alternatives are no better.
Many teachers get it wrong and worse, a lot have no idea of why we have such conventions in the first place. “Implied Brackets” is perhaps a better phrase, but only in the sense that VCE is superior to SACE.
I haven’t tried to speak to teachers about it. I don’t feel prepared. I don’t understand the high school mathematics culture well enough yet. Some teachers refer to evaluating arithmetic expressions in general as ‘performing BODMAS’. What are they thinking?
That’s so sweet that you regard “What are they thinking?” is the appropriate question.
What do you think is the appropriate question?
“Why aren’t they thinking?”
In defense of teachers (but only briefly, because I really do think this is a simple issue to overcome, or at least improve), some of these battles are not worth it.
There are so many year 7/8 teachers who are not “Math” teachers. They see themselves as a different type of teacher (yes, apparently there are different types) and their one Mathematics class is just something they have to do to fill up their load. So they do what any teacher does who is asked to do something they are not confident about: go to the textbook.
If the textbook is wrong… but what the textbook says is what the students were told in primary school… then the fight is a bit too difficult. I’ve asked my colleagues to never use the phrase BODMAS in my presence with mixed results and I’ve asked my Year 7 students to never use the phrase either with less success (parents, it would seem, are using the phrase a lot at home)
So, why aren’t they thinking? Because there is no pressure on them to do so and there is plenty of pressure on them to do other things, such as “be a nice colleague”. The battle continues, but in the shadows a lot of the time.
No. The teachers don’t even contemplate there is a fight/issue.
I’ve seen mathematics-as-their-main-subject teachers encourage students to use BODMAS/BIDMAS and hang posters of it, and it’s been in the unit planning documents for Year 7 at both schools I’ve taught at. I don’t think it’s just out-of-field teachers.
Exactly.
You may well be correct. If so, this is quite a large problem and I wish I knew where it all begins (laziness is my guess, but I have been known to be wrong a lot)
As Marty has said a few comments ago, the simple reality that many (perhaps a majority, I’m not sure) teachers do not contemplate there is an issue is… sad.
Why do they not contemplate? Laziness comes to mind again but… I have no idea.
Who knows how much time and effort is wasted ‘teaching’ this BODMAS, PESTILENCE, etc. crap. I’m sure there are hidden agendas somewhere. As has been said elsewhere, the simple use of brackets to convey without ambiguity the intent of the calculation is sufficient. How hard to teach and reinforce this simple thing.
My guess is that calculators have caused a lot of this nonsense. And yes, never underestimate sheer laziness and stupidity. And hidden agendas (maybe DODGYMAS posters would have to be replaced with times tables posters).
I don’t think it’s calculators. Old textbooks have this nonsense as well.
Clearly, I’ve seen this “problem” about 9^9^9 times, but very rarely (at least in the English speaking world) have I seen the “actual” problem(s) being addressed.
Of course, at the basic level, it is just a boring issue regarding a convention, i.e. operator precedence. In every reference manual of every programming language in this universe, this convention is clearly stated on one of the first pages. For obvious reasons. And this leads to the first (but only minor) point responsible for the virality of the question. There is no (at least no widely used and contemporary) “dialect” of mathematical notation (programming language) that uses the obelus as a division symbol and no symbol as a multiplication symbol. So, it’s like writing a line of code that looks like a mixture of python and lisp, and asking for the “correct” output. This alone is stupid, but somehow people (math teachers?) seem to believe that there exists a “universal” / “natural” / “true” interpretation algorithm (parser) for strings (code) that roughly look like mathematical expressions.
But the major point is something completely different. It’s the “parentheses first” rule, which does not work the way most (US-socialised?) people (teachers?) use it. The rule says: Compute what’s inside the parentheses first, right? So let’s do it.
3·(2+1)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=3·(3)=…
Oh, something goes wrong. That’s because we did not set up the rule correctly. The actual rule must be: Compute what’s inside the parentheses and remove them. Without the “and remove them” part, the rule makes no sense. But now … abracadabra … it works.
3·(2+1)=3·3=9.
And NOW you can start formulating the rules that come after the parentheses rule. For example you can address the question as to whether a notation with “no symbol” as the only avialable symbol for multiplication is compatible with the decimal notation for natural numbers. It’s obviously not. And THAT’s the actual point. All the (never ending) discussions about this “viral” problem are doomed from the very beginning, because superficially the “experts” disagree on operator precedence, but no one ever realizes that they all try to use an inconsistent notation and a nonsensical “parentheses-rule”.
Oh, and, by the way, what is 9^9^9?
Thanks, Hans. Obviously I agree with your first point, the flaw in taking conflicting (and bad) conventions and treating them as God-given rules. I don’t get you second point about removing the parentheses. Yes, of course they must be removed to perform the next bit of arithmetic, and this is not explicitly stated in any of the conventions. But how does this lack of explicit statement lead to active confusion or conflict? Lastly, I would never write
, because I can never remember the convention (even though there is a good argument for the convention used).