I wrote a (very long) blog post about those viral math problems and am looking for feedback, especially from people who are not convinced that the problem is ambiguous.
It’s about a 30min read so thank you in advance if you really take the time to read it, but I think it’s worth it if you joined such discussions in the past, but I’m probably biased because I wrote it :)
The ambiguous ones at least have some discussion around it. The ones I’ve seen thenxouple times I had the misfortune of seeing them on Facebook were just straight up basic order of operations questions. They weren’t ambiguous, they were about a 4th grade math level, and all thenpeople from my high-school that complain that school never taught them anything were completely failing to get it.
I’m talking like 4+1x2 and a bunch of people were saying it was 10.
My years out of school has made me forget about how division notation is actually supposed to work and how genuinely useless the ÷ and / symbols are outside the most basic two-number problems. And it’s entirely me being dumb because I’ve already written problems as 6÷(2(1+2)) to account for it before. Me brain dun work right ;~;
I found a few typos. In the 2nd paragraph under the section “strong feelings”, you use “than” when it should be “then”. More importantly, when talking about distributive properties, you say x(x+z)=xy+xz. I believe you meant x(y+z)=xy+xz.
Otherwise, I enjoyed that read. I’m embarrassed to say that I did think pemdas meant multiplication came before division, however I’m proud to say that I’ve unconsciously known that it’s important to avoid the ambiguity by putting parentheses everywhere for example when I make formulas in spreadsheets. Which by the way, spreadsheets generally allow multiplication by juxtaposition.
It’s actually fine to do multiplication before division, you just have to be sure about which numbers are intended to be included in the divisor of your fraction!
Thank you so much for taking the time and reading the post. I just fixed the typos, many thanks for pointing them out.
There is nothing really to be embarrassed about and if you look at the comment sections of such viral math posts you can see that you are certainly not the only one. I think that mnemonics that use “MD” and “AS” without grouping like in “PE(MD)(AS)” are really to blame here.
An alternative would be to drop the inverse and only use say multiplication and addition as I suggested with “PEMA” but with “PEMDAS” one basically sets up students for the problem that they think that multiplication comes before division.
A fair criticism. Though I think the hating on PEDMAS (or BODMAS as I was taught) is pretty harsh, as it very much does represent parts of the standard of reading mathematical notation when taught correctly. At least I personally was taught its true form was a vertical format:
B
O
DM
AS
I’d also say it’s problematic to rely on calculators to implement or demonstrate standards, they do have their own issues.
But overall, hey, it’s cool. The world needs more passionate criticisms of ambiguous communication turning into a massive interpration A vs interpretation B argument rather than admitting “maybe it’s just ambiguous”.
The problem with BODMAS is that everybody is taught to remember “BODMAS” instead of “BO-DM-AS” or “BO(DM)(AS)”. If you can’t remember the order of operations by heart you won’t remember that “DM” and “AS” are the same priority, that’s why I suggested dropping “division” and “subtraction” entirely from the mnemonic.
It’s true that calculators also don’t dictate a standard but they implement what conventions are typically used in practice. If a convention would be so dominating (let’s say 95% vs 5%) all calculator manufacturers would just follow the 95% convention, except maybe for some very special-purpose calculators.
Calculators do not implement “what conventions are typically used in practice.” Entering symbols one by one into a calculator is a fundamentally different process from writing them in a sentence. A basic traditional calculator will evaluate each step as you enter it, so e.g. writing 1 + 2 * 3 will print 1, then 3, then 6. It only gets one digit at a time, so it has no choice. But also, this lends itself to iterative calculation, which is inherently ordered. People using calculators get used to this order of operations specifically while using calculators, and now even some of the fancy ones that evaluate expressions use it. Others switched to the conventional order of operations.
The expression 6/2(1+2) involves both multiplication and division. According to the order of operations (PEMDAS/BODMAS), you should perform operations inside parentheses first, then any multiplication or division from left to right.
Certainly! The expression 6/2(1+2) is ambiguous due to the implicit multiplication. Let’s solve it in both ways:
Implicit multiplication with higher priority:
[ \frac{6}{2}(1+2) ]
First, solve the parentheses:
[ \frac{6}{2}(3) ]
Now, perform the division:
[ 3 \times 3 = 9 ]
Implicit multiplication with the same priority as division:
[ \frac{6}{2(1+2)} ]
Again, solve the parentheses:
[ \frac{6}{2(3)} ]
Now, perform the multiplication first:
[ \frac{6}{6} = 1 ]
So, depending on the interpretation of implicit multiplication, you can get different results: 9 or 1.
I think it’s funny that ChatGPT figured out 1 and 9 but has the steps completely backwards. First it points out what has high priority and then does the exact opposite, both times 🤣
It’s hilarious seeing all the genius commenters who didn’t read the linked article and are repeating all the exact answers and arguments that the article rebuts :)
I’m still not used to having combined image and text posts so I usually don’t notice the text portion if it isn’t a big ol’ wall and I hope I’m not the only one.
❤️ True, but I think one of the biggest problems is that it’s pretty long and because you can’t really sense how good/bad/convining the text is it’s always a gamble for everybody if it’s worth reading something for 30min just to find out that the content is garbage.
I hope I did a decent job in explaining the issue(s) but I’m definitely not mad if someone decides that they are not going to read the post and still comment about it.
“when in doubt” is a bit broad but left to right is a great default for operations with the same priority. There is actually a way to calculate in any order if divisions are converted to multiplications (by using the reciprocal value) and subtractions are converted to additions (by negating the value) that requires at least a little bit of math knowledge and experience so it’s typically not taught until later to prevent even more confusion.
For example this: 6 / 2 * 3 can also be rewritten as 6 * 2⁻¹ * 3 and because multiplication is commutative you can now do it in any order for example like 3 * 6 * 2⁻¹
You can also “rearrange” the order without changing the meaning if you move the correct operation (left to the number) with it (should only be done with explicit multiplication)
6 / 2 * 3 into 6 * 3 / 2 (note that I moved the division with the 2)
You can even bring the two to the front. Just remember that left to the six is an “imaginary” (don’t quote me ^^) multiplication. And because we can’t just move “/2” to the beginning we have to insert a one (empty product - check Wikipedia) like so:
1 / 2 * 6 * 3
This also works for addition and subtraction
7 + 8 - 5
You can move them around if you take the operation left to the number with it. With addition the “imaginary” operation at the beginning is a plus sign and the implicit number you use is zero (empty sum - check Wikipedia)
8 - 5 + 7
or like this
0 - 5 + 8 + 7
because with negative numbers you can use the minus sign to indicate negative numbers you can even drop the leading zero like this
-5 + 8 + 7
That’s not really possible with multiplication because “/2” is not a valid notation for “1/2”
Build two cases, calculate for both, drag both case through the entirety of both problems, get two answers, make a case for both answers, end up with two hypothesis. Easy!
When I used to play WoW years ago I’d always put -6 x 6 - 6 = -12 in trade chat and they would all lose their minds. Adding that incorrect solution usually got them more riled up than having no solution.
What’s especially wild to me is that even the position of “it’s ambiguous” gets almost as much pushback as trying to argue that one of them is universally correct.
Last time this came up it was my position that it was ambiguous and needed clarification and had someone accuse me of taking a prescriptive stance and imposing rules contrary to how things were actually being done. How asking a person what they mean or seeking clarification could possibly be prescriptive is beyond me.
Bonus points, the guy telling me I was being prescriptive was arguing vehemently that implicit multiplication having precedence was correct and to do otherwise was wrong, full stop.
Without any additional parentheses, the division sign is assumed to separate numerators and denominators within a complete expression, in which case you would reduce each separately. It’s very, very marginally ambiguous at best.
👍 That was actually one of the reasons why I wrote this blog post. I wanted to compile a list of points that show as clear as humanity possible that there is no consensus here, even amongst experts.
That probably won’t convince everybody but if that won’t probably nothing will.
When I went to college, I was given a reverse Polish notation calculator. I think there is some (albeit small) advantage of becoming fluent in both PEMDAS and RPN to see the arbitrariness. This kind of arguement is like trying to argue linguistics in a single language.
Btw, I’m not claiming that RPN has any bearing on the meme at hand. Just that there are different standards.
Ambiguity is fine. It would tedious to the point of distraction to enforce writing math without ambiguity. You make note of conventions and you are meant to realize that is just a convention. I’m amazed at the people who are planting their feet to fight for something that what they were taught in third grade as if the world stopped there.
You’re right though. We should definitely teach different conventions. But then what would facebook do for engagement?
I think this speaks to why I have a total of 5 years of college and no degree.
Starting at about 7th grade, math class is taught to every single American school child as if they’re going to grow up to become mathematicians. Formal definitions, proofs, long sets of rules for how you manipulate squiggles to become other squiggles that you’re supposed to obey because that’s what the book says.
Early my 7th grade year, my teacher wrote a long string of numbers and operators on the board, something like 6 + 4 - 7 * 8 + 3 / 9. Then told us to work this problem and then say what we came up with. This divided us into two groups: Those who hadn’t learned Order of Operations on our own time who did (six plus four is ten, minus seven is three, times eight is 24, plus three is 27, divided by nine is three) Three, and who were then told we were wrong and stupid, and those who somehow had, who did (seven times eight is 56, three divided by nine is some tiny fraction…) got a very different number, and were told they were right. Terrible method of teaching, because it alienates the students who need to do the learning right off the bat. And this basically set the tone until I dropped out of college for the second time.
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