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 :)
I don’t have much to say on this, other than that I appreciate how well-written this deep dive is and I appreciate you for writing it. People get so polarized with these viral math problems and it baffles me.
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.
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.
Damn ragebait posts, it’s always the same recycled operation. They could at least spice it up, like the discussion about absolute value. What’s |a|b|c|?
What I gather from this, is that Geogebra is superior for not allowing ambiguous notation to be parsed 👌
Your example with the absolute values is actually linked in the “Even more ambiguous math notations” section.
Geogebra has indeed found a good solution but it only works if you input field supports fractions and a lot of calculators (even CAS like WolframAlpha) don’t support that.
Yeah! That’s why I mentioned it, it was a fresh ambiguous notation problem that I’ve never encountered before. Discussions of “is it 1 or 9” get tiring quickly.
At least WA and others tell you how they interpret the input, instead of being a black box (until you get to the manuals). Even though it is obvious in hindsight, I didn’t get why two calculators would yield different results; thanks!
While I agree the problem as written is ambiguous and should be written with explicit operators, I have 1 argument to make. In pretty much every other field if we have a question the answer pretty much always ends up being something along the lines of “well the experts do this” or “this professor at this prestigious university says this”, or “the scientific community says”. The fact that this article even states that academic circles and “scientific” calculators use strong juxtaposition, while basic education and basic calculators use weak juxtaposition is interesting. Why do we treat math differently than pretty much every other field? Shouldn’t strong juxtaposition be the precedent and the norm then just how the scientific community sets precedents for literally every other field? We should start saying weak juxtaposition is wrong and just settle on one.
I tried to be careful to not suggest that scientist only use strong juxtaposition. They use both but are typically very careful to not write ambiguous stuff and practically never write implicit multiplications between numbers because they just simplify it.
At this point it’s probably to late to really fix it and the only viable option is to be aware why and how this ambiguous and not write it that way.
As stated in the “even more ambiguous math notations” it’s far from the only ambiguous situation and it’s practically impossible (and not really necessary) to fix.
Scientist and engineers also know the issue and navigate around it. It’s really a non-issue for experts and the problem is only how and what the general population is taught.
“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”
I feel like if a blog post presents 2 options and labels one as the “scientific” one… And it is a deserved Label. Then there is probably a easy case to be made that we should teach children how to understand scientific papers and solve the equation in it themselves.
Honestly I feel like it reads better too but that is just me
I’m not sure if I’d call it the “scientific” one. I’d actually say that the weak juxtaposition is just the simple one schools use because they don’t want to confuse everyone. Scientist actually use both and make sure to prevent ambiguity. IMHO the main takeaway is that there is no consensus and one has to be careful to not write ambiguous expressions.
“If you are a student at university, a scientist, engineer, or mathematician you should really try to ask the original author what they meant because strong juxtaposition is pretty common in academic circles, especially if variables are involved like in $a/bc$ instead of numbers.”
I’m a scientist and I’ve only ever encountered strong juxtaposition in quick scribbles where everyone knows the equation already. Normally we’re very careful to use fraction notation (or parentheses) when there’s any possibility of ambiguity. I read the equation and was shocked that anyone would get an answer other than 9.
My comment was directed to the blog post and the claims contained in it.
The blog post claims it is popular in academy, if that is a deserved label, then I don’t understand how the author of the post lands on “there is no good or bad way, they are all valid”. I am in favor of strong juxtaposition but that is not the case that I am making here. Sorry for the confusion.
The order of operations is not part of a holy text that must be blindly followed. If these numbers had units and we knew what quantity we were trying to solve for, there would be no argument whatsoever about what to do. This is a question that never comes up in physics because you can use dimensional analysis to check to see if you did the algebra correctly. Context matters.
It’s also clearly not a bug as some people suggest. Bugs are – by definition – unintended behavior.
There are plenty of bugs that are well documented. I can’t tell you the number of times that I’ve seen someone do something wrong, that they think is 100% right, and “carefully” document it. Then someone finds an edge case and points out the defined behavior has a bug, because the human forgot to account for something.
The other thing I’d point out that I didn’t see in your blog is that I’ve seen many many people say they need to evaluate the 2(3) portion first because “parenthesis”. No matter how many times I explain that this is a notation for multiplication, they try to claim it doesn’t matter because parenthesis. screams into the void
The fact of the matter is that any competent person that has to write out one of these equations will do so in a way that leaves no ambiguity. These viral math posts are just designed to insert ambiguity where it shouldn’t be, and prey on people who can’t remember middle school math.
Regarding your first part in general true, but in this case the sheer amount of calculators for both conventions show that this is indeed intended behavior.
Regarding your second point I tried to address that in the “distributive property” section, maybe I need to rewrite it a bit to be more clear.
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.
It’s not ambiguous, it’s just that correctly parsing the expression requires more precise application of the order of operations than is typical. It’s unclear, sure. Implicit multiplication having higher precedence is intuitive, sure, but not part of the standard as-written order of operations.
I’d really like to know if and how your view on that matter would change once you read the full post. I know it’s very long and a lot of people won’t read it because they “already know” the answer but I’m pretty sure it would shift your perception at least a bit if you find the time to read it.
My opinion hasn’t changed. The standard order of operations is as well defined as a notational convention can be. It’s not necessarily followed strictly in practice, but it’s easier to view such examples as normal deviation from the rules instead of an implicit disagreement about the rules themselves. For example, I know how to “properly” capitalize my sentences too, and I intentionally do it “wrong” all the time. To an outsider claiming my capitalization is incorrect, I don’t say “I am using a different standard,” I just say “Yes, I know, I don’t care.” This is simpler because it accepts the common knowledge of the “normal” rules and communicates a specific intent to deviate. The alternative is to try to invent a new set of ad hoc rules that justify my side, and explain why these rules are equally valid to the ones we both know and understand.
They weren't asking you if there are two sets of rules, we're in a thread that's basically all qbout the Weak vs. Strong juxtaposition debate, they asked you which you consider correct.
Giving the answer to a question they didn't ask to avoid the one they did is immature.
I can't have stopped because I never started, because I'm not even married... See, even I can answer your bad faith question better than you answered the one @onion asked you.
But I will give it to you that my comment should've stipulated avoiding reasonable questions.
The difference is that there are two sets of rules already in use by large groups of people, so which do you consider correct?
However I still think you need your eyes checked, as the end of this comment by @onion is very clearly a question asking you WHICH ruleset you consider correct.
Unless you're refusing the notion of multiplication by juxtaposition entirely, then you must be on one side of this or the other.
“Which ruleset do you consider correct” presupposes, as the comment said, that there are 2 rulesets. There aren’t. There’s the standard, well known, and simplified model which is taught to kids, and there’s the real world, where adults communicate by using context and shared understanding. Picking a side here makes no sense.
When the @onion said there were two different sets of rules, you know as well as I do that they meant strong vs. weak juxtaposition.
You're right that in reality nobody would write an equation like this, and if they did they would usually provide context to help resolve it without resorting to having to guess...
But the point of this post is exactly to point out this hole that exists in the standard order of operations, the drama that has resulted from it, and to shine some light on it.
Picking a side makes no sense only if you have the context to otherwise resolve it... If you were told to solve this equation, and given no other context to do so, you would either have to pick a side or resolve it both ways and give both answers. In that scenario, crossing your arms and refusing to because "it doesn't make sense" would get you nowhere.
In all honesty, I think you're acting like the people who say things like "I've never used algebra, so it was worthless teaching me it as a kid" as though there aren't people who would learn something out of this.
I'll just say it again, you're the one saying this problem is completely unambiguous, with your only explanation as to why being that real people communicate as though that solves every edge case imaginable.
I'm just saying, if you really believe that to be the case, Good luck.
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