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 :)
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.
Assumed by whom? Clearly not everyone.
It’s what is actually taught in high school, so there are those who remember and those who don’t.
You are correct with your definition - Terms are separated by operators and joined by grouping symbols - and it’s consequently not ambiguous at all (using so-called “weak juxtaposition” breaks that rule).
👍 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.
I wanted to compile a list of points that show as clear as humanity possible that there is no consensus here, even amongst experts
And I wrote a bunch of fact checks pointing out there is consensus amongst the actual experts - high school Maths teachers and textbook authors, the 2 groups who you completely ignored in your blog post.
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.
This comment is left by the HP50g crew.
It would be better if we just taught math with prefix or postfix notation, as it removes the ambiguity.
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?
enforce writing math without ambiguity
It already is written without ambiguity.
were taught in third grade
This is actually taught in Year 7 - the people who only remember the 3rd Grade version of the rules are the ones getting it wrong.
There isn’t ambiguity to begin with - just people who have forgotten the rules of Maths.
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 :)
No, it doesn’t. It never talks about Terms, nor The Distributive Law (which isn’t the same thing as the Distributive Property). These are the 2 rules of Maths which make this 100% not ambiguous.
I disagree. Without explicit direction on OOO we have to follow the operators in order.
The parentheses go first. 1+2=3
Then we have 6 ÷2 ×3
Without parentheses around (2×3) we can’t do that first. So OOO would be left to right. 9.
In other words, as an engineer with half a PhD, I don’t buy strong juxtaposition. That sounds more like laziness than math.
How are people upvoting you for refusing to read the article?
I did read the article. I am commenting that I have never encountered strong juxtaposition and sharing why I think it is a poor choice.
You probably missed the part where the article talks about university level math, and that strong juxtaposition is common there.
I also think that many conventions are bad, but once they exist, their badness doesn’t make them stop being used and relied on by a lot of people.
I don’t have any skin in the game as I never ran into ambiguity. My university professors simply always used fractions, therefore completely getting rid of any possible ambiguity.
You probably missed the part where the article talks about university level math,
This is high school level Maths. It’s not taught at university.
I have never encountered strong juxtaposition
There’s “strong juxtaposition” in both Terms and The Distributive Law - you’ve never encountered either of those?
Because those people also didn’t read the article and are reacting from their gut.
are reacting from their gut
As was the person who wrote the article. Did you not notice the complete lack of Maths textbooks in it?
Because as a high school Maths teacher as soon as I saw the assertion that it was ambiguous I knew the article was wrong. From there I scanned to see if there were any Maths textbooks at any point, and there wasn’t. Just another wrong article.
Lol. Read it.
Why would I read something that I know is wrong? #MathsIsNeverAmbiguous
Mathematical notation however can be. Because it’s conventions as long as it’s not defined on the same page.
Mathematical notation however can be.
Nope. Different regions use different symbols, but within those regions everyone knows what each symbol is, and none of those symbols are in this question anyway.
Because it’s conventions as long as it’s not defined on the same page
The rules can be found in any high school Maths textbook.
Let’s do a little plausibility analysis, shall we? First, we have humans, you know, famously unable to agree on an universal standard for anything. Then we have me, who has written a PhD thesis for which he has read quite some papers about math and computational biology. Then we have an article that talks about the topic at hand, but that you for some unscientific and completely ridiculous reason refuse to read.
Let me just tell you one last time: you’re wrong, you should know that it’s possible that you’re wrong, and not reading a thing because it could convince you is peak ignorance.
I’m done here, have a good one, and try not to ruin your students too hard.
Great write up! The answer is use parentheses or fractions and stop wasting everyone’s time 😅
That’s actually a great way of putting it 🤣
No it isn’t dotnet.social/@SmartmanApps/110819283738912144
FACT CHECK 1/5
If you are sure the answer is one… you are wrong
No, you are. You’ve ignored multiple rules of Maths, as we’ll see…
it’s (intentionally!) written in an ambiguous way
Except it’s not ambiguous at all
There are quite a few people who are certain(!) that their result is the only correct answer
…and an entire subset of those people are high school Maths teachers!
What kind of evidence/information would it take to convince you, that you are wrong
A change to the rules of Maths that’s not in any textbooks yet, and somehow no teachers have been told about it yet either
If there is nothing that would change your mind, then I’m sorry I can’t do anything for you.
I can do something for you though - fact-check your blog
things that contradict your current beliefs.
There’s no “belief” when it comes to rules of Maths - they are facts (some by definition, some by proof)
How can math be ambiguous?
#MathsIsNeverAmbiguous
operator priority with “implied multiplication by juxtaposition”
There’s no such thing as “implicit multiplication”. You won’t find that term used anywhere in any Maths textbook. People who use that term are usually referring to Terms, The Distributive Law, or most commonly both! #DontForgetDistribution
This is a valid notation for a multiplication
Nope. It’s a valid notation for a factorised Term. e.g. 2a+2b=2(a+b). And the reverse process to factorising is The Distributive Law. i.e. 2(a+b)=(2a+2b).
but the order of operations it’s not well defined with respect to regular explicit multiplication
The only type of multiplication there is is explicit. Neither Terms nor The Distributive Law is classed as “multiplication”
There is no single clear norm or convention
There is a single, standard, order of operations rules
Also, see my thread about people who say there is no evidence/proof/convention - it almost always ends up there actually is, but they didn’t look (or didn’t want you to look)
The reason why so many people disagree is that
…they have forgotten about Terms and/or The Distributive Law, and are trying to treat a Term as though it’s a “multiplication”, and it’s not. More soon
conflicting conventions about the order of operations for implied multiplication
Let me paraphrase - people disagree about made-up rule
Weak juxtaposition
There’s no such thing - there’s either juxtaposition or not, and if there is it’s either Terms or The Distributive Law
construct “viral math problems” by writing a single-line expression (without a fraction) with a division first and a
…factorised term after that
Note how none of them use a regular multiplication sign, but implicit multiplication to trigger the ambiguity.
There’s no ambiguity…
multiplication sign - multiplication
brackets with no multiplication sign (i.e. a coefficient) - The Distributive Law
no multiplication sign and no brackets - Terms (also called products by some. e.g. Lennes)
If it’s a school test, ask you teacher
Why didn’t you ask a teacher before writing your blog? Maths tests are only ever ambiguous if there’s been a typo. If there’s no typo’s then there’s a right answer and wrong answers. If you think the question is ambiguous then you’ve not studied enough
maybe they can write it as a fraction to make it clear what they meant
This question already is clear. It’s division, NOT a fraction. They are NOT the same thing! Terms are separated by operators and joined by grouping symbols. 1÷2 is 2 terms, ½ is 1 term
BTW here is what happened when someone asked a German Maths teacher
you should probably stick to the weak juxtaposition convention
You should literally NEVER use “weak juxtaposition” - it contravenes the rules of Maths (Terms and The Distributive Law)
strong juxtaposition is pretty common in academic circles
…and high school, where it’s first taught
(6/2)(1+2)=9
If that was what was meant then that’s what would’ve been written - the 6 and 2 have been joined together to make a single term, and elevated to the precedence of Brackets rather than Division
written in an ambiguous way without telling you what they meant or which convention to follow
You should know, without being told, to follow the rules of Maths when solving it. Voila! No ambiguity
to stir up drama
It stirs up drama because many adults have forgotten the rules of Maths (you’ll find students get this right, because they still remember)
Calculators are actually one of the reasons why this problem even exists in the first place
No, you just put the cart before the horse - the problem existing in the first place (programmers not brushing up on their Maths first) is why some calculators do it wrong
“line-based” text, it led to the development of various in-line notations
Yes, we use / to mean divide with computers (since there is no ÷ on the keyboard), which you therefore need to put into brackets if it’s a fraction (since there’s no fraction bar on the keyboard either)
With most in-line notations there are some situations with conflicting conventions
Nope. See previous comment.
different manufacturers use different conventions
Because programmers didn’t check their Maths first, some calculators give wrong answers
More often than not even the same manufacturer uses different conventions
According to this video mostly not these days (based on her comments, there’s only Texas Instruments which isn’t obeying both Terms and The Distributive Law, which she refers to as “PEJMDAS” - she didn’t have a manual for the HP calcs). i.e. some manufacturers who were doing it wrong have switched back to doing it correctly
P.S. she makes the same mistake as you, and suggests showing her video to teachers instead of just asking a teacher in the first place herself (she’s suggesting to add something to teaching which we already do teach. i.e. ab=(axb)).
none of those two calculators is “wrong”
ANY calculator which doesn’t obey all the rules of Maths is wrong!
Bugs are – by definition – unintended behaviour. That is not the case here
So a calculator, which has a specific purpose of solving Maths expressions, giving a wrong answer to a Maths expression isn’t “unintended behaviour”? Do go on
