6Γ·2(1+2)
π‘πππΊππππΊπ π°ππππ± @ SmartmanApps @programming.dev Posts 22Comments 591Joined 2 yr. ago
π‘πππΊππππΊπ π°ππππ± @ SmartmanApps @programming.dev
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You havenβt provided a textbook that has strong juxtaposition
I told you, in my thread - multiple ones. You haven't provided any textbooks at all that have "weak juxtaposition". i.e. you keep asking me for more evidence whilst never producing any of your own.
At best I can search the title of the file youβre in that you also happened to screenshot and hope that I find the right text
I didn't "just happen" to include the name of the textbook and page number - that was quite deliberate. Not sure why you don't want to believe a screenshot, especially since you can't quote any that have "weak juxtaposition" in the first place.
BTW I just tried Googling it and it was the first hit. You're welcome.
What does matter is that I shouldnβt have to go treasure hunting for your sources.
You don't - the screenshots of the relevant pages are right there. You're the one choosing not to believe what is there in black and white, in multiple textbooks.
with differing rules
Yeah, I wrote about inconsistency in textbooks here (also includes another textbook saying you have to expand brackets first), but also elsewhere in the thread is an example where they have been consistent throughout. Regardless of when they remove brackets, in every single case they multiply the coefficient over what's inside the brackets as the first step (as per BEDMAS, and as per the screenshot in question which literally says you must do it before you remove brackets).
people donβt agree
People who aren't high school Maths teachers (the ones who actually teach this topic). Did you notice that neither The Distributive Law nor Terms are mentioned at any point whatsoever? That's like saying "I don't remember what I did at Xmas, so therefore it's ambiguous whether Xmas ever happened at all, and anyone who says it definitely did is wrong".
no such complaint.
So what do you think he is complaining about?
Citation needed.
So you think it's ok to teach contradictory stuff to them in Maths? π€£ Ok sure, fine, go ahead and find me a Maths textbook which has "weak juxtaposition" in it. I'll wait.
Your comments only reference βmath textbooksβ, not anything specific
So you're telling me you can't see the Maths textbook screenshots/photo's?
outside of this link which you reference twice in separate comments but again, itβs not evidence for your side, or against it, or even relevant
Lennes was complaining that literally no textbooks he mentioned were following "weak juxtaposition", and you think that's not relevant to establishing that no textbooks used "weak juxtaposition" 100 years ago?
Weβre talking about x/y(b+c) and whether that should be x/(yb+yc) or x/y * 1/(b+c).
It's in literally the first textbook screenshot, which if I'm understanding you right you can't see? (see screenshot of the screenshot above)
you have an uncited refutation of the side youβre arguing against, which funnily enough you did cite.
Ah, no. Lennes was complaining about textbooks who were obeying Terms/The Distributive Law. His own letter shows us that they all (the ones he mentioned) were doing the same thing then that we do now. Plus my first (and later) screenshot(s).
Also it's in Cajori, but I didn't find it until later. I don't remember what page it was, but it's in Cajori and you have the reference for it there already.
you should probably make it easy to find the evidence you speak of
Well I'm not sure how you didn't see all the screenshots. They're hard to miss on my computer!
That doesnβt prohibit them from teaching additional material
Correct, but it can't be something which would contradict what they do have to teach, which is what "weak juxtaposition" would do.
a single reference
I see you didn't read the whole thread then. Keep going if you want more. Literally every Year 7-8 Maths textbook says the same thing. I've quoted multiple textbooks (and haven't even covered all the ones I own).
mathematics is a science
Actually you'll find that assertion is hotly debated.
Regarding your second point I tried to address that in the βdistributive propertyβ section,
When I discovered this comment I went to read it, and yes, it's true you discussed the Distributive Property, however, what these people are talking about is The Distributive Law which isn't the same thing (though people often call it the wrong name), and makes the question completely unambiguous. You literally can't move on from the "B", Brackets, in the rules until there are no brackets left - the B is literally short for "solve Brackets" (every letter is "solve (something)"), and so anyone who does the division before solving the brackets has just violated the order of operations rules.
I'll take that as an admission that you're wrong then. Bye now.
P.S. someone else just provided me with even more things which are wrong in it. Even more glad I didn't waste time reading the rest of it.
Notation isnβt semantics
Correct, the definitions and the rules define the semantics.
Mathematical proofs are working with
...the rules of Maths. In fact, when we are first teaching proofs to students we tell them they have to write next to each step which rule of Maths they have used for that step.
Nobody doubts that those are unambiguous
Apparently a lot of people do! But yes, unambiguous, and therefore the article is wrong.
But notation can be ambiguous
Nope. An obelus means divide, and "strong juxtaposition" means it's a Term, and needs The Distributive Law applied if it has brackets.
In this case it is: weak juxtaposition vs strong juxtaposition
There is no such thing as weak juxtaposition. That is another reason that the article is wrong. If there is any juxtaposition then it is strong, as per the rules of Maths. You're just giving me even more ammunition at this point.
Read the damn article
You just gave me yet another reason it's wrong - it talks about "weak juxtaposition". Even less likely to ever read it now - it's just full of things which are wrong.
How about read my damn thread which contains all the definitions and proofs needed to prove that this article is wrong? You're trying to defend the article... by giving me even more things that are wrong about it. π
I stopped reading it when I found it was wrong., and said what was wrong about it. You have still not said where mine is allegedly wrong. I'll take that as an admission that you're wrong then.
Ok so youβre saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today
You asked me what I do if my students show me 2 different answers what do I tell them, and I told you that has never happened. None of my students have ever had one of the calculators which does it wrong.
that both behaviors are intentional and documented
Correct. I already noted earlier (maybe with someone else) that the TI calculator manual says that they obey the Primary School order of operations, which doesn't work with High School order of operations. i.e. when the brackets have a coefficient. The TI calculator will give a correct answer for 6/(1+2) and 6/2x(1+2), but gives a wrong answer for 6/2(1+2), and it's in their manual why. I saw one Youtuber who was showing the manual scroll right past it! It was right there on screen why it does it wrong and she just scrolled down from there without even looking at it!
none of these calculators is βwrongβ.
Any calculator which fails to obey The Distributive Law is wrong. It is disobeying a rule of Maths.
there is no ambiguity where there actually is.
There actually isn't. We use decimal points (not commas like some European countries), the obelus (not colon like some European countries), etc., so no, there is never any ambiguity. And the expression in question here follows those same notations (it has an obelus, not a colon), so still no ambiguity.
i recommend reading this one instead: The PEMDAS Paradox
Yes, I've read that one before. Makes the exact same mistakes. Claims it's ambiguous while at the same time completely ignoring The Distributive Law and Terms. I'll even point out a specific thing (of many) where they miss the point...
So the disagreement distills down to this: Does it feel like a(b) should always be interchangeable with axb? Or does it feel like a(b) should always be interchangeable with (ab)? You can't say both.
ab=(axb) by definition. It's in Cajori, it's in today's Maths textbooks. So a(b) isn't interchangeable with axb, it's only interchangeable with (axb) (or (ab) or ab). That's one of the most common mistakes I see. You can't remove brackets if there's still more than 1 term left inside, but many people do and end up with a wrong answer.
By βweβ do you mean high school teachers, or Australian society beyond high school?
I said "In Australia" (not in Australian high school), so I mean all of Australia.
Because, Iβm pretty sure the latter isnβt true
Definitely is. I have never seen anyone here ever use a colon to mean divide. It's only ever used for a ratio.
Do you have textbooks where the fraction bar is used concurrently with the obelus (Γ·) division symbol?
All my textbooks use both. Did you read my thread? If you use a fraction bar then that is a single term. If you use an obelus (or colon if you're in a country which uses colon for division) then that is 2 terms. I covered all of that in my thread.
EDITED TO ADD: If you don't use both then how do you write to divide by a fraction?
You canβt prove something with incomplete evidence
If something is disproven, it's disproven - no need for any further evidence.
BTW did you read my thread? If you had you would know what the rules are which are being broken.
the article has evidence that both conventions are in use
I'm fully aware that some people obey the rules of Maths (they're actual documented rules, not "conventions"), and some people don't - I don't need to read the article to find that out.
Indeed Duncan. :-)
his rule could be replaced by the strong juxtaposition
"strong juxtaposition" already existed even then in Terms (which Lennes called Terms/Products, but somehow missed the implication of that) and The Distributive Law, so his rule was never adopted because it was never needed - it was just Lennes #LoudlyNotUnderstandingThings (like Terms, which by his own admission was in all the textbooks). 1917 (ii) - Lennes' letter (Terms and operators)
In other words...
Funny enough all the examples that N.J. Lennes list in his letter use
...Terms/Products., as we do today in modern high school Maths textbooks (but we just use Terms in this context, not Products).
I have never encountered strong juxtaposition
There's "strong juxtaposition" in both Terms and The Distributive Law - you've never encountered either of those?
unable to agree on an universal standard for anything
And yet the order of operations rules have been agreed upon for at least 100 years, possibly at least 400 years.
unscientific and completely ridiculous reason refuse to read
The fact that I saw it was wrong in the first paragraph is a ridiculous reason to not read the rest?
Let me just tell you one last time: youβre wrong
And let me point out again you have yet to give a single reason for that statement, never mind any actual evidence.
you should know that itβs possible that youβre wrong
You know proofs, by definition, can't be wrong, right? There are proofs in my thread, unless you have some unscientific and completely ridiculous reason to refuse to read - to quote something I recently heard someone say.
try not to ruin your students too hard
My students? Oh, they're doing good. Thanks for asking! :-) BTW the test included order of operations.
Noted that you were unable to tell me what The Distributive Law relates to (given your claim it's not brackets).
Skimmed your comment and itβs wrong
So tell me where it's wrong.
Let me know if you ever decide to read the article instead of arguing against an imagined opponent
There's nothing imaginary about the fact that he claimed it's ambiguous, and is therefore wrong. Tell me why I should read a wrong article, given I already know it's wrong.
P.S. if you DID want to indicate "weak juxtaposition", then you just put a multiplication symbol, and then yes it would be done as "M" in BEDMAS, because it's no longer the coefficient of a bracketed term (to be solved as part of "B"), but a separate term.
6/2(1+2)=6/(2+4)=6/6=1
6/2x(1+2)=6/2x3=3x3=9
It isnβt, because the βcurrently taught rulesβ are on a case-by-case basis and each teacher defines this area themselves
Nope. Teachers can decide how they teach. They cannot decide what they teach. The have to teach whatever is in the curriculum for their region.
Strong juxtaposition isnβt already taught, and neither is weak juxtaposition
That's because neither of those is a rule of Maths. The Distributive Law and Terms are, and they are already taught (they are both forms of what you call "strong juxtaposition", but note that they are 2 different rules, so you can't cover them both with a single rule like "strong juxtaposition". That's where the people who say "implicit multiplication" are going astray - trying to cover 2 rules with one).
See this part of my comment... Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler)
Yep, saw it, and weak juxtaposition would break the existing rules of Maths, such as The Distributive Law and Terms. (Re)learn the existing rules, that is the point of the argument.
citation needed
Well that part's easy - I guess you missed the other links I posted. Order of operations thread index Text book references, proofs, the works.
this issue isnβt a mathematical one, but a grammatical one
Maths isn't a language. It's a group of notation and rules. It has syntax, not grammar. The equation in question has used all the correct notation, and so when solving it you have to follow all the relevant rules.
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 itβs conventions as long as itβs not defined on the same page
The rules can be found in any high school Maths textbook.
Yes, the guy who should mind his own business.
How about his reference for historical use
Are you talking about his reference to Lennes' letter? Lennes' letter actually completely contradicts his claim that it ever meant anything different.
Elizabeth Brown Davis
Haven't seen that one. Do you have a link?
He also references a Slate article by
...a journalist. The article ALSO ignores The Distributive Law and Terms.
I wouldnβt disagree with that.
Thank you. And also thank you for being the first person to engage in a proper conversation about it here.
Iβve heard Presh respond to people in the past over questions like this
I've seen him respond to people who agree with him. People who tell him he's wrong he usually ignores. When he DOES respond to them he simply says "The Distributive Property doesn't apply". We're talking about The Distributive LAW, NOT the Distributive Property. It's called "law" for a reason. i.e. ALWAYS applies. I've only ever seen him completely unwilling to engage in any conversation with anyone who points out he's wrong (contradicting his claim that he "welcomes debate").
I have a lot of respect for him
Really?? Why's that? I'm genuinely curious.
Iβve never heard the variant where there was a clear change in 1917
Me either. As far as I can tell it's just people parroting his misinterpretation of Lennes' letter.
Instead, it seems there was historical vagueness until the rules we now accept were slowly consolidated
I can't agree with that. Lennes' letter shows the same rules in 1917 as we use now. Cajori says the order of operations rules are at least 400 years old, and I have no reason to suspect they changed at all during that time period either. They're all a natural consequence of the way we have defined the symbols in the first place.
The Distributive Law obviously applies
Again, thank you.
Iβm seeing references that would still assert that (6Γ·2) could at one time have been the portion multiplied with the (3)
If it was written (6Γ·2)(1+2), absolutely that is the correct thing to do (expanding brackets), but not if it's written 6Γ·2(1+2). If you mean the latter then I've never seen that - links?
It's not.
Agree completely! Notice how they ALWAYS leave out high school Maths teachers and textbooks? You know, the ones who actually TEACH this topic. Always people OTHER THAN the people/books who teach this topic (and so always end up with the wrong conclusion).
Literally no-one in education uses so-called "weak juxtaposition" - there's no such thing. There's The Distributive Law and Terms, both of which use so-called "strong juxtaposition". Most calculators do too.
It is. In fact it's the rules (The Distributive Law and Terms).
Maths teachers already DO say it's wrong.
No, this is mostly a Maths teacher argument. You started off weak (saying its ambiguous), but then after that almost everything you said is actually correct - maybe you should be a Maths teacher. :-)