SmartmanApps

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 isn’t, because the ‘currently taught rules’ 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 isn’t 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 isn’t 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 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.

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 wouldn’t disagree with that.

Thank you. And also thank you for being the first person to engage in a proper conversation about it here.

I’ve 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.

I’ve 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.

I’m seeing references that would still assert that (6÷2) could at one time have been the portion multiplied with the (3)

If it was written (6÷2)(1+2), absolutely that is the correct thing to do (expanding brackets), but not if it's written 6÷2(1+2). If you mean the latter then I've never seen that - links?

The distributive law has nothing to do with brackets

BWAHAHAHA! Ok then, what EXACTLY does it relate to, if not brackets? Note that I'm talking about The Distributive LAW - which is about expanding brackets - not the Distributive PROPERTY.

a(b+c) = ab + ac

a(b+c)=(ab+ac) actually - that's one of the common mistakes that people are making. You can't remove brackets unless there's only 1 term left inside, and ab+ac is 2 terms.

ab+c = (ab)+(ac)

No, never. ab+c is 2 terms with no further simplification possible. From there all that's left is addition (once you know what ab and c are equal to).

brackets are purely notational

Yep, they're a grouping symbol. Terms are separated by operators and joined by grouping symbols.

did you stop after realizing that it was saying something you found disagreeable

I stopped when he said it was ambiguous (it's not, as per the rules of Maths), then scanned the rest to see if there were any Maths textbook references, and there wasn't (as expected). Just another wrong blog.

What will you tell your students if they show you two different models of calculator, from the same company

Has literally never happened. Texas Instruments is the only brand who continues to do it wrong (and it's right there in their manual why) - all the other brands who were doing it wrong have reverted back to doing it correctly (there's a Youtube video about this somewhere). I have a Sharp calculator (who have literally always done it correctly) and most of my students have Casio, so it's never been an issue.

trust me on this

I don't ask them to trust me - I'm a Maths teacher, I teach them the rules of Maths. From there they can see for themselves which calculators are wrong and why. Our job as teachers is for our students to eventually not need us anymore and work things out for themselves.

The truth is that there are many different math notations which often do lead to ambiguities

Not within any region there isn't. e.g. European countries who use a comma instead of a decimal point. If you're in one of those countries it's a comma, if you're not then it's a decimal point.

people don’t often encounter the obelus notation for division at all

In Australia it's the only thing we ever use, and from what I've seen also the U.K. (every U.K. textbook I've seen uses it).

Check out some of the other things which the “÷” symbol can mean in math!

Go back and read it again and you'll see all of those examples are worded in the past tense, except for ISO, and all ISO has said is "don't use it", for reasons which haven't been specified, and in any case everyone in a Maths-related position is clearly ignoring them anyway (as you would. I've seen them over-reach in Computer Science as well, where they also get ignored by people in the industry).

"The obelus is treated differently,” Church said. "It could mean ratios, division or numerator and denominator, and these all tweak the meaning of the symbol.”

This is the only symbols I've ever seen used (but feel free to provide a reference if you know of any where it isn't - the article hasn't provided any references)...

Ratio is only ever colon.

Division is obelus (textbooks/computers) or slash (computers, though if it's text you can use a Unicode obelus).

Fraction is fraction bar (textbooks) or obelus/slash inside brackets (computers). i.e. (a/b).

read the article instead of “scanning” it.

I stopped reading as soon as I saw the claim that the right answer was wrong. I then scanned it for any textbook references, and there were none (as expected).

You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up

Funny that you use the word "term", since Terms are ONE of the things that people are referring to when they say "implicit multiplication" - the other being The Distributive Law. i.e. Two DIFFERENT actual rules of Maths have been lumped in together in a made-up rule (by people who don't remember the actual rules).

BTW if you think it's not made-up then provide me with a Maths textbook reference that uses it. Spoiler alert: you won't find any.

Implicit multiplication is not the controversial part of this equation

It's not the ONLY controversial part of the equation - people make other mistakes with it too - but it's the biggest part. It's the mistake that most people have made.

shitty blog

So that's what you think of people who educate with actual Maths textbook references?

Read. The. Article.

Read Maths textbooks.

The first step in order of operations is solve brackets. The first step in solving unexpanded brackets is to expand them. i.e. The Distributive Law. i.e. the ONLY time The Distributive Law ISN'T part of order of operations is when there's no unexpanded brackets in the expression.

The examples I gave were that the expansion of brackets would be done differently if the order of operations was “PESADM”

Yep I read it, and no it wouldn't. Expanding Brackets - or in the case of this mnemonic Parentheses - is done as part of B/D (as the case may be). i.e. expanding brackets isn't "multiplication" (no multiplication sign), but solving brackets (there are brackets there), which always come first in all the mnemonics.

reverse polish notation exists

...but is not taught in high school.

your level of qualification on this topic is not above mine

Maybe not, but it means it's not an "appeal to authority" (as per screenshot). Maths teachers ARE an authority on Maths. The most common appeal to authority I see from people is claiming that someone (not them) is a University professor, and "they would know". No, they wouldn't - this topic isn't taught at university - it's taught in high school.

why you were so engaged in this.

I'm a teacher. You say you're on the same level as me - don't you like to teach people what's correct?

3 month old post

Which will show up in search results for all eternity (it's how I found it - I was looking for something else!).

probably won’t be a lot of engagement in this thread from this point on

Got another 12 responses after yours. But the point is I'm not even LOOKING for responses, just to correct misinformation. As a teacher (a Maths teacher?) have you not had people say to you "But Google says"? I certainly have. It's the bane of my professions.

it seems like you’re on your own

Did you read my thread? Maths textbooks, calculators, proofs, etc. Also, someone else said what you just did, asked a Maths teacher, and was told I was correct, then was man enough to go back and edit his posts and admit I was correct and specifically said "SmartmanApps is not on his own with this".

clear examples against what you are saying

Which are where, exactly? You haven't presented any. You haven't, for example, shown how one can make (2+3)x4=14.

re: appeal to authority

I believe you’re conflating the rules of maths with the notation we use to represent mathematical concepts.

You think a Maths teacher doesn't know the difference?

There is absolutely nothing stopping us from choosing to interpret a+b×c as (a+b)×c

Yes there is - the underlying Maths. 2x3 is short for 2+2+2, which is therefore why you have to expand multiplications before doing additions. If you "chose" to interpret 2+3x4 (which we KNOW is equal to 14, because 3x4=3+3+3+3 by definition) as (2+3)x4, you would get 20, which is clearly wrong, since 20 isn't equal to 14.

We don’t even have to write it like that at all

No that's right, because it IS written differently in different languages, but regardless of how you write it, it doesn't change that 2+3x4=14 - the underlying Maths doesn't change regardless of how you decide to write it. Maths is literally universal.

× before + is a very convenient choice

It's not a choice, it's a consequence of the fact that x is shorthand for +. i.e. 2x3=2+2+2.

it is still just a choice

It is a consequence of the definitions of what each operator does. If x is a contraction of +, then we have to expand x before we do +. If it were the other way around then we'd have to do it the other way around. Anything which is a contraction of something else has to be expanded first.