The mathematical value of "Of"

In my mind this falls in the same catagory with the fractured English. While it can be understood it promotes inaccuracies. Which when people get in the work force can be deadly. "But the needle was only part way in the red zone. I did not know the nuke power plant would meltdown."

Actually these are quite old usages that are apparently being brought back...perhaps in an effort to shift back from the cookbook approach to math, to a think numbers in your head approach. The use of "of" as opposed to other prepositions (for example, "in") could simply reflect parochial customs regarding diction. Use of the mental approach to calculations might also be intended to inculcate the understanding that division is nothing but a special form of multiplication; and multiplication a form of addition.... Another operation using "of" would be functions. Quite probably the curriculum has in mind to prepare the daughter for that later on. Another use of "of" is for subtraction (and direction)...when we refer to, say, 8:45 o'clock as 15 minutes of 9, or 15 of nine, or quarter of nine. Linguistically speaking, it would be my guess that the daughter is attending school in the midwest? Close?

Those are some really interesting observations. I hadn't even thought of the use of "of" for subtraction – but yes, "a quarter of nine" etc. is pretty common usage. Your guess re location is pretty close. Actually, we are in the southeast, but the book from which the worksheet was taken is used nationally, and (I think) published in Chicago.

A favourite hobby horse of mine. It's 'different from', 'similar to' and certainly neither of them take than - ugh! even the BBC doesn't understand any more

My mother, who thought English at business college, had a pet hate for the use of the phrase

"All of a sudden"

instead of

"Suddenly".

Her usual response was something along the lines of

"What does a sudden look like?"

or

"Really, I have never even seen part o a sudden let alone all of one."

One of my pet hates is the use of the plural s suffix with the word you in sentences like this.

"Will all of yous be quiet."

I usually respond with something like

"I don't think that will work because there are no sheep here"

Or

"How did the sheep get into this conversation?"

The usual response it total and complete confusion and a really dumb look on the face of the person I have directed the statement at.

I have a feeling that 'yous' came to us from a Northern Ireland dialect, though precisely where part I don't know.

Never thought of 'all of a sudden' but can easily see how it annoyed now it's pointed out. A list of all of these, I fear, would be longer that the dictionary or thesaurus!

Hi Wrenched,

• A list of all of these, I fear, would be longer that the dictionary or thesaurus!

I think your 100% correct on that one.

By the way, hello, I havn't been able to get to the usual threads on CR4 for a while and I don't believe we have conversed before, so, a warm gidday from the land down under.

I like the avatar. You would probably like my brothers job. He works for one of those exclusive car clubs that rent out those hyper desirable super cars. His job is to teach and examine the members to make sure they are capable of handling the machines they are driving so they don't go out and kill themselves or damage the cars. One of the perks is he gets to take whatever isn't being used home and last weekend he had an Aston Martin DB9. During the week he had a Bentley Continental GT but this weekend he has to slum it because the only car left was a miserly Porsche.

He's a lucky, little bugger and to think he actually gets paid to drive these cars. Having a never ending precession of super cars parked in the drive way certainly gets the neighbors fairly envious.

Sounds just the job to me and in Australian sunshine. I could cuss! The Avatar (so far as I remember) is was some sort of proposal for a Russian exotic. Yes, not a misprint! I think it's supposed (like so many recent 'retro' designs) to evoke the Bugatti coupe Atlantique. A car which I grew up with and will never be able to forget.

Hi,

I can definitely see the Bugatti lines particularly when you compare it to the Bugatti Veyron. However the front grill looks somewhat like it has had a BMW influence somewhere along the line.

The car club my brother works for tried to get hold of a Bugatti Veyron but it only comes in left hand drive and in Australia we drive on the left so vehicles need to be right hand drive. The New South Wales government no longer allows you to drive left hand drive vehicles on public roads so if they imported one it could only be driven on race tracks and private roads which makes it a not go, particularly when you look at the price.

One thing they do however do is replace the cars as soon as they hit the 30,000 km mark so they turn over the vehicles pretty quickly. I believe they have a standing order with people like Lamborghini, Ferrari, Aston Martin, Bentley etcetera for whatever is their latest vehicle.

A couple of months back he brought a top of the range hummer home. The quality of the finish was excellent but as my brother said it was sort of like a back the front TARDIS from Dr Who. The TARDIS is supposed to be larger on the inside than the outside, but the Hummer's interior was way, way, way smaller then the outside and had less leg room in the back than many medium sized cars. Not only that but it is too wide to fit on any of the road or rail vehicle transporters so it has to be driven everywhere that it is required. This often means driving several hundred kilometres to deliver it then flying home only to do the reverse in a couple of days to get it back.

His favorite and I must agree with him, is the Bentley Turbo RT. Not only is the attention to quality and detail superb but it goes like greased lightning and handles fantastically well especially when you consider how big it is. It also fantastically comfortable and easy to get in an out of unlike cars like Ferraris, Lamborghini, etcetera. It's sort of like a blend of where they took the best of a sports cars and limousine to build a new vehicle. Normally such a compromise would result in something that wasn't good at anything but Bentley seem to have pulled if off extremely well.

His second favorite is the Aston Martin DB9 and again I must agree with him. The super cars like the Ferrari etcetera are more along the lines of F1 cars with electronic transmissions, paddle shift etcetera while the DB9 is more like a sports car with manual gearbox etcetera. Because it's not as flashy and technically complex it is less popular than the others so he gets to take it home for the weekends more often and even on the occasional short holiday.

Talk about fringe benefits, the lucky little s…t. Then again I did have a company plane for a while so I guess I cant be too gealous.

I can understand why "of" would mean multiplication, but I don't think there is a formal definition and I know for a fact that there are cultural differences between the US and India in these sort of things.

For square root, I would say the square root of 2, but my Indian friends say root 2.

For division, I would say 5 into 6, or 6 divided by 5, 5 over 6, but the Indians I know say 6 by 5 for the same thing.

So for:

√2/2

I would say "The Square Root of 2 divided by 2"
And they would say "Root 2 by 2"

It gets really confusing when I ask them for a formula, but they've figured out by now that they have to say it slowly since I have to translate the by's and root's and other operators into American English.

Interesting insights, Roger. Thanks.

In English English, we would use root 2 as an abbreviation of squart root of 2 - and I know of a few people, myself included, who refer to it as squirt 2, the phonetic version of computer/calculator function SQRT(2)

"By" however, always implies multiplication in English - as in a piece of 2 by 4 (or more usually 2 be 4, phonetically!) for a piece of wood of cross-section 2" x 4". I believe this comes from the phrase 2 multiplied by 4 (hmm...2 divided by 4 would follow the same logic).

For division, we've always used the word "over" so √2/2 would be "root 2 over 2"

In the US, I think we say "tuba 4."

Ahah! - it may depend on the age of the speaker and the location. "4 by 2", or phonetically, "4 b' 2" (with very little in the way of a vowel sound after the 'b') is also used by the crumblies in the population. The cross-section of the piece of wood is the same.

'4 x 2' has the following meanings over here:

• It refers, naturally, to the aforementioned useful cross-section of wood.
• It can refer to the drive configuration for the wheels of what may be known as a 'Chelsea Tractor', a 'Ute', an 'SUV' or any other vehicle with an off-road capability and style.
• It can refer to someone who follows the Jewish faith, a term derived from Cockney Rhyming Slang.

Confused yet?

Of has been a mathmatical operaator for many years. I am now older than I easily care to admit but remember in my school days "BOD" as a reminder of the order in which to carry out operations, where B = Brackets, O = Of and D = divide and for "Of" read multiply.

As regards your hypothetical dentists thats just correct english usage 6 of 10 in that context is just lazy speak, 6 out of 10 is correct and unambiguous

We see 1 of 10, 2 of 10, 3 of 10, all the time at the bottom of pages. When we go online to buy a book, we see "4 of 16 reviewers found this review helpful." I think in both cases there is an implied division: on page 2, we are 2/10 or 20% through the document. Regarding the reviewers, we would not think "64 reviewers found this review helpful." We'd think 25% or 1/4 found it useful.

I think that using "of" alone re pages or reviewers is not lazy. It has a slightly different meaning than "out of". If we say "1 out of 10 reviewers" we are suggesting that there may have been several hundred reviewers, and that 10% found the review helpful. If we say "1 of 10 reviewers" then we are somewhat more clearly (but not entirely unambiguously) saying that there were, in fact, just 10 reviewers (of which 1, or 10%, found the review useful).

But it is interesting that you were taught BOD. Personally, I'd prefer to use the standard operation symbols in calculations because they are entirely unambiguous. In word problems, ambiguity naturally creeps in, but context helps disambiguate. In my obsessive-compulsive view: if it is a word problem, use words. If it is a calculation for which you want an unambiguous answer, use symbols. (Granted, in our heads we may be saying "two plus two" or "two and two"… or some of my least favorite: "three take away 2" "three less two" …)

I was led to believe that the universal language that everybody understands is math. (using everyone rhetorical)

True, when symbols instead of words are used as operators.

When you do the reverse, insert a word where a symbol belongs, then the language is no longer universal. "Of" has many meanings in English, no meaning at all in some languages, and completely unintended meaning in yet other languages.

"OF" course, I wonder if there are any more ambers burning on this subject.

Maybe it is all part of the effort to find "new" math. New math is very conveinent for schools since it slows down parents, by making them believe "Maybe I missed something"

Perhaps English is now being taught in mathematics to free up time for teaching Spanish.

I remember, a few years back, with my son, the word 'and' was being taught to mean addition. All though Clinton pointed out that 'IS' remains undefined.

Sorry for the rant-

The use of "of" as a mathmatical operand is subjective. Subtle, but further evidence of objectivity being removed from education.

I always thought 'of' meant multiplication (just as you say in words: seven tenths of 24), but with 'new math', I don't know

But what about 2 of 10? Do you then think 20 (multiplication) or 2/10 (division)? Or "4 of 16 reviewers found this review helpful." Isn't that a ratio -- an implied division? If you simplified the expression, wouldn't you come up with 1/4, rather than 64?

For an insight into the New Math of the 1960s see here

This is not new, it's a strategy for word problems.

Fifty percent of one hundred is fifty.

0.50 * 100 = 50

it is dependent on the context, but some words ("is", "are") are dependable operators.

They teach the above example so students can quickly figure out percentages in a word problem. eg. what percent of pi is 90 degrees? ->
x * pi = pi/2 -> x = pi/2pi = 1/2 = 50%

you wrote:

eg. what percent of pi is 90 degrees? -> x * pi = pi/2 -> x = pi/2pi = 1/2 = 50%

---------------

I am REALLY bothered by your example; this is typical of some of the grade school math problems I've seen... ie... incorrect formulation of problems!

Pi is 3.1428.... and that is all it is... so your example makes no sense. If you did not mean the use of 'pi', as I have, then you chose a very poor variable name in this problem example!

If I were to define pi as other than the irrational number, then it is the number of radians in 180 deg, since there are 2*pi radians in a complete circle!

I agree that "is" and "are" are dependable operators. But I also agree that "it is dependent upon context", and in the case of "of," I think there is too strong a dependence to allow it to be used without additional words to establish that context.

"Fifty percent of one hundred is fifty.": Entirely unambiguous because there are enough other words for context .

0.50 * 100 = 50: Taken in context, utterly and totally unambiguous. (Only a computer programmer would say: Doesn't it need to parsed to determine if it is a string or not?)

"4 of 16 reviewers found this review helpful." Pretty unambiguous, given enough other words for context. Very unlikey to simplify to 64 (multiplication)

"4 of 16." Almost completely ambiguous. Could be ¼ or 64, but I would lean toward ¼, given no other info. In the worksheet my daughter was doing, they leaned in the other direction.

For this reason, I rant that word problems should be word problems, and that math calculations should use the standard symbols. I know there are well-intentioned, bright students how could answer the question "What is 4 of 16?" in either of two ways. To mark some of those students "wrong" seems to discourage them from learning the real concepts and encourages memorization of "tricks" or mnemonic devices that sometimes fail.

Isn't it pretty standard to say "what's half of ten"? Or "you earn what? I only earn a third of that"? Makes sense in Australia anyway.

Maybe it only seems strange when it's written down in a maths question (or "math question" for those few Americans who frequent this forum) where you might expect "times" or "multiplied by."

It reminds me of a story I heard from a woman who said her primary school child had got a maths question wrong. The question was "what is the difference between 7 and 4?" He had written "one is odd, the other is even". Great answer - he should have got full marks!

I assume the "right" answer was:

The 7 has one acute angle; the 4 has two acute angles and four right angles.

This sort of thing worries me. I would hazard to say that most of the major posters here on CR4 have a better than average understanding of mathematics and would use it on a fairly regular basis. I know participating has inspired me to dust off the text books and try to get my calculus up to scratch.

My point is that it appears that somebody has arbitrarily decided that this is the way it shall be taught without actually asking the people that use mathematics for a living.

It seems to be, like the teaching of irrelevant English literature, yet another example of the education system being out of touch with the real world.

Masu, thanks for saying that. I'm glad that other people agree with me. I too think that WAY too many schools are teaching useless and stupid things that are only there for the sake of teaching. In my calculus classes, I've found that there are many topics that are just there "for the sake of teaching" and not because it's useful, and sadly some of my professors have even admitted that. One of them said, "you will never use this, but we're going to trudge right through this anyways". Maybe it's there to lay down certain processes in your mind, for later things, but still it's not logical to teach and test something useless! This annoys me even though this "of" operation is in a sixth grade textbook. I see "of" not as being an operation, but as part of speech, which puts meaning to quantities of things. I was tutoring a junior high student, and I was helping with math a while back. There were some things that were so vague and just plain odd, I wasn't able to give him 100% certainty that the answers were right, and I just couldn't believe it! I felt insulted almost that I couldn't 100% figure something out, while I've taken harder math courses than that students teacher! It's like these things have stupidity built in, and pure logic doesn't help you solve them... wtf?

-Nick

To a certain degree that is correct. But remember you only use a fraction of what these institutions teach you. They just don't teach you only what you'll be using. They also include fundamentals and materials that can support what you need.

I've work with green engineers that can solve problems out of book, but give them a problem in industry, and they'er floundering. They are given tools, but its up to the individual to use them.

At one point I was also inexperienced, and as conservative as I am, I believe one has to pay ones dues, because nothing worth having is free.

Oh, I do aggree with you 100%. I realize that most of what I actually learn will be something that I use. I should've used a better example: when we have to learn the syntax of a program that my prof wrote that completely changes the syntax of Mathematica, that's just too far. This seems to make the gained knowledge of Mathematica completely useless.
I want to learn Mathematica, not my Prof's derivation of Mathematica, which will never EVER be seen again, outside of my school. Even though Mathematica isn't really used in the engineering fields, it is still a great program.
There are quite a few things that we are taught that are totally different from what industry uses, and that just bugs me. i.e. we learn the notations and processes that the math department uses, but who besides math teachers uses that? No one but them. At my school, it's something that the Engineering departments hate, because they have students learning "math for math teachers", instead of "math for engineers" and it makes it harder for the engineering departments. Sorry I suppose this is a little off topic.

That is what is good about this forum, there are problems and sometimes one has to be reminded of them to step back a little. It makes one think. I myself included.

You hit it on the nose, I think. Her current math teacher is a little better than a previous one, so I am hoping that there is some appreciation for the myriad of ways in which math problems can be solved. The tendency seems to be to enforce rigid thinking because it makes teaching simpler, or possibly better for the majority of students at the expense of those at either end. My daughter really "gets it" re math (and scores at the 99th percentile on the standardized tests) so I listen carefully when she says "What do they mean here? I can come up with two completely different answers." To have her marked wrong if she doesn't happen to pick the "right" one seems completely counter to what we should be doing in school. (As it happens, she was not marked wrong, because it was a multiple choice test, in which the "wrong" answer was not available. But what if it had been fill-in-the-blank – which is far more "real world.")

Math has the strength that once it is in symbols, it is completely unambiguous. You can solve it in Italy, France, the US, etc. and come up with the same answer. To add ambiguity unnecessarily, seems crazy.

Virtually all real world problems are "word problems." To solve them you need to apply linguistic skills and logic skills first. Single words taken out of context are virtually useless.

Every time you see "of," multiply. Nonsense.

Every time you see "of", think. Much better.

Word problems can be difficult, even in this simplest form.

Actually, I think they can be difficult especially in this simplest form. On a complete word problem, the ambiguity is usually removed. But without more context, and especially in a string of problems with the standard symbol operators, then "of" seems to me to be ambiguous.

Stand corrected,

thanks

I hope you didn't take it as a correction. It was intended as just a comment among many in my long rant here.

Actually it is how things are worded, and one can not talk loose, and your commented was taken as that.

Thank you,

I (we) understand in the real world, words are used in math problems. What is needed is for schools (pre-college) to stick to the fundamentals of mathematics. (and all subjects)

Once the chidren grow and leave schools they will each have different life experiences, which will mold their vocabulary naturally. Attempting to teach 'for life after school' is risky at the public school level, since so much is going to change before the children even enter their careers; and each child is going to have vastly varying lives.

The only common ground, at school age, is the fundamentals. Once fundamentals are mastered, absorbing the career/life specific, concepts will be easy.

Good luck with your next 6 years of homework...I feel for you.

PS>I am still upset that our school offers virtually no computer science/programming classes until 10th grade, yet from 1st grade on they have 'computer' class. Why is my child learning MSWord?

You say:

PS>I am still upset that our school offers virtually no computer science/programming classes until 10th grade, yet from 1st grade on they have 'computer' class. Why is my child learning MSWord?

Yes, isn't that nuts!!! These kids could be doing meaningful programming in grade school, and having a great time learning, making computer games, programming robots, etc. etc. And they are stuck typing. Go see a First Lego League competition to see the kind of programming grade schoolers can do, when turned loose to learn.

It sounds like "of" makes the most sense when the first number is less than one, in which case multiplication is like division, in that you get a number smaller than the second operand... a part "of" it. So, "0.5 of 10" is like asking for "50% of ten", which is quite natural.

I wonder if all of the problems in the test followed that less-than-1 rule?

Okay, I was following the thread okay until that bit about pi - I think that threw a few of us. Anyway, I think Nickjd hit it dead-on - a lot of what is being taught now is crap! My son is in middle school now, and they use a lot of english in the math(s) classroom as well - for the life of me I can't figure out why. I thought for a while that it was the old school coming back, but after talking to the teachers I have decided that they are lazy and sadly, not very smart. We seem to have a lot of teachers that teach because it gives them their summers off, with no real love of teaching or of kids for that matter. I know there are still real teachers out there, but we don't seem to have many nearby.

The maths teachers are probably teaching English in maths classes because the English teachers are too busy teaching poetry and drama instead of grammar and spelling in English classes.

"Those who can, do; those who can't, teach"

Can't remember who said that...

and Those who can't teach, teach teachers to teach.

Ok, that's it. As an English teacher I can't let that go unchallenged (and you have left your flank open.)

The word "shore" pertains to the boundary between land and sea. Your pithy quote needs to read "it sure is the way to bet."

And don't blame your old English teacher; just turn off your spellchecker and get a dictionary.

"The race may not always be to the swift nor the battle to the strong but it shore is the way to bet."

Congratulations RobertOz and go to the head of the class. I have been waiting for about 6 weeks for somebody to notice that is should be "sure". The previous deliberate mistake only took 4 weeks to get noticed. Either I am becoming more subtle or people aren't reading what's in front of them. If the latter is the case then it tends to reinforce my point on poor spelling an grammar.

By the way the English department at my high school was abysmal and the only reason I managed to get through was private tutelage. Many of my fellow students weren't so fortunate. I know of at least one student that spent four years under their supervision and the didn't notice he was illiterate. I myself would be one of the worst spellers out and use spell checkers extensively prior to proof reading. I find them very helpful. I do however only use them to show me where I have erred and give me possible choices rather than letting them run wild and mutilating what I am trying to say.

You do realize that this means I need to think up some other subtle or maybe not so subtle error to include in my signature.

Hi Masu,

Haven't been to this thread in a long while... but I just reread my comment to you and found it was out of line - one of those heat of the moment things. My apologies.

Robert

G'day RobertOz,

No worries. No offence was taken so no apology was necessary.

Regards MASU

My spelling and grammar is probably some of the worst on CR4. Unbelievably I received zero teaching in English at school ! It's only through being an avid reader and lover of words that I can communicate at all. Much as I've tried to read up on written English, I just have a mental blind spot for it. To cap it off, I have almost unreadable handwriting. Still, I just about manage to convey my meaning which is the main thing. I'd be lost without word-processors.

Wow! I had some great English teachers, most of whom were very adept at reading between the lines. Has that ability been lost entirely in today's crop? I suspect that next you will criticize my tagline for its apparent missing word and punctuation? There are those of us who have fun with language... and, I suppose, those who do not.

By the way, I'd suggest The Elements of Style as a good source for guidance in punctuation. There, you will find that your second sentence, beginning "As an English teacher..." should have a comma after the word "teacher." I hope that is helpful in your continuing quest to learn more about the language.

There are a great many people who misspell words in this forum, both intentionally and unintentionally. Your time might be better spent in making comments having something to do with the subject matter, rather than simply criticizing spelling. If your intention is simply to insult, then there are loads of other forums elsewhere on the Web where your contributions will be better received.

...whereas your criticisms are perfectly adored here? Somehow, I don't think so.

Maybe you better relax a bit before telling someone they are not welcome on this web site. You might be surprised to learn what is and isn't acceptable around here.

I'm sure someone already answered, but I've always thought "of" as mulitplication.

i.e 1/4 of 100 is 25, 1/4x100=25

Lol, "of" is not an operator per se, this is a word problem, it is a elementary practice in u nderstanding word problems and converting/epressing them as mathematical problems, . The idea is that you can take a verbal description of something and express it as in mathematical terms to obtain a result. I do not know any normal people who speak using mathematical terminology to describe everyday occurrences, e.g. 35 divided by 40 engineers on this site lacked sufficient english comprehension develop a mathematical expression from a elementary word problem, therefore 87.5% illiteracy (this is only one possible interpretation to obtain a result, could be 35-40 therefore 5 do, there are many operation that may be preformed here depending on the results sought).

I went through this same exercise with my daughter just a few years ago (she's now in high school, dealing with that mess).

We went back through each section of her pre-algebra textbook to see how the word "of" was used in every case we could find.

If I remember correctly, we decided that if "of" was used CORRECTLY by itself, it always referred to a multiplication operation.

If it was correctly used only with a modifier like "ratio of" or usually "OUT of" then she should divide, as if "out" meant the reciprical.

Similarly, with "total of" or the obvious "sum of" she should add, and the "difference of" was a subtraction problem.

It's all in how the word "of" is modified by other surrounding words...or not.

Then she missed a problem because the teacher wrote his own test and did not correctly modify/use the word "of"...probably why he wasn't teaching English, though it was an interesting discussion, since he claimed "most of" the other kids got the answer correct, and wouldn't budge. Homeschooling never looked better - just wish I could do it for her, time-wise.

Would you believe they are even taking PE time away from actual physical activity so they can teach "nutrition and healthy living activities" because the HEALTH class has too much time devoted to state-required anatomy, sexuality, self-esteem, relationship, and psychology subjects for them to actually teach HEALTH in HEALTH class? Thank you, NEA. What a waste. Don't even get me started on the NEA definition of "proper" parental involvement...

Thanks for your insights.

I was able to pull my son out of school for one term and home school him (because the public school here, where we'd moved in the middle of the term, was completely unable to provide the courses he needed. (Ironically, the local school raves about how wonderful it is and how the high school it feeds is one of the ten best in the country – something I have yet to verify despite several web searches. The courses he needed were available in the small, backwoods southern town from which we'd moved. There, given a little push, they transported him to the local high school for science and math classes and back to middle school for others. Here, in this fancy-schmancy school district, they could do nothing.)

On the bright side, he is now in a science and math magnet that is phenomenal. The teachers are really engaged, and the administration bends over backwards and does summersaults to make sure the kids get what they need. If only all schools showed the same enthusiasm for helping kids to learn and stay engaged. Same county… both public schools… but a night-and-day difference between his high school (not the one fed by the middle school) and her middle school (which was temporarily his middle school.)

Thank you very much. Not! (Used here as a retrospectively applied negative). My little girl's only two and I'm already lying awake at night dreading all the up and coming battles with schools. I've seen enough "education" in action on friends kids to be worried. What I read here just confirms it!

You wouldn't think it'd be a lot to ask for a school to teach them to read write and add up but it seems the hardest thing of all?

Fear not. (Maybe.) Although I rant and rave about this stuff, until this year, my daughter has had really great teachers, and pretty good text books. (On the other hand, she briefly had one teacher who was jaw-droppingly poor. She was convinced that if there was a single kid in her class with a love of learning, she'd have that cured in the first month! But we were able to get her into a different class. We've lived in states not known for great schools, but we have nevertheless found some very good teachers.

If you remember that you know your child better than the school administrators, and occasionally push hard if you need to, you'll probably find some very good teachers too. I'd have to say that Michelle has loved school for 5.5 of six years. (That would be 33 years right?)

Thanks Ken. All reasurance gratefully received. Trouble is that I was not impressed by the education system thirty years ago when I was the victim. Now I understand that they are refered to as the good old days!

Personally if it wasn't for all the OH&S issues and concerns about socialisation I'd be taking my daughter to work with me. Reckon she'd learn more things that are useful trotting along behind than she'll ever learn in our PC education system. "0.7 of " indeed!

Ahem. Division is multiplication. And multiplication is addition. And subtraction is division….and so on. It's okay to tell a daughter—or son—that something is not familiar to you. After all (you could say to her), your parents didn't help you like you're helping her; and your teachers never taught you about that; so you're glad to be learning later rather than never…even if with your very smart daughter's help! As the daughter learns, in school, she can bring you up (or should I say back?) to basics as they were once commonly known. (I can assure you there's no better way to tutor a child than to encourage her to teach the tutor! and no better way to teach a child to think clearly.)

If a child sees 0.7 of 10, it (reinforces the concept of decimals as well as) helps her to quickly (in her head, even) translate 0.7 to 7/10; so she says to herself, not "point seven of twenty-four but, rather, seven tenths of twenty-four. If she must proceed with pen and paper, she will arrive at a division/multiplication form--as opposed to a mere multiplication (of decimals) form. Accordingly she will come to readily grasp—or recognize on standard tests—or when talking to others—(not only that decimals are fractions expressed another way, and vice versa, but also) that there are multiple ways (some easier or more facile based on what presents itself) to express mathematical (and, in due course, algebraic, analytic, etc.!) calculations.

[Note to reader— in the expository that follows, the reader will have to imagine that the number problem examples are written in "school book" form (i.e. vertical, or block, form rather than paragraph, or left-to-right form). Actually, this posting was originally composed in school format, complete with fraction simplification mark outs. However, the CR4 boards are quite limited in ability to actually process & display any but rudimentary word processor format features; and, as yet, still don't include displaying of strikeouts—a site deficiency the correction of which one hopes will not be too long in coming. Anyway, it was found—even after using "format cleanup," that the examples as originally written would not be easily deciphered unless re-submitted in paragraph format.]

Post continues…

So, in the example at hand: whereas if she was given

.7 times (or "x") 24, she would (be prone) to write

0.7 x 24 = 16.8 (or, sixteen and four fifths). [Remember this answer.]

If, on the other hand, she was given .7 ("seven tenths") of 24 (twenty-four), she would (be encouraged [not discouraged] to) think (that's right, think) (&or write)

7/10 x 24

= (7 x 24)/10

= (7 x 12)/5

= 84/5

= 80/5 + 4/5

= 16 + 4/5

= 16-4/5 (sixteen & four fifths).

Or, she might think to skip the reduction step and (more readily) come up with

168/10

= 160/10 + 8/10

= 16/1 + 8/10 or 16 + 4/5

= (16 and 4/5) or (16 & 8/10)… [Remember that last one]…

…which she will come to recognize at the same thing, 16.8, as she could have gotten by multiplying decimals—instead of dividing (or should we say multiplying?) fractions (which themselves are multiplication (or should we say, division) operations.

So you see, you should be thankful (at long last) that your daughter is receiving a comprehensive tutelage in mathematics; that the tide of dumb-ing down of math (i.e., the "new" math for ordinaries of the 70's – 80's), and using buttons and displays to (not) think, has finally been reversed. You should be thankful also that your daughter and her peers might be better fit to reverse the hitherto growing trend of declining competitiveness in the world of American-born workers, including engineers…provided, one might hope, that such educational gains can be maintained in spite of countervailing "dumb-ing down" trends wrought by the internet.

And, I might add, we should hope, that if the new order of things in Congress is such that the teacher's unions—to whom the Dem's are beholden—succeed in pressuring for reversion back to dumb-down education (as they've been trying so hard to do until now), then the current Executive, and his successors in office, will wield their veto pens with a vengeance.

Oops, let me correct the typo, in bold face below.

Ahem. Division is multiplication. And multiplication is addition. And subtraction is division….and so on. It's okay to tell a daughter—or son—that something is not familiar to you. After all (you could say to her), your parents didn't help you like you're helping her; and your teachers never taught you about that; so you're glad to be learning later rather than never…even if with your very smart daughter's help! As the daughter learns, in school, she can bring you up (or should I say back?) to basics as they were once commonly known. (I can assure you there's no better way to tutor a child than to encourage her to teach the tutor! and no better way to teach a child to think clearly.)

If a child sees 0.7 of 24, it (reinforces the concept of decimals as well as) helps her to quickly (in her head, even) translate 0.7 to 7/10; so she says to herself, not "point seven of twenty-four but, rather, seven tenths of twenty-four. If she must proceed with pen and paper, she will arrive at a division/multiplication form--as opposed to a mere multiplication (of decimals) form. Accordingly she will come to readily grasp—or recognize on standard tests—or when talking to others—(not only that decimals are fractions expressed another way, and vice versa, but also) that there are multiple ways (some easier or more facile based on what presents itself) to express mathematical (and, in due course, algebraic, analytic, etc.!) calculations.

[Note to reader— in the expository that follows, the reader will have to imagine that the number problem examples are written in "school book" form (i.e. vertical, or block, form rather than paragraph, or left-to-right form). Actually, this posting was originally composed in school format, complete with fraction simplification mark outs. However, the CR4 boards are quite limited in ability to actually process & display any but rudimentary word processor format features; and, as yet, still don't include displaying of strikeouts—a site deficiency the correction of which one hopes will not be too long in coming. Anyway, it was found—even after using "format cleanup," that the examples as originally written would not be easily deciphered unless re-submitted in paragraph format.]

Post continues…

So, in the example at hand: whereas if she was given

.7 times (or "x") 24, she would (be prone) to write

0.7 x 24 = 16.8 (or, sixteen and four fifths). [Remember this answer.]

If, on the other hand, she was given .7 ("seven tenths") of 24 (twenty-four), she would (be encouraged [not discouraged] to) think (that's right, think) (&or write)

7/10 x 24

= (7 x 24)/10

= (7 x 12)/5

= 84/5

= 80/5 + 4/5

= 16 + 4/5

= 16-4/5 (sixteen & four fifths).

Or, she might think to skip the reduction step and (more readily) come up with

168/10

= 160/10 + 8/10

= 16/1 + 8/10 or 16 + 4/5

= (16 and 4/5) or (16 & 8/10)… [Remember that last one]…

…which she will come to recognize at the same thing, 16.8, as she could have gotten by multiplying decimals—instead of dividing (or should we say multiplying?) fractions (which themselves are multiplication (or should we say, division) operations.

So you see, you should be thankful (at long last) that your daughter is receiving a comprehensive tutelage in mathematics; that the tide of dumb-ing down of math (i.e., the "new" math for ordinaries of the 70's – 80's), and using buttons and displays to (not) think, has finally been reversed. You should be thankful also that your daughter and her peers might be better fit to reverse the hitherto growing trend of declining competitiveness in the world of American-born workers, including engineers…provided, one might hope, that such educational gains can be maintained in spite of countervailing "dumb-ing down" trends wrought by the internet.

And, I might add, we should hope, that if the new order of things in Congress is such that the teacher's unions—to whom the Dem's are beholden—succeed in pressuring for reversion back to dumb-down education (as they've been trying so hard to do until now), then the current Executive, and his successors in office, will wield their veto pens with a vengeance.

You write:

"Ahem. Division is multiplication. And multiplication is addition. And subtraction is division….and so on. It's okay to tell a daughter—or son—that something is not familiar to you. After all (you could say to her), your parents didn't help you like you're helping her; and your teachers never taught you about that; so you're glad to be learning later rather than never…even if with your very smart daughter's help!"

I suppose your post is not intended to be insulting, although you can probably understand how I might find it to be so. You've made many assumptions about me and my family. Both my mother and father were valedictorians of both their high school and college classes, and my mother graduated high school at 16, college at 19. Like my daughter, I scored at the 99th percentile on standardized tests. I scored 100% on every New York State math regents test and well into the 99 percentile on the SATs back in the days of the Sputnik, before the SATs were re-centered by 50 points to adjust for the continuously falling scores since those days. My parents did help whenever I needed it, and we lived in hideously expensive neighborhoods specifically to take advantage of great schools. (We weren't there to hob knob with the rich and famous; my dad was much too frugal for that. For example, we mixed milk 50/50 whole/dry, because it was cheaper that way, but didn't taste quite as awful as 100% dry milk). So you can probably see that I might be offended that you would suggest that my beloved math teacher "never taught me about that" or that I should be glad to be "learning later rather than never". No, I am not a nitwit who cannot evaluate math expressions seen in 6^th grade texts. Rather, I propose that teaching rote interpretation of certain words as math operators encourages students to give up on thinking for themselves. Had my daughter failed to see the ambiguity in the use of the word "of" as a math operator (when used in precisely the same context as the use of x or / or + or (k), I would have been sorely disappointed. To have her basic ability to think logically pushed aside in favor of thinking in lock step is sad.

Although your statement "Division is multiplication. And multiplication is addition. And subtraction is division….and so on" is catchy, it is equally true and sophomoric to say that everyone lives in his or her own reality. 2x3 evaluates to 6 (not 1.5) throughout the world. That's one of the beauties of symbolic math. To throw that aside, and suggest that certain English words should be taken in the same literal fashion as a math operator does a disservice to thinking students. Further, this same statement, (that "division is mult…") seems counter to your argument that "Of' has a specific meaning. By your own argument, "Of" should then mean multiplication, division, addition, and subtraction. This is precisely the sort of logic I hope my daughter will not buy. I would hope she would counter: "Of" means multiplication. Multiplication is division. Therefore, "Of" means division.

Consider this: My son is also pretty bright, and has had some superb math teachers both in private school and, now, in public school. As a sophomore in public high school, he is taking AP Calculus, AP Physics, AP Chemistry, and a programming course, just this semester. He is far ahead of where I was at his age. He should be able to come up with a simple answer for what "of" means mathematically, if your premise that it has some rigid, specific mathematical meaning is correct. His first answer: exponentiation, as in "the cube of 2, or "the third root of 6". On further questioning, he allowed as how he could also take it to mean multiplication or division. To teach any student -- bright, dim or otherwise -- that "of" has some specific meaning, aside from its various meanings in linguistic context, does a real disservice to all those students. If this is what math has come to today in the US, no wonder the US is loosing out to other countries in terms of test scores, innovation, jobs, etc. etc.

As far as Dems are concerned: when I was a kid, wildly liberal California was the envy of the nation in education at all levels. New York was well up there, as was Wisconsin. As a result of Reagan's policies, and subsequent conservative policies, California is now at very near the bottom of the heap. Wisconsin, still liberal, is still near the top. There is no evidence whatsoever that states with strong teachers unions perform worse than those without.

If you (or others) remain convinced that "of" has a specific inviolable mathematical meaning, then try entering it into the Google calculator, which is far more sophisticated than most people realize. "5 times 4", for example, evaluates to 20. Then try "4 of 5". It evaluates as a string. Try "(mass of earth times G)". Then try "(mass of earth of G)". Consider that Google is probably now the world's largest hirer of math majors. (BTW, don't enter the quotation marks.)

Am I ranting? Of course. We are talking about fundamental issues in education, something I care dearly for. Do we want kids who can think, or kids who accept simple, rote answers where the issues may be complex or ambiguous? Do we want a country at the bottom of the heap or at the top?

From California, via east, west, north, and south--

Do you not realize, that while enveighing against simple, rote answers, you are espousing simplified, rote thinking?

Properly understood, the core of my message was: that .7 times 24 can be expressed as seven tenths of 24 without giving offense, no matter how rigorous the listener deems himself or herself to be.

Suggestion: Read everything I wrote but assume the opposite of what you did, and assume (the fact that) I correctly anticipated your station and age in life (and sought strenously not to insult you by careful choice of words and examples) then perhaps the substance of my post will be clearer. If that doesn't work, show it to your daughter and she will surely grasp what was said.

Our discussion so far seems to be a case--in mathspeak sense--of reducing a (mathematical or any) argument in hopes of attaining the solution, only to have it expanded again, and then some, before the solution is recognized. There, you now have an insult (where simpatico would normally be perceived) if you choose to see it that way.

As to your pronouncements (innocent as they are) about California, politics and such, I can only advise that the perceptions you have are based on supposed facts (every state likes to look better than it actually is, and "propagandizes" accordingly) that never were true. Consider, for example, that transferees from California public schools have, as long as you and I could remember, been placed back a year when enrolling in schools in many, if not most, other states. Or that California has only recently adopted the SAT entrance exam in place of CAT...because CAT is held in lesser esteem by many, if not most, institutions--except in California, in which until recent times, the objective was to admit all comers, in spite of qualifications. Or ask why it is that, among California families who have the means, it is typically the choice to send one's children out of state for college education?

Let me conclude with an anecdote--one which may help demonstrate that this whole thing is really a discussion not about math, or about teaching, but simply about semantics. I once encountered a teenage girl who had heard (from me) the word "latrine" but did not know its meaning. She asked its meaning and received an answer: that it was a hand-dug ditch used for a certain purpose, say, when camping. With snarly, teenage contempt, she retorted: Why not just call it a ditch? The question for forum participants, then, would be: Who was missing the point? And what poiint(s) was(were) being missed? Or should we just call it the single hemisphere vs. double hemisphere dilemma? Or the tyranny of the dictionary (and of the school text book) in action?

Here's another tidbit. Let us look at the operator, x (times). But for the fact that one learns to express an equation, say 2 x (times) 2, in a classroom (it could be a home class room as well), could we not say, that a number TIMES another number makes little sense...or could mean numerous things, multiplication of which would not necessesarily be the first chosen? Would not the multiplication meaning be one that must be reserved, rigorously, for application only in the classroom...but avoided elsewhere for fear of confusing someone? Well, yes, and no, provided semantics are not taken into account. Provided that, for purposes of mathematical symbolization (shorthand expression) we fail to realize that, say, 2 times 2 meant something else before it came to be expressed as it is written mathematically, 2 x 2: namely, that when one sees or says, 2 x 2, one actually is saying and meaning, two (of something) taken two times; or 2 2 x, which for uniform consistency and clarity is reordered to avert confusion--in the classroom or elsewhere. Similarly one might use times or of (or times the inverse) interchangeably depending on context and necessity for clarity in communication. So, one might ask for 1/2 of a gallon (half'a gallon, same thing), or 3 of this or that, in the grocery store (that's market for New Yorkers) rather than for 1/2 times a gallon or 2 times this or that...and receive a funny stare in return.

My compliments on your and your parents' spendid achievements and capabilities. If your daughter has similar perplexites in the future, I would be more than happy to help provide an answer she can readily understand. Just have her post it, or send PM. No offense intended...only to be helpful.

You wrote:

"Properly understood, the core of my message was: that .7 times 24 can be expressed as seven tenths of 24 without giving offense, no matter how rigorous the listener deems himself or herself to be." I agree that a portion of your message stated essentially that. No one here would disagree with that. Few third or fourth graders would disagree. Many teens and preteens would say "Well, Duh." If that was the core of your message, then you need not have rambled on for so long. No one here would disagree with your long list of equivalent expressions. Most fifth graders could see very quickly that they all evaluate to the same number. Nothing I said suggested that there are not an infinite number of expressions that equate to any given expression.

My point is that "of" completely lacks the unambiguous nature of a symbolic math operator, and that to treat it as if it is unambiguous is unfair to students. (Sure, it belongs in phrases and word problems – again… Duh) "Of" can connote multiplication. Virtually no one would argue against that. Of can also connote division: "2 of 16 reviewers." Few, other than you, will argue against that. To suggest that "2 of 16" should somehow always evaluate to 32 is nonsense. As one reader pointed out, "of" can connote subtraction, as in a "quarter of nine". "Of" is sufficiently ambiguous as an operator that Google will not parse it as such. Google, will, however, parse "multiplied by" and "times" as math operators because they are less ambiguous, certainly in the US, but also in most of the English-speaking world.

My daughter's teacher says that 2 of 16 always evaluates to 32. If my daughter, or any other student in the class, suggests that it might evaluate to 1/8, she is told she is wrong, and is marked wrong on tests. To me, that fundamentally conflicts with good educational practice. It is a classic example of dumbing things down so that simplistic rules can be applied, right or wrong. In order for each member of the class to get the "right" answer, they must ignore all they've learned about math and logic, and simply, by rote, apply the incorrect rule: if it says "a of b" it always evaluates to ab.

You wrote: "Let me conclude with an anecdote--one which may help demonstrate that this whole thing is really a discussion not about math, or about teaching, but simply about semantics." You seem to have missed the point entirely. It is not about semantics; it is about the imposition of overly simplistic rules, and the stifling of both inside-the-box and outside-the-box thinking. What we say when we think through problems has nothing to do with it. What we say when we annunciate a math problem or ask for a half gallon of milk has nothing to do with. The issue is that "of," when printed in an expression to evaluate in exactly the same way that / or x would be printed, is inherently ambiguous. That makes it completely unlike the standard operator symbols which are entirely unambiguous for anyone around the world remotely skilled in math. Therefore, it is unfair and demotivating to create tests which treat the ambiguous word exactly like an unambiguous operator symbol.

I suspect this will have little effect on your thinking. After all, you have clearly stated "Division is multiplication. And multiplication is addition. And subtraction is division….and so on" Then, clearly, "2 of 4" can mean (by your own logic), 2/4, 2x4 2+4, or 2-4. Your logic and mine are quite different. Apparently yours has worked for you. Mine works for me. Reality is subjective.

I am late joining this because I took a couple of days vacation. This does touch on one of my pet peeves. I tutored my children through math in the public schools. I had a high school math teacher once tell me that one of the tips the gives to his students for interpreting word problems is, "If the problem contains the word "of", you divide." This is clearly an attempt to circumvent true understanding of how the known information must be expressed in the format/language of correct mathematical notation. These simplistic rules will only work if the problem is stated in a canned format similar to what will show up on test day.

Until/unless we can teach students to recognize and express the relationships given in the word problem, we cannot produce mathematically literate students. In the case sighted, you actually are multiplying 24 by 7, then dividing that result by 10 -- or divide by 10, then multiply by 7 -- it doesn't matter. I know that engineers know that, but junior high students may not. A simplistic rule (if the problem contains the word "of") will never allow them to achieve that understanding.

A related shortcoming of modern education is excessive free thinking. Free thinking does have value, but only on your own time after the homework is done. This is where you prove for yourself that a/b = a x (1/b) and that a x (b + c) = (a x b) + (a x c). You are not inventing new notations, you are understanding the notations coming down from centuries of scholarship. I went through three courses of calculus/analytic geometry, then differential equations and what I needed to understand was always the same. "What information do I have, how can these known relationships be expressed in rigorous mathematical logic, and what manipulations have been proven effective since at least 1640 AD (the time of Rene Descartes)?" Save the free thinking until you reach graduate school.

I concede there is another level up there. I took partial differential equations in graduate school. That was the first time I realized there were engineers, and then there are mathematicians -- and I was the former; but I sure cleaned up in calculus using only "stuff" already in practice by 1700 AD.

Very well put!

Just to add confusion to and otherwise clear interpretation of the word "of". How come on drawings when a particular part is used more than once they use the word "off" to mean multiplication? For example:

"3 off 12mm Bolts"

Meaning that there are three 12mm bolts required in the assembly. Seems that they are trying to avoid the misinterpretability of the word "of".

I've always assumed it's was becuase you asked the bloke on the machine to 'run three off for me' .....to finish this widget.

Long ago I really wondered about how well standardized test writers understood what they were doing. At that time most of my concerns were with the math, for similar reasons, but later I started too wonder about the language sections. Maybe there were more factors to consider in the ___ is to ____ as _____ is to ____. questions. It has been rewarding to see the SAT and ACT correcting scores for THEIR wrong answers.

RchH

Amen!

This has been a good thread which I couldn't help recalling in a recent discussion. Whether my little anecdote is off thread I'll leave for others to judge.

In Australia we have GST. This is a 10% tax added onto the final figure in any financial transaction. A friend, who is in business, had his son present the following problem extracted from his homework: If two bricklayers earn $1000 after GST, how much did they each earn before GST? My friend who shares my regard for the education system told his son that he was going to get the question wrong and that his teacher would also get it wrong. To prove his point he wrote out the correct answer along with the calculations and put it in an sealed envelope to be handed to the teacher at the end of class.

The teachers response after reading this missive was to say "Yes your dad is right but that was not the answer I was looking for"!!!

What hope is there for an education system that, not only can't recognise that 10% on is different to 10% off, but entrenches this into incorrect homework handouts?

Oh no! Please don't tell me the teacher was suggesting that the bricklayers earned $900 before GST. If so, I will gladly testify in your friend's trial that his reaction was justifiable homicide.

You guessed it. What's worse is that is wasn't really the teachers fault as it was Education Dept' supplied material. I believe he is still pursuing it up the line to find whom to practise his justifiable homicide on. The "correct" (read "official") answer was $450 each instead of the more tricky $454.55.

The concern is that the teacher didn't immediately spot the trap when setting the homework. When I was told the question I immediately spotted it because when I went to school my teachers ( being sadistic offspring of unmarried parents) slipped things like that intentionally into otherwise harmless homework and exams just to make sure everyone was awake. I actually quite like the question. It's a very neat way to illustrate a point. Unfortunately the point appears to beyond the "educators"

I should add that the problem of people knocking off a 10th rather than an 11th is a very common frustration with GST so it's doubly aggravating to have the next generation corrupted at the one place they should be learning better practice.

Pretty clearly the answer to the thread title is whatever the sum of the letter scores are for O and F in the game of scrabble.

Terribly sorry but haven't a board in the office, though I do note a lot of colleagues in a corner playing 'Countdown' quite often.

Yeah, yeah, I know there's bound to be loads of games going on all over the internet.

Of a similar nature, I once saw a shop sign that proudly proclaimed " 20 % off or less ! " .