Really? Using a calculator for 6*7 wastes valuable seconds when one could just think, 42, and have the bigger problem half solved by the time the calculator has the answer.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Probably not the best example, as Galileo determined the period of a swinging light at the Sistine Chapel (in heartbeats) and that it was independent of displacement, without math. Now it took Newton to show, with math, why that was the case. I think Galileo did come up with the formula for period versus length with math.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Not anymore, as I learned when I TA'd calculus at UVA. We had to reteach a fair amount of algebra in the sections for non-math-science majors.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

When I was a grocery sacker, the scanners were new. The older workers knew the prices of the most commonly bought items, but the younger workers did not, and if the scanner failed, had to type, very slowly with one finger, the price in.

Oh, and errors DID have to be canceled (we didn't require supervisor approval, though I know that opens up an avenue for fraud by customer-cashier collusion--though best we could tell, that wasn't where the bleeding was. It was shoplifting.) mainly to reduce the number of times a by-hand inventory had to be done.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

I just thought of something--isn't University Level something different in Europe than in the US?

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

I tend to take the 2*(3*7) approach.


Or it can waste you far more time when you misremember and use the incorrect result for a page full of calculations, or when you're constantly cross-checking yourself to avoid such errors.

Plus, the calculator is a useful tool for learning that 6*7 = 42. If you're unsure, it's a couple moments of button-pressing to find the correct answer, which is far more effective at reinforcing that association than dropping what you're doing and working out the answer through other means, or ploughing ahead in the hopes that you're right.

You can and certainly will make errors while entering stuff into a calculator as well, of course. But you do get a feel for what the correct answers should be, and without cramping up your hand, using up sheets of paper, and constantly making trips to the pencil sharpener. Blind trust in the machine is in large part a sign of inexperience with the machine...or perhaps more likely in this case, just not particularly caring if the result is correct. Neither of these is going to be helped by banning calculators from classrooms. The idea that mechanical aids are somehow a cause of innumeracy is just pure nonsense.

Calculators should not be banned, not at any grade level. In fact, I think they should be provided at the earliest preschool level. Specialized for the audience, of course, with things like verbal feedback, games and exercises for teaching numbers and math, etc., but powerful ones, complete with programmable and with graphical capabilities.

I've never used the term to describe any level. There's college level, which is the four years it takes to get a Bachelor's, and there's grad school, which is anything above that.

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Gillian

"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"

"You can't erase icing."

"I can't believe it doesn't work! I found it on the internet, man!"

I never heard this on the West Coast, but out here "college physics" is at the algebra and trigonometry math level while "university physics" is at the calculus level. That, to me, implies a "university level," which might be synonymous with "college level" for the most part.

Errr, we had all three in High School (in Massachusetts).

The major difference between the US and Europe is that in Europe one only studies the major and minor subject at the university, from the first semester onward. Accordingly, "high school" lasts one to two years longer. The "general curriculum" subjects associated with (parts of) the freshman and sophmore years at US colleges/universities are all completed beforehand. In the US, this can be called the "Associate of Arts" degree.

This is why, in general, a US high school diploma is not enough qualification to be admitted to a European university.

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Ach, mein Sinn, wo willst du endlich hin, wo soll ich mich erquicken? Bleib' ich hier, oder wünsch' ich mir Berg und Hügel auf den Rücken?
Bei der Welt ist gar kein Rat, und im Herzen steh'n die Schmerzen meiner Missetat, weil der Knecht den Herrn verleugnet hat.

What about all the people who took physics in high school? (Yes, I knew some people who took AP physics, though I don't know who took the test or who passed it. But still.)

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Gillian

"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"

"You can't erase icing."

"I can't believe it doesn't work! I found it on the internet, man!"

uh...who's going to misremember 6*7? I'd say punching a wrong button on a calculator is FAR more likely (I've graded so many papers with silly nonsensical answers--answers that defy common sense if you even think while you work--that were obviously punched into a calculator with a digit incorrect somewhere--calculators have their uses, but using them to avoid thinking is a real problem among Freshmen, as I know from seeing it in action).

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Sure - it means your CB has been overtuned and is above the legal power limit of 4W on your SWR meter!

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Anyone and everyone who does a lot of calculations. Are you saying you've never screwed up basic arithmetic? Yeah, right.


Perhaps more likely for simple problems, but still more easily detected, and more quickly corrected.


A student giving an answer does not mean that student believed that answer to be correct. Even if they are interested and want to give a correct answer, there's not always time to find the error. Making them do it all by hand certainly isn't going to help with that.

I also don't believe you can easily spot errors made due to blind reliance on a calculator. It's too easy to make the same errors grinding out numbers with pencil and paper. From writing the wrong digit down or mis-reading a poorly written digit to outright omitting or doubling digits, to screwing up borrows and carries, etc...it's a notoriously slow and error prone process. Drills on mental arithmetic with small numbers are useful, but beyond that it's both a waste of time better spent learning real math, and conditioning of students to avoid anything that resembles math.

I like to call real maths - mathematics

and simple addition and subtraction - arithmetic.

Are you guys familiar with the term arithmetic?

I don't here many people from North America using this term.

Right. Arithmetic is where you perform the basic operations on numbers.

When you start working with equations using variables into which you can insert numbers, then math. This is mostly algebra.

Then when you start with calculus you start really using math.

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“He who asks a question is a fool for five minutes; he who does not ask a question remains a fool forever”
Chinese proverb
"All you need in this life is ignorance and confidence - and then success is sure." - Mark Twain.

No you can't, it is in a spaceship which is accelerating at 2g, a long way away, it is 2 metres long, and the mass per metre down its length is given by mass per metre = (distance^2)+1

So, what's the period of that, without using any maths?

I'm not sure if it is different at the beginning, but I'm pretty certain that the courses here give a more in-depth knowledge than in the US, as the course tends to be a coherent whole, where you must cover the subject for the whole time (we don't have any "minor", and we don't get any credits for unrelated subjects), so, for example, if you do a physics degree, it is four years of physics (with, of course, the maths that you need as well), and nothing else.

Er...?
This seems a very odd question, considering that I used it earlier in the sentence you quoted and specifically contrasted it with "real math". But yeah, arithmetic = glorified counting, memorization and practice of algorithms for performing hand computations. While both useful in and a product of math, arithmetic itself barely qualifies as math. It's that symbolic manipulation where the real interesting stuff is. (algebra, trigonometry, calculus, geometry, logic, etc)

I think it'd be very helpful to introduce students to algebra and formal reasoning much earlier, right alongside basic arithmetic, and focus less on mindless churning through page after page of of addition, subtraction, multiplication, etc. Not only would it give them an idea of what real math is like, but those reasoning skills are valuable in mental shortcuts and cross-checks for arithmetic...like the examples given above for breaking 6*7 into simpler operations.

So you did my mistake

Okay, I'm confused, actually. When I grew up, "arithmetic" was basically treated as a subset of "mathematics." Is that not true? We had math books, not arithmetic books. (Or if the cover called it arithmetic, certainly possible, everyone, teachers included, just called it math.)

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Gillian

"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"

"You can't erase icing."

"I can't believe it doesn't work! I found it on the internet, man!"

Absolutely. Especially 'formal reasoning', or simple mathematical logic. I think young kids would love ~B->~A.

And yeah, an early introduction to and long familiarity with just the rudimentary idea of algebra should make it much more broadly palatable when students get to Algebra I.

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Everyone is entitled to his own opinion, but not his own facts.

Yes I can.

No it isn't. There is no such pendulum or spaceship.

You are describing an imaginary pendulum in an imaginary spaceship.

It has no period because it doesn't exist.

You are asking for a prediction of what the period would be if
it were built and subjected to the conditions you specify, as I said.

Your first post was just poorly worded. You meant to say that one
cannot predict what the period of a pendulum would be, given a set
of physical parameters, without using maths. If one can see the
pendulum swinging, there is no need of maths to determine its period.

-- Jeff, in Minneapolis

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http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

Gillian, and everyone,

The very first volume of the fabulous Life Science Library is 'Mathematics',
published in 1963. If you haven't seen it, look for it in your library.
(The home version has a weak binding. The library binding is much better.)
Chapter one is "Numbers: A Long Way from One to Zero". Chapter two
is about geometry, chapter three is about algebra, chapter four is about
analytic geometry and trigonometry, chapter five is about calculus, chapter
six is about probability and statistics, chapter seven is about mathematics
developed in the 19th century-- what some here might call "real" math,
chapter eight is about math of the twentieth century. Lots of pictures,
as you'd expect from a Life publication.

I looked in it hoping to find something to quote that would help the
discussion, pertinent to Gillian's comment, but I haven't found anything
specific. The book as a whole is excellent.

-- Jeff, in Minneapolis

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http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

It is the height of arrogance to tell me what I meant. Please don't do it.

And I don't agree that my post was poorly worded. I said that you cannot work out the period. "Work out" is a synonym for calculate. I still say that you cannot do this without maths.

You may disagree, but that isn't of particular concern to me.

Absopositively correct.

Reliance on calculators is most revealing of the lack of thought among some students. Appalling is the word.

They are pretty useful for eliminating a need to look up trigonometric functions in tables (same for logarithms), but they are too often used as surrogate for thinking. For instance I have seen things along the line of this : given a triangle with two sides length 3 and 5 and an included angle of 27 degrees find the length to the remaining side -- following some key punching a length of 9,327,435 is offered, with not a hint of a thought that such an answer is ridiculous.

Calculators are useful in physics and engineering classes, and a help in eliminating a need for table look-ups. But fundamentally calculators are counter-productive for most mathematics classes.

Frankly, I would prefer slide rules for trig and log functions, if they were available any longer. They get the job done, but the student has to think enough to know where the decimal point goes.

True slide rule story. When the HP 35 calculator first came out (first calculator that could handle trig and logarithms) a young physics professor, who figured he was pretty hot stuff, bought one for about $350 (1970 dollars, and $350 then was quite a bit of money). Figuring that some "lesser light" could make use of his old Pickett slide rule he put a note on the bulletin board and set it out for sale. An older gentleman saw the note and purchased the slide rule, figuring that it would meet his needs. It did. That "lesser light" was Eugene Wigner.

Here's another problem with calculators, in addition to how they can substitute for thinking about the meaning of numbers: they can substitute for thinking about the meaning of formulae too. If we say the force of gravity is GMm/d², in calculatorese that formula says nothing more than "look up G, insert M, hit multiply, insert m, hit multiply, insert d, hit square, and divide". Like a recipe for cooking stew. But what a formula like that is really trying to tell us is, what does the force of gravity depend on, and how does it depend on it? What will happen to the force if we change various things in various ways? If you ask a student what happens to the force if you double the distance, and they say, "OK, well according to my recipe, the first thing I need to know is G", you know they are thinking in calculatorese. I think we should allow students to use calculators as needed, but make sure that they are only using the calculator to calculate for them-- not to think for them.

You know, people who fail to think because they have calculators are probably not experts at thinking full stop. As it happens, there are a few very simple multiplication figures I regularly forget; I don't look them up on a calculator, but it delays me a minute while I stop to remember them. No, I'd never give some wildly improbably answer because I'd entered it into a calculator wrong, but that's because I'm a basically intelligent person who doesn't rely on others to think for me. That's true in other fields, as well.

As another example, spell check is a valuable tool for the average person. Everyone makes typos or just can't spell certain words, and it's helpful for that. You can tell when someone's relying on it too much, however, when their paper/post is full of homophones and various words that are spelled similarly to what they mean but aren't anywhere close in, well, meaning. The problem is not with the tool.

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Gillian

"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"

"You can't erase icing."

"I can't believe it doesn't work! I found it on the internet, man!"

It isn't the height of arrogance. I've seen far worse. In any case,
what I said was correct.

Obviously one cannot calculate a mathematical value without using
mathematics. However, one can construct a pendulum and measure
its period without using any mathematics. That is working out the
period every bit as much as doing the calculation.

I could tell.

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

agreed, this is why my son does maths at 2-3 years above his grade level. It isn't because he know 67 * 45 = 3015 but that 67 * 45 =( (7*45) * 10) - 135. There are a few kids doing higher level maths in his class (they are in a class for gifted kids), but he's still does better then all but on kid when doing math competitions because if the others don't know a formula for something they are kind of stuck while he has learned how to solve the underlying problems himself. His teacher says he's got the best divergent skills of the bunch. This isn't a "gift" he had as much as much of a talent that was nurtured.

I to believe that more kids should be exposed to concepts like algebra early on. Less time spent on multiplication tables and more time on what multiplication actually means. I'd say a good 30% of the students out there could probably "perform" at the the top 5%-10% of current students if they did this. Of course then you'd have people complain about the "divide" that would be caused by those students and the ones that naturally struggle with simple addition.

Can anyone tell I'm passionate about childhood education?!?!?!

That may work for some very good students. But for most it would be nice if they knew enough by rote to be able to do arithmetic easily and not have arithmetic be a stumbling block to learning more advanced mathematics.

For instance, it is not uncommon for students to have trouble with fractional exponents, not because they don't understand powers and exponents, but because they can't add fractions.

You need to be able to do arithmetic in your sleep before trying to really learn and understand algebra and more advanced mathematics. Otherwise there is a tendency to just learn "symbol pushing" -- this is a common problem with students who learn their calculus in high school.