Problem switching back and forth between the diffrerent arithmetics

[Tyrion101]

I do really well if I have one form of arithmetic like addition, or subtraction. But as we all know math doesn't work like that sometimes. I will do silly things like 10/2 = 8, or 10*5 = 3. If you can understand what I am saying. I mix up my tables when I'm dealing with a long problem, and I was wondering if there was any trick to avoiding something like this when you're doing a long complicated problem?

 

[jedishrfu]

The best strategy is to identify exactly when you make this mistake under what operations and with what numbers.

Are you confused about operator precedence like multiplication and division before addition and subtraction and parentheses have higher precedence over arithmetic operations?

I used to have difficulty with 9 x 6 vs 8 x 7 and by reinforcing the 9 x 6 answer and remembering that the sum of the digits would also add upto 9 then I was able to master it.

You need to watch yourself doing the problem and develop a style. As an example, do the algebra first then plug in the numbers to the simplified equation and solve.

You need to drill yourself by redoing the problems that gave you trouble several more times to be sure you don't make the same mistake.

 

[Tyrion101]

I've been working on linear inequalities but for some reason no matter how hard I work I just hit a wall is there any common mistakes that I can watch out for? Also how do I know my answer is right, with equations there is just one answer.

 

[Mark44]

I do really well if I have one form of arithmetic like addition, or subtraction. But as we all know math doesn't work like that sometimes. I will do silly things like 10/2 = 8, or 10*5 = 3. If you can understand what I am saying.

Do you know the "times table" up to, say 10 * 10 or 12 * 12? The mistakes you're making suggest to me that maybe you don't. When I was a teacher, there were a few teachers who derided the idea that students should have a good grasp on basic arithmetic, by saying that the students could just use calculators.

IMO, those teachers should be sued for malpractice.

I mix up my tables when I'm dealing with a long problem

What do you mean "mix up my tables"?

, and I was wondering if there was any trick to avoiding something like this when you're doing a long complicated problem?

 

[Tyrion101]

So I don't know my tables because I mix them up when I have a long problem to do? I can do short ones in my head. If you read the actual post I gave an example.

 

[jedishrfu]

there's a nintendo ds game that can help with strengthening your recall of these tables:

https://www.amazon.com/dp/B001LNYM90/?tag=pfamazon01-20

The basic idea is to fill in the table where the the row and columns are scrambled and then allow you to fill in the squares. Its known as the Kageyama hundred-cell method.

I wasn't able to find a web version of it but did find a wikipedia reference for it:

http://en.wikipedia.org/wiki/Profes...Training:_The_Hundred_Cell_Calculation_Method

 

[Tyrion101]

Thank you to those that tried to help, when I am fully caught up with all of my classwork I will have to try out the cell method thing. Until then, I've decided, calculator, and being very careful to check the numbers I write down vs the numbers that the calculator, and the problem on the screen gave me.

 

[symbolipoint]

I do really well if I have one form of arithmetic like addition, or subtraction. But as we all know math doesn't work like that sometimes. I will do silly things like 10/2 = 8, or 10*5 = 3. If you can understand what I am saying. I mix up my tables when I'm dealing with a long problem, and I was wondering if there was any trick to avoiding something like this when you're doing a long complicated problem?

Memorize the basic multiplication facts up through 10 or 12. I too still forget a few of them but I just think carefully and fill in any simple forgotten fact since I know how multiplication works. 8 x 6 ? OH! I forget... 8 x 3 is 24, so 8 x 6 is just twice that, so just double 24 to get 48. 9 x 8 ? Ohhh! Forgot. Well, I know 9 x 4 is 36, and if needed, I can add 36 to 36 because this should be 9 x 4 twice, so it is 72.

 

[Mark44]

Thank you to those that tried to help, when I am fully caught up with all of my classwork I will have to try out the cell method thing. Until then, I've decided, calculator, and being very careful to check the numbers I write down vs the numbers that the calculator, and the problem on the screen gave me.

I would strongly advise memorizing the multiplication facts (as symbolipoint recommends, below) as soon as possible over relying on a calculator. It's very easy to enter a number incorrectly, which is a guarantee of a wrong answer. Also, not knowing how to do simple multiplication makes it impossible to get a rough estimate of what an answer should be.

Memorize the basic multiplication facts up through 10 or 12. I too still forget a few of them but I just think carefully and fill in any simple forgotten fact since I know how multiplication works. 8 x 6 ? OH! I forget... 8 x 3 is 24, so 8 x 6 is just twice that, so just double 24 to get 48. 9 x 8 ? Ohhh! Forgot. Well, I know 9 x 4 is 36, and if needed, I can add 36 to 36 because this should be 9 x 4 twice, so it is 72.

 

[lurflurf]

Whether students should have a good grasp on basic arithmetic depends on what is meant by good, grasp, basic, and arithmetic. It is clear is that doing hand calculations are an error prone waste of time. Anyone who thinks otherwise should do a thousand divisions with ten digit numbers by hand then and score their time and accuracy. It is also amusing that hand calculation fans always assume that hand calculations never error, when they error often. They are also oddly particular in there examples. Things like 45*19 and such. Where are hand calculation fans when serious calculating is to be done? Some day some one will factor RSA-1024 = 135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563

I believe that person(s) will be machine assisted, but go ahead hand calculation fans, prove me wrong.

 

[Mark44]

Whether students should have a good grasp on basic arithmetic depends on what is meant by good, grasp, basic, and arithmetic.

There's no need to parse the words individually. "Grasp" is pretty much self-explanatory. "Basic arithmetic" is commonly understood to consist of addition, subtraction, multiplication, and division. By "good grasp of basic arithmetic" what I had in mind is being able to add any two single-digit numbers and to know the times table up to 10 * 10 or 12 * 12.

It is clear is that doing hand calculations are an error prone waste of time.

They might be error prone for some, but I disagree that they are a waste of time. People who are unable to do simple arithmetic as described above, are likewise incapable of recognizing when they have entered a number into a calculator incorrectly, by being unable to do a quick order-of-magnitude check on their work.

Anyone who thinks otherwise should do a thousand divisions with ten digit numbers by hand then and score their time and accuracy.

This is well beyond what anyone would consider basic arithmetic.

It is also amusing that hand calculation fans always assume that hand calculations never error, when they error often.

Who assumes this? People make mistakes in everything they do. This is the reason for double-checking (or even triple-checking) your work.

They are also oddly particular in there examples. Things like 45*19 and such. Where are hand calculation fans when serious calculating is to be done? Some day some one will factor RSA-1024 = 135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563

And back in 1991 when the Pentium processor came out, how was it that someone discovered that some division operations were incorrect in the 5th and beyond decimal places? If your device is giving incorrect results, how else to tell this if you don't know how to do the computation by hand?

I believe that person(s) will be machine assisted, but go ahead hand calculation fans, prove me wrong.

And if they are totally dependent on that machine, and it breaks, the battery goes dead, or they lose it, what then? That person will be unable to do anything. A person who knows how to add, subtract, multiply, and divide, is still able to perform a wide range of computations.

 

[SteamKing]

It's a sad fact of modern mathematical pedagogy that insufficient attention is paid to ensuring that students learn and become familiar with basic arithmetic operations and numerical facts like addition and multiplication. I think this is why the OP seems confused that there are two arithmetics: one for addition, and another for multiplication. In my schooling, back when giants roamed the earth, we were introduced to the multiplication table by the process of counting by twos, threes, etc. After this was drilled into us sufficiently, it was only natural to proceed to constructing and understanding the multiplication table.

Sadly, in only a couple of generations, you now see heated and bitter debates within the educational community about the need for students to practice in becoming competent at arithmetic and whether the basic algorithms (long division, manipulation of fractions, etc.) should be taught at all. Just throw the kids a calculator and don't bother the teacher with a lot of questions about math.

 

[lurflurf]

There's no need to parse the words individually. "Grasp" is pretty much self-explanatory. "Basic arithmetic" is commonly understood to consist of addition, subtraction, multiplication, and division. By "good grasp of basic arithmetic" what I had in mind is being able to add any two single-digit numbers and to know the times table up to 10 * 10 or 12 * 12.
They might be error prone for some, but I disagree that they are a waste of time. People who are unable to do simple arithmetic as described above, are likewise incapable of recognizing when they have entered a number into a calculator incorrectly, by being unable to do a quick order-of-magnitude check on their work.

I do not think your "good grasp of basic arithmetic" is commonly understood. One digit multiplies and two digit adds should be within most peoples ability, but that's a pretty limited toolbox. If it is fine to reach for a calculator for 45*19 I don't see why it is so bad to reach for one for 9*7, or to think 9*7=(8+1)(8-1)=8^2-1^2=64-1=63. It might take a few tenths of a second longer, but it demonstrates understanding that rote memorization does not. And again order-of-magnitude and other checks can be used with a calculator, in fact it is easier to check for errors. For example a pentium chip user could catch the error in the well know example 4195835 / 3145727 by computing (4195835 / 3145727) * 3145727 - 4195835.

This is well beyond what anyone would consider basic arithmetic.
Who assumes this? People make mistakes in everything they do. This is the reason for double-checking (or even triple-checking) your work.
And back in 1991 when the Pentium processor came out, how was it that someone discovered that some division operations were incorrect in the 5th and beyond decimal places? If your device is giving incorrect results, how else to tell this if you don't know how to do the computation by hand?

If a hand calculation is impossible it cannot be used to compute or check a result. Machine calculations are used to check machines. A person equipped with a twenty year old computer and ignorant of its well known flaws is still better off than William Shanks. You have admitted 10 digit division is impractical for hand calculation, so even with some lost accuracy we are better of with a pentium, even more so if we know about the error and avoid it.

And if they are totally dependent on that machine, and it breaks, the battery goes dead, or they lose it, what then? That person will be unable to do anything. A person who knows how to add, subtract, multiply, and divide, is still able to perform a wide range of computations.

Pilots, doctors, carpenters, and others are totally dependent on having appropriate tools. They are wise to maintain them and have spares available. Most important is the fact that their tools allow them to do things they could never do without them. Some amount of practice for equipment failure might be wise, but only a small amount.

 

[Mark44]

I do not think your "good grasp of basic arithmetic" is commonly understood. One digit multiplies and two digit adds should be within most peoples ability, but that's a pretty limited toolbox.

They should be within the abilities of most, but in the case of the OP, I don't think they are. As SteamKing reported, there is a big controversy in the math education establishment about whether these simple concepts of basic arithmetic are important. One side in this controversy uses bumper sticker phrases such as "drill and kill" to downplay the importance of having a basic set of arithmetic skills on hand (i.e., memorized).

It seems to me that only in education do we find this silly argument. If you look at other endeavors, such as sports or music, it is well understood that you have to put in many hours of practice before you can get to a level of competence, let alone a mastery of the sport or instrument. The hours spent at practice are analogous to the hours spent mastering arithmetic facts.

If it is fine to reach for a calculator for 45*19 I don't see why it is so bad to reach for one for 9*7, or to think 9*7=(8+1)(8-1)=8^2-1^2=64-1=63.

I don't have any problem with someone using a calculator to multiply 45 and 19, provided that this person would be able to do this problem using only paper and pencil. As far as writing 9*7 as (8+1)*(8-1), that's a creative way to do things, but a person who doesn't know the multiplication table would likely be as stumped by 8*8 as by 9*7, so I'm not sure that's a valid argument.

It might take a few tenths of a second longer, but it demonstrates understanding that rote memorization does not. And again order-of-magnitude and other checks can be used with a calculator, in fact it is easier to check for errors.

Agreed, it is much easier, but that's not my point. As I already mentioned in an earlier post, a person who is completely dependent on a calculator for all arithmetic problems will be completely unable to function if the calculator is lost, stolen, broken, or otherwise unavailable. A person with some basic skills in arithmetic will take more time, but has a chance of completing the problem.

For example a pentium chip user could catch the error in the well know example 4195835 / 3145727 by computing (4195835 / 3145727) * 3145727 - 4195835.


If a hand calculation is impossible it cannot be used to compute or check a result. Machine calculations are used to check machines.

Who checks the machines that are checking the machines? The quote attributed to the poet Juvenal comes to mind: "Quis custodiet ipsos custodes?" Who watches the watchmen? What if your calculator happens to be using the same flawed chip?

As far as the well-known example you cited (of which I am very well aware, having written an article that was published in a popular computer magazine at the time), any person who is competent at long division could carry out that division. It would take some time, but it's doable.

A person equipped with a twenty year old computer and ignorant of its well known flaws is still better off than William Shanks. You have admitted 10 digit division is impractical for hand calculation,

But I didn't say it was impossible.

so even with some lost accuracy we are better of with a pentium, even more so if we know about the error and avoid it.

But at the time the error was discovered, most people didn't know about the problem, and therefore couldn't avoid it. By the time the problem was discovered, Intel had shipped a large number of Pentium processors. The recall cost Intel somewhere in the neighborhood of $1,000,000,000.

Pilots, doctors, carpenters, and others are totally dependent on having appropriate tools. They are wise to maintain them and have spares available. Most important is the fact that their tools allow them to do things they could never do without them. Some amount of practice for equipment failure might be wise, but only a small amount.

We're not talking about pilots, doctors, or carpenters here. It's safe to assume that pilots and doctors have many years of academic training, from which we can be reasonably sure that they are compentent at arithmetic. We can also assume that carpenters and others in the trades have received training in their areas. Again, I'm not concerned about these people - I'm concerned about kids in the primary grades who manage to get through the first six years of school without becoming competent at ordinary arithmetic. In my view, this is an indictment on our (US) education system. The so-called "educators" who push the nonsense that knowledge of arithmetic is unimportant are deeply misguided, IMO.

 

[lurflurf]

^The so-called "educators" who push the nonsense that knowledge of arithmetic is so important that it should be practiced to the exclusion of other things are deeply misguided, IMO. The main thing a cost benefit analysis. Being rather moderate on the issue I believe that most children should practice arithmetic for a few hundred hours between the ages of seven and eleven or so. This is more than enough for practical purposes and more time spent would be wasted. If this practice is missed for some reason, it is may be done later. In this time a few will (for reasons such as disability) not have much skill, but should move on anyway. Many like myself who do not "know the times tables" do not need to. It is quite rare that I do a thousand one digit multiplies in a day, so bringing my time down from say five minutes to two is a waste of time. I am not "stumped" by any multiplication facts I do not know, as I can figure them out as needed. In fact if I did "know" them my speed would not improve much as I would still check mentally for correctness. Those that "know them" do things like 5*7=55 all the time. Even so most errors are in multiplies with more digits anyway.

I would say practicing hand calculations is more like practicing air guitar than guitar. It is pointless even if you become quite good. When checking calculations, they should be checked in a different way. This is true of hand and machine calculation. My example of (4195835 / 3145727) * 3145727 - 4195835 reveals the pentium chip error, a different chip is not needed. I computed 4195835 / 3145727 by hand today to ten digits, it took me half an hour and my answer was off by more than the pentium. My error checking found and corrected the error in ten minutes, but still the pentium is better, as flawed and old as it is. Hand calculation fans love to add and multiply (maybe because they are easy). They do not seem to like solving equations or computing special functions. If it is alright to do those things with a machine why not add and multiply too?

 

[lurflurf]

I just remembered Asimov on Numbers by Isaac Asimov has a humorous take on useless arithmetic, in particular you may be interested in the now public domain A new and complete system of arithmetic, composed for the use of the citizens of the United States by Nicolas Pike which is chock full of useless arithmetic. I just think given limited time it is better to learn a little arithmetic and many other things rather than lots of arithmetic.
http://archive.org/details/newcompletesyste00pikerich

 

[Tobias Funke]

The difference between 45*19 and 9*7 is that the latter is made up of single digit factors. It's not an arbitrary distinction between "big" and "small" numbers; it's the crux of the matter. Memorizing the single digit multiplication table let's you theoretically do any multiplication without any further memorization. That's the whole beauty of the multiplication algorithm, and it's a shame that "knowing arithmetic" is interpreted as simply memorizing tables and not understanding the idea behind the algorithm, or that arithmetic itself is not though of as "real math". What a perfect, age-appropriate example of mathematical thinking we have in the arithmetic algorithms (whatever variant you choose) that just goes to waste.

By the way, I don't have every single entry in the table memorized, especially the sevens for some reason. But I can do something like symbolipoint described and still have my times tables written almost as fast as my hand will go. It doesn't have to be instant recall, but not being able to do 12*4 within 5 seconds without a calculator is pretty bad. Despite mathematicians wildly exaggerating by claiming that they can't multiply or figure out a tip, I think they're almost all pretty good at arithmetic and severely underestimate just how bad a lot of other people are, especially when it comes to fractions.

Also, your choice of nine times seven is quite a coincidence, since Asimov also wrote "The Feeling of Power", about a future where nobody can do any arithmetic without the help of machines. The rediscovery of this ability isn't exactly beneficial, though.

 

[Mark44]

^The so-called "educators" who push the nonsense that knowledge of arithmetic is so important that it should be practiced to the exclusion of other things are deeply misguided, IMO.

Of course you are entitled to your opinion. Whether mastery of arithmetic is less important that "other things" depends on what those other things are.

The main thing a cost benefit analysis. Being rather moderate on the issue I believe that most children should practice arithmetic for a few hundred hours between the ages of seven and eleven or so. This is more than enough for practical purposes and more time spent would be wasted. If this practice is missed for some reason, it is may be done later. In this time a few will (for reasons such as disability) not have much skill, but should move on anyway. Many like myself who do not "know the times tables" do not need to. It is quite rare that I do a thousand one digit multiplies in a day, so bringing my time down from say five minutes to two is a waste of time.

This is yet again another straw man argument. I never once advocated for doing a thousand single-digit multiplies.

I am not "stumped" by any multiplication facts I do not know, as I can figure them out as needed. In fact if I did "know" them my speed would not improve much as I would still check mentally for correctness. Those that "know them" do things like 5*7=55 all the time.

Kind of belies their claim to knowing them, doesn't it?

Even so most errors are in multiplies with more digits anyway.

I would say practicing hand calculations is more like practicing air guitar than guitar. It is pointless even if you become quite good. When checking calculations, they should be checked in a different way. This is true of hand and machine calculation. My example of (4195835 / 3145727) * 3145727 - 4195835 reveals the pentium chip error, a different chip is not needed. I computed 4195835 / 3145727 by hand today to ten digits, it took me half an hour and my answer was off by more than the pentium.

Not surprising in light of your admission that you don't know the times table.

My error checking found and corrected the error in ten minutes, but still the pentium is better, as flawed and old as it is.

As I mentioned before, Intel issued a recall of those chips, so it is very unlikely that there are very many of the flawed chips still out there.

Hand calculation fans love to add and multiply (maybe because they are easy). They do not seem to like solving equations or computing special functions.

?
Based on what evidence? In any case, the point of this thread was a minimum level of competence at arithmetic, which is limited to the operations of addition, subtraction, multiplication, and division of numbers. In my 21 years of teaching experience, people who had trouble with arithmetic had even more difficulties in algebra and trig, not to mention other areas of mathematics.

If it is alright to do those things with a machine why not add and multiply too?

 

[lurflurf]

^It is not opinion, hand arithmetic is less important than algebra, calculus, geometry, logic, and statistics. People use those things for many productive purposes. There is a use for an expert in nonassociative ring theory say, there are no opportunities for an expert in hand arithmetic, I cannot name even one. If you do not advocate doing thousands of single-digit multiplies, it weakens your case. Repetition is needed to develop the skill you value, and the skill is useless if it not used. If one does not multiply much, a second is fast enough. Why waste time getting faster? You do not need to "know" the times tables. Your "drill and kill" movement devalues understanding, which I object to philosophically, but as a practical matter understanding is a safety net, rote learning leads to strange mistakes.

Don't back off your pentium chip now, it was your best point despite the fact that the mistake is easily avoided by either using appropriate error checking or using a chip less than twenty years old. The reason it was even a story is it was such a surprising error, hand calculation errors do not make the newspaper. As to my own error it had nothing to do with times tables, at the 10501080-9437181=1063899 step I got 1073619. You did not even credit me for catching it, too bad I was over confident, if I checked as I went instead of at the end I would not have propagated the error. You have still not told me how to avoid being a modern day William Shanks should I take up hand calculation.

The evidence is that all the hand calculation fan rants are about long division and addition and such. Please link to some rants that encourage more complex hand calculation if you know any. I see no reason that dividing by hand is noble and taking square roots is not. Just use a calculator for both. You have not commented on my above linked A new and complete system of arithmetic, composed for the use of the citizens of the United States by Nicolas Pike. Back then basic arithmetic was useful (and more inclusive). Your experiences may have biased you; I have known many people including scientists, tenured mathematics professors, and disabled people; who who had trouble with arithmetic and few difficulties in algebra and trig, not to mention other areas of mathematics. Much older research on the topic had flawed methodologies. Subjects with arithmetic difficulties were either not instructed or not tested in other areas of mathematics.

The OP cites 10/2 = 8, or 10*5 = 3 as typical mistakes. I do not know what causes those mistakes, but they could well be logic errors rather than arithmetic.

 

[Mark44]

^It is not opinion, hand arithmetic is less important than algebra, calculus, geometry, logic, and statistics.

I never claimed that arithmetic was more important than algebra, etc. You said it was less important than "other things", and I said it depended on what those other things are.

People use those things for many productive purposes. There is a use for an expert in nonassociative ring theory say, there are no opportunities for an expert in hand arithmetic, I cannot name even one. If you do not advocate doing thousands of single-digit multiplies, it weakens your case. Repetition is needed to develop the skill you value, and the skill is useless if it not used.

You seem to throw up a great many "straw man" arguments, with objections to points I didn't make. I did not advocate doing thousands of single-digit multiplications. What I am saying, since you seem to have difficulty understanding my point, is that it is important, in my opinion and that of many others, for students in the primary grades to know how to do arithmetic. This entails, in part, being able to add, subtract, multiply, and divide numbers of a reasonable size - without the use of calculator or other computing device. I didn't define "reasonable size" but it doesn't include 1024-bit numbers. It probably would include numbers with 10 or fewer digits.

If one does not multiply much, a second is fast enough. Why waste time getting faster? You do not need to "know" the times tables. Your "drill and kill" movement devalues understanding, which I object to philosophically, but as a practical matter understanding is a safety net, rote learning leads to strange mistakes.

Another straw man. I am not advocating for the memorization of, say, the arithmetic facts and the times table at the expense of understanding. What I'm saying is that mastery of the basic operations of arithmetic is the foundation on which much of the more advanced areas of mathematics depends. Just as when a house is built, if the foundation is weak, the house won't last as long.

As I mentioned earlier, I was a teacher for 21 years. Toward the end of that time I started hearing the same arguments you are making, including a movement in Portland, OR, to issue calculators to kindergarteners. It amazes me that some of these "educators" are not able to draw parallels in other life endeavors such as sports and music, to name just a couple. To excel in these areas requires a lot of practice of basic operations, even for those who have a natural talent for these pursuits. Someone who has to reason through how to catch or hit a ball (in baseball), or grab a B⁷ chord (guitar) is not likely to become accomplished in that endeavor. These skills need to be ingrained in "muscle memory" just like 6*8 needs to be in memory, so that the brain can take on more complicated tasks.

Don't back off your pentium chip now, it was your best point despite the fact that the mistake is easily avoided by either using appropriate error checking or using a chip less than twenty years old.

I'm not backing off what I said about the Pentium chip, and don't know why you think I was.

As far as the problem being easily avoided, I don't think so. The Pentium chips that had the problem didn't consistently give incorrect answers. For the vast majority of possible divisions, they produced the correct answer. If a device gives consistently wrong answers, it doesn't take long for someone to realize that they are erroneous. However, if the incorrect answers come infrequently, it's much more difficult to notice them. Does your appropriate error checking include checking every single floating point arithmetic operation performed by the chip? That's costly in terms of performance.

The reason it was even a story is it was such a surprising error, hand calculation errors do not make the newspaper.

As to my own error it had nothing to do with times tables, at the 10501080-9437181=1063899 step I got 1073619. You did not even credit me for catching it, too bad I was over confident, if I checked as I went instead of at the end I would not have propagated the error. You have still not told me how to avoid being a modern day William Shanks should I take up hand calculation.

The evidence is that all the hand calculation fan rants are about long division and addition and such.

What's your point?

Please link to some rants that encourage more complex hand calculation if you know any. I see no reason that dividing by hand is noble and taking square roots is not.

I have no problem with people using a calculator to perform arithmetic operations, provided that they are able to do them by hand, for those times when a calculator is not available. On the other hand, all of those students who are the beneficiaries of the "enlightened" pedagogy that you favor, and who can't do arithmetic, will be dead in the water.

As far as taking square roots, many of us learned how to do this with paper and pencil, and I still remember it, even though it's been a good long while. If it comes down to it, I can calculate, using only paper and pencil, the square root of a number to any desired precision, something the vast majority of calculators can't do. Of course, if I need to do a square root, I reach for a calculator.

Just use a calculator for both. You have not commented on my above linked A new and complete system of arithmetic, composed for the use of the citizens of the United States by Nicolas Pike. Back then basic arithmetic was useful (and more inclusive).

It's still useful for lots of people, such as carpenters, machinists, surveyors, and many others.

Your experiences may have biased you; I have known many people including scientists, tenured mathematics professors, and disabled people; who who had trouble with arithmetic and few difficulties in algebra and trig, not to mention other areas of mathematics.

So what? The fact that these people had trouble with arithmetic, algebra, trig, and so on does not seem to me to be a good argument for dispensing with arithmetic in the lower grades.

Much older research on the topic had flawed methodologies.

Examples?

Subjects with arithmetic difficulties were either not instructed or not tested in other areas of mathematics.

The OP cites 10/2 = 8, or 10*5 = 3 as typical mistakes. I do not know what causes those mistakes, but they could well be logic errors rather than arithmetic.

Which makes them no less errors. When I saw them, I inquired into his arithmetic capabilities, thinking he might be one of those unfortunates who made it all the way through the US education system without being able to instantly recognize that 10/2 is NOT 8 or that 10*5 is NOT 3.

 

[lurflurf]

I may misunderstand your position, but if I do it is a sincere misunderstanding. I will clarify my own. Hand arithmetic is about the least useful skill in the world. I have tried to frame my opinion with factors of ten. A disagreement less than a factor of ten is not significant. Still my position is quite moderate. My experience and typical educational practices suggest reasonable guidelines. A student should practice arithmetic for somewhere between 25.7 and 257 hours in their formative years. A reasonable amount of practice for a day is between 100 and 1000 1 digit multiplies (some days other skills than multiplication would be practiced). Single digit multiplies (ie 7*8=56) are a fair metric, they are practiced alone at first then used to effect more digits (ie 97243*345=33548835 would require 15 single digit multiplies), fractions, algebra and so forth. I think 1000 1 digit multiplies (by hand) in a day is silly, but I have seen it assigned, still it is hard to object strongly as it takes less than ten minutes without "knowing your times tables". Improving on this seems not worthwhile. I thought speed was the point of all this "drill and kill" talk, but you make an interesting point that speed is not the goal but rather to make multiplication automatic so that the brain can take on more complicated tasks. I would be interested in any studies on this point. I think the analogy is flawed sports and music involve timing, speed, and automatic response in a way mathematics does not. A few second stumble can ruin a musical or sport performance, taking 74 seconds to solve a 70 second math problem is inconsequential. Also mathematics is a thinking pursuit, thinking for a few seconds even about a trivial detail is very in keeping with the spirit. Arithmetic is not of key importance in solving problems.

I think we can agree students should practice arithmetic a bit and if they meet your standards whatever they are all the better. If a student practices arithmetic a bit and was still bad I would give them a calculator and tell them to move on to more important subjects without giving it a second thought. As I understand it you would tell them to keep working on it and wait until they get it to move on. That seems crazy to me. You claim such practice is not at the expense of understanding, but time spent is rote practice is not spent gaining understanding. I cannot see how the two are not in opposition. In the past many results could only be reached by a select few who calculated fast, accurately, and for a long time. Machines allow the average person to reach those heights as well as higher heights that no human could reach by hand calculations. Arithmetic facts are not the foundation of mathematics, they are trivial. No deep results follow from 4*7=28.

 

[lurflurf]

Pentium chip proper practice (after first considering if division could be avoided all together or if ignoring the error was safe) would be to catch divides with errors and shift them into an error free zone. Sure there is a performance penalty, but sometimes using a known defective twenty year old chip has consequences. In general calculations should be performed at least twice as independently as possible (preferably on different computer lines). Sometimes results can checked cheaply. Machine arithmetic is a skill worth learning unlike hand arithmetic. In any case hand calculations need to be checked as well and machine calculations are very cheap. In terms of performance a defective Pentium beats any human. You have no problem with people using a calculator to perform arithmetic operations, provided that they are able to do them by hand? Why would one learn a skill to not use it? There is a cost to learning a skill. You buy a book, take a class, spend time, give up learning some other skill instead. Calculating square roots by hand is not a very practical skill. "Those times when a calculator is not available" is a straw man. Get a spare, get a spare for the spare, carry extra batteries, work in an office with extra computers, have a backup generator, and so on. Very few calculations can be done by hand in reasonable time. Even an expert in hand calculation will be dead in the water when serious work is to be done.

As far a examples of flawed methodologies this article sumarized well the problem "Is it possible that their main difficulties are restricted to numeracy and in mental computation and do not involve the entire domain of mathematics? Little can be said about mathematical skills that require few computations, but more logic because almost nobody tries to teach these kinds of skills to students with Down syndrome. On the contrary, teachers insist on building up "the basics", which fall down as soon as the pupil gives an answer that makes no sense and then the teachers start all over again." The obvious thing to do when a student has problems in arithmetic is to teach them other things, yet this is not done.

Also A new and complete system of arithmetic, composed for the use of the citizens of the United States by Nicolas Pike and William Shanks.

 

[pwsnafu]

Question to lurflurf: IYO should we teach integration techniques despite the fact that a computer is superior to a student?

 

[lurflurf]

I would say integration techniques should be taught in that each illustrates an important idea and they facilitate easy examples like sin(3x). I think practicing them much (especially complicated examples) is a waste of time. For example calculating

$$\int \! (\tan (x))^{(37/59)} \, \mathop{dx}$$

Is not very instructive.

 

[Mark44]

Pentium chip proper practice (after first considering if division could be avoided all together or if ignoring the error was safe) would be to catch divides with errors and shift them into an error free zone. Sure there is a performance penalty, but sometimes using a known defective twenty year old chip has consequences.

Why would anyone use a 20-year old CPU that doesn't work right?

In general calculations should be performed at least twice as independently as possible (preferably on different computer lines). Sometimes results can checked cheaply.

You have completely misunderstood my point. The flaw in the first Pentium processors was discovered in 1994 or thereabouts, and was widely publicized at the time. The manufacturer, Intel, spent ~$1,000,000,000 recalling these chips. I would be very surprised to hear that anyone is still using one of these problematic chips, so advice on how to work around the problem is not useful, IMO.

If we bring things up to the current time, and find that we have a CPU that is unable to perform some kinds of arithmetic, your advice still doesn't seem very good. If an application user requires some division to be performed, how can it be avoided? Certainly a/b is the same as a * (1/b), but we're still going to need to divide 1 by b.

Since the error in the old Pentium chips produced some results that were incorrect in the 5th decimal place and beyond, if your precision requirements were loose enough, then yes, you could ignore the error.

As far as shifting errors to an "error-free zone" I have no idea what you mean by this.

Machine arithmetic is a skill worth learning unlike hand arithmetic. In any case hand calculations need to be checked as well and machine calculations are very cheap. In terms of performance a defective Pentium beats any human.

This is silly. Your assertion that 1,000,000 incorrect answers is somehow better than 1 correct answer obtained in that time is nonsensical. You don't seriously believe this, do you?

You have no problem with people using a calculator to perform arithmetic operations, provided that they are able to do them by hand?

Correct.

Why would one learn a skill to not use it?

The point is that they would use it. For example, when I put gas in my motorcycle this morning, I did a mental division of the miles I had gone since I filled up the last time, and the number of gallons I put in. For those people who are so ill-educated that they don't know single-digit products, this would be an impossibility. In fact, most order-of-magnitude estimates would be over their heads.

There is a cost to learning a skill. You buy a book, take a class, spend time, give up learning some other skill instead. Calculating square roots by hand is not a very practical skill.

It isn't now, but it used to be when I was taught it.

"Those times when a calculator is not available" is a straw man. Get a spare, get a spare for the spare, carry extra batteries, work in an office with extra computers, have a backup generator, and so on. Very few calculations can be done by hand in reasonable time. Even an expert in hand calculation will be dead in the water when serious work is to be done.

Speaking for myself, there are many times when I don't have a calculator, so that is NOT a straw man argument. (I suspect that you don't understand what this terms means.) Despite our best intentions, there are times for most people when there is no device available or charged or working.

As far a examples of flawed methodologies this article sumarized well the problem "Is it possible that their main difficulties are restricted to numeracy and in mental computation and do not involve the entire domain of mathematics? Little can be said about mathematical skills that require few computations, but more logic because almost nobody tries to teach these kinds of skills to students with Down syndrome. On the contrary, teachers insist on building up "the basics", which fall down as soon as the pupil gives an answer that makes no sense and then the teachers start all over again." The obvious thing to do when a student has problems in arithmetic is to teach them other things, yet this is not done.

The quote above is talking about teaching people with Down syndrome, so how is it germane to the discussion? I am talking about standards that would apply to most students, not a small minority of folks with lower mental capacity.

Also A new and complete system of arithmetic, composed for the use of the citizens of the United States by Nicolas Pike and William Shanks.

You seem to find this funny, as I think you have mentioned it three times. The actual arithmetic in it seems OK to me. The parts that are less useful, IMO, are the parts that involve conversion from pints to pecks and the like.

 

[Mark44]

I think you have a misplaced trust in the ability of computing devices to do arithmetic. Here's an example in C.

float sum = 0.0;
for (int i = 0; i < 10; i++)
{
   sum = sum + 0.1;
}
if (sum == 1.0) printf("All is good");
else printf("What happened?");

The code above adds 1/10 ten times, so one would think that you would end up with 1.0 in sum. That is not the case, due to the fact that real numbers are not represented exactly.

Here's another example that is fairly current, dating back to 2010 - http://productforums.google.com/for...bsearch/unexpected-search-results/BqWHFKg6dQc. The thread in this link reports that an online calculator from Google reports the wrong Celsius value for 1 deg. Kelvin.

Here's a page that details potential problems with Excel calculations (and with most calculators, as well) http://support.microsoft.com/kb/78113

Add 0.000123456789012345 and 1. What does your calculator display? Someone who can do simple arithmetic can get this with no problems, but calculating devices not so much.

Another example from Excel:
(43.1-43.2)+1
Output: 0.899999999999999
Correct value: 0.9

Here's a problem reported by someone using a Google Android device http://code.google.com/p/android/issues/detail?id=26026
Calculation: 9 - 8.89
Result: 0.109999999999
Correct value: 0.11

Here's an error reported in AutoCAD in 2009 - http://www.cad-notes.com/2009/11/autocad-performs-incorrect-calculation/
Calculation: 750 - 693
Result: 63
Correct value: 57
This error is stunning in how far off it is, considering that it's doing integer subtraction. A competent fourth grader could get this one right.

Another Excel problem, reported in 2012 - http://answers.microsoft.com/en-us/...yed-when/2fbe7cbe-eda0-42d8-9d44-cf3b568fb887
Calculation: 111,111,111 by 111,111,111
Result: 12345678987654300
Correct value: 12345678987654321

A competent 6th grader could probably get the correct answer to this one.
I could go on...

 

[SteamKing]

I just remembered Asimov on Numbers by Isaac Asimov has a humorous take on useless arithmetic, in particular you may be interested in the now public domain A new and complete system of arithmetic, composed for the use of the citizens of the United States by Nicolas Pike which is chock full of useless arithmetic. I just think given limited time it is better to learn a little arithmetic and many other things rather than lots of arithmetic.
http://archive.org/details/newcompletesyste00pikerich

Lurflurf:
I can see you are quite the sophist. The tome you cite was written in the 18th century when all math had to be done by hand, whether one was a scientist, a merchant, or a tradesman. The book was chock full of hand calculations because there was no other practical way to obtain a result. Even if one had the knowledge to use logs for advanced arithmetic, then a certain amount of training would have been required.

Even if one accepts your thesis that mistakes like 7*5 = 55 occur frequently, all that means is that the one doing the calculation was either distracted or not very good at arithmetic. and really, by avoiding the memorization of a multiplication table, how much time did you really save? What other fields of knowledge did you explore? What sensual delights did you experience which would have been out of your grasp but for skipping out on your math homework?

 

[lurflurf]

^The Autocad one is interesting because there is not explanation provided. The others are well known issues with number base conversion, rounding, and limited precision. That is why it is important to know what output to expect. At least machine calculations are more consistent than hand calculations.

^(longer one)
Interestingly in computer arithmetic identities like a* (1/b)=a/b,a+(b+c)=(a+b)+c,and b -(1/a)b*a do not hold.

Shifting errors to an "error-free zone" mean that computers like humans use stored numbers to effect calculations. The Pentium error was like a misprint on a page of a table.
(4195835*15/16)/(3145727*15/16)
can be substituted for 4195835/3145727
and the chip will read a not defective area of the table.

1,000,000 incorrect answers can be better better than 1 correct answer in several situations.
-The incorrect answers are sufficiently accurate for the task.
-The incorrect answers can be used in additional calculation to produce useful results.
-The incorrect answers reveal something.
-Most interestingly an exciting concept has been proposed that it might be possible to build a computer very fast and very inaccurate. Its output would be corrected by some process achieving a net gain in speed.

Your oops my dog ate my calculator stories are not convincing, I feel a calculator or computer is essential to calculating. If you do not have one your capabilities are greatly reduced. Sure sometimes a hand calculation would be handy, but those times are few. What I want to know is the times you go for a motorcycle ride and forget your motorcycle, what do you do?

I think there is a group of people with arithmetic problems that includes some people with Down syndrome and many others as well. These people have low skill in arithmetic, additional practice gives little improvement, their skill in other areas (including other mathematical areas) are much better. It is a disservice to these people to inflict more arithmetic, negative opinions, and loss of exposure to other areas of mathematics. A calculator and some modified lessons are a better response. People with Down syndrome are often identified and there is some idea of the modifications they require. Others with arithmetic problems have no idea what causes there problems or what should be done. Anyone with arithmetic problems should be offered modifications, the chance to explore other areas of mathematics, and freedom from prejudice.

 

[lurflurf]

^SteamKing
Many skills that were valuable in the past are no longer. I think a lot of arithmetic like square roots, long division and compound addition are no longer useful. I think there is a donut hole in arithmetic ability, that is upon reaching a particular proficiency further improvement requires not only much time and effort, but a risk of ending up worse off. I do not know that I saved time not memorizing a multiplication table, I wasted much time trying to. At some point I felt that further efforts were not worthwhile. Memorization of a multiplication table increases ones speed, but if one ever forgets an entry he slows down or makes an error. A great many things can be learned without being good at arithmetic. By doing calculations on a machine one can calculate more in few minutes that can be done by hand in a lifetime and gain the insights previously unavailable. Also it is more fun, cheaper, less boring, faster and less painful and frustrating. Machine calculations are just better than hand calculations.

 

[Tobias Funke]

The Asimov essay you refer to, Forget It!, says nothing about the kind of arithmetic we're talking about. It's much more about useless units of measurement than arithmetic anyway, and he even gives the example of the "simple addition" 15+17+19 not to say that it's useless, but to compare it to the examples from Pike that he does consider useless. It seems to me that he takes it for granted that his readers should be able to do this addition by hand.

 

[lurflurf]

^I take the main idea of the essay to be that instead of teaching each generation the same mathematics we should continuously discard the topics we find to no longer be useful and replace them with more useful ones. That seems a reasonable Statement, but there could be disagreement as to which topics are most important and how often changes should be made. I would say working (add subtract multiply divide) with two digit numbers (but not from memory) is useful. Students who want to can learn to work with larger numbers up to say ten digits, but there is no reason to emphasize such uselessness or reward or punish people based on them. Even more so powers, roots, special function evaluations by hand are even more silly. Most people would not calculate
exp((sin(e))^(1/7)) = 2.4124584105724454
by hand
4195835/3145727
is no more enlightening.
Though you make a good point that a useless thing done in a stupid way is even more useless. The essay mentions using multiple units at the same time like saying "Yesterday I was 112 miles 50 fathoms 86 furlongs 50000 yard and 9024 feet from London. How ever you feel about those units using them simultaneously require compound addition and is silly. Arithmetic is very specific if one must know long division which one? Hand arithmetic fans usually like confusing methods with a lot of "guess and check". They hate it when others use a different method especially if it is less confusing as that would be "new fangled". They also like to say that they don't want the calculator to be doing things they don't understand. This is silly as calculators and computers use different methods than people do. Methods that are much better.

 

[Tobias Funke]

True, the overarching theme was discarding useless knowledge, but his examples were all about units and the following essays are about the superiority of the metric system, so I don't know if he would discard hand arithmetic. From what I've read of his robot series, he doesn't seem to think it's a good idea to rely too much on machines, but I don't want to get further off topic by trying to predict whether Asimov was a fan of arithmetic or not, or would be today.

I'll give you one thing: there are a lot of people who think doing arithmetic by hand without actually understanding the process behind it should be the totality of elementary school math education. They probably don't understand any math themselves and don't want this exposed, so they stick to the purely mechanical processes they do know and avoid any unfamiliar methods because they can't figure them out, despite being only superficially different. I can't speak for anyone else here, but I don't think anyone's pushing that idea. Despite what people who weren't alive in the 1950s or before might say, that was tried already and it never resulted in some golden age where everyone could do math.

I also don't think we all disagree as much as it first seems. Does anyone really think shaving a hand calculation time from 74 to 70 seconds is particularly important (that's why Mark44 was mentioning straw men)? Conversely, are you really saying that it's ok for someone to not be able to do, say, 2308-1819 without a machine? I don't mean in 74 seconds instead of 70, I mean literally can't do it (at least not without drawing 2308 lines, crossing some out, and recounting) Surely there's a happy medium where students can do hand calculations, understand what they're doing, but be allowed to use calculators as aids. A few hundred hours should be enough to get reasonably fast--what's reasonable is up for debate--at single digit arithmetic and at least get a good feel for how the algorithms work.

So...how did this thread begin, and have we been sort of off-topic for like 90% of it?

 

[lurflurf]

Maybe I misunderstood the time thing, hand arithmetic fans mention time a lot like above about doing something in 5 seconds, talk of timed tests being important, and what not. I never understood why time was important. I have now been informed that it is more important to do arithmetic automatically without thinking. That does not seem to be a good reason either. I thought only those rote folks who were faster than me were better off, now I learn that so are those that are slower, how depressing.

Sure it should be easy to do 2308-1819 by hand. Say a child does this
let
3=1000
2=100
1=10
0=1
3322200000000-3222222221000000000
3-2222210
2222222222-2222210
22222-10
22221111111110000000000-10
222211111111000000000-
489
That look good to me, hand arithmetic fans are disappointed and would not want this child going to the next grade with the other boys and girls. Not to mention if they got it wrong. I think this method or a calculator are both fine they get one to the answer. Sure it is better to know more ways and standard ways, but one way is enough. Mark44;4399187 reminded us that "It's very easy to enter a number incorrectly, which is a guarantee of a wrong answer." That is supposed to be a fair criticism of calculator use, but "It's very easy to write a number incorrectly, which is a guarantee of a wrong answer." is equally true of hand calculation as is "It's very easy to make any number of mistakes, which is a guarantee of a wrong answer." Better advice would be "Try to avoid entering a number incorrectly."

The OP seems to have moved on, but I would like to see the work that leads to 10/2 = 8, or 10*5 = 3. Others disagree, but I find that understanding the cause of an error is helpful in correcting it.

 

[Mark44]

Maybe I misunderstood the time thing, hand arithmetic fans mention time a lot like above about doing something in 5 seconds, talk of timed tests being important, and what not. I never understood why time was important.

Time is important in the real world. There is even a saying, "Time is money."

One of the traditional goals of elementary education, in which arithmetic and other subjects are taught, is for students to become reasonably competent at doing problems in arithmetic. Competence is measured primarily by the accuracy of the computations, but there are also expectations about how many of these calculations can be done in some time interval.

In the workplace, if two candidates for a job can produce whatever output is required for the job with the same level of accuracy, but one candidate can do this work in a third of the time the other one can, the quicker one will probably get hired.

I have now been informed that it is more important to do arithmetic automatically without thinking.

It is more important to be able to do simple operations (e.g., single digit adds, subtracts, and multiplies) with little conscious thinking. Knowing the single-digit addition and multiplication facts by heart enables the brain to stay focused on the broader problem rather than the details. A book I found several years ago makes this case very convincingly, I believe - "The Schools We Need, and Why We Don't Have Them," by E. D. Hirsch (http://www.barnesandnoble.com/w/schools-we-need-e-d-hirsch/1100619825?ean=9780385495240).

Hirsch likens the mind to a computer, with part of the memory functioning as cache ("working set") and part function as ordinary RAM (long-term memory). The working set is useful for storing information with a short lifetime, while long-term memory is useful for storing the addition facts, for example. The working set can be used to analyze the problem, to determine which operations need to be performed, and the actual operations, such as 6 + 7 or 9 * 4 can be done by retrieving this information from long-term memory.

That does not seem to be a good reason either. I thought only those rote folks who were faster than me were better off, now I learn that so are those that are slower, how depressing.



Sure it should be easy to do 2308-1819 by hand. Say a child does this
let
3=1000
2=100
1=10
0=1

Each equation above is wrong. I understand the thinking, where the number on the left is the log of the number on the right, but anyone who uses "=" should understand that the quantities on either side must have the same value. 0 and 1 are obviously not equal.

3322200000000-3222222221000000000
3-2222210
2222222222-2222210
22222-10
22221111111110000000000-10
222211111111000000000-
489

That look good to me, hand arithmetic fans are disappointed and would not want this child going to the next grade with the other boys and girls.

Not necessarily. Speaking for myself, I admire the creativity of such a student, but would show him a way that is more efficient than this way, which seems to be a combination of ordinary subtraction and counting on your fingers.

Not to mention if they got it wrong. I think this method or a calculator are both fine they get one to the answer. Sure it is better to know more ways and standard ways, but one way is enough.

Not if that way is not acceptable. You do know that some college math courses don't allow calculators of any kind?

Mark44;4399187 reminded us that "It's very easy to enter a number incorrectly, which is a guarantee of a wrong answer." That is supposed to be a fair criticism of calculator use, but "It's very easy to write a number incorrectly, which is a guarantee of a wrong answer." is equally true of hand calculation

In the arena we're talking about, students are given a test paper with the problems already written on it, so they don't need to transcribe the number on the paper to another sheet of paper. If they are doing the problem using a calculator, they have to transcribe each number into the calculator, so your argument loses its validity.

as is "It's very easy to make any number of mistakes, which is a guarantee of a wrong answer." Better advice would be "Try to avoid entering a number incorrectly."

Well, duh! Some equally useful advice - "Try to avoid getting run over by a truck today."

People are human, and humans are prone to making mistakes.

The OP seems to have moved on, but I would like to see the work that leads to 10/2 = 8, or 10*5 = 3. Others disagree, but I find that understanding the cause of an error is helpful in correcting it.

 

[lurflurf]

^Sure it is better to be able to do arithmetic fast, more accurately, using more digits, and so on; but only if other things are equal. Things are not equal as to get better one needs to practice more and the practice has cost in terms of misery, a book, a teacher, money, time, and lost opportunities. I do not know anyone who is particularly good at arithmetic, who spends much time doing arithmetic, or who obtained any significant advantage in life due to arithmetic skill. It is simply not worth while.

It is not too bad to show students some hand arithmetic, I think time has already been wasted. Some of them have not achieved the desired proficiency, no problem let's show them again. Maybe some have gotten it now, but others still have not, well better keep showing them. At some point the slow progress should be a signal to stop, hand out calculators, and wish them well. I recently met some students in an extra arithmetic class. I wish now I would have asked there opinion on it. The idea was student determined to lack arithmetic skill would take the regular math class and the extra one as well. This seems misguided in a few ways. If the extra arithmetic is in fact helpful the students might benefit more if they complete it before the regular class, being required to take two math classes might make them dislike and resent math, it meets for 150 hours, and they miss out on another class.

Thanks for the book recommendation, I will check it our. I like reading education rants like

https://www.amazon.com/dp/067152934X/?tag=pfamazon01-20
https://www.amazon.com/dp/0312878672/?tag=pfamazon01-20
https://www.amazon.com/dp/0394719816/?tag=pfamazon01-20
https://www.amazon.com/dp/B0041OT8EK/?tag=pfamazon01-20

Here is a pro memorization blog post. Several of the points are interesting. "leaving room in their working memory" this seems to be a problem with problem solving, breaking a prob;em into step should prevent the steps from being bewildering. "recalculating the facts several times while solving the problem" sure like I said above if there are 100 1 digit multiplies, if we do 1000 we are doing them at least 10 times each, but we can write them down. This raises questions about the use of memorization in math. Much memorization is required, but each student should decide for herself what to remember and what to derive or look up. If ones has a view of the big picture it is much easier to remember facts. Ones memory tends to fail, so it is good if a moments thought allow one to reconstruct a few facts. Memory depends upon practice, one remembers more trigonometry if she uses it frequently, infrequently used facts are harder to remember, but also less important.

I see calculation in terms of experience and equipment. Let's consider each in powers of ten. For equipment let's consider the device cost, although there are more expenses like electricity, books, training, and so on.

$0 hand calculation
$1 Basic calculator (does square roots)
$10 Scientific calculator (does logarithms)
$100 Graphing calculator (graphs, is programmable)
$1000 Personal computer (very powerful, many software options)
$10^4 Workstation/Cluster (professional capabilities)
$10^5 Large cluster
$10^6 Small supercomputer
$10^7 Medium supercomputer
$10^8 Large supercomputer

Hand calculation is a pretty poor value proposition.