Problem switching back and forth between the diffrerent arithmetics

[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.