# The Changing Way That Math Is Taught To Children

### from the *carry-the-1* dept

NPR has a fascinating story about how the methods for teaching basic mathematics have been changing in schools. For example, they show the following comparison for teaching multiplication:

The other interesting thing that hit me was the article's explanation for why things have shifted:

This fascinates me because I was definitely taught that first method as a kid, but what really gets me is that I ended up teaching myself the second method, because it seemed like a fun trick that made it easier to multiply larger numbers in my head (shocking news: I was a bit of a nerd). But once I had taught myself the latter method, I could never figure out why that wasn't more common. Apparently, I was just ahead of my time.The Way We Used To Multiply

The old way to multiply required a student to add the products of 36 x 4 and 36 x 2. The trick is to add that 0 at the end of the second product.How Kids Learn To Multiply Now

These days, students add four products to get the answer.

The other interesting thing that hit me was the article's explanation for why things have shifted:

So for all the times kids claim that they shouldn't need to learn mathematics because they'll never need aspects of it in real life, it's nice to see that the education system is actually adapting to make the"That's largely to reflect the different needs of society," he says. "No one ever in their real life anymore needs to -- and in most cases never does -- do the calculations themselves."

Computers do arithmetic for us, Devlin says, but making computers do the things we want them to do requires algebraic thinking. For instance, take a computer spreadsheet. The computer does all the calculations for you automatically. But you have to write the macros that tell it what calculations to do -- and that is algebraic thinking.

"You cannot become good at algebra without a mastery of arithmetic," Devlin says, "but arithmetic itself is no longer the ultimate goal." Thus the emphasis in teaching mathematics today is on getting people to be sophisticated, algebraic thinkers.

*process*of how you think about math much more practical in today's world.
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(Flattened /

Threaded)ChurchHatesTucker(profile),Mar 11th, 2011 @ 6:28pm## Huh

This fascinates me because I was definitely taught that first method as a kid, but what really gets me is that I ended up teaching myself the second method...So did I, but I never really thought about it.

I wonder how common that is/isn't.

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Anonymous Coward,Mar 11th, 2011 @ 6:58pm## Re: Huh

http://www.vintagecomputing.com/wp-content/images/precomputer/precomputer_large.jpg

I wrote simple programs on it, stuff that would ask your age (and I mostly copied it from the book it came with at that), but there were games that ask you math questions and you would answer. While I long lost the ability to do math in my head quickly, I remember I reached a point where you don't even think about the answer, you just know it no matter what is thrown at you. It's like your brain just calculates it subconsciously, you can do complicated multiplication, addition, division, etc... without a second thought. But it takes practice to do and really there is no need for it in this day and age. Any math class that I take these days lets you use a graphing calculator, but it won't help you at all if you don't understand the material.

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ChurchHatesTucker(profile),Mar 11th, 2011 @ 7:05pm## Re: Re: Huh

I remember I reached a point where you don't even think about the answer, you just know it no matter what is thrown at you.Yeah, but is that rote memory? Like memorizing the times table?

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Anonymous Coward,Mar 11th, 2011 @ 7:23pm## Re: Re: Re: Huh

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Anonymous Coward,Feb 6th, 2013 @ 9:33pm## Re: Re: Re: Huh

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Almost Anonymous(profile),Mar 14th, 2011 @ 9:49am## Re: Huh

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N,Nov 29th, 2013 @ 7:35am## Re: Huh

Why change something we have been doing for generations? Imagine having to multiply 1,234 x 5,678; you're going to telle it's easier to do this the 'new' way? By the time a kid writes this out, they could be on the next problem, if done the old way. You all are crazy.

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Mar 11th, 2011 @ 6:47pm[ reply to this | link to this | view in chronology ]

Mike(profile),Mar 11th, 2011 @ 6:47pm## Seems harder

If you're multiplying a 3 digit number by a 4 digit number, you'll either end up adding 12 numbers or 3 numbers. Adding 3 numbers is easier to deal with.

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ethorad(profile),Mar 12th, 2011 @ 1:31am## Re: Seems harder

1) 36*24 = 36*20 + 36*4 = 720 + 144

2) 36*24 = 36*20 + 36*4

= 30*20+6*20 + 30*4+6*4

= 600+120 + 120+24

Depending on the numbers involved and how comfortable you are doing mental arithmetic, there would be more levels you could break them down until you get to a set of sums that are in the 10 times tables.

One thing I think is important is to teach children reasonableness checks. After all, because of calculators you rarely have to do big multiplications in your head (or even on paper). However it is useful to know if you've got the right answer. For example you could mistype and get 63*24 = 1,512.

However if you were taught to check for reasonableness you might say that 36*24 is just less than 36*25, which is 36*100/4 or 9*100 so you should get a number just less than 900 - so your calculation was obviously wrong. Of course there are many ways to check for reasonabless, the way I do it is to round numbers so that they are easier to multiply (I didn't round the 36 since it's divisible by 4 and that helps when multiplying by 25)

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David Liu(profile),Mar 12th, 2011 @ 6:55am## Re: Re: Seems harder

In the "old" method, you'd be doing each base of 10 in one line:

The only reason why the two methods look so different is because the numbers are so short and 2x36 is an easily remembered product. But in reality, no one remembers 5x5436 off the top of their head, so they go through the process of 5x6, 5x3, etc. just like in the second method. The only difference being that they write it on a single line instead of one for each individual product.

In other words, the article is really about how people can't tell why they're the same thing.

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Mar 12th, 2011 @ 9:59pm## Re: Re: Re: Seems harder

In other words, the article is really about how people can't tell why they're the same thing.

I disagree. The thing is, with multiplication, almost every method is "effectively" the same. It pretty much has to be since you're doing the same operation on the same numbers. But the way you approach is quite different from the mental perspective, and I think that's important.

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Anonymous Coward,Mar 11th, 2011 @ 6:48pm## derp

one is easier to do in the head with small numbers, the other is easier to do on paper with big numbers.

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Spaceman Spiff(profile),Mar 11th, 2011 @ 7:14pm## Whatever works!

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Anonymous Coward,Mar 11th, 2011 @ 7:28pm[ reply to this | link to this | view in chronology ]

Anonymous Coward,Mar 14th, 2011 @ 4:17am## Re:

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Nicolasp(profile),Mar 11th, 2011 @ 7:41pmWax on, wax off.

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Jose_X,Mar 12th, 2011 @ 1:59pm## Re:

The essence is just: (1+1)*1 = 1*1+1*1 = 2

From this (after many steps and regroupings and applying commutativity.. ie, the axioms of addition) you can show, eg, that (10+7)*6 = 10*6+10*7 = 102.

Eg, 6=5+1 (and eventually 6=1+1+1+1+1+1), 10=9+1 (.. 1+1+1+1+1+1+1+1+1+1), 7=6+1 (.. 1+1+1+1+1+1+1), and we would end up simply applying many levels of distributive property if we carried out a step-by-step detailed proof. [In mathematics, we'd use more sophistication that would remove the need to carry out every single step.]

Both of these two methods, each a variation of the other, are simply different ways to compose and apply the distributive property.

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Anonymous Coward,Mar 11th, 2011 @ 7:53pmSo to do 36 x 24, I'd go:

36 x 10 = 360

360 x 2 = 720

So that's the 20 sorted and then for the 4 I go:

360 x 10 = 360

360 / 2 = 180 (so that's 360 x 5)

180 - 36 = 144

And then finally

720 + 144 = 864

I can *not stop doing it this way*. It actually helps me get bigger numbers easier than other people seem to, but it slows me way down on smaller numbers because I have a lot of superfluous steps.

And now you know!

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Hellbent,Mar 12th, 2011 @ 8:55am## Re:

I don't do a lot of maths, except in stores, where I just sum up prices to the nearest 50 cents (always rounding up, so 20c gets rounded up to 50c). That way paying cash will always be less than what I'm willing to spend or have at hand. And even when I pay with a card, you get the nice feeling you got off with less than you expected.

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Mar 11th, 2011 @ 8:24pm## Old-- And Older

36 * 24 = 36 * (25-1)

= (36 * 25) - 36

=(36*100/4)-36

=(3600/4)-36

=900-36

=864

There is an old joke about multiplication. The versions on the internet are all spoiled because they have been paraphrased by people with no sense of Old Testament literary style.

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

When the Ark landed, Noah bade the animals to go forth, and multiply. Most did so, but two snakes remained, gazing coldly at Noak.

"Why are you not multiplying," he asked.

"We're adders," they replied.

So Noah sighed, and went forth, and cut down some trees, and trimmed the trunks, and built them into platforms.

Because everyone knows that adders cannot multiply without log tables.

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aperson(profile),Mar 11th, 2011 @ 9:00pm[ reply to this | link to this | view in chronology ]

Brian Schroth(profile),Mar 14th, 2011 @ 6:59am## Re:

"How did you get this answer for 5476 * 126?"

"I calculated it in my head."

"I don't believe you."

"OK, ask me to multiply 2 different numbers."

"3748 * 424"

"1589152"

"Oh, I guess you can do it. OK."

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Almost Anonymous(profile),Mar 14th, 2011 @ 9:54am## Re:

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Anonymous Coward,Apr 3rd, 2014 @ 9:42pm## Response to: aperson on Mar 11th, 2011 @ 9:00pm

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Greevar(profile),Mar 11th, 2011 @ 9:13pm## Interesting

4*6=24

30*4=120

24+120=144

20*6=120

120+144=264

30*20=600

600+264=864

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Anonymous Coward,Mar 11th, 2011 @ 9:14pmI think I prefer the 'old way,' but that's probably because I have lousy short term memory and am happy to use a pencil.

Though when I do multiplication mentally, I have tendency to construct mental rectangles in multiples of ten and then chop off the extra as a mental method. Very similar to Andrew D's method. Main reason for it is that a rectangle is only 1 object to remember, which is easier than multiple numbers.

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The eejit(profile),Mar 11th, 2011 @ 11:50pm## Re:

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Mar 11th, 2011 @ 10:18pm## That's Funny

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Anonymous Coward,Mar 12th, 2011 @ 1:36amI ended up adding two numbers together, as in the first image, but the two numbers came from four products, as in the second image. (I only have the multiplication tables memorized through 12; the idea of memorizing them through to 36 is new to me.)

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Miff(profile),Mar 12th, 2011 @ 1:55am## Feels kinda strange

If it makes me feel any better, I did use a similar method for addition:

36 + 24 = 30 + 6 + 20 + 4 = 30 + 20 + 10 = 30 + 30 = 2*30 = 2*3*10 = 6*10 = 60

The thing from 30+30 to 60 is mainly how I saw adding those up.

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Anonymous Coward,Mar 12th, 2011 @ 2:42am## Puzzled by question

i would:

36x2=72

2x72 = 144 (thats 36x4)

72 + 144 = 216

why is everyone saying its 36x24?

very definitely anonymous coward as i must be being very stupid.

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Anonymous Coward,Mar 5th, 2014 @ 9:16am## Re: Puzzled by question

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grumpy(profile),Mar 12th, 2011 @ 4:03amI pity the people unable to do simple estimation in their heads. They are the ones likely to be taken in by scams like "Special offer: Buy 1 for $30 or 3 for $100!". Being able to do simple arithmetics on the fly without breaking out the smartphone should be a prerequisite for being allowed to vote...

Is my age showing? ;-)

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Anonymous Coward,Mar 12th, 2011 @ 6:14am[ reply to this | link to this | view in chronology ]

D,Mar 12th, 2011 @ 7:57am## changing ways

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D,Oct 12th, 2011 @ 10:06am## Re: changing ways

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abc gum,Mar 12th, 2011 @ 8:09amThere is usually more than one way to do anything and this is certainly true in math. Demonstrating the equivalency of various problem solving methods, their benefits, efficiencies and potential uses should be part of the curriculum.

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Anonymous Coward,Mar 12th, 2011 @ 8:37am## Same Thing

Typically, what happens when people are learning to do math is that they start out with small problems and explicitly break everything down into very small steps. Then, as they become more proficient, the smaller steps become implicit and only the larger ones are explicitly thought out. The main difference between the two processes you highlighted is that the first one might be used by someone who is more proficient in arithmetic than the second one as it uses more implicit and fewer explicit steps than the second one.

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Anonymous Coward,Mar 12th, 2011 @ 8:51amLater what I found most useful was when I learned algebra, to conceptualize multiplication this way:

(10a +b)(10c + d)= 100ac + bd + 10bc + 10 ad

For me, understanding this was especially helpful for multiple choice tests - I could eliminate some answers by inspection, without doing arithmetic at all.

So for instance:

17 x 72 = 700 + 14 +20 +490 = 1224

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Richard(profile),Mar 12th, 2011 @ 9:10am## Look at the numbers

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Richard(profile),Mar 12th, 2011 @ 9:15am## If you use the same method every time then you don't understand what you are doing

Every sum has it's own quick method.

If you use the same method every time you don't really understand what you are doing - people who really know numbers have a who variety of different methods in their arsenal - and use the most appropriate on for the job in hand

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Jose_X,Mar 12th, 2011 @ 2:14pm## Re: If you use the same method every time then you don't understand what you are doing

I use different "paths" partly because it takes the pain out of multiplying. I rely on 2s and 5s (10s) or something else that is used in my head frequently.

More importantly, taking different paths allows for (as was mentioned earlier) the ability to cross check whatever other method I will use (eg, a calculator).

If you work with numbers frequently, it also helps you refresh your "intuition skill" when you can use a different path each time.

[Note that all of these paths are different instances of applying the distributive property to different breakdown of the numbers.. since the distributive property captures all new that is important in defining multiplication from addition. (subtraction being readily converted into addition of negative numbers).]

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Bengie,Mar 12th, 2011 @ 9:15amWhat is 2 + 3 * 4

We had lots of call ins saying our software was *incorrectly* saying 14 was the answer and the *teachers* said 20 was correct.

WTF?!

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abc gum,Mar 12th, 2011 @ 12:39pm## Re:

Yup - I'm sure that all math teachers are fairly represented by the one to which you refer.

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Bengie,Mar 12th, 2011 @ 1:09pm## Re: Re:

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Jose_X,Mar 12th, 2011 @ 2:22pm## Re: Re: Re:

This is a matter of remembering the convention of adding in parenthesis. Even people who know this well, but who don't perform the calculation frequently, momentarily forget sometimes.

Now, you might argue that a person teaching this should remember that, but the reality is that a grade school teacher usually has to cover lots of material in diverse topics (as well as skills of a more general nature applicable to young minds growing up in the world) and may forget the details until they see the lesson plans (that they bought or created in earlier years).

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Almost Anonymous(profile),Mar 14th, 2011 @ 9:56am## Re: Re: Re: Re:

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abc gum,Mar 12th, 2011 @ 4:41pm## Re: Re: Re:

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Not That Chris(profile),Mar 13th, 2011 @ 8:41am## Re:

I can't even being to guess what the issue behind this is (poor teaching methods, over reliance on parenthesis for denoting order, etc.). Really, I'm not sure which one is scarier: teachers in charge of teaching the concept to our kids, or engineers in charge of designing parts for things like cars and airplanes.

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kstahmer(profile),Mar 12th, 2011 @ 11:24am## Distributive law

36 * 24 = (30 + 6) * (20 + 4) = 30 * 20 + 30 * 4 + 6 * 20 + 6 * 4

In general: (a + b) * (c + d) = ac + ad + bc + bd

That's all well and good as long as kids are also taught the distributive law and thoroughly understand it.

And that's the problem. Too many kids are taught the method, but don't understand the underlying theory which justifies the method. Either they were poorly taught or they were never taught.

There many times where dyscalculia resides within the teacher, not the student.

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Mar 12th, 2011 @ 11:25am## Paper vs head

The old way is much faster for large numbers - and also has a degree of certainty as you go along that is kind of comforting to me. Maybe it's ingrained prejudice, but my thinking is that arithmetic is about taking the guess work out of the basics.

Obviously both methods work, and of course explaining *why* something works has always been better for me. I just learned that rote has a purpose - get the scales down and the rest of the music theory will be *much* easier....

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John Duncan Yoyo,Mar 12th, 2011 @ 11:37amYou draw a box with the digits that you are multiplying along the top and the right side with boxes for each column and row. Look at the example here for a good explanation.

http://www.superteacherworksheets.com/lattice.html

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Cipher-0,Mar 12th, 2011 @ 11:52am## Then there's me...

36 * 24 = X

(6*4) + (3*4 + 2*6)*10 + 6*100 = X

I drove math teachers crazy like this, and had to show each of them that yes, those single digits around the original problem are in fact my work.

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Shadow-Slider,Mar 12th, 2011 @ 12:53pmWhat about factoring first like 2^5 * 3^3=

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Darryl,Mar 12th, 2011 @ 1:38pm## Arithmetic is algebra, is an algorithm.

If you think you can write software without a strong knowledge of mathematics, arithmatic, and algebra, you have no chance of being able to even write a "hello world" program.

Algebra, is a middle eastern term, who developed it, and much other science as well, algebra, and algorithm are derived from the same word.

They are the same, algebra is an algorithm that is simply a method of manipulating symbols, at some stage (at end) you might convert those symbols into numbers, but it is not a necessary requirement.

The algormithm itself is what is important, "arithmatic" is and application of an algorithm, therefore it is algebra.

Get it ??? :)

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Jose_X,Mar 12th, 2011 @ 2:29pm## Re: Arithmetic is algebra, is an algorithm.

This is why cooks make great computer programmers. In fact, algebra comes from an African delice: boiled zebra a la algae.

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abc gum,Mar 12th, 2011 @ 4:44pm## Re: Arithmetic is algebra, is an algorithm.

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Darryl,Mar 12th, 2011 @ 7:03pm## Re: Re: Arithmetic is algebra, is an algorithm.

Its not algebra, algebra would be:

Let A = 1

Let B = 1

Let C equal the product of A and B.

therefore (1+1)=2 is arithmetic

algebra is

(A+B)=C

1+1=2 is a recipe (take one egg and 1 cup of milk)

(A+B)=C is an algorithm

that says "Take ANY number or thing, call it A and take any other number or thing and add those two together the result will be the product of the first two values.

If I said the value of Pi was 4.72 you would think I am wrong.

But in Octal it is 4.72

But if I said:

Pi = C/d

where C = the circumference of a circle

and d is the diameter of the circle.

Approx 22/7

then it does not matter what number system you apply, as long as you apply the correct algorithm.

once again the US education system must leave ALOT to be desired !!.. what a shame.

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Jim O(profile),Mar 12th, 2011 @ 10:19pm## Re: Re: Re: Arithmetic is algebra, is an algorithm.

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Anonymous Coward,Mar 13th, 2011 @ 5:20am## Re: Re: Re: Arithmetic is algebra, is an algorithm.

>> (A+B)=C is an algorithm

I am not sure what you are trying to get at by limiting what "recipe" can mean.

The definition I quoted said a recipe can mean a "formula", and I gave the name of a known book which referred to algorithms as recipes.

A recipe:

Accept a two values and add them up. This is A+B=C.

This is a set of instructions. It's not the kitchen cookbook kind, but it can be called a recipe as well. At least that is the association I have with "recipe" in my mind and the online dictionary definition seems to be consistent with it.

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abc gum,Mar 13th, 2011 @ 9:07am## Re: Re: Re: Arithmetic is algebra, is an algorithm.

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D,Mar 14th, 2011 @ 8:59pm## Re: Arithmetic is algebra, is an algorithm.

You need math to write hello world?

In what language?

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HarryS(profile),Mar 12th, 2011 @ 1:40pm## New Method

The left to right does not work for subtraction or long division. So now instead of learning 2 methods you need to learn 4.

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Jose_X,Mar 12th, 2011 @ 2:32pm## Re: New Method

What I came to realize was that the more ways you learn, the easier it is to pick any particular way of doing it. At this point, I just let my whims dictate.

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Darryl,Mar 12th, 2011 @ 1:41pm## or

But an algorithm with work with any number base system, or no number base system. Even imaginary numbers.

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Jose_X,Mar 12th, 2011 @ 7:49pm## Re: or

Or are you saying something else?

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Anonymous Coward,Mar 13th, 2011 @ 7:09am## Re: or

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Darryl,Mar 12th, 2011 @ 1:45pm## magic squares.

It's a very old method of math, not a "new" method.

Maybe you are a bit behind where you were schooled.

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Darryl,Mar 12th, 2011 @ 1:51pm## Hmm,, any programmers here ?

I guess you guys are too busy trying to pirate music to learn things, even about computers !!!!.

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Jose_X,Mar 12th, 2011 @ 3:06pm## Re: Hmm,, any programmers here ?

Below is the remainder of the recipe ("algorithm" in computer-speak) exactly as I learned it:

[It's important that this be a song everyone wants to steal. I don't know of any other method to execute the remaining steps otherwise.]

Then you strip away the DRM.

Then you share the file with about 100,000,000 people.

Then each of these people prints out the entire file on a ticker tape using a monospace font, with the ticker tape shortened to exactly the right length. [cut off left-over ticker tape paper]

Then each person comes together in a partial fulfillment of world peace and carefully links each end of his or her unwound fully used ticker tape with the corresponding ends of 2 neighbors.

Then the group gets into 1000 (approximate) concentric circle and starts running around so as to widen out and turn the circles like a merry-go-round.

Then all the kings horse and everyone really just breaks out into an grand ole party.. At this point an announcer formally announces the result.

Post notes:

Make sure the announcer has a legible copy of the wall-sized poster of pi to 1,000,000 places such as one sold here http://mathematicianspictures.com/PI/Pi_One_Million_Decimal_Places_Pi_1_Million_Decimal_Places_poste r.htm

The poster does NOT have DRM, but, out of so many people, we should be able to find someone willing to buy it and trot it out in public. If not, someone else would almost surely be willing to buy it for their neighbor, intercept the mail, pirate the relevant digits, and return the poster back to sender.

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Darryl,Mar 12th, 2011 @ 6:44pm## Re: Re: Hmm,, any programmers here ?

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Jose_X,Mar 12th, 2011 @ 7:35pm## Re: Re: Re: Hmm,, any programmers here ?

Again, I was having fun with you for taking a jab at this website ("piracy") in a comment and discussion that otherwise had absolutely nothing to do with copyright issues.

You also mentioned needing to have a "strong knowledge of magic squares" (or of 'new' math) in order to program a mathematics routine. I understand you probably need to understand at least the essence of basic math, but not any specific algorithm (like magic squares).

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Anonymous Coward,Mar 12th, 2011 @ 4:42pm## Old way is better in many cases

In any case the "new way" isn't new... in fact it's a very common way for people who are decent/good at math to do multiplication in their head. I see no reason to teach it as "THE WAY" to do multiplication. It's a useful trick for people with a good algebra background

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Anonymous Coward,Mar 12th, 2011 @ 4:47pm## BREAKING NEWS

Mike Masnick pats himself on the back!More on this exciting topic every day, only on TechDirt...

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abc gum,Mar 12th, 2011 @ 4:50pm## Re: BREAKING NEWS

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Stuart(profile),Mar 12th, 2011 @ 4:50pm## can you say calculator

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Jose_X,Mar 12th, 2011 @ 7:36pm## Re: can you say calculator

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Anonymous Coward,Mar 13th, 2011 @ 5:30am## Re: can you say calculator

Do you think if one of your eyeballs fell out and one of my eyeballs fell out, that we could give full sight to someone with .. um... how many eye sockets would we be filling? .. umm....

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Darryl,Mar 12th, 2011 @ 6:40pm## A recipe is not an algorithm

Sorry, you are WRONG.

An algorither is somthin like "Take A and add it to B and divide the relulst of that addition by C".

I does not matter at all what A, B and C are, the algorithm will always work.

A Recipe is "add butter to flower, add milk, mix it up and heat treat it for a specific time and temperature.

You cannot substitute 'butter' for cheese, or flour for sand.

That is the difference between an algorithm and a recipe, here endith today's lesson. Kindagartgen will commence again tomorrow.

Notice how in an anlogithm A,B and C can me anything, but in a recipe that is not the case.

But you allready know that right Jose, you are such a smart person and all. You must have been joking, because not many could be deliberately that stupid.

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Jose_X,Mar 12th, 2011 @ 7:27pm## Re: A recipe is not an algorithm

rec·i·pe (rs-p)

n.

2. A formula for or means to a desired end: a recipe for success.

I can use the word recipe to refer to an algorithm.

>> An algorither is somthin like "Take A and add it to B and divide the relulst of that addition by C".

Right, a formula. (see definition of recipe)

>> You cannot substitute 'butter' for cheese, or flour for sand.

You also can't substitute a copyrighted work for an uncopyrighted work and get the same results.

You can't substitute one piece of data for another and necessarily expect to get something that "tastes good" to a computer and the human waiting for the result.

I was mocking you a bit since you were sort of attacking people here. http://www.techdirt.com/articles/20110310/03354413427/changing-way-that-math-is-taught-to-children.s html#c571

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Jose_X,Mar 12th, 2011 @ 7:45pm## Re: Re: A recipe is not an algorithm

Numerical Recipes in C, Second Edition (1992) and several other books many have heard about and used.

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mirradric,Mar 15th, 2011 @ 1:33am## Re: A recipe is not an algorithm

>>I does not matter at all what A, B and C are, the algorithm will always work.

err.. what if C is zero?

What if A is a matrix and B is a scalar?

Can you not grasp that an algorithm can place limits on the inputs just as a recipe can limit what ingredients you can use.

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John Cipolletti,Mar 12th, 2011 @ 8:52pm## How to Multiply

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MD,Mar 13th, 2011 @ 3:10am## New Math Strikes Again

This was the push in the 1960's. "We're professional educators, you parents know nothing, get out of the way and let us show your kids the RIGHT way..." The way that worked for decades or centuries can't be right, because it's old.

Anyone who has done higher math will recognize FOIL - "first outside inside last". The trouble is this is an n-squared algorithm, whereas the traditional way is an n-algorithm. 3x3 digits, 9 ops. 4x4 digits, 16 ops; etc. Plus it says that a child is incapable of doing a 1 digit times n-digits multiplication on one line, which most not-too-dull kids can within a few months of learning. So why teach this method, or worse, teach it then try to teach the real way a few years later. (If you can keep track of 9 or 16 ops and all the implied zeroes, and can add a column of

16 numbers - you can do 1xn-digit multiplication on one line!)

No wonder American kids are the dumbest in the world. Too many education "professionals" with their fingers in the pie.

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abc gum,Mar 13th, 2011 @ 9:14am## Re: New Math Strikes Again

This is obviously not true.

In addition, I would like to point out that the "New Math" seems to be all about selling new books. Math is math, there has been very little change in its basics since long ago but publishers need to buy new yachts, sports cars and vacation homes. The good teachers push back on this and some even allow college students to use books from last year. The publishers hate the second hand book market.

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Shon Gale(profile),Mar 13th, 2011 @ 8:01amYou know 1*9 = 9 or 9 + 0 = 9, 2*9 = 18 or 1 + 8 = 9, 3*9 = 27 or 2 + 7 = 9, 4*9 = 36 or 3+6=9, etc. Works up to 9*.

We were also taught something else that is starting to catch on again. We call it Mental Math. You approximate the answer first and then solve the problem, or you approximate the entire process and come to the answer quicker. That way you are always working with what you know and are more confident. I taught that method to my kids and they all excelled in math. Even with having immigrant Indian teachers who don't speak English very well.

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Palle,Mar 13th, 2011 @ 8:11amAnd now they teach the kids to be dumb and lazy. i don't think math should be taught the more "practical" way. Because it's about the theory, and people learn a lot more, if they get math taught like it's meant. So they can understand the theories behind.

The aim of schools isn't that children can multiply prices and amounts when they stand in the store and have several items, school's job is to build up an elemtary understanding of how things work.

Hope this doesn't carry on.

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Mar 13th, 2011 @ 5:39pmSo, for example, we'd do this:

4 * 6 = 24 (write down 4 in column 1, carry the 2, add to the next)

4 * 3 + 2 = 12 + 2 = 14 (write down 4 in column 2, carry the 1)

(write down 1 in column 3)

That gives us 144. Then we'd write a 0 in column 1 on the next row and repeat.

2 * 6 = 12 (write down 2 in column 2, carry the 1)

2 * 3 + 1 = 6 + 1 = 7

That gives us 720, and then we just do the addition or 144 + 720.

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Tim,Mar 14th, 2011 @ 10:54am[ reply to this | link to this | view in chronology ]

Mar 14th, 2011 @ 12:37pm## Binary

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Mar 14th, 2011 @ 8:45pm## How I did it

3 6

x2 4

---

0624

24

----

0864

less steps to take

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Syzygy,Mar 19th, 2011 @ 12:59pmnx5 = nx10 / 2 which is simple decimal shunting then doubling

nx9 = nx10 - n

put these two concepts together and you get nx4

nx4 = nx10/2-n

nx7 = (nx10/2)+(nx2)

nx3 = nx2+n

36 x 24 =

((36 * 10) * 2) + ((36 x 10)/2)-40+4

or put simpler:

+

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D,Oct 12th, 2011 @ 10:08am## So many different ways

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Anonymous Coward,Jan 1st, 2014 @ 5:34am## Re: So many different ways

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Mathlover,Nov 1st, 2011 @ 11:49pm## Math

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Erin,Feb 27th, 2013 @ 10:16am## Tricks are for math

36 x 4 =12 x 3 x 4 = 12 x 12 = 144

36 x 20 = 144 / 2 x 10 = 72 x 10 = 720

144 + 720 = 864

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Erin,Feb 27th, 2013 @ 10:21am36 x 24 = 12 x 3 x 12 x 2 = 12 x 12 x 6 = 144 x 6 = 864

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Dec 18th, 2013 @ 3:15pm## Vtgdwsdtgujijij

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steve,Apr 13th, 2014 @ 6:45am## rediculous

Who allowed this to be put forward should have their head examined...

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