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 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.
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 other interesting thing that hit me was the article’s explanation for why things have shifted:
“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.
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 process of how you think about math much more practical in today’s world.
Comments on “The Changing Way That Math Is Taught To Children”
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.
Re: Huh
While I can’t really do it because it’s largely not necessary, if you practiced you can do math in your head very quickly without visually thinking of the numbers. It’s sorta like you can train yourself to automatically know the answer to any problem, hard to explain. When I was little I used to have a small ‘pre’ computer, it was a blue computer with a single textbox as its screen. It can only have a couple of characters on the screen. In fact, I think this was it
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.
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?
Re: Re: Re: Huh
I have no idea. I think at first it kinda is, but after a while it kinda isn’t. Or maybe a combination of figuring it out and memory, like you can remember the general patterns and apply them?
Re: Re: Re: Huh
The times tables should be memorized, would you ever allow a student to not memorize the basics of english grammar?
Re: Huh
Well, it’s really just a natural extension of estimation. I also “taught” myself the second method, but more often than not, unless an exact answer was required, I would just round upward or downward appropriately to give myself a “close enough” answer.
Yep, somewhere along the way I taught myself the same shortcut. It’s definitely easier to do math in your head with the second method.
Seems harder
Well I learned the 1st way as well, and I have to say that this new method still seems harder to me.
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.
Re: Seems harder
It seems that the second method is the same as the first method, but taken to extremes. The operation in (1) of breaking a number into it’s component digits and their power of 10 is the key concept.
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)
Re: Re: Seems harder
Yeah, the second method is actually the exact same as the first, except you just use more lines and write everything out.
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 2×36 is an easily remembered product. But in reality, no one remembers 5×5436 off the top of their head, so they go through the process of 5×6, 5×3, 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.
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.
derp
its a matter of scaling.
one is easier to do in the head with small numbers, the other is easier to do on paper with big numbers.
Whatever works!
I unconsciously taught myself this method as well as a youth so I could amaze people with my ability to multiply large numbers in my head quickly, usually faster than they could input it in a calculator. However, when I did it on paper, I would do it the old way, as I was taught in school! Interesting enough, I was teaching a class recently on data communication networks to AT&T techs and had to get them to do binary->octal->decimal->hex arithmetic in order to deal with IPv4 network address issues (network class, subnets, etc). I was still using the old methods, formally (as the syllabus dictated), but to get them to “see” the process more easily in their minds, I went to the “newer” process. They learned, and did well, but it was still a struggle!
The concern I would have is whether or not the kids understand why both methods are equivalent. To me that seems much more important than whether or not the system favors base 10.
Re: Re:
all methods of multiplying two numbers together are equivalent, otherwise you’d get different answers.
I can do both (or any other method such as 36*20 + 36*4 or whatever) because I learned the essence of multiplication. And you learn that by learning the tables. I also learned Latin but never spoke it. What I learned was the essence of language.
Wax on, wax off.
Re: Re:
The essence comes down to the distributive property. This property defines multiplication from addition.
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.
I’ve always done multiplications in my head in a ridiculously long way that involves using multiples of ten because they’re easy. I’ll put it here! Because people are talking about math and stuff.
So 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!
Re: Re:
I’ve always used the multiples of ten method for doing multiplication in my head, it’s slower on smaller numbers, yes, but it really eases the calculations on higher ones, which usually come up when you need to do a bit of fast arithmetic in your head. I don’t know when I first got into the habit of doing that, probably when the math problems in school started involving numbers greater than 100.
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.
Old-- And Older
I was a kid back in the 1960’s, and I got taught to do that kind of arithmetic in my head, according to the following process:
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.
i thought i was the only one…doing math like this once caused me to fail an algebra test, i couldn’t explain to the teacher how i’d done it in my head. I remember being pissed off, he obviously thought i’d cheated but couldn’t prove it.
Re: Re:
You couldn’t prove it?
“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.”
Re: Re:
Sorry, but I agree with your teacher here. Showing your work is often more important than the actual answer, that way your teacher knows if you understand the concept and how to reach the answer. My best teachers would give you more than half credit if you used the correct process but gave a wrong answer due to a silly addition or subtraction error.
Interesting
I use a similar method, but I sum the products right away so I don’t have to keep so many numbers in my head. I have a deficient working memory. On paper, I do it the old way.
4*6=24
30*4=120
24+120=144
20*6=120
120+144=264
30*20=600
600+264=864
Am I missing something, or does this ‘new’ method become somewhat insane for numbers longer than 2 digits?
I 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.
Re: Re:
No, it doesn’t become insane. Lengthy, yes, but not insane.
That's Funny
I taught myself to chunk the numbers the same way in my head. It drove my teachers to exasperation that I never showed my work.
I guess the method I learned would be kind of halfway between the two. I would’ve done 6*4=24, written down the last “4” in “144” and a tiny “2” above the “3” in “36” (this was known as “carrying the two”), then done 3*4=12, added the 2 to that, and finally written down the “14” in “144”. Same system for the next digit after putting in the 0: 6*2=12->20, 3*2=6, 6+1=7->720.
I 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.)
Feels kinda strange
Being one of the few who never came up with their own method for this.
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.
Puzzled by question
if i wanted to ‘add the products of 36 x 4 and 36 x 2’
i would:
36×2=72
2×72 = 144 (thats 36×4)
72 + 144 = 216
why is everyone saying its 36×24?
very definitely anonymous coward as i must be being very stupid.
“Computers do arithmetic for us”
I 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? 😉
The easiest way is to round the numbers to something easy to multiply – find the product, find the product of the difference, and the subtract the latter from the former – either that or find a infinite series estimate.
changing ways
Well I have a fifth grader and in our school district “Everyday Math” is the curriculum. It would be good to have our “Math Specialist” explain the why to parents so they see the benefit in learning it themselves not as a “fun way to do math” but as necessary to help their children grow and enjoy math throughout their lives. Algebraic thinking and higher forms of critical thinking are what will give us the next crop of home grown wonders populating the R&D departments for companies like Apple and others that will invent the next best thing in carbon neutral energy production. Besides what kid doesn’t ask “why”.
Re: changing ways
Amen!
They called it “New Math” – What a misnomer.
There 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.
Same Thing
The two “methods” are the essentially the same, Mike. The second one is just showing more of the details of the first one by explicitly breaking each of the two steps of the first one down into additional steps. Those steps are just implicit in the first method. You could then then break those steps down into even more steps if you wanted to: 36 * 24 = 144 + 720 = (24 + 120) + (120 + 600) = [(4 * 6) + (30 * 4)] + [(60 * 2) + (30 * 20)} = [(4 * 6) + (3 * 4 * 10)] + [(6 * 2 * 10) + (3 * 2 * 10 * 10)]. It could be broken down even further, but I’m not inclined to do so.
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.
I too learned the ‘near easy multiples of hundreds’ method that Andrew Tood described.
Later 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
Look at the numbers
Re: If you use the same method every time then you don't understand what you are doing
36×24= 3x12x2x12=6×144=6×150-6×6=900-36=864
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
Re: Re: If you use the same method every time then you don't understand what you are doing
You are benefiting from experience, where you know the factors of these numbers (the composition into smaller numbers that multiply out to the original number) and various easy ways to combine these factors.
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).]
What gets me is how stupid math teachers are. My company sells educational software and one of the questions we had was
What 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?!
Re: Re:
“What gets me is how stupid math teachers are.”
Yup – I’m sure that all math teachers are fairly represented by the one to which you refer.
Re: Re: Re:
It wasn’t just “one”. It was an on going tech issue. Lots of calls.
Re: Re: Re: Re:
Well if by “stupid” you mean that they don’t remember that there is a standard default syntax rule in algebra/arithmetic that multiplication takes priority over addition when reducing the operations. Ie, there are implied parenthesis: 2+3*4 actually equals (by convention) 2+(3*4) and not (2+3)*4.
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).
Re: Re: Re:2 Re:
Uh uh. Order of operations should be Lesson 1. Or Chapter 1. Or however you want to say it.
Re: Re: Re: Re:
Oh, I’m sure you believe you’re right.
Re: Re:
Order of operations seems to elude a lot of people. At one point in time my father had taught some statistics classes for engineers and had so many people get the order of operations stuff wrong he started including it as a refresher for the class.
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.
Distributive law
It’s the distributive law. In this particular case:
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.
Paper vs head
I was taught the old way, but figured out the “guesstimate” way for multiplying things in my 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….
My Daughter came home with yet another different method for Multiplication- the Lattice Method. It works really well for pencil on paper problems but I think it may be limiting for math in your head.
You 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
Then there's me...
Who combines everything when forced to write it out.
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.
What about factoring first like 2^5 * 3^3=
Arithmetic is algebra, is an algorithm.
You guys must not be programmers, that is for sure.
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 ??? 🙂
Re: Arithmetic is algebra, is an algorithm.
I don’t know the origin of the word “recipe”, but everything we do in life can boil down to a recipe.
This is why cooks make great computer programmers. In fact, algebra comes from an African delice: boiled zebra a la algae.
Re: Arithmetic is algebra, is an algorithm.
arithmetic – Get it ??? 🙂
Re: Re: Arithmetic is algebra, is an algorithm.
alot of things are algebra, its arithmatic when it involves numbers like (1+1=2), that is arithmatic. Simple.
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.
Re: Re: Re: Arithmetic is algebra, is an algorithm.
I can’t tell if your post is a troll or not. You call out the shortcomings of an educational system with a gross misspelling. In caps to boot.
Re: Re: Re: Arithmetic is algebra, is an algorithm.
>> 1+1=2 is a recipe (take one egg and 1 cup of milk)
>> (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.
Re: Re: Re: Arithmetic is algebra, is an algorithm.
context is important
Re: Arithmetic is algebra, is an algorithm.
Ok, I’ve been trolled.
You need math to write hello world?
In what language?
New Method
Along with the change in Multiplication, there is a change to do addition is a similar way from left to right. The reason I was given is that it makes it “Easier” for the students to learn.
The left to right does not work for subtraction or long division. So now instead of learning 2 methods you need to learn 4.
Re: New Method
I have learned over time over 10 different ways of getting out of bed.
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.
or
or to put it another way, the arithmetic for binary is different from that of octal or decimal or hex.
But an algorithm with work with any number base system, or no number base system. Even imaginary numbers.
Re: or
I think you are comparing what can be called a somewhat specific algorithm with a hypothetical less specific algorithm. You are saying that some algorithms are more specific than others.
Or are you saying something else?
Re: or
Darryl please use replies instead of just changing the title. For those of us who read this threaded your posts just don’t make sense because they are not in context.
magic squares.
look up “magic squares” that is where this “new” method is derived from.
It’s a very old method of math, not a “new” method.
Maybe you are a bit behind where you were schooled.
Hmm,, any programmers here ?
try to write a computer program that will calculate Pi to say 10,000 decimal places and see if you can do it without a strong knowledge of magic squares or your ‘new’ method of math.
I guess you guys are too busy trying to pirate music to learn things, even about computers !!!!.
Re: Hmm,, any programmers here ?
The best method I learned of calculating pi to 10,000 decimal places starts off by taking a digital download of a really popular, highly sought-after, mega-triply-meta-copyrighted DRM’s song.
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_poster.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.
Re: Re: Hmm,, any programmers here ?
What the hell drugs are you ON Jose ??? gezzzz.. !!!
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).
Old way is better in many cases
The “old way”, which is basic algorithmic arithmetic is better if you are using a paper and pencil and/or working with large numbers. I’ve used the 2nd “trick” for years as a quick way to do multiplication in my head… HOWEVER, I feel strongly that the “old way” should still be taught first… the new way can be learned as part of algebra
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
BREAKING NEWS
This just in …
Mike Masnick pats himself on the back!
More on this exciting topic every day, only on TechDirt…
Re: BREAKING NEWS
Thank you for your well thought out contribution, you must have been taking magic mushrooms with Darryl.
can you say calculator
I don’t think I have added any thing in my head for years.
Re: can you say calculator
for more than 1 year?
Re: can you say calculator
>> I don’t think I have added any thing in my head for years.
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….
A recipe is not an algorithm
Jose_X you surprise me, thinking and even stating a recipe is ‘just 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.
Re: A recipe is not an algorithm
http://www.thefreedictionary.com/recipe
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.shtml#c571
Re: Re: A recipe is not an algorithm
http://www.nr.com/oldverswitcher.html
Numerical Recipes in C, Second Edition (1992) and several other books many have heard about and used.
Re: A recipe is not an algorithm
>>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.
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.
How to Multiply
There are many ways to approach a problem. But some of these ways take more time then others. As a mathematician with a science background I can say that time is money in the real world. The new method of taking the multiplican and multiplier apart and producing a longer problem with more partial answers to add together takes longer and creates a greater chance of mistakes. Does the example give any kind of an advantage over the old method. I don’t think so. Why sweat the small stuff, just learn the short version and be done with it. Oh, and one more point. Using the new method will further cause problems when applying it to division. And people have enough problems when dividing.
New Math Strikes Again
Anyone who remembers the educators pushing “New Math” or “Whole Word Reading” in the 1960’s and 70’s will recognize what is happening. Someone who says “we’ve been doing it right all along” gets no air time or recognition. SOmeone who comes out and says “we’ve been doing it all wrong, here’s the right way” gets air time, gets a few advanced degrees and scholarly papers out of it, then makes a tidy living going out and teaching the educators how to teach children. If it leaves a generation even more confused and illiterate/innumerate, so what?
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. 3×3 digits, 9 ops. 4×4 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.
Re: New Math Strikes Again
“No wonder American kids are the dumbest in the world.”
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.
I just turned 64 and I was taught the second method. We memorized our multiplication tables from 2 through 9. We were even taught a great trick by our 4th grade teacher on how to do 9’s fast. They numbers in the answer add up to 9.
You 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.
Mah, this really doesn’t impress me, because both methods are actually the same. Look at each and you will see, that if you work with the first method, but can’t figure out 24*30, you just split that up again and voila, you get the “new” method.
And 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.
In Australia when I was learning this in school 20 years ago, I was most definitely taught a method similar to method one, but the the multiplication itself still required us to multiply single digits and carry the remainders, more like method 2. We didn’t multiply the big numbers in our heads directly.
So, 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.
This isn’t anything new, they’re just using the FOIL method at an earlier age.
Binary
This fascinates me because it’s how binary is converted to decimal. It makes me think that instructors who come from a computer age realize it is a better way to teach multiplication and at the same time prepare students for working with binary.
How I did it
I always used to do it in a triangular way : so I would write
3 6
x2 4
—
0624
24
—-
0864
less steps to take
I use a different multiplication trick to calculate in my head based on breaking the problem down until im left with only ‘easy’ operations
nx5 = 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:
+
So many different ways
What is great about this thread is that so many people explained their thought processes in so many different ways on ONE problem. That’s what mathematics should be all about! There is not one way to figure out a math problem. Conversation needs to take place in the classrooms and in the world around us. It was fascinating reading through this thread. Thanks for all of your input!
Math
I would like to say, please you all guys can have a look at Vedic Mathematics tricks which teach you a totally different way of multiplication. Whatever mentioned here are whatever we derived from the first method. Or just know about the V-MAT mathlearning program that I came to know about.
Tricks are for math
36 x 24
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
Or…
36 x 24 = 12 x 3 x 12 x 2 = 12 x 12 x 6 = 144 x 6 = 864
Huh
I really don’t understand how any of you think this is easier. You’re adding 4 lines to get an answer (new way) instead of 2 lines (old way).
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.
Vtgdwsdtgujijij
Good
So many different ways
Thank u
Puzzled by question
There’s a typo in the problem. It’s missing a zero.
Response to: aperson on Mar 11th, 2011 @ 9:00pm
rediculous
this is absolutely rediculous.this will not help in business, or their future day to day life…..
Who allowed this to be put forward should have their head examined…
yeah you are way,way,way,ahead of your time mike. NOT!
Math
does anyone know what 36 breaks down to when you multiply fractions?
Easiest way to look at it from left to right 3×2 add 2 zeros 2×6 add one zero 4×3 add one zero then 6×4 and then add each of them. The main benefit to left to right method is to promote better number sense in a faster manner, so that our children can advance mathematically beyond our past failures. We read left to right only makes sense to do math left to right. However a lot of public schools are still teaching kids to do it right to left. I’m trying to teach my daughter both ways.
Math
It is basically the equation: (a+b)(c+d) = ac +bc+ad+bd. We learned that sometimes in High School
Huh
NO
Math
Tom is right, this is the expansion of equation:
(a+b)(c+d) = ac +bc+ad+bd.
By the way you can get math based topics and tutorial from studypivot.
Here is a tutorial for logarithm rules with examples.
2nd method was taught to the kids that couldn’t properly calculate the first method in their head.
1st method and 2nd method are exactly the same but 2nd method is just broken into more steps, to make it easier to calculate mentally as you have smaller mental calculations that you can take note of rather than larger ones.