How come mental math is so hard?

[kyrax12]

What is the scientific reason why human have such a hard time holding numbers in their head?

 

[OverVolt]

People do alot of calculations on the fly.

http://scienceblogs.com/builtonfacts/2013/02/27/quick-hit-the-brakes/

Its not that mental math is hard its that numbers suck :awe: Our brains just aren't wired for proofs and derivations so much as catching a glass you dropped mid-air before it hits the ground which if you made it into a math problem would take 15 minutes to figure out.

 

[Colt45]

not enough registers. I think that's my problem, anyway.

 

[DrPizza]

I'm assuming you were raised in the U.S. One reason it's hard to do is the algorithms that children are typically taught in this country for doing things like multiplication. For example, when multiplying 34 times 57, US children would multiply the 7 and 4 first, get 28, have to carry the two, multiply the 7 and 3, add the two. Then put a zero down to the right on the next line as a place holder, and multiply the 5 times the 4, carry the two, 5 times 3 and add the 2.

Other places would approach it like this: 30 times 50 = 1500. Easy to remember 1500. 4 times 50, add to the 1500. Now you're up to 1700. 7 times 30, add to 1700 is 1910. Just have to remember 1910. Lastly, 4 times 7, and add that to the 1910 = 1938.

In the first algorithm, it takes multiple steps to arrive at the first number and multiple steps to arrive at a 2nd number before you add them together. In the latter algorithm, you're keeping a running total while still doing one multiplication at a time.

Cue about middle school age. The brain has developed to the point where it can handle more abstract concepts (like variables.) Now we're teaching Algebra, and it takes a couple days of teaching before the kids get good at multiplying something like (2x+3) and (4x-5). Kids in other places think, "shit, this is what I've been doing for 4 years. This is nothing new." This, in part, explains some of the differences on standardized testing. Through, I think it's 4th or 5th grade, U.S. children compare quite favorably, then they fall way behind.

-----

Also, fucking calculators. If you had a little robot that carried absolutely everything for you, you never even had to lift up a glass to your mouth to drink, in a couple of years, you'd be asking, "why is it so hard to carry a bag of groceries." Without practice, you can't do arithmetic very quickly. The more often you use a calculator for simple arithmetic operations, the slower your brain gets at doing that mental arithmetic.

Further, if you don't have a calculator, you start learning tricks. 87 times 93? Why, that's just 8100 minus 9. Why? Because it's (90-3)(90+3); the product of conjugates which is 90²-9. Or, in a calculus class, faced with finding the area under a curve using the midpoint method (midpoint of the base of a rectangle determines the x-value at which you determine f(x) for the height of a rectangle). It's fairly common on those problems to have to find 6.5², 7.5², 8.5², etc.
6.5 squared: 6 squared (36) plus 6 = 42. Tack on a .25 to the end: 42.25
7.5 squared: 7 squared (49) plus 7 = 56. Tack on a .25 to the end: 56.25
8.5 squared: 8 squared (64) plus 8 = 72. Tack on a .25 to the end: 72.25
At which point you might see a pattern. "Hey, it goes up by 14, then 16, I'll bet the next one is 18 higher." Though the pattern might be a little harder to discern if f(x) is something like 2x²+x+5; but no more difficult to do in your head.

Why it works? Again with the algebra. (x + 0.5)(x + 0.5) = x² + x + .25
Note: if you did (10x + 5)(10x + 5), you get 100x²+100x+25. So, the pattern is the same without the decimal.

As your mind learns more of these tricks, you look absolutely amazing to people who can't do mental arithmetic. You might say, "but you're doing..." Doesn't matter - it's still mental arithmetic, it's just that with a fluency in Algebra, you have a lot more algorithms at your disposal. And, with practice (not relying on a calculator from 4th grade on), you get quicker and quicker at identifying which tricks work on which problems.

Fun problem: I'll get a class to give me digits of a 9 digit number. Once I have a lot of kids shouting out numbers, it's easy to be selective for the 2nd number to make it 142857143. I can multiply any 9 digit number times 142857143 faster than the students can type it into a calculator (only to find out that the display on the calculator isn't large enough. So, when they get a rounded number in scientific notation, we have to turn to a computer to verify my answer. But it's simple: I multiply by 142857143 indirectly. First I multiply the 9 digit number by 1000000001 (which just makes it an 18 digit number with the first 9 digits repeating. So, if the first 9 digits that the kids give me are 234897561, then the new number is 234897561234897561. Then, divide by 7. Long division - in your head. (omg, it's soooo hard, since you haven't done long division in your head in 20 years.)
7 goes into 23 3 times (write down the 3, mentally carry the 2; 7 goes into 23... )


So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

 

[ViRGE]

So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

Go on, DrP, tell us how you really feel.

 

[pmv]

I'm assuming you were raised in the U.S. One reason it's hard to do is the algorithms that children are typically taught in this country for doing things like multiplication. For example, when multiplying 34 times 57, US children would multiply the 7 and 4 first, get 28, have to carry the two, multiply the 7 and 3, add the two. Then put a zero down to the right on the next line as a place holder, and multiply the 5 times the 4, carry the two, 5 times 3 and add the 2.

Other places would approach it like this: 30 times 50 = 1500. Easy to remember 1500. 4 times 50, add to the 1500. Now you're up to 1700. 7 times 30, add to 1700 is 1910. Just have to remember 1910. Lastly, 4 times 7, and add that to the 1910 = 1938.

I'm not sure about that. I tried doing 34x57 and did indeed do it the second way. But I'm pretty sure I was taught the first way. I don't think you have to be 'taught' the second way, its just an obvious shortcut that people are going to think of for themselves precisely _because_ people are lazy. The first way is probably more natural if you are using pen and paper.

On the other hand, I got part-way through the calculation and ran out of motivation for remembering the intermediate figures in my head ("now add 4x7, but, er what was the previous total again, and why the hell am I doing this anyway?")

I'm definitely far, far less good at mental arithmetic than I was when I was younger, not because I'm stupider (though I suspect my memory isn't as good), but because I have so much less motivation than I used to to practice it! I only do it now when I really, absolutely, have to.

I reckon this is why IQ scores decline with age - its not that us old people are dumb, its that we just can't be bothered any more unless there's something in it for us! I used to be very anxious about being good at maths, now I just don't care as much.


Mind you - some of the other tricks you mention are interesting. But I'm still not sure its worth the effort of remembering them!

 

[Juddog]

I'm assuming you were raised in the U.S. One reason it's hard to do is the algorithms that children are typically taught in this country for doing things like multiplication. For example, when multiplying 34 times 57, US children would multiply the 7 and 4 first, get 28, have to carry the two, multiply the 7 and 3, add the two. Then put a zero down to the right on the next line as a place holder, and multiply the 5 times the 4, carry the two, 5 times 3 and add the 2.

Other places would approach it like this: 30 times 50 = 1500. Easy to remember 1500. 4 times 50, add to the 1500. Now you're up to 1700. 7 times 30, add to 1700 is 1910. Just have to remember 1910. Lastly, 4 times 7, and add that to the 1910 = 1938.

In the first algorithm, it takes multiple steps to arrive at the first number and multiple steps to arrive at a 2nd number before you add them together. In the latter algorithm, you're keeping a running total while still doing one multiplication at a time.

Cue about middle school age. The brain has developed to the point where it can handle more abstract concepts (like variables.) Now we're teaching Algebra, and it takes a couple days of teaching before the kids get good at multiplying something like (2x+3) and (4x-5). Kids in other places think, "shit, this is what I've been doing for 4 years. This is nothing new." This, in part, explains some of the differences on standardized testing. Through, I think it's 4th or 5th grade, U.S. children compare quite favorably, then they fall way behind.

-----

Also, fucking calculators. If you had a little robot that carried absolutely everything for you, you never even had to lift up a glass to your mouth to drink, in a couple of years, you'd be asking, "why is it so hard to carry a bag of groceries." Without practice, you can't do arithmetic very quickly. The more often you use a calculator for simple arithmetic operations, the slower your brain gets at doing that mental arithmetic.

Further, if you don't have a calculator, you start learning tricks. 87 times 93? Why, that's just 8100 minus 9. Why? Because it's (90-3)(90+3); the product of conjugates which is 90²-9. Or, in a calculus class, faced with finding the area under a curve using the midpoint method (midpoint of the base of a rectangle determines the x-value at which you determine f(x) for the height of a rectangle). It's fairly common on those problems to have to find 6.5², 7.5², 8.5², etc.
6.5 squared: 6 squared (36) plus 6 = 42. Tack on a .25 to the end: 42.25
7.5 squared: 7 squared (49) plus 7 = 56. Tack on a .25 to the end: 56.25
8.5 squared: 8 squared (64) plus 8 = 72. Tack on a .25 to the end: 72.25
At which point you might see a pattern. "Hey, it goes up by 14, then 16, I'll bet the next one is 18 higher." Though the pattern might be a little harder to discern if f(x) is something like 2x²+x+5; but no more difficult to do in your head.

Why it works? Again with the algebra. (x + 0.5)(x + 0.5) = x² + x + .25
Note: if you did (10x + 5)(10x + 5), you get 100x²+100x+25. So, the pattern is the same without the decimal.

As your mind learns more of these tricks, you look absolutely amazing to people who can't do mental arithmetic. You might say, "but you're doing..." Doesn't matter - it's still mental arithmetic, it's just that with a fluency in Algebra, you have a lot more algorithms at your disposal. And, with practice (not relying on a calculator from 4th grade on), you get quicker and quicker at identifying which tricks work on which problems.

Fun problem: I'll get a class to give me digits of a 9 digit number. Once I have a lot of kids shouting out numbers, it's easy to be selective for the 2nd number to make it 142857143. I can multiply any 9 digit number times 142857143 faster than the students can type it into a calculator (only to find out that the display on the calculator isn't large enough. So, when they get a rounded number in scientific notation, we have to turn to a computer to verify my answer. But it's simple: I multiply by 142857143 indirectly. First I multiply the 9 digit number by 1000000001 (which just makes it an 18 digit number with the first 9 digits repeating. So, if the first 9 digits that the kids give me are 234897561, then the new number is 234897561234897561. Then, divide by 7. Long division - in your head. (omg, it's soooo hard, since you haven't done long division in your head in 20 years.)
7 goes into 23 3 times (write down the 3, mentally carry the 2; 7 goes into 23... )


So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

Wow... impressive! I also have used the method you mentioned when doing multiplication and always wondered why they didn't teach it that way in school.

 

[clamum]

Great post Dr. Pizza! I of course was taught the first method to multiply numbers together but I'll try and remember to make use of your other algorithm in the future.

 

[Arcadio]

They do teach the partial-products method of multiplication in most public schools in NYC, but teachers don't encourage its use.

 

[Fritzo]

People do alot of calculations on the fly.

http://scienceblogs.com/builtonfacts/2013/02/27/quick-hit-the-brakes/

Its not that mental math is hard its that numbers suck :awe: Our brains just aren't wired for proofs and derivations so much as catching a glass you dropped mid-air before it hits the ground which if you made it into a math problem would take 15 minutes to figure out.

This is actually true. Our neural net is built on a "root" system, making it more adept to "if/then" logic than mathematical groupings. The good news is a portion of your brain can be rewired for math through training.

If I remember my biology the angular gyrus and supermarginal gyrus regions of the brain handles things like language, cognitive skills, and number processing. Mathematical geniuses tend to have these areas enlarged (I think Einstein was a good example of this).

I have no idea why I remember this. Useless trivia is my superpower.

 

[cbrunny]

What is the scientific reason why human have such a hard time holding numbers in their head?

Hard relative to what? I guarantee we hold numbers in our heads better than any other species on earth, by a wide margin.

 

[Mixolydian]

Because math sucks, plain and simple.

 

[uclaLabrat]

I'm assuming you were raised in the U.S. One reason it's hard to do is the algorithms that children are typically taught in this country for doing things like multiplication. For example, when multiplying 34 times 57, US children would multiply the 7 and 4 first, get 28, have to carry the two, multiply the 7 and 3, add the two. Then put a zero down to the right on the next line as a place holder, and multiply the 5 times the 4, carry the two, 5 times 3 and add the 2.

Other places would approach it like this: 30 times 50 = 1500. Easy to remember 1500. 4 times 50, add to the 1500. Now you're up to 1700. 7 times 30, add to 1700 is 1910. Just have to remember 1910. Lastly, 4 times 7, and add that to the 1910 = 1938.

In the first algorithm, it takes multiple steps to arrive at the first number and multiple steps to arrive at a 2nd number before you add them together. In the latter algorithm, you're keeping a running total while still doing one multiplication at a time.

Cue about middle school age. The brain has developed to the point where it can handle more abstract concepts (like variables.) Now we're teaching Algebra, and it takes a couple days of teaching before the kids get good at multiplying something like (2x+3) and (4x-5). Kids in other places think, "shit, this is what I've been doing for 4 years. This is nothing new." This, in part, explains some of the differences on standardized testing. Through, I think it's 4th or 5th grade, U.S. children compare quite favorably, then they fall way behind.

-----

Also, fucking calculators. If you had a little robot that carried absolutely everything for you, you never even had to lift up a glass to your mouth to drink, in a couple of years, you'd be asking, "why is it so hard to carry a bag of groceries." Without practice, you can't do arithmetic very quickly. The more often you use a calculator for simple arithmetic operations, the slower your brain gets at doing that mental arithmetic.

Further, if you don't have a calculator, you start learning tricks. 87 times 93? Why, that's just 8100 minus 9. Why? Because it's (90-3)(90+3); the product of conjugates which is 90²-9. Or, in a calculus class, faced with finding the area under a curve using the midpoint method (midpoint of the base of a rectangle determines the x-value at which you determine f(x) for the height of a rectangle). It's fairly common on those problems to have to find 6.5², 7.5², 8.5², etc.
6.5 squared: 6 squared (36) plus 6 = 42. Tack on a .25 to the end: 42.25
7.5 squared: 7 squared (49) plus 7 = 56. Tack on a .25 to the end: 56.25
8.5 squared: 8 squared (64) plus 8 = 72. Tack on a .25 to the end: 72.25
At which point you might see a pattern. "Hey, it goes up by 14, then 16, I'll bet the next one is 18 higher." Though the pattern might be a little harder to discern if f(x) is something like 2x²+x+5; but no more difficult to do in your head.

Why it works? Again with the algebra. (x + 0.5)(x + 0.5) = x² + x + .25
Note: if you did (10x + 5)(10x + 5), you get 100x²+100x+25. So, the pattern is the same without the decimal.

As your mind learns more of these tricks, you look absolutely amazing to people who can't do mental arithmetic. You might say, "but you're doing..." Doesn't matter - it's still mental arithmetic, it's just that with a fluency in Algebra, you have a lot more algorithms at your disposal. And, with practice (not relying on a calculator from 4th grade on), you get quicker and quicker at identifying which tricks work on which problems.

Fun problem: I'll get a class to give me digits of a 9 digit number. Once I have a lot of kids shouting out numbers, it's easy to be selective for the 2nd number to make it 142857143. I can multiply any 9 digit number times 142857143 faster than the students can type it into a calculator (only to find out that the display on the calculator isn't large enough. So, when they get a rounded number in scientific notation, we have to turn to a computer to verify my answer. But it's simple: I multiply by 142857143 indirectly. First I multiply the 9 digit number by 1000000001 (which just makes it an 18 digit number with the first 9 digits repeating. So, if the first 9 digits that the kids give me are 234897561, then the new number is 234897561234897561. Then, divide by 7. Long division - in your head. (omg, it's soooo hard, since you haven't done long division in your head in 20 years.)
7 goes into 23 3 times (write down the 3, mentally carry the 2; 7 goes into 23... )


So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

I have a PhD in chemistry. I want to sit in your class for a few weeks.

 

[Jaskalas]

So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

:'( Such beautiful efficacy, at owning most the population. Dr. Pizza for President.

 

[surfsatwerk]

I was told there would be no math.

 

[Wet Squirrel]

The chinese method of rote learning seems to help alot.

 

[mmntech]

I'm terrible at math but I think a large part of it is because I was never taught how to do basic maths properly. This is why I keep saying the education system fails by not enforcing mastery of a subject before moving onto another.

 

[AstroManLuca]

I was listening to a radio program recently that got into certain aspects of how our brains work:

http://www.radiolab.org/2008/nov/17/

Go to about 4:40 and they talk about a study from way back in the 50s that tried to determine how much we could memorize in the short term. They found that it was around 7 digits, +/- 2 digits. Holding a lot of numbers in your head at once requires training. You can improve your ability to recall numbers in the short term but you have to use it constantly.

 

[kyrax12]

Hard relative to what? I guarantee we hold numbers in our heads better than any other species on earth, by a wide margin.

machines.

 

[HamburgerBoy]

Further, if you don't have a calculator, you start learning tricks. 87 times 93? Why, that's just 8100 minus 9. Why? Because it's (90-3)(90+3); the product of conjugates which is 90²-9.

This one I happened upon in 6th grade out of boredom/luck, and I consider that to by the high-point of my mathematical self-discovery. The rest I will eagerly practice and aspire to learn.

 

[SphinxnihpS]

Hard relative to what? I guarantee we hold numbers in our heads better than any other species on earth, by a wide margin.

Not bees though, somehow.

 

[Tweak155]

SNIP <wall of text>

So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

And exactly how does that make you smarter? What does the ability to do math on the fly actually gain you? Has practically 0 real world usage and benefits nothing to you other than your arrogance.

Anything I need math for on a day to day basis I can easily do in my head. Your examples are pretty useless, although interesting.

 

[Fritzo]

And exactly how does that make you smarter? What does the ability to do math on the fly actually gain you? Has practically 0 real world usage and benefits nothing to you other than your arrogance.

Anything I need math for on a day to day basis I can easily do in my head. Your examples are pretty useless, although interesting.

You're in a dungeon, chained to a wall. You see Dr. Pizza across the room also chained to the wall. An ominous voice begins speaking through an intercom:

"Education in America is crippled, and soon you will be as well if you can not meet my challenge. There is a bomb attached to each of your legs, and they are set to detonate in the next 3 minutes. You can deactivate the bomb by simply reciting every number between 0 and 1000 that is divisible by 7."

Dr. Pizza walks away 15 seconds later.

 

[exdeath]

I'm assuming you were raised in the U.S. One reason it's hard to do is the algorithms that children are typically taught in this country for doing things like multiplication. For example, when multiplying 34 times 57, US children would multiply the 7 and 4 first, get 28, have to carry the two, multiply the 7 and 3, add the two. Then put a zero down to the right on the next line as a place holder, and multiply the 5 times the 4, carry the two, 5 times 3 and add the 2.

Other places would approach it like this: 30 times 50 = 1500. Easy to remember 1500. 4 times 50, add to the 1500. Now you're up to 1700. 7 times 30, add to 1700 is 1910. Just have to remember 1910. Lastly, 4 times 7, and add that to the 1910 = 1938.

In the first algorithm, it takes multiple steps to arrive at the first number and multiple steps to arrive at a 2nd number before you add them together. In the latter algorithm, you're keeping a running total while still doing one multiplication at a time.

Cue about middle school age. The brain has developed to the point where it can handle more abstract concepts (like variables.) Now we're teaching Algebra, and it takes a couple days of teaching before the kids get good at multiplying something like (2x+3) and (4x-5). Kids in other places think, "shit, this is what I've been doing for 4 years. This is nothing new." This, in part, explains some of the differences on standardized testing. Through, I think it's 4th or 5th grade, U.S. children compare quite favorably, then they fall way behind.

-----

Also, fucking calculators. If you had a little robot that carried absolutely everything for you, you never even had to lift up a glass to your mouth to drink, in a couple of years, you'd be asking, "why is it so hard to carry a bag of groceries." Without practice, you can't do arithmetic very quickly. The more often you use a calculator for simple arithmetic operations, the slower your brain gets at doing that mental arithmetic.

Further, if you don't have a calculator, you start learning tricks. 87 times 93? Why, that's just 8100 minus 9. Why? Because it's (90-3)(90+3); the product of conjugates which is 90²-9. Or, in a calculus class, faced with finding the area under a curve using the midpoint method (midpoint of the base of a rectangle determines the x-value at which you determine f(x) for the height of a rectangle). It's fairly common on those problems to have to find 6.5², 7.5², 8.5², etc.
6.5 squared: 6 squared (36) plus 6 = 42. Tack on a .25 to the end: 42.25
7.5 squared: 7 squared (49) plus 7 = 56. Tack on a .25 to the end: 56.25
8.5 squared: 8 squared (64) plus 8 = 72. Tack on a .25 to the end: 72.25
At which point you might see a pattern. "Hey, it goes up by 14, then 16, I'll bet the next one is 18 higher." Though the pattern might be a little harder to discern if f(x) is something like 2x²+x+5; but no more difficult to do in your head.

Why it works? Again with the algebra. (x + 0.5)(x + 0.5) = x² + x + .25
Note: if you did (10x + 5)(10x + 5), you get 100x²+100x+25. So, the pattern is the same without the decimal.

As your mind learns more of these tricks, you look absolutely amazing to people who can't do mental arithmetic. You might say, "but you're doing..." Doesn't matter - it's still mental arithmetic, it's just that with a fluency in Algebra, you have a lot more algorithms at your disposal. And, with practice (not relying on a calculator from 4th grade on), you get quicker and quicker at identifying which tricks work on which problems.

Fun problem: I'll get a class to give me digits of a 9 digit number. Once I have a lot of kids shouting out numbers, it's easy to be selective for the 2nd number to make it 142857143. I can multiply any 9 digit number times 142857143 faster than the students can type it into a calculator (only to find out that the display on the calculator isn't large enough. So, when they get a rounded number in scientific notation, we have to turn to a computer to verify my answer. But it's simple: I multiply by 142857143 indirectly. First I multiply the 9 digit number by 1000000001 (which just makes it an 18 digit number with the first 9 digits repeating. So, if the first 9 digits that the kids give me are 234897561, then the new number is 234897561234897561. Then, divide by 7. Long division - in your head. (omg, it's soooo hard, since you haven't done long division in your head in 20 years.)
7 goes into 23 3 times (write down the 3, mentally carry the 2; 7 goes into 23... )


So, in summary, why is mental math so hard for you? Because you're a lazy, stupid American.

This is what I do and what I teach kids.

Do it like a computer in fast simple iterative steps.

99x99 becomes 100x99-99. 9900-99 becomes 9900-100+1.

If you want to be as fast as a machine, start acting like a machine. Lots of simple steps in parallel.

 

[Nebor]

Dr. Pizza's post is one of the best things I've read this week.