The Great Mental Calculators (Mental Math)

[PokerDad]

I'm starting this thread to serve as a quasi-book report  on the above mentioned title. Consider it a celebration of my 100th post.

Perhaps the biggest goal I've seen on Brillkids regarding math is to somehow give the gift of mental math to our children. I'm right there with you, and if this describes you, then this is the thread to read. On another thread, I was recommended The Great Mental Calculators by Steven B Smith. Portions of this book are discussed in Mathew Syed's bestseller Bounce.

The portion discussed in Syed's book has to do with a study done comparing mental math prodigies with cashiers at the Bon Marche. Effectively, the cashiers could do the same multiplication that the prodigies could do up through 3 digit by 3 digit multiplication. The best cashier answered the following question using mental math: 7286 x 5397, though it took the cashier 13 minutes! One prodigy took 21 minutes to do it, and another took only 2 minutes and 7 seconds. Syed's conclusion was that the prodigies spent loads of time working with numbers because they loved it. The cashiers spent loads of time because it was their job. In the end, all that mattered was how much practice they had. If I'm not mistaken, this challenge took place around 80-90 years ago, but I could be wrong about that. This is in chapter 8 of Mental Calculators.

I've currently finished chapter 9 and will update the thread as I move through the remainder of the book. These first chapters really only explain minutiae, and I mostly found it dry. The crux of the author's argument can be found early in the book on page 4, "The thesis of this book is that this ability is based upon the same faculty as that for speech." I'm not sure if he means that mental calculation uses the same exact process, especially structures in the brain, as speech, but more that the two phenomena function in like manner. There is a portion of the book where he discusses Broca, and therefore I'm quite confident he doesn't mean this literally.

The early portions of the book discuss some of the common characteristics of the mental math prodigies. Some that I can sum off the top of my head (without citing a specific chapter or paragraph), are that:

• prodigies learned math early; they learned about numbers in many cases before they learned their alphabet
• prodigies spent a lot of time by themselves; the time alone was spent doing mental math
• prodigies lacked distractions, or pretty much stuff that you'd find in a common home (such as television and toys)
• in some cases, prodigies actually became less proficient as they got older
• prodigies did not learn their skill in school, and most (but certainly NOT all) learned before going to school
• rarely did prodigies begin to excel in their early teens, but it did happen.
• almost every prodigy calculates from left to right.

I'm sure there is other content discussed in the first 8 chapters that you'd enjoy, but I just thought it was dry and fairly pedantic. One prodigy in particular, Jacques Inaudi, gave some strong opinions that I mentioned in another thread. Among them were that mental math was something anyone could do, and that the average person could get up to 3 digit by 3 digit multiplication without the use of memory assistance. I'm not sure if this opinion came from the Bon Marche study or not (where the cashier proved it was possible).

I just finished chapter 9, and decided to write up a summary thus far because the chapter was cogent and thought provoking. Smith begins the chapter, "... any normal child is a potential calculating prodigy, just as every normal child learns to speak. But the existence of a potential does not mean that it can be easily realized. Any 6 year old child is capable of acquiring a foreign language of his peers. On the other hand, a few hours a week of Spanish lessons given to English speaking grammar-school children typically lead to only the acquisition of some garbled mispronounced phrases -- the children do not in any sense become speakers of Spanish... What then are the prospects for converting an ordinary child into a calculating prodigy? I think that it could be done, but the requirements are such that I doubt that it will be"

That's perhaps the gist of the chapter. He summarizes his reasons for this disappointing conclusion throughout the remaining paragraphs. I'll bullet some of them

• Prodigies are rare because of the lack of motivation. There is motivation to learn to talk, there's lack of motivation to learn mental math
• Children are not surrounded by others that perform mental math
• Becoming proficient requires a lot of isolated time, which is counter-culture and perhaps even alienating
• It takes a lot of practice and a lot of time, neither of which the typical person is willing to do

Just from the short list above, you should already have an idea of what needs to happen to foster mental math acquisition. First, as a parent wanting it, you've gone further than almost everyone else because you have the desire to instill. Second, environment is pivotal if you're to imbue the love of numbers and math. This would include a peer group as well as adults (parents mostly). Finally, the isolation mentioned is due to the unstructured nature of mental math as learned and exhibited by the prodigies. These guys learned it themselves pretty much (though one of the main prodigies discussed at length in the book was the son of a mental math prodigy and said his father was far superior than he was). This last sentence reminded me that mental math seemed to be exclusively male, perhaps because males tend to think mathematically more in a general sense, but as discussed in the book, another variable is the isolated time.... very few parents would let their little daughter roam about town all by herself. These same stipulations weren't as strict with boys who were often left to tend to the sheep or something and only had their thoughts to occupy themselves.

George Bidder (I believe this is the guy with the father prodigy), gave recommendations for someone looking to teach mental math to their child. According to him...

• Numbers should be taught before their written figure (knowing numbers before you know how to write them, or read them!)
• "probably" should be taught before the alphabet
• Teach the child how to count to 10, then how to count to 100.
• Facilitate the child to construct their "own" multiplication table. I found this portion enlightening. Back then, the common toy was a marble, and so he suggested using that due to their affinity for it and associating those positive feelings with it. However, it can be anything. Creating their own multiplication table means, on the floor or desk or table, constructing a ___ x ___ rectangle of the item. For illustration, we'll say 5 x 5: you'd have 5 across and 5 up making a square of 25 marbles. You can then see the rectangle of other numbers multiplied. You can only look at the first row of 4, and see 5 high, and then count them. You can see 4 x 4, etc... this allows the child to discover for themself the meaning of "square" and why it's called that.
• Bidder recommend facilitating the child to create their own table up to 100 (marbles total that is, 10 x 10).
• Then teach how to count to 1,000 by 10s and then by 100s.
• Once this is done, 2 digit by 2 digit multiplication is easy (but he presumes you can add obviously)
• "by patience and constant practice... he would gradually be taught to multiply 2 figures by 2 figures" he says.

Then the author steps in with his own recommendations. he talks about factoring numbers at random, and mentions a guy that had an annoying habit (the prodigy's feelings, not mine) of seeing a license plate and factoring it... such as, "oh, that plate is #731, why, that's 17 times 43". I will point out that they don't automatically "know" this just by looking at a number, but rather by habit try to break the number down into smallest denominator type of... they arrive into knowing that it's 17 x 31 by dividing the number up in their head to get the simplest parts.

Most of the prodigies acquire mental math to the point of it being automatic, like language. In one sequence, the author talks about how ridiculous it is to ask a prodigy "how do you that?" when referring to a calculation because it's like asking someone "how did you just speak like that? how do you speak?" -- a person can not answer how it is we speak. We have knowledge or the words in our head and they just pop out.... that's a big portion of the early chapters is setting up how the author knows that they calculate and not memorize (of course eventually they'll acquire patterns over time, just like we know popular phrases and such).

The author, still in chapter 9, then discusses the mixed results of teaching mental math in regular schools. His point is that a school would spend maybe an hour doing this new method of mental math (that almost certainly clashes with the traditional pencil and paper algorithms taught), and that after an hour, the students aren't all that good at it... and therefore, people look to the experiment as a relative failure.

For anyone that has read Mathew Syed's book, or anyone thinking critically, you can spot the flaw in that logic. An hour on the basketball court or an hour behind the piano doesn't make you Michael Jordan or Wolfgang Mozart anymore than an hour doing mental math in school would make you a proficient mental calculator. It takes time and devotion to become good at anything! The author discusses some marvelous successes of teaching mental math in Samoa. "After seven weeks and 20 contact hours (the children were encouraged to practice on their own) they were able to multiply mentally up to 99 times 19 with 90 percent accuracy. The project was abandoned at that time because of other demands on the children's time."

That's where I'm at now. Sorry it's so long, but I am summarizing the first 9 chapters of a book (and in a scatter shot manner mind you) 

My impressions at this point (before he gets into methods) is that time and motivation are the two biggest variables. Efficiency of teaching and efficiency of method might be two other factors to consider. Schools such as Jones' Geniuses or any of the good Soroban schools would seem to fill the biggest hurdles nicely. A room full of kids learning and valuing mental math cannot be understated for its value. A proven method also cannot be undervalued. Lacking either of these is a huge reason why there aren't loads of adults out there performing mental math all the time... I can speak for myself, I reach for a calculator several times per day.

Will update the thread as I read more. Let me know what you think. I'll try to answer questions if the question is addressed in the book

[Mandabplus3]

Wow poker dad what a great summary. In truth I wanted to know but really didn't want to read the whole book  I have read Bounce loved the first half and thought the second half was a waste of my time.
So the beginning suggested ideas for creating a mental math genius seem quite attainable. That's a releif for many I am sure.
 Interestingly we are working on making our times tables here! Why? Well my oldest needs to know them instantly as part of this years schooling and my other two are fascinated by the big numbers  so we create them on a grid pattern so they learn them not just by rote but understand them too. (hence why I posted a while back looking for Little math/readers times tables files) Subitizing would do the same thing wouldn't it?  If you happened to be lucky enough to achieve it...
 Natalya seems to prefer to calculate left to right ( even though the school insists on her doing it right to left most of the time) so I would add that perhaps the reason they need to be good at math earlier is because the training they get in school could ruin their chances! She is no math genius, but has made dramatic improvements as MY knowledge increases, which allows us to work at her level and work in a way we can build on easily.   For parents treading this path, left to right calculations require a strong sence of place value and no fear of using big numbers. So talk in hundreds and thousands early in life. Show big numbers often.
You mentioned that they assume kids can add at one point. I wonder if kids can count very well, by ones, by tens, by 100 and do it quite naturally, would they consider it adding or would they consider it to be just counting on?

[PokerDad]

A quick update on the book:
Frankly, I don't think this is a book for me. Perhaps when I was immersed in mathematics all day, I would have enjoyed the nitty gritty of mental math calculations, but now, I just find it over the top. Personal opinion aside... I've gone through the chapters on how the prodigies did their mental math.

It seems obvious in retrospect, but I found it enlightening to understand why we're taught to multiply right to left and why the prodigies will pretty much always perform their calculations the other way around. In the early days of education, there was a paucity of supplies. Conservation was a key requirement. To save on paper and ink, calculating right to left became the norm. It really reduces the number of steps (or intermediate answers to be precise); what this does, however, is put the intermediate answers out of context for many children learning cross multiplication (and by derivation long division).

The mental calculator, without the assistance of a written record, must rely upon his (sorry for the gender usage, these prodigies are nearly 100% male) memory as he goes through the calculation. If you have a problem such as 383 x 279, calculating 300 x 200 is really easy and is really only 1 bit of memory among the theoretical 7 +/- 2 digits of information that a typical short term memory can handle. In adding up the number as he goes, he keeps a running total. You will see that the total is only 6 digits, well within the typical memory limitations.

To summarize the point succinctly, the written algorithm is designed to streamline the process and minimize ink and paper usage as well as minimize the steps involved in solving a mathematical problem. The person without paper and pencil can only use their memory which has a limited capacity, and so they're more than willing to trade off number of steps (or intermediate answers) in favor of being able to achieve the task.

I never really had thought about it that way.

While I'm not going to walk through how these guys calculate cubed roots of 10 digit (or more) numbers, I will comment on them. Like Richard Feynman said in his book, "these guys don't know numbers" (when talking about the soroban user). Well, I read Feynman's entire book (and I haven't enjoyed a book similar to that, ever, in my life) and this is a bit out of context. It's in the chapter where he started spending loads of time doing mental math just for giggles - and then would challenge people to ask him a question that he couldn't get. In so doing, and with his life long tenure as a top tier physicist, this guy KNEW numbers. I don't know how else to say it, he really really knew numbers. Well, these prodigies are all similar.

As mentioned in my OP, these guys can see a number such as 7551 (or any other random number) and immediately factor it out. In so doing, they can tell you whether it's a prime number or not, whether it's a sum of squares, what it's square, cube, or even larger root is... some of this is mental shortcut/tricks that have been discovered or learned, BUT these guys are so good that when their mental short cuts fail in the middle of a calculation, they can navigate their way out. To do this means that they KNOW what they are doing and have an incredible grasp for numbers beyond which I can relate to or understand.

In my 7551 example, they'd know it's divisible by 3, which is about as far as I can get. They can then take 3 x 2517 and figure out that the square root is around 50 and then deduce if there's any other factoring that can be done OR if it's a sum of squares (which I don't think I could do in a million years without significant training). They can do this within 2 or 3 seconds typically.

One thing I've noticed is that in doing these mental calculations, the prodigies are really good with place value.

I've now entered the third section of the book which appears to be some sort of mini-biographies of the prodigies. The first section was a summary of them, the second detailed their methods, and now the third.

I'm doubtful that there will be too much for me to add in this thread.

Okay, now for Mandabplus3's post.... the prodigy will say that ALL of math comes from multiplication in a sense. I think they view adding or subtracting as nothing more than counting up or down in a big chunk. What seems to attract them to multiplication is THE SERIES. It took me a bit to understand that COUNTING by 2s is actually moving you through the multiples of 2. Same with counting by any other number (5, 10, 100, etc). The only thing we don't normally do when we count by twos or whatever is keep track of how many we've done in the series. I think these guys perhaps do (but it's not mentioned).

Bidder's idea to have the child create their own times table was just so that they could SEE the math involved and understand the SHAPE of it. It becomes multi-sensory and in creating their own they can use objects that they have an affinity for (such as marbles for the kid that likes to play with marbles). Bidder couldn't think of how to do it, but once you go cubic functions, you build a height on top of the times table - nowadays there are manipulatives that can achieve this.

Because the prodigy gets good at counting by whatever number by doing a series, they develop place value and a good sense of numbers. In a typical calculation, they can tell you instantly how many digits the final answer will have - and so much of this is from their practice practice practice in everything number oriented.

Okay, so... having read as far as I have, I'm a bit overwhelmed of the thought of actually creating or fostering such a view of the world. It's no small undertaking to train a mental math child.... if done left to your devices that is. If anything, I appreciate the duplicable nature of some of these programs that have emerged since the writing of the book. My recommendation to a parent looking to bring their child up as someone that can perform mental math is to 1. acquire the skill yourself so that you can teach it and nurture the skill or 2. find someone that CAN perform mental math to serve as a mentor.
That's not to say that without a model there won't be success... it's just that it wouldn't be all too likely to randomly have a child that can do this stuff when you're actually trying NOT to have it happen randomly (in other words, a system, a love, etc must be imparted)

[Maquenzie]

Oooh!  Thanks!  I think this will be my next book.

I'm absolutely not (and never was) a math prodigy, but I was socially isolated in elementary school (being a "gifted trouble-maker" with ADHD) so I had loads of time to myself AND was never taught the standard algorithms either because I couldn't pay attention that long, didn't want to, or physically was not allowed to view the chalkboard.  So I taught myself, and enjoyed it.  And I did all of my work from left to right.  I still cannot fathom why subtraction is not taught this way...it's significantly easier and faster.  But I got in loads of trouble for it.

I was on math team in high school and had a personal ban on calculators until I hit Calculus.  I brought a slide ruler to school...in the 90s.  I eventually dropped out of math team because I could not compete with the coached kids.  I HATED that, it made me feel like a failure.  I truly believe I had a deeper understanding of math than most of the peers I competed with, but I didn't have their training. (and I usually didn't memorize all the "tricks" either, because I like the puzzles...this made me slower).  I'm also a person who does mental math with all the numbers I see.  I get figidity waiting for my next number when I drive alone.  :P  But I certainly cannot do some of what you described that quickly.

Anyway, if and when my kids have the "fire", I want to ignite it and help them with some of the early coaching I never got. We play math a LOT at home. 

So, were Montessori methods ever mentioned in this book?  The marbles sound like Montessori bead chains and squares (and cubes...Maria Montessori solved that problem over 100 years ago). And this is currently being taught to preschoolers all around the globe.  My son came home earlier this week having memorized 3 squared, 4 squared, and 6 squared, and knew several multiplication facts in those families.  He's in preschool, not yet kindergarten, and this was seen as a completely normal event.   The next steps in that sequence (counting by 10's and 100's to 1000) is not a common Montessori work...at least not that I'm aware of, I am not trained in the philosophy.

I've been thinking a lot about how I'd design a math program from birth up if I could.  I was actually coming on to post some of those ideas.  I had begin to wonder if flashing patterns to babies would be more useful than working on subitizing....everything from simple ones like the numbers on dice and dominoes to larger numbers lumped together in groups, utilizing pictures of materials you'd use with the child as their fine motor skills develop. I'm not convinced that working on subitizing is all that useful...but working on chunking might be VERY useful.  (and I say that as someone who purchased and used Little Math with my youngest, only beginning at 11 months). 

That was probably long and babbly and a little off-topic, but I'm interested in checking this book out as well.  Thank you for sharing.

[Mandabplus3]

Mackenzie I have yet to be convinced on subtilizing but I haven't seen a kid actually do anything with the ability... I kind of think it may not be worth the time if the skill is lost so soon. But if they can learn multiplication and squares and cubes while still practicing subitizing then I would change my view. If they could actually mentally visualize what 8x9 looks like then that would be worth it. I just feel that lots of parents can teach subitizing but not many go much further than that...oh that reminds me of another post i commented on about early crawling anyway....
So Pokerdad, I am surprised that after reading this you are overwhelmed about going ahead with teaching or creating a mental math prodigy. After reading your post, (and combining it with my teaching knowledge I suppose) I was enthusiastic about the simplicity and possibility actually of it all being within the reach of most!.
So what am I missing? Or have you gotten bogged down in the details perhaps?
So basically the start line is simple and easily achievable by all the brillkids mums.
First teach counting ( with or without subitizing) lots of counting. Up and back.
Then skip counting by all numbers. Skip count by 2s, 3,4,5-9, 10,100,1000...
Then learn times tables visually. Build your times tables until they are memorized.
Teach addition and subtraction from left to right (before the kids get to school!!!!!)
My kids learnt it like this
 23 + 45
20 + 3 + 40 + 5
60+ 8
68
I think this is a simple way to get the idea of left to right addition in a young child's head as it clearly expands on place value, but its not the end way you want them to do it. Eventually you want them to go strait to 20+40 and 3+5
I will mention again, and again and again ( yes poker dad I saw you mentioned it too) place value is very important to left to right addition, subtraction, all multiplication and division. So teach that til the cows come home!
 I see the possibilities as quite open to us all. Perhaps the level achieved may be a bit lower than prodigy without professional training, but I still think mental calculating ability on a reduced level is a worthwhile skill to chase. After all I am not teaching my kids to be the best readers in the world just very good will keep me happy

[arvi]

Wow, a wonderful post, thanks for sharing PokerDad! I was eagerly waiting for this post of yours as I have no access to that book. I have few questions:
1. Has any of those mental math experts used any standard mental math methods like Soroban or Trachtenberg or any other system.
2. I am also interested to know about the mental math methods discussed in the book. I know it might be overwhelming to skim through all. But can you just give us a basic strategy.

Thanks once again.

[PokerDad]

Mandabplus3,

I agree that setting up the foundation is easily within reach of the average parent, and probably within the reach of a below average parent 

I guess what I was getting to was once I started reading the details of some of these calculations, I just had an overwhelming feeling. The people highlighted in the book are often from 100 years or so ago. There are few that might still be alive today (though I haven't started reading the biographies, which will give me a really good grasp of some of this). Being able to take a 10th root of a 30 digit number just seems overwhelming to me.... or geez, I can't recall off the top of my head, but one guy practiced a bunch and went from taking 20 minutes to do a problem about that difficult to taking a minute to do something crazy like a 30th root from a 100 digit number! (Or something mind blowing like that, I'm probably not accurate in the exacts here)

Having said that, the basics can give anyone some simple mental math, just maybe not the mental ability to go all the way through school using only mental math. Perhaps I'm wrong about that. Part of my concern is the lifestyle or environment if you will; mental math ought to be loved, etc, if it's going to be fostered. That could be difficult if the parent can't do it, too. But, I could be wrong about that.

I think the main point I want to emphasize is that it will take PRACTICE, and lots of it. If the parent is into it and that enthusiasm shines through to the child, then yep, it will be a piece of cake! Practice, practice, practice!

And I guess I was sort of responding to Arvi's concerns about math; looking for a method that that instills the love and KNOW of numbers. I don't believe in myself enough to believe I could create the sort of "KNOW" that these prodigies exhibit. It's really WOW.... but, BUT.... I did recently find Scott Flansburg's Mega Math at the Goodwill and bought it for like $3, ha ha ha... I don't know how extensive it will be, but I mentioned previously that I read one of his books back in the 90s. Flansburg is NOT discussed in this book, probably because he didn't really show up until well after the book was written.

For that reason, I haven't YET read about any system that is geared for mental math. Soroban and Anzan have been around a long time, but with their Eastern roots, perhaps the author of the book never thought to include some of the highest ranked anzan masters in his book.... really, I never heard of anzan before and I think there was a recent author that brought the concept to the USA in one his books just within the last few years really.... so the typical person on the street in the USA has never heard of it before. Maybe some of them remember seeing Flansburg on an infomercial in the 90s... but I digress, sorry for that.

Arvi, I will copy down some of the methods. They're a bit tedious for a skim or light read. I found the root extraction fascinating! So, I'll try to write up some on that and then some multiplication and division.
Multiplication will be fairly straight forward (you already know the steps, it's a question of what order to do them in)... division I really skimmed over quickly, the root stuff, like I said, was really interesting, so I'll make sure to write up that.

[Mandabplus3]

So pokerdad and Arvi, the end question becomes do you want your children to be able to find the 30th root of a 100 digit number in 1 minute or less? How high are you aiming?
I would be happy with mental math ability up to the end of high school...that would be great. I don't think it's all that hard to achieve. Like you say though I am the one who has to learn it to teach it...I doubt there is a class near here..or even a tutor who can do mantal math at that level. How many of our population can?
 My kids seem to enjoy doing math problems, like I have said they aren't all that great at math, but that fault is clearly mine.  I havnt enjoyed teaching numbers until recently when the kids got to a level where I had to start thinking.  so the groundwork was rocky. Now, knowing they enjoy it I am more enthusiastic to teach it.
I think I had better read that book by Bill Handley..it could be the answer or it could be just another hyped up math program. I did see a video on the program used in schools that made me go buy the book though. Will let you know what it's like.

[PokerDad]

"Calculators have always preferred extracting cube roots to square roots, the reason being that cube roots are more impressive because  a higher power is involved, and cube roots are easier to do, since the last digit of the power unambiguously determines the last digit of the root (not true of squares)...
Root/Cube
0/0
1/1
2/8
3/7
4/4
5/5
6/6
7/3
8/2
9/9"

You'll see in the chart that in cubes, there are certain digits that are the same: 0, 1, 4, 5, 6, and 9. The others have "friends" - a concept I learned about in researching soroban use. The chapter goes on to talk about the need to memorize the cubes of each digit 1-9. In so doing, you can derive the first digit when figuring out a cubed root.

"no great sophistication or computing skill is required to find cube roots of perfect cubes less than a million"

So effectively, the calculator uses some short cuts and trickery to obtain the seemingly impressive answer (at least perfect cube roots of 7 digits or less). I don't want to type out the whole chapter about this, but it did fascinate me. They did talk about some larger rooting and non-perfect rooting. They also talk about taking a square root to many digits.

"let x be the number whose square root is sought. Choose the first approximation the integer (call this y) nearest the square root of x. Then x/y is another approximation. Take the mean of the two:
(x+y^2)/2y = z
now z becomes the new approximation and is substituted for y in the formula to get a still better approximation"
one of the prodigies goes into detail about getting the square root of 51:
"An expert would know very well that 7.1414141414 is 707/99 and dividing 51 by this we have 5049/707, easily accomplished by dividing 101 first, yielding 49.99009900... and then by 7, so that we have 7.141442715700 and the mean of this and 7.1414141414 is 7.14142842857 to 12 digits where as the true value of the square root of 51 is 7.14142842842854 to that degree of accuracy"

most of the book goes on and on about minutiae like this; very important stuff if you're to learn the strategies, but as a curious looker as I am, it makes for a garrulous read. The roots was the most interesting chapter for me in the strategies department.

Here's a sample multiplication problem in the book. As I mentioned earlier, it's really a function of the ORDER you do the steps; you should already know the steps from your basic math classes:
348 x 461 would be computed as follows (pg 110)
300 x 400 = 120,000
300 x 60 =  18,000
sum = 138,000 <-- this is the number we'll tally as we go.
300 x 1 = 300
sum = 138,300
40 x 400 = 16,000
sum = 154,300
40 x 60 = 2,400
sum = 156,700
40 x 1 = 40
sum = 156,740
8 x 400 = 3,200
sum = 159,940
8 x 60 = 480
sum = 160,420
8 x 1 = 8
sum total = 160,428

There's also a shortcut discussed in the following chapter using algebraic equivalents
ab = (a + c)b - cb
ab = (a - c)b + cb
where c is a relatively small number, and a + c or a - c are easy numbers to multiply.
For example, Bidder would take a problem such as 173 x 397 and would just do 173 x 400 and then subtract 3 x 173; this saves a few steps than that drawn out example above.

I doubt that I wrote anything here earth shattering, or what you may not have known otherwise. I did think it was interesting reading how some of these guys learned how to do sum of squares, prime numbers, or huge root problems.
And perhaps I misspoke when I said they'd do some of this stuff in a few seconds; often the problems they did were really large and took several hours to compute in their heads!

[Dr Miles R Jones]

PokerDad,

This is a fascinating thread!  I have studied Smith's book in detail but have never read Matthew Syed's book Bounce.  I will have to get it.  Thanks for the reference.

 I teach rapid mental mathematics for kids of all ages in our Jones Geniuses materials and I have accomplished that by studying many of the methods of the great mental calculators and simplified them so that even young children can learn them.  You may not want to teach your child to do the 10th root of 30 digit numbers but deriving the cube roots of numbers up to one million is quite simple, square roots are only a bit more complex to derive. Everyone should know this as powers and roots are the core of higher math and those used the most are squares & cubes.  So don't worry about the esoteric heights of mental math calculation but don't ignore the practical levels that can be scaled rather easily.  Usually these mental calculation methods will not be taught in school which seems wedded to paper, pencil, and calculators.  Typical long multiplication is right to left so you do not have to retrace your steps and change numbers when sums are carried over.  Mental math starts to the left with the biggest product in a multiplication then adds the smaller products to it one by one keeping a running total in their head while doing the next multiplication.  It takes a little concentration but anyone can do it. Concentration is perhaps the most valuable skill we can teach.   Everyone should be able to do 2 digit by 2 digit multiplication mentally.  You came up with a great observation that all mental calculators proceed left to right.  Here were your excellent observations:

prodigies learned math early; they learned about numbers in many cases before they learned their alphabet
prodigies spent a lot of time by themselves; the time alone was spent doing mental math
prodigies lacked distractions, or pretty much stuff that you'd find in a common home (such as television and toys)
in some cases, prodigies actually became less proficient as they got older
prodigies did not learn their skill in school, and most (but certainly NOT all) learned before going to school
rarely did prodigies begin to excel in their early teens, but it did happen.
almost every prodigy calculates from left to right.

Most calculating prodigies did indeed start early, pre-school age.   It is not essential to learn numbers first before the alphabet.  However, many famous stage calculators were illiterate shepherds like Mondeux, Mangiamele, Pierini, and Inaudi.  Yes, clearly their massive amounts of free time plus the need to constantly count the animals in their charge stimulated some of them to a fascination with numbers.  Bidder was taught by his brother putting pellet shot arranged into squares to practice multiplication which is why he recommends it.  It is a good technique but not essential.  Prodigies tend to lose their skills, much of which is subconscious, when they begin to attend school and all their time is taken up with mundane tasks in literacy and mathematics that are far beneath them but occupy all their time so they regress in their skills quite dramatically over time.  This happens in reading as well as math so beware of putting your child into the public school mill unless you want them homogenized. Idiot savants, usually autistics, simply do mental math all the time because they find it comforting.  Anyone who does mental math a lot will get good at it. 

For every child who develops these skills spontaneously, thousands if not millions, can develop rapid mental calculation when taught the skills competently.  After Mozart's father published a book on how he developed his son's musical ability there suddenly appeared hundreds of musical prodigies worldwide.  George Bidder did not have a father who was a prodigy but he did go on to pass his abilities down to his children and grand children.  There is a meme out there (ala Doman) that only children can do these things but this is quite untrue.  There are notable cases of adults like Millie Osaka and Maurice Dagbert who learned prodigious math calculation skills as adults.  Plus, think about it, many geniuses like Richard Feynman are the product of a gradual acquisition of skill over a lifetime.

The point is that current math education and materials are typically so lacking that calculation skills that are essential cannot even be taught well, like the times tables and fractions.  Lack of skill in fractions is mainly caused by a lack of underlying skill in factoring.  Factoring is called the Fundamental Principle of Mathematics it is so important.  It is quite easy with only a little practice and knowledge (you must know the factors and primes of all numbers to 100) to do seemingly amazing feats.

PokerDad, you mentioned the number 7551 in your factoring discourse.  It took me about a minute to work it out.  7551 is divisible by 3 since all the digits add up to a multiple of 3, so we divide it by 3 to get 3 x 2517 which is also divisible by 3, so we now have 3 x 3 x 839.  839 is a prime number.  I know this because we teach our kids to memorize all the primes to 1000.  It is crucial to know high primes (at least to 1000) in calculation.  For example, if this were a fraction denominator you would be stuck there not knowing if it could be simplified further.  With memory training it is not a difficult task to memorize the primes to 1000, takes an hour or two.  We use picture mnemonics, very simple once you know the basics of memory training.  If you had asked to factor a much bigger number with zeros like 7,551,000 it would have been just as easy:  2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 839  (each zero is a ten - its factors are 2 x 5 so with 3 zeros add three 2s & 5s).

Why do this when we have calculators?  That is the question really.   Number one, there is nothing wrong with using calculators.   I just recommend using them to expand the mind rather than replace it.  Neurophysiological development is dependent upon doing pushups for the brain.  Mathematical ability forms a key part of the structure of thought, our ability to comprehend the world we live in (how many people don't much recognize the exponential difference between the government spending a million dollars or a billion dollars versus a trillion dollars).  It is all just fuzzy zeros.  Brain research is quite explicit that exercising our, and our children's, minds mathematically makes them smarter.  Number two, doing metal math up to a certain level is much faster than being calculator dependent and builds number sense that prevents errors.  Those who can do mental calculation (for example of SAT type problems) are typically 3 to 5 times faster that those with calculators!  This is the difference between the engineer (or carpet salesman) who can think on his feet, do the calculation in his head and see right through to the solution to the problem versus the guy who has to go crunch the numbers for half a day.  Time is money. The defining qualities of a professional are SAS - Speed, Accuracy, and Skill.   That is why you pay the accountant $100 an hour rather than hire someone at minimum wage to do your books. 

Anyway, do not despair!  There are many simple, effective mental calculations that can and should be taught to great effect. And they should be taught far earlier than they are.  Unfortunately you won't fine them in math textbooks.  (Never say never, but rarely)   Our kids will spend massive amounts of time learning arithmetic, algebra, geometry just for starters;  might as well make it easy and enjoyable for them.  Kids tend to like doing things they can do well and tend to not like doing things they cannot do well.  Sort of like us adults.  Once they get fascinated with a subject like mathematics they may choose to fly very high indeed.  We need people who can do that. 

 

[PokerDad]

Great post Dr. Jones. Thank you.
And for what it's worth, going through this book really made me appreciate your courses that I've seen referenced on old news reports posted on youtube.

[Dr Miles R Jones]

Thanks.  You must be a speed reader.  You replied to the post almost as soon as I put it up.

What really excites me about this work is the synergistic effect of using good techniques from various fields like rapid mental calculation, speed reading, and memory training.   The amazing thing is that teaching these skills takes less time than an ordinary curriculum would take.  The time spent teaching rapid learning is more than made up for because a child learns, say, the times tables in a few weeks instead of a few years giving them ample time to go on to more advanced levels of mathematics and/or expand their interests into other fields. 

It find it very stimulating all the things Brillkids parents have worked with and I often follow up on their posts to buy books or check out links or methods they recommend.  Never fail to learn something new.  I wish public education displayed that kind of dynamism.

[Mandabplus3]

So I have spend a few days practicing calculating in my head and yep it's true....you do get better with practice I have a long way to go    I am still on times not even close to square and cube yet

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