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