What is the integral part of the square root of 3x10^40? What is the product of 79532853 and 93758479? What are the first 2000 digits of Pi? These are all questions posed by mathematics. Some people cannot answer any of them. Others may be able to answer some of them with the help of a calculator. A select few can answer all three with the assistance of a calculating device. Believe it or not, there are some, though, who can not only answer all three questions without any assistance, but can do so accurately and relatively quickly. These individuals certainly would classify as genius, if not odd.
What actually separates those who can perform these feats from those who cannot? Is the genius a product of the ability, or vice-versa? Is the mental agility involved in the calculations a skill that can be learned, or is it an innate trait that only few will ever possess? Is this type of ability requisite for being successful at math, or does it have anything to do with math at all?
These amazing feats of memory and extraordinary mental powers have been exhibited by a select few well-educated mathematicians including Ampere, Hamilton, and Gauss, as well as a number of people who had no mathematical skills, little or no education, and yet were able to perform the same feats. Von Neumann, particularly, not only was a master at complex mental arithmetic, but had extraordinary mental capacity and displayed amazing feats of memory, as described by his colleague Herman Goldstine.
As far as I could tell, [he] was able on once reading a book or article to quote it back verbatim; Moreover he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English.
Only one mathematician, A.C. Aitken, has ever described in detail how he was able to perform his incredible feats of memory and calculating. His methods will be analyzed later in an attempt to answer some of the questions posed above. One thing that is known is that his “nearly perfect” memory played a large part in his ability to calculate.
In order to understand memory, it is first beneficial to discuss other elements related to memory. Intelligence is the capacity to learn. Each individual has a limitless capacity to store information. Learning is based on the acquisition of new knowledge about the environment. Memory is simply the retention of things learned. This model implies that anything that is learned is stored in our vast and endless memory bank. The problem most individuals face is recalling the information they have learned appropriately. Research has shown that memory can be improved through enhanced learning techniques and a variety of recall cues. Memorization is enhanced in the course of acquiring understanding. The more interest one has in a subject, the better one is able to later recall information about the subject.
Modern research has concluded that memories are stored electrochemically. The exchange of information takes place at the synapses. The more of these connections in the brain, the quicker and easier it is to store and transfer information. These connections are developed in the early years of childhood. Some evidence supports that the more young children are exposed to positive, intriguing inputs, impulses, and information, the more nerve cell connections are made in the brain. The extent or implications of these findings are arguable, but it is known that the first five years of life are critical to the development of the brain, although connections in the healthy brain continue to be made throughout life. In other words, someone who had a positive experience with math, showed early interest and ability, and was supported throughout, would have a better chance of developing into someone who could perform quicker calculations.
Researchers acknowledge two types of memory systems: locale and taxon. Locale memory is based upon natural learning or learning in context. For example, remembering last night’s dinner is very easy, although no formal thoughts occurred to remember the menu. It is based on schema, or mental maps. It is the memory that stores our life experiences and helps us become who we are. The items stored in locale memory exhibit a complex set of relationships among each other. Schema formation is motivated by novelty, curiosity, and expectations. Because it is intrinsically oriented, those who become curious about math at an early age, for whatever reason, begin learning using the locale memory. It becomes insufficient by itself, though, to produce the abilities described at the opening.
Taxon memory is based on the information-processing model. It is the area of the brain where lists, categories and numbers are stored, including the majority of mathematical tables and formulas. The most pervasive characteristic is that taxon memories must me rehearsed. Information stored here requires rigorous learning. It is more difficult for information to be stored in this matter, requiring long repetitions and rehearsals. However, once information is stored in the long-term taxon memory, it is very difficult remove or change and tends to last much longer.
The individuals mentioned earlier who have displayed the amazing feats of memory have attributed their skills to practice, practice, and practice. Zerah Colburn is interesting because his abilities diminished when he underwent education. This may be due to the simple fact that his abilities required continual practice for hours a day, which he was unable to afford with his studies. A demonstration of the permanence of taxon memory is illustrated by an anecdote from the life of A.C. Aiken. When he was proceeding to recite the second 500 digits of Pi, he uncharacteristically hesitated and sometimes corrected himself. When asked why he found the second 500 much harder than the first, he had an interesting answer.
Before the days of computing machines there was a kind of competition in seeing how fare they could calculate Pi. In 1873, Shanks carried this to 707 decimals; but it was not until 1948 that it was discovered that the last 18 of these were wrong. Now, in 1927 I had memorized those 707 digits . . . and naturally I was rather chagrined to find that I had memorized something erroneous. [Later], I re-memorized it, but I had to suppress my earlier memory of those erroneous digits.
Aiken’s problem was that he could not forget the incorrect 180 digits. (By the way, the current record for pi recitation is 100,000 digits, by Japan's Akira Haraguchi. a feat which took him 16 straight hours.)
It is best to study the methods for storing information and calculating through the methods used by Aiken, for he is the only one who could actually described some of the methods he employed. But first, it is appropriate to discuss general methods of improving memory and recall.
There are three requirements for information storage: registration, consolidation, and retrieval. Registration is the input of information from our environment received by our senses. Once we have registered the data, we must organize or consolidate it into either our short or long-term memories. After it is stored, we must be able to have access to it when we want it. There are several ways to improve the way we store information and retrieve it, ranging from mnemonic devices, chunking, association, rehearsal, and even eating certain foods.
It should not be surprising that the individuals who possess the amazing mental abilities share the characteristic that they actually ENJOY doing the calculations. This implies the cooperation of the locale and taxon memories. Each began developing their skills as either a hobby, pastime, or for enjoyment. The genetic predisposition they may have possessed for calculations would never have developed if they were not given the chance to practice. The question whether this “predisposition” for calculations is specifically math related or simply a learned respect for diligence and detail is unclear, Aiken himself offers some sort of explanation. “Familiarity with numbers acquired by innate faculty sharpened by assiduous practice does give insight into the profounder theorems of algebra and analysis.” Without the discipline, no ability would ever develop into talent.
One of the most commonly employed calculating strategies and memory aids is regrouping or chunking. It is believed that most human calculators use this technique to some extent. Regrouping involves the synthesis of numbers and rewriting sequence of operations in terms of equivalent expressions. Chunking requires the association of equal values and expressions taken together, rather than singularly. Aitken used these techniques to aid his calculations. He did not begin to develop this skill until age 15, when he would continually practice until he acquired enough taxon memories that these calculations became easier and easier. For example, when Aiken was presented with the number 1961, he instantly saw a variety of ways of expressing it.
1961 = 37 x 53 = 44^2 + 5^2 = 40^2 + 19^2
By doing this, he was able to use the property of the smaller numbers and combine them to achieve the desired result. When asked to multiply 123 by 456, Aiken provided the following explanation.
I see at once that 123 times 45 is 5535 and that 123 times 6 is 738. Then 5535 plus 738 gives 56088. Even at the moment registering 56088, I have checked it by dividing it by 8, so 7001, and this by 9 gives 779. I recognize 779 as 41 by 19. And 41 by 3 is 123, while 19 by 24 is 456. A check, you see; and it passes by in about one second.
These types of algorithms provide insight into the shortcuts involved in mental calculations. Most students of mathematics rarely develop a memory for such sophisticated tasks. Some more common mathematical memory aides are those used in number sense competitions, such as divisibility by multiples of 3, multiplying by eleven, or squaring powers of five.
It is interesting to note that most of the mental abilities associating with calculating and memorization relate to sound rather than sight. Many of the individuals capable of these feats have attributed their storage and recall as having a particular rhythm or cadence that helps them “keep the beat” as they perform these feats. Much attention and praise has been given to those individuals who seem to have a “photographic memory.” The truly astounding people claim that visualizing the information slows them down and prefer not to “see” the problem. Indeed, Aiken was slower when forced to visualize. When asked to recite Pi backwards, he was forced to visualize the numbers and recite them from his visual image. The speed was still impressive, though. The last 50 digits required 18 seconds to recite them forward using his audio cadence and 34 seconds to recite them backward using his mental image.
The conclusions that can be drawn from research and the explanations of such calculating wonders as Aiken offer insights into math, memory, and calculating ability. The pattern is sequential. The early years are the most crucial to the development of the brain and establishing the necessary neural connections for the best memories. Without these early experiences, it is doubtful if one would ever be able to reclaim the lost opportunity to produce the necessary connections. The interplay of both the locale and taxon memory is important for the continuation and pursuance of learning and development. The locale provides the initial interest and intrinsic motivation, the taxon provides the storage capacity and recall algorithms required for the amazing memory feats. Diligence and discipline are required to continually improve and practice calculating and memory-improving skills. Hard work and continual effort ensure that one stays sharp and able to recall quickly and effortlessly.
As these skills progress, one begins to gain insight into the complex workings of mathematics and develops a deeper understanding. He can begin to form his own specific memory aids and calculating algorithms, capable of making authentic contributions to the field, regardless of any formal training, using only his own immense mental abilities. Memorization and Calculation are both skills. Skills can be learned and forgotten depending on the amount of practice. The prolific mental calculators will always be few in number, but there is hope for all individuals who wish to increase their mental capacity. I cannot remember when I heard anything so hopeful and promising.
Perhaps with a bit of practice you can come across 1,588,533 and say, "Hello there, 589 times 2697!"
* Final Note on Aiken: His perfect memory was his glory as well as his demise. His horrific memories of his youth in the battle of Sommes, in which he fought, vividly haunted him all of his days, as he was unable to forget about them. His journals reveal that the bad memories contributed to his ill health near the end of his life and eventually led to his death. Oh, the curse of genius.
I realize I haven't blogged in a while, but perhaps that's a good thing. I haven't felt the need to rant or rave, or quietly speak my mind. Add to that the fact that I'm swamped at work, a result of taking too much of a break over spring break (YES, teachers have work outside the classroom.) Why, just yesterday (and early this morning) I had to create two new tests for my BC calculus and PreAP Precalculus classes. Because we have brand new textbooks in all math classes this year, everything I do is from scratch. I don't even have time to work the keys to the exams prior to running them off and administering them, instead, working the key when the first class takes the exam. This means I must work faster than my smartest student so that students are alerted of any typos and (not "or") errors on the exam. Today, I caught one on each test. A BC student had beaten me to the error near the end of their exam (albeit they started on the back page.) I was dully humbled.
