This article describes five world records of mental calculation and memorization to better understand working memory and image schemata from a cognitive perspective set by Paolo Fabiani. The purpose of the records is to demonstrate that the potential and limits of working memory are closely connected with the image schemes1.
On January 19, 2019, at a conference open to everyone, Paolo Fabiani gave a lecture on mental calculation and memorization. The conference took place in the public library of Rufina in the metropolitan city of Florence, Italy. Inside the conference, in front of an independent jury composed by public officials and experts, Paolo Fabiani performed successfully these five different mental calculation exercises: set time to mentally sum ten random binary numbers of seven digits each and give the result in a decimal number (18.45 seconds); set time to mentally sum seven random square numbers (10.93 seconds); set time to mentally calculate the average of five random square numbers (10.17 seconds); set time to mentally sum two random numbers of 30 digits each (calculus: 1.59 minute) doing calculations aloud, column by column and writing the result without looking at the addends; set time to mentally sum five random dice seen for only 0.1 (a tenth of a) second (3.06 seconds).
The tests were designed to highlight some specific features of image schemata. Some of these aspects are:
1) use specific mental images to facilitate the calculation of large numbers;
2) the conversion from binary numbers to decimals; count from bottom to top;
3) make arithmetic sums proceeding from left to right;
4) convert images into numbers and then “add” the images thinking about numbers.
We can considerably improve the performance in mental calculations by using the knowledge of the operating mechanisms of the schemes. Furthermore, we can note that these records consist in mentally performed arithmetic operations without any kind of material support or other form of help. They are made with numbers consisting of many random digits (up to thirty). It is not a matter of simple "mechanical" (rote) mental calculations, nor of pure memorization of numbers, but rather they are the integration of the calculation with the memorization of the partial results that are obtained column by column up to the solution of very complex arithmetic operations, much more complex than what has been done until now.
The purpose is to demonstrate that mnemonics strategies can be a valuable aid for mental calculation. Until now, they have been considered only as a subsidy to memorize theorems, principles, formulas, etc. Mental images are often considered as slowing down and weighting down of mathematical thought. With this experiment, I intend to show that it is not always true and that, if properly used, they can replace both the sheet of paper and the computer in basic arithmetic calculations. With these techniques I (but almost anyone could) can calculate sums made up of numbers of 30 digits each and, with a little of commitment, I could even go beyond.
The records that have been established in the field of mental calculation and in that of memorization are all founded on “procedural” memory. Obviously, not only, but especially procedural memory is involved. In particular, the speed of calculation and storage is attributable to certain cognitive processes designed to solve specific tasks very quickly. All that leads only to a very mechanical process of learning and calculation. My method is very different. To demonstrate the most important differences, in this paragraph I refer above all to the fourth attempt (mental calculation of the sum of two numbers of thirty digits each). However, nobody is able to add up such great numbers. Above all, in this task nobody is able to say out loud every single part of the calculation column by column. With my techniques and my record, I can demonstrate this too, that is I can redo (repeating out loud) every single step of the arithmetic operations while I am resolving them. This is perhaps the fundamental point: the interference. As already mentioned, the mental calculation records are based on procedural memory, which is partly true also for the records in the field of mnemonics. More generally, we can assert that both memory records and mental calculation records are based on procedural thinking rather than only memory and/or calculation. But again, my method and my records are different since they required a true memorization and a true calculation, not a rote one. In other words, they require – given the greatness of the numbers used – to have to go beyond the simple assimilation and repetition of mechanically produced thoughts.
My record requires that both “calculation and memorization” operations are performed at the same time. This necessarily implies that the calculation disturbs the memorization and that the storage slows down the calculation. Calculation and memorization interfere with each other and, above all, one breaks the chain of cognitive processes of the other. Being able to overcome this obstacle is itself a record and doing this with very large numbers is like establishing two records together. In other words, the validity of the method that I use to establish this record also allows to overcome the disturbance that the calculation makes to the memorization and viceversa. Indeed, in the calculation of huge numbers, these mental operations must necessarily be performed simultaneously and the processes of the one tend to block the execution of the other. It is important to repeat this concept since it is pivotal. Everybody can experience it. Try to add up each column of a long arithmetic sum and, at the same time, memorize the result of each of these. The calculation of the next deletes the result of the previous column from the memory. This is always true unless you stop the calculation and spend a lot of time memorizing. To do the calculation aloud is useful also to verify that the calculation doesn’t stop to make room for memorization. But, above all, doing calculations aloud also allows you to prove that I’m really doing calculations and I’m not using other techniques to simplify arithmetic operations. The tests I carry out involve writing the result without being able to look at the numbers I had to calculate. First, I perform the calculation and then, once the sheet (or the spreadsheet) on which the task is written is removed, I rewrite the result on another sheet (or on another spreadsheet). Otherwise said: while I am calculating, I am forbidden to write or perform any other action aimed at storing the calculation. Only after the numbers to calculate have been taken away from me, I can (and I must) write the result.
In all mental calculation performances I've seen, mental athletes write the results while watching the numbers (if the numbers are very long) to be calculated. I’m able to do this without watching the numbers. All mnemonists who remember large numbers never perform at the same time mental calculations on them. I can do that. I did it.