Mental Calculation Skills
Mental Calculation
What Is Mental Calculation
Mental calculation is a compromise between “mental arithmetic” — a common, competitive extracurricular activity in many Asian countries, especially Japan — and “competition math.” While “mental arithmetic” typically emphasizes the development of skills and speed in computationally simple questions (addition/subtraction, A x B, A / B), “mental calculation” expands the types of questions to include more complex formats — for example, roots and exponents, multi-step, multioperation questions, factoring numbers, solving systems of equations etc. Consequently, quantitative and logical reasoning — in addition to arithmetical ability — are tested in mental calculation competitions.
Mental calculation is designed to be more representative of “real world” numeracy. In Calculation League, the formats are deliberately messy and unpredictable. Additionally, many of the questions have a non-integer or irrational answer and the emphasis is on ability to estimate the answer to a certain required accuracy. While many questions in Calculation League do not have a rational answer, or would be impossible for a human to precisely answer in a short period of time, the ability to rapidly estimate the answer to within 1% (or 0.1/0.01%) of the exact answer requires developing practical “number sense” skills, using creative quantitative reasoning, and being comfortable applying a variety of mathematical principles.
Definition of Mental Arithmetic
GMCA defines the term “mental arithmetic” as an activity limited to the following traditional, computationally simple questions formats:
-Addition of a series of numbers (possibly combined with subtraction);
-Multiplication of two numbers (A x B);
-The division of one number into another, generally without a remainder (A / B).
While “mental” arithmetic of course contemplates solving the questions without the use of any devices, in many communities around the world the use of a physical device (such as a soroban or other form of abacus) is commonly used with younger students while they are developing their skills. For many, this spatial representation of numbers is a valuable tool for developing elementary number sense that will later be critical when doing arithmetic mentally.
Approaches to Mental Arithmetic
When practicing mental arithmetic, one of the first decisions a student has to make is to determine whether to emphasize becoming competent at a variety of approaches or the mastery of a single, specific approach. It is important to consider the objectives and goals prior to determining
The Skills of Mental Calculation
Mental calculation — as defined by GMCA and as practiced in Calculation League — requires the development of a variety of different abilities and traits. Some of these relate to overall cognitive ability and would be translatable to non-quantitative fields. Others relate to mathematics and would be translatable to school mathematics. The following is an overview of the primary attributes tested and trained in mental calculation:
Information Processing Speed:
Information processing speed is the most general — and probably least significant — skill emphasized in mental calculation. Information processing speed simply refers to how quickly someone can process the information presented. The form of the presentation would be visual since oral presentation is not going to occur fast enough to strain a typical person’s information processing speed.
There is one common form of mental arithmetic where information processing speed plays a large role — Flash Anzan. In Flash Anzan, numbers are flashed on the screen at a defined interval and the user has to add the numbers. When done at the highest level, the numbers are flashed fast enough that even processing what numbers were flashed becomes difficult for untrained individuals. Here is a video from Japanese super prodigy Rinne Tsujikubo demonstrating the addition of fifteen three-digit numbers:
Information processing speed is distinct from — but closely related to — reaction time and the time it takes to provide the answer (typically typing or writing speed). Simple or easy “rapid fire” questions are unlikely to be meaningfully beneficial to the improvement of information processing speed because the limiting factor will be how fast the answer can be provided, rather than how fast the question is actually processed.
Information processing speed can be improved in a variety of ways — even playing fast-paced video games is an effective way to improve information processing speed. In mental arithmetic, however, Flash Anzan is the traditional activity linked to information processing speed. For those beginning with mental calculation, simply having the numbers flash on the screen — without trying to add them — and then attempting to recall them afterwards would be a helpful way to emphasize information processing speed.
You can try GMCA’s Flash Anzan simulator, select a fast speed, and begin with a small number of one-digit numbers, gradually increasing the count of the numbers or the digits per number.
Ultimately though, information processing speed does not play a significant role in mental calculation and, unless your information processing speed is significantly below average, it should not be a meaningful detriment to your mental calculation performance.
Algorithimic Processing Speed:
Algorithmic processing speed refers to the speed at which one applies a known process to solve the question. Essentially the terms corresponds to the question “how fast can you solve the question once you know the most efficient way to solve it?”.
The speed at which a known process can be improved can be dramatically improved with practice. One analogy is what I am doing right now — typing. Even if you knew where all the keys on a keyboard were, if you had never typed a sentence before, it would be a difficult task. With practice, however, typing speed can easily be dramatically improved, only limited by the need to physically move the fingers.
In mental calculation, the limitation imposed by the brain’s mental processing mechanisms is far less. In fact, in many situations, it is the limiting factor of moving the fingers (i.e. typing, or writing) that controls the speed of the answer. Neuroscience research has suggested that mental calculation may actually be the fastest “high level” mental activity, although this observation is more appropriately an observation of mental arithmetic than mental calculation.
In mental calculation generally, the objective is to break down the question into manageable steps. For common, or standard, question formats (especially those used in mental arithmetic) repetitive practice of a fixed format can lead to the implementation of a specific process occurring at an incredibly fast rate. For example, here is a video from Japan’s prestigious Meijin competition showing Rinne Tsujikubo and Takuya Yakami solving basic operations at an incredible speed:
It is important to note that algorithmic processing speed is only a secondary attribute in mental calculation. Certainly, many simple formats are used often in mental calculation and mastering a specific approach can be highly advantageous. For example, most experienced competitors will have predetermined approaches that they will use for basic multiplication (A x B) that will only depend on the number of digits in A and B. In this simple format, even if an alternative approach may be more efficient for a specific question, it is likely that the time it would take to identify that observation will exceed the time saved by the more efficient approach. This is especially true for highly experienced competitors who may only be limited by typing or writing speed, and thus unable to take advantage of a theoretically more efficient approach.
GMCA recommends avoiding placing too much emphasis on algorithmic processing speed when beginning mental calculation practice — especially for younger students. There are a few reasons for this:
(1) The most preferable methods when starting out may not be the most efficient methods later. When a student learns the most effective approach for beginners — or quick tips — they will likely have to unlearn those techniques after building skill or their potential will be limited. As a result, it is advisable that students learn a variety of approaches to specific questions or that they focus on an approach that would be appropriate at the level they want to get to rather than at the level they currently are at.
(2) A focus on algorithmic processing speed can create the illusion of progress and lead to a narrow, obsessive focus. Continually practicing the same task using the same process will naturally lead to some progress — which is exciting! — but not necessarily indicative of building overall skill. Younger students’ minds are gradually developing, getting stronger and faster, naturally. Apart from that, muscle memory, rather than overall skill, is gradually strengthened (see below).
(3) Algorithmic processing speed is highly muscle memory dependent and often inflexible. Many competitors who focus on algorithmic processing speed often see significant decreases in ability when the format is only modestly changed from what they accustomed to.
A couple analogies may help illustrate this last point. For older competitors who drive a car, one may have the experience of navigating a common route — home from work or school — without really even thinking about it. After repeated exposure, the mind is used to the series of turns. This process is quite analogous to an experienced competitor practicing an algorithmic task — there is no thought left, and the process is applied rapidly and subconsciously.
But what happens when there is construction or a road is blocked? Even the smallest change is likely to knock the driver out of auto-pilot mode and force them to consciously think about the drive. In Calculation League, we have created variety for the most standard tasks — making the multiplications and divisions unbalanced and varied in size, unpredictably mixing subtractions in with the additions — so that competitors are unable to rely solely on algorithmic processing speed in approaching the questions.
Reasoning Speed or Quantitative Reasoning:
In the form of mental calculation used in Calculation League, “reasoning speed” is a critical component of the required skill. “Reasoning speed” is used here to denote the ability to rapidly select an efficient approach to the question when there are a variety of possible approaches to the question. Reasoning speed in this context could also be referred to as “quantitative reasoning” or conceptualized as a combination of “number sense” and “logical reasoning.”
There are three primary — and distinct — areas where quantitative reasoning plays a role in Calculation League:
(1) Choosing between different types of approach that can be used to solve a single question format;
(2) Choosing the order in which to complete operations or steps; and
(3) Understanding how the required accuracy affects the approach to the question —or the degree to which calculations need to be performed.
The above illustrated is illustrated most clearly by Calculation League’s multioperation format, a format that can look like this:
A@B@C@D@E@F@G@H,
Where A through H are randomly generated numbers and @ is any basic operation (addition, subtraction, multiplication, or division). Let’s randomly fill in the equation with integers:
Example:
96@41@85@83@36@92@67@43, and with operations:
((96*41)+85-83+36+92-67)/43
Here, I note that because of the predominance of addition and subtraction, the question is relatively easily — to answer to the nearest integer.
I would complete the simple additions/subtractions — resulting in 63 first, and turning the question into:
((96*41) + 63) / 43
I could perform the calculations and solve the question, but I don’t really need to. Instead, I can see 63/43 as slightly less than 1.5.
Then for (96*41)/43 I note that this is equal to
96 (41/43), or
96 (20.5/21.5) or
96 - 96 *(1/21.5) or
96 - (96/21.5)
In estimating 96/21.5, I note that this is slightly less than 4.5.
Therefore, 96 - (96/21.5) is slightly more than 91.5
Adding in 63/43, I get an estimated answer of 93. In fact, the answer happens to be exactly 93.
In solving these questions, I have to make a quick determination regarding:
(A) The extent to reduce the question using common factors, if applicable;
(B) The extent to use estimations rather than exact calculations; and
(C) The order in which the calculations should be performed.
The GMCA Guides related to multioperation & fraction questions provide more insight into using quantitative reasoning to approach multi-step, messy calculations.
Working Memory:
Outside of speed, having a strong working memory is very important in mental calculation. There is no single agreed upon definition or theory behind working memory currently. Illustrating the capability by way of example may be helpful:
123 x 321
(a) One-digit
1 x 3 —>3
(1 x 2) + (2 x 3) —> 8, 30+ 8 is 38
(1 x 1) + (2 x 2) + (3 x 3) → 14, 380 + 14 is 394
(2 x 1) + (3 x 2) → 8, 3940 + 8 is 3948
3 x 1 → 3, 39480 + 3 is 39483
(b) Three-Digit
1 x 321 → 321
2 x 321 → 642, 3210 + 642 is 3852
3 x 321 → 963, 38520 + 963 is 39483
Example (a) is the solution for 123 x 321 using single-digit cross multiplication going left to right. Example (b) is the solution for 123 x 321 using the traditional method for multiplication.
Example (b) is a more efficient approach than example (a) if you are comfortable with the visualization necessary to smoothly do the calculations in example (b).
The average person’s discomfort with applying their muscle memory to numbers is the primary obstacle to doing mental calculation. Take the question 66 x 17. Can you do this question quickly and confidently? If so, you are probably considered a “numbers person” by people around you. But what actually is required to solve this question?
Write the first number → 66
Do 7 x 6 → 42, write the number twice, shifted one unit to the right the second time.
660 +
420 +
042
1122
So the actual calculations required to solve this question are — in their entirety:
-7 x 6
-6 + 4, and
10 + 2
That’s it. Certainly, for some people the lack of knowledge about how to do arithmetic is the primary problem. But for many, it would be difficult — or, more accurately, uncomfortable — to apply the above process. And that discomfort relates to working memory or visualization.
Not having a strong enough working memory hinders mental calculation ability in two ways. First, it forces one to break down the question into the smallest chunks possible which is not always the most effective and efficient way to solve the question. Second — more importantly — in multi-step questions it can cause one to lose one train of thought during the question, making solving the question difficult. If you were trying to solve
(7202595 x 4119762) / 23911
even if you knew the process for digit-by-digit multiplication and division, solving this question mentally would be very difficult due to limitations with working memory.
For “real world number sense” — developing your ability to work with the numbers you encounter all around you — working memory is quite important. Questions are not clean and simple, but rather messy and often multistep. Calculation League emphasizes working memory more than mental arithmetic competitions or even earlier mental calculation competitions by focusing more on multi-step, complex questions where working memory can be strained.
Knowledge of Milestones:
Knowledge of milestones refers to the other type of memory — long term memory. While mental calculation is not a memorization contest — and the variety and size of questions used in Calculation League would preclude a heavy reliance on memorization — memorization of some information can be helpful.
The only thing that you truly need to have memorized to begin doing mental calculation is your multiplication tables up to 9. Unfortunately, if you do not already know the answer to 8 x 7 then doing mental calculation would be a very frustrating task.
Most mental calculation competitors do not spend any time intentionally trying to memorize significant amounts of numerical information. The information slowly just becomes known over repeated practice and exposure. Some common examples include:
-Expanded multiplication tables (for example, 2 digit by 1 digit multiplications);
-Squares, cubes, and sometimes larger exponents;
-Prime numbers (or the factorization of certain numbers);
-Logarithmic tables
These would be the four most common types of milestones, in descending order of importance. 2 x 1 multiplications are of significant benefit when engaging with higher level mental calculation. Knowing your multiplication table past 2 digit by 1 digit or knowing any of the other items on the list to a significant extent, is simply a luxury that would allow one to more quickly perform specific types of calculations.
Knowledge of Mathematical Principles:
Through playing with numbers, many mental calculation competitors end up deriving — or approximating, in more complex cases — certain algebraic principles and sometimes principles from more advanced mathematics. Of course, the mathematics that is familiar to most people — the mathematics taught in primary school up through early university — was derived long ago, before modern technology, by people who had to work on the arithmetic themselves.
Being comfortable working with numbers allows one to confidently experiment or play with numbers and ultimately see the patterns and principles that are fundamental to algebra. Below are a few basic examples, in order of increasing complexity. These examples are simply meant to give an overview of some mathematical principles used in mental calculation. More detailed summaries of these principles can be found in the respective GMCA Guides sections.
Cross-multiplication
32 x 27 —> There are a variety of ways this may be taught in grade school. You might be taught to double 32 (do 32 x 2) to get 64. Since the “2” in “27” is in the tens place, you would then add a zero to 64 to get 640. Then 32 x 7. Well 32 x 7 is more difficult for an inexperienced student, so you might need to break that down further, into 30 x 7 and 2 x 7.
After doubling 32 and doing 30 x 7 and 2 x 7, what you have in effect done is a series of crosses. You multiplied each of the two digits in “27” by each of the two digits in “32”. In performing those multiplications, you have to be careful to retain the proper units by adding zeroes where appropriate. The number of zeroes to be added is the sum of the units to the left for each of the two single-digits multiplication. For example, when doing 3 x 2, you must add two zeroes because both 3 and 2 are one unit to the left.
This process can be expanded to numbers of any size. If I want to multiply two 8-digit numbers together, it may be more efficient to solve the question “unit by unit.” To directly solve a specific unit of the answer —- for example the one-hundred thousand digit (which has 5 zeroes) — I will have to multiply together the digits whose distance to the left adds to 5.
With further experimentation, I can get comfortable applying this process from left to right (more practical) or using multi-digit blocks of numbers (more advanced). If a student only works with small or simple numbers, however, they likely will not be able to see this principle.
Binomial Theorem
Consider the following pattern:
100, 121, 144, 169, 196
These are, of course, the squares of the numbers from 10 to 14. If l look at the difference between these squares, I get 21, 23, 25, 27 — a very predictable pattern.
After noting the difference between 10^2 and 11^2 is 21 and that the difference between 11^2 and 12^2 is 23, it should become clear that
11^2 = 10^2 + 10 + 11, and
12^2 = 11^2 + 11 + 12.
So (x+1)^2 = x^2 + x + x + 1 = x^2 + 2x + 1, the first iteration of the Binomial Theorem.
Similarly, 12^2 = 10^2 + 10 + 11 + 11 + 12, so
(x+2) ^2 = x^2 + x + x + 1 + x + 1 + x + 2 = x^2 + 4x + 4, the second iteration of the Binomial Theorem.
I could also work out similar expansions for higher powers.
1^4 = 1
2^4 = 16
3^4 = 81
4^4 = 256
The difference between 16 and 81 is 65. What can I do with 2 and 3 to make them equal 65? 2 and 3 multiply to 6. They add to 5, which is a factor of 65. Let’s start with that. 65/5 is 13. What can I do with 2 and 3 to make them equal 13? Well 2^2 + 3^2 is 13.
So now I have 3^4 = 2^4 + ((2+3) * (2^2) +(3^2)). I could substitute x=2 and x+1=3 and create the first iteration of the fourth power of the Binomial Theorem.
Factorization Tests:
Using modular arithmetic, we can devise certain tests to factor numbers (or test for divisibility generally) without performing the full divisions. The classical example of this is the number 1001, which is: (a) close to 1000; and (b) broken into the following prime factors: 7, 11, 13.
First, we can note that if
X mod y = 0
And z mod y = 0
Then x mod z (or z mod x) must be a multiple of y.
Using numbers, this can be illustrated by:
1001 mod 7 = 0
1008 mod 7 = 0
Then 1008 mod 1001 (and 1001 mod 1008) must be multiples of 7
Here, because 1001 is a multiple of 7, 11, and 13, we can check whether a certain number “x” is divisible by 7, 11, and 13 by doing x mod 1001 instead of dividing by each of the three primes separately.
By creating “magic numbers” (like 1001) that are near each other or where the difference is an easy number to work with, a student can check divisibility by a number of primes efficiently. The particulars of the use of magic numbers is highly dependent on the desired context (i.e. the size of the number to be factored, the time permitted, speed vs. thoroughness, etc.). Notable magic number tests were devised by recreational mathematics enthusiast and Princeton mathematician John Conway (Conway’s 150 Method and Conway’s 2000 method).
The second property of 1001 is referred to as a “rip-off” test. The summary of a rip off test is that you break a number into two components: (1) the last “n” digits; and (2) the rest of the number (we will refer to this as “m”). Then, using modular arithmetic, you can construct a formula that will check for divisibility without checking the whole number. In the case of 1001, the formula is:
M * 1 minus N , where “n” is the last three digits. Here, “n” is three-digits because 1001 is close to 10^3. Because 1001 is greater than 1000 the sign is negative (minus) instead of positive. And because 1001 is 1 greater than 1000, M only has to be multiplied by 1.
Take the number 735774. Instead of dividing by 1001, we can do 774 minus 735 → 39. Then we can test whether 39 is divisible by 7, 11, and 13, and we can quickly see that 735774 is divisible by 13, but not 7 and 11.
In thinking about rip off tests, we can quickly note that 735774 mod 1000 is 774. If we multiply 1001 by 735, we are now decreasing the remainder (774) by 735 x 1, or just 735 since 735 x 1001 is 735 x 1000 + 735 x 1. The new remainder will be 39. [While factorization tests usually refer to mod, it does not matter whether the calculated “remainder” is negative or positive. Rather the concern is with absolute values not strictly remainders.]
Irrational Exponents
Consider the following question:
222990^.687
Often times, the most efficient way to address an exponent or root question is going to be through the application of logarithmic principles. In a question format of the type:
A^B, one would start with log(a). As outlined in the GMCA Guide section, log(a) can usually be estimated with reasonable accuracy only knowing two digits of log(2), log(3), and log(7). As a rough estimate, we would get 5.35 here.
We now perform the step B * log(a) —> 3.675, which can be sometimes difficult.
Finally, we take 10^3.675 — and attempt to estimate the result as precisely as possible using our knowledge of logarithims.
This works due to elementary logarithmic principles. The definition of a logarithm is that A=10^(log(A)). If we want to calculate A^B, then we can raise both sides to the power of B.
Now we have
A^B (the original question) = 10^B log(A)
Thus calculating the answer to A^B is simply composed of the three operations on the right-hand side of the equation above — (1) log(A) (call that “X”); (2) B*X (call that “Y”); and (3) 10^Y
Alternatively, we could use basic principles of exponents:
A^(B+C) = A^B * A^C, and
A^(B-C) = A^B / A^C
Here, I could note that .687 is relatively close to .667. We can try to do 222990^(⅔) + 222990^(1/50).
6^3 is 216 —> Therefore, the cube root of 222990 is going to be just above 60.
If I square 60, I get 3600. So we have 3600 as our rough estimate for 222990^(⅔).
For 222990^(1/50), we already calculated 222990^(⅓) — and estimated 60. Now I have to estimate 60^(3/50), which is close to 60^(1/16). 16 is 2^4 so this step involves taking the square root (a core mental calculation task) 4 times.
60^.5 → 7.8
7.8^.5 → 2.8
2.8^.5 → 1.7
1.7^.5 -> 1.3
So let’s take 3600 * 1.3 → 4680.
Simply using very rough estimates and only performing relatively modest calculations, we have estimated the answer to within 1% (the correct answer is about 4723.6).
Conclusion
The above examples are merely designed to illustrate a representative group of examples, of varying difficulty, of how knowledge of mathematical principles —- usually algebraic principles of varying complexity — is important to refining the approach used in mental calculation.
An individual’s competitor’s prioritization of the six skills listed above will depend on the types of questions they desire to solve. Algorithmic and information processing speed are of high importance for mental arithmetic — simple and clean formats — but their importance decreases as the question difficulty increases. As the question types become messier and the amount of discrete steps is increased, the importance of reasoning speed increases, while the importance of working memory is, of course, correlated with the number of “unseen” digits a competitor needs to work with. Knowledge of mathematical principles can be important for any question type, but the more complex and messy the type of question, the more important it will likely be. And, finally, a well-developed “number library” can be strategically employed in certain question formats to skip steps or assist the quantitative reasoning process.