Guide to Becoming a Mental Calculation
One of the most common questions asked by people who are interested in mental calculation is not a technical question, but simply “how do I approach mental calculation?” Importantly, there is no single correct answer to this question. Nevertheless, after having corresponded in detail with many of the most skilled mental calculators throughout the world in my role as organizer of the Global Mental Calculators Association and Calculation League, I am going to offer an imperfect set of recommendations for beginning your journey. Generally, I believe these recommendations are supported and approved by the top competitors in mental calculation, although not all individuals would agree with all suggestions. And it is important to approach mental calculation in a manner that you think is enjoyable, practical, and sustainable.
Note that the below are general recommendations for how to approach, practice, and engage with mental calculation. This is not a technical discussion for solving specific questions —- please consult other guides for that.
Step Zero: Unlearn Bad Tendencies
If you already have some exposure to mental calculation, but only informally and rather limited, I would recommend, as a preliminary step, to review the list of things you already do and know. Specifically:
How do you approach addition and subtraction?
How do you approach multiplication?
How do you approach division?
If your answer to the above is that you don’t really have a specific approach, then you should feel ready to start practicing. If, on the other hand, you do have specific approaches that you consistently use when doing arithmetic, it is time to think about whether these approaches are desirable or limiting. Most commonly, a “bad tendency” arises from one of the following situations:
Overreliance on Comfortable Processes
You have learned a trick or an approach that works well in specific situations but you attempt to apply that approach far more often than appropriate. Mental calculation is highly varied and dynamic and a willingness to keep developing and broadening your skills and approach is critical. Calculation League champion Wenzel Gruss even spent Season 1 of Calculation League unsettled on his approach to basic A x B multiplication of a fixed format!
Just because one is comfortable and experienced with a specific approach, does not mean that one should avoid learning new approaches. This can be difficult in the context of mental calculation because of the objective feedback constantly provided — if you are faster with your old approach today, it can be a challenge to accept that relearning a new approach is desirable.
If you are concerned that this subsection may apply to you and find it difficult to challenge yourself to learn new approaches, my suggestion would be to identify types of tasks that you cannot quite do using your existing approaches. For example, make the question more difficult by making it larger or adding additional steps. By focusing on something that you are not capable of doing now, you will force yourself to learn a new process, in the process developing a new skill. In the future, you may find that that new approach is more effective at solving the questions you can do now than your current approach.
An over-concern with short term progress is a common impediment to developing skill in mental calculation and mental arithmetic. It is often the case that the most effective way to develop your skill over the next few days is very different from — even opposed to — the most effective way to develop your skill over the next few months or years. Even some of the best competitors in the world retain some inefficiency in their approach, especially if they were exposed to mental calculation informally as a child and had no expectations of becoming a world class competitor.
Over-reliance on specific methods is most commonly associated with multiplication and division because there are a variety of ways to approach multiplication and division and no “one size fits all” approach. Chances are that if you have a method that allows you to do multiplication and division well in some situations, that approach truly is a valuable approach, even the ideal approach in certain circumstances. In order to most effectively develop your overall skill, however, you should work on familiarizing yourself with a variety of different approaches, keeping in mind when the different approaches might be useful.
Use of Wrong Methods
A second common problem is simply that the student has been taught poor approaches, usually in math class at a young age. This could occur simply because the math class was taught by a teacher without meaningful knowledge of arithmetic. It can especially occur in the Western World — where arithmetic is often de-emphasized or ignored — and can be the result of teaching methods intended to demonstrate principles when those methods are not a reasonable approach to actually doing arithmetic.
Use of improper methods can be associated with any type of mental calculation, but is likely to be most destructive if associated with addition/subtraction. Quite simply, some early education techniques designed to help students understand the basics of numbers are a hindrance if not quickly replaced by more effective methods.
The GMCA YouTube page has instructional videos relating to adding and multiplication that can be used to get a quick course on approaching those operations. Additionally, the written instructions contained on this website can help you identify whether some unlearning of inefficient methods or approaches is necessary before you get started with mental calculation.
Step 1: Assess Goals/Objectives
As with anything, it is helpful to begin by asking: what is your goal? Of course, you don’t need a precise answer to this question, but it is important to note that because mental calculation is quite multi-dimensional, your general answer to the question may highly influence your approach to the field. For example, here are some examples of quite different priorities:
I want to learn how to solve highly difficult or complex questions;
I want to learn how to rapidly do simple addition and multiplication;
I want to learn how to work with the numbers and arithmetic around me in every-day life;
I want to learn how to interpret quantitative information presented in a school or work setting;
I want to use mental calculation for brain training in order to improve my cognitive abilities;
I want to work towards being a competitive mental calculator.
Generally speaking, there are three different categories of skills that are relevant in mental calculation — each discussed more in the introduction section — (1) mental processing speed; (2) memory; and (3) methods / quantitative reasoning. The more “simple” and “artificial” the task, the more “mental processing speed” is important, with “methods / quantitative reasoning” being less valuable. The more “complex” and “messy” the tasks, the more that “methods / quantitative reasoning” are important, and the less that “brain processing speed” is critical. Memory tends to be of moderate importance in both categories, although in a different manner. In many simple/artificial tasks, repetitive practice may result in the building of muscle memory that results in a variety of arithmetical information becoming known. In complex/messy tasks, memory is more likely to revolve around “working memory” or visualization ability.
The above distinction should be familiar to competitors at the Mental Calculation World Cup where individual results were recorded in two separate types of tasks: (1) standard; and (2) surprise. In standard tasks, the formats were fixed, known in advance, simple, and susceptible to being repetitively practiced. In surprise tasks, competitors are unaware of the types of questions prior to the competition and — typically — the questions are more complex or include multiple steps. In general, a clear pattern was observable in the results: younger competitors — with faster minds and less exposure to methods — tended to perform better relatively speaking in standard tasks, while older participants performed better in surprise tasks.
As a result, it is worth considering your objectives and the list above before determining how to approach practice. What results do you want to see? What are your goals? A disconnect between a student’s priorities and their goals is — unfortunately — a common frustration in practicing mental calculation or mental arithmetic.
The following recommendations attempt to strike a balance between different objectives, but, overall, are geared towards (6) — how one could efficiently work towards becoming a competitive mental calculator. This objective is the most well-rounded of the different goals listed, incorporating each of the other goals, without placing special emphasis on anything specifically.
Step 2: Develop Foundational Skills
While mental calculation does not require extremely high skills with basic arithmetic, it is important to be relatively good and comfortable with basic arithmetic. There is no need to memorize the multiplication table to an extraordinary extent, or to endlessly practice simple arithmetic questions. Yet, mental arithmetic plays a similar role in mental calculation as vocabulary would play for a writer. Possessing a strong vocabulary does not mean you will be a successful writer, but if you want to be a successful writer, a strong vocabulary is likely beneficial. In rare cases, a unique individual may be able to become a successful writer without a strong vocabulary, but those situations are the atypical exception.
In mental calculation, you need to know your multiplication table up to 9 x 9. If you do not know your single-digit multiplications, you need to memorize them thoroughly before starting mental calculation practice. The other foundational skills that are important and are worth practicing before you get started are the following:
(Two-digit) additions/subtractions;
2 digit by 1 digit multiplication; and
3 digit by 1 digit division.
The multiplication/division requirements listed above are critical; the addition requirement is important and helpful, but there are occasionally highly skilled competitors who are still not truly comfortable rapidly doing two-digit additions. Being able to rapidly do single-digit addition/subtraction is, of course, critical.
The GMCA Guides on the website contain more detailed information regarding how to learn addition/subtraction, multiplication and division, including how to start beginning to develop the foundational skills for each operation.
Step 3: Assess Strengths/Weaknesses
Once you have developed some comfort with the foundational skills, it is time to start objectively evaluating your performance. Fortunately, in mental calculation/arithmetic, this is relatively easy to do thanks to the objectivity of the activity and the ability to precisely determine speed. For serious competitors, the GMCA website can be used to consult statistics for a variety of question formats used in Calculation League. These statistics would be instructive even if a competitor is not yet particularly close to being capable of qualifying for Calculation League. For true beginners, a variety of online mental arithmetic or educational platforms contain detailed leaderboards that would provide helpful feedback for students.
There are two important and separate components of determining strengths and weaknesses: (1) assessing one’s performance in the “skills of mental calculation”; and (2) assessing one’s performance in different standard operations or core calculation tasks. The second category is more obvious and easier to do. By evaluating one’s performance on common task formats in a standardized setting — for example, the Japanese Soroban Simulator or the Standard Task Simulator — one can determine if there is a specific type of task where additional effort would be more beneficial.
The important consideration is that practicing one’s weaknesses is more effective for progress than practicing one’s strengths. As an example, if a student’s addition and multiplication performance clearly exceeds their division performance, then further focus on division is going to yield more productive results than practice focused on addition or multiplication. Because mental calculation and real world quantitative analysis is messy & mixed together, one simply cannot simply focus on “what one is good at” and ignore one’s weaknesses and hope for ideal results.
In order to assess one’s overall performance in the “skills of mental calculation,” a student is going to need to have some experience practicing a variety of formats and tasks and some basis for comparing that performance to others. The Calculation League competition app and the statistics pages on the GMCA website are uniquely equipped to provide feedback covering a wide variety of mental calculation. Additionally, match videos are posted on the GMCA YouTube page and can be utilized by skilled competitors in order to achieve a more detailed assessment.
Quite simply, prior to developing a practice plan for approaching working with numbers mentally, it is important to diagnose what your skills and limitations are. In the same way that a rational student would focus their efforts in school on subject areas where they needed further work, a student of mental calculation whose priority is improvement needs to work on addressing those areas that are limiting their performance.
Step 4: Create a Versatile Practice Routine
The critical recommendation here is that a promising plan to approach mental calculation should be varied and thoughtful. This may sound obvious but there are powerful psychological obstacles that result in many mental calculation students adopting an ill-advised practice routine that prevents them from achieving the desired results.
The most common bad habit is finding a specific task that one enjoys or excels at, and excessively practicing that task, sometimes referred to as “drilling.” Drilling a task has certain advantages that appeal to our psychological natures: (1) having a speciality increases the sense of having achieved something; (2) detailed and precise feedback about a consistent, uniform task makes it feel like you are in control and seriously evaluating your performance; and (3) consistent, even if slow, progress attributable to muscle memory provides the illusion of progress. It is easy to get addicted to “drilling” a specific task — continued progress, detailed results, and a sense of achieving or striving for something specific can trap one in an ineffective training regime.
Ultimately, however, if one’s practice routine feels too routine, it is time to change. Exercising the brain is no different than exercising the body — progress is going to come through struggling and through overcoming challenges. Or to use a different analogy — if one wanted to develop their basketball shot, the ideal approach would be to take a variety of shots, and even vary the lead up to taking the shot. The ideal approach would not be standing at the same place on the court, and taking the same shot in the same way over and over and over — as tempting as that is to do. This bad habit seems to be more common in practicing mental arithmetic than in other fields — likely because having a uniform and consistent practice routine provides numerical information that the student interested in mental calculation appreciates.
Instead, after one has internalized that the goal is to address weaknesses rather than emphasize strengths, it is appropriate to develop a routine that emphasizes certain types of problems or cognitive skills in proportion to the amount of progress desired in those areas. Periodically performing the assessment in Step 3 is important to: (a) determine whether the practice routine is actually working; and (b) determine whether modifications to the routine are warranted.
Importantly, this is not to say that one should avoid practicing one’s strengths entirely. Muscle memory, however, is best built up over time and there are diminishing returns to excessively practicing the same thing over and over. A comfortable, consistent “warm-up” routine prior to a more challenging and uncomfortable session aimed at learning something new or addressing a weakness, is a reasonable approach for balancing: (a) slowly building your existing skills with (b) learning new things.
Step 5: Focus on Developing Cognitive Abilities & Practical Skills
As you develop your skills, you will likely notice two things occur that may disrupt — or prevent — your practice. First, you will likely notice a decrease in the rate of progress. Second, after developing noticeable skills and when confronted with a decreasing rate of progress, you will likely question whether you are “already good enough” or whether further effort is worth it. Both of these tendencies are natural obstacles after having put in a moderate amount of effort to develop skills in any new activity.
If you reach this juncture and are faced with this problem, it is worth considering whether to “mature” your approach. If you choose to do so, I would recommend making a conscious effort to increase your emphasis on developing two different things: (1) the underlying cognitive abilities; and (2) practical arithmetical skills. Regarding (1), directly emphasizing developing the underlying cognitive abilities may: (a) continue to improve your mental calculation ability; while (b) providing clearer practical utility and application to other pursuits. Regarding (2), working on mental calculation in a more practical manner — rather than in a competition manner — can improve your problem solving skills, understanding of mental calculation, and arithmetical versatility.
As briefly outlined in the skills of mental calculation article, the cognitive skills underlying mental calculation can generally be classified as consisting of three kinds of speed (information processing, algorithmic processing, and reasoning), two kinds of memory (long term and short term), and one category of “mathematical knowledge.” Directly emphasizing the development of a specific cognitive skill can be done through the pairing of a mental calculation task and a non-mental calculation task that both heavily emphasize that ability. The options are endless, but here are a few examples:
Flash Anzan & fast paced video games (information and/or algorithmic processing);
Complex, multi-step calculation questions & logic puzzles (reasoning speed);
Doing verbal or unseen calculations & memorizing cards (without formal reliance on advanced memory techniques) (working memory);
Complex questions with exponents or roots & memorizing numerical facts (long term memory);
Analyzing topics in algebra or number theory & creating mental calculation formats where that knowledge would be helpful (mathematical knowledge).
Additionally, there are an increasing number of websites — an early example is Human Benchmark — that are used to evaluate specific cognitive abilities or generally combat “brain rot” through encouraging “brain exercise.” These websites can often be used to test and emphasize specific cognitive abilities, although often using an artificial task.
When pairing two tasks that rely on similar abilities, one can use mental calculation to develop a cognitive ability while also directly practicing that cognitive ability — in a different way than through mental calculation — hopefully resulting in improved mental calculation performances.
Regarding the development of “practical skills,” this is simply a suggestion that one practice some (mental) quantitative analysis in areas or fields one enjoys. Specifically, data-heavy real world fields offer a great opportunity to do practical mental calculations in a manner that may: (a) encourage new mental exercise; (b) provide insight or feedback that could be used to further modify or refine one’s approach to practicing mental calculation.
Some common “hobby fields” that are data rich and provide ample opportunity to practice “practical mental calculation” include: (a) sports; (b) politics/elections; (c) business & the stock market; and (d) prediction markets generally. One can apply mental calculation, however, to anything where detailed data exists and quantitative analysis is of some value.
It is worth noting that — generally speaking — emphasizing cognitive skills may be the more appropriate option for younger participants, while emphasizing practical application of skills may be more appropriate for older participants, for a couple reasons. First, up until 25 to 30 years old, a human’s brain is still naturally developing and thus the relevant cognitive abilities are still quite susceptible to being developed further. After that age, however, individuals could begin to find it difficult to make significant progress in developing their cognitive skills — similar to an athlete engaging in physical exercise, they may be working against the natural aging process. On the other hand, emphasizing the practical application of mental calculation usually will require developed reasoning skills and significant knowledge in the relevant field, and, therefore, may be more reasonable for older participants.
Step 6: Emphasize Multiple Approaches to Solving Questions
In contrast to traditional soroban teaching, GMCA encourages developing multiple types of approaches to solving a single question format rather than working on perfecting the application of a specific algorithm to a specific type of question.
The most convenient illustration of this recommendation involves multiplication. Consider the following types of multiplication questions:
79 x 22
8128 x 4568
13546540 x 76430989
3847 x 9188 x 2246
46 x 91 x 23 x 96 x 56 x 16
There is not a one-size-fits-all approach to all the above questions. If one is only comfortable working right to left — or even left to right — or if one is only comfortable with the traditional school method, or only with cross multiplication, then one will have to resort to less efficient approaches to solving some of the above questions.
Unless one is practicing a specific format for a specific competition, it is advantageous to be comfortable switching between a variety of approaches. Both the development of practical skills and the development of cognitive abilities are furthered by doing so. Generally, if one changes the number of digits in the integers, the number of integers or operations, or the form of the question (i.e. written answer, spoken answer, oral question etc.), one is likely to be forced to develop calculation flexibility —- and, in doing so, develop reasoning skills to select between different options.
A competitor does not even need to change the format however to practice different approaches. Simply forcing oneself to try a new approach to a well-known format may prove beneficial. It also may provide insight into the utility of the approach. For example, if the new approach — Approach B — is significantly worse than the old approach — Approach A —sometimes, but in other cases is almost the same, then what is the reason? And what does that reason say about whether Approach B would be advantageous in certain circumstances if practiced to the same extent as Approach A?
When you are experimenting with different approaches to mental calculation, it is especially useful to document your experiences. Once you’ve reached Step 6, you should be relatively experienced with mental calculation. Approach it scientifically and evaluate it as an “expert” would evaluate their field of expertise.
Step 7: Modest Daily Practice
One of the primary obstacles to practicing mental calculation — or mind sports generally — is an erroneous belief that impressive results require extraordinary amounts of practice. Accounts of Japanese soroban exports — or Aaryan — practicing multiple hours day after day contribute to a perception that the field takes extreme commitment.
But as the top American mental calculator, I can say that an extreme commitment is not necessary to achieve impressive results. I, myself, generally practice 15-30 minutes per day. The top American memory competitor Alex Mullen — who holders numerous records and world championships — also practices 15-30 minutes per day. Extraordinary results, but only modest practice. A critical qualification, however, is daily — not some days, not 15-30 minutes per day averaged over a week or a month, but almost every day.
Your mind — just like your other muscles — needs consistent, regular exercise to improve. Large blocks of irregular practice is not an adequate substitute, but extreme daily practice is also unnecessary. If you were preparing for a race, which of these sounds like the most reasonable way to prepare:
Running for two to three hours every day;
Running for two to three hours once a week;
Running for fifteen to thirty minutes per day.
Obviously, the third approach is the normal, preferred approach. The first approach is unreasonable and unnecessary, and may even end up causing harm. The second approach is not an adequate replacement for the third approach. Similarly, one could replace running with sit-ups, push-ups, or other exercise in the above analogy, and the observation would hold.
Modest daily practice is the key to exercising — whether you are exercising your mind or your body. Ideally, this would be supplemented by an increased effort to have “mentally healthy” habits — adequate sleep, hydration, diet, even exercise — all have some effect on your cognitive capabilities. World memory champion — and former Calculation League participant — Vishvaa Rajakumar, when asked to provide a recommendation for training, emphasized the importance of staying hydrated. I personally double or triple my daily water intake during periods of more serious training. And GMCA board member Jeonghee Lee has previously emphasized the importance of physical conditioning on mental calculation performance.
Finally, one cannot mention lifestyle habits without acknowledging that a major impediment to mental calculation in our modern times is the effect that modern technology has on attention spans and ability to focus. This topic has been written about by scientists and experts in detail, but it is worth noting that it is my observation that top-tier mental calculation competitors — in general — use technology significantly less than average. Whether you hope that mental calculation practice improves your attention span and focus, or you merely want to practice within the limits of your current characteristics, it is worth considering the effect that technology has on your overall cognitive abilities.
Step 8: Incorporate Numbers & Quantitative Reasoning in Daily Life
While lifestyle habits certainly have an effect, the degree to which you can incorporate working with numbers in your daily life is more important. Quite simply, it is critical if you wish to make high quantitative skill a permanent characteristic rather than a temporary skill. For example, if one considers physical ability, if you want to remain in strong physical shape you are likely going to need to either: (a) exercise extensively; or (b) incorporate significant physical activity into your daily life. Merely practicing a modest amount — while engaging in a physical inactivity lifestyle — is unlikely to keep you in good physical shape for very long.
Fortunately, it is much easier to incorporate working with numbers into your daily life than it is to incorporate physical exercise. Mental calculation being merely a mental task, you can do it anywhere, any time, without the need for any resources.
In fact, I attribute my own mental calculation skills to this step more than any other. When I was a small child, almost immediately upon memorizing the alphabet, I also learned each letter’s numerical position. For example, a=1, b=2, c=3 . . . z = 26. This relationship became so ingrained in my mind that I can add letters nearly as fast as I can add numbers.
The advantage of this is that we encounter letters almost constantly throughout daily life, while only encountering numbers in specific situations. Using letters as numbers provides an endless opportunity to perform calculations. Even the mere act of converting the letters to numbers –without performing any calculation — seems to briefly activate the “numbers” part of the brain.
This “light” mental calculation exercise for me seems to occupy the same place as merely “walking” or “standing” may occupy in relation to physical health. By keeping active throughout the day —- even if the exertion is minimally — the muscles (whether physical or mental) remain in better shape. Memorizing the relationship between letters and numbers is not going to transform you into an elite mental calculator. But it would help maintain your “mental calculation fitness” so that when you do practice, you are more ready to do so. And there are plenty of challenging calculation questions one can use letters for as well — if you have some time.
Otherwise, as mentioned earlier, there are plenty of common activities or hobbies where reviewing data or numbers is of value. Sports, politics, economics/business — and now, prediction markets — all provide great opportunities to regularly engage in numbers. Challenging yourself to do some (modest) data analysis (mentally) may simultaneously be of value to developing your number skills, while also providing insight into the field.
Finally, there are various apps — some functional on your phone — where you could take out the app and practice, even if only for a minute or two. New applications are going live every year, and they are constantly evolving. The critical consideration with these applications is not to be too repetitive. The law of diminishing returns necessitates that your progress will decrease, but, more importantly, repeatedly exercising your brain in the same way will eventually just result in maintaining your existing ability rather than developing anything new. Additionally, if you reach the point where you are primarily relying on muscle memory (you have the answers semi-memorized), you may not actually be exercising the mental calculation part of your brain at all.
Naturally, however, there is going to be a correlation between the frequency of the engagement with numbers and the impact of that engagement. While doing data analysis in a real world field may be appropriate for blocks of time (time that you already use engaging with those quantitative-rich fields) and practicing on an app may be appropriate when you have a short amount of time, I have yet to encounter any recommendation other than my “letter activity” that allows one to effectively “practice numbers” when one has merely a few seconds of time. In fact, I can practice some numbers while simultaneously typing this sentence.
Step 9: Keep Challenging Yourself to Expand Your Skills & Abilities
Embrace the struggle. Keep shifting your mental calculation practice focus so that you are learning new things and engaging in a different way.
Earlier, it was recommended to implement a versatile practice regime. Step 9 goes a bit further and recommends that once you are a very experienced competitor, the practice regime should continue to grow to incorporate entirely new things — rather than simply a wide range.
For the same reasons that people usually naturally gravitate towards “drilling” a small range of tasks, it can be psychologically difficult to get yourself to continue doing new things. Retreating to a small range of tasks where you are comfortable — and where your skill is likely very impressive — seems to be the default action. Yet this will likely eventually result in boredom, burnout and — at a minimum — ineffective practice.
Yet, the interconnectedness of your mind — and of arithmetic — mean that you may actually be able to improve your core skills by practicing something new. If a certain category of your brain processing speed or visualization ability is no longer meaningfully improving with “normal” practice, exercising that same part of your a different way may produce improved results.
As one modest example, after having practiced normal “competition addition” for a significant period of time, my own results seemed not capable of improving and — as a result — I discontinued that part of my practice. Some time later, however, I decided to make a conscious effort to directly practice subtraction. After only a week or two of modest practice with subtraction, I returned to addition and realized that my addition scores were now noticeably improved from my peak performance — despite not having directly practiced addition in approximately one year.
The various cognitive skills and mathematical processes that are involved in mental calculation contain so much overlap and interconnectedness that continually practicing a modest number of things a specific way — even after your progress has become minimal or stopped altogether — is simply an inefficient use of your energy. Not only could you be building new skills, but the exercise of your mental ability in a different way could develop your cognitive abilities such that your results on your core skills improve as well. Quite simply, if you artificially limit how you engage with mental calculation, you will also limit how you perform in mental calculation.
Step 10: Prioritize Speed Over Accuracy & Accept Reasonable Estimates
People make mistakes. Our minds have computational abilities — extraordinary potential even — but your mind is also always busy doing all kinds of things. It does not matter how much you practice, you will make mistakes.
I brought home the full score sheets from the first time I competed in the Mental Calculation World Cup, in Germany in 2018. In the multiplication task — multiplication of two eight-digit numbers — the score sheet reflects the following accuracy for the top six competitors:
Tomohiro Iseda, Japan (83%);
Marc Jornet Sanz, Spain (76%);
Freddis Reyes Hernandez, Cuba (77%);
Jeonghee Lee. South Korea (77.5%);
Samuel Engel, USA (76%)
Mohammad El Mir, Lebanon (78%)
That is quite a narrow range!
It is common for mental calculation competitors to do sets of 10 questions, but regardless of what you are practicing, I have always subsequently used the observation above as my guide when practicing — 8/10 or better, I more or less ignore the mistakes. If it is only 6 or 7 out of 10, I should probably briefly take a look at what is occurring or maybe slow down a little bit, or try to increase focus. If the result is 5/10 or worse, there is a problem, and I need to take a break, review what happened, maybe stand up or drink some water, etc.
Certainly, the type of format matters. The multiplication format above is the most difficult standard format used at the Mental Calculation World Cup. It is likely unreasonable for any top competitor to expect 10/10 most of the time. If you are doing 2 digit by 2 digit multiplication, then, of course, 80% is likely too low of an error target.
The reason that I recommend to prioritize speed over accuracy is not because speed is more important (obviously quickly getting an answer wrong is not helpful), but because emphasizing speed is important for mental calculation to be impactful as brain training and result in increased processing speed. Few top mental calculation competitors practice formats where they routinely exceed 95% (or even 90%) accuracy. If your accuracy on a large sample size in a certain practice format clearly exceeds 80%, then my opinion is that you should either shift to more difficult questions or push yourself to go faster. Your practice should be challenging — don’t practice what you can already do, practice what you would like to be able to do.
The second part of Step 10 — “accepting reasonable estimates” — is almost inevitably the final step for an accomplished and experienced mental calculator. When there is a requirement for completely accurate and full results in mental calculation, the variety of possible formats is substantially reduced. In this reduced universe of mental calculation formats, there is only so much progress and development you can hope to achieve. Once you redefine the goal as “partial solutions” or “reasonable estimates,” then the number of possible formats substantially increases to cover highly complex or multi-step or mathematically involved programs. Your ultimate goal as a mental calculator should be to never feel overwhelmed or incapable of interpreting numbers. Obviously, that does not mean never having to use a calculator (or computer) to calculate anything. Instead, it means always having the confidence and ability to reasonably interpret numerical information.
Accepting reasonable estimates also forces a mental calculator to develop new approaches to questions. The strategy for rapidly estimating a question to three significant figures may be very different from the strategy for solving the question to nine significant figures. Generally, it is going to be the former that is the more practical method for working with numbers in daily life. Additionally, the partial solution — the incomplete result — may actually end up encouraging you to develop your “number library,” your reasoning skills, and even your memory more than the full result. How? Because many approaches for fully solving difficult questions rest on the premises that: (a) one’s visualization is inadequate; or (b) no set of of algorithms or mathematical principles will allow one to derive a complete solution efficiently; or (c) one’s number library will not — and cannot be — developed enough to consistently assist with solving the question.
To illustrate this, simply consider the simple case of A x B. At some point, all competitors are going to have to shift to an approach that is cross-multiplication (or a variant of cross-multiplication). Attempts to use visualization will eventually fail, and mathematical principles or a developed number library will have limited effectiveness, likely only complicating the attempt to solve the question. Yet, when solving A x B questions that are only two-digit or three-digit numbers, all of these alternative strategies are potentially effective.
The unlimited depth of mental calculation comes from using the whole suite of tools to attack as great a variety of questions with as much precision as possible. One can take, for example, an 8 digit x 8 digit multiplication question and test how many significant digits one can answer using alternative approaches. Not only can this lead to development of those alternative approaches, but the process may reveal additional insights about when each of your approaches is most effective. Reflecting on these results will assist in developing the reasoning to not only calculate, but to reflexively know how to calculate.
Of course, this final step in the development of a mental calculator is likely to occur during adulthood, after relatively substantial time engaged in mental calculation. Yet, this iterative, reflective process — when combined with more traditional practice — is what opens up endless possibilities for mental calculation — and what lays the groundwork for the extraordinarily ability only rarely achieved (i.e. Wenzel Grüß).