When studying mental math in the classroom, or training for mental calculation competitions, the focus is usually on writing the answers to written questions.
But in everyday life we encounter situations where the numbers to calculate are spoken to us, or generated in our head—this is verbal mental math or verbal mental arithmetic.
Typical Situations Requiring Verbal Mental Math
- Somebody asks you a mental calculation question, e.g. “Hey! What’s forty-nine times eighty?”
- In the office, a colleague is discussing some numbers with you, and you need to respond appropriately without interrupting the conversation.
- When travelling in a foreign land, someone quotes you a price, and you need to convert it to a familiar currency.
- In an interview, the interviewer asks you a numerical question involving some arithmetic steps.
Challenges of Verbal Mental Arithmetic
As we have seen elsewhere, the biggest difficulty we have in any sorts of mental calculation is the ability to keep all of the numbers in our head. Even under the best circumstances, it’s difficult to keep more than 10 digits in working memory. Calculations such as \(23 \times 34 \times 45\) are especially difficult for precisely this reason!
In the extreme case, if an arithmetic question contains too many digits, then it doesn’t fit in our working memory leaving any space for us to perform the calculation! Then it is impossible to answer the question without either writing down some working, or using advanced memory techniques.
For example, if someone asked you to answer (translated into your native language) “four hundred and three million, seventy thousand, two hundred and six plus ninety million, one hundred and five thousand and eighty”, you couldn’t do it unless you had trained specifically for sort of challenge.
However, when written, the question becomes very easy:
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How to Improve at Verbal Mental Math
Much of the standard techniques work well regardless of whether the numbers are written, spoken, or generated in your head. However the following ideas will help enormously:
(1) Learn more mathematical facts
The times tables (from 2×2 to 9×9) are absolutely fundamental for mental calculation. But everyone who finds it easy to operate withe numbers has learned—either intentionally or incidentally—many more of these.
For example, 17 × 24 is much easier to calculate if you know that 7 × 24 is the number of hours in a week, the answer being 168 hours. Then 17 × 24 becomes simply 240 + 168 = 408! One of the first things I recommend to most of my students is to expand their knowledge of these with some efficient training. Both verbal calculations and written calculations become much easier as more of these shortcuts become familiar.
(2) Know techniques to simplify calculations
Typically a question like 96 × 22 would best be solved by cross-multiplication.
However, when presented verbally it is easier to express the 96 differently as follows:
(100 – 4) × 22
(100 × 22) – (4 × 22)
2200 – 88
2112
This is still difficult! But once the method is understood, each step requires relatively little internal memory, so it is much more practical for humans.
Another specific technique that can help is doubling and halving. This technique works when multiplying an even number with a number that ends in ‘5’.
For example, 24 × 35 can be simplified by doubling the number that ends in ‘5’, and halving the other number. This has no effect on the answer, since doubling and halving are opposites.
12 × 70 is then easy to calculate, since 12 × 7 = 84 should be familiar from our basic knowledge of times tables.
There are many more shortcuts like this, but they are a topic for another time.
(3) Know when to estimate
Sometimes an approximate answer is enough! Suppose we have to convert US$65 to British pounds. Today the conversion is £1 = $1.3667.
The calculator answer is £47.56, but for most practical situations, this high level of accuracy is unnecessary and probably distracting. Instead, you could arrive at an estimate such as “about £47”, “about £48” or “slightly less than £50”. There are several techniques for making efficient estimates, but they are also a topic for another time.
(4) Train your visual handling of numbers
Most of our processing of numbers during calculation is performed in the visuospatial sketchpad (VSS)—one of the two major components of working memory according to Baddeley’s model.
Recall, the primary difficulty of verbal mental math is the limited capacity of our working memory, If you can improve your ability to represent numbers in your VSS, verbal mental math will become much easier.
One way of doing this is by training to use the soroban—a form of abacus—to represent numbers. Young students who have trained extensively with the soroban have excellent visual processing of numbers, and have been very successful in competitions. However despite its advantages, there are some disadvantages:
- It is very time-consuming to learn as it is an entirely new way of thinking about numbers
- Teenage and adult learners find it much harder to become accomplished in, similar to how adults find it harder to learn accents and gymnastics.
- Estimation techniques cannot be performed on the soroban—as Richard Feynmann famously described.
The good news is that there are alternatives:
- Solve calculations while listening to a conversation, or even while talking with someone.
- Practise solving calculations that stretch the capacity of your visual memory.
- Discover more patterns between numbers.
- Visualize numbers changing, such as on a digital stopwatch.
(5) Practise performing verbal mental math
Finally, we all know that improvement comes through practice. Combine some drills for verbal mental calculation into your training and you’ll be able to apply your number skills outside of the classroom more effectively.