What is Mental Math?

Mental math is the ability to perform mathematical calculations in your mind. It is also known as mental arithmetic, mental calculation, quick math, fast math, or doing math in your head. In German, it is known as Kopfrechnen, which translates into English as “head calculation”.

Have you ever studied mental math? For many people, the last time they developed their ability to do math in their head was at primary school, when they could still count their age on their fingers! At this age, Mathematics classes include counting, addition facts, memorizing the times tables, as well as simple methods for addition, subtraction, multiplication and division.

Mathematics vs. Mental Math

Later Mathematics classes teach many topics, such as algebra, trigonometry and calculus. Mathematics is a huge, broad subject, and mental math is only one small part of it. But if you never studied more mental math, your calculation abilities might be stuck at the basic level of primary school arithmetic, regardless of your ability in other areas of Mathematics.

In a strict sense, pure mental math should all be done mentally—in your head. This means without needing pen and paper, or calculators. This gives you the most versatility. However, in some practical situations, you might prefer to write some steps if you are not confident doing the calculation purely in your head.

Basic mental math from primary school typically involves calculations with only one step, such as 7 × 8 or 30 + 50. These can be done essentially from memory, and therefore are easy if you know your addition facts, times tables, and so on. Pen and paper is needed only for stating the question, and for writing the answer for the teacher to check.

The Main Challenge of Mental Math

More advanced arithmetic at school involves using written methods to add e.g. 375 + 289. This is not purely mental math, as parts of the method are usually written down—such as the “carry digits”.

It is much more challenging to solve a similar question when the question is spoken aloud rather than written!

 

The main difficulty in mental math is needing to hold several digits in your mind without writing them down. For example, when solving 34 × 8, you might calculate 30 × 8 = 240 and 4 × 8 = 32. But the final addition of 240 + 32 must be performed purely mentally, as these numbers are not written. This will be difficult if you don’t have experience with such calculations, because our minds are only able to hold a few digits in working memory at a time. You can read more here about the limited size of human working memory and how that relates to mental calculation.

Improving Mental Math

Most of the people I’m coaching in mental math are professionals. Even if they have a strong background in Mathematics, or a proficiency with calculators and spreadsheets, they have realized that mental math is a separate skill, and want it to become a strength of theirs. Typically, the most relevant area to work on is the intermediate mental math, with high levels of speed, accuracy and confidence.

Intermediate and Practical Mental Math

  • adding and subtracting numbers of up to 3 digits
  • multiplication of 1- and 2-digit numbers, such as 24 × 7
  • division of numbers by simple 1- and 2-digit numbers, such as 15 ÷ 8 or 3700 ÷ 12
  • work with percentages using the four operations, such as 4.5% of 12 million
  • multiplying whole numbers by a simple fraction, such as 3/4 of 25
  • estimating multiplications and divisions (where the numbers are too difficult for exact calculation)

Because mental math is a skill, the good news is that with the right training, you can improve your mental math to the level that gives you confidence with numbers.

As with most skills, there is no upper limit to your potential ability—there are also techniques for more advanced mental calculation! However, these are seldom done outside of specialist competitions.

Further Resources