Working Memory

Modern calculators (including smartphones, laptops and most digital devices) are great at performing calculations.

These devices use a type of electronic memory known as RAM to store the temporary data during these calculations.

When you buy such a device—such as a laptop—the manufacturer will tell you how much RAM is available. For example, a cheap modern laptop might have 8 GB of RAM. This is enough memory to store 2.5 billion decimal digits.

As humans, when we perform calculations with our brains, we also need something like RAM to store the temporary data during mental math. Psychologists call this working memory rather than RAM, but it has the same function.

However, our brains have very, very different hardware to the microchips in laptops and smartphones! And our working memory has vastly less memory capacity—only enough to hold about 10 digits at a time! This makes mental math much more difficult for us, compared to calculations on any modern computer.

Note: of course it is possible to remember more than 10 digits, but only when using long-term memory (e.g. remembering multiplication tables or your own phone number) or short-term memory (perhaps using mnemonic techniques such as the method of loci and/or the Major system). But calculations are not possible outside of our working memory, so this is not relevant.

Because we have very limited working memory:

It is important to use mental math methods that don’t require too much working memory.
It is important to concentrate, so that we don’t waste space in working memory with unrelated thoughts.
It is helpful to have certain common calculations memorized (e.g. 4 × 7 = 28) so that they can be used directly without requiring space in working memory.

Structure of Working Memory

Psychologists use Baddeley’s Model to describe the structure of our working memory. According to this model, working memory has two components:

Visuospatial Sketchpad (VSS): this stores temporary visual data in a highly compressed form. This could be someone’s face, a 4-digit number or a place that you have seen before. The VSS also stores temporary data about movement.
Phonological Loop (LP): this stores about 2 seconds of temporary sound data—such as a short sentence or phrase, a fragment of a melody, or about 6–8 spoken digits.

As you can see, we have two options for storing numbers: in the VSS and in the PL. That is, if someone told you the number “one hundred and forty-eight”, you could either hear that phrase in your mind, and/or imagine the digits 148 as if they were in front of you.

Working Memory for Mental Math

Mental math requires quickly modifying numbers stored in working memory.

But this is typically much, much easier when the numbers are stored in the VSS, rather than the PL. For example, calculating one tenth of three hundred and fifty is easiest when you imagine the “350” and simply chop off the terminal zero to get “35”. Trying to do this without imagining the numbers is much more complicated.

Furthermore:

Every time you access data in the PL, it takes time to imagine hearing the data. So it is slower.
Sometimes data in the PL gets corrupted to something similar (but now incorrect) without you being aware. So you need to maintain high concentration to prevent this. Data in the VSS persists in its correct form much more reliably.
If you need to do mental math during a conversation, your VSS can handle the numbers without getting distracted by the conversation.

In conclusion, we will mostly use the VSS and represent numbers using this visual memory.

Phantasia & Aphantasia

Try to imagine the 3-digit number four hundred and ninety-nine. Can you “see” that in front of you? Do the digits seem real? Do the digits persist or do they evaporate as soon as you imagined them?

For some people, visual imagination appears very real and vivid. These people have a high degree of phantasia (a high degree is known as hyperphantasia). Other people (such as me) perceive these numbers less vividly—a lower degree of phantasia. Personally, the digits are perceived for only a moment. Very weak phantasia is known as aphantasia.

If your phantasia is relatively weak (like mine), do not worry about it. Your perception of the data you imagine does not affect whether you can process it effectively in the VSS 🙂

Capacity of our Working Memory

How many digits can we store in our working memory? Watch this video to find out: TODO video

It would be nice to think that we could improve the capacity of our working memory with some sort of training. Unfortunately, the evidence shows that this is not practical:

I’ve personally tested the working memory of various top mental calculators (including myself, plus other highly-trained individuals in various types of mental calculation) and our results are similar to the untrained population. This indicates that extreme skill in mental math is not a result of expanding the capacity of the working memory.
Various training exercises exist—notably Dual N-Back—that claim to increase working memory capacity. However, the conclusion from various research (notably a 2016 meta-analysis by Melby-Lervåg et al.) is that doing these exercises makes you better at the specific exercise, but doesn’t actually increase working memory capacity.

In conclusion, our strategy for getting more comfortable with mental math should be to practise using our VSS for mental math, in order to use our fixed capacity more effectively. Since this strategy demonstrably works, we use it during this course.