The Counting Cloth and Tudor sums.

Introduction

In the dark ages and early Tudor times, in Europe, one of the main ways to do sums was by using a counting board or cloth. Writing out sums would have used up precious ink and paper, so devices like the counting board, abacus and just simple piles of pebbles were used - portable, reusable, frugal.
Initially, piles of pebbles would have been used to represent numbers. The counting board and abacus were methods to extend this without having to use quite so many pebbles. The idea is that pebbles on the ones line stand for just what they are, one pebble, but pebbles on other lines stand for different numbers of pebbles, 10, 100, 1,000 pebbles. (Notice the powers of ten progression.) A counting board can be scratched into the dirt, chalk on a table, marked on slates, or made rather more permanently.
Until comparatively recent times, most merchants in Europe used it, and even when the methods we commonly teach for arithmetic were introduced, many teachers of arithmetic thought it was useful to first learn it to help with understanding. The first English language maths text books, brought out in the 16th century, introduced the concepts of "pen reckoning" and using the arabic numerals we are so familiar with today, until then Roman numerals and counting cloths were used. It was only for sums with larger numbers that a scholar of pen reckoning was faster than an adept with a counting cloth.

This image shows a competition between the counting board and arithmetic done by pen reckoning (we might call it "paper and pencil", but they didn't have pencils.) It is from G Reisch, Margarita Philosophica (1508).

Roman Numerals

The "Arabic" numeral system wasn't introduced to England until the 16th Century (I am uncertain about the dates for the rest of Europe); prior to this people used the old Roman system (and it took a fair while to change over.) They didn't use it exactly as the Romans did though, or what we 21st century people think of as the Roman system. These days, we are mostly likely to see them on a clock face or at the end of a TV programme to denote the year.
Roman numerals are not a positional system, they don't have a symbol for zero. They are an additive system, with each symbol representing a value no matter where it appears.
Here are the symbols and their values:

An easy way to remember this is that V is half of X (when cut across the middle), C is a century (from the Latin "centum" which means 100), M, millenium (from the Latin "mille" which means 1,000), is like two facing Cs (|) with a line between and D is half of that, |).

To go to higher numbers, there are a couple of conventions. One is that we use a bar over a symbol for 1,000 times, so V overline would be 5,000. It is more common to see it with an underline too, though.
The other way is to use deep C brackets around, so that 1,000 is ( I ) which looks curiously like a curly M, and 10,000 would be ( X ). I have not seen any examples where the 5 based letters are used with the deep brackets.

Although the system is value based rather than position based, it is conventional to have the bigger values to the left and the smallest to the right. This makes it easier to read and use.
Technically DM would be the same as MD and equal to 1,000+500=1,500 but only MD would be used. The biggest possible symbols would be used to make up a number. While CCCCCCCLXIIIIII is 100+100+100+100+100+100+100+50+10+1+1+1+1+1+1=766 it would have been more efficiently written as DCLXVI

Subtractive Rules

An even more efficient method was developed. 4 takes up a lot of characters, IIII. The convention for this method is to "subtract" a symbol representing a 1, 10 or 100 from only the next two high symbols respectively, by writing the smaller to the left. The only subtractive forms are:

The Tudor method

However, the Tudors and late mediaeval period didn't use the subtractive method, and they didn't use the serif capitals we are used to, but used a lower case notation of i for I (and quite often without a dot on top.) They also used a "terminal j" - the last in a sequence of ones would have been written as j. The reason for the j being the end of the ones is so that you can see it's the end of the ones and thereby the end of the number. As a result, the numbers representing 1 to 12 would have been written thus:
j, ij, iij, iiij, v, vj, vij, viij, viiij, x, xj, xij
Not having to leave a big gap between numbers saves paper. Not a big saving, you think? Paper was so precious that they saved every scrap they could, scraped the top off and reused it.
Why lower case? I don't know, but having used this system, it's a lot faster to write i and j than I. So they might well have written 1-12 thus:
jijiijiiijv vjvijviijviiijx xj xij
and understood it just fine. You can see the amount of space saved compared with the line above, too!

Here are some example representations of numbers:

The basic counting board

This image shows my Tudor style counting board.

The bottom-most line is for the 1s, then the 10s, 100s, 1,000s and 10,000s. On my board, I haven't used "1", "10", "100" or "1,000", I've used J, X, C and M.

The little x in the 1,000s row marks the 1,000s. You would find one in the 1,000,000 row too. It is thought that this mark is why we use a mark, or comma in the UK (. is used in Europe) to separate the groups of three digits in large numbers, although more recently it has become a convention to use a space to remove confusion between countries. If you have no "lower value" numbers, then you can start your 1,000s on the 1s row, to save space. You might chalk your x in on that row temporarily.

The board has two main sections, which are subdivided into further sections. I typically use three of them, only going to the fourth for more complicated sums.

This image shows my Tudor style counting cloth. Nice and portable, but because of the raised lines due to the sewing, I am using spaces rather than lines to represent the 1s and 10s etc. There are dark blue lines to mark the vertical divisions, but they are pretty hard to see in my photo. It is made from dark green well felted wool and some linen thread.

For practise, it's perfectly quick and easy to draw it out on paper, thus:

Note that standard A4 or letter paper is a bit small, A3 or bigger is easier to work with. I used paper from a roll, and I have taped it down for ease.

The Tudors would not have drawn a counting board on paper because paper was too precious. However, they might have chalked a table or scratched in the ground for a quick temporary board.

The counters were called "Jetons", and were about the size of a 1p piece. Many were made in Nuremberg. They were quite detailed, as you can see from these which were found in Haddon Hall in Derbyshire. This is so that a merchant would be certain none were sneaked onto his cloth during calculations; he could identify his own at a glance. For practise, I find 2p sized tiddly winks are handy. In this tutorial, I am using different colours so that you can see which counters I am currently working with.

Getting started with representing numbers on the board

Here is 4686 represented on the board. This has 4 counters in the 1,000s row, 6 in the 100s, 8 in the 10s and 6 in the 1s. As you can see, this takes a fair few counters and we're trying to save dealing with quite so many.

The fact that Roman numerals have V, L and D as part of their system might give you a hint as to what we do next. Can we get rid of some of these extra counters? How about the ones I have turned red...

We use the spaces between the lines to represent those multiples of 5. We don't typically write the names in but I have shown them here for ease. I've swapped the red ones which were on the lines in the above picture for the ones in the gaps below. Five counters on the 1s line are replaced by one counter in the 5s space, similarly five counters on the 10s line is replaced by one in the 50s space, and five on the 100s line by one in the 500s space.

As a rule, for a finished number you may not have more than four counters on a multiples of 10 line, and you may not have more than one counter in a space. Practise representing some numbers on the board before having a go at the next steps - the sums. If you have five on a line, take them away and put one in the space above. If you have two in a space, take them away and put one on the line above. This, I generally call "tidying up". To do this, work from the bottom of the board upwards. While working with the counters for sums, you may need to have more than the rule mid-calculation.

Practise

Try representing these numbers on the board. Have a go at writing it out Tudor style as well. Click on each link to see a picture of the answer.

Text books

I mentioned text books earlier. The main one which is the first to introduce "pen reckoning" in the English language was "An Introduction for to Lerne to Recken", it was written in 1539 and the author is unknown. It has a section on using the counting cloth at the back. It, and other early text books, have been carefully reprinted by Renascent books and are fascinating to peruse. There are also a number of scanned books available through the "Early English Books Online" project. These can be difficult to read, and in some cases the scans pick up details from the other side of the page, but they are a super resource.

Go to:
Addition
Subtraction
Multiplication
Division