(One of the oldest surviving fragments of Euclid’s Elements, dated to circa AD 100)

Prehistoric Mathematics. Sounds boring, right? Wrong. Now I’m not one for history but after our workshop last week on the origin of number systems, I decided to look into the history of maths and number systems a bit more. If you (like me) think that prehistoric maths is something to that will put you to sleep, be prepared to be amazed as it really opened my eyes. So if you thought our current understanding maths was born overnight, you’re wrong as it actually toke several ages and to this day we are still discovering new mathematical concepts as maths research is very much ongoing and ever-growing. In fact, numerals are approximately 5,500 years old!

Back in the day, our prehistoric ancestors actually knew the difference between say one stick and two sticks, all through instinct as believe it or not school wasn’t around back then. However, the massive leap in intelligence from the idea of having 2 objects to the invention of a symbol (i.e. 2) or a word for this abstract idea (i.e. two) did take numerous centuries. Even in this day and age there are a few remote hunter-gatherer tribes in Amazonia whose number sequence simply consists of “one” “two” and “many”. Similarly, there are other tribes who only have words for up to five. How bizarre is that? Imagine trying to bake a cake using a recipe that just says ‘many’ eggs. However, without settled argiculture and trade there is, naturally, no need for a formal system of numbers like ours.

The earliest evidence of mankind thinking about numbers that we have today is the Ishango bone. The Ishango bone consists of notched bones from central Africa dating back to 35,000 to 20,000 years ago. However, as you can see from the image below, it was mere counting, what we would call tallying rather than mathematics.

What’s interesting is this bone was found in 1960 by Belgian Jean de Heinzelin de Braucourt among the remains of a small community in Africa that had been buried in a volcanic eruption. Whilst some interpret the bone to have that of tally marks, others suggest that the scratches may in fact be for getting a better grip on the handle or some other non-mathematical reason, it’s impossible to know and is down to individuals beliefs. Personally, I’d like to believe that the implement was used the construct a numeral system.

When analysed you can see that the bottom row (from right to left) begins with three notches and then doubles to 6 notches. The process is repeated for the number 4, which doubles to 8 notches, and then reversed for the number 10, which is halved to 5 notches. These numbers may not be purely random but could suggest some understanding of multiplication and division by two. Therefore, this bone may have been used as a counting tool for simple mathematical procedures.

In the book *How Mathematics Happened: The First 50,000 Years*, Peter Rudman disputes that the development of prime numbers could have only arouse after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. Additionally, he writes that “no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10.”

When Alexander Marshack examined the Ishango bone, he believed that it may represent a 6-month lunar calendar. However, Judy Robinson argued that he over interpreted the data and the evidence doesn’t suggest a lunar calendar. Furthermore, Claudia Zaslavsky suggested it that this might indicate the bone was created by a woman tracking the lunar phase in relation to the menstrual cycle.

Moving on, mathematics properly developed as an acknowledgement to bureaucratic needs as a result of settlement and agriculture development (i.e. measurement of plots of land, tax, etc). The first time this occurred was in the Sumerian and Babylonian civilisations of Mesopotamia and in ancient Egypt.

To link back to more modern day mathematics, I started reading ‘Alex’s Adventures in Numberland’ which I highly recommend as it provides some eye opening moments and shows you just how maths isn’t boring and mind-boggling but in fact is fascinating and underpins our everyday lives. I can guarentee you will learn something new. For example, I didn’t know that every number can be winnowed down to a product of prime, try it for yourself! There is some more interesting theories as well that are still to be proven such as the Goldbach Conjecture (every even number bigger then 2 is the sum of 2 primes).

So to link this all back to teaching as I’m sure you are wondering why on earth any of this will be relevant if I will never teach it in the classroom. Well, it all links to Liping Ma’s 4 interrelated properties. This post refers to ‘longitudinal coherence’ as by not limiting your knowledge to whatever is in the curriculum (i.e. prehistoric maths is not in the curriculum) you allow yourself to use every opportunity to learn about these crucial elements of maths and you never know you might even enjoy it! Additionally, there was a little bit of multiple perspectives in here as I considered all the various approaches to the origin of the Ishango bone which allowed me to have a flexible understanding of this concept. Surprisingly, I actually enjoyed discovering the understanding of maths long before my time and it’s encouraged me to open my eyes to a range of different mathematical topics that are out there rather than just the curriculum as I believe it will strongly benefit my teaching in later years as well as my overall understanding of mathematics.

**References**

Bellos, A. and Riley, A. (2011) *Alex’s adventures in numberland*. London: Bloomsbury Publishing PLC.

Mastin, L. (2010) *Prehistoric mathematics – the story of mathematics*. Available at: http://www.storyofmathematics.com/prehistoric.html (Accessed: 10 October 2016).

*How mathematics happened: The first 50, 000 years*. United States: Prometheus Books.

Richard HolmeThis is a really insightful analysis with several sources, and ideas, I haven’t seen before. This will certainly feed in to how I approach this module in future. I also learnt a new word – ‘winnow’ – so thanks for that too!

I suppose my next question is how can your understanding of how children learn be informed by the way very early civilized people made sense of mathematical concepts. On face value this might seem a strange question but the way human constructed meaning might help you understand the way children develop understanding. What do you think?

Ailsa MackiePost authorThank you very much for your comment and I am delighted that I have been able to contribute to the way you will approach the model next time! I think that is a very good question as it will indeed help me understand the way children develop understanding. Before I looked into how early civilised people made sense of maths, I believed that children learn well when given the opportunity to discover a concept/theory themselves and then having the teacher explain it afterwards as this will pull them in and get them interested from the start as they are finding things out for themselves. For example, I watched a video on teachers.tv about constructivist model of learning in which a teacher taught Pythagoras by getting the pupils to discover it by themselves by asking them to figure out how they would find the other side of a triangle and not revealing it as Pythagoras’ theorem till the end of the lesson, therefore they are taking ownership of their learning. My understanding of this was strengthened when I looked into prehistoric maths as when they first tried to make sense of mathematical concepts there was no teacher there to tell them and therefore they learned through discovery and trial and error such as the theory beyond the Ishango bone being used like a tallying system. So I suppose my understanding of how children learn has been informed by the way that very early civilised people made sense of mathematical concepts through discovery and creating there own theories. I hope this answers your question and I’d be grateful if you let me know what you think.

RichardHi again,

I think this is a really interesting explanation. This idea of learning, thousands of years ago, well before formal schools or teachers existed is a great theory. Although it seems fairly obvious I had never considered this.

And thanks for the reply!

Richard