Richard’s last two inputs about number systems and place value have left me perplexed to say the very least.

Binary, a counting horse and indigenous tribes…

All these aspects were covered in two inputs and they definitely broke down my structured beliefs on what mathematics really is. A key point that I took away from the lessons was to think beyond the confinements of what we know about the subject of mathematics and our 10-based numeral system.

It really is Discovering Mathematics all over again in a much deeper-rooted manner.

Rather than getting bogged down in the complexities of the possibilities of differing number systems and giving up, I embarked on reading Alex’s Adventures in Numberland in order to find an everyday answer:

*“Without a sensible base, numbers are unmanageable”* (Bellos, 2010, pg. 44).

Base systems of five, ten and twenty have been the most commonly used through the various cultures of mankind (Bellos, 2010) and it’s a pretty straightforward answer of why:

What is the most common tool a child (or anyone for that matter) would use in order to count? They use their fingers! In Early Years, “fingers are used in a range of ways and with varying levels of sophistication.” (Wright et al. 2006, pg. 13) Well, this instinctive notion towards mathematics has a rich meaning in terms of how we represent our numbers because, in reality, that is all a numeral system is: a way in which we express numbers and quantities of those numbers.

However, Richard introduced us to different variations on number systems that go beyond our commonly known systems. Not only that, but we were also shown the other number systems that were influenced by the culture that they were used within.

Number systems, in reality, are ways in which we give identity to a quantity. 1,2,3,4,5 are all just the symbols we have given to a quantity. Delving deeper into this concept of a numeral system, we need to first realise, how did we create such a vast amount of numbers?

Lets take an indigenous tribe like the Arara tribe in the Amazon for example; they only have base 2 number system, where they only have 2 words for 1 and 2, and anything after that is a combination of the two (anane =1, adake = 2, adake anan = 3, adake adake = 4 etc.) (Bellos, 2010).

Why? They have no real use for numbers beyond that. Their lives revolve around survival. A reserved community in the amazon are never going to need thousands or even hundreds of something, so they just don’t have it.

Farmers have also been shown to have their own number system where Base 20 is used. Farmers would count up (yan, tan, tethera) until they got up to 20 and then they would either pick up a stone or make a mark on the ground in order to indicate that he had got up to one set of 20 sheep and then he would begin again.

Yan. Tan. Tethera.

Could you imagine trying to quantify, say, a population of a whole country using these formats of number systems? The representations would be very time consuming! Once again, the tribes and farmers would not have a population that could equal the populations we have across the modern nations.

The fact that we have so many numbers is down to the fact that we have advanced to the point that we *need *a huge amount of numbers. We are beyond just surviving as a species, like the indigenous tribes or the independent farmers of the past. Similar to my post about the advancement in agricultural, we’ve adapted in order to advance and, in doing so, adopted a number system that allows us to easily distinguish between place value when putting a quantity on something (particularly large quantities). As we have multiplied, so have our quantities of population, food, cars, houses and so many more factors. An indigenous tribe does not need a number system that goes up to a million because that number has no right to exist. When are they ever going to need a million things of anything?

Here is an interesting video by TED about the history of our numeral systems:

Binary, another spanner thrown into the math-works, was also something difficult to understand at first, due to it using the original place holder symbols of 1 and 0… and that’s it. Similar to the Arara’s, binary only uses two symbols to define various quantities. I vaguely remember aspects of binary being used way back in high school IT lessons; however, I didn’t really know the whole purpose behind it. Computers do not work the same way our brains do. Binary is used because a computer can only work through programming with a state of on or off. This is where the 2-based number system of binary comes into practice well:

The circuits in a computer’s processor consist of billions and billions of transistors. A transistor is basically a tiny switch that is initiated by signals of electricity passed through the computer. The digits 1 and 0 used in binary can reflect the on and off states of a transistor (BBC, 2017). So, computer-literate people can program commands into a computer using binary and the computer will be able to translate these codes (much quicker than the human brain could) into processes.

James May explains binary numbers within this video:

Now, if indigenous tribes, binary and abstract number systems weren’t enough to comprehend across two inputs, then this question that we were faced with will surely perplex you:

**Can animals count?**

Many opinions and theories circulated the room but the main thinking was… not really. An animal can maybe understand a form of quantity but they probably don’t know *why *they understand this.

An interesting video Richard showed us was about the enigmatic counting horse called Clever Hans. In the 1900s in Germany, Hans was taken around the country to demonstrate to people his great ability to work out arithmetic that his owner asked him to calculate… Could this possibly be true?!

Unfortunately, it was too good to be true. What Hans was *actually *doing was reacting to the positive praise through body language of his owner when given a sum. He would learn from cues when to facilitate an answer through tapping his hoof. Psychologist Oskar Pfungst investigated this and even discovered that the owner of the horse didn’t even know he was giving these positive cues, which revealed another theory years later known as observer-expectancy effect. This means that Han’s owner subconsciously gave the answer that he wanted through visual hints like a nod of the head.

Animal cognition is not the same as human cognition. Milius (2016) wrote an article about the topic of animals and mathematics and stated that “some nonhuman animals — a lot of them, actually — manage almost-math without a need for true numbers” and she explores how the argument has varying perspectives from psychologists and scientists alike. One theory is that animals just so happened to gain aspects of mathematical thinking through convergent evolution from similar ancestors as us. This evolution is similar to how bats and birds can fly however, are from completely different families and their wings derived in different pathways of evolution (Milius, 2016). It is also similar to sharks and dolphins both having to gain the best possible traits and abilities to survive in the ocean, yet neither are related in any format. Animals have gained the ability to understand some form of quantity in order to judge if there is 1 or many predators in front of them, however, they don’t have a numeral system to define this understanding.

In reality, much like the tribe, animals have no real use in knowing numbers because they do not think conceptually, like we do as a modern society.

Returning to the concept of place value within numeral systems, teachers need to be able to comprehend what the underlying meaning behind what place value really is. As Ma (2010) found in her studies, the students that excelled the most in mathematics in terms of comprehending number systems were the ones that were taught the appropriate measures when dealing with higher digit numbers when it comes to differing place value with subtraction and addition, for example.

Therefore, as educationalists, we need to know what the best methods for students to tackle number systems are. The answer? Preference is really down to the student. However, we need to be there to facilitate the various learning styles, challenges and boundaries that come our way in terms of learning mathematics – **in a positive manner. **This correlates well with Ma’s basis of **multiple perspectives: **teachers should be “…able to provide mathematical explanations of these various facets and approaches. In this way, teachers can lead their students to a flexible understanding of the discipline.” (Ma, 2010, pg.122). Giving children multiple avenues to explore problem solving, in terms of arithmetic, will only benefit their independent evaluation in terms of dealing with mathematical problems. It will benefit them far greater than giving them a formula.

In conclusion, Mathematics has various avenues when it comes the representing quantities and exploring huge amounts of quantities. Knowing the basics of 1,2,3 as teachers will only get us so far. It will also hinder our children greatly… Even discussing the great horse Clever Hans would be an interesting lesson to explore how different mathematics is between them and their pet peers. Being open to mathematics as a vast subject can only bring about great things within the classroom.

Reference:

BBC (2017) Bitesize: Binary [Website] Available at: https://www.bbc.co.uk/education/guides/z26rcdm/revision (Accessed 19th of October 2017)

Bello, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Milius, Susan (2016) Animals can do ‘almost maths’ [Article] Available at: https://www.sciencenewsforstudents.org/article/animals-can-do-almost-math (Accessed 17^{th} of October 2017)

Wright Martland Stafford Stranger (2006) Teaching Numbers: Advancing children’s skills and strategies 2^{nd} edn. London: Sage Publishing Ltd.

Ma, Liping (2010) Knowing and Teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States New York: Routledge.