Richard said to us that in order to look at some of what we would look at in Discovering mathematics, we would have to forget everything we know. And he was right. One of our lectures we looked a place value and binary and my mind boggled. So much so that Richard went over binary in greater detail in another lecture – thank you!

I never knew the existence of any other number system until this module. Well, I knew time was not a base 10 however as I said in a previous post, time along with a base 10 system and place value a base 10 system is taught from the very beginning of school. In the use of units, tens, hundredths and so on. One of the first things we learn with numbers, and that I have taught my daughter I to count to ten (even though technically a base 10 system is 0-9, but who learns to count this way?) I do this every day with her, from counting going up stairs and reading books. 1-10 and so on is the way it is, she does not know why, and I did not think about it in depth until this module either.

The idea of different base systems because of how easy I think the one we use is seems bizarre. We looked at “yan, tan, tethera…” a base 20 system Lincolnshire shepherds used to count their sheep. If a shepherd had more than 20 sheep, he would record one cycle of 20 by putting a pebble in his pocket or marking a line in the ground and start again (Bellos, 2010). So five marks or pebbles would represent 100 sheep. This base 20 system works well for what it was used for – counting sheep. If a base 10 system was used for this example, then there would be a lot of notches or pebbles to carry (depending on the size of course) and could possibly become more confusing. However, using this base 20 system for anything other than herding and counting sheep, does not seem the most sensible option.

Bellos suggests the trick of a good base system is that the base number needs to be large enough to be able to express numbers such as 100 with ease. Obviously this why a base 10 system works so well in his option but I also think the base 10 system works so well as it is in tuned with the human body. I still use my fingers occasionally to count. But if we weren’t born with 10 fingers – who knows if we would be using a base 10 system or not.

Binary

So looking at a base 20 system wasn’t too difficult but then we moved on to binary. Cue the utterly puzzled feeling and look on my face! In the first lecture I didn’t really get it and we quickly moved on. I’ve gone through my whole life not knowing what binary is and how it works so I wasn’t too concerned at the time. But, that’s not what this module is about is it! Cue BBC Bitesize website which explained Binary is a base 2 number system that uses 1 and 0 and is processed by computers! WHAT!? Yep, back to not ever knowing or needing it or wanting to look at it again. Thankfully Richard did show us the YouTube video below and showed us a different table to what he had shown in the previous lecture and alas, as I said at the start of this post, in order to understand, we need to unlearn all that we know about numbers. As there is only two digits that can represent values in binary (0 and 1) this what I found hardest, the difference between number and numeral. We are so used to the number 1 meaning 1 and 2 meaning 2. I still don’t feel confident enough to explain how binary works but I can use this table myself and have included an image of my handy work! (I just hope it is correct!)

I don’t suggest going in to great detail of different base number systems with children but perhaps delving in to them to make them aware that we should count ourselves lucky that the one we use is easy compared to “yan, tan tethera”. But, the fact that binary is used our digital world and Code Club was something was popular at my placement school, it is important for at least as a teacher to know one of its existence but to be able to explain it in simple terms. I’m still not entirely keen on Binary, due to it being so different from all I’ve ever known. However, appreciating how difficult it was to get my head around certainly gives me an understanding of how learning something new for a student is and how easily fragmented it could become.

Connectedness

And of course, all of this links to Ma’s “connectedness” of Profound Understanding of Fundamental Mathematics (PUFM). In order to understand different number systems, I had use prior knowledge to link different mathematical concepts together. That is why it is so important for elementary stage to have the simpler “basic ideas” instilled within students so that they can be used instinctively when learning new topics. However, it is the duty of the teacher to show the connections between what they have learned and how that knowledge is implicit to learning new topics and for their future. I feel that by learning about different number systems and binary in particular, I was able to draw on previous knowledge and with that in mind apply it to a new concept which together become the “unified” body of knowledge that Ma describes (Ma, 2010).

References

Bellos, A. (2010). Alex’s Adventures in Numberland. London. Bloomsbury Publishing Plc.

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. London: Routledge.