Monthly Archives: October 2018

Nothing Matters!

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There’s no denying zeros existence within our number system. 0 represents nothing, e.g. there’s zero (0) of something. But this representation leads me to question whether 0 is really necessary at all! How important is it to teach children about 0? Do we even need to?

Its pretty well known that zero is far from an intuitive concept. Throughout history mathematical problems started as ‘real’ problems rather than abstract problems and so numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today (O’Connor and Robertson, 2000). There are huge mental leaps from counting 5 horses to 5 abstract ‘things’ and then to the cardinal idea of ‘five’. If ancient history solved a problem about how many horses a farmer needed then the problem was not going to have 0 as an answer. This seems to suggest that zero is not needed within our mathematical language, if we are making maths relatable and making connections to real life situations, there seems to be no need for a zero – you’d just have nothing left.

One crucial (an undeniable) purpose of zero is that it works as a placeholder in our number system. For example, take the number 603, without the number zero, we wouldn’t be able to have a placeholder in the tens place, and we wouldn’t be able to tell the numbers 603, 63, 630, 603000, and 6300 apart. So this seems to counter previous points and suggest that zero is in fact important.

However, Babylonians also had a place value number system without this feature for over 1000 years and there is absolutely no evidence that they felt that there was any problem with the ambiguity that existed. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge shaped appearance. It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate 216 or 21 ” 6 (Lamb, 2014). We can see from this that the early use of zero was to show an empty place and did not really use zero as a number at all! Instead, it was more like a type of punctuation so that the numbers were interpreted correctly. Again, this seems to question the need for zero as an actual number – why not swap it for any other shape or symbol?

Now I’m sure most people are aware of the great Fibonacci (especially if the history of maths interests you as much as me!). And it was in fact Fibonacci who was one of the main pioneers to bring new ideas about the number system to Europe. Even he however was not bold enough to treat 0 in the same way as other numbers and he spoke of zero as a ‘sign’ while the other symbols he spoke of as numbers. (Poliani, Randic and Trinajstic, 2001). Furthermore, counting from zero is actually not a very natural thing to do! It takes a certain amount of sophistication to realise that you can have a set of numbers containing zero elements, or that you need a number to describe such a set in the first place. As a result, most number systems from history (e.g. Roman numerals) don’t contain a symbol for zero, or any way to represent the concept (Mulwitz, 2012). The most ‘natural’ way to count is by making a mark for every item counted. If you have seven sheep, you write ||||||| If you have no sheep at all, what do you write? Nothing at all! If there were nothing to keep track of, why would you need a number to keep track of it? In our modern day schools, children are not taught to count with zero. Instead, they start with 1. This is because it would be confusing to use a zero as it stands for nothing and has no numerical weight (Mulwitz, 2012). Think about it – count aloud right now. Did you count 0,1,2,3 or 1,2,3?

I guess you could say that things aren’t looking good for zero. But fear not! Its place within our number system is actually more valued than is realised (realised by me, mainly).

According to Bellos (2010) India was the perfect setting for the creation of the 0 as a number in its own right: “The idea of nothing being something was already deep in their culture. If you think about ‘nirvana’ it’s the state of nothingness – all your worries and desires go. So why not have a symbol for nothing?” The symbol was called ‘shunya’, a word still used to mean both nothing as a concept, and zero as a number. According to research into Indian mysticism, zero is round as it signifies the circle of life! (Fry. 2016). Seems pretty important now doesn’t it!

Zero is an important aspect of our number system for many reasons; one being that it is the gateway between positive and negative numbers. From this use, it is essential, for example, in a thermometer and therefore measuring the temperature of the human body. On it’s left side – negative numbers and on it’s right – positive. This helps the evaluation process of extremely low temperatures (Weiss, 2017). Zero is also important in rounding off numbers. If the digits beyond the decimal are greater or equal to 5, zero replaces it. Without a 0, this concept would not be achievable, a concept which has real world consequences for approximations of big numbers. There are also areas of importance that I haven’t delved into such as algebra and calculus. While crucial in the use of 0, the maths is too complicated to go into in too much detail and I won’t make you sit and read the nitty gritty elements of each. But I promise, 0 is important! A fundamental element to each of these mathematic equations.

So, what is my conclusion? Well, it seems that zero is pretty relevant to say the least. One symbol that signifies literally nothing actually signifies a lot within maths. Whilst it is still unwise to teach primary three pupils algebra, it is still important to teach them the importance of 0 and not dismiss it as nothing (even though, technically, it is).

 

 

 

Need a hand with maths? You can count on me!

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Should children be allowed to use their fingers to help them count?

Children are often made to feel silly for using their fingers to count. As a child, I struggled with numbers – they seemed completely alien and frankly they intimidated me. I was told numerous times by numerous teachers to ‘stop using your fingers! You’re 8!’ I hid my embarrassment well and my fingers for that matter, which spent most of their time under my desk (manically trying to continue the calculation without anyone seeing). If I am being totally honest, I’m still partial to using my fingers to count (don’t tell Mrs Burton!).

So what are the actual pros and cons of finger counting?

If pupils are counting on their fingers, their calculations can become slower. This is not ideal for quick fire mental maths where speed is imperative (though I’m temped to argue here that maths isn’t really about the speed in which children can complete work – but that’s for another blog post). There is also the argument that if you have your fingers with you all of the time, and use them all of the time, there is no incentive to try to memorise and figure out how to do more than just basic arithmetic and it can actually be harder for children than using other visual or memory based techniques (Mukisa, 2010). You can even say that using fingers can give the perception that children are working at a lower level than they actually are and therefore may be given additional support that could be better given to someone else (I’m sure Mr Burton would agree!).

Although these are valid points, they seem to dim in importance when held against the pros of finger counting. Many studies have been completed around the topic, including Berteletti and Booth (2015). They analysed a specific region of our brain that is dedicated to the perception and representation of our fingers (called the somatosensory finger area if you were wondering). Remarkably, brain researchers know that we ‘see’ a representation of our fingers in our brains, even when we do not use fingers in a calculation. The researchers found that when 8 to 13 year olds were given complex subtraction problems, the somatosensory finger area lit up, even though the students did not use their fingers. This suggests that finger counting is not simply a small debate to be made at the teacher’s own discretion but is in fact a major factor for children’s’ development. Butterworth (2016) even goes as far to say that if pupils aren’t learning about numbers through thinking about their fingers, numbers “will never have a normal representation in the brain.”

This means that although there are a small number of reasons why children should not use finger counting, the reasons why still heavily outweigh them. Stopping children from using their fingers when they count could, according to new brain research, be halting their mathematical development (Boaler and Chen, 2016)

So what about people that can’t use their fingers?

There are in fact a number of cases of individuals who lose the ability to use their fingers and simultaneously lose the ability to work with numbers in their head. Henry Polish was 59 when he suffered a small stroke in the back of his left parietal lobe; an area that has an important role in number understanding. He had lost the ability to coordinate the movements of his hands and had lost ‘finger awareness’. Henry was unable to do simple calculations of addition in his head or even dial a phone number (Beilock, 2015). This suggests that allowing children to count on their fingers has less to do with confidence building (which my 8 year old self would tell you is still important) and is actually proven by research on the brain. So next time someone tells you to stop using your fingers because you look ‘silly’, tell them that you don’t want to end up like Henry Polish and let them figure the rest out (maybe even on their fingers).

Did you know that 93.7% of statistics are actually false??

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No, not really, I just made that number up. However, if I had put that percentage into a graph or colourful pie chart then most people would not question it (and I am including myself in that!).

If someone had told me a week ago that I would be sitting down to write a blog post on statistics because it interested me, well I’d probably have laughed and carried on with my day, but that is exactly what I am going to do. My memories of statistics in school are of staring at bar charts and line graphs with a blank expression and wondering how this can ever be worth learning.

Kennelly (2010) suggests that we have been raised to think that numbers represent absolute fact, and that in a maths, there is one and only one correct answer. Much less emphasis is put on the fact that in the real world numbers don’t give any information without units, or some other frame of reference. We are almost hard wired into believing that the numbers we see in everyday life were measured with infinite precision, and that can lead to some major misinterpretations! I remember sitting in a class in high school with my tutor, who (as an English teacher was very vocal about his issues with maths) said, “If you go to the Stonehenge one year, and the tour guide says it’s 5,000 years old, if you went back a year later, the tour guide won’t say it’s 5,001 years old.” Although arguably bitter about how numbers are sometimes dubious, he is (in hindsight) correct. We can’t ever really know the age of the Stonehenge that precisely because of the methods used to determine its age.

This therefore leads me onto the actual reliability of statistics and how simple maths can have wide reaching affects such as changing our perceptions and determining our unconscious biases. Statistics are undoubtedly persuasive. So much so that people, businesses, and whole countries base some of their most important decisions on this data. But any set of statistics might have something lurking inside it that can turn the results completely upside down. One particular study suggested that smokers were shown to have a higher survival rate over 20 years. The cynic in me suggests that this study is entirely fictional. However, it was a genuine study. What they didn’t tell us? The group of non-smokers were significantly older by average. It is therefore important that we carefully study the statistics and the groups that they describe before determining our perceptions.

While numbers don’t actually lie, they can be used to mislead with ‘half-truths’ (Kennelly, 2010). This is known as the “misuse of statistics.” It is often individual people or companies wanting to gain a profit from distorting the truth. For example, I am an avid user (and promoter) of good old hand sanitiser. My mum even used to suggest using it instead of touching the dirty taps and handles in public toilets (gross). Why? Because they kill 99.9% of germs of course! But in fact (to my horror) when reading up about statistics, I found out that it only kills as few as 46% in real world settings rather than a laboratory. This means only one thing. I am going to have to break the news to mum. Just kidding, this is one statistic that I am just going to have to pretend I did not see, not every Monday morning has to be filled with bad news.

Unfortunately, the murky web of statistical deceit only gets deeper from here. A 1998 real-life journal, ‘The Lancet’, claimed that the vaccination for measles, mumps, and rubella for babies was causing an increase in autism. It was later confirmed that the author lied about his findings. Unfortunately, this statistical misinformation led to a huge increase of people who were afraid to immunise their babies, which led to an increase in diagnoses of measles across the US. As well as this, a 2009 investigative survey by Dr. Daniele Fanelli from The University of Edinburgh found that 33.7% of scientists surveyed admitted to questionable research practices, including modifying results to improve outcomes, subjective data interpretation, withholding analytical details and dropping observations because of ‘gut feelings’ (Lebied, 2018).

So how can statistics be made to be misleading, either on purpose or by accident?

Faulty polling

The ways that questions are phrased can have a huge impact on the way a participant answers them. Specific wording can have a persuasive effect and cause individuals to answer in a way that the question asker requires for their study. For example, being asked ‘Do you believe that you should be taxed soother people don’t have to go to work?’ and ‘Do you think that the government should help people who cannot find work?’ These are examples of loaded questions. A better question would be ‘Do you support government’s assistance programs for unemployment?’

Flawed correlations

The big issue with correlations is that if you measure enough variables, eventually it will appear that some of them correlate. Studies can be manipulated (with enough data) to prove a correlation that does not exist or that is not significant enough to prove causation.

Misleading data visualisation

Graphs and charts can be misleading if they do not; use an appropriate scale, start at 0 and have an appropriate method of calculation such as a time period. Basically, if the graph looks good people may believe it.

Selective bias

This is about the nature of the people surveyed, for example asking only young people about lowering the legal age limit for drinking.

So what can I conclude from these surprising findings? (Other than to trust no one!). If anything, this has made me more determined and passionate about teaching maths. It is obvious that it is so important to teach children the real world value of numbers and to engage them, because if it is not something as small as misunderstanding how many germs you’re actually killing, it could be as large as a dangerous perception of immunisation. If I were taught with this knowledge, perhaps I would have paid more attention to these seemingly ‘boring and useless’ graphs.