During one of our most recent inputs with Richard we discussed a bit about time which eventually led us into a discussion about timetables and how they work.

We began talking a bit about the study of time- also known as Horology, and it’s importance to humans. The more I thought about it the more I began to realise that pretty much our whole lives are influenced by time, from the moment we get up to the moment we go to sleep at night our whole day is based on this mathematical concept without bus even thinking about it. If we really think more into it it’s not just humans that base their lives on time, what about animals? do they have any idea about time?

If we firstly think about household pets such as our beloved dog, many of us can say that when it reaches a certain time in the day the dog goes wild demanding food or their daily walk. This could be referred to as a “body clock” so we could say that animals have some concept of time, although it is hard to say whether its their internal clock or just when their tummy starts to rumble. When we think about other animals, if their routines aren’t influenced by humans, for example an owl, how do they know that they should come out at night? They have special eyesight for night time so we could say that it is just in their genetic makeup to know when to go to sleep and when to wake up. Hibernation is a similar concept, how do animals know to stock up on food and when to go to sleep for the winter?. Birds also know when to migrate, most of this could just be down to the weather being simply too cold for them to cope with so every year they come up with the same ideas of hibernation and migration? Or we could say that time is embedded into their instincts telling them when it is time to change their behaviour to suit their surroundings.

So before we had created our mechanical clock that we still use today how did we tell the time? was it just embedded into our instincts? Well Marie, N. (2016) helped with my understanding of the early use of time by speaking about sundials. A sundial uses shadow to track the movement of the sun which then gives us the time of day according to its position. Within our input we discussed just how much maths really applies to this, before using it people would have needed to be clear about what speed the sun moved at, did it travel at the same speed all day? This shows us that there was an understanding of maths and how to link mathematical concepts to apply it to real life.

Overall this input helped me to understand what it means to have a “profound” or “specialist knowledge of fundamental mathematics” as it has given me yet more examples of how maths can be used beyond the classroom without us even thinking about it!

Marie, N. (2016) When Time Began: The History of Science and Sundials. Available at: https://www.timecenter.com/articles/when-time-began-the-history-and-science-of-sundials/. [Date Accessed: 10/11/16]

During one of our workshops Richard presented us with the question “In the dark, how many socks do we need to take out of a drawer to make a pair?” Not to my surprise, the group’s first answer was “two” because who really has the time to rummage through their sock drawer to find a matching pair of socks? However, once it had been cleared up that the socks had to be matching it soon became obvious that there was more to this problem than I first thought.

So, if we have 2 different colours of socks in our drawer, pink and blue for example(guaranteed visit from the fashion police if one of each is worn) and we have three of each colour, how many do we need to take out before we find a matching pair? it took me quite a while to get my head around this but I finally managed it.

So basically, we would need to take 3 socks out of the drawer to guarantee a pair! just 3! because if there are six socks and three of each of the two colours we could either pull out one of each colour or a pair within the first two, in the case of the first two being two different colours no matter what colour we pull out next we will make a pair! my mind is blown!

So what about writing a formula to help all of our fellow sock wearers out there? basically what we did was pull out a number one more than the number of colours of socks that we have, still with me? So if we represent the number of colours of socks with the letter “n” then the formula should be n+1.

So if were to try it with different numbers, if I have 3 colours of socks and 3 of each colour I would need to pull out 4 socks to guarantee that I had a pair, the extra sock is to support me if I am unlucky enough to pull out one of each colour within my first 3, the back up sock has to be one of the colours therefore this guarantees a matching pair!

While this is a fun concept, it is actually a great example of why having a “profound understanding of fundamental mathematics” is so important. It is easy to look at the question and come up with the answer “2” but if we really unpick the question it soon becomes obvious there is more to it. I used my previous knowledge of maths and applied it to this problem, this highlights that when teaching children about maths we should build up their conceptual understanding rather than just their procedural knowledge. This will give them better foundations for attempting problems without a set procedure and solution, it will build confidence and generate innovative thinking.

Eastaway, R.(2010) How Many Socks Make a Pair? Surprisingly Interesting Everyday Maths. London: JR Books.

Since starting the Discovering Mathematics module I have been completely shocked and amazed at how interesting and relevant to real life the subject actually is!

For one of our tutor directed tasks we were asked to take a look at Liping Ma’s Knowing and Teaching Elementary Mathematics chapter 5. This chapter really made it clear to me what it takes to be able to teach Maths successfully in the primary school. Ma, L(2010) frequently uses the term “Profound Understanding in Mathematics” or “PUFM” and in chapter 5 she explains what it means to have PUFM and how to promote it through teaching.

A teacher’s knowledge of mathematics should go beyond the set topic areas, it should be extremely wide and thorough Ma, L.(2010, p.133). There should be clear connections between topics that also link directly to everyday life. I strongly agree with this as I believe that children find it harder to stay engaged with a lesson if they can’t see themselves using it in the future. Whereas if it is linked to real life situations or even just to an area of interest such as football or any other hobbies that the various pupils may have then they are far more likely to tune in and gain more from the lesson.

Ma, L.(2010, p.134) presents four key elements to teaching in this way.

1. Firstly she uses the term connectedness, a teacher should be able to create and display links between various concepts within mathematics, the subject should not be learned as isolated topics but as a combined chunk of knowledge.

2. Secondly, we should provide pupils with multiple perspectives, encouraging them to find various approaches to reaching a solution, however it is important to put emphasis on the importance of justifying their methods, this will then lead to a more flexible understanding of mathematics.

During one of our Discovering Mathematics inputs we were lucky enough to be visited by someone from the Dundee Science Centre. This was extremely interesting a greatly backed up Liping Ma’s idea of multiple perspectives. As humans on Earth we have a pretty big influence upon our environment. However if we look at ourselves as humans on a planet in our solar system within space, we are tiny. Even compared to the Sun the Earth seems pretty insignificant! This displays that at first when we reach a solution in maths it is important to explore a little more so that we can gain a wider understanding of the concept and not just start celebrating because we got the number that’s in the answers at the back of the book.

3. The third aspect Ma, L.(2010, p.134) highlighted was that teachers should reinforce basic ideas within mathematics, this will then encourage pupils to carry out real mathematical activity instead of just finishing a topic, moving on and forgetting everything that has been learned.

4. Finally, she talks about longitudinal coherence which to me, seems to be one of the most important. We should drop the attitude that certain topics are taught at certain stages in a child’s primary school life. We should always be referring back to previous learning and seizing opportunities to lay the foundations for the future. This one particularly stood out to me as many teachers may just teach the class what they need to know until they are no longer their responsibility.

I feel as though Liping Ma has really encouraged me to look at Mathematics in an entirely different way, in the future when I am about to teach a Maths lesson I will think more about what concepts are behind the topic that I am teaching, what parts of the children’s previous knowledge can I pull from to make it clearer for them and how can I link it to everyday life to make it more relevant?

References

Ma, L.(2010) Knowing and Teaching Elementary Mathematics. New York: Taylor and Francis.