# Reflecting on Discovering Maths

When beginning the module, I wasn’t exactly sure what to expect from it. In fact, when picking it at the end of first year I expected I’d be going into another maths class like at school – which filled me with excitement. I have to say I was a little disappointed when I found out this wasn’t the case.
However, the module has allowed me to explore so many areas of maths that I didn’t think of before. Who would have thought maths was necessary when riding a motorbike?
At the beginning of the module Liping Ma and her Profound Understanding of Fundamental Mathematics confused me terribly. I just couldn’t get my head around the meanings and explanations of the 4 principles she used to describe what a PUFM was. When reading her book, I was submerged into this and found out how important it really is. When looking back onto my first placement I’ve realised even more how necessary this is. One of my first maths lessons didn’t exactly go to plan and I ended up getting myself mixed up and confused by the questions the pupils were doing. The lesson was on area – straightforward to someone who has done Higher Maths – but when the pupils were coming to me confused about one particular question, I was beginning to get flustered when my explanations weren’t helping them to understand. I have now realised how important it is to have PUFM as it allows you, as the teacher, to explain much more to the pupils in depth and breadth. With this deep understanding, you are less likely to get confused yourself, which is vital when trying to instil confidence and understanding in the pupils.
Throughout the different inputs I found it fascinating to unpick all the different areas where maths is used. My favourite input was the one on Logistics and Supply Chains, which included information about food miles and how much it costs to fly or ship things to us, how things are packaged to maximise space when shipping and retailers having to choose what to stock in their stores. We then did a role play exercise and game to see how demand planning in stores worked. We had to work in pairs to choose what 5 items and how many of each item we would buy per quarter of the year, without going over budget, to see who could make the biggest profit. This allowed us to see how effective stores have to be when choosing what to stock, as if they buy too much of something and it goes off or people don’t buy enough of it, this will all be money wasted. They have to be able to budget their money efficiently and buy what will sell most. A tricky process even in a make-believe game.
Overall, I think the module was very effective at providing me with more maths knowledge and helping me to see how important it is that I do have a good maths knowledge for going out into schools. I need to be able to give the pupils in my class the best opportunity to learn maths in a fun and challenging environment that pushes them to the best of their ability. I believe the interaction in this module has helped me realise how important that is.

# Maths and Sport

As a child I wasn’t particularly sporty. However, I did try. In my 6th year of school I did crash Higher PE which was certainly an eye opener to how bad at sport I really was. I was forever put in the good team, instead of the really good team but this didn’t mean to say I didn’t try. It just wasn’t meant for me. For my higher performance I did dance, as this is the one sport – even though I was constantly reminded by the boys in my class that it wasn’t a sport – I really excel in. As part of this I had to analyse and evaluate my performance looking at how I stayed in time with the music and with the others I was dancing with etc. And now I think about it, I was using some maths that I didn’t realise before. Although this is the basics of maths such as counting to 8 to make sure I do each move on the right beat, it still counts and is one of the Basic Ideas talked about by Liping Ma.

In the session yesterday with Richard, we were looking at how maths is used in sport in particular football. It is used in the score, timing so you know the length of the match, formations, points, the league table and I’m sure there could be more. We then went on to look at a table of the first ever premier league season, which is very different to how we see it today. We then had to convert it to show how it would be shown today. This shows how maths evolves and how things can be simplified to make it easier for the consumer/customer to observe and understand. It’s also important to be able to understand many Basic Ideas of maths in order to be able to enjoy the things we do. You wouldn’t think maths had a lot to do with football but if you couldn’t count, read numbers, tell the time etc. you wouldn’t be able to enjoy such simple things in sport as the majority of us can.

The very first league table.

The updated version of the table.

We also created our own/adapted current sports to have new rules/equipment in order to make some of the mathematical features more prominent/important. I think this helped identify why some things are in place in sport e.g. changing a win in football from 2 points to 3 points as it can make for a more exciting game or a more exciting league/title race. It also helped identify what mathematics is actually involved within different sports when you wouldn’t really notice them otherwise.

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

# Maths and Art

In last week’s input with Anna we were looking into the relationship between maths and art. We looked at Mondrian, Fibonacci and the Golden Ratio – Phi.

Piet Mondrian was a Dutch artist who focused on Pointillism and Cubist art. His art was very geometric and is seen a lot in modern day art. – https://www.guggenheim.org/artwork/artist/piet-mondrian?gclid=CjwKCAjwssvPBRBBEiwASFoVd_XWXluR5gKRvWAB9ZhvVXuoW08LlVXAJYz3SyyIHQROWYoqqD5VpBoCdMYQAvD_BwE

My take on Mondrian’s art.

When creating my Mondrian style art I wasn’t exactly sure how it related to maths as I wasn’t focusing on any maths concepts when doing it. I think this would need to be brought in explicitly so that children were actually learning some key maths concepts for example using the art to look at angles. However, I do think this is a good way to relax the pupils as when doing it I felt very calm and at ease which can enable learning (Hayes, 2010).

Next we went on to look at Fibonacci. This was something I had heard of before as I had done it in school myself. The sequence begins at 0 and 1, naturally, and goes on by adding the two numbers before it. E.g. 0 1 1 2 3 5 8 13 21 34 etc. The numbers of the Fibonacci sequence can be used to draw squares on graph paper in a spiral or circular direction. As you can see on the picture (ignore the blue lines – mistakes made by me along the way) the sequence begins in the centre with 1cm square then to the right another 1cm square is added. This is because in the Fibonacci sequence 0 + 1 = 1. The sequence is then continued, 1 + 1 = 2 so a 2cm square is added and so on.

The Fibonacci sequence drawn onto graph paper.

This can then be used to draw a spiral from the middle of the first square and through the centre of every square on the page. This is known as the golden spiral or the golden ratio which can be symbolised by Phi. This video clip explains the spiral and how it’s seen in nature.

Phi denotes a special ratio of line segments. This ratio results when a line is divided in a special way. The lines are seen in the rectangles created on the graph paper. For example, the squares of 1, 1, 2 and 3 beside each other show a rectangle. Pickover (2009) said “we divide a line into two segments so that the ratio of the whole segment to the longer part is the same as the ratio of the longer part to the shorter part”.

The line which can be used in the formula.

(a+b)/b = b/a

The formula can be used with any rectangle in the sequence and the answer will come out to 1.6 every time or very close. This answer is seen by artists to be the number of beauty. This might explain why it’s seen in so many beautiful natural and living things on the planet, or maybe it’s another phenomena with no explanation which makes our planet so wonderful.

Hayes, D. (2010). Learning and Teaching in Primary Schools. Exeter: Learning Matters.

# Can Animals Count?

Yesterday in an input with Richard we were thinking about our number system and how we teach children everyday how to use it, how to count from 1-10 and so on.

The subject then progressed to if humans can count can animals count? We were told of different case studies including lions, ants, chicks and chimpanzees, which were the most interesting to me.

Here is the link to the video Richard showed us in class:

When watching this in class I found it fascinating! It was unbelievable how quick the chimpanzee could process where the numbers were in the correct order and get them all right even once covered up. We then tried this as a class through an online game. We were only successful one time out of three attempts, which I think was partially down to a bit of luck and guessing. Personally, I could only remember three numbers in each game which was definitely nowhere near as impressive as the chimp.

Although this video appears to show the chimpanzee is counting up to 9, I’m not convinced if it’s really counting or he just knows the symbols and shapes of the numbers and that’s what he has to do to get food. Either way, it certainly shows how much quicker the chimp’s brain can process the numbers and store the order and picture on the screen in his brain to be able to complete the task so quickly, impressive.

# Making Maths Fun

Last week in an input with Eddie we explored how to make maths fun in our classrooms.

The activity we did involved looking at tessellation – fitting shapes together to make a pattern with no gaps. We had to cut out the shapes we wanted – from a choice of triangles, squares and pentagons – and design a pattern which involved the shapes fitting together with each other. We were then to stick our patterns onto card and could paint them if we wanted to.

My tessellation

When completing the task the atmosphere in the room was very calm and relaxing, everyone was focused on what they were doing and the basic processes of cutting, sticking and designing a pattern were somewhat therapeutic. I could feel zen in the room, or in myself at least.

So, the objective of the task was to show the importance of making maths activities fun for pupils. I personality wouldn’t use the word fun to describe it, but I have a different outlook of what fun is compared to a child in primary school, although it was enjoyable to complete the task. When I was doing something that I actually took pride in and wanted to finish I wasn’t thinking about how this was actually doing maths – which made me think it’s always important to relate to the maths side of the task and not just a fun arts and crafts activity, but both. It’s necessary to show the links to the mathematical concepts so that pupils are still learning.

I tried to take this on in my own practice and created 6 different angles stations in my formative observation in 1pp1b. Some stations included an angle tarsia, an angle treasure hunt and an angle poster which the pupils had to measure and name the angles on the poster created with bright tape. I could really see the concentration from the pupils when doing something active for a change, as there was a lot of textbook maths in my class. Doing tasks in different ways and allowing pupils to see there is not only one method or one way to do something is very important, bringing change into maths makes it exciting and not the same thing every day. (Boaler, 2009).

Boaler, J. (2010). The Elephant in the Classroom: helping children learn and love maths. London: Souvenir Press.

# What is maths?

On Monday in our third Discovering Maths input we were asked the questions what is maths and why do we teach it? This got me into thinking so what actually is maths, other than the basic formulas and calculations we always think of?

We use maths on a daily basis without even knowing and when we do use it we don’t think about how we do it or why we do it, it just happens. There’s countless occasions from the moment you wake up to the moment you go to sleep where maths is used without even thinking about it e.g. checking the time when you wake up and throughout the day, using money to buy things, even looking at a parking space and checking the size and angles to see if you’ll fit. All these things use the maths skills we are taught from day 1 to make our day to day life that bit easier, so of course we need to teach it!

Some of the main problems with teaching maths in school are maths anxiety and the idea of maths being boring. To me, a lover of maths, it could never be boring. It’s logical, there’s a problem that can always be solved and it requires strategic thinking which comes with great satisfaction when getting an answer right. So why is it that some people find it so stressy and plain?

A main factor which contributes to “maths anxiety” can be the idea that people are just not born with a maths brain. You hear it all the time in schools and work places that someone’s brain just isn’t wired up properly in order to be able to be good at maths. I used to think this was true to some extent, as I believed I had a maths brain but not a language one – which by the way, was never an excuse allowed to be used in my school’s English department – but now having gone into a class on my first placement, I have seen some people either just don’t try or don’t have the confidence to push themselves. Being a maths enthusiast made me really try and give the pupils different strategies and activities to help them to be successful and realise that they all can be successful even when they find something tricky. I think it’s so important for teachers, and us as student teachers, to really facilitate the learning of maths in an active and creative way, and encourage pupils to find what works for them so they know that it is possible to be good at maths and do well even when they seem to be struggling or think they just can’t do it.

Albert Einstein said “Do not worry about your difficulties in mathematics. I can assure you, mine are still greater.” showing that even the smartest minds find things hard and confusing, but you have to persevere and keep trying in order to make a breakthrough and succeed.