Maths and Play is the Best Way.

Bringing play into the classroom, especially for mathematics, is a passion of mine. Too many times in primary I remember copying sums off a blackboard or using maths textbooks to answer questions. In my eyes this is the most unstimulating way to teach a child. I don’t think that there is anything wrong with copying sums off a blackboard or even using a text book but we HAVE to mix things up. To do this you need to have a Multiple Perspective. This is one of Liping Ma’s (1999, pg 122) four properties meaning teachers need to teach the topic from multiple perspectives.

Jean Piaget a theorist believed that children can benefit from their learning best when they interact with the subject in an active way. This way children get the experience to become hands on and explore. Bird (1991, pg. 3) agrees as she encourages that children “calculate, record, order, search, compare” which are all typical mathematical features, that should be produced an active way. It enables them to experiment as they have the ownership and control of their own learning.

I experienced the advantages of this first hand when I was in placement in first year. I remember the children were learning their times tables, however they were given the questions out of a textbook. It soon came to my attention that as the multiplication became harder the children would lose interest. I decided to use a lesson plan we had learned in a lecture to see if it would work. I wrote down sums on the inside of a folded piece on paper. Each piece of paper would be placed around the room. I then wrote the answers on the outside of the folded sums. This meant that when a child had answered a question they would look for the answer then open it up to see what the next question was. This created almost like a maze that the children had to follow. They were also put in pairs so that the children were motivated to work as a team to reach the finish line. After the game, I received feedback from the children that proved they had fun whilst learning. Every child had also completed the question trail, proving it was a success. This promoted social learning as well as learning their maths in an active way.

When thinking back to when I was at primary I don’t remember a lot about maths lessons that I received. Despite this, the ones I do remember, are the ones where I got to make a connection with what I was learning. It was a lesson based on weight, where we were allowed to explore the outdoors. We were asked to find three objects that we would later bring to the classroom to weigh. We would then list the weights of our objects and compare them from the heaviest to the lightest. What would usually take me a couple of lessons to understand, from this I understood first time. I feel my fundamental knowledge of this subject area was created through the stimulating play and the encouragement to interact by exploration of my own interests. Therefore, I feel we need to do the same for children so that they view maths as a fun curricular area that they will take great interest in.

Bird, M. (1991). Mathematics for Young Children. New York: Routledge.

Liping, M. (1999). Knowing and Teaching Elementary Mathematics. New Jersey: Lawrence Erlbaum Associates.

Fibonacci Sequence

Before this module, I had heard of the Fibonacci sequence but I didn’t realise how much it’s based on the nature that surrounds us. The sequence goes like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. This pattern is made from adding all the numbers up step by step. For example, 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, 13+21=34, 21+34=55. This sequence was first introduced in the West in 2102 by Leonardo of Pisa (Fibonacci).

This sequence, as said before can be found in nature as well as art. The spiral shown below can be similarly recognised in The Nautilus Shell.fibonacci-spiralnautilus-shell

The numbers grow by continual expansion. This tells us that the Fibonacci sequence underlines internal harmony. It’s the play of life itself surrounded by maths. As the shell matures the numbers gets larger, building a widening shell with open sequence of balance. You also see this pattern in the growth of many organic plants. Trees can get enormously large but the tree still manages to balance. This proves that any organic expansion would benefit from this sequence (Schneider, undated).

The golden ration is also connected with this. The golden ration is the division of a quantity into two equal parts which always equals 1.618. This is known as Phi Φ and the equation can be viewed as (a + b)/ b = b/a).

Everyone is always looking for perfection and in ancient Greece the golden ratio was applied to find perfection or beauty in a person. In class, we were asked to try this out ourselves. I got my partner to measure all sections of my body then apply them to the equation. To our surprise all sections of my body came back as 1.6. According to ancient Greeks this would be seen as “beautiful”. Lucky me!

I really enjoyed learning about Fibonacci’s sequence and it really has opened my eyes to the world around me. I was watching the weather on television the other day and even on there I was coming across this sequence. It is truly fascinating that by learning the basics behind mathematics we can truly begin to connect with it in the real world.

 

 

Schneider, M, (undated). Fibonacci Sequence Documentary – Golden Section Explained – Secret Teachings. Proper Gander. Available at: https://www.youtube.com/watch?v=4ToUaU4vPks [Available from: 3rd December]

 

Maths and Sport? Surely not?

I see myself as a very sporty person and for some sports know everything about the activity. However, I still failed to notice the mathematics that surrounds them all. Take football for example. I play as a striker for a Sunday league team and through this module I have become aware of how many times maths is used. Firstly, we must be at the match on time. This may seem easy but when you are playing another team outside of Dundee we must estimate how long it takes to get there. This is followed by how costly the journey will be by calculating the miles travelled by each hour. Secondly, each person has a number on the back of their shirt, myself being number 14. footballThirdly, we need 11 players for a team plus substitution players which means the manager must make sure they have exactly enough to play. The BBC (undated) states the length of a pitch must be between 100 yards (90m) and 130 yards (120m) and the width not less than 50 yards (45m) and not more than 100 yards (90m). I would not like to be the person having to measure this out. There is also the time of a match which the referee has to count. 90 minutes for a full game with half time being after 45 minutes. So, the next time you watch a game you may start to appreciate that without mathematics you can’t have a game of football. You may even want to think of any mathematics that are involved in any sports that you enjoy to play or watch. I bet you will be surprised just how big a part mathematics plays.

We can also link sport with one of Liping Ma’s (1999. Pg 122) 4 principles which is Connectedness. Connectedness means that teachers teach children maths in a way that can link different mathematical topics together. By weaving them together children can make the connections rather than being confused by individual topics. A good way to do this according to, Goodman and Williams (2000, pg 108) is to set children up in their own classroom sports day. “Many games require keeping score and scoreboards” which can be a good way to get children reading and writing numerals as well as adding and subtracting scores. This gets children involved in maths in a practical way. Mathematical equipment can also be mixed in for good practice, as stop watches can be used for races and measuring tapes can be used for activities such as throwing or long jump. By doing practical maths and linking it with the outside world instead of classroom textbooks, children will begin to enjoy the subject. Especially when they know maths is linked with something as fun as sports. This is something a lot of children do find pleasure in.

 

BBC. (undated). Pitch Dimensions. Viewed at: http://news.bbc.co.uk/sport1/hi/football/rules_and_equipment/4200666.stm [Available from: 22nd November 2016]

Goodman, S and William, S. (2000). Helping Young Children with Maths. London: Hodder and Stoughton.

Ma, L. (1999). Knowing and Teaching Elementary Mathematics. New Jersey: Lawrence Erlbaum Associates.

1, 2, 3, 4, 5, Many..

Looking back, it seems that numerals are around 5500 years old and some anthropologists believe that because of trading, numerals were created. Numerals are important in mathematics as they are the symbol that represent the number. For example: there are four people living in my household. “Four” is the spoken representation of the number whereas “4” is the symbol used for written representation (Barmby, pg 12, 2009). As we know, English have a base ten number system which makes numerals a lot easier to remember. The following for the first ten written representation of numerals are:

1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.

These first ten numbers are called units. We then begin to see a pattern with the next ten numbers that appear in the system:

11, 12, 13, 14, 15, 16, 17, 18, 19 and 20.

The second numbers are repeated where as a “1” is put in front to represent the tens. This would be “ten and one = 11, ten and two =12, ten and three =13” and so on. Likewise, twenty is based on “two tens”. For example: two tens and one = 21, two tens and 2 = 22, two tens and three = 23. Although this may seem complicated at first, it is much easier than a number system that has different symbols and spoken words for each number up to 100. This would be impossible which is evidence that proves a patterned number system, like our own, is the sensible way to go.

During a lecture with Richard we were asked to try write our own number system. Without even realising it we had borrowed our idea from our own base ten number system. It became clear how difficult it is to move away from base ten and try another form.15050045_1204511272948173_1328498317_n

However, do we need that many numbers? An English philosopher in the 1690s called John Locke believed that although numbers “were helpful in learning to count and calculate” it is not completely “necessary in the possession of numerical ideas” (Butterworth, 2004).

A great example of the would be the Munduruku tribe. The tribe were studied by a French team that found this tribe only used numbers up to 5 (BUTTERWORTH, 2004). Any number after 5 would be used by the word “many”. This would be impossible for us to use in today’s world. Imagine In a maths exam and any answer over 5 would have to be written as “many”. Now that’s a maths exam I would like to sit. However, the French found that Munduruku rarely needed to use counting in their everyday lives which is why they don’t use a vast number system like our own. They did not need symbols of written or spoken representation. All they needed was the small knowledge of mathematical concept of numbers 1-5 in their head. Proving that Locke did speak some truth when saying we do not need numerals in written or spoken representation.

This links with my profound understanding of fundamental mathematics as it allows me to see how mathematics works in a world of different societies. I now see the importance of learning the backgrounds of different societies and their connection with mathematics. In our society, it shows the importance of numerals in written and spoken terms, as we depend on them a great deal. Without them it would be hard to do simple, day to day tasks. On the other hand, if I lived in the amazon with the Munduruku tribe I would have learnt to live without this fascination of mathematics. Making it possible to live without. Despite this I am quite happy using the number system the English have developed. It saves the confusion when somebody asks how many students are studying primary education. Many, of course

 

B, Butterworth. (2004). Viewed at: https://www.theguardian.com/education/2004/oct/21/research.highereducation1 [Accessed on: 11th November 2016]

P, Barmby. (2009). Primary Mathematics. Teaching for Understanding. The McGraw Hill Company: London

Maths in the Surrounding World.

On placement I remember we took a lesson on money and the different ways that we can pay for items in a shop such as: notes, coins, cheque, bank card, credit card ect. So that the children could see the connection with the wider world, we had a class discussion on the use of each. Many children were able to join in saying they had maybe spotted their parents using different forms of payment. Children love to be able to see the relation of their work with the wider world so I could really notice a difference in the quality of work that was being done. 45 minutes of the lesson was also based on a class discussion which meant the last 15 minutes was used to complete their worksheets. By having a group discussion it got all the class engaged and showed them that mathematics did not always have to be calculations out of a text book, which is something they were used to.

However, prior to this lesson, one of the main questions that was raised, when taking a maths lesson had to be “what do we need this for”. It is hard to answer this without saying “everything”. Not only are we dealing with simple usage of mathematics throughout our boneday but the fact that it is a subject which is needed to achieve acceptance into many jobs or universities. Not to mention going back 22000 years ago when the Ishango Bone was discovered (Wolfram Research, 1999-2016). This bone was found in the Congo, with sets of markings carved into it. The sequence of numbers being “3, 6, 4, 8, 10”. This is now one of the oldest objects dating mathematics back to thousands of years, giving us a great starting point to where it may have all begun. A great topic to touch upon with children.

Noyles (2007, pg 8) states that teachers “rarely engage with their subject outside their work or know little about how it is used in the world around them”. This therefore proves that if adults have not discovered the mathematical world that surrounds then, then the link is not being made for the children. As a result of reading this quote it signified the importance of making the connection, which I therefore included into my lesson. Children need to understand why mathematics is important and useful for everyday life. It is impossible to do this if we do not back this view up in the classroom.

In one lecture our tutor directed task was to take pictures of any places that we spot mathematics. It became clear to me just how many times we use some sort of fundamental mathematics without even realising it. Here are some of my examples:

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This gave me the idea that we could ask the children to do this themselves. The class could then create a wall full of pictures where they have found something with some mathematical thinking behind it. This then creates an importance for the subject using real-world context. It also shows them the ideas and relation behind our teaching. This creates a more positive attitude and therefore children can see the subject as something fun rather than something that has to be endured. The Scottish Governments, CfE (2009, pg 39) defines numeracy as a skill for “life, learning and work.” It also states that by being numerate we can function responsibly in everyday life. I have never really thought about this before this module however now it has become clear that without a fundamental understanding of mathematics we would struggle to do even basic tasks.

 

 

Noyles, A. (2007). Rethinking School Mathematics. London: Paul Chapman Publishing.

Scottish Government. (2009) Curriculum for excellence, experiences and outcomes for all curriculum areas. Available at: http://www.educationscotland.gov.uk/Images/all_experiences_outcomes_tcm4-539562.pdf

Wolfram Research, (1999-2016). Ishango Bone. Available at: http://mathworld.wolfram.com/IshangoBone.html

Beginning of Mathematics..

In one input a quote describing mathematics clearly stood out to me: “a problem-solving activity supported by a body of knowledge” (SOED, 1991, p.3). During my time at primary school I found mathematics very interesting. The notion that each question has a definite answer intrigued me. I liked the idea that stepping stones of investigation, guided us to finding the answer, which lead to us either being right or wrong. However, despite enjoying the subject, some factors of the mathematical lessons I received did create an anxiety towards parts of the subject within me.

To this day I still use my fingers to count, mainly surrounding my times tables. I distinctly remember back in primary 5 being asked “6 x 7” in front of my whole class. As I proceeded to count using my fingers I was told to sit down as “I should not be using my fingers to count at that age.” I have never understood to this day why using my fingers was a problem? As long as I got the correct answer in the end then there would be no need to humiliate me in front of the class like I had been. Therefore, creating a fear within me that I wasn’t learning in the correct way.

Until lately I never realised that my teacher was restricting my ways of learning. Going out on placement made me aware that each child learns in a different way and it is our job not to limit this. We need to explore different ways to teach a subject including maths. Again in primary I struggled very much with the concept of fractions. I just could not grasp an understanding of this element due to my teacher only teaching it in one way. As I moved up the school other teachers just assumed I knew fractions, so it was never went over again. Due to embarrassment I never raised the issue that I wasn’t sure what I was doing, as everyone else in my class seemed to get it. As I went on to high school I remember praying in maths class that the lesson wouldn’t be surrounding fractions and going on to university I only knew the very basics. Many people would think “what kind of student, wanting to be a teacher doesn’t understand fractions?” but it’s true. My mind was blank anytime they were mentioned. I then made it my own responsibility before placement to teach myself and find different ways to teach the children so they weren’t limited like I was.thbvvdr7rn

In a recent lecture we were told the importance of developing a profound understanding of fundamental mathematics. Meaning we need to know the basic ideas before we build on to mathematical problems and this is where I feel my learning went wrong. Bruner (1964) created a scaffolding theory where we (the teachers) are the stepping stones of a child’s learning. They start at the bottom of the ladder learning the basics and as they become confident they move up a step. In my experience I feel I did not have a profound understanding of the basics and therefore was not ready to move on to the next steps of my learning. Despite having this negative experience with mathematics it has not put me off and I am eager to learn in this mathematical module. It has also made me acknowledge the teacher I want to become when teaching the subject myself.

Behaviour Management

Behaviour Management is something that scares me the most about going on placement to our primary schools. With past experience I have only ever worked with younger years from primary 1 to primary 3. The behaviour management strategies that are usually put in place for this age group tends to be the popular traffic light system. All children start on green due to positive behaviour. If the child starts to misbehave the child is asked to move their name to yellow. This acts as a warning and signifies to the child that their behaviour needs to be changed. If the child continues to misbehave then they are then asked to move their name to red. A punishment can then be put in place to show the child that the behaviour they are displaying is not acceptable. This may be loss of Golden Time. traffic-light-allHowever, I am aware that this strategy is more age appropriate for the younger years. As I have been given a primary 5 class, when I first go I will ask the teacher what behaviour management they use in the class so I can go home and do further reading. This will allow me to feel confident when going in to the classroom and using the system. As a teacher I also need to be aware of other issues that may effect a child’s behaviour such as: home issues or children with additional support needs. We need to be aware that every child is different and that not all children in the class will benefit from the class system that is put in place. Some children may need their own behaviour systems in order for them to understand what is expected of them. I know how important literature is and I know it will expand my knowledge when dealing with behaviour management. From this I may be able to use my own strategies and realise what works best for me. This will help massively when I go on to be a teacher in my own classroom.