So, this elective is finished, my essay is done, i’ve blogged like I’ve never blogged before and I feel a little bit like I’ve stepped off of a roller-coaster or something; the whistle-stop tour of Discovering Mathematics.
I chose Discovering Mathematics as my elective simply because despite my anxiety about the subject, I somewhat actually enjoy maths with all it’s numeracy, calculating and problem solving. I just didn’t feel confident in doing maths and I thought the module would help me boost my confidence and give me a better understanding of the concepts and skills in mathematics and enable me to teach it in a positive, fun way to others.
Discovering Mathematics wasn’t what I thought it was going to be but I am by no means disappointed by it and my learning, Richard’s constant enthusiasm was both infectious and motivational, he made learning easier through fun tasks and the interesting topics he covered within the elective. My favourite was the input on data & statistics with Dr Ellie Hothersall. However, I feel the elective did highlight the fact that what I knew about maths really wasn’t very much after all and I had no idea about just how much mathematical concepts are in my everyday life, nor was I aware of the underpinning maths in areas other that education. I had never really thought about that properly before.
In regards to my trepidation at learning and teaching maths, I don’t now feel quite as anxious as I did before this module. I can see that if I put the work in, my knowledge and understanding will of course develop, which will enable me to be more confident in the subject and be able to teach it more effectively to my own class. I want to ensure that every pupil I teach feels none of the maths anxiety I felt both as a child and an adult.
I am committed to developing my own understanding of maths by continuing to read and build upon my existing knowledge of mathematical concepts and skills. I will make every effort to ensure I am up to date on new developments and skills which will enhance my learning and teaching and I will put into practice what I have learned about fundamental mathematics. Over the past few months whilst covering this elective, I have found myself looking more for mathematical concepts in my everyday life and I think this is something I will not be able to help doing still – I enjoy it now and I think it is safe to say I am on the right path to Discovering Mathematics.
The Fibonacci sequence is a series of numbers where a number is found by adding up the previous 2 numbers, starting with 0 + 1.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. For example, the 5 was found by adding the previous 2 and 3 together. It can also be written as a rule ×n-1 + ×n-2. The sequence is named after an Italian man named Leonardo Pisano Bogollo who was alive between 1170 and 1250. The sequence can also be illustrated as a sspiral when squares are made within the widths of the sequence numbers.
The Fibonacci sequence also relates to the Golden Ratio (PHI) which can be described as the ratio between any two consecutive numbers in the Fibonacci sequence. This ratio is considered to be the most aesthetically pleasing to the eye and for this reason it can be found to be used in many forms and places. For example, it can be found in architecture within the building design as they are considered more attractive to look at using this ratio.
Fibonacci in Nature.
Fibonacci in a galaxy.
It can even be found in your face!
With mathematics being the least favourite subject in primary school with children, it is important to make their learning of maths fun so they can enjoy it and see it as an opportunity rather than a hindrance. A fun way to incorporate mathematics into everyday learning is through the popular game Minecraft.
Minecraft offers a variety of mathematical concepts for exploration and children LOVE it! For example, addition and subtraction, to find out how many blocks are needed to build, or if is a large area then multiplication can be used instead. Spatial awareness is needed because the amount of space needed to build is important and overlapping is not possible. Time must be considered because you only have a certain amount of it to build; a half finished house is no use to live in! Volume is another concept integrated into Minecraft.
With multiple possibilities of mathematical understanding contained in one game, Minecraft can only enhance a learner’s understanding of mathematical skills and concepts.
Whilst I was on placement in my primary 5 class during my first year of university, I took a maths lesson and decided to include a division exercise which was Minecraft themed. The children were to solve the division sum in each square on the page and each answer would give them a corresponding colour which they would use to colour that particular box. Once finished, a picture would be revealed relating to Minecraft. The children loved it and it changed their mind about learning maths – for that lesson anyway!
The fundamental factors of mathematics are considered as the being the basic ideas in which a learner should have an understanding of in the first instance before progressing on to attempt more complex mathematical problems and processes. In essence, this is knowledge is used as the foundations of which knowledge can be built upon thus enabling learning and understanding to evolve. Liping Ma (2010) suggests that we must develop ‘a profound understanding of fundamental mathematics’ in order to be able to promote effective mathematical learning and in turn, teach it effectively. This involves having a concrete knowledge of the structures embedded in mathematics and how they are used.
Primary school teachers require a certain level of mathematical knowledge to ensure that effective teaching is taking place. Liping Ma (2010) identifies four key elements which all contribute to a person gaining a profound understanding of mathematics. These four elements are ‘Connectedness’ which is the ability to be able to relate topics to others thus enabling the existing foundations of knowledge and understanding to be built upon allowing new concepts and operations to be processed. ‘Multiple perspectives’ which can be described as the ability to vary the approach used in solving mathematical problems in which Ma (2010) suggests that if a person is successful in managing this process then their knowledge and understanding of that particular topic is considered complete. The next element is ‘Basic ideas’ being the ability to be able to identify the basic concepts and attitudes of mathematics and utilise these when informing future pathways in mathematical problem solving. ‘Longitudinal coherence’ relates to the content learned from the beginning of the understanding of mathematical processes and how it influences the current understanding, regardless of how limited this may or may not be. This concept is essential to teachers as they are essentially influencing the learners understanding from the start of learning and onwards through their learning journey. Ma (2010) supports that teachers ‘are able to provide mathematical explanations of approaches’ and ‘can lead their students to a flexible understanding of the discipline’ (pg.122). She also states that these four key elements ‘are the kind of connections that lead to different aspects of meaningful understanding of mathematics’ which is essential to teaching. Teachers in possession of a vast and thorough understanding of mathematical concepts are able to constitute it through different rules and concepts.
According to Skemp (1989) there are two kinds of learning in mathematics; Instrumental or relational understanding. This is my understanding of the two kinds.
Instrumental understanding – having a mathematical rule and being able to apply and manipulate it.
Relational understanding – having a mathematical rule, knowing how to use it AND knowing why it works.
From this, I can see that relational understanding is a deeper, more complex understanding of instrumental understanding. While instrumental understanding is knowing and applying the rule, relational understanding is the same but also knowing why it works and how it connects to other rules. Using both of these understanding you will arrive at the same correct answer but relational understanding is way more extensive. Of course each has its own strengths for example, Instrumental understanding is often much easier to comprehend; some concepts in maths are difficult to follow but can be grasped quicker through the use of rules rather than knowing the ‘ins and outs’ of why it works. Results are instant; once you have learned the rule, it can be applied to many mathematical concepts in the same format to achieve correct answers. Relational understanding is pretty much the opposite of this as it is much more difficult to understand and it is so much more time consuming that just applying a rule. However, because relational understanding is already present, the learner can take what they have learned and easily adjust it to a new mathematical concept when a new task is introduced. This approach can also be its own goal since having the understanding of complex mathematical concepts can be rewarding in the first instance.
Ok, so you probably don’t know that I studied hairdressing and salon organisation when I left school at the tender age of 16! But I did – for 3 years. I learnt how to book appointments and plan time, cut hair, colour hair, perm hair (heaven forbid!) and style hair. So you might say, I am aware of the underlying mathematics that are central to the world of being a hairstylist. I thought I would note down a few points.
1. Planning your time. Even tasks like booking appointments involve having some sort of basic understanding of how numbers work. This ensures you set aside the correct amount of time for each client. No-one like to be kept waiting!
2. Stock & Supplies. Being able to manage stock successfully and being able to order enough stock in for use is integral. A basic knowledge of sums and counting is involved in this and although we didn’t actually have to control the stock we were taught thee importance of it.
3. Payments & Handling Change. This is probably the most obvious way maths is used within the industry. An understanding of how much to charge clients is important, it must be enough to cover the overheads and remember to give the correct change!
4. Colouring Hair. Measuring the right proportion of product and developer to achieve the look is important. Volume and ratio is essential to mastering the art of hair colouring.
5. Cutting Hair. It requires more knowledge and understanding than just being able to hold a pair of scissors! An understanding of angles, length and shape are essential mathematical concepts to getting the cut right. For example, an inverted bob or layering. (Unless you like a bowl cut in which case it’s relatively easy!)
I still love hairdressing second to teaching and occasionally still do hair. I could watch hair programmes and tutorials on styling for hours if I had the time.
Mathematics is everywhere in the world of Art for example, in symmetry and patterns, in architecture and even clothing. Islamic art is a good example of tessellation as it comprises of pattern and repeats this throughout which creates a wonderful image to look at. It can be found in many forms such as religion, architecture and pictures. During this workshop for Discovering Mathematics, we were encouraged to try some paper folding to create various different shapes in order to promote the potential learning that could be had in our own classroom. It is also an effective way to make leaning concepts in maths different and fun!
Tessellation is the term known for shapes that fit perfectly together without any gaps showing or overlapping occurring. We spent some time figuring out what shapes we were supplied with would tessellate and which ones wouldn’t. Over the course of the hour workshop, I learnt that shapes would only tessellate if the angle they made added to 360 degrees upon the vertices meeting. A vertice is where two or more straight lines meet – I learnt that on placement when my class sang the wee song!
Typical examples of tessellation include regular shapes (where all sides are the same), pieces of a jigsaw, and tiles in the form of animals which cover the surface in a symmetrical way.