Nature of mathematics

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Nature Of Mathematics And The Need To Overcome Misconceptions For Improved Learning In Schools

Section 1

Aim

The purpose of this investigation is to discuss the nature of mathematics and how its implementation in school has evolved over the years. I will also look into the different kinds of misconceptions that pupils develop in the early years and how best to identify and overcome them. Through the planning and delivery of lessons, some aspects of this will be discussed and evaluated.

Introduction

Many definitions of Mathematics as a discipline abound. Some of these include:

“science of patterns and relationships”

“derivation of theorems (consequences) from axioms (rules)”

“study of abstractions”

“abstract science of space and number”

My understanding of mathematics is that it is a universal language which can convey meaning and ideas without the use of words. Mathematical ideas and applications have benefitted and improved our lives in countless ways.

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We study maths because it teaches us a way of thinking. It provides us with a method of solving a host of life's problems away from the classroom.

There are obvious reasons like making sure you have enough money for the cinema ticket, deciding whether the piece of clothing on sale is actually a bargain, calculating whether 5kg of washing powder is better value than the 2kg packet.

But there are bigger and more important problems that require the use of mathematics to solve them. For example, when deciding which mortgage policy to take out, how much to save over a given period to be able to buy the car you want or the holiday you want to go on. All these problems may appear to have nothing to do with the mathematics we learn at school. But they do. All the problems we come across have something in common. They all have some information that needs to be weighed up, sorted out and then processed in a certain way. Once this has been done, it must be interpreted so that an intelligent decision can be made. All this requires planning, logical thinking, perhaps some experimentation and then evaluation and testing to make sure that the decision reached is the best one.

Many of these skills are needed and developed when studying mathematics. For example, consider this problem: ‘A farmer has some ducks and some cows. He finds that together these animals have 19 heads and 60 legs. How many ducks and cows does he have?'

Now, let's just think about what you would need to do to get the answer. First, you would have to weigh up all the information and decide what kind of problem it was. Then, once you are happy it is a simultaneous question you then need to present it in a simple, manageable way. It is here where the ability to think abstractly becomes very useful. By representing ducks and cows by letters and writing out two simultaneous equations requires a skill that can only be developed through the use of one's imagination and a lot of practice.

There is much involved in answering such a question but studying mathematics teaches you to do all of them automatically without really thinking too much about what you are doing. Studying maths contributes to the development of analytical and reasoning skills and trains you up to be an expert problem solver.

Smith (2004) identified the importance of:

  • Mathematics for its own sake: universal and intellectual tool - kit for abstraction, generalisation and synthesis; logical reasoning; analytical problem solving; it ‘trains the mind';
  • Mathematics for the knowledge economy : science, technology, engineering, finance;
  • Mathematics for the workplace: mathematical literacy which encompasses interpretation and use of different representations of data; data entry and monitoring; related communication skills; recognition of errors and anomalies; knowledge of what, how and when to calculate; use of relevant degrees of accuracy and plausibility;
  • Mathematics for the citizen: access to labour market and general social and political inclusion.

From the above list, it is self evident that there are enormous advantages in becoming proficient in the subject. But there are other benefits that are not mentioned. Mathematics allows one to appreciate the beauty and grandeur of nature itself and the variety of patterns and order present therein. It develops within oneself the confidence and self-esteem that comes from being able to overcome seemingly difficult problems and situations.

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As you come to master a branch of mathematics, it's as though you've grown a new abstract organ of perception through which you may then view the world. You've grown a new "mind's eye" that can perceive realities literally inconceivable without this new organ of perception.

Rafael Espericueta
Professor of Mathematics
Bakersfield College

Concise History Of Mathematics

In Mathematics: The Science of Pattern, Keith Devlin describes Egyptian and Babylonian mathematics up to 500 B.C. as "the study of number," and the era of Greek mathematics from 500 B.C. to 300 A.D. as "the study of number and shape." In the mid-1600's, with the invention of calculus by Newton and Leibniz, mathematics became "the study of number, shape, motion, change, and space." As interest grew in mathematics itself, "by the end of the nineteenth century, mathematics had become the study of number, shape, motion, change, and space, and of the mathematical tools that are used in this study." In the past century, mathematical activity and knowledge has increased more than a thousand fold. Devlin describes mathematics as the "science of patterns... patterns of counting, patterns of reasoning and communicating, patterns of motion and change, patterns of shape, patterns of symmetry and regularity, and patterns of position."

History Of Mathematics Education In The U.K.

At the beginning of the 20th century Local Education Authorities (LEA) were set up through the Balfour Act which allowed for all schools to be funded by local taxation. Later the “Butler Act” of 1944 brought in a tripartite system in which bright and able pupils (determined by some sort of selection exam) were taught in grammar schools. The less able pupils attended secondary modern schools or technical schools.

Technical schools were closely linked to the world of industry and commerce. It provided a general education with specific emphasis on technical subjects. It was definitely more in touch with reality than Grammar Schools and more specifically geared to preparing the pupils for their trade after leaving school. However, there was a shortage of qualified teachers which might have been the cause for its lack of success.

In 1965 comprehensive education was introduced which incorporated everything in the tripartite system. These schools take all pupils regardless of ability and cater for children from a variety of backgrounds. However, this has not eliminated distinctions. There is still “setting” according to learning ability. This means that students are grouped together to achieve a degree of uniformity in classes.

The Education Act of 1988 brought in the National Curriculum which made certain subjects compulsory in all schools, including mathematics and English.  The first version of the mathematics National Curriculum appeared in 1989. It contained fourteen Attainment Targets, each with ten levels. This made it possible to see what students were expected to learn in mathematics.

Since the introduction of GCSE in 1988 it has been examined at three levels: higher, intermediate and foundation.  But from 2006, a two tier GCSE was introduced eliminating the intermediate level. This now allow all students the possibility of attaining a grade C. Not long after this the National Strategy and Framework for Teaching Mathematics was brought in which gave guidance on the pedagogy to the curriculum.

Schools in England have begun the modularised course where the workload is spread over two and sometimes three years (Years 9, 10 and 11). The advantage of this is that pupils are highly focused and motivated on the short-term goal in hand and therefore achieve better results.

Identifying And Overcoming Misconceptions

Some of our mathematical ability comes naturally, but the majority is learnt. There is a basic human number sense in all of us. It seems from birth we have the ability to understand the difference between one, two, three and many. But as we grow we naturally begin to construct our own understanding of how the laws of nature function.

Pupils, when learning mathematics, sometimes interpret the information they are presented with different to what the teacher might expect. There are many reasons for this. As learners we continually try to make sense of our surroundings from our own experiences, not just from school but from the outside world.

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Every mistake that a pupil makes does not necessary indicate a misconception but may be due to many other factors. It is important to distinguish between an error on the part of the pupil and an inherent and deep-rooted misconception that needs to be identified and overcome.

Errors can occur for many reasons. For example, there could be a lapse in concentration, intense pressure, boredom or misunderstanding in what the question is asking. In such cases pupils recognise their mistakes and this does not necessary impede their progress in any way.

However, there are other mistakes that occur due to a totally wrong interpretation of mathematical concepts. In such cases there is something fundamental that is not understood by the pupils. For example, according to Key Stage 3 National Strategy, most pupils will generalise from their early experiences that:

  • you cannot divide smaller numbers by larger ones;
  • division always makes numbers smaller;
  • the more digits a number has, the larger is its value;
  • shapes with bigger areas have bigger perimeters;
  • letters represent particular numbers;
  • ‘equals' means ‘makes'.

Each pupil is different and comes with its own set of theories constructed from experience. Therefore the first step towards overcoming misconceptions is to identify and understand them. This can be carried out by regularly checking pupils' class work for mistakes that occur consistently and analysing them for any underlying assumptions. Many misconceptions can be explained by Tall's generic extension principle: “If an individual works in a restricted context in which all the examples considered have a certain property, then, in the absence of counter-examples, the mind assumes the known properties to be implicit in other contexts” (Tall, 1991).

From my experience I have noticed that when misconceptions are discussed and pupils' experiences and thoughts are shared there is deeper and lasting change in their understanding.  I have taken such an approach in the delivery of some lessons that will be discussed now.

Section 2

This section includes a series of three lessons which were delivered to Year 8, Year 11 and Year 10 classes at my first school placement. The guidelines for the lesson plans were taken from the Scheme of Work that the mathematics department provided. One of the most important tasks to be performed by a new teacher before delivering a lesson is to ascertain the current level of attainment of the class to be taken.

Lesson Structure

A Learning Plan has to be completed for each lesson which indicates the different objectives that all, most and some will reach according to their ability. Students with special needs and those who are gifted and talented are highlighted and additional resources provided for their needs. A thorough risk assessment is carried out to prevent any avoidable accidents. Also any resources required for the lesson are highlighted and any learning links to websites that would be used during the lesson are also indicated.

The second page of the Learning Plan gives a breakdown of the lesson starting with Preparation/Entry lasting about 5 minutes when students put away their belongings and take out their planners. Then follows a starter activity which generally has some connection to the main subject to be covered. As the school has a 100min per lesson policy, there is the possibility of covering two or even three main topics per lesson. Any differentiation in work for low and high ability pupils has to be written into the plan, naming pupils for whom special work has been prepared. There are instances where, for example, pupils with dyslexia need worksheets with large print in order for them to read. The last 10-15mins is used to recap all the main objectives and to address any misconceptions and misunderstandings that may remain. This is also the time when homework is normally set.

There is provision in the Learning Plan to indicate the types of groupings, learning styles, assessment and thinking skills that would be catered for and incorporated in the lesson. A key is provided for the different types of learning styles, groupings, assessment and thinking skills. The Learning Plan is a ‘working document' in that it should indicate all the resources, needs and cater for all possible eventualities that may occur during the delivery of the lesson.

As per the rules of the school, the teacher at the outset of the lesson writes out on the board the learning objectives, keywords, the date and the title for all pupils to copy. This gives students an overview of what is being taught. As part of a good learning and teaching pedagogy it is important to grab the attention of the target audience by providing some shocking statistics related to the topic being taught. This can be in the form of pictures, diagrams, audio or video clip. This creates an environment in which pupils want to know more and are willing to pay attention. This starter should set a positive mood for the rest of the lesson. The next step is to provide the benefits of listening to you. It could be related to gaining high grades or the benefits it would bring in real life. The rest of the time is used in the teaching of the planned topic, building on and using previously learned knowledge. Here the teacher employs a variety of techniques to engage pupils and to help them understand new materials. This can include a video clip, picture, a game or a diagram. This will ensure that the needs of pupils of all learning styles are met. The teacher also keeps explanations simple and provides as many varied examples as possible to make sure pupils follow and have a clear picture of the concepts. In the last quarter of an hour or so the teacher recaps the lesson objectives and clears up any misunderstandings.

LESSON 1 (See appendix 1)

Class: Year 8 Lower Set

The lesson was planned with the different learning styles and abilities of the pupils in mind. I had prepared a PowerPoint presentation on median and range and decided to use tally charts and bar charts as an introduction to the topic. In order to grab their attention and interest, I informed them of my intention to carry out a survey of the class. I told them that I wanted to know what type of sandwiches pupils liked. I also said that this information could be used to provide better choice at the school canteen and therefore it was important to be honest. This created an excitement in the class and brought everyone together on the topic. I informed them that they can only have one favourite filling. A tally of the results was created as pupils shouted out their favourite sandwich. Once this was complete, I made sure that the total added up to the number of pupils in the class. I then created a new column and labelled it ‘Frequency' and asked what they thought the word meant. It became obvious to them that it must mean the number of times a particular category occurs. After completing the table I told them that the information now needs to be presented in a bar chart.

I decided to discuss with the class how best to present data on a bar chart - whether the x-axis should represent the frequency or the categories of sandwich fillings. I drew both types of bar charts and asked which one they thought was the most common way of presentation. They all agreed that it is better to place the frequency on the y-axis and categories of fillings on the x-axis. At this point I thought it best for pupils to consolidate and reinforce their understanding by attempting some exercises on the handout that was now being given out.

As the pupils worked on the exercises, I made myself available to anyone who needed clarification or explanation on what to do. Many questions were raised as I moved around the room. It was gratifying to see that all of them were on task, some discussing with one another and others working individually.

Once I saw that all of them had completed a few exercises, I decided to introduce the next topic. I asked everyone to put down their pens and listen to new instructions. I asked one pupil to read what I had written on the board: “Median means the most common” and “Range = largest number - smallest number”. I asked them what the most common sandwich filling is for the class and some shouted “bacon sandwich”.  As for the range some of the pupils were able to work out the answer by themselves. Others needed to be shown. Once I was satisfied that they had understood, I asked them to find the median and range for the exercises they had already completed from the handout. The final ten minutes of the lesson were spent on plenary - asking them what a tally chart is, how data is represented on a bar chart and defining the terms median and range. I also asked the class to learn the spellings of the keywords for the next lesson.

On reflection, the lesson went well and pupils were well engaged. With the use of additional handouts (with more difficult work) it was well differentiated. Even though minimal technology was used in this instance, I felt that the pupils, nonetheless, grasped everything that they were told and demonstrated this in their class work.

LESSON 2 (See appendix 2)

I was asked to team teach Year 11 top set, those who had recently sat their GCSE exams and were now considering taking A-level mathematics. The topic chosen for study was sine and cosine rules. The lesson objectives were clearly stated on the board and were copied by all. The lesson was started off by recapping previously learned rules for right angle triangles. Some examples using the Pythagoras Theorem were then carried out as a starter activity. In the meanwhile, I drew a dozen or so triangles with various angles and sides given. For some the sine rule was appropriate and for others the cosine rule. I started off by asking pupils for the cosine rule and hinted that it was similar to Pythagoras Theorem but with something taken away. One pupil was able to recite the rule from memory. I then wrote two possible instances where the cosine rule can be used. Firstly, when all the sides of a triangle are given and an angle needs to be calculated. And secondly when two sides and the included (in between) angle is given and the third side or another angle needs to be calculated. With this in mind, I asked the class to go through the set of triangles I had drawn and decide on which of them they would use the cosine rule. A tick or a cross was placed besides each triangle to indicate whether they would or not. For each one I asked why and was given the correct response. Taking that as evidence of understanding, I decided to give some exercises and see how they would get on. To my delight all of them were able to work through the exercises without any major problems. Only one was struggling, he had not realised that cos130 is a negative number and that taking away a negative number is same as adding the number.

At this point my colleague took over the teaching of the class and my role shifted to that of support and assistance. He began by asking pupils the sine rule, which a lot of them knew, and then went on to say that the sine rule should be used whenever possible since it is easier of the two to apply. He also explained that the sine rule can be changed according to what they are trying to find. If the unknown is a length then the formula can be ‘turned on its head' so that the unknown is now a numerator, making calculations that bit easier. If the unknown is an angle then the formula can be used as it is. He then went through a couple of examples on the board and got the class involved as much as possible. In order to expel any misconceptions about reciprocals, he also explained, for example, = then if we turn both of them on their heads we get = . He then brought up a PowerPoint presentation with some questions for the class to attempt.

As part of the plenary, the importance of knowing which formula to use was stressed and also the value in developing a systematic and logical approach to problem solving.

Looking back on this lesson, I sometimes wonder whether it would have been better if the sine rule had been introduced first. My colleague is of this opinion that it would have been. However, I did not experience any adverse reaction from the pupils nor any dissatisfaction as to why it was taught that way.

LESSON 3 (See appendix 3)

Class: Year 10 Higher Set

As part of my ongoing development it was decided that I should prepare a lesson for Year 10 higher set. The topic chosen was prime factor decomposition, HCF and LCM. I was advised to identify those pupils with special needs and then go over to the SEN department and ascertain how best to help them in their learning. Using the school database I was able to search for all the pupils and realised that there were no major learning issues with any of them. However, I did identify couple of gifted and talented pupils for whom I would have to prepare additional work.

As soon as the pupils came in they were all handed a mini white board with the express instructions to use it only when told to do so. They all sat according to the seating plan drawn up beforehand by the head of department. The seating plan incorporated the school's boy-girl seating arrangement so as to minimise chatter and disruption. After introducing myself I asked everyone to write down the learning objectives and the associated keywords for the lesson that I had written on the board.

As a starter activity I began with integers and prime numbers. I asked everyone to write down on their white boards the only even prime number and then the first 4 prime numbers. I discussed with them that these numbers can only be divided by one and themselves. Once I was confident that this was understood by all, I began my main topic.

I started by saying that any number can be broken down into a product of prime numbers. Using the seating plan I chose someone to give me a number between 50 and 100. I then explained that using only the prime numbers 2, 3, 5, and 7 any number can be represented as a product of these numbers. Using the tree diagram I showed that 84 = 2x2x3x7 or 22x3x7. I chose a different pupil to give me another number and then worked out its prime factor decomposition. In order to find out how much they understood I decided to test them by giving them three numbers, one at a time, and getting them to show me the answers on their white boards as quickly as possible. From the responses I realised that some were dividing the numbers by 4 and others by 9. I referred to those answers and asked the class what was wrong with them. Having clarified this misunderstanding I moved on to HCF and LCM.

I began by finding the prime factors of two numbers and then asked the class to write down on their white boards the factors that were common to both. I then asked ‘What is the product of those numbers?' That is the HCF for those two numbers. I decided to give the class some exercises to do on HCF before introducing LCM.

Having gone through the answers for the exercises with the class and clarified any misconceptions, I said that to find the LCM it is best to first find the HCF. Using a Venn diagram I explained that the factors that are common to both would go in the middle where the circles overlap. The factors that are not common will go in their respective bits of the circle. To calculate the LCM one only has to multiply the HCF with the rest of the factors. After having gone through a couple of examples on the board I decided to give them another exercise involving both HCF and LCM.

For the last 10-15 minutes I went over the prime numbers that we need to use for the decomposition and got them to do some real exam questions. I stressed the importance of practice and the need to try to complete as many exercises as they can in their own time.

Analysing the lesson, most pupils were able to reach the learning targets that I had set for them. There was good class participation and peer-to-peer support.  The plenary imbedded the importance of the topic to the final objective - passing the exams. It is always important, in my view, to express the importance of practice to deepen understanding and improve skills.

Conclusion And Professional Development

This essay has allowed me to reflect upon my own teaching style and philosophy. I have learned a huge amount about all aspects of teaching from misconceptions, pedagogy, time and behavioural management, planning, delivery, pace and creativity.

During my own teaching the aspect that I most struggled with was always behavioural management. Being soft spoken did not lend itself well with those pupils who took it as a weakness. I had to quickly develop a strong presence in the classroom. This process is ongoing and the effort is beginning to pay dividends.

Through the analysis of pupils' work I was able to identify common misconceptions that they had about some mathematical operations. Through incorporating the use of probing questions and counter-examples I have tried to make pupils think about the mathematical operations they are performing. This has forced them to reflect on any misconceptions they may have had and begun the process of overcoming them.

From my experience, I have noticed that it is not enough to have good curriculum knowledge, behavioural management skills and pedagogy, important as they are, but a teacher needs to develop an ability to identify the types of misconceptions that pupils have and come up with effective strategies to overcome them. This is no easy task and it is a skill that comes with time and experience. It is a journey well worth undertaking and one that I have begun.

Bibliography

Keith Devlin (2000) The Math Gene. New York, NY: Basic Books.

Devlin, K (1994) Mathematics: The Science of Pattern. New York, NY: Scientific American Library.

Smith, A (2004) Making Mathematics Count. London: The Stationary Office.

DfES, Key Stage 3 national Strategy, (2002) Learning from mistakes, misunderstandings and misconceptions at Key Stage 3. London: DfES.

Haigh, G (3 February 2006) The answer's in the mistakes. TES magazine.

Watson, Anne (2005) Maths 14-19: Its Nature, Significance, Concepts and Modes of Engagement University of Oxford.

Tall, D (1991) The Psychology of Advanced Mathematical Thinking. In: D. Tall (Ed.) Advanced Mathematical Thinking (pp. 3-21). Dordrectht: Kluwer.