# The nature of mathematics

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### Overview

This research paper is to discuss the nature of mathematics and how the teaching and learning of this unique subject has changed over the years in England. I will assess this in relation to its use in class room teaching while the national curriculum is implemented. The main question is why the changes are taking place and is it heading in the right direction.

### Introduction

What is the nature of mathematics? University of Georgia's learning frame work defines as follows;

"Mathematics by nature is both a pure, theoretical adventure of the mind and a practically applied science which relies on both logic and creativity." This sounds like an easy and understandable way to describe it in just a brief sentence.

Furthermore, the Cockroft Reports confirms the following;

"There can be no doubt that there is general agreement that every child should study mathematics at school; indeed, the study of mathematics, together with that of English, is regarded by most people as being essential."

(The Cockroft Report (1982) Mathematics Counts)

I can confidently say, that it is a well known fact that Knowledge of mathematics is important for every one to function successfully. Even though this fact is stressed by educationalists and employers, the younger generation in a school based learning environment find it difficult to learn this important subject. (Martin Hughes, Children and Number 1986). As a result, large numbers of children leave school lacking both competence and interest in mathematics. As a matter of fact pupils and adults try to distance themselves from mathematics. In "Children and Number" Dr Martin Hughes proposes a new perspective on children's early attempts to understand mathematics. He describes the surprisingly substantial knowledge about number which children acquire naturally before they start school, and contrasts this with the difficulties presented by the formal written symbolism of mathematics in the classroom. He argues that children need to build links between their informal and their formal understanding of number, and shows what happens when these links are not made.

I believe that Mathematical thinking is vital to everyone in a modern society. Mathematics is required for its use in the workplace, business and finance and for personal decision-making. Furthermore, mathematics is not only the foundation to a nation's prosperity but it provides tools for understanding science and engineering. It is an important tool to understand economy as well as the financial and banking industry.

### Street Mathematics and School Mathematics.

The mathematical activity that is learned and carried out outside school is called Street Mathematics (Nunes 1993). The school and street mathematics are two forms of mathematics that is learned in and outside the school which is based upon the same mathematical principles.

Imagine you are in South America. You are walking through a noisy street market, full of activity. You're actually in the city of Recife in Brazil. You walk up to one of the stalls, selling coconuts. It is manned by a largely uneducated twelve-year old boy from a poor background.

"How much is one coconut?" you ask.

"Thirty-five," he replies.

You say, "I'd like ten. How much is that?"

The boy pauses for a moment, "Three will be 105; with three more, that will be 210. (Pause) I need four more. That is . . . (pause) 315 . . . I think it is 350."

It is clear the boy isn't doing it in the quickest way. To multiply by ten you simply add a zero to 35 which becomes 350. He did not know this rule because he did not have traditional knowledge of mathematics which one learns in the school. His arithmetic skills are self taught at the market stall. The following illustration shows how he solves the problem.

He often sells his coconut in groups of two or three coconuts, and he knew the cost is 70 for two and 105 for 3 coconuts. With an unusual request of 10 the youngster splits the 10 into groups that he can handle.

Now he needs to calculate 105 + 105 + 105 + 35. First he adds 105 and 105 which is 210 and then adds the next 105 and gets 315, and finally works out 315 + 35 = 350. This is an excellent performance of a young boy with poor education. The same experiment was carried out with some more children in different stalls. These experiments were carried out as part of a study to compare the Street Mathematics and School Mathematics by Nunes and her colleagues over a period of 10 years.

About a week later when the same market stall youngsters were "tested" again with a pencil and paper with questions that included straightforward arithmetic questions in written form and the same worded problems same as those carried out at their stalls. Although the arithmetic was excellent when they were at their stalls, but in this informal test they were successful only 74% when presented with market stall worded problems, and only 37% when the exactly same questions were given in the straightforward arithmetic test.

The children demonstrated that they could do the arithmetic when it meant something to them. Meaning plays a major role to do arithmetic. The children found meaning in the calculations and the operations they performed. The questions the children were unable to do were the ones that were entirely with symbols. In the school mathematics you learn a set of procedures or methods for addition, subtraction, multiplication, or division. You carry out these operations without an understanding of what the numbers actually represent or the meaning of them. The method taught in the school is international, and once you learned them you can apply them to any particular circumstances involving any number. For some one who could understand the symbols and apply appropriately, the procedures are helpful and indeed they are necessary. As a matter of fact all our science, engineering and technology depend on the mathematical knowledge. However this does not make any easier to understand or learn mathematics.

I believe that mastering school mathematics involves knowing some kind of meaning for the objects involved and the procedures performed on them. In my view I don't think that the human brain can perform meaningless operations. It cannot be compared to a computer hard drive to perform, without understanding, meaningless calculations. To understand and give a meaning to the mathematical operations "Functional Skills are being introduced in the National Curriculum.

"Functional mathematics requires learners to be able to use mathematics in ways that make them effective and involved as citizens, able to operate confidently in life and to work in a wide range of contexts."

In my view the Functional skills will give the School mathematics an understanding or meaning that is necessary to use it in the real world. Today's major problem is the pupil's lack of understanding and application of mathematics in everyday life.

### GCE

General Certificate in Education (GCE) was introduced as part of the British educational system in 1950. There were two levels namely ordinary level (O- Level) which is a subject based qualification replaced the summative "School Certificate" and the other academically challenging advanced level (A - Level). Some educationalists were critical about the GCE O Level, due to lack of proof of pupils overall academic ability. Mathematics was learned as a subject on its own merit. Pupil learned the subject without a clear understanding of how, where and when it could be used.

### GCSE

In 1986 General Certificate of Secondary Education (GCSE) replaced the GCE O Level and the Certificate of Secondary Education (CSE) ( Tanner, 2000) and the Central Government took over the responsibility of education. In Mathematics only the top 20% passed during the period of O Level and GCSE aimed to increase this to 50%. The only practical way to achieve this was by making the mathematics syllabus "easy" in the National Curriculum. This had an effect on the pupil's mathematical knowledge (lack of) when they completed their GCSE. Furthermore GCE A'level syllabus also needed to be made easy to keep the gap closer. This had a further effect on the Universities where undergraduates did not have the adequate knowledge in Mathematics to complete their chosen programmes especially in Engineering, science, economics and Mathematics. Universities had to introduce new modules or in some cases lengthening the course from 3 to 4 years. This introduction of GCSE, increased the pass rates in Mathematics and encouraged pupil to learn and like the subject.

### Advanced Supplementary, Advanced Subsidiary and A2 Levels

Advanced Supplementary levels in the KS5 curriculum were introduced in 1987. The purpose of the introduction of AS level was to allow, if they wish to, learn and expand their knowledge in different subject areas. In 2000 the government introduced the Curriculum 2000, which dealt with splitting A level into Advanced Subsidiary and A2 examination. The difference between the advanced supplementary and advanced subsidiary is they both covered the same amount of subjects, though the previous Advanced Subsidiary levels differ from the previous Advanced Supplementary levels in that they cover the same breadth of the subject as in the full A level but in less depth. The old AS-levels were also not a prerequisite for the corresponding A level and were examined separately. The Advanced level mathematics was becoming easier and easier and the mathematical knowledge of pupil was becoming weaker.

### New A* grade in A2 Level

Over the years there has been a steady rise in the A level grades and this may be due to the fact that the exams are becoming easier or on the other hand it may well be the changed or improved teaching methods. Another view may be that since the implementation of Curriculum 2000 students are able to drop subjects they found difficult after As level, and could retake modules a number of times from which the best marks and factors that improved the results are considered for the final grade, particularly mathematics in which six module exams determine a single grade. With the rise in "A" grades at A'level, Universities are finding it difficult identify the suitable candidates for degree courses. As a result of which Universities have started to have entrance exams such as "BMAT", "LNAT" and "UKCAT" for some courses and interviews for selection purposes. In view of the above from 2008 the highest As grade will be "A" but for the A level it will be A* for those achieving on average 90% or more in the A2 modules. Some differentiation is going to emerge over this, but the material covered for Mathematics will remain the same, and universities may have to continue with their entrance examinations.

### Three/Five part Lessons

Traditionally a teacher's maths lesson involved "explain - then - practice", where students will follow procedures rather than problem solving. In 1998-1999, as part of the new Literacy and Numeracy strategy three part lessons were introduced. In numeracy the lessons were divided into starter or warm up (10 minutes at the beginning) then the main part (40 minutes in the middle) followed by the plenary (up to 15 minutes) time. In my view this structure of a lesson allows pupils to switch between the three parts and helps them to be focused on task and thus maximising their learning. However during revision or investigative work the structure may not always be suitable and teachers should be able to be flexible regarding it. Some advanced skilled teachers consider the structure as a five part lesson including pupil's entrance to and exit from the class room as additional parts. This is a very efficient way of teaching mathematics. Pupils tend to use variety of fun mathematical resources such as mini white boards, puzzles, voting pods and quizzes to enhance learning and teaching.

### Cognitive acceleration through Mathematics Education CAME

Michael Shayer and Philip Adey from Kings College London designed a set of lessons for Year 7 and 8 pupils to motivate the thinking skills and this is called the Cognitive acceleration. CAME is widely known as thinking maths. The approach is built mainly on Vygotsky's theory of Zone of Proximal Development (ZPD), where a learner is stretched to the ZPD, with the help of a facilitator.

I analysed a lesson plan and a two Note sheets on the topic of Distance and Direction, and I was amazed by the way the proposed lesson was planned. The teacher is performing as a facilitator and the children are thinking and learning by themselves in small groups and deciding on a suitable method depending on the situation. Furthermore the children are asked to compare distance and direction with real life situations. The Note sheet is helping pupil to apply the knowledge of mathematics gained in the class to the outside world. As the pupils learning takes place through analytical thinking, I believe the knowledge stays with them. In my view this is the way forward to teaching and learning mathematics.

### Cross curricular mathematical teaching

Why is it important to embed Maths in other curriculum areas? Pupils' learning depends on their prior knowledge, understanding and what they can do in Mathematics. Pupils learning abilities are best, when they are needed and meaningful. Embedding the skills in other subjects such as Science will make a real difference to learning. Lessons will become interesting, enjoyable and the pupil will become motivated and see the connection of their learning across the whole range of areas. While this approach may not be possible across the whole range of subjects, consideration needs to be given where it is possible. The scheme of work needs to be planned in a manner which familiarises similar topics with different topics. This approach is becoming popular in many schools in England in learning and teaching of mathematics. The learning and teaching of mathematics is heading in the right direction.

### Assessment for Learning (AFL) and New Technology

Assessment in the class room should be used to raise pupils' achievement. Most of the Pupils will improve their learning if they understand their aim of the learning and how it can be achieved. Every assessment should include or identify a pupil's progress and comments to provide further or extended learning. This is a powerful way to increase pupils learning. My personal experience in questioning, in marking pupils work and providing AFL comments, made me realise the impact this may have on their future learning.

Information and Communication Technology is one of the main class room resources at present time in education. Starting with Interactive White Boards, Voting pads, power point presentations, Bowland Maths and Maths watch are some of the common classroom resources that are used in England. In my view Virtual Learning Environment should be the most effective resource, offering online access for 'anytime, anywhere' learning. This promotes and encourages home learning for pupil while teachers assess progress on line or in some cases assessed by the technology itself.

### 2. Planning and Teaching

According to the National Strategies for mathematics, their guidance for planning shows "the links between the strands of mathematics as an interconnected set of ideas. Good planning ensures that mathematical ideas are presented in an interrelated way, not in isolation. Awareness of the connections helps pupils to make sense of the subject, avoid misconceptions, and retain what they learn."

### Mathematics learning in the classroom

My first placement school is "The Bromfords School" in Wickford in Essex, where I observed KS3, Ks4 and KS5 mathematics lessons taught by different teachers. I also observed KS3 teaching of Biology, Chemistry and French which gave me an insight into how the teaching has changed since my time at school. This gave me a flavour of teaching strategies used by various teachers. There is a "No Hands Up" policy in force during the lesson time to answer questions. There were many kinds of ways teachers randomly chose pupils, but two of the methods were interesting and caught my attention. The first was a computer programme called "Random Name Selector", which brings a name with the person's photo on the Interactive White Board (IWB) when the teacher taps on it with a special pen. The second was, tables in this classroom had two "Playing Cards", stuck using clear back plastic, while another pack of cards were held by the teacher and chose cards randomly which relates to a particular pupil sitting at a particular table to answer a question. Both ways were an effective way of nominating, while every pupil had to pay attention during the lesson, not knowing when they could be called upon to answer. This method of selecting encourages and maximises the learning while keeping them motivated and focused. This method also provides an equal opportunity for all to take part in the learning process. This engaged pupils throughout the lesson making them successful learners thus fulfilling the national curriculum requirements/aims.

The table arrangements differed classroom to classroom, to maximise the learning and teaching. In one of the classrooms the tables were arranged in twos, where four pupils were able to sit in a small group, the teacher had easy access to every pupil in the classroom. Another example to be noted here would be, that Special Educational Need (SEN) pupils are seated at the end of a row of tables, again allowing, once again easy access to particular pupils by the teacher and the "Learning Support Assistant"(LSA) implementing inclusion and differentiation.

Mini White boards are used successfully as class room resources in the starters, main part and in plenary to assess all pupils learning during the lesson. All pupils actively take part in this kind of mini assessment, especially if they are competing against a clock, bringing a happy and fun atmosphere into the classroom. The situation is more competitive and enjoyable when voting pads are used. Both of these resources ensure the participation of all pupils while encouraging the learning to take place in an enjoyable happy environment while addressing the Every Child Matters (ECM) outcome.

During my observations I came across a pneumonics that teacher's used, to help pupil to remember the underlying mathematics as and when required. Example to include such as "SOH CAH TOA" in Trigonometry. Furthermore some teachers used unusual words such as Fangle, Zangle and Cangle to identify Z angle, F angle and C angle. These abbreviations are very important, powerful and enjoyable way of learning, remembering and embedding the knowledge of mathematics.

The lesson which I am going to analyse is a yr.10 set 2 lesson. The pupil lined up outside the classroom and the teacher invited them into the class room. Before they went and sat down they were told to collect from the "Help Desk" a mini white board, a board pen and a board rubber. The clear learning objective of the lesson was on the boards that was to be able to plot graphs of y = x3 and y = 1/x. This was one of the topics in the Departmental Scheme of Work.

The lesson started promptly. The starter activity was recognizing graph types of y = mx + c, y = x2, y = x3 and y = 1/x. The teacher asked the class to respond to on MWB the "y" values in the equation y = x2 for given values of x. The pupil responded actively and enthusiastically. During which misconception such as when x = -4 the x2 = -16 was identified which was explained and corrected with further examples. The Teacher made sure that "All" had the answers on the MWB to make sure every one is working on task to accommodate inclusion in the lesson. He also commented and encouraged peer help addressing the AFL concepts.

Main part of the lesson started with first explaining and defining the terminology of Reciprocal of a number as 1/number. Some questions were asked from pupils to make sure that they had a clear understanding (reciprocal of 4, 8, 3, 1/33, 5/8 etc.). The pupil actively joined in and answered enthusiastically. The missing values in the table were found by the pupil using the MWB. 1/0 is a very "huge undefined number" also brought into the pupils understanding. This was a surprise to pupil but accepted (when proved) against their initial thought of 1 and zero.

Pupils achieved the objectives of the lesson and drew the graphs. Plenary was exciting. Pupils were made into groups of 3 and 4 and the task was to match the graphs with the correct equations. Even though the class became little noisy but this was due to mathematical talking. Pupil argued among peers and agreed on a solution. This is a great way to learn mathematics. During the plenary, teacher asked to demonstrate a linear equation. The pupils extended their arms either northeast to southwest direction or with a negative gradient northwest to south east indicating a straight line equation. For a quadratic equation of the graph of y = x2 , children put their arms up in the shape of a statistical symbol for union "È", and for the graph y = - x2 , the arms were bent looking down wards in the shape of the statistical symbol for and "Ç ", for y = x2 -1, pupils with arms up and bend downwards to show the movement of the graph one unit downwards and similar movements to other quadratics graphs. y = x3 was demonstrated similar to linear equation but curved and more demonstrations were carried out. This was a fun activity where every one including the teacher joined to demonstrate the shape of the graphs embedding important mathematical concepts in pupil. In this type of a class room atmosphere the pupils were enthusiastic about every activity that took place keeping a positive attitude towards learning. There was a safe and fun environment provided in the class room from the beginning to the end where pupils achieved their potential.

In my view the lesson above included all the aspects that was given below by the DES in 1985 which considers the way a pupil to learn mathematics;

- Fascination with the subject;
- Interest and motivation;
- Pleasure and enjoyment from mathematical activities;
- Appreciation of the power, purpose and relevance of mathematics;
- Satisfaction derived from a sense of achievement;
- Confidence in an ability to do mathematics at an appropriate level.

According to New Brompton College "Learning should be fun, engaging, meaningful, rewarding and accessible to all" (New Brompton College - Times Connect).

### Lesson1

In this section I am going to be analysing the important steps taken in the planning of a lesson and will be carrying out an evaluation of the starter which I delivered. The planned lesson was for a Yr8 Set 4 (8MA4) lower ability (level 4) class on the topic of Area and Perimeter within the field of Mathematics namely Shape Space and Measures with a lesson objective of "To calculate the perimeter and area of compound shapes".

At first I looked at the National Curriculum, Ma3 Shape Space and Measure. According to the Framework document, "Pupils should be taught to deduce and use formulae to calculate lengths, perimeters, areas and volumes in 2-D and 3D shapes" (Framework for teaching mathematics, DfEE, p234), identified what exactly needs to be taught. This related to yr 7 pupils, but in view of the lower ability of the Yr 8 it was considered appropriate. The Frame work also identified the literacy element of the topic and gave the words that pupils should be able to read , write and spell correctly. Examples include area, surface, surface are, perimeter, and use of the units such as square centimetre (cm2). Furthermore level ladders (www.kangaroomaths.com, 2008) on Area, Perimeter, Volume Pythagoras, & Trigonometry was also used to identify suitable "level" materials in planning the lesson.

With regards to prior considerations, the pupils have learned area and perimeters of basic shapes such as square rectangle and triangle. However due to their lower ability they needed recapping and embedding this knowledge again as a starter activity. While agreeing, a starter activity does not have to relate to the main activity, it was agreed that this approach was suitable to this particular class. There were 11 pupils from the Special Educational Needs register who required differentiation and inclusion. Learning support assistant was briefed about the lesson in advance for effective learning during the lesson.

To promote positive behaviour class room expectations were reminded with the no hands up policy while names were picked randomly from a pack of "named" cards.

Three basic questions needed answers?

- Where are my pupil's heading?
- How they are going to reach the destination?
- How will I know that they have arrived?

All my pupils should be able to calculate perimeter and area. Most of the pupils will also be able to recognise shapes to split up compound shapes to calculate perimeter and area. Some students will also be able to calculate the area and perimeter where an aperture appears.

Once the starter is completed where area and perimeter calculations are reminded using a power point presentation, the main activity to lead to the destination which was explaining two examples on the IWB, while interactions to take place with teacher to pupil as well as pupil to pupil. Further exercises were to be completed in the lesson from text books while going around the class room and helping those needed. To have a clear understanding of a compound shape for very low ability pupils, card boards cut out of rectangles and squares were given with measurements and were to make shapes and stick them in the books and calculate the properties of the shapes. The schools traffic light system method was used to identify pupils who needed help. Pupils who displayed green cards from their planners on the table indicated that they have an understanding and were happy to progress, while Red required help to progress further. Extension questions were planned to those that would require assistance.

Part of the plenary was a timed question and answer session on the IWB as well as functional skills questions bringing the street mathematics into the class room.

The Bromfords school's scheme of work was also looked carefully before the lesson was planned.

Perimeter was an easy enough topic for even a lower ability pupil to grasp the concept. However to understand and apply area was considered to be a difficult task. To over come this I considered the relevance of this topic to real life situations and gave examples such as wall papering and changing the carpet in a room was used to embed this knowledge. Furthermore the units of measurements were reiterated many times to over come any possible confusion. There was a fear that there may be a possibility at the end of the lesson for pupils to get confused, area with perimeter or vice versa. Repetition was used to eliminate this problem successfully.

### More Lessons

I have planned and delivered whole lessons on "finding the nth term of a sequence and Calculating fraction of an amount" for Yr 7 set 1 (lesson plans are in the appendix). Both lesson planning went through a similar process explained as above in Lesson. For the first lesson after going over few examples I asked pupils to come up with a method to find the nth term and the constant. Knowing the ability of the pupil in the class I was confident some one will be able to give adequate explanation. To embed the concept, a "Tarsia Puzzle" was given to grouped pupils to solve and make a poster. At the end I will show some of the completed puzzle to the class with some exit questions.

The second lessons timed starter activity was to complete a table to indicate divisibility of numbers. This should have sharpened their brain in dividing numbers and the completed task to be marked by peers. Further simple to harder examples to be practiced on the MWB. Another Tarsia puzzle was given to grouped pupil to complete on sugar papers. In both lessons names to be picked randomly using named cards.

### Evaluations

Area & Perimeter; I arrived at school very early to check my power point presentation on the IWB as well as familiarise with the IWB. I only delivered the starter of the lesson.

I introduced my self and started to present my power point. It was a question and answers session using MWB's. I received a very good response to my questions and established a very good relationship. Pupil felt comfortable to react with me in the lesson. Even though I had the name cards to choose pupil randomly, at the beginning I chose the pupil who put their hands up to answer the questions, but later on I chose pupil randomly. I felt questioning was appropriate and differentiated according to their ability and managed to get the meanings of Area and Perimeter from the pupils themselves. I insisted on units and made sure every pupil in the class room had an understanding of it by the end of the starter. I praised them whenever required and promoted peer support during the starter. I should make sure in future lessons "No hand up to answer questions" and give adequate time for all the pupils to answer on the MWB.

nth term; I met the pupil at the door and invited them into the class room while reminding the classroom expectations. Pupils were more engaged in the set work and I insisted on no hands up and picked randomly. Pupil willingly participated in the lesson and even came to the board to answer and explain answers. I gave clear instruction and pupil did the task methodically and some managed to finish earlier than I anticipated for whom I had extension work in hand. Some pupil needed support in completing the poster and I was there to help, motivate and praise them.

Calculating fraction of an amount; Similar to the previous lesson but this second time around I was more comfortable with pupil and I remembered some pupil's name. Pupils were able to understand the concept of equivalent fractions as well as fraction of amount. The keywords such as Numerator and Denominator were explained and repeated many times. I also tried to connect the mathematical concept with the every day life such as money, weight and volume to embed the conception. I also gave constructive praise to encourage, facilitate and include all students to learn without any intimidation in the relationship between me and the pupil. In future lessons I will provide more differentiated work and plan my lesson accordingly to cater for more able pupil.

I did not plan and deliver a set of lessons continuously for a class at the Bromfords School.

### Conclusion

This assignment gave me an opportunity to investigate and learn about the ways mathematics teaching is changed and changing. It was very interesting and exciting to find out the changes as well as the reasonings behind it. I do believe that street mathematics needs to be brought into the class room in terms of functional skills.

I also had an opportunity to analyse a set of questionnaire responses by pupils, on successful functional skill project called "Ground Force" which gave me an insight into the positive views expressed by them.

I believe in teaching of functional skills within the school curriculum and I will include it within my lesson plans. During my investigations the mathematical thinking skills commonly known as CAME is another area that greatly interested me most and would recommend its wide use in schools.

The assignment gave me an opportunity to learn and see the National Curriculum in practice and I have a positive view that the learning and teaching of Mathematics is going in the right direction.

### References.

- Nunes,T, Schliemann,A & Carraher, D.W (1993) 'Street Mathematics and School Mathematics' Cambridge University Press & http://www.maa.org/devlin/devlin_05_05.html
- Johnston-Wilder, S & P, Pimm, D, Westwell, J (1999) ' Learning to teach Mathematics in the Secondary School' Routledge
- 'Mathematics counts' The Cockcroft Report (1982). Available at http://www.dg.dial.pipex.com/documents/docs1/cockcroft.shtml
- Hughes, M& Donaldson M (1986) 'Children and Number' WileyBlackwell publisher
- National Curriculum KS3 & KS4; Available at; http://curriculum.qcda.gov.uk/key-stages-3-and-4/index.aspx
- Tanner H & Jones S (2000) Becoming a successful teacher of Mathematics Routledge Falmer
- Article on TES Connect on three part lesson; Available at http://www.tes.co.uk/article.aspx?storycode=2247419
- Article on CAME, Available at https://www.ncetm.org.uk/cpd/2581
- Level Ladders; Available at www.kangaroomaths.com
- A Level; Available at ; http://en.allexperts.com/e/a/a/a-level.htm#hd6