The Aims Of Teaching Mathematics

The Aims of Teaching Mathematics

1. Introduction

The mind is an instrument, you first sharpen it, and then use it the acquisition of the power of solving a quadratic equation is part of the process of sharpening the mind , Whitehead (1967). Having this phrase as one of the top-of-mind quotes, from the very first moment I started to study for my degree - almost thirty years ago I realised that, the way to self-improvement can undoubtably pass by expert knowledge but - first and foremost it is dependent on a general attitude - a philosophical perception of life - which I label as the ability to manage it .

Today, after deciding to take this sharp turn in my career - choosing to become a teacher - the abovementioned opinion is the first thing depicted in my Personal Philosophical Statement (PPS) it is my first objective in terms of what kind of teacher I want to be.

Throughout the following essay, I shall try to give the answer about the aim of education, focusing - whenever this is possible - on mathematics. I shall try to support my beliefs, as they are conveyed in the PPS, by providing proves about the importance of imparting knowledge and not just skills creating problem solvers (students who will face challenges including mathematical ones as a mean to sharpen their mind and become better students who will enjoy the procedure of finding multiple ways to answer questions without remaining focused on the result). I shall try to illuminate possible real-life impediments that can be observed towards this desired end state - as they are identified throughout my tiny experience in this profession - and to indicate possible ways of overtaking them.

2. Personal Philosophical Statement

After 26 years of service as a navy officer having completed two more years of experience working for a multinational firm I identified the internal force that guided me to a teaching training programme the inspiration that persuaded me to abandon a job that just suits my financial needs and follow the path that can for once again - deliver the essence of happiness and can keep me passionate and driven. It is the same force that defines, what kind of teacher I want to become: I want to be the teacher who educates and not the one who just trains - his students.

My vision is to invest appreciation, energy and time, so my students to understand that, success does not come only in the form of just passing exams first and foremost, I intend to make them understand that the key for achieving their strategic goals passes by the improvement of their ability to manage it . I want to impart lessons which make them realise that, deriving and illustrating is not just a subject knowledge task, but one more drill towards real-life problem-solving . I intend to make them appreciate the value of perseverance and hard-work to persuade them about the fact that, in many occasions an effort that does not come to fruition, is more valuable than an easy job done .

In terms of classroom management, my intention is to promote intrinsic discipline that comes naturally, as a result of mutual respect and not as an outcome of sanctions and punitive methods. I aspire to bring them to the point of embracing the necessity of relentless politeness and order, not only as school regulations but as fundamental principles of every flourishing society.

I aspire to make my students love learning, not just because they come to school every morning but because this is the way for self-improvement and self-completion. I aspire to impart lessons that are intended to create questions and provide kick-off for further research I want their knowledge to deepen intrinsically so as to find their way in life, devising later on their guidance for even younger pupils.

In a similar vein, I shall invest effort, so as my colleagues to accept me as soon as possible as a good teacher and professional like what happened in my previous career, I want to be respected to leave my indelible mark at every school I am assigned to. I want to make new friends and trustful associates to be accepted by people who share the same passion: to serve a higher cause and provide service to the society.

Knowing that this is not a rosy path, I want to be the person who does not crack under pressure I am planning to strive for greatness and I shall do whatever it takes to be the expected role model for my students, my associates and eventually my family I shall support my decision to the end I am convinced it s worth it.

Part A

3. Education versus Training

Trying to demystify the pretty vague difference between these terms, I shall invoke the opinion of Chris Barnes as expressed in 2014 about the difference between education and training. Education is all about learning the theory it reinforces knowledge that you already possess (you have command of the English language, but you still learn English) or serves as a baseline for knowledge that you will never use (you learn about splitting the atom, but you don t do it). And that s where training differs: it creates or improves skills (the ability to use one`s knowledge effectively and readily) so as to perform something, rather than knowing about it, Merriam-Webster (2003). Training is specified and tailored to bridge gaps or improve specific abilities.

That s not someone to say education has no place in training or vice versa lawyers must understand the principles of law before they learn to deploy it and a surgeon must have a detailed knowledge of anatomy before picking up the scalpel. In a similar vein, both of them have to spend a huge number of hours practicing, before their academic studies are completed, Barnes (2014).

It is obvious that the two terms can be used and this is typically the case interchangeably it is also self-proven that education is something broader, which - to an extend - includes training. Practice (the act of working at something repeatedly, so as to become proficient) is very much included in educational procedures as well, Merriam-Webster (2003). So, what is the critical difference that can clarify the two meanings with regard to their place inside such a system and - first and foremost why someone to admit that one of them is better than the other, as a method of bringing pupils up?

Education is actually a procedure through which, mind is encouraged and fostered to think independently while instilled knowledge introduces it to the physical and cultural world. It is able to produce a mindset not currently owned or to deepen a mindset already possessed. Understanding the theory behind topics, is actually a procedure that harness the sense of values it promotes judgement and reasoning and extends far beyond by developing thinking abilities, attitudes and behavior patterns. It creates self-discipline, improves self-esteem and broadens the horizon for further research and exploration, Bruner (1966). Nevertheless, there is always some overlap - practice is always inevitable and necessary but education is the one that provides insights, mental enjoyment and an appreciation of beauty, Rickman (2004).

To an extent, the aim of education is not just to create high achievers: evidently, good grades are important indicators of high-quality in teaching, hard work and increased understanding. However, such achievements could only be measured in relation to specific circumstances, that have to do with personal choices and - first and foremost are time dependent: a student can achieve well at GCSE examinations, but might not be able to have a similar success under some future educational regime, aiming at different objectives, Ernest (1991).

Consequently, the multifold role and aim of education is to promote subject knowledge in conjunction with the construction of a mindset a set of values, attitudes and traditions that leads to intrinsic discipline, creative reasoning and finally, self-improvement. The abovementioned combination (knowledge, values, critical thinking, creative reasoning and appreciation to beauty) was exactly what Alfred North Whitehead had in mind when he stated that the aim of education is to create culture . But how can we adhere to such an opinion? Is it feasible to concentrate something so vital in one single word? What is culture and is it big enough to include the aims of something so important, like education?

Culture can be considered as one of those terms that have peculiar properties: it is universally appreciated as something particularly important and yet, when the question arises of what this is, the term have varying connotations. I shall adhere to the opinion of Edward Sapir (1921) and Regna Darnell (2010): culture is not an object or something that can be signposted it is a pattern of behaviour. Under the light of such an explanation - yes - the aim of education is to create culture. And by the time Mathematics is one of the core subjects in our educational system - yes - its aim is to promote students culture to create pupils pattern of behaviour that produces expert knowledge through critical thinking, instilling curiosity for further and deeper research, championing creative reasoning and forging the appreciation to the beauty and the relevance of mathematical applications in almost every area of modern life.

4. Mathematics and Understanding

By the time teaching mathematics includes instillation of expert knowledge, the interconnection with understanding has to be investigated. Leonard Joseph Waks proved through The Monist (1968 p.104 to 119) that, these two terms (knowledge and understanding) cannot be considered as separate educational aims they have to be examined in parallel and under no circumstances to be contrasted: If knowledge is to be taken seriously as an educational aim, then we must realise that not only can knowledge not profitably be contrasted with, but must be conceived as presupposing, understanding .

Under the scope of the conclusion mentioned above, I shall try to expand my beliefs about teaching mathematics, towards this direction I shall try to outline the desired form of understanding that should be achieved, in order the aims as mentioned above to be met.

According to the definition of the word, understanding is the knowledge about a situation, a topic or how something works . Richard Skemp was one of those who wrote about this subject (first published in Mathematics Teaching , 1976). According to his opinion, there are two types of understanding: instrumental and relational. The first can be considered as, the ability someone to know the rule that applies in the occasion, the way this rule is used and how to manipulate it. In contrast, relational understanding is deeper: knowing why the rule works and how it connects to other rules is imperative so as this kind of understanding to be achieved.

Despite the fact that both of them lead to the same correct answers, relational understanding is way more extensive. In order to present the differences between the two terms, he used the example of a person who moves into a town and tries to find fixed routes in order to cover specific destinations. Instrumental understanding can be considered as the method of using landmarks and benchmarks, so as to cover particular locations (e.g. in order to get to the post office, start from the police station, take the third right and then the first left). In contrast, relational understanding is the state of being able to know alternative routes, so as to cover occasions like a blocked road. It is obvious that the first can be achieved in a few days the second needs much more time but, it is self-proven that it illustrates mastery of the topic.

According to the same opinion, the main advantage of the second form is its adaptability to new tasks . Additionally, deep knowledge that can be transmuted to relational understanding, creates no obligations someone to remember things and of course, the ability to be effective as a goal in itself . Despite the fact that, both forms of understanding can lead to knowledge, mathematical education according to Richard Skemp should aim for relational understanding.

5. Social Aims of Mathematical Education

By the time education does not take place in a vacuum and since it always reflects - and at times challenges - the social, economic, political and intellectual context of its age, it would be wise to examine the aims of education - and to an extent, the aims of teaching mathematics - through the prism of society. And I could not think of a more appropriate public figure to express this opinion, than the person who is responsible for the strategic planning in education: the Schools Minister.

According to Rt Hon Nick Gibb MP delivered on 9 July 2015 education has been identified as the vehicle to social fairness, the engine of the economy, the foundation of our culture, and the essential preparation for adult life .

Presenting a political opinion, this approach cannot be considered as something less than contextual and normative, Harris (1999). It displays the desire of a Conservative Government to prepare the next generation of informed citizens, introducing them to the best that has been thought or said , instilling them intrinsic love of knowledge and culture it can be identified as a relativistic approach of the desired intellectual development of students, as this is defined by William G. Perry (1970), which nonetheless includes commitment to values that are dictated by politicians.

And here is the point where Harry Brighouse disagrees (2006): the aim of education should not be the preparation of students to become sufficient contributors in the economy, as young adults. With all the respect to this opinion, I shall agree with the opinion of the politician since, before anything else, a person has to cover the lowest stages of the hierarchy of needs, which are the physiological and those that have to do with safety, Maslow (1943).

As an additional objective, education is expected to aim towards the direction of preparing these people to have the desired vocational development and to equip them with the resilience and moral strength needed to overcome challenges. The whole position combines constructivism with social justice, in a framework that includes knowledge and values related to development, following the ideological principles of Relativistic Fallibilism, Ernest (1991).

Despite the fact that this opinion was expressed very recently and that there are major arguments about the topic, the modern politician comes to agree with Alfread Whitehead and Peter Rickman: the education is expected - for once again - to create mindset and promote culture. It is also expected to forge the characteristics expected by a modern citizen (socially fair, well informed, resilient, persevering) in order to be ready to overtake challenges and promote liberal and economical values of our multinational, demanding and highly competitive modern age.

Another important point of this posture is the fact that, mathematics and other subjects like physics, chemistry and biology are consider topics of essential significance in our modern economy and that mathematical education is set at the forefront of the effort. Maybe, at this moment, the abovementioned element cannot be considered as added value and we cannot ignore the opinion that, potentially, every subject could provide, what Michael Young and Johan Muller described as powerful knowledge (2013). In any case, mathematics was always important in every curriculum throughout history and its aims - as they are illustrated under the prism of social needs cannot differ from whatever mentioned earlier.

6. Conclusion

As can be seen from those mentioned above, the ideas about the aims of education either in general or in particular in mathematics have a unique point of intersection: the utilisation of knowledge in life. And since the particular knowledge of finding the roots of a quadratic equation cannot be considered as an everyday application (unless you are teacher), the aim of education has to be transmuted into something broader: into ideas, values or skills that could provide the ability, a young person to become better, to achieve self-improvement and flourish into a society which changes through the time.

All in one, the aim of teaching mathematics can be considered as the utilisation of knowledge, so as a specific mindset and attitude towards life to be created. As a result, the students will be inculcated with the curiosity to be informed, the motivation to achieve self-improvement, the self-discipline to underpin values and laws, the capability to promote liberal ideas, the sensitivity to face modern challenges and the eagerness to promote the efficiency of the society they live in.

Part B

7. From Theory to Practice

During my days in the engineering school almost 30 years ago I was taught to find answers to everyday questions, based on elegant formulas and idealised preconditions: the friction was almost always negligible ropes were inextensible and so forth. Conditions were always predictable so there was always an appropriate solution in accordance with the rules of mechanics. And here is when the impediment of reality appears, most of the times at the blind spot after a sharp turn.

For once again, I ran into it during one of my projects while working for the armed forces my thesis ran afoul of one of what we call real life discontinuities, which destroyed all the elegant theories based on mechanics. The designer had installed a backstop to prevent the over-stressing of a valve, which even though was extending the reliability of the system could not fit in a computerised mathematical model of the engine.

This is exactly what happened when I started to realise - pretty recently - that there is also another impediment (that has to be overtaken with the use of another non-fitting backstop ), between studying or investigating how teaching should be in theory, and how actually is on the battlefield what the aim of education is in theory and in practice.

From this point on, I will focus on the question mentioned above, staying attached to the subject I teach: all the experiential positions and opinions will be dependent on my tiny experience in teaching mathematics in a mainstream school. Keeping this precondition top of mind, the reader will be introduced to my opinion about what the aims of teaching mathematics are, under the imperfections and discontinuities of every day practice.

8. Understanding in Action

The necessity of investigating understanding as part of the educational procedure, is something that has been proven throughout previous paragraph. The thesis that has been expressed is that, the desired understanding we should strive for - if we want to achieve the abovementioned aims of education - is relational.

What I realised from the very first lesson was that, the understanding actually achieved in a school - especially in mathematics - is deeply instrumental. Pupils are exposed to a plurality of empirical explanations, a number of different methods and a variety of schemes of work, without any opportunity to deepen into interconnections or possible expansions into other rules.

Typical example of the common strategy used in schools is, the way students are taught how to divide fractions: in order to divide two fractions, Keep the first as it is, Change to multiplication and Flip the second KCF rule (the level of standardisation has reached the point, of using an anagrammatism of a famous junk food brand as an abbreviation of the rule , so as pupils to remember it).

Any effort of mine to change the way topics like this are taught, received comments like increased talking time or slow pace . The unanimous advice I got by experienced teachers during my first lessons as a trainee - was: if it works, do not change it, otherwise students will be confused .

Obviously, the abovementioned technique works and by the time the head of Key Stage or the Department gets the results of students outcome during the assessment, everything is fine. Pupils are satisfied because they have found some kind of rule for getting the answer as soon as this is reached, they latch on it and ignore the rest , Skemp (1976) the teacher is happy because he does not need to spend additional time and effort to give a different explanation normally based on a textbook that students do not possess so, additional workload is avoided in order these notes to be photocopied and distributed to them the management board of the department approves, since targets are met and finally, school is delighted because GCSE results convey an obvious increase in A to C category . Subconsciously, the aim of (mathematical) education is sidetracked to impart knowledge with any possible way, so as targets to be met .

As a proof of those mentioned above - Ofsted plans to overhaul the way schools are inspected. According to Chief Inspector Amanda Spielman (11 October 2018), the organisation is downgrading the influence of exam results in favour of a closer look at pupil behaviour and at the breadth of subjects being taught exam performance has increased the pressure on school leaders, teachers and indirectly on pupils to deliver perfect data above all else , Adams Weaver (2018). According to Ofstead s announcement (GOV.UK / Press Release 11^th October 2018), schools will no longer receive a separate grade for outcomes of pupils while, the may effort will be on the substance of education and a broad curriculum. A consultation on draft framework has been announced to be launched in January 2019 so as the framework to be in effect in September.

But even if the backwash effect of examinations could no more be the major impediment towards relational understanding, the over-burdened syllabus is. It is a fact that mathematical statements can condense into single lines mathematical ideas are extremely concentrated so non-experienced people - and of course the students - are not able to realise them easily, Skemp (1976).

In agreement with those mentioned above, my effort after a couple of weeks in class was focused on delivering topics that could be understood by the majority of students, in a pace that could provide them adequate time to digest and practice them thoroughly. Nevertheless, the Term Assessment that took place at the end of December was very disappointing because, even though my class did well in topics that we had covered, advanced topics which a top set had covered, were not accessible to them at all. Despite the fact that the class had a very good average, the filling of not giving them any chance at all to be more competitive, is something that will make me reconsider my overall strategy during Spring Term.

9. Teaching in Action

I am indebted to my father for living, but to my teacher for living well , Alexander the Great (336 BC) no other quote can concentrate in a better way the importance of the teacher in every educational procedure. Teachers have the unique opportunity to impart knowledge and create characters - if they are well qualified - or to mislead, disappoint and discourage, if their cognitive skills are not well developed. As a result, it is equally important, the quality of teaching to be examined, so as its influence on the theoretical aim of education to be estimated.

It is known that the educational system suffers from a severe shortage of teachers, leading to bigger classes and many subjects to be taught by staff without a relevant degree. According to the Education Policy Institute (independent think tank, the aim of which is to raise the standards in education) the government faces a significant challenge to recruit enough teachers - particularly in subjects such as maths and science . The abovementioned problem has triggered an increase to teacher-pupil ratio by 1.5 units in one year (17 students per teacher in 2018), BBC (2018).

It is self-proven that such a shortage in experienced teaching personnel has an equally severe impact on the quality of the lessons and the efficiency of the educational system to meet its aims. And it is not just the number of vacancies that indicates the actual size of the issue the vacancies and the timetables will be filled one way or the other. It is the knowledge and - even above knowledge - the experience that makes the difference. Eventually, pupils find themselves being taught by temporary staff, teachers who perhaps teach outside their specialist subject - not entirely familiar with the curriculum - or supply teachers who provide a short-term solution and cannot create any psychological connection with their classes.

This frustration and disappointment were the reactions I received when I entered my first Year 8 class during Day 1. Students were annoyed understanding that a new trainee teacher would be with them this year and - even worse - was about to leave them after a while. It is the same reaction I see whenever I observe supply teachers as well students demonstrate a very disruptive behaviour and poor performance throughout the lesson.

And here is the point, where teaching in action confronts and undermines one of the theoretical aims of education: to promote equal opportunities and improve the fairness of our society. By the time not all schools are affected or at least, not in the same way a considerable sense of unevenness of opportunity is created. By the time some schools are able to snap the small number of well-trained and highly experienced teachers, a sense of inequality in provision is forged, undermining one of the aims of the educational system: to provide equal opportunities in order to create a fairer society, CEER (2000).

10. Aims and Curriculum

The last area of investigation towards the direction of finding the differences between the aims of teaching mathematics in theory and practice, is what actually being taught in our schools. Through these paragraphs, the curriculum is about to be examined, so as a conclusion to be produced, about its effectiveness to achieve the original aims of teaching this subject.

It is self-proven that, if you want to sell a product, you have to create two conditions: first is the need a good reason for someone to buy it. The second is a suitable ratio of cost over benefit, so as this product to be attractive and preferable. The situation in mathematics - especially in terms of what is been taught in mainstream schools - does not cover any of these conditions.

Despite the fact that mathematics is a subject with massive applications in everyday life, almost none of them is taught in schools. Basic operations, percentages, ratios, approximations, elevations and fundamental principles of using a map are items that can be found in the national curriculum. Nevertheless, all of them require limited amount of mathematics and apparently, do not promote any mathematical thinking or the curiosity for exploration and further research their contribution to students discovery learning as described by Jerome Bruner - is limited.

My observations throughout my lessons so far, have led to the conclusion that pupils are not interested - and consequently, the level of understanding is lower when the examples used are not relevant to everyday life: what is the point of finding the Lowest Common Multiple in order to see when two busses will be at the specific stop at the same time? All I need is when the next bus arrives was one of the most disarming questions I ever had.

Things were much different when I combined mathematics with knowledge from other subjects, engaging them more richly with what if ? Students were much more interesting when the question about percentages was combined with the advice of a dietologist about body weight: how important for our health is to be 10% underweight or overweight? Students were interested in calculating how much greener it is, if we take the train from London to Edinburgh rather than to fly?

Perhaps, someone will oppose the plurality of topics to be covered and perhaps this is true. But isn t it a way to create curious future researchers? Isn t it a way to achieve the theoretical aims of teaching mathematics or educating in general?

Further observations of mine have to do with the fact that students - and Ministry for Education as well, according to those mentioned in previous paragraph - are very interested in vocational development. The level of understanding was much higher when topics were related with specific professionals. Students were interested to know that engineers use derivatives and integrals during their everyday activity or that bank managers, typically have good knowledge of statistics. By the time topic was related with potential salary, further motivation was created.

Of course, a lot of ideas have been produced towards this direction. The idea of an aim based curriculum instead a subject based one, is something that has been introduced a few years ago. According to this posture, Department for Education - the organisation which has the responsibility for strategic planning in terms of education - has produced a National Curriculum which creates the main framework for every subject. According to this opinion, the main framework for an aims-led curriculum should be the needs and wants of students, Reiss White (2013).

11. Conclusion

Perhaps, my experience does not allow me to express a valid opinion about which one is better maybe the aims of education, as stated by the Schools Minister, can serve as a baseline for an effort towards the direction of an aims-based curriculum. The point is that, so far and for a variety of reasons (pressure for good results, overloaded curriculum, inexperienced teachers, the curriculum itself) students strangle to keep up with mathematics, society is facing serious difficulties in finding STEM graduates and the educational system - and consequently, the way mathematics are taught faces challenges to achieve its theoretical objectives. Apparently, this situation calls for immediate change to the course of action.