Education within the field of mathematics is the enhancement of knowledge and understanding, and for me, there is a strong link between the two, there seems little point in acquiring knowledge without understanding, and therefore, in order to raise achievement we must address this fundamental issue, however, whilst a deep understanding is important students must first acquire the appropriate knowledge to unlock the mysteries that lie within a problem in order to find the solution. Fritz Schumacher, the author of ‘Small is Beautiful’, illustrates this point by using the analogy of the shoemaker. An excellent shoe maker needs to know about feet, the lives of his customers, where the feet go and what loads they will be expected to carry. The knowledge of how to make excellent shoes is only truly useful when linked with an understanding of the lives of those who wear the shoes. This illustration of the link between knowledge and understanding has always struck me as profound, a link which can easily be broken and difficult to restore. This, therefore, begs the question of how can I as a teacher of mathematics increase knowledge and understanding and in so doing raise standards of achievement.
Given that the national average for GCSE results for 2006 was 57% 1. I consider myself to have been extremely privileged to have been placed in two schools that were both high achieving, popular, over subscribed and well disciplined schools, both had GCSE A* to C pass rates for mathematics of over 90%,with one school receiving a commendation from OFSTED. As a result my Initial Teacher Training has focused on continuing to maintain those high standards, and has encouraged me to actively seek out new and alternative ways of ensuring that all pupils, including those that have not necessarily been identified as failing but are not learning at their optimal level, are encouraged to meet their full potential. Whilst my two placement schools were very similar in terms of their academic success, the schools individual demographics were, at first glance, dissimilar. At my first school placement the pupils were predominately Asian, many of whom were from a working class backgrounds, whilst at my second school placement the pupils are all Christian or Jewish and predominately white, that said, there is a significant ethnic minority of mainly Afro-Caribbean pupils and according to the IDACI2. tables there is an even spread between working and middle class backgrounds.
On the face of it my two school experiences could be said to be very different, however, upon reflection the schools are more similar than they are dissimilar, for both have the same hard working ethos and belief that they are excellent schools with exceptionally high expectations from staff, pupils and parents. Both promote a strong sense of ownership within the school, a greater sense of belonging with the emphasis on pupils taking individual and personal responsibility for their own learning, general safety and well being. Both implemented policies which, supported by the results of current educational research, ensured that every pupil would be encouraged to fulfill their full potential, and thus both my schools work in harmony with latest educational theories, ( the headmaster of one is currently actively involved in consultations with the UK government with regard to raising achievement). Strict policies with regard to discipline were rigorously enforced whilst maintaining a caring and nurturing environment where everyone from pupils to cleaning staff, right through to head teacher and including members of the public mattered, thus taking the idea of ‘ every child matters’ to a new, higher plateau.
Class sizes varied with some of my classes being as large as 34 children in one class, however, as long as pupils are well behaved then class size becomes less pertinent. I was encouraged at all times to ensure I used lots of discipline techniques, such as insisting on silence before I spoke, hands up when pupils wanted to ask a question, pens down and looking at me and no going to the toilet in the middle of my mathematics lesson! Gradually I became much more confident and familiar with these techniques which are now slowly becoming second nature where I no longer need to consciously make the effort to implement them. I found that shouting at a class, particularly an older, teenage class such as a year 10 only served to antagonize and aggravate them; whilst there remains the occasional need to raise my voice in order to gain their attention I ensure that my voice remains calm and steady at all times, with no hint of impatience or anger.
I found that in so doing I gained much more respect and they were more willing to concede and become manageable, thus creating a more preferable, pleasanter and stronger learning environment. Nevertheless, regardless of class size there was always the premise that every student is an individual and therefore needs individual attention. As a result both schools, in particular my second school placement, had an open staff room door policy, so long as a pupil is not disturbing a scheduled class they are free to ask for help at any time, hence lunch is often very rushed.
I often take my lunch in the dinning hall along with several other members of staff and the pupils, I found this makes me appear more accessible and approachable not only to the children but to other members of staff, in particular classroom assistants and staff from other departments. I am always fascinated to hear their stories, trials and tribulations they face with some of their pupils; it is so interesting how some pupils can behave very differently depending on the teacher and the subject lesson! This interaction and exchange of ideas often stimulates me to improve upon my approach with some of the children I have found a challenge. I have also volunteered to perform lunchtime duties which has enabled me to become better acquainted with all staff and children alike, very often I am stopped and asked to help with difficulties pupils have encountered in their mathematics. Form time often lends itself to offer further assistance with mathematics problems, especially as I get the opportunity to explain and teach in an informal way to children that I do not normally teach within my scheduled timetable.
Truancy is not a big problem in either of my two schools as all staff are actively seen around school at all times so pupils are much less likely to attempt to leave the premises outside of school hours and there are fewer incidents of misbehavior, furthermore, both my schools have large playing fields and lots of lunchtime clubs which actively encourage children to burn off any excess energy. Pupils, both girls and boys alike always find it much harder to concentrate and stay on target during lessons when the weather is bad and they have not been allowed the freedom to run about outside. The mathematics department in both schools ran lunchtime mathematics clubs which offer more fun and interactive mathematical games. I helped to run some of these which I thoroughly enjoyed as I learnt something new as well! I am also involved in after school booster classes which are open to all, not just those who have been identified as weaker and in need of help. The feedback from the pupils themselves has been very positive as all find the sessions very helpful, particularly as it gives an opportunity for teachers to offer tips and advice on exams as well as allowing pupils greater time to practice their exam techniques with immediate advice on hand should they need it.
Achievement in all areas can only be truly accomplished when there is a strong emphasis upon learning the ‘3 Rs’ and having a solid understanding and knowledge of the fundamentals. Whilst I agree that IT is a highly useful tool I tend to use it with some caution. IT can hide and mask fundamental errors, misunderstandings and misconceptions within children’s learning. Whilst teaching a year 8 class statistics I felt it necessary for them to spend some time calculating averages manually in their exercise books in order for them to gain a clear understanding of the mechanics behind the numbers. When I was satisfied that they were clear and could calculate the averages using pen and paper I then progressed on to using excel on their laptops. All of my class became absorbed in getting the computer to perform all the complicated calculations for them; at the end of term when they needed to revise for their tests I was confident that they would be able to answer a question on ‘averages’.
Self–evaluation plays an enormously important part in raising achievement. By reflecting upon my lessons not just afterwards but also during lesson time I am far better able to adjust, amend and tweak my lessons in order to progress my class forward. Recently I have been teaching my year 10 class ‘factorising simple quadratic equations’, unfortunately they really struggled with the idea of finding two numbers that multiplied together to give the constant and that added to give the coefficient of x. I realised quickly that their weakness lay in their lack of confidence with adding and multiplying negative numbers. For the next lesson their starter for the lesson was completing an addition and multiplication number grids which included negative numbers. Once they were all proficient at these the mechanics behind factorising simply fell in to place. I often find that revising topics, especially the fundamentals is of vital importance when attempting to push a less able class on, although, there are times when it has been necessary to remind top sets the basics. In addition to self –evaluation CPD (continual professional development) is considered essential, if pupils are to raise their game then so must that of the teacher themselves.
Formative assessment and the use of AFL (assessment for learning) techniques are crucial for raising achievement. I was introduced to several tools which can assist and benefit children’s learning enormously, many of which I now consistently use as part of my lesson planning. My favourites are include:
1. Time to think – allows students to think about an answer for a few moments, I like to combine this with
2. Call a Friend – another student can help and answer the question, I try to encourage the pupil being helped to repeat the correct answer and explanation
3. Writing an example or answer from a student that contains an error – my pupils love to spot a teachers error, sometimes even trying to convince me that a correct answer is wrong which creates discussion and subsequent clarification
4. No hands up – a great way of ensuring I as the teacher selects who answers the question, and not from those who usually put up their hands
5. Pupils set the question – all my year 7s love this. They come up with a question either in pairs or by themselves and come up to the front of the class and pose the question to the rest of the class
6. Different ways to answer the same question – after accepting a correct solution to the problem, ask if there are alternative methods to solving the problem
7. Smiley faces – I introduced this technique to my second placement as it allows me upon marking pupils work to see how well they feel they have understood the topic. As it is fairly private I usually find I get an honest/reliable answer
8. Traffic Lights – helps students to self – assess their work by holding up a red, amber or green light, red meaning pupil is unhappy has not fully understood the topic and is not ready to move on. It enables the teacher to assess understanding quickly so that lesson planning can be adjusted
9. Timed activities – give set time such as 5 minutes to complete a task
10. Group work – is a particular favourite of mine. Pupils working together either in pairs or small groups promote collaborative working and improvement of pupil’s thinking skills. I find pupils enjoy teaching one another and feel happy to learn from one another.
My second placement school does not expect me to mark every piece of work that my class produces. I have 34 children in every class and so it is impossible to mark everything, all my pupils are given two pieces of homework per week, one we class mark and one I mark. Pupils swap with the person who sits next to them who in turn marks the work whilst I call out the answers; I review these marks when I collect their exercise books in. When marking the schools policy is to give a grade for effort along with an informative comment provoking thoughtful improvement rather than ‘right first time’. The school operates a credit system and so I often reward outstanding work, either class work or homework, by awarding a credit. All homework is expected in on time, failure to do so will result in a detention where the work will need to be done, and for persistent offenders there will be a phone call home.
Homework is written in school planners which are reviewed and signed weekly by parents and form tutors to ensure any issues a put right as quickly as possible.
All would agree that in order to raise pupil achievement the close and continued involvement of parents with school is of vital importance. Both the schools I was placed at had exceptionally strong relations with the parents. Parents were expected to be involved and were notified and included in discussion at all times and in return there was substantial parental support for the school. They believed in the school system and felt that their voice was considered an important and crucial one. If children are not being supported and encouraged to learn at home it makes it harder to persuade teenagers to consider their education as being of importance. Children need to believe that they need an education and that being clever is far ‘cooler’ than being a disinterested trouble maker. My second placement school has an active parent association which, through social events, raises funding for the school to purchase additional resources, as well as promotes the strengthening of links between home and school.
Through regular parent’s evenings parents and children are kept informed of progress, given targets and told specifically what is required in order for them to attain those targets. I was fortunate to be involved with parent’s evenings at both my schools; I enjoyed meeting everyone, and it was lovely to see the beaming faces of my ever so clever year 7s and their ever so proud parents. When pupils performance is not going as well as could be expected maintaining a tactful balance between wanting to encourage better performance and not demotivate, whilst still being honest and giving clear, unambiguous advice as to what needs to be done in order to achieve improvements can be difficult. However, if pupils and their parents believe that they are in a good school which they feel proud to be apart of, and are reassured and trust that the school will do everything possible to help improve their child’s performance, the system will only fail if all parties do not uphold their responsibilities, and so by all working together progress is surely inevitable.
As a new teacher I am always being asked to come up with new and inventive ways to teach lessons. Most pupils regardless of age love games; I found that the promise of an extended plenary of the battleship game ensured I had a completely engaged class who remained on task throughout the whole lesson. Obviously it is not possible to play games in every lesson, but a lively engaging starter or the promise of a fun end to the lesson often ensures compliance and good behaviour, with all remaining on task as otherwise they will not have the necessary skills to enjoy the fun part. One of the ingredients to raising achievement has to be ensuring that pupils are actively engaged throughout the lesson. Giving pupils the opportunity to present their own ideas within the structured lesson is important.
For one of my observations I taught circumference of circles to a year 8 class, the lesson went well, they were all happily involved in measuring the circumference of various different circles and were able to come up with the concept of pi by themselves. Among one of the most challenging lessons I had was recently. My second placement school believe, and quite rightly so, that regardless of whether it is the first or the last day of term, learning should still take place. I had a middle set year 10 last lesson of the last day of term, and none of them were in any mind to do ‘circumference of a circle’! I had suspected that this may well be the case, and promised that if they concentrated for at least half the lesson and did some work I would show them something a little more fun on circles. You Tube had supplied me with a fun, colourful cartoon animated song on pi. They loved it, and for one student it consolidated their understanding of pi in a way that up until then I had failed to do!
Any discussion on raising achievement often includes issues such as gender and class.
Interestingly, both my schools contend recent research that suggests that girls are out performed in mathematics by boys; they both have high proportions of girls taking mathematics at both GCSE and A’ level and gaining high, A* grades. At my first placement school the highest scoring pupils for both Further Mathematics, Statistics and Mechanics were all females, whether the almost all female teaching staff have any impact upon this is difficult to tell, however, it does not appear to be disadvantaging the boys. Most of the children I taught respected me for my ability to teach and I connected with them regardless of my or their gender or class.
In my limited experience pupils responded to high quality teachers who put in the effort, you get out exactly what you put in. If I am enthusiastic and keen to teach, show an interest and care whether they learn something then pupils will respond positively and mirror image that enthusiasm back. As a teacher I try to ensure I praise as much as I can, pupils need to be told that they can do mathematics, I remember the beaming face of a young year10 boy who once told me he could not do algebra. I assured him that by the end of the lesson he would, I chose him several times to answer my algebraic questions, and by the end of the lesson not only could he do algebra, he was brilliant at it and I told him so!
There are so many wonderful resources available today that make teaching mathematics so much easier; ten tick worksheets, mymaths, You Tube, interactive worksheets, user-friendly text books with lovely illustrations and explanations, however, I still believe that the heart of the issue of raising achievement is raising the expectations of all within the education system. Teachers need to build better rapports with the classes they teach, pupils need to feel the desire to learn more keenly. Some blame falling standards on the lack of appropriate role models, whilst others contend that the white middle class success masks the increasing failure of white working class children3. I am sure that there are many lesson to be learnt from the success of my two placement schools, nevertheless, I would argue that never before has education been so widely available to so many. No system, however well designed and implemented, will ever work unless all parties believe, want and make it work; and at the forefront must be the fundamental belief that ‘every child has the right to learn, every teacher has the right to teach4.
1. DfES Government Statistics
2. The IDACI tables categorise 10 levels of depravation with 1 being the lowest level. At my second placement school 13% of pupils lived in areas categorised as being level 1 to 3, with the majority (48% of pupils) living in areas categorized as being between levels 3 to 7 (they is an even spread between these four levels).
3. DfES Research & Statistics Gateway 2005 – identified 83% of white working class students are failing to achieve 5 A* – C GCSEs.
4. Berkeley Parents Network-Cragmont Elementary
Sunday Times Articles:
1. Cutting class sizes ‘is waste of money’ – Alexandra Frean 23 March 2008
2. Pupils from immigrant communities overtake white Scots in achievement. A new study underlines the key role of parental support – Jason Allardyce
3. Gordon Brown’s Black Wednesday- 22 Jan 2008
4. Stop helping boys, says equality watchdog – September 10, 2007Mollycoddle curse of the middle class – 22 Mar 2008
5. Every child in school numbered for life – 13 Feb 2008
6. Tell those pushy parents to push off – and take their Chelsea tractors, too – 12 April 2006
7. Business executives could save badly run schools, says Ofsted – 23 Nov 2006
8. Helping black youths to achieve may bring £24bn boost – 9 Aug 2007
9. Black pupils trail in GCSE test – 22 March 2005
10. We are failing too many pupils in state education’ – 27 Sep 2007
1. Raising standards
2. An Education System That Works For All
3. Raising the bar on Education
4. No Child Left Behind
KS3 Attainment – Raising Aspirations
DCSF Standards Site: Raising Achievement
DCSF Standards Site: Raising the Aspirations of Year 11 Students
Secondary Assessment – Formative Assessment
Improving Core GCSE Results – Maths
White Under-Achievement: Putting Class into the Classroom
Teaching With Bayley – Showing Them Who’s Boss
All the series from ‘Teaching with John Bailey’ and associated research:
Below are links into series in to series:
1. Teaching With Bayley – Friendly But Firm
2. Teaching With Bayley – Girl Talk
3. Teaching With Bayley – In at the Deep End
4. Teaching With Bayley – The Trouble with Girls: Winning Them Over
Articles from ATM:
1. Impact of assessment on learning and teaching
http://www.atm.org.uk/professional-services/policy/ATM-070304-Impact-of-Assessment-Report.rtf – 79k – 2007-03-17
Raising Standards – Patricia Pine
Startegies to Promote Inclusion – Rose & Tilstone
Principals Guide to raising mathematical achievement – McKewan
Becoming a successful teacher of maths – Tanner & Jones
Issues in maths teaching – Gates
Teaching maths in secondary school – Haggarty
Learning to teach maths – Wilder, Wilder & Westwell
Raising achievement in maths – Watson
The right to learn – DarlingHammond