In order to locate the findings relating to lesson design in a context I now present a very brief overview of the school’s mathematics department. This will then be followed by short biographies of the four participants involved in the research whose beliefs about teaching mathematics and lesson plans I discuss later
.
4.2.1 The Research School’s Mathematics Department.
The school’s mathematics department consists of 11 teachers, 8 females and 3 males (tables 3.3.1 and 4.2.2). Only two of the teachers have mathematics related degrees and 10 teachers are younger than 35 years of age. The
department has worked hard to increase the status of mathematics with parents, pupils and the wider community. In the last five years it has developed from being an inward looking department who found difficulties recruiting suitably qualified teachers to being a stable, fully staffed outward looking department engaging
161 with the wider aspects of the profession; such as initial teacher training (Roberts and Foster, 2015).
The participating school is an active member of the University Initial Teacher Training Partnership. Given the proximity of the school to the University and the fact that the school is actively involved with initial teacher training, at all levels, it is therefore not surprising that the vast majority of the mathematics teachers are post graduate alumni of the University. Surprisingly none of the eleven teachers were undergraduates at the University. It is also interesting to note that seven of the nine teachers needed to undertake a full-time 36 week mathematics
conversion course, prior to starting their teacher training as their degrees were deemed to contain insufficient mathematics.
4.2.2 Individual Participant Biographies
From the nine teachers involved in the research (table 4.2.2 – teacher F and K were not involved) five self-selecting participant teachers (teachers C, D, G, H and I) all had the same reason for wanting to be included in the classroom research. Each teacher was about to embark on the final third of their post graduate level qualification which involved a dissertation based on a classroom piece of research. All had recently studied the research methods module and had during their initial training course undertaken a small piece of classroom based research as part of the post graduate qualification. Since completing this research study all five teachers have successfully concluded their studies and gained a post graduate level qualification. Their mathematical backgrounds and teaching experience can be found in table 3.3.1.
At the time of the research all five members of staff taught classes across the years and attainment range. Eventually, by mutual consent of all involved, three teachers decided to commit to the classroom research. The classroom research used a class taught by two of these teachers. Professional biographies of these three teachers (Andy, Sarah and Tim pseudonyms) and my personal biography are included as a means of defining participants’ educational backgrounds and expertise. The biographies are also examples of the range of experiences and backgrounds of the participating teachers and how their views and beliefs have changed since gaining qualified teacher status.
162 The three selected teachers, who are representative of the department, are used as examples to demonstrate the differences in views and beliefs held by
departmental members. The three teachers had very different school mathematics experiences; this was due in part to the time when they were educated. The one teacher in the department (Tim) with a mathematics degree did not want to be fully involved in the classroom study, however he was happy to participate in a limited way such as being interviewed and providing lesson plans and a biography for analysis.
4.2.3 Biography – Andy
Andy, a mature career changer, qualified as a mathematics teacher six years ago after a very successful career in the music industry. He has a degree in theatre studies. As a direct result of his highest mathematics qualification being GCSE he was required to take a full-time mathematics course immediately prior to his teacher training course; he therefore took two years to qualify. On qualifying as a mathematics teacher his first post was a one year appointment in an academy which immediately preceded his current post in the research study school. Andy’s own mathematical experiences at school were during the period when public examinations included extended coursework tasks. These tasks were used to assess one component of the then newly introduced GCSE examination. Having completed his compulsory education at the age of sixteen he left school and began work only to return to part-time study for a degree.
His views and beliefs about teaching mathematics, expressed at an early stage during the teacher training course were that, the practising of mathematical skills should be the occasional focus of lessons and that these newly acquired skills almost always led to pupils gaining mathematical insights. He did not view tasks as coursework in disguise, but he did think that tasks could be time consuming to organise and for pupils to complete.
From a piece of Andy’s academic work, written during his teacher training year, his beliefs were that good mathematics teaching should be based on five key principles:-
1. Create links between mathematical topics and other subjects 2. Encourage understanding
163 4. Use effective questioning that promotes discovery rather than
teaching tricks
5. Allow and expect learners to use prior knowledge
When questioned again during the classroom research, in 2014, Andy had modified his views indicating that he now believed that the practising of
mathematical skills should be the focus of all lessons even if this did not result in pupils gaining mathematical insights. He also at this stage viewed tasks as coursework in disguise which did not fit easily into the one hour lesson structure and that they were time consuming. He had completely changed his views on the usefulness of sharing lesson objectives with pupils from this being a highly
positive to a highly negative strategy. Even though he had modified his views his five core principles for good mathematics teaching remained unchanged.
4.2.4 Biography – Sarah
Sarah qualified as a mathematics teacher four years ago after studying for an accountancy degree and moving from a short career in the finance industry. Whilst at school she studied mathematics to GCE ‘A’ level, but because her degree contained very little mathematics she was also required to take a full-time mathematics subject knowledge course immediately prior to the teacher training course, she therefore took two years to qualify. On qualifying as a mathematics teacher her first post was a one year full-time permanent appointment in an academy. This immediately preceded her current post in the research study school. Sarah’s own secondary school mathematical experiences excluded the long extended coursework type tasks. This was because the public examinations at the time when she took them were modular courses where she had only been required to study the subject in relatively self-contained topics.
Sarah’s views and beliefs about teaching mathematics, sampled during October of the teacher training course, were that tasks were not coursework in disguise, nor were they time consuming to organise. She thought that coursework type tasks were a very positive strategy for teachers to use when teaching
mathematics. She also thought this was a direct result of the full-time subject knowledge mathematics course she had studied prior to beginning her teacher training. The predominant pedagogical and learning approach of the subject knowledge course had been through the extensive use of coursework type tasks. Sarah had therefore experienced two very different approaches to learning
164 mathematics, one based on a formal modular content driven curriculum and the other based on an investigative open-ended task approach to teaching and learning mathematics.
From a piece of Sarah’s academic work written during the teacher training course her rationale for good mathematics teaching was that the subject should be considered to be a tool for solving problems and that problem solving skills are essential for everyday decision making processes. She considered mathematics to be an essential part of our lives which involved the acquisition of basic skills and knowledge.
When questioned again during the research, in 2014, Sarah had completely modified her views indicating that she now believed that tasks were coursework in disguise which were time consuming and did not result in good pupil learning. She had therefore completely changed her views on the usefulness of tasks, but surprisingly during the research she made no reference to usefulness of
practising skills which was the central tenant of her initial academic rationale.
4.2.5 Biography – Tim
Tim qualified as a mathematics teacher five years ago after gaining a first class honours degree in mathematics from a redbrick university. His own school experiences of mathematics were in a selective grammar school where he achieved 12 A* grades at GCSE and 3 ‘A’ level all at grade ‘A’ which included mathematics and further mathematics. The public examinations were modular in nature which excluded the long extended coursework type tasks. Immediately prior to beginning his teacher training year he travelled the world for two years. On his return to the UK he spent six months working in a number of local secondary school mathematics departments.
Tim’s views and beliefs about the teaching of mathematics, sampled during October of the teacher training course, were that tasks are time consuming to organise, whereas exercises that gradually increase in difficulty build learners mathematical confidence better than tasks. However, there were some
inconsistencies in his views as he also believed that pupils should tackle tasks all the time.
165 Tim’s rationale for good mathematics teaching was from a purist stance where
the love and beauty of the subject were the main reasons for the inclusion of the subject in the curriculum. He considered mathematics to be an essential
communication tool with a unique and unambiguous language.
When questioned again during the research Tim held a firmer view that tasks take up too much teaching time and that exercises that gradually increase in difficulty were almost always better than tasks. He had changed his view about pupils tackling tasks all the time to pupils should only occasionally tackle tasks. He had therefore, after four years of teaching, resolutely remained true to his original rationale expressed during his training year. His beliefs concerning pupils experiencing tasks were also changing to be more in line with his rationale.
4.2.6 Biography – Mike (me)
I qualified as a mathematics teacher in 1973 immediately after studying for a mathematics degree. Whilst at school I studied pure mathematics, applied mathematics, further mathematics, music, physics and history at ‘A’ levels after ‘O’ levels. On qualifying as a mathematics teacher I took a one year post in a primary school before moving to a full-time appointment in a bilateral (grammar and secondary modern) high school. My own grammar school mathematical experiences were traditional in that I was prepared for examinations by being required to complete long, repetitive exercises of questions which gradually increased in difficulty. The lessons followed a format of being shown how to solve some examples by the teacher and then sitting in absolute silence to complete set exercises with no interaction between members of the class or with the teacher.
My initial views on how mathematics should be taught began to formulate during my training year where practical work, which involved dialogue and discussion were the norm and were encouraged as the expected pedagogical approach by the lecturers. During this period schools could ask their examining boards to examine pupils on a syllabus of their own design. The School Mathematics Project (SMP), started by Bryan Thwaites, at Southampton University, and The Midlands Mathematics Experiment (MME), organized by Cyril Hope, at Worcester College of Education, were just two such experimental mathematics curricula. Adapting my initial views on how mathematics should be taught is not surprising
166 as my teacher training lecturers were the team, under the leadership of Cyril
Hope, who developed and wrote the MME curriculum.
During my time spent teaching the landscape of school mathematics was changed and influenced by a number of important reports (Cockcroft, 1982; Swann, 1985); and major curriculum initiatives School Mathematics Project (1961); Curriculum Matters – Mathematics (HMSO, 1985a, 1985b); Cognitive Acceleration in Mathematics Education (Adey, 1988) and changes to the examination system with the introduction of GCSEs and the SATs (Dearing, 1994). All of these helped to shape my views and beliefs that mathematics should be taught in a completely different way to that in which I had experienced at school.
I eventually came to the belief, which I currently still adhere to, that good mathematics teaching and learning should be based on a number of key principles:-
1. Mathematics teaching needs to use a variety of pedagogical strategies and worthwhile tasks; appropriate activities and exercises are needed to develop mathematics skills.
2. Mathematics teaching has to build on pupils’ thinking.
3. Mathematics teaching exemplifies and develops links between mathematical topics and other subjects
4. Mathematics teaching develops the careful and precise use of mathematical language by both the teacher and pupils
5. Mathematics teaching exposes misconceptions and enables pupils to reformulate their understanding
6. Mathematics teaching encourages mathematical understanding rather than the selective recall of facts
7. Mathematics teachers need both good subject knowledge and pedagogical knowledge.