An Roinn Oideachais agus Eolaíochta
Department of Education and Science
Subject Inspection of Mathematics
Coláiste Choilm Christian Brothers’ School
Swords, County Dublin
Roll number: 60383I
Date of inspection: 14 October 2008
Report on the Quality of Learning and Teaching in Mathematics
This report has been written following a subject inspection in Coláiste Choilm Christian Brothers’ School (CBS), Swords. It presents the findings of an evaluation of the quality of teaching and learning in Mathematics and makes recommendations for the further development of the teaching of this subject in the school. The evaluation was conducted over two days during which the inspector visited classrooms and observed teaching and learning. The inspector interacted with students and teachers, examined students’ work, and had discussions with the teachers. The inspector reviewed school planning documentation and teachers’ written preparation. Following the evaluation visit, the inspector provided oral feedback on the outcomes of the evaluation to the principal and subject teachers. The board of management was given an opportunity to comment in writing on the findings and recommendations of the report; a response was not received from the board.
Coláiste Choilm CBS, Swords has an enrolment of 616 boys for the current year. Timetable provision for Mathematics is adequate in the junior cycle and very good in the senior cycle. First and second year groups are timetabled for four class periods of Mathematics per week. Third year students receive five periods of Mathematics per week. In the senior cycle, transition year (TY) students are allocated three periods of Mathematics per week. Six mathematics lessons are provided for fifth and sixth year students. It is good that there are plans to increase the timetable allocation for first year, second year and TY students by one lesson per week in the coming academic year. When this plan is implemented, overall timetable provision for Mathematics will be very good throughout. Mathematics lessons are evenly spread across the day and week which is good practice. Concurrent timetabling of Mathematics occurs from second year through to sixth year. This good practice provides for a high degree of flexibility when students need to change levels. Commendably, there are plans to concurrently timetable first year mathematics lessons next year; this is intended to facilitate team teaching as a mode of learning support provision.
The mathematics department comprises twelve teachers. It is mathematics department policy that classes retain the same teacher from year-to-year for the duration of a cycle, which is good practice. Higher level Mathematics at both junior and senior cycle is rotated among most members of the teaching team. This good practice contributes to the retention of a high level of expertise within the mathematics department and will enable the department to meet the challenges of the forthcoming revisions to the mathematics syllabuses.
Students are assigned to mixed-ability classes in first year. In second and third year there is a higher level band and an ordinary level band. Within each band, students are allocated to mixed ability class groups. Students are assigned to class groups on the basis of performance in formal examinations at the end of first year. TY is optional and, currently, there is one mixed ability class group. In fifth and sixth year there is a higher level class and an ordinary level band comprising a number of mixed ability class groups; students are placed in class groups on the basis of performance in the certificate examinations. Student preference and teacher advice also play a part in level selection and concurrent timetabling provides the flexibility to change level throughout the year. Any level change takes place in consultation with students, teachers and parents. This is good practice.
The mathematics department is in the early stages of establishing a mathematics library. This very good development is intended to facilitate the sharing of resources from a central location. These resources include geometry sets, overhead projectors, calculators, and a bank of useful handouts, worksheets and examination material. It is good that there is an annual budget for the mathematics department and it is suggested that this be used to acquire further resources over time. These could include relevant mathematics software, reference books of general mathematical interest, probability kits and algebra tiles. It is also suggested that students be encouraged to produce 3D models that can be used as resources for teaching and learning in volume and area, geometry and trigonometry. In addition it is suggested that teachers might collect everyday objects that can be used as geometric solids such as boxes and containers of different shapes and sizes. The creation of a shared mathematics library is very beneficial as it provides an informal forum for the sharing of ideas and methodologies and this can have a positive effect on classroom experience for students.
Mathematics teachers have access to information and communications technology (ICT). There are two computers available for staff use which can be used to create supplementary classroom materials. Some classrooms are fitted with ceiling-mounted data projectors and there are mobile data projectors and laptop computers available for use by the mathematics department. At present ICT is used for teaching and learning in Mathematics by a minority of teachers. It is recommended that teachers make more use of the ICT resources available to incorporate ICT in teaching and learning in Mathematics on a more widespread basis. Teacher continuing professional development (CPD) is fully facilitated and teachers are encouraged to attend in-service courses.
Students who are in need of general learning support are identified through pre-entry assessment, diagnostic testing and information from feeder primary schools. Students who require learning support in Mathematics are usually identified through ongoing teacher observation and assessment. Support in Mathematics is provided through small group withdrawal where emphasis is put on individual help from the learning support teacher and on peer and group learning. Team teaching is also used to incorporate learning support into mainstream classes. This approach is good as it enables any student requiring additional help with Mathematics to discretely receive the help he needs in a timely and accurate way while remaining with his peers. It is good that students receive extra support for the duration of their need and are returned to their mainstream groups when it is felt that they can cope well with the mainstream lessons. There are two computers and a range of computer software suitable for the provision of learning support available for teaching and learning in the resource room. All of this is very good practice. A high level of support is provided to students who experience difficulty with Mathematics and this support is provided in a range of ways designed to suit the individual needs of students.
In the current year, team teaching is also used to provide additional support to the fifth year higher level class. This provision is an indication of the school’s commitment to the subject and is commended. However, since this involves the allocation of a second teacher to that class and since the teaching of higher level Mathematics provides limited opportunities for this type of in-class support to benefit learners it is recommended that alternative uses of this additional resource be explored and some changes be implemented as soon as possible. Consideration could be given to making two separate class groups so that higher level students could progress at different rates that are appropriate to their ability. Alternatively team teaching could be used in an ordinary level group with students who might also benefit from extra help.
Students of Mathematics participate in the PRISM mathematics challenge, the Irish Junior Mathematics Competition and in the past have been involved in training for the International Mathematical Olympiad. Encouraging students to engage with Mathematics outside of the classroom is very good practice as it provides opportunities for students to experience Mathematics for pleasure and contributes to their appreciation of the subject.
Formal planning time is allocated once per term as part of the whole school planning process. Records are maintained of all formal planning meetings and minutes are kept. Mathematics teachers also meet on a regular informal basis to discuss the day-to-day issues that arise. The position of mathematics department co-ordinator is currently held by an experienced member of the mathematics teaching team. This is viewed as a long-term appointment by members of the mathematics department. It is recommended that this arrangement be reviewed in time. Consideration might be given to rotating the position or to the appointment of an assistant.
It is evident from the review of planning documentation that considerable work has been completed on the mathematics plan. The plan contains mathematics department policy on division of classes, student access to levels, assessment, record keeping, homework, and provision of learning support. The plan also includes lists of resources available for teaching and learning in Mathematics and of in-service courses attended by members of the mathematics department. The plan is at the stage of development where focus can be put on planning for classroom activity. It is recommended that collaboration takes place around planning for variety in teaching methodology and student learning experience. It is suggested that the incorporation of alternative methodologies such as active, discovery and investigative methodologies in teaching and learning in Mathematics be explored. The sharing of experience, expertise and ideas that this collaboration should generate will contribute to the positive effects that the planning process can have on student experience in the classroom. Furthermore it is recommended that the long term planning process for Mathematics include modes of revision and review, so that the plan can become a living document that puts the student at the centre of planning and informs everyday teaching and learning in Mathematics.
The mathematics plan contains schemes of work for each year group in Mathematics. These consist of a list of topics to be covered within an agreed timeframe. In some cases these are in terms of learning intentions, methodologies, necessary resources, and modes of assessment. This is very good practice. It is good that the plans for ordinary level and higher level for each year group are co-ordinated; this will ensure that any student who changes level mid-year will have covered the same material as the class he is joining.
The TY plan is in keeping with the underpinning principles of a good TY programme. The content of the TY plan is very well chosen and is designed to be taught in a mixed-ability setting. Every opportunity is provided for students to experience Mathematics on an interactive and enjoyable level. Students study a range of topics that are not on the Leaving Certificate course. For example, students complete projects on the lives of famous mathematicians and on designing a mathematical board game. Students also study the history of Mathematics, Fibonacci sequences, puzzles and ‘Airline Mathematics’. The Leaving Certificate course material included in the TY plan covers probability, statistics, graphs, algebra, geometry and calculus. In addition to whole class teaching, the methodologies used to teach these topics include group work, pair work, project work and a variety of active and practical exercises. An example of best practice in this regard is where a survey is carried out by TY students and the resulting data is used to inform their study of statistics. This involves a visit to the neighbouring girls’ school to research the girls’ opinion and then to compare these to the findings of a similar survey carried out in the boys’ school. This active methodology is an example of excellent practice because it adds to the level of interest in the findings from the research. ICT is incorporated into teaching and learning in TY Mathematics wherever possible. Since road safety forms part of the TY programme in the school, a module of Applied Mathematics which would include the study of linear motion and collisions is suggested as an addition that might be considered in the future.
In all cases lessons were purposeful and appropriate to the syllabus. Lessons had a clear focus and were well structured. Most teachers shared the learning intentions with the students at the beginning of lessons. Best practice in this regard was observed where the lesson objectives were written on the board at the start of the lesson and checked at the end to ensure that they had been achieved. This encourages students to take personal responsibility for their own learning and can help to increase motivation and a sense of achievement on reaching the lesson goals. It can also alert the teacher to areas that may have proven difficult for some students and would need to be revisited in the next lesson. It is therefore recommended that the good practice of sharing the learning objectives explicitly be extended to all lessons.
In all cases teachers were well prepared for their lessons and had all the necessary resources available. All handouts and worksheets used were well designed and supportive of learning. In a lesson observed on the fundamental principle of counting this was of particular note. In one exercise the students were expected to calculate the number of possible pizzas that could be ordered from a menu. Another similar exercise involved the selection of cars. These handouts were very well designed and the subject matter was well chosen to appeal to boys. This is very good practice because it can increase the level of student interest while providing a very comprehensive explanation of the underlying mathematical concept.
The pace of the lessons observed was lively yet appropriate to the ability level of the students. In all cases teacher instructions and explanations were very clear. A good example of this was observed in a lesson on fractions where each explanation included a picture so that the students could associate the visual explanation with the mathematical one in each example. Teachers were careful to relate explanations to students’ prior learning and to their personal experience. This very good practice helps students to situate new ideas and identify with their course material. In most cases lessons began with the correction of homework. Teachers made good use of questioning to engage students and to assess learning. Most teachers routinely use higher-order questions, requiring reflection and consideration to help students explore difficult concepts and ideas. This is good practice because the use of open and probing questions allows the teacher to guide the student to full understanding by expecting him to focus on his own interpretation of each concept and his own thought process. The use of higher-order questions is very beneficial to teaching and learning in Mathematics and it is recommended that this practice be incorporated into all lessons.
Teaching mainly consisted of teacher example followed by student exercise. This worked best where teachers were careful to vary the lesson activity to keep lessons lively and the students engaged. In some lessons observed the variety in lesson activity was a significant factor in the lesson’s success. This variety ensured that the students were very interested and engaged in all of the learning activities. In some cases teachers expected students to independently work on progressively more difficult exercises. This was reinforced with individual assistance from the teacher where necessary. Allowing students plenty of time to solve mathematical problems independently is very worthwhile since it can enable students to develop confidence in their own problem-solving skills and can lead to a great sense of personal satisfaction. It is recommended that the good practice of varying the lesson activity and of providing independent learning opportunities for students be adopted by all teachers
In the fifth year higher level lesson observed, as mentioned earlier, a second teacher was available to provide support to students throughout the lesson. The lengthy explanations that characterise higher level leaving certificate Mathematics naturally meant that individual attention was not necessary for the majority of the lesson. The availability of extra individual attention when students were working on exercises in class contributed to students looking for that help too early, before they had made sufficient effort themselves on solving the problems. It is therefore recommended that the provision of additional in-class support to fifth year students at this level be discontinued. Consideration could be given to dividing the class group so that the small number of students who require the extra support to continue at higher level Mathematics could work at a slower pace than the rest of the class group.
The evident good rapport that exists between students and their teachers has allowed for the development of learning environments where students can engage with Mathematics with confidence. This was of particular note in an ordinary level junior cycle class visited. The excellent relationship between the teacher and the students in this case, through a conscious effort to build student confidence with Mathematics, has contributed to the high levels of student interest and engagement demonstrated. In all cases standards of student behaviour were observed to be high and classroom management was good. Teachers are very encouraging and affirming of student effort.
All students except third and sixth years are formally assessed at Christmas. Third and sixth year students are assessed in November. Summer examinations are held in May for first, second, fourth and fifth year groups. Students taking the certificate examinations sit ‘mock’ examinations in February. Common examination papers are set wherever possible. It was evident from the review of the last common examination paper set for first year students that each question and its parts were of a similar standard of difficulty. It is recommended that each question should comprise a number of parts of graduated difficulty as in the certificate examinations. This measure would ensure that each examination question contains a section that can be attempted by all students, one that can be attempted by most students, and a more difficult section that requires a higher level of expertise. This type of differentiation contributes to the quality of the information provided by examination results and helps to ensure that the majority of students experience some level of success in formal examinations in Mathematics. Reports are sent home on foot of all formal assessments and parent teacher meetings take place annually.
It is evident from the review of student copybooks that the standard of presentation of students work is high and that the majority of students are making good progress in Mathematics. In keeping with good practice homework is set regularly and is usually corrected as part of the following lesson. There is a homework policy for Mathematics and the school operates a homework club for students. All of this is very good practice.
Learning is routinely assessed through teacher observation, oral questioning in class and end-of-topic tests. Some teachers are using this valuable opportunity to set tests that serve to motivate and encourage students; this is achieved by carefully choosing test questions to ensure that every student experiences success. This success contributes to student confidence with Mathematics and encourages further engagement with the subject. This is an example of very good assessment practice. It is good that some of the teachers observed during the evaluation are using assessment for learning (AfL) principles in the correction of student work by using comment-based marking. This is very worthwhile as it provides students with critical feedback and can be a source of positive reinforcement. The extension of the use of AfL practices is therefore recommended. Further information on AfL is available on the National Council for Curriculum Development website (www.ncca.ie)
The following are the main strengths identified in the evaluation:
· The mathematics department is developing a mathematics library which is intended to facilitate the sharing of resources and ideas.
· A high level of support in Mathematics is provided and a variety of approaches is used to deliver this.
· The mathematics department provides opportunities for students to experience Mathematics for pleasure by participating in extra-curricular mathematics related activities.
· Good progress is being made on planning for Mathematics.
· The TY plan is in keeping with the underpinning principles of a good TY programme and students are given opportunities in TY to deepen their appreciation for the subject.
· The majority of students are making good progress in Mathematics.
· The valuable assessment practice of using tests to motivate and encourage students and to build student confidence in Mathematics was in evidence.
· High standards of student behaviour were observed and the relationships between students and their teachers are very good. Teachers are affirming of student effort.
As a means of building on these strengths and to address areas for development, the following key recommendations are made:
· Teachers of Mathematics should explore ways in which active, investigative and discovery methodologies and ICT can be further incorporated into lessons.
· The position of co-ordinator of the mathematics department should be rotated or an appointment of an assistant should be considered.
· The use of open and probing questions to encourage students to explore difficult mathematical concepts should be extended to all lessons.
· The good practices of varying the learning activity and providing students with independent learning opportunities should be adopted by all teachers.
· The questions on common formal examination papers should comprise a number of parts of graduated difficulty as in the certificate examinations.
· The provision of additional in-class support to higher level fifth year students should be reviewed and alternative uses of this resource should be explored.
Post-evaluation meetings were held with the teachers of Mathematics and with the principal at the conclusion of the evaluation when the draft findings and recommendations of the evaluation were presented and discussed.
Published March 2009