An Roinn Oideachais agus Eolaíochta
Department of Education and Science
Subject Inspection of Mathematics
Gormanston, County Meath
Roll number: 64420I
Date of inspection: 28 April 2009
Report on the Quality of Learning and Teaching in Mathematics
This report has been written following a subject inspection in Franciscan College, Gormanston, conducted as part of a whole-school evaluation. 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 acting deputy principal and subject teachers.
Timetable provision for Mathematics is very good. In the junior cycle five mathematics lessons per week are provided for first, second and third-year students. Four class periods of Mathematics per week are provided for transition year (TY) students. In fifth and sixth years, students are allocated seven mathematics lessons per week. In keeping with good practice mathematics lessons are evenly distributed across the day and week.
The mathematics department comprises eleven teachers. Levels in the junior cycle are rotated among all members of the mathematics teaching team. Currently three mathematics teachers are involved in teaching higher-level Leaving Certificate Mathematics. It is department policy that teachers retain the same class group from year-to-year for the duration of a cycle. The school’s practice in relation to teacher allocation to class groups is good.
At the beginning of first year, students are assigned to one of four mixed-ability classes. This is good practice. However, the arrangements for level choice, in the junior cycle, are very unsatisfactory. For second and third years, the base class groups to which students are assigned are determined by their subject choices. For the current cohort of second and third-year students, the choice between Technical Graphics and Music formed the basis on which students were assigned to classes. Mathematics is taught to students in their base class groups. Mathematics lessons in neither second nor third year are scheduled at the same time; therefore, students cannot move to class groups which are taking syllabus levels suited to their particular needs. Currently, four mathematics classes are formed in each of second and third years. Consequently, students taking the higher-level and ordinary-level Mathematics courses are taught together in mixed-ability class groups. An extra teacher has been allocated to withdraw the ordinary-level students, for three class periods per week, from one of the mixed-ability class groups in third year. It is recommended that concurrent timetabling of all mathematics classes in both second year and third year should be introduced as a matter of urgency. This arrangement would allow the mathematics department to assign students to higher and ordinary-level class groups on the basis of their needs and achievement in Mathematics and would facilitate students in changing levels when the need arises.
Senior cycle students are assigned to Mathematics class groups on the basis of progress and achievement in Mathematics. In fifth year there are three higher-level and two ordinary-level class groups. In sixth year there are two higher-level and three ordinary-level groups, one of which includes foundation-level students. Most mathematics lessons in fifth year and sixth year are concurrently timetabled which allows for flexibility in level choice for students in these class groups. It is recommended that in fifth year and in sixth year, the remaining classes be concurrently timetabled also. This would facilitate change of level for all fifth and sixth-year students.
Teachers have access to a wide range of learning resources. These include probability kits, algebra tiles, jumbo playing cards, relational geosolids, geometry equipment, wall charts and class sets of geometry equipment and angular rulers. The resources are kept in a central location and are shared among the teaching team. It is evident that teachers use learning resources in teaching and learning in Mathematics whenever it is possible; this is a positive finding.
The supports in place to enable the use of information and communications technology (ICT) in the teaching and learning in Mathematics are quite good. A number of digital projectors and laptop computers are available. These are stored in a resources room. The computer room is also available and this can be accessed by means of a booking system. Computer software such as ‘Digital Grinds,’ has been acquired to support student revision and its success was reported during the evaluation.
Classrooms are student-based therefore materials, resources and ICT equipment have to be carried from the storage areas to the classroom. The possibilities for incorporating ICT regularly and frequently in teaching and learning in Mathematics are limited. This situation is not ideal. If classrooms were teacher-based it would be possible to have materials, resources and ICT facilities to hand so that they could be used to enhance and clarify mathematical explanations on a day-to-day basis when the need arises. It would also create the opportunity for teachers to display wall charts, posters and student work. It is therefore recommended that, as far as possible, mathematics teachers should be based in their own classroom.
Teacher participation in continuing professional development (CPD) is encouraged by school management and all mathematics teachers have recently attended a workshop on ICT. Some teachers are members of the Irish Mathematics’ Teachers Association (IMTA).
Maths Week is celebrated each year in the school. Students are encouraged to participate in training for the Irish Mathematical Olympiad and in past years students have achieved a very high level of success. Students take part in the courses organised by the Centre for Talented Youth in Ireland (CTYI) that take place in Dublin City University. Participation in extracurricular mathematics-related activities is very good practice as this encourages an interest in the subject and allows students to experience Mathematics for pleasure.
Formal meeting time for Mathematics is allocated by school management once per term. Records are maintained. It is recommended that a copy of the minutes of mathematics department meetings be kept within the planning documentation. Mathematics teachers also frequently meet on an informal basis to discuss the day-to-day issues that arise. The mathematics department is currently co-ordinated by an experienced member of the teaching team. It was evident during the evaluation that mathematics teachers provide a high level of collegial support for each other.
Some progress has been made on planning for Mathematics and the department plan was made available during evaluation. The plan includes mathematics department policy on assessment, reporting procedures and access to ICT. The plan also includes a list of methodologies that are suitable for teaching and learning in Mathematics. It is recommended that the mathematics department continue to work on the development of the mathematics plan. A reflective approach is recommended whereby policies are discussed and, when changes are made, the plan can be amended accordingly.
The mathematics plan contains programmes of work for each year group. These are written in terms of topics to be covered within agreed time frames. It is evident from the review of the minutes of department meetings that considerable discussion and effort has gone into their creation and this is commended. It is recommended that teachers use the planning process to share ideas for variety in classroom experience for students. This collaboration should aim to incorporate investigative, discovery and active methodologies and ICT in mathematics lessons. Lessons should be described in terms of learning outcomes, methodology, resources and assessment. Overall, the mathematics teaching team should work towards a plan that reflects and informs classroom practice, puts the student at the centre of planning and is subject to regular revision and review.
The TY plan comprises a list of Leaving Certificate topics to be covered for the year. This is not in keeping with the spirit of TY and therefore needs to be reviewed. It is important to take advantage of TY to provide students with a mathematical experience that is different from that of the Junior and Leaving Certificate courses. This can be achieved by studying different Mathematics or by adopting a different approach to the teaching and learning of Leaving Certificate course material. It is therefore recommended that the current TY plan be extended to incorporate a wider range of course content and of teaching methodologies. A research project on the lives of famous mathematicians and a study of some of the Mathematics that made them famous would be ideal for TY. It is suggested that students carry out a survey and use the results to inform their study of statistics. A module of Applied Mathematics could also be considered for TY. The TY offers a valuable opportunity for students to engage with Mathematics on an enjoyable level and to gain a greater appreciation for the subject; it is therefore recommended that every effort be made to optimise the potential of this opportunity.
Teacher explanations and instruction were clear in all of the lessons observed. In all cases lessons were well structured and progressed at a pace that was lively yet appropriate to the ability level of the students. In most cases teachers were careful to relate the current work of the lesson to previously learned material. Some of the lessons observed opened with a brief but comprehensive recapitulation of the previous lesson. This very good practice helps students to situate new ideas and to understand the interconnections between concepts and procedures in Mathematics. Where it was appropriate teachers chose examples that related to students’ own personal experience or used everyday materials such as domestic bills to explain mathematical ideas. In a lesson on volume and area 3-D models were used effectively to clarify explanations. This is all in keeping with best practice.
The predominant methodology used was teacher example followed by student exercise. The good balance that was achieved between teacher input and student input made lessons lively and kept the students engaged. In some lessons observed students were encouraged to complete worked examples on the blackboard for their classmates. This is a simple but effective way of encouraging more student involvement in lessons. Teachers made good use of questioning both global and directed to assess learning and engage students. Students were allowed ample time to answer teacher questions and in many cases very good class discussions allowed students to explore lesson content. This very good practice contributed to the high level of student participation that was evident in all of the lessons observed. It was evident from the review of planning documentation that some teachers incorporate some variety in methodologies such as pair work and active learning in their lessons. However, it is recommended that, over time, teachers incorporate more active, investigative, discovery, research methodologies and ICT in teaching and learning in Mathematics.
In some cases teachers encouraged students to focus on the strategic elements of solving mathematical problems. This is very beneficial to learning in Mathematics because it contributes to the development of the critical thinking and problem solving skills that are essential for success at all levels. Teachers have good and regular opportunities to emphasise this when students are expected to apply learning in unfamiliar situations. Problems in area and volume that involve recasting provide good examples; it is important that students gain plenty of experience of working through the strategic aspects of these types of problems for themselves. It is appropriate in these situations to focus on general problem solving techniques rather than providing students with a step-by-step conceptual breakdown of the problem and expecting them to then apply the formulae to work out each step outlined. It is therefore recommended that teachers provide opportunities wherever possible for students to develop strategic problem solving skills.
Teachers established clear logical connections between the concepts presented in most of the lessons observed. This was not universal practice however. It is recommended that explanations start at the concept’s most basic point and that teaching concentrate on skills that are transferable as the concept expands and becomes more complicated. Students should be encouraged to work from a position of solid understanding, so that learning can gradually progress as concepts become more difficult. Factorisation provides a good example of this, the idea of a factor could be explained using factors of a natural number. This could then progress to factorising simple quadratic equations, while establishing the skills that will be necessary to factorise more complicated quadratics. The skills acquired should then be used to explain the difference of two squares, and so on. It is recommended that this approach be extended to all teachers.
In some of the lessons observed teachers differentiated learning carefully. This was achieved by careful questioning and by providing opportunities for students to work independently, and at their own pace, on tasks. The support of individual attention was provided where necessary. Some lessons began with the correction of homework and in keeping with very good practice some teachers allowed those students who had correctly completed their homework to work ahead on exercises from the textbook. This is a simple but effective way of maintaining student engagement and participation while homework is corrected on the blackboard. These are examples of good differentiation practices. It is recommended that all teachers include teaching and learning strategies that ensure the lesson content and activities are accessible to all students while providing challenge for the more able student.
Classroom management was observed to be very good and the students were very well behaved in all classrooms visited. There is a good rapport between students and their teachers. Teachers are encouraging, affirming and supportive of student effort. In many cases, where it was appropriate humour was used to great effect. In interactions with the inspector the students demonstrated an interest in the subject. Teachers have created learning environments where students can engage with Mathematics with confidence.
All class groups sit mid-term tests in October and formal examinations at Christmas. Summer examinations are held in May for first, second and fifth-year groups. Students preparing for the certificate examinations sit ‘mock’ Junior and Leaving Certificate examinations in spring. In addition to the reports sent home on foot of formal examinations, parents receive a report on their son’s or daughter’s progress at Easter. In keeping with good assessment practice, common examinations are set in Mathematics for students taking the same level. Parent-teacher meetings take place annually.
It was evident from the review of students’ copybooks that the standard of student work is high and the majority of students are making good progress in Mathematics. All teachers monitor student work in class through oral questioning and observation. Homework is set regularly and usually corrected as part of the following lesson. This is good practice. Some teachers are using the opportunity of checking students’ work to provide constructive feedback and encouragement for students. In some cases teachers use stickers with encouraging comments and in others teachers write positive comments. This very good practice is in keeping with assessment for learning (AfL) principles. The National Council for Curriculum and Assessment (NCCA) website (www.ncca.ie) provides further information on AfL. It is recommended that this good practice be extended to all teachers.
The uptake of higher-level Mathematics, in both the Junior and Leaving Certificate examinations, is high. The school is justifiably proud of its students’ achievements in the certificate examinations.
The following are the main strengths identified in the evaluation:
· Timetable provision for Mathematics is very good.
· Mathematics teachers provide a high level of collegial support for each other.
· Teacher explanations and instruction were clear in all of the lessons observed. In all cases lessons were well structured and progressed at a pace that was lively yet appropriate to the ability level of the students.
· There is a good rapport between students and their teachers. Teachers are encouraging, affirming and supportive of student effort.
· In keeping with good assessment practice, common examinations are set in Mathematics for students taking the same level.
· Teachers use the checking of students’ work to provide constructive feedback and encouragement for students.
· The uptake of higher-level Mathematics, in both the Junior and Leaving Certificate examinations is high. The school is justifiably proud of its students’ achievements in the certificate examinations.
As a means of building on these strengths and to address areas for development, the following key recommendations are made:
· Concurrent timetabling of mathematics classes for second and third years should be introduced as a matter of urgency.
· In order to facilitate the use of materials, resources and ICT in lessons, as far as possible mathematics teachers should be based in their own classroom.
· The mathematics department should continue to work on the development of the mathematics plan and it should take a reflective approach.
· The current TY plan should be extended to incorporate a wider range of course content and of teaching methodologies.
· Over time, teachers should incorporate more variety in methodologies and ICT in teaching and learning in Mathematics.
· Teachers should provide opportunities wherever possible for students to develop strategic problem solving skills.
· All teachers should use teaching and learning strategies that ensure the lesson content and activities are accessible to all students while providing a suitable level of challenge for the more able students.
Post-evaluation meetings were held with the teachers of Mathematics and with the acting deputy principal, at the conclusion of the evaluation when the draft findings and recommendations of the evaluation were presented and discussed.
Published March 2010