An Roinn Oideachais agus Scileanna
Department of Education and Skills
Subject Inspection of Mathematics
St David’s Christian Brothers’ School
Artane, Dublin 5
Roll number: 60471F
Date of inspection: 5 October 2009
Report on the Quality of Learning and Teaching in Mathematics
This report has been written following a subject inspection in St David’s Christian Brothers’ School, Artane. 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. 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.
St David’s Christian Brothers’ School has a current enrolment of 453 boys. Timetable provision for Mathematics is good.
There is a higher level class, two mixed-ability classes and a small foundation level class group in first year. Students are assigned to one of these groups on the basis of pre-entry assessment. Mathematics lessons in first year are not concurrently timetabled. This means that students who have been identified as requiring a change of level either change base group mid-year or change level at the beginning of second year. Both of these options could present difficulties for students. Changing base group mid-year can mean a change of class group for some other subjects which could prove hard on students who are just settling into first year. The programme of work for the higher level group currently comprises all of the mathematical topics to be covered by the mixed-ability groups and a significant amount of additional material. Because of this arrangement any students who move to the higher level class at the beginning of second year are at a disadvantage. The students assigned to the foundation level class group follow the foundation level course from the outset. Any student from this group identified as requiring a move to ordinary level would not have covered the same material as the students of the mixed-ability class groups. It is therefore recommended that concurrent timetabling of mathematics lessons be introduced for first year class groups and that, as far as possible, all first year mathematics class groups follow the same programme of work. These measures would greatly contribute to ensuring that each student studies Mathematics at a syllabus level appropriate to his needs.
In all other year groups the arrangements for student access to levels are good. The majority of mathematics lessons in second, third, fifth, sixth and transition years are concurrently timetabled. There was much evidence during the evaluation that the mathematics department encourages students to study the highest level possible for as long as possible. Students are assigned to levels on the basis of performance in class tests, formal school assessments and the certificate examinations. Teacher judgement, student and parental preference also play a role in level choice; this is good practice.
The mathematics department comprises twelve teachers. Teachers are allocated to class groups and levels in accordance with their qualifications and experience. There is good rotation of levels in the junior cycle; however Leaving Certificate higher level Mathematics is the responsibility of one member of the teaching team. It is recommended that more teachers become involved in teaching higher level Leaving Certificate Mathematics; this is advised to ensure that the capacity to teach all levels is maintained within the subject department.
Teachers make use of a wide variety of resources in teaching and learning in Mathematics. These include geometry equipment, overhead projectors, ‘Pascal Triangle’ model, student laminate boards, and 3-D solids. In keeping with very good practice, teachers have collected everyday items such as objects of various shapes and sizes, chess boards, ‘Smarties’, and rubber balls which are used in the teaching of volume and area, trigonometry, statistics, and functions and graphs. Teachers use games to encourage and motivate students. A number of mathematics teachers are also members of the science department and they take full advantage of the access that they have to science facilities and resources to enhance their mathematics lessons. Other resources such as household bills, newspapers and internet websites provide a real life mathematical context for learners. In addition, students have created lesson resources such as clinometers. The student projects and commercial posters that are displayed on classroom walls help to create mathematical environments and to encourage student interest in Mathematics. The range and variety of resources used in lessons is evidence of the teachers’ enthusiasm for the subject and of the department’s aim to develop student interest in Mathematics.
The resources for information and communications technology (ICT) that are available for teaching and learning in Mathematics are very good. There are two computer rooms available on a booking system for mathematics lessons. All classrooms have broadband internet access. Some classrooms are fitted with fixed data projectors and there is a mobile data projector on each floor of the school. There is a personal computer in most mathematics teachers’ classrooms. There are two Smart Notebooks, which are remote versions of the interactive white board. These are used to provide students with the opportunity to interact with the ICT in the classroom. Teachers have prepared a range of PowerPoint presentations for use in mathematics lessons and material from a variety of internet websites is also used. Geometry software is available on one of the school’s computers and it is recommended that it be extended to each classroom-based computer. It is encouraging to note that ICT is regularly integrated into mathematics lessons and it is very good that teachers are continually seeking ways to further its effective use.
Overall, the provision for students who experience difficulty with Mathematics or have been identified as requiring support with Mathematics is very good and students are very well supported. The procedures for identifying students who require learning support include pre-entry assessment, communication with feeder primary schools and with parents, ongoing teacher observation and class testing. Support is provided through the creation of smaller class groups, team teaching, and individual and small group withdrawal where necessary. Students also benefit from the supports provided as part of the Junior Certificate School Programme (JCSP) programme. The school has an autism unit and this was visited as part of the evaluation. It was evident that a very high level of care is being provided for the students in this unit and it was clear that they are making very good progress in Mathematics. In addition, teachers provide high quality support to students through the provision of individual attention in class on a day-to-day basis.
Students of mathematics are encouraged to participate in the PRISM Mathematics Challenge. Maths Week is celebrated each year as a significant event in the life of the school. In addition, visiting speakers are invited to address Leaving Certificate Applied (LCA) class groups, on a range of mathematics-related topics. Participation in extracurricular mathematics activities is very worthwhile as it provides students with opportunities to experience Mathematics for pleasure.
The mathematics department hold formal meetings once per term as part of the whole-school planning process. Much informal planning also takes place on a day-to-day basis. Minutes are kept of all formal planning meetings and these are included in the planning documentation. The position of mathematics department co-ordinator forms part of a post of responsibility and is currently held by an experienced member of the teaching team. The members of the mathematics department work well together as a team and provide strong support for each other. This is of particular note and value where sharing of experience around classroom practice takes place. This currently happens informally and it is recommended that it be formalised by allocating a section of planning meetings to discussion of teaching methodologies.
Good progress has been made on planning for Mathematics. The subject plan contains mathematics department policy on planning for students requiring additional support in Mathematics, student access to levels, and planning for cultural diversity. It is good to note that a list of effective teaching methodologies which includes the use of concrete materials and the use of active methodologies forms part of mathematics department policy. There was good evidence throughout the evaluation that the mathematics department’s aims and objectives in terms of making mathematics accessible and enjoyable for students are lived out in classroom practice. In this regard the plan reflects the actual work of lessons. The plan is available on the school’s computer system and is reviewed and revised on an ongoing basis which is good.
Programmes of work, for each year group and level form part of the mathematics plan. The programmes for first, second and third year are set out in terms of chapters of the class text-book to be covered within defined timeframes. The programmes for fifth and sixth year consist of a list of topics to be covered for the year. It is recommended that timeframes for the implementation of the fifth and sixth year programmes of work be agreed; this would facilitate the setting of common formal assessments. Programmes of work should be described in terms of expected learning outcomes, resources necessary, methodologies used, and modes of assessment. A section should also be included for reflection and review. Furthermore, the syllabus documents should be used as the main reference material in planning for Mathematics. While bearing in mind that this will involve considerable effort on the part of the members of the mathematics teaching team, it is recommended that they engage in this process. Over time, with the collaboration of the entire team the programmes of work for mathematics should facilitate a very worthwhile sharing of experience and expertise. This approach will contribute to plans that reflect the variety of very valuable approaches and methodologies used and the student-centred approach that is currently a hallmark of the work of the mathematics teachers in the school.
The programme for Transition Year (TY) comprises a list of Leaving Certificate topics to be covered for the year together with topics such as, stock markets, Fibonacci and the ‘Golden Ratio’, Pascal’s Triangle, time zones, and investigating pi. In keeping with the spirit of a good TY programme active methodologies, investigation, research, discovery, ICT and project work are used to teach these topics. The study of trigonometry involves participation in a project to construct clinometers and use them to calculate the heights of various structures around the school. The TY programme establishes links between Geography, Science, Physical Education and Mathematics. ICT is regularly used to support the TY mathematics programme. The central aim of the school’s programme for TY is to promote a positive attitude to Mathematics. By designing a course that focuses on the use of activities that encourage enjoyment of the subject, TY teachers help students to develop an appreciation for Mathematics.
Nine teachers and eight lessons were observed as part of the evaluation. In all cases the quality of teaching and learning was observed to be very high. Teachers’ work on the board, explanations and instructions were very clear. Teachers were well prepared for lessons and the resources used, for example PowerPoint presentations, concrete materials, handouts and worksheets, served to clarify explanations, involved students in lessons and added variety of approach. Lessons were very student-focused and this allowed for learning to be differentiated in order to suit the variety of abilities present. All of the lessons observed had a clear focus, were well structured and progressed at a pace that was lively yet appropriate to the ability levels of the students. In most cases, a good balance was achieved between teacher input and student activity as teachers were careful to vary the learning activity regularly throughout lessons. This very good practice should be extended to all lessons.
Teachers made very good use of questioning to facilitate students’ understanding of the underlying ideas being presented in lessons. In most cases higher order questions, that required students to examine and explain their reasoning, were used to achieve this. Questioning focused on the strategic elements of the work in class. This meant that much time was spent working on the decision-making processes in solving the problems presented rather than on the actual numerical workings that followed. This approach, together with the very high levels of student involvement in making these decisions, contributed to the development of students’ confidence in solving problems for themselves. A good example of this was observed in a higher level Leaving Certificate lesson. In this case the students were asked to look at a number of questions and to distinguish between combinations and permutations, by identifying particular words used. In addition, in order to clarify explanations and to frame questions, the teacher frequently referred to activities that had been undertaken in previous lessons. This very good practice used the experience gained from participation in concrete activities to build understanding as the topic became more abstract. This lesson provided evidence of best practice in relation to teaching for understanding.
Students are encouraged to take responsibility for their own learning. A good example of this was observed in a TY class on volume of a sphere. The lesson opened with a quick discussion on previous days’ work on investigating the relationship between pi, the circumference of a circle and the diameter of a circle. The students were then presented with the formula for the volume of a sphere using a short video clip presented via the data projector. An assortment of spheres including golf balls, rubber balls, tennis balls, and glass marbles was then presented to pairs of students along with several rulers and a piece of string. Working in groups of two, the students were asked to find the radius of the golf ball. After much discussion and trial and error the students established the best way to find the radius and used this to calculate the volume of the sphere. The teacher then produced vernier callipers for each pair of students with a supporting handout explaining how to read the scale. Again the ball was measured and the volume calculated. Finally, to round off this investigative lesson, the students found the volume of the sphere using a displacement cylinder and a graduated cylinder. This was an excellent lesson because the students were encouraged to think for themselves and the variety of methodologies used contributed to very high levels of student engagement with the learning activities.
A lesson involving team teaching was observed during the inspection. The sixth year LCA class was taught by one teacher with appropriate support from the second teacher. The learning activities for this lesson on Pythagoras’ theorem were well planned and this contributed to the success of the lesson. The roles of main teacher and support teacher were reported to be regularly switched, from lesson to lesson, which is good practice as it gives both pupils and teachers some variety from day to day; to build on this it is suggested that the teachers alternate roles as the lesson progresses.
A very interesting variation on the ‘traffic lights’ system was seen in a lesson on the graphing of equations. Each student was provided with a square of hardboard and a whiteboard marker at the start of the lesson. The teacher wrote a linear equation on the board and began to call out numbers to the class. The students were then asked to write the solution to the equation on their whiteboards when the number called out was substituted into the equation. The boards were then held up for the teacher to see, were wiped clean and the next number was called out. This provided the teacher with a quick means of assessing understanding and a very effective way of covering material. The lesson continued with the teacher throwing a small rubber ball to students around the room. The ball was then bounced from student to teacher. Through discussion the idea that the ball followed a curved path was developed; this path was drawn on the board and named a parabola. This provided a good introduction to the graphing of a quadratic equation which tied the whole lesson together. When providing examples, the teacher sat at the back of the classroom and used the Smart Notebook to write on the white board. This was effective in placing the teacher among the students and supported the collaborative approach taken to learning in this lesson. Overall, the range and variety of strategies used facilitated much student involvement and made for a lively and enjoyable lesson.
A wide variety of methodologies was observed throughout the evaluation. These included teacher exposition, group and pair work, active methodologies, investigation and discovery, team teaching and ICT. This variety is evidence of the mathematics department’s enthusiasm for the subject. A range of teaching and learning strategies was used in almost all lessons which made lessons lively, interesting and engaging for students. By adopting this student-centred approach teachers have created very positive mathematical experiences for students.
The relationship between students and their teachers was observed to be very good. This was demonstrated by the respect and enthusiasm shown by teachers and students alike to the work of lessons and the team spirit that was evident in each classroom visited. In all cases very high levels of collaboration and participation were observed. Teachers were encouraging, supportive and affirming of student effort. In all cases teachers have created secure learning environments where students can engage with Mathematics with confidence.
All groups are formally assessed at Christmas. Summer examinations are held for first, second, fifth and transition year students. ‘Mock’ examinations take place in spring for students preparing for the certificate examinations. The current practice around the setting of in-house examination paper is that each teacher sets the papers for his or her own class group. Since this approach limits the potential to compare students’ performance in Mathematics, it is recommended that common examination papers be set within levels for formal in-house assessments. This measure would contribute to the quality of the information used by mathematics teachers in planning for students.
Ongoing assessment takes the form of questioning in class and teacher observation. Students’ work is well monitored by teachers through the provision of individual attention in class. It was evident from the review of student copybooks that the standard of student work is generally high. Most teachers provide students with valuable feedback by including comments in the correction of tests and homework. In some cases, teachers use ‘stickers’ and positive comments on written work. Students receive ‘merit’ notes from some teachers for doing well in class tests, these are kept in student files and a copy is sent home. These are examples of very good practices in relation to providing students with valuable sources of advice, encouragement and motivation through assessment. LCA student assignments are well maintained. Homework is set regularly and usually corrected as part of the following lesson and this is good practice.
Each year the mathematics department carries out an analysis of the school’s performance in the certificate examinations and compares it to the national norms. In keeping with good practice this analysis is used to inform planning for Mathematics.
The following are the main strengths identified in the evaluation:
· Timetable provision for Mathematics is good.
· Mathematics is very well supported in terms of resources and ICT.
· The provision for students who have been identified as requiring support with Mathematics is very good.
· The members of the mathematics department work well together as a team and provide strong support for each other.
· In all eight lessons observed the quality of teaching and learning was very high.
· A wide variety of methodologies was observed throughout the evaluation.
· Teachers made very good use of questioning to facilitate students’ understanding of the underlying ideas being presented in lessons and encouraged students to take responsibility for the own learning.
· By adopting a student-centred approach teachers have created very positive mathematical experiences for students and the relationship between students and their teachers is very good.
· There is very good practice in relation to providing students with valuable sources of advice, encouragement and motivation through assessment.
As a means of building on these strengths and to address areas for development, the following key recommendations are made:
· Concurrent timetabling of mathematics lessons should be introduced for first year class groups and, as far as possible; all first year mathematics class groups should follow the same programme of work.
· More teachers should become involved in teaching higher level Leaving Certificate Mathematics.
· A section of planning meetings should be set aside for the sharing of experience and expertise and for discussion around teaching methodologies and lesson ideas.
· The programmes of work for each year group should be described in terms of expected learning outcomes, resources necessary, methodologies used, and modes of assessment.
A section should also be included for reflection and review.
· Common examination papers should be set in formal in-house assessments, wherever feasible.
A post-evaluation meeting was held with the principal at the conclusion of the evaluation when the draft findings and recommendations of the evaluation were presented and discussed.
Published May 2010