Supporting continuity and progression in the learning of Mathematics in the transition from primary to post-primary education: insights from inspections

Chapter 6 - Supporting continuity and progression in the learning of Mathematics in the transition from primary to post-primary education: Insights from inspections (PDF)

Transitions between educational settings and stages can have a significant influence on continuity and progression in the educational experiences of children and young people. The Department of Education and Youth Inspectorate has a unique insight into these transitions, given its evaluative and advisory roles regarding educational provision in early learning and care settings and schools. The Inspectorate’s cross-sectoral vantage point is being used with increasing frequency, as inspectors with specialist knowledge of different sectors of the education system work together more regularly in schools and settings.

When making policy decisions about curriculum, assessment and teacher professional learning, the department looks at a range of evidence. This includes considering student achievement data from national and international assessments alongside what inspection reports tell us about teachers’ practice and students' experiences. The Department of Education Inspectorate Report (2021-2023) and Thematic Review,[1] for example, considered how learner experiences in primary school might affect achievement patterns of post-primary students, as reported in the Programme for International Student Assessment (PISA) 2022[2] and the Trends in International Mathematics and Science Study (TIMSS) 2023.[3] TIMSS, in particular, invites a cross-sectoral perspective, as it assesses achievement in Mathematics of students in both primary and post-primary education.

This chapter focuses on the learning and teaching of Mathematics across the ‘bridging’ years of the transition from primary to post-primary education. Primary inspectors and post-primary inspectors of Mathematics worked together to take a closer look at the learning and teaching of Mathematics as children move from the senior classes in primary school to the junior cycle of post-primary education. These inspectors carried out a series of inspections with a focus on curriculum continuity in Mathematics in eleven post-primary schools and twenty of the primary schools from which those post-primary schools typically receive their students. Consideration of the findings of these thirty-one evaluations, together with the findings of curriculum evaluations of Mathematics in seventy-one primary schools and subject inspections of Mathematics in twenty-two post-primary schools, provide the basis for the reflections in this chapter.

[1] Charting new directions: supporting the enactment of Ireland’s Primary Mathematics Curriculum, in Department of Education Inspectorate Report (2021-2023) and Thematic Review. Available at: https://www.gov.ie/en/department-of-education/publications/the-department-of-education-inspectorate-report-2021-2023-and-thematic-review-pdf-version/

[2] Available at: https://www.erc.ie/

[3] Available at: https://www.erc.ie/studies/timss/

Prior to the most recent cycle of curriculum redevelopment, the Junior Cycle Mathematics Syllabus of 2013 included a bridging framework, which was developed to facilitate a smooth transition for children between primary school and junior cycle.[4] The bridging framework was designed to support continuity and progression in the content of the syllabuses between primary and post primary and in the teaching approaches used.

Recent curriculum redevelopment has strengthened continuity and progression across the curriculum specifications for Mathematics at primary level and in junior cycle. The current junior cycle curriculum for Mathematics was introduced in 2018.[5] The Primary Mathematics Curriculum was introduced in 2023.[6]

The curriculum specifications for Mathematics at primary level and in junior cycle have a shared aim of developing mathematical proficiency, which is recognised in both specifications as having five interconnected and interwoven components:

• conceptual understanding
• procedural fluency
• strategic competence
• adaptive reasoning
• productive disposition

It is widely recognised, however, that alignment of specifications does not guarantee continuity and progression in learning. The most recent cycles of curriculum development are informed by an understanding of curriculum as something that is constructed through interactions between the published materials, teachers, students and others. Curriculum making is seen as a social process of shared sense making among those involved in enactment, as well as the learner experiences that emerge from that process.[7]

Redeveloped curriculum specifications in primary and post-primary education provide broad learning outcomes that allow for key decisions to be made, at the level of the school and classroom, about what to teach, when to teach it, and how to sequence and pace learning. This acknowledges the centrality of teacher agency to the process of curriculum making, and promotes an active role for teachers and school leaders in shaping the curriculum in response to the context in which they work.

Reflection on inspection findings suggests that students’ experience of continuity and progression in Mathematics, in the transition from primary to post-primary education, is underpinned by two key factors. These are

• teachers’ knowledge of the students and their prior learning
• teachers’ shared understanding of curriculum and pedagogy.

The remaining sections of this chapter discuss each of these areas in turn.

[4] Department of Education and Skills (2013). Junior Certificate Mathematics Syllabus. Available at: https://www.gov.ie/en/department-of-education/collections/junior-certificate-syllabuses-and-prescribed-material/#mathematics

[5] Department of Education and Skills (2018). Junior Cycle Mathematics. Available at: https://assets.gov.ie/static/documents/mathematics-junior-certificate-curriculum-specification-2018.pdf

[6] Department of Education (2023). Primary Mathematics Curriculum for Primary and Special Schools. Available at: https://www.curriculumonline.ie/primary/curriculum-areas/mathematics/

[7] Hayward, L., Spillane, J., Walsh, T. and Wyse, D. (2022). From Purpose to Practice – Primary Curriculum Developments in Ireland: Reflections from the Advisory Panel (Primary). Developed in collaboration with and funded by the National Council for Curriculum and Assessment. Dublin: National Council for Curriculum and Assessment. p17

Availability and use of information from primary schools on prior learning in Mathematics

The National Council for Curriculum and Assessment (NCCA) has developed an Education Passport and associated guidance materials to support the transfer of student information from primary to post-primary school.[8] Schools are required to use the Education Passport materials to support the reporting and transfer of student information at the end of sixth class. The first step in the information transfer process is a request from the post-primary school to the primary school after the child’s enrolment in the post-primary school has been confirmed.

Inspectors found that practice with regard to the Education Passport varied considerably from one school to another. In a small number of instances, the post-primary schools had requested the Education Passports, and these had been provided promptly, and the relevant information had been factored into the preparation of learning experiences and programmes of work in Mathematics for first-year students. In most cases, however, information from the Education Passport did not inform teaching and learning of Mathematics in the post-primary school. Typically, this was because the relevant information had not been requested or had not been transferred within the required timeframe. In some cases, assessment information had been transferred effectively but was not taken into account in planning for teaching and learning in Mathematics for first-year students.

Even where the Education Passport is transferred, its capacity to support continuity and progression in learning in Mathematics is relatively limited at the present time. Currently, the assessment information for Mathematics is confined to an overall Standard Ten (STen) score. It does not include information about attainment in the various curriculum strands and process skills, although this information is available from standardised attainment tests conducted up to and including sixth class in the primary school. Therefore, in its current format, the Education Passport is of limited use to teachers of Mathematics who may wish to build on students’ prior knowledge and address areas of relative weakness.

Classroom assessment practices

The relatively limited availability of information on students’ prior learning on entry to post-primary school is concerning. Furthermore, the use of assessment to support learning in post-primary Mathematics has been identified as an area for development for some time. Classroom assessment practices provide an important way for teachers to gain knowledge of the students and their prior learning. The recent inspection visits to primary and post-primary schools, with a focus on curriculum continuity in Mathematics, found that the quality of assessment was lower than the quality of practice in other areas of teaching. The proportion of primary schools in which the quality of assessment was good or better was significantly higher than the proportion of post-primary schools in which this was the case.

Where classroom assessment practices were effective in both primary and post-primary schools, they informed preparation for the next steps in teaching and learning and supported the progression of the whole class and individual students. Teachers demonstrated a good knowledge of where the students were in their progression towards the learning outcomes of the relevant curriculum strands and strand units. Textbooks were among the resources used to support the student on the learning path set out in the curriculum. In these classrooms, however, the teachers’ knowledge of the learning path to be followed in their subject meant that they were not depending on the students’ textbooks for the sequence and the pace of learning.

A key feature in all instances of effective assessment practice, across primary and post-primary levels, was that the teachers measured success in terms of evidence of progression in learning. Assessment information (a combination of the teacher’s knowledge of the student and of the curriculum) informed the teacher’s decision about the learner experience needed to support students on the next steps of their learning journey.

Where assessment practice was particularly good, feedback from the teacher was used to foster the students’ capacity for self-assessment and the development of their ownership of, and responsibility for, their own learning. This inclusion of the student in the assessment process supports the development of key competencies and skills in primary education, junior cycle and senior cycle. The need for more effective provision of feedback featured frequently in the recommendations by inspectors to both primary and post-primary schools during the inspections of Mathematics.

The recommendations made by inspectors to individual schools highlighted the need for teachers to improve practice with regard to the following key strategies of formative assessment identified by Professor Dylan William[9]:

• clarifying, sharing and understanding learning intentions and success criteria
• eliciting evidence of learning
• providing feedback that moves learning forward
• activating students as instructional resources for one another
• activating students as owners of their own learning.

As stated by William (2017), “Formative assessment involves getting the best possible evidence about what students have learned and then using this information to decide what to do next”[10]. The limitations of current arrangements in the Irish education system for the transfer of assessment information from primary to post-primary school mean that there is often very little evidence on which post-primary teachers of Mathematics can build a knowledge of the student and their prior learning. As a result, teacher decisions about ‘what to do next’ are not always as well informed as they should be. Evidence from inspection indicates that this can lead to the allocation of unnecessary time and resources to learner outcomes already achieved in primary school, and levels of challenge that do not take account of strengths and weaknesses in the prior learning of individual students or groups of students. This is a significant obstacle to students’ experience of continuity and progression in the learning of Mathematics. Classroom assessment practices following the transition to post-primary education do not appear to make up for this deficit.

The inspections also found that an awareness of students’ future curriculum needs was an important support for continuity and progression in the learning of Mathematics. In this regard, it was noted that there was very limited evidence of calculator use in the senior classes of the primary schools visited, even though calculators are a useful resource for teaching Mathematics in these classes, and have become essential in post-primary education.

In recent years, there has been an increased awareness of the need for teaching and learning to respond to the needs of more able students. Ireland’s Literacy, Numeracy and Digital Literacy Strategy 2024-2033[11] includes actions aimed at providing appropriate challenge and support for more able students in Mathematics. The inspection findings show that the learning experiences and outcomes of these students were particularly affected by shortcomings in the transfer of knowledge about students’ abilities and their prior learning, and deficiencies in classroom assessment practices in the post-primary schools. The inspections also found that an over-reliance on the students’ textbooks, to support the pacing and sequencing of learning, affected these students disproportionately.

Overall, there is evidence from inspections that teachers’ knowledge of the prior learning of first-year students is not sufficient to provide effective support for continuity and progression in students’ learning in Mathematics. It would be beneficial to explore potential for greater sharing and use of the assessment information held by primary schools and for ongoing development of classroom assessment practices in the early years of post-primary education.

[8] Available at: https://ncca.ie/en/primary/reporting-and-transfer/education-passport/

[9] William, D. (2017). Embedded formative assessment: strategies for classroom assessment that drives student engagement and learning. Solution Tree. 10.1007/978-981-10-3654-5_31.

[10] Ibid.p.56

[11] Available at: https://www.gov.ie/en/department-of-education/publications/literacy-numeracy-and-digital-literacy-strategy/

As well as teacher’s knowledge of students and their prior learning, continuity and progression in Mathematics also depend upon a shared understanding of curriculum and pedagogy among teachers in both sectors. This shared understanding was evident to varying extents in relation to three components of mathematical proficiency, the development of which is the over-arching aim of the curriculum specifications for both primary and junior cycle education. These are productive disposition, adaptive reasoning and strategic competence.

Productive disposition

The Primary Mathematics Curriculum defines productive disposition among students as “the tendency to see Mathematics as practical, useful and worthwhile”[12]. This definition is extended in the junior cycle mathematics curriculum as the “habitual inclination to see Mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence, perseverance and one’s own efficacy”[13].

A lack of “belief in … one’s own efficacy” can be a significant obstacle to learning and progression in Mathematics. In focus groups conducted as part of inspections, students referred to their attitude to Mathematics being formed from the middle classes in primary school and to their perception that Mathematics became more difficult at that stage. In post-primary schools in which students were allocated to Mathematics classes by ability, students made frequent reference to class or grade level allocation, and there was evidence that it formed an important part of their identity as students of Mathematics. Research indicates that practices such as streaming and setting of students can have a negative impact on the development of attitudes, beliefs and expectations associated with the formation of a productive disposition[14]. Furthermore, the limited information available on the prior learning of first-year students means that the evidence base for allocating students to classes by ability is questionable.

During several inspections, teachers and school leaders shared their perception that boys tend to have a greater belief in their own efficacy, even when their assessed mathematical ability might not support such a belief. They also observed that boys demonstrated a greater willingness to take risks and attempt answers even if they might be incorrect. Girls were perceived by teachers and school leaders to be more likely to double-check their work for accuracy before answering. In some cases, teachers surmised that a belief in the importance of neatness and structure prevented girls from experimenting or exploring alternative solutions.

These observations by teachers and school leaders are relevant to the findings of the two most recent large-scale international assessments in which Ireland participated. In PISA 2022, the difference in achievement between males and females in mathematical literacy in Ireland was greater than the OECD average. TIMSS 2023 showed that there were significant gaps in achievement between male and female students in Mathematics in second year of post-primary school, with males significantly outperforming female students. While there was no significant difference in overall achievement by gender in fourth class of primary school, more boys than girls reached the advanced benchmark in Mathematics. It is important, therefore, that we consider if learner experiences in primary and post-primary schools are contributing in any way to differences in achievement in these assessments by gender.

Concerns expressed by teachers and school leaders about the productive disposition of girls in Mathematics did not appear to result in any co-ordinated whole-school actions to address this. Overall, inspectors found that schools did not have a coherent approach to identifying and addressing gender-specific attitudes and outcomes in Mathematics and were unclear as to how to approach such issues.

The ‘tendency to see Mathematics as practical, useful and worthwhile’[15] is supported by classroom environments that provide opportunities for inquiry-based learning. In classrooms where inquiry-based learning was used effectively, inspectors observed that students demonstrated a high level of independence in investigating questions set by the teacher. Furthermore, students also had opportunities to generate and explore their own questions. In discussing their practice with inspectors, some teachers in these classrooms observed that practical, learner-led, inquiry-based activities were more effective, in the development of overall mathematical proficiency, than textbook-based approaches.

In some post-primary classrooms, while hands-on practical activities took place, there was scope to link these more explicitly with the relevant learning intentions and success criteria. Some teachers noted that most post-primary students associated the use of manipulative resources in the learning of Mathematics with the early years of primary school and that this was an inhibiting factor.

Learning experiences that involve collaboration with other students can have a positive impact on productive disposition. However, there is evidence from inspection that opportunities for collaborative learning are much less frequent in post-primary than in primary education and occur with decreasing frequency between first year and third year. It was found that the seating arrangements and layout of the learning environment typically provided in primary classrooms allowed for more collaboration and students working in groups than those that were typically provided in post-primary classrooms.

Overall, there is scope for improvement in practice regarding the development of productive disposition in Mathematics. There is a need for greater awareness in both primary and post-primary schools of factors affecting the disposition of girls in relation to Mathematics. There is a need also for greater recognition of the impact of organisational and pedagogical practices on the development of productive disposition in post-primary schools.

Adaptive reasoning

The Primary Mathematics Curriculum defines adaptive reasoning as “the capacity to use logic to understand, explain and justify one’s thinking”[16]. The junior cycle mathematics specification provides a similar definition.

The inspections highlight three particular areas in need of improvement in many of the schools inspected in order to support students’ adaptive reasoning: the development of mathematical communication; the use of questioning by teachers; and the use of digital technologies.

Inspections in both primary and post-primary schools often found a need for the development of students’ mathematical language in order to assist them in reflecting on and communicating their own learning and in reasoning, justifying and generalising their strategies.

Some teachers in both primary and post-primary settings demonstrated highly effective questioning techniques. These teachers frequently asked well-considered, open-ended questions that encouraged higher-order thinking and required students to explain their reasoning and justify their solutions. These teachers typically encouraged students to take time to consider possibilities, and to share those possibilities with a peer before answering. They also encouraged students to identify the mathematical concepts and procedures that were relevant to the problem to be solved.

Where practices to support adaptive reasoning were less than effective, opportunities for dialogue with, and among, students were limited. In some instances, teachers missed opportunities to challenge students' thinking by not probing incorrect answers or asking follow-up questions. In others, questioning strategies were focused on procedural recall rather than conceptual understanding.

Digital technologies have significant potential to enable students to visualise and experiment with mathematical models at their own pace. The use of multimedia resources, interactive exercises and adaptive assessments provide great opportunities to enhance and reinforce learning. There was evidence that primary school students had regular opportunities to reinforce their learning in Mathematics using digital technologies. As students transitioned to post primary such opportunities became less frequent and were often confined to the use of organisational and reference tools. Inspectors found that there was scope in both sectors for more frequent opportunities for students to use digital technologies to support adaptive reasoning.

Strategic competence

The junior cycle mathematics specification defines strategic competence as the “ability to formulate, represent, and solve mathematical problems in both familiar and unfamiliar contexts”[17]. The Primary Mathematics Curriculum has a similar definition.

One of the particularly positive findings of the inspections was that problem solving was a core focus in many primary and post-primary classrooms. In most instances, teachers’ practice was informed by a recognition that there can be multiple, equally effective approaches to solving a particular problem. Where practices to develop strategic competence were most effective, teachers supported students in using concrete materials, the problems were linked to real-life situations and there were opportunities for students to collaborate with one another. Concrete materials such as number lines, counting sticks and fraction manipulatives supported students in conceptualising mathematical ideas. Some teachers successfully linked mathematical problem solving to everyday situations, such as measuring distances in hallways, analysing shopping catalogues and engaging in construction-related tasks. The facilitation by teachers of peer discussion and group problem-solving required students to articulate and justify their reasoning and modify it, if necessary, having listened to the observations of peers.

In a small number of lessons, problem-solving activities placed an exclusive emphasis on procedural fluency and could have been used more effectively to enhance conceptual understanding.

[12] Department of Education (2023). Primary Mathematics Curriculum for Primary and Special Schools. Available at: https://curriculumonline.ie/getmedia/484d888b-21d4-424d-9a5c-3d849b0159a1/PrimaryMathematicsCurriculum_EN.pdf p.12

[13] Department of Education and Skills (2018). Junior Cycle Mathematics. Available at: https://curriculumonline.ie/getmedia/4f6cba68-ac41-485c-85a0-32ae6c3559a7/JCSEC18_Maths_Examination-in-2016.pdf p.5

[14] Francis B., Taylor, B. and Tereshchenko, A. (2019). Reassessing ‘Ability’ Grouping: Improving practice for equity and attainment. Taylor and Francis.

[15] Department of Education (2023). Primary Mathematics Curriculum for Primary and Special Schools. Available at: https://curriculumonline.ie/getmedia/484d888b-21d4-424d-9a5c-3d849b0159a1/PrimaryMathematicsCurriculum_EN.pdf p.12

[16] Ibid

[17] Department of Education and Skills (2018). Junior Cycle Mathematics. Available at: https://www.gov.ie/en/department-of-education/collections/junior-certificate-syllabuses-and-prescribed-material/#mathematics

Continuity and progression in the teaching and learning of Mathematics are underpinned by teachers’ knowledge of the students and their prior learning, and a shared understanding of curriculum and pedagogy. Inspection findings indicate that there are opportunities for improvement in both of these areas.

There is considerable scope to enhance teachers’ knowledge of the prior learning of first-year students in order to support more targeted allocation of teaching and learning resources and supports. This might be addressed effectively through system-level guidance on greater sharing of assessment information between primary and post-primary school, as well as the further development of classroom assessment practices in early post-primary education.

In relation to developing a shared understanding of curriculum and pedagogy among teachers in both primary and post-primary education, inspection findings indicate that there is scope for a more explicit focus on mathematical proficiency, the development of which is an over-arching aim of curriculum specifications in both primary and junior cycle education.

Considerations

Transfer of information on prior learning in Mathematics from primary to post-primary schools

Under the current Education Passport arrangements, an overall STen score in Mathematics for each student is shared by the student’s primary school with the post-primary school in which they have enrolled, along with some teacher observations. Primary schools hold considerably more data on students’ strengths and weaknesses across content strands and process skills. The department should consider steps at system level to facilitate and provide guidance on the sharing of more detailed information from standardised assessment tests and other sources to provide a more rounded picture of the student as a learner of Mathematics. As part of this work, the department should consider engaging with NCCA to explore possibilities for the development of the Education Passport.

Promoting cross-sectoral professional learning

With a view to supporting continuity and progression, consideration should be given to the development of cross-sectoral resources to support effective practices in teaching, learning and assessment for primary teachers and post-primary teachers of Mathematics, with an initial focus on the following:

• the development of the five components of mathematical proficiency, a shared aim of the primary and junior cycle curriculum specifications
• whole-school and classroom practices to address gender patterns in achievement in Mathematics
• implementation of NCCA guidance on assessment to support differentiation of teaching and learning
• supporting more able students in fulfilling their potential in Mathematics
• effective use of digital technologies to support learning in Mathematics.