The beginning of the school year is an appropriate time to consider change in how and what mathematics is taught in our schools. The need for mathematics education to evolve is not recent, however changes to educational practices are typically much slower than the rate of change in the world we live in. For example, digital technologies have dramatically changed how we receive and process information, how we communicate, and how we learn, yet it wasn’t until the COVID-19 pandemic forced schools into remote learning mode that many educators recognised the value of digital resources in mathematics education. Similarly, student engagement with mathematics has continued to challenge educators for many decades and while educators are continually presented with new programs, strategies, and initiatives, student attainment and retention in mathematics courses beyond the compulsory years remain in need of improvement. Put simply, the students in our classrooms in 2023 are different to those we taught two, five, or ten years ago, but for many, teaching practices have not evolved to the same degree. So, what should we change, and what should remain the same? How can we reduce the number of times we hear the question, “Why do I need to learn this?”, or the comment, “I’m just not a maths person”? How do we adapt and evolve our practices to make mathematics comprehensible to all students?
To improve student engagement with mathematics we first need to understand that it is a multi-dimensional construct, consisting of cognitive, operative, and affective domains (Attard, 2014; Fredricks et al., 2004). In other words, students who are engaged with mathematics are more than just ‘on task’. Rather, they are ‘in task’, thinking hard (cognitively engaged), working hard (operatively engaged), and feeling good about mathematics (affectively engaged). In addition, students who are engaged value their learning and view themselves as users of mathematics both within and outside the classroom, and in their future lives. The Framework for Engagement with Mathematics in Figure 1 (FEM) (Attard, 2014) provides details of the elements and actions that contribute to student engagement.
|Pedagogical Relationships||In an engaging mathematics classroom, positive pedagogical relationships exist where these elements occur:|
|PK||Pre-existing Knowledge: students’ backgrounds and pre-existing knowledge are acknowledged and contribute to the learning of others|
|CI||Continuous Interaction: interaction amongst students and between teacher and students is continuous|
|PCK||Pedagogical Content Knowledge: the teacher models enthusiasm and an enjoyment of mathematics and has a strong Pedagogical Content Knowledge|
|TA||Teacher Awareness: the teacher is aware of each student’s mathematical abilities and learning needs|
|CF||Constructive Feedback: feedback to students is constructive, purposeful and timely|
|Pedagogical Repertoires||Pedagogical repertoires include the following aspects:|
|SC||Substantive Conversation: there is substantive conversation about mathematical concepts and their applications to life|
|CT||Challenging Tasks: tasks are positive, provide opportunity for all students to achieve a level of success and are challenging for all|
|PC||Provision of Choice: students are provided an element of choice|
|ST||Student-centred Technology: Technology is embedded and used to enhance mathematical understanding through a student-centred approach to learning|
|RT||Relevant Tasks: the relevance of the mathematics curriculum is explicitly linked to students’ lives outside the classroom and empowers students with the capacity to transform and reform their lives|
|VT||Variety of Tasks: mathematics lessons regularly include a variety of tasks that cater to the diverse needs of learners|
The Framework is divided into two separate but inter-related sections: Pedagogical Relationships and Pedagogical Repertoires. For engagement to occur, positive pedagogical relationships must be developed which then allow for engaging pedagogical repertoires to be planned and taught. These engaging repertoires, or practices, are what we should be considering in relation to the needs of contemporary mathematics learners.
Consider the following questions:
Do you and the teachers at your school:
- Make connections to previous lessons and prior assessment of students’ learning?
- Make connections to a topic covered in another area of the curriculum?
- Make connections to students’ current interests?
- Make connections to a previous topic where they may have discussed a similar idea in another form?
- Cover one idea in a variety of ways?
- Aim to make connections with students?
These questions are a good place to start when reflecting on current practices and considering new ones. The emphasis on connections is an important one, particularly when we are enacting the mathematics curriculum. Traditionally, mathematics content was taught in isolated topics with the view that this assisted in ‘covering’ the curriculum. However, this does not assist in the development of conceptual understanding. In her book About Teaching Mathematics, Marilyn Burns (2022), talks about making a shift from covering the curriculum to ‘uncovering’ it. She makes the point that teachers often stress and become anxious about ticking all the curriculum boxes, resulting in a return to deficit practices that treat topics in isolation and teaching in a procedural rather than conceptual manner. ‘Covering’ the curriculum does imply a teacher-centred approach, where ‘uncovering’ the curriculum implies a student-centred, sense-making approach that allows students opportunities to make mathematics connections and engage in deeper learning. Tasks that promote connections and student engagement in mathematics include:
- Open-ended tasks
- Rich tasks
- Problem solving and investigation
- Inquiry-based learning
- Tasks that promote critical and creative thinking
- Tasks that spark curiosity.
An excellent example of a task the promotes connections is this investigation: Do right handed-people have bigger left feet? To successfully engage with this investigation students need to draw on and make connections with knowledge and skills from number, measurement, statistics and probability. They would also be accessing a range of big ideas. In addition, the physical nature of the task and the fact that it is about the students makes it particularly engaging.
Of course, a task is only as good as the pedagogy it is embedded within. For example, you could use a rich task that promotes connections across a range of mathematical concepts and incorporates high levels of problem-solving and mathematical reasoning. However, if students are not grouped appropriately, not provided with opportunities to share their thinking or use appropriate materials (digital and/or concrete) and engage in deep reflection, the potential for deep learning within the task my not be reached. Pedagogies that promote engagement and learning include:
- Purposeful and flexible grouping of students
- A balance of formative and summative assessments
- Effective use of digital resources (pedagogy driving technology rather than technology driving pedagogy)
- Opportunities for differentiation
- Contextualised learning
- Dialogic practices.
CHANGING TECHNOLOGY-RELATED PRACTICES
As mentioned earlier, COVID-19 was a catalyst for many teachers to begin or further develop their skills in using digital resources in the teaching of mathematics. However, due to the inconsistencies in resourcing and teacher technology-related confidence and experience, not all teaching and learning using digital resources was successful. Regardless of the degree of success, the opportunity provided by the pandemic to use digital resources is one that should be taken as a valuable opportunity to evolve how we teach mathematics.
Acknowledging that no two school contexts are the same, the Technology Integration Pyramid (Mathematics) (TIP-M) (Figure 2) was developed to assist teachers in planning for technology-enabled mathematics education (Attard & Holmes, 2020). The base of the pyramid details the complexities of the influences on technology-infused instruction that should be acknowledged and if appropriate, addressed, for teachers to effectively use digital resources (Figure 3).
|School Leadership Professional Development Collaboration (teachers working in teams/individual) Innovation||Parents|
Other local stakeholders (business, media, government) ColleaguesStudents
System (policy, type, restrictions)
|Support (technical and instructional)|
Individual Beliefs (mathematics, technology, teaching & learning)
Teacher self-efficacy with technology
Willingness to innovate
When teachers and school leaders fully understand the influences on how digital resources are accessed and used in the mathematics classroom, they are then more likely to realise the potential of those resources. To assist them, the sides of the TIP(M) represent four critical classroom considerations for planning. These are pedagogy, mathematics, engagement, and tools (Table 1).
|Content (topics in isolation vs. connected)|
Process – problem solving vs. fluency
Representations (dynamic) (making connections between)
Computation and/or higher order thinking
|Devices (BYOD, type, number, affordances, constraints)|
Software (type, affordances, constraints)
Administration (connectivity, updates, downloads)
Number of devices (shared or 1-1)
FlippedTech for teaching and/or learning
Student as consumer vs. producer
Develops Positive Pedagogical Relationships
Expands pedagogical repertoires
The four considerations are inter-related and congruent, implying equal importance amongst the elements, and while the base of the pyramid provides a starting point to understanding the nuanced technology landscape that exists within each school, the four sides of the pyramid provide a focus for teacher decision making, ensuring student engagement is paramount. The TIP(M) provides teachers with a model that will assist in developing ‘best practice’ in individual mathematics classrooms.
IS THERE A BEST PRACTICE IN MATHEMATICS?
When considering if and how mathematics education practice needs to change, we must also then reflect on whether there is, in fact, a universal best practice in mathematics education. Should all mathematics classrooms look the same? Should all students experience the same mathematics education? The unique nature of each school and classroom is evidence that the only best practice in mathematics education is the education that best addresses the needs of your students, at your school, in this moment. When contemplating how to develop and adapt your practices to align with changing times, know your students, know your content and how to teach it, and know the resources you have at hand. Only then can you develop practices that will improve student engagement and achievement in mathematics.
Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34, 1–14.
Attard, C. & Holmes, K. (2020). Technology-enabled mathematics education: Optimising student engagement. (2020). Routledge.
Burns, M. (2022). About teaching mathematics. (4th Ed.) Heinemann.
Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74, 59–110.