A vision for mathematical expertise and excellence
Peer reviewed article
Dr Christine Mae is the Education Officer for Mathematics in Sydney Catholic Schools. Her research article focuses on the effective teaching of mathematics.
Mathematics is a fundamental aspect of student learning. It stimulates students’ capacity for logical thought and action and teaches them to reason and make sound judgments (NSW BOS, 2012). Feedback from universities has heightened awareness that students who study higher levels of mathematics are more likely to persist in tertiary courses and gain employment in related fields. By contrast, low levels of numeracy are associated with lower levels of social, emotional, financial and physical well-being in life beyond schooling (Bynner & Parsons, 2006). While mathematics is a discipline, and numeracy involves recognising the role of mathematics in the world and having the capacity to use mathematical knowledge and skills purposefully (ACARA, 2010), effective mathematics teaching is essential for increasing both levels of numeracy and participation in higher levels of mathematics.
It is time to go beyond recognising the need to improve mathematics education, to developing and implementing strategies that maximise student engagement, achievement and aspiration. Changing the curriculum, articulating teacher standards and implementing national testing make expectations clear, but they do not of themselves improve mathematics education or students’ levels of numeracy. Teachers and quality teaching make the difference.
This article presents insights into the Mathematical Expertise and Excellence (MEE) project, which commenced in 2018 to improve mathematics education across a system of schools. First, a brief overview of the aims and structure of the project is presented. Then, early impacts from deepening the mathematical knowledge for teaching of over 600 primary teachers who completed the MEE Proficient Course during the first two years of the project, are shared. During the COVID-19 crisis, many teachers have expressed that knowledge gained through the course has assisted them in sustaining their students’ interest in learning mathematics via remote learning. However, as a large-scale, eight year project, currently in its third year of implementation, the longer-term impacts of the project are unknown at this point.
The Mathematical Expertise and Excellence project
The MEE project was designed and implemented in response to a seven year research study into relationships between teachers’ understandings of mathematics, the tasks they provide for student learning and the ways in which they respond to students’ thinking (Mae, 2019). The findings of the research regarding how teachers’ understandings of the mathematics they teach influence the nature of the tasks they provide for student learning and the ways in which they interpret students’ thinking, led to the design of the project. The variability of teachers’ subject matter and pedagogical knowledge and its implications for equity in mathematics education, were key findings of the research. Hence, realisation of the following two long-term outcomes of the MEE project is reliant upon every teacher, rather than some teachers, developing the expertise to teach mathematics effectively:
- maximising the levels of numeracy attained by all students, and
- increasing the proportion of students studying, and aspiring to study, higher levels of mathematics and mathematics-related subjects.
Improving the mathematics education available to all students is a substantial undertaking. It requires a sustained, strategic effort towards a coherent vision for quality learning and teaching, a systematic approach to sustain interest, motivation and improvement, and well-considered support and resourcing. Yet, if we truly believe in the goals of equity and excellence, the question we must ask is not whether we need to improve mathematics education, but how, when and in what ways will improvement take place?
Global research has identified the need for Australia to work strategically to maintain a base of mathematical knowledge and skill through increased opportunities for students to solve more complex, unfamiliar, non-routine problems, higher expectations for communicating and reasoning and greater exposure to alternative solution approaches (Thomson, Hillman & Wernert, 2016). While concerns regarding Australia’s falling rankings in PISA were raised more recently in 2019, recommendations regarding what we need to do to improve mathematics education have been reasonably clear and consistent for some years. It is time to implement the recommendations!
The MEE project is founded on recommendations for teaching and learning mathematics in all schools in Australia. The design and implementation of rich, cognitively challenging tasks, and the ways in which teachers respond to students’ thinking as they engage in them, are examples of important, practical, recommendations for mathematics teaching and learning.
Rich, cognitively challenging tasks
The selection, design and implementation of rich tasks with appropriate levels of cognitive demand is crucial for effective mathematics instruction, because tasks form the basis of the lessons that students experience. However, teachers’ expectations influence the tasks they provide for students. To shift teaching beyond exercises that focus on learning procedures, teachers need a repertoire of powerful examples, problems, analogies and illustrations through which their students can explore and understand concepts.
Noticing student thinking
To maximise student learning, teachers need to be able to interpret and respond to students’ mathematical thinking. Teacher noticing provides the connection between students, learning tasks and the content (National Council of Teachers of Mathematics [NCTM], 2014). This requires skilful perception of how, rather than whether, students respond, make calculations, or reason when solving problems. To ‘scrutinize, interpret, correct, and extend’ (Ball, Hill & Bass, 2005, p 17) students’ mathematical thinking, teachers need to be able to represent ideas in multiple ways and ‘carry out and understand multi-step problems’ (Ball, Hill & Bass, 2005, p 21).
For these reasons, effective mathematics teaching is affected by teachers’ proficiency with the subject matter, inclusive of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition (Kilpatrick, Swafford & Findell, 2001). In the research study that led to the development of the project, analysis of relationships between teachers’ subject matter and pedagogical knowledge revealed that teachers understandings of the content were highly significantly predictive of the levels of cognitive challenge in the tasks they designed and significantly predictive of their noticing of students’ thinking (Mae, 2019). While many teachers know about mathematics, not all teachers possess the knowledge and confidence to design cognitively challenging tasks, solve unfamiliar problems and interpret students’ thinking.
It is with this background that the MEE project was developed and implemented following the findings of the research study, the evaluation of a pilot project to trial and refine professional learning, support and resourcing, and an extensive review of Australian and international literature. The project sets out to deepen teachers’ understandings of mathematics, the NSW Mathematics K-10 Syllabus and mathematics pedagogy. It involves a four-year commitment from each school to strategically develop, embed and sustain expertise and excellence in the teaching of mathematics, with each year pertaining to a project phase (Phase 1, Phase 2, Phase 3 or Phase 4). As each school identifies the year in which to commence the project, the duration of the project is greater than the time commitment of any individual school.
To develop the mathematical expertise and excellence of all teachers, Phase 1 focuses on ensuring the knowledge of the leaders and teachers who will lead mathematics and the project in their school through completion of the proficient professional learning course. In Phase 2, participants who have successfully completed all workshop and in-situ components of the Proficient Course are invited to engage in the Highly Accomplished Course to deepen their knowledge of mathematics and learn how to support colleagues commencing the Proficient Course. This course includes learning how to support colleagues through effective in-class modelling, co-teaching, observation, analysis of practice and feedback. In Phase 3, teachers who have completed the Highly Accomplished Course can engage in a Lead Course in which they further deepen knowledge of mathematics and develop the skills and confidence to lead mathematics, including professional learning, across the school. By the end of Phase 3, all teachers of mathematics in the school should have completed the proficient level course and the ratio of teachers who have completed the Highly Accomplished Course should be sufficient to provide in-class modelling, co-teaching, observation and feedback for all teachers in the school. Phase 4, the final year of the project for each school, is a crucial period of the project during which schools embed and sustain mathematical expertise and excellence across the school.
Phase 1 – Professional learning: Proficient mathematics teaching
The proficient professional learning course blends workshop-style learning that introduces theoretical elements and makes them tangible through personalised in-situ learning. In-situ components involve Leaders of Learning and Numeracy Coaches working alongside each course participant to model, co-teach and observe mathematics teaching in their classroom. Each session commences with a pre conversation about the design of the lesson, the learning task and the anticipation of pedagogical decisions to maximise learning in the lesson. Following each lesson, participants engage in a professional conversation that provides the time and space for them to reflect on and analyse teaching practice and its impact on student learning. Translating theory into practice in each teacher’s classroom is an important feature of the project that complements the deepening of content, syllabus and pedagogical knowledge gained in workshops. The graphic that follows communicates the pedagogical and content focus of each workshop in the Proficient Course.
As a consequence of the Proficient Course, teachers deepen their knowledge of syllabus content for teaching four syllabus sub-strands K-8, and develop pedagogical knowledge to:
- articulate clear learning intentions, design rich, cognitively challenging tasks and develop differentiated success criteria to provide challenging yet inclusive learning for all students;
- select and use a variety of strategies to attend to, interpret and respond to students’ mathematical thinking and engage them in productive mathematical discussion;
- balance opportunities and time for students to learn new concepts with opportunities to practise, master and apply their learning; and,
- use deep knowledge of the syllabus to sequence learning effectively across the year and within program units.
The image that follows offers an example of a task as it might appear in a classroom when applied to Stage 3 content introducing the language of increase and decrease. Elements of the task, such as the learning intention, task and success criteria, are animated so that they can be introduced as needed and in ways that focus students’ attention on concepts and meaning. This task is posed as a ‘challenge’ supported by differentiated success criteria. The first criteria are written so that any student with a low starting point can start independently. By contrast, the last criteria are designed so that the most capable student in the class needs to exert significant cognitive effort to succeed. Teachers learn to articulate learning intentions, design these types of tasks and develop success criteria with the scope to facilitate learning for the full range of students in their class: they design low entry – high ceiling learning tasks.
Once teachers are confident in designing these types of tasks, they focus on implementing tasks using an array of strategies to support and empower students with different starting points for the learning on any given day. We refer to these strategies as ‘pedagogical moves’. They are teaching and learning strategies that can be selected to maximise mathematical learning by maintaining classroom environments that focus on improvement, challenge and support. Pedagogical moves are practical adaptations of the three noticing skills of attending, interpreting and responding to student thinking (Jacobs, 2010) and the five practices of anticipating, monitoring, selecting, sequencing and connecting described by Smith & Stein (2011).
Examples of strategies that teachers have reported as being effective for increasing students’ interest, confidence, communication, reasoning and effort include the Fishbowl, Gallery walk and Showcase space.
The Fishbowl is an ideal strategy for addressing misconceptions, taking learning to the next level or explicitly teaching important points based on students’ current responses to a task. By monitoring the class and posing questions, teachers identify students who can model or explain an idea in ways that will contribute to the learning of other students in the class. The class form a circle around the student who is sharing so that they can clearly see and hear the mathematical thinking being highlighted. Then, the teacher uses the student’s thinking as a starting point for explicitly teaching or clarifying an important point. In the words of a Year 3 student:
… when I am the person in the middle of the Fishbowl I feel excited, like I’m a teacher. The teacher is like the fish food that feeds the fish to make them grow. When another student steals one of my ideas I feel great because it means that they are learning something from me.
A Gallery walk is a useful way for teachers to clarify expectations in relation to a task or success criteria, encourage students to consider other possibilities and show students what good work could look like and how they might improve their work. The teacher invites students leave their work displayed and move quietly around the room viewing the work of others in relation to the task and the success criteria. The role of students is to identify examples of work that are interesting or that they can learn from. Then, the class discuss their observations of different responses to clarify ideas and increase their understanding of what the teacher is looking for. They are encouraged to use their observations to improve their own work. Teachers and students refer to a variation on the Gallery walk as a Spy walk, where the teacher invites one or more students to move around the room to find examples of work that they can learn from while the rest of the class continue working.
A Showcase space is a great strategy for preparing students to share their insights into the mathematics or the task they are working on. Early in the lesson, some students are selected to work in special spaces that will be ideal for sharing with the class. For example, two students might work on a large whiteboard to record written strategies or a diagram in response to a task while another student might build a model in a central location that will make viewing possible for all members of the class. Together, their responses provide the class with multiple representations of the same idea that can connect all students to the goal of the lesson.
The art and purpose of selecting pedagogical moves provides substantial opportunities for teachers to know their students as learners of mathematics – not just the correctness of their answers, but their communication, reasoning, problem solving, understanding and fluency, together with their interest and effort. The strategies also aim to increase students’ metacognitive awareness. By selecting and using strategies effectively, teachers teach students to reflect on their learning and set goals to improve their work. Teachers also set time aside for students to engage in the type of purposeful practise required to develop fluency and mastery in mathematics.
Early impacts of the project
Upon commencing the Proficient Course, teachers complete a mathematics teacher efficacy survey to provide baseline data. They then complete a survey each term regarding their confidence in relation to different aspects of teacher knowledge that have been addressed through the professional learning at that point in time. We now have the data from teachers in the first two cohorts completing the Proficient Course, observations of over 600 teachers in action in classrooms and survey data from the students in these classrooms.
When examining teachers’ ratings for the survey items that are repeated each term, there are notable shifts in teachers’ confidence in articulating clear learning goals, designing cognitively challenging tasks with a low entry point and high ceiling and writing differentiated success criteria. The chart that follows illustrates changes in teachers’ ratings for designing tasks across the four terms of professional learning in the Proficient Course across 2018 and 2019. All items use a rating scale from 1 (I cannot do this) to 10 (I am 100% confident in doing this). The chart highlights the variability in teachers’ confidence for designing tasks at the start of the course (ratings from 2 to 10), as well as increased confidence in designing tasks by the end of the course, with 83% of teachers rating their confidence as 8, 9 or 10 out of 10.
Student surveys reveal that most students in the classrooms of the teachers completing the Proficient Course perceive themselves as successful and capable in mathematics. Most students in the classrooms of teachers who have had one or more years of professional learning as part of this project, enjoy learning mathematics and look forward to it each day. Students’ perceptions are supported by other sources of data, including NAPLAN numeracy and standardised assessments such as the ACER Progressive Achievement Tests (PAT-M).
In the Learning mathematics student survey, designed for the purposes of the project, the final question is an open response item asking students to describe how they learn mathematics in their school. In the pre project survey, the average number of words per student response is just four words, with the most common responses being generic ones such as, ‘I don’t know’. By comparison, students’ responses to the same survey after their teacher has completed the Proficient Course demonstrate confidence and preparedness to articulate how they learn mathematics in their class, with most students writing a short paragraph of around 4 to 5 lines. The following response from a Year 6 student in 2019 captures many of the sentiments commonly expressed by students of the same age:
In my class we learn maths very openly, like we are a team. Our teacher will sit and explain the goal for the lesson. We read through the task together and talk to the person next to us to share what we are thinking and get some ideas. After this we go and do our work. Some students choose to work on the whiteboard and others like to work in their books. Some students prefer to work alone, and others like to work in a group. Sometimes we get to go on a gallery walk and share our work with others. When we are stuck, this helps us to see what others might do to solve the same problem. After we finish the lesson, we talk about what we did well and what we need to work on. I love learning like this so much better than the other way in maths. I would love to learn this way in high school.
The shift in students’ enjoyment of mathematics is paralleled by increases in teachers’ knowledge for teaching it. In 2019, the mean rating of around 400 teachers in the Proficient Course regarding the extent to which their knowledge for teaching mathematics had increased, was 9.2 on a scale from 1 to 10. In open response items, teachers report that they have never previously understood the syllabus, or the progression articulated by it, as well as they do now. Increases in knowledge are helping teachers to design and implement effective, engaging lessons that respond to where each student is in relation to syllabus outcomes. Importantly, mathematics teaching practice has become a shared endeavour, characterised by reflective, collaborative teachers and leaders who openly model, observe, and give and receive feedback to improve the teaching and learning of mathematics for all. The learning culture among teachers is mirrored in classrooms where we are realising the belief that it is possible for every student to struggle, grow, succeed and contribute to the learning of the others, even though their starting points for learning on any given day or topic may vary.
Perhaps the most important early impact of the project has been changes in teachers’ dispositions to mathematics. Initially, many teachers were anxious about ‘doing mathematics’ as part of the course because their own education had led them to fear mistakes. Through professional learning, teachers see themselves as problem solvers who can solve mathematics problems, as well as teaching problems, making the work of the project positive and productive. We know this because we gather baseline data prior to teachers commencing the professional learning and then monitor changes in teachers’ efficacy ratings throughout the project. The following comment, reflects sentiments commonly expressed by teachers as a consequence of the Proficient Course, regardless of their teaching experience:
I am teaching mathematics with a fresh pair of eyes. When my class fist pumps and cheers when it is maths time, I definitely know that my teaching practice has improved for the better. The tasks we use to stimulate learning are challenging yet fun, and every single child in my class is able to experience success at their own level and pace. I feel much more confident with the content and even more so now that I can make those seemingly obvious, but previously unnoticed, connections between the content strands. This is because I am no longer afraid of the mathematics and because of this I can respond flexibly and creatively to each student.
The primary project is comprehensive. Yet, it is only a starting point for increasing the proportion of students studying higher levels of mathematics and maximising the levels of numeracy attained by all students. Ultimately, these aims will be achieved when every student experiences mathematical expertise and excellence in each successive year of their educational journey. With a critical mass of students impacted by the primary project now moving into secondary schools, we need to sustain their enthusiasm, success and readiness to engage in challenge, practice and mastery.
References and further reading
Australian Institute for Teaching and School Leadership (AITSL). (2014). Australian Professional Standards for Teachers. 1st ed. Melbourne: AITSL.
Ball, D. L., Hill, H.C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1).
Bynner, J. & Parsons, S. (2006). New light on literacy and numeracy. London: National Research Development Centre for Adult Literacy and Numeracy.
Kilpatrick, J., Swafford, J., Findell, B. (Eds) (2001). Adding It Up: Helping Children Learn Mathematics. Washington DC: National Academies Press.
Mae, C. (2019). In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics Education Research: Impacting Practice (Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia) pp 476-483. Perth: MERGA.
National Council of Teachers of Mathematics (NCTM). (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM, National Council of Teachers of Mathematics.
NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales. (2012). Mathematics K-10 syllabus.
Sullivan, P., & Australian Council for Educational Research. (2011). Teaching Mathematics: Using Research-Informed Strategies. Camberwell, Vic: ACER Press.
Sullivan, P., Borcek, C., Walker, N. & Rennie, M. (2016). Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks. The Journal of Mathematical Behaviour. 41, 159-170.
Thomson, S., Hillman, K., Wernert, N. (2012). Monitoring Australian Year 8 Student Achievement Internationally: TIMSS 2011. Melbourne: Australian Council for Educational Research (ACER).
How to cite this article - Mae, C. (2020). ' A vision for mathematical expertise and excellence', Scan, 39(6).