Mark Waters and Pam Montgomery are both Numeracy Leaders in Hume Region, DEECD, Victoria. They have worked extensively in numeracy education in Victoria and the USA. They wrote and presented the principals’ training described in this article: DEECD Hume Region ‘Common Curriculum: Leading Numeracy’ Modules 1 & 2.

In August last year, 167 principals from the Hume Region of the Department of Education and Early Childhood Development (DEECD) gathered in Shepparton in north-eastern Victoria to participate in the first of twelve modules of leadership training entitled the Common Curriculum. This is the Hume Region’s strategy to reaffirm principals as instructional leaders in their schools. Over two years, all principals from the region will participate in eight one-day workshops devoted to educational leadership, and four one-day workshops dealing with transformational leadership. Each workshop will be followed by school-based action and reflection. Core leadership training for principals was originally proposed in 2006 by Stephen Brown, Hume Regional Director. This proposal was endorsed by Hume Region principals, and designed by Stephen Brown and Leon Kildea. The Common Curriculum was launched in 2007.

Leading Numeracy: The Learner

The first two educational leadership modules deal with numeracy education. Each module consisted of presentations by Hume Region Numeracy Leaders Mark Waters and Pam Montgomery, followed by small group discussions among principals arranged by school clusters, and question and answer sessions guided by Mark and Pam.

The theoretical underpinning for both of the Leading Numeracy modules was drawn from How People Learn (Bransford, Brown & Cocking cited as The Skills and Knowledge of an Effective Teacher in Professional Learning in Effective Schools, Department of Education & Training, Victoria, 2005), and Breakthrough (Fullan, Hill, & Crévola, 2006).

This paper covers the content from Module 1, Leading Numeracy: The Learner (DEECD Hume Region, 2007). Module 1 was devoted to examining how effective teachers assist individual learners to become numerate.

Drawing out the pre-existing understandings of the learner

As Bransford, Brown and Cocking argue, effective teachers draw out and work with the pre-existing understandings that their students bring with them. To illustrate this process, principals were introduced by video to Alex, a Year 7 student. They first examined his numeracy test scores from a pen and paper test, and were asked to predict what Alex understood in numeracy. This highlighted that many assessment tools will rank students, but do not inform classroom action.

Principals then watched and listened to Alex as he participated in a one-to-one ‘interview’ assessment. This assessment required him to solve a range of mental calculations and verbally explain his thinking strategies. Snippets of the transcript from Alex’s interview follow:

Teacher: Start at 273. Count forward by 10s.
Alex: 283, 293, 203, I mean 303, 313, 323 … (fluently)
Teacher: Start at 336. Count backward by 10s.
Alex: 336 (pause) … 326, 316 (pause) … 326, 316, 306 (pause) … I’m going down by 10s aren’t I? (pause) … 394, (pause) …388 (pause) … 378 (pause) … 368.
Teacher: Add 32 and 99.
Alex: (Pause) … 131. You take one from the 32 and add it to the 99 to make 100.
Teacher: 101 subtract 26.
Alex: (Pause) … 85. Make it 100 and take away 25. That’s 80 (pause) … it’s 85.

In small group discussions, principals were then asked to analyse Alex’s numeracy understandings and to situate his performance on a developmental pathway, or continuum of numeracy development. This flagged that some assessment tools do inform classroom action, and these can be used by teachers to plan for student learning.

Principals were asked to reflect on numeracy assessment practices at their own schools. Some responses included:

• ‘Alex’s multiple choice test gave us his correct answers – which he may have guessed – but observing him solving some maths problems showed what he really could and couldn’t do.’
• ‘This was like taking a running record for maths. You get to see how a particular student figures out maths problems – you don’t see that thinking with a regular maths test.’
• ‘We put a lot of time into individual literacy assessment at our school, but we don’t put similar time and effort into finding out what our individual kids understand in numeracy.’
• ‘We do a lot of numeracy assessment but the results sit on the shelf and we don’t really use these to plan tomorrow’s teaching.’

The implications for future school improvement were to consider more effective numeracy assessments that provide teachers with rich information about students’ understandings, that reveal students’ thinking strategies, and that can be used to inform teachers’ day-to-day lesson planning.

Teach the learner in depth with a firm foundation of factual knowledge

The identification of the learner’s pre-existing understandings is essential to inform subsequent teaching. This enables the teacher to pinpoint the student’s zone of proximal development (Vygotsky as cited in Fullan, Hill & Crevola, 2006, p 33), the level of instruction that is ‘just right’ for this student. Principals were reminded that this zone is unique to each learner, and cannot be assumed to fall within the curriculum level assigned to the student’s ‘year level’ at school.

Within this zone for teaching, the teacher needs to design learning tasks that embody the numeracy concept in a variety of rich ways. ‘Effective teachers teach some subject matter in depth, providing many examples in which the same concept is at work and providing a firm foundation of factual knowledge.’ (Bransford, Brown & Cocking, cited as The Skills and Knowledge of an Effective Teacher in Professional Learning in Effective Schools, Department of Education & Training, Victoria, 2005). Teachers also need to provide the student with sufficient practice with ‘just right’ learning tasks to develop fluent understanding.

Principals watched Joel, a Year 5 student, on video. His teacher had conducted interview assessment to determine Joel’s numeracy understandings, and had found that Joel had some difficulty bridging hundreds when adding 10 to any number. His teacher designed a task that Joel practised with Base 10 blocks. He used a starting number such as 251 and then continually added a ten (‘rod’ or ‘long’ block) and named the new number. This task built both conceptual understanding and visual imagery. The teacher showed Joel that the ‘ones’ digit remained unchanged, the ‘tens’ digit increased each time, and the ‘hundreds’ digit increased only when the ‘nineties’ moved into a ‘new hundred’.

On video the principal group observed Joel working on a follow-up task called Predict-a-count, designed to build Joel’s fluency with bridging the hundreds (Williams, 1987, p 67). Joel used playing cards to generate his starting number, eg 479, and then progressively added 10. Each time Joel predicted and wrote the next number, he confirmed his prediction with a calculator. As Joel practised this, his teacher watched and strategically prompted when necessary, for example: ‘Which digit won’t change?’ and ‘What will be the trickiest part?’.

In small groups, principals then worked through several numeracy tasks designed for students to build fluency with numeracy. Each task used a random generator such as dice or playing cards to generate examples to practice. This method allows students to continually repeat the task without having to depend on the teacher for examples to practise. Each task used a model that provided conceptual support for the student, such as a tens frame or a number line. And each task had mechanisms that allowed the student to check their own results, such as a 100s grid, tape measure or calculator. The point was made that tasks such as these must be matched to individual students based on their learning needs. This ‘close matching’ is a practical example of instructional precision and personalisation called for by Fullan, Hill and Crévola (2006).

Principals were once again involved in reflecting on numeracy practices at their schools. This time the focus was on the ability of their teachers to select conceptually rich instructional tasks that closely match the needs of an individual learner, and to effectively scaffold the numeracy learning with that student. Some responses included:

• ‘Our teachers often select tasks just because they are part of the year level curriculum, not because they fit the learner’s zone of numeracy learning.’
• ‘In literacy we put a lot of thought into matching texts to individual students. This has shown me that we need to think about how to more accurately match numeracy tasks to individual students.’
• ‘It was fascinating to watch a skilful teacher customise a task that was ‘just right’ for a student, and then strategically draw attention to the most relevant teaching points.’
• ‘Explicit teaching of numeracy concepts and strategies is a real challenge for my teachers. Their own confidence in mathematics is low. I think we need to do a lot of ‘upskilling’ in numeracy teaching.’

The implications for future school improvement were to consider the quality of numeracy instruction in classrooms. Teachers need to be well equipped to teach numeracy in depth, with attention to developing students’ conceptual understanding and mathematical strategies. They need to understand how to closely match learning tasks to individual students, and know how to provide students with purposeful numeracy practice to develop fluency.

Develop independent learning

Bransford, Brown and Cocking point out that effective teachers focus on the teaching of metacognitive skills, integrating those skills into the curriculum in a variety of subject areas (cited in The Skills and Knowledge of an Effective Teacher in Professional Learning in Effective Schools, Department of Education & Training, Victoria, 2005). This can be characterised as developing independent learning.

Typically, mathematics teaching has not encouraged students to develop independent learning. Instead, the teacher sets the goals and tasks, the students complete given exercises, and the teacher looks for students’ mistakes and then corrects their attempts. Yet the work of Bransford and others tells us that effective teachers help students to reflect on their thinking, plan their own work, monitor their own understanding, and evaluate their own progress.

To illustrate the teaching of metacognitive skills, principals viewed a final video of Joel’s teacher working one-on-one with Joel practising a numeracy task. Attention was drawn to the way the teacher asked the student to reflect on their learning through questions such as:

• ‘What maths are you learning by doing this?’
• ‘What did you think about to help you solve this?’
• ‘How could you check your answer?’
• ‘What did you do really well?’
• ‘How will you know that you’re getting better at this?’

In this context principals were asked to reflect on numeracy teaching at their schools. This time the focus was on the ability of teachers to help students to monitor their own numeracy learning. Some responses included:

• ‘This really made me think about better, thinking-based ways to get kids to check the sense of their own work – not just looking up answers in the back of the textbook.’
• ‘It was impressive to hear students saying what maths they were learning, and why. Some of my teachers clearly explain the ‘how to’ steps in the activity but they don’t actually tell the kids what maths they’re going to learn from this.’
• ‘Lots of my teachers take great "hands-on" maths lessons but they don’t bring the students back to reflect on what they have learned.’

The implications for future school improvement were to consider the integration of metacognitive skills within numeracy learning. This requires a shift towards students knowing what they are learning in numeracy, students selecting appropriate tools for solving problems, students managing their own numeracy tasks, and students self-correcting when strategies or answers appear suspect.

Action and reflection

This first module of the Common Curriculum engaged principals in thinking about the quality education that each individual learner deserves in numeracy. It raised awareness that the mathematics curriculum must be personalised around individual students’ learning needs rather than be delivered at a set level determined solely by a student’s age. This means that educators need to:

• find out what mathematics each student knows
• teach in-depth mathematics content that is precisely matched to each student’s zone of learning, and
• integrate independent learning skills into numeracy instruction.

In collegiate groups, all Hume Region principals rated their own school’s performance in these three areas listed above, and then discussed the gap between current and ideal practice. For each principal this highlighted particular school improvement actions in numeracy that would be undertaken with their staff, and would form part of their own Performance Plan.

To reinforce learning from the module, principals committed to making classroom visits at their school (or cluster of schools) during numeracy instruction. They would look for evidence of teachers using students’ pre-existing understandings to inform their instruction, of teachers striving to provide conceptually rich and personalised numeracy instruction, and of teachers developing independent learning during numeracy. To gain evidence, principals were asked to discuss the mathematics learning with various students in class, and to take photos of teachers and students as they worked together on numeracy.

Findings from these classroom visits were shared when principals reconvened for the second DEECD Hume region Common Curriculum module, Leading Numeracy: The Classroom. This module covered how to apply the pedagogy in a class of 25 learners, and will be described in a future edition of Curriculum Leadership.

References

Bransford, J, Brown, A & Cocking, R (Eds) 2000, How People Learn, National Academy Press, Washington DC.

Department of Education & Training Victoria 2005, Professional Learning in Effective Schools, Department of Education & Training, Victoria.

Fullan, M, Hill, P & Crévola, C  2006, Breakthrough, Corwin Press, Thousand Oaks, California.

Hume Region 2006, Leadership Capacity Building in the Hume Region, discussion paper, Hume Region Department of Education and Early Childhood Development, Victoria.

Hume Region 2007, Leading Numeracy: The Learner, Hume Common Curriculum Module 1, Hume Region Department of Education and Early Childhood Development, Victoria.

Hume Region 2007, Leading Numeracy: The Classroom, Hume Common Curriculum Module 2, Hume Region Department of Education and Early Childhood Development, Victoria.

Vygotsky, L, cited in Fullan, M, Hill, P & Crévola, C 2006, Breakthrough, Corwin Press, Thousand Oaks, California.

Williams, D 1987, Calc.U.Lator: Your Classroom Companion, Oxford University Press, Melbourne.