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Curriculum & Leadership Journal
An electronic journal for leaders in education
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Leading numeracy: the classroom

Mark Waters
Pam Montgomery
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.

Under the leadership of Stephen Brown, Regional Director, all primary and secondary principals from Hume Region DEECD Victoria are currently participating in leadership training entitled Common Curriculum (Hume Region, 2006). This program is designed to reaffirm principals as educational leaders of their schools, equipping them to guide teaching staff in the delivery of learning. Two modules focus on numeracy: Leading Numeracy Module 1: The Learner, and Leading Numeracy Module 2: The Classroom. A recent article in Curriculum Leadership reported on the content of Module 1. This article describes the content of the second module – Leading Numeracy: The Classroom.

Module 1 was based on the practices of effective teachers:

  • Drawing out the learner’s pre-existing understandings;
  • Teaching appropriate subject matter in depth to promote connected learning; and
  • Integrating metacognitive processes.
    (Bransford, J., Brown, A., & Cocking, R., 2000.)

These practices enable teaching to be precise and personalised when working with an individual learner (Fullan, M., Hill, P., & Crévola, C., 2006). Module 2 Leading Numeracy: The Classroom translated this personalised learning to a classroom of learners.

Drawing out the pre-existing understandings of all learners

Teachers need to conduct rich assessment for learning to establish what each student already knows, and to determine the ‘next step’ for each student. This means using assessment tools that give descriptions of student understandings rather than just numerical scores. Interview assessments – where students explain their strategies for solving numeracy problems – are preferred, as they give teachers rich insights into each student’s thinking (e.g. the Mathematics Online Interview, 2007).

During the training, principals were shown how to plot the assessment results of all students from one class on a developmental pathway to form a class profile (see Figure 1). The class profile shows the range of understanding, and also natural groupings of students with similar knowledge levels. With this range in mind, teachers can then select or design an appropriate lesson.

When teachers select a mathematics lesson they should ask, “To be able to do this lesson, what understandings would students need?” This should be compared to the class profile to see the ‘match’ between student understandings and the proposed lesson (see Figure 1). Teachers can then ask, “Which of my students is this lesson ‘just right’ for? Which of my students is this too hard for? Which of my students is this too easy for?” For example, in Figure 1, a proposed lesson on ‘doubles strategies’ in number is matched to the students from this class profile. The lesson matches only nine of the students. If the lesson is not modified, it will be too easy for four students, and too difficult for fourteen students.

Not apparent

Count all

Count on

Count back

Basic strategies

Derived strategies

Multi-digit strategies

Not yet able to combine and count two collections of objects

 

Counts all to find the total of two collections

Counts on from one number to find the total of two collections

Given a subtraction situation, chooses from strategies including count back, count down to and count up from

Uses strategies such as doubles, commutativity, adding 10, tens facts, and other known facts

Uses strategies such as near doubles, adding 9, build to next ten, fact families and intuitive strategies

Can solve multi-digit calculations mentally, using the appropriate strategies

 

Jack

 

Ali
Hayley
Ruby

Luis
David
Trent
Amy
Candice
Arika
May Ling
Sandeep
Jasmine
Paul

Sergio
Katelyn
Jorge
Tristan
Ben
Vanessa
Josh
Tyshawn
Mira

Liam
Serena
Ahmed
Trina

 

 

            Figure 1  Lesson prerequisites matched to a class profile for Addition and Subtraction Strategies. The highlighted yellow column indicates the level of understanding needed for a doubles strategies’ lesson in number. Adapted from Early Numeracy Research Project Framework, 2002

Principals discussed assessment for learning practices at their schools. This resulted in the following realisations:

  • Many of our schools use numeracy assessment tools, but the results give numerical scores that only rank students into ‘high, medium, and low’ groups. We need to use assessment tools that describe the numeracy understandings of students.
  • A class profile mapped onto a developmental pathway provides a clear way to decide on appropriate lessons. Rather than selecting a lesson based on the students’ year level, teachers would use the class profile to choose lessons that more closely match their learners’ needs. Very few of our classrooms currently develop class profiles to inform their planning. 
  • Constructing class profiles and then matching numeracy lessons to the profile is a technique that could be learned by teachers. This would assist them to select appropriate numeracy lessons.

Teaching all learners in depth with a firm foundation of factual knowledge

Module 1 (Leading Numeracy: The Learner) raised the need to teach numeracy in depth, attending to conceptual development, strategies, visualisation, and relationships. In Module 2, principals were shown how to apply this to a classroom of learners with diverse mathematical understandings. Principals viewed a video of Tony, a primary teacher, as he worked with his class.

Tony was conducting a series of lessons to explore relationships between addition, multiplication, and division. He chose to use an investigative lesson from Maths 300 entitled ‘Consecutive Sums’ (Curriculum Corporation, 2004). Tony wanted all his students to investigate adding three consecutive numbers, and to link this to multiplying the central number by three. He eventually wanted his students to consider a given number and determine which three consecutive numbers would add to give that number.

Tony had previously conducted interview assessments with all of his students to determine their pre-existing understandings in number, and had plotted this to form a class profile. He then matched the requirements of the ‘Consecutive Sums’ lesson to his class profile by asking, “To be able to do this lesson, what understandings would students need? Which of my students is this lesson ‘just right’ for? Which of my students is this too hard for? Which of my students is this too easy for?”

Tony found that this lesson would only match to the highest achieving students in his class, so he differentiated the content of the lesson to more closely match his class profile. He made the following decisions:

  • To conduct the lesson as a whole class investigation with all students generating their own examples of consecutive numbers that illustrate the relationship between addition, multiplication and division;
  • To seat his students in three homogeneous groups and give a different number range for each group to work within; and
  • To have one group work mentally, to have one group use number lines and the 100 chart to assist their calculations, and to provide connecting cubes to one group for constructing physical models.

Through viewing Tony’s lesson, principals were introduced to the range of tools that teachers can use to differentiate a mathematics lesson, such as small group instruction, use of open-ended tasks, and strategic use of concrete and visual supports. These differentiation tools enable teachers to ‘stretch’ lessons to fit the class profile more precisely.

Attention was also directed to the many ways that Tony promoted deep understanding. Throughout the lesson he:

  • Had students clarify mathematical vocabulary such as ‘consecutive numbers’;
  • Encouraged students to discuss and share their strategies for calculation;
  • Drew attention to efficient mental calculation strategies;
  • Built on students’ ideas and observations about consecutive numbers;
  • Prompted students using inefficient strategies to “Try that a quicker way”; and
  • Worked with each group to explicitly promote links between addition and multiplication.

Following focused observation of Tony’s lesson, principals discussed current teaching practices at their schools. This raised the following points:

  • Many teachers select lessons based solely on the students’ year level. It is quite a shift in teacher thinking to select lessons based on their students’ pre-existing knowledge.
  • It is an expectation that our teachers differentiate instruction in literacy. This needs to also become an expectation for numeracy instruction.
  • Many teachers believe they currently differentiate mathematics lessons by asking students to complete less exercises on a worksheet, or by sitting a struggling student with a more competent student for assistance. Few teachers actually differentiate by focusing on strategy development, modifying the lesson content, or providing concrete and visual supports to more closely match student abilities. The differentiation tools presented would assist teachers to develop skill in personalising numeracy instruction.

Integrating metacognitive processes for all learners

Effective teachers help students to reflect on their thinking, plan their own work, monitor their own understanding, and evaluate their own progress (Bransford, Brown & Cocking, 2000). Module 1: The Learner had previously shown how effective teachers help individual students to understand these metacognitive processes. In Module 2, participants viewed a video of Laura, a secondary teacher, as she involved each of her students in setting individual learning goals for numeracy. The students explained what they were learning and how various tasks helped to develop these understandings. They collected and conducted the tasks themselves whilst Laura roved and assisted various individuals. ‘Self-checking’ devices such as calculators were used by students to check their own solutions. As she roved, Laura asked students to evaluate their progress, and to decide when to move to more challenging numeracy tasks.

Discussion indicated that principals currently see great diversity in teacher practice in this area, ranging from teachers who continually involve students in metacognitive processes in numeracy, through to teachers with very low knowledge of how to teach students to self-manage and self-monitor their numeracy learning.   

Pathways for numeracy improvement

The final section of Module 2: The Classroom outlined a range of tools and processes for numeracy improvement. These included:

  • Professional Learning Teams

Teachers meet to discuss students’ numeracy performance and generate ‘best practice’ ways to teach numeracy. Lesson study (Stigler & Hiebert, 1999), action research and case discussions are ideal models to use within professional learning teams.

  • Numeracy Walks

This process is based on the LearningWalk (Institute of Learning, 2007). Numeracy teachers nominate an area for numeracy improvement, and a small group of teachers (including the Principal) visits several classrooms to look for evidence within classroom practice. Findings are shared with the whole staff.

  • Indicators of Effective Classrooms

This is a menu of indicators of effective numeracy teaching being developed and trialled by Hume Region. It can be used either for schools to select areas for whole school numeracy improvement, or for teachers to select a goal for their own teaching.

  • Numeracy Observable Practices & School Numeracy Improvement Tool

These are rubrics being developed and trialled by Hume Region describing teacher practices in numeracy. Teachers can locate their current practice on the rubric and identify descriptions of more effective practice. These can then be used for goal-setting.

School based action

Common Curriculum: Leading Numeracy training for principals has been the first step in aligned numeracy improvement across Hume Region. This has promoted an awareness of the need to improve numeracy teaching, developed shared language, and provided improvement tools and processes for schools.

All Principals have included numeracy improvement within their own Performance Plans, and are embarking on numeracy improvement projects within their schools. Hume Region is supporting the implementation of these numeracy improvement projects by providing Common Curriculum training for Numeracy Leaders in all schools.

References

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

Clarke, D., Cheeseman, J., Clarke, B., Gervasoni, A., Gronn, D., Horne, M.,   McDonough, A., Montgomery, P., Rowley, G., Sullivan, P. (2002). Early  Numeracy Research Project Final Report Executive Summary. http://www.education.vic.gov.au/studentlearning/teachingresources/maths/enrp/execsu      mmindex.htm

Department of Education & Training, (2005). Professional Learning in Effective Schools. Department of Education & Training, Victoria. www.sofweb.vic.edu.au/blueprint/fs5/default.asp

Department of Education & Early Childhood Development, (2007). Mathematics Online Interview. Department of Education and Early Childhood Development: Victoria.http://www.education.vic.gov.au/studentlearning/teachingresources/ma  ths/assessment.htm#2

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.

The Institute for Learning, University of Pittsburgh (2007). The LearningWalk. http://ifl.lrdc.pitt.edu/ifl/index.php?section=learningwalk

Maths300, (2004). Lesson 139, Consecutive Sums. Curriculum Corporation. http://www1.curriculum.edu.au/maths300

Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: Free Press.

Key Learning Areas

Mathematics

Subject Headings

Mathematics teaching
Professional development
School principals
School leadership
Victoria