Knowledge of mathematics learning (archived)

Knowledge of mathematics learning is one of a set of professional development modules designed to support school leaders as they lead professional learning about the National Standards for years 1–8 within the New Zealand Curriculum. The modules are suitable for use during the cycles of professional inquiry that leaders and teachers engage in to improve outcomes for their students.

See Knowledge of literacy learning, which is an equivalent module for literacy.

Introduction to the module
Key outcome of the module
Reflective questions for school leaders and teachers
Leading shifts in practice through focused activities
Resources and references

Introduction to the module

The focus of this module

This module focuses on the knowledge of mathematics, and of mathematics teaching and learning, that teachers need in order to work with the mathematics standards for years 1–8.

The structure of this module

This module has three main sections.

Key outcome of the module, which:

• states what the module aims to help schools leaders and teachers achieve
• lists indicators that describe what to look for as evidence that they have achieved the outcome
• provides a rationale for the key outcome.

Reflective questions for school leaders and teachers, which:

• helps determine the professional learning needs of the whole staff, syndicates, or individual teachers or leaders
• can be used within activities for leaders and teachers (see next section).

Leading shifts in practice through focused activities, which:

• outlines some professional development activities that relate to the reflective questions
• can be used flexibly to help meet identified needs
• draws on existing resources and professional development opportunities.

A final section, Resources and references, lists texts cited or quoted in the module along with resources that include useful information about mathematics learning.

How to use this module

School leaders can use this module to identify and explore shifts in practice that might be needed as their school works with the National Standards.

Teachers can use the reflective questions and/or activities to guide them through any changes they might need to make as they work with the National Standards.

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Key outcome of the module

The key outcome for this module is that school leaders and teachers develop a shared understanding of the mathematics demands of their school curriculum and of how these reflect the expectations of The New Zealand Curriculum and the mathematics standards.

Indicators

Indicators that this outcome is being achieved include the following:

• Teachers are developing the content and conceptual knowledge they need in order to support mathematics learning at all levels of the school and across the curriculum.
• Teachers and leaders can articulate the key concepts and characteristics of learning and teaching in mathematics and statistics for a range of year levels in their school.
• School documents, assessment, and classroom practices consistently support students with building their understanding of mathematics and statistics as they progress through the year levels.

Rationale for the key outcome

The central role and pervasive nature of mathematics in all aspects of life demand a sustained focus on professional growth for teachers and school leaders. To support students’ learning within their school context, teachers need to be secure in their knowledge of mathematical content and familiar with how it is organised within the school curriculum. They need to know how to generate learning opportunities to enable their students to make useful connections between the mathematical and statistical ideas they encounter. Depth of teacher knowledge significantly impacts on classroom teaching:

Enhancing teachers’ understandings about teaching and learning mathematics also enables them to gain confidence in this area and to address underlying misconceptions, producing better outcomes for students.

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Reflective questions for school leaders and teachers

The following reflective questions are designed to help school leaders and teachers understand their school’s current practice in relation to mathematics learning.

1. How secure are teachers in their mathematical content knowledge? How well can they demonstrate proficiency and articulate key concepts within each strand at each level?
2. How well do teachers understand how best to present, support, and develop mathematical ideas with diverse learners? Are they aware of misconceptions that may develop and of how to build connections between mathematical ideas? Which areas require further development within our school?
3. Do teachers regularly use reflecting-in-action to support quality mathematics teaching? How can we encourage teachers to focus on specific areas of knowledge in order to generate and support rich mathematical learning opportunities?

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Leading shifts in practice through focused activities

Select activities that will help to deepen understandings about mathematics teaching and learning. Further exploration may be needed to reach the outcome for this module. For example, discussions may reveal a need to explore a specific aspect of mathematics learning in greater depth.

The activities can be used in a variety of ways for whole-staff, syndicate, group, or individual inquiry.

The activities in all the modules, including this one, are based on the core resources listed in the Overview. Refer to these as appropriate when exploring practice through the activities.

Activity 1: Content knowledge

1. Choose one of the three curriculum strands as a focus: Number and Algebra, Geometry and Measurement, or Statistics.

• Select an activity from Figure It Out that lies within the chosen curriculum strand. To find activities, use the resource database on NZmaths.
• Describe some possible key concepts for students implicit in this activity. To do so, it may help to identify the knowledge and strategies that students need in order to complete the activity and any misunderstandings that may arise while they do so.
• Explore the learning progressions for one of these key concepts by looking through the different levels of the curriculum and discussing how the concept is developed.
• Consider other concepts in mathematics, and from other learning areas, that are related to this key concept.

2. Turn to the mathematics standards. Find the standard that matches the curriculum level of the Figure It Out activity.

• Which expectations (statements) within the standard link to the key concept? How do these expectations develop through previous and subsequent standards?
• How does our school’s mathematics programme address these expectations from the standards? Do the expectations in our programme and the standards co-incide at the different year levels?
• As a result of this activity, what changes or additions might we make to planning documents, types of tasks, and teaching and learning support? What changes or additions might we make to our own learning to ensure a closer match to the expectations of the curriculum and the standards?
• What barriers to teaching this concept do we foresee? How will we deal with these?

Activity 2: Pedagogical content knowledge

1. Using NZmaths, choose a rich mathematical task or activity to explore as a group of teachers.

• Individually, complete the task or solve the problem. Share your strategies with a partner.
• Analyse the knowledge and strategies required to solve this problem. Consider your own approaches as well as drawing on the example provided.
• Consider other forms of knowledge that may come into play as students engage with the task, for example:
  • general knowledge
  • prior experiences
  • knowledge of other strands
  • cultural and sociocultural knowledge
  • knowledge of language, text, or diagrams.

2. Using the accompanying teacher support material for the activity, discuss how you could use this activity in class.

• What mathematical ideas – and misconceptions – might emerge for students as they engage with the activity?
• What related activities or resources could we use to support this learning?
• How could we extend or develop this activity to create further learning opportunities?
• How would we make links to other learning or to relevant mathematical ideas?
• How can we ensure that the mathematics knowledge and understandings of various cultural groups within our classes are recognised, respected, and valued as students undertake the activity?
• Are we confident that we have the content knowledge needed to encourage productive discourse and critical thinking? If not, how could we develop it?

Activity 3: Reflective practice – using evidence to improve teaching

This activity helps teachers to establish individual or collaborative inquiries into aspects of their teaching practice and to reflect on their pedagogical content knowledge.

• Identify a focus for inquiry that has emerged from discussions during activities 1 and 2. To refine the focus, refer to the Dimensions of Quality Teaching (described in the Numeracy Development Projects, Book 3: Getting Started, pages 3–4) or the effective classroom example on the NZmaths website.

• Analyse a short episode (approximately ten minutes) of your teaching and learning in the classroom. To do the analysis, use videotaped footage, transcripts, or a colleague’s observation notes of the episode. Make sure that directly after the episode, you and/or the observer talk with students to check their understanding and to gauge the impact of your teaching. Before the observation, establish a focus for the data gathering (relevant to the focus of the inquiry), negotiate the process with the observer, and arrange for a feedback session after it. You may wish to use or adapt: Table 1: Observation of mathematics learning (Word, 44 KB)

• Following the observed or recorded lesson, note your initial responses in relation to the focus. Then, working with a colleague, use the observation notes, transcripts, or footage and the notes from discussions with students as artefacts to provide evidence relating to the focus. Discuss the extent to which your understanding of the mathematical ideas helped you to provide opportunities for deepening students’ engagement with key concepts and to support a 'press for understanding'. (The press for understanding is the insistence by the teacher that students engage cognitively. For further discussion on press for understanding, see Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration (BES), page 120.)

• Reflect on the content and pedagogical content knowledge that informed your actions, in particular, the decisions you made 'in the moment'. What further knowledge could have helped you to deepen the students’ understanding? Note the ways in which you will use this learning to inform your future practice.

Activity 4: Reflective practice – school-wide coherence

As a staff or in groups, ask:

• Are we well prepared (at whole-school, syndicate, and class levels) to support all of our students to meet the mathematics standards? For example:
  • How proficient are we in our own understanding of the mathematical concepts we teach?
  • How often do we reflect on our own knowledge and practice as it relates to student outcomes?
  • How can we best use the expertise within the school to support teachers who want to develop their mathematical knowledge?

• What do we know about the home environments and numeracy practices and experiences of our students? How can we build on our knowledge to support students’ mathematics learning at school?
• How do we ensure common understandings and approaches to teaching and learning in mathematics across the school?

For a useful example, refer to Learning stories, story 5 in the QTR&D learning materials on the New Zealand Curriculum webite.

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Resources and references

This section includes details of texts that are cited or quoted in the module and/or that will be helpful to users of this particular module. The full list of core resources is available in the Overview.

• Anthony and Walshaw (2007). Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration (BES). Wellington: Ministry of Education.
• Ferguson, P. B., Gorinski, R., Wendt Samu, T., and Mara, D. (2008). Findings from the Literature review on the experiences of Pasifika learners in the classroom. Wellington: Ministry of Education.
• Macfarlane, A. H. (2004). Kia Hiwa Ra! Listen to Culture – Māori Students’ Plea to Educators. Wellington: New Zealand Council for Educational Research.
• Ministry of Education (Pasifika Education). Connections and Conversations: Making Links for Learning. Wellington: CWA New Media Limited.
• Numeracy Development Projects, Books 1–9. Wellington: Ministry of Education.
• Picking up the Pace
• Robinson, V. and Lai, M. K. (2006). Practitioner research for educators: A guide to improving classrooms and schools. Thousand Oaks, CA: Corwin Press.

Effective practice

Professional learning modules

• Overview
• Establishing shared expectations
• Building effective partnerships
• Knowledge of the learner
• Knowledge of literacy learning
• Knowledge of mathematics learning
• Engaging learners with texts
• Engaging learners with mathematics

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