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Using “stories of practice” as a mathematics leadership tool

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Retrieved from https://unsplash.com/s/photos/stories

This post explores “stories of practice”, specifically, mathematics leaders’ own stories of mathematics teaching and positions them as a leadership tool for supporting practice development.

Practice development is the intentional and collaborative work of improving mathematics teaching in ways that stay close to “what matters” most. For mathematics leaders, it means critically examining and transforming the ways we lead, teach, and learn mathematics, and it’s about surfacing “what’s not there yet” with mathematics teaching practice. Leading practice development involves co-creating new, research-informed teaching approaches with colleagues to enhance the impact of mathematics teaching.

I believe that “stories of practice” have the potential to serve as a powerful leadership tool to support the leadership of practice development.

Being close to practice: an excellent opportunity!

As mathematics leaders in primary schools, you occupy a valuable space as you enact middle leadership, working between classroom teachers and the principal. This positioning offers a unique opportunity to stay close to the mathematics teaching and learning happening in classrooms. It also, hopefully, means you have opportunities to teach mathematics yourself, whether in your own classroom or alongside colleagues.

These teaching experiences, and the stories of practice that grow from them, are more than moments to reflect on. They can serve as powerful leadership tools to support the collective understanding of the mathematics teaching practice being developed through your mathematics leadership. When shared thoughtfully, stories of mathematics teaching can become important resources for leading professional learning and supporting practice development in your school.

The power of story

For as long as time, stories have played an important role in influencing our feelings and emotions (well-being), shaping our behaviours and actions (practices), and developing what we come to understand and know (knowledge). Stories have the potential to shift the way we act in the world. Part of the reason for this is that we remember stories, particularly ones that have an impact on us in some way.

One person who has influenced my thinking in the area of teacher professional learning is Emeritus Professor Doug Clarke (ACU). If you have participated in any of his professional development, you will know how Doug uses stories of practice to highlight essential points about effective mathematics teaching. Doug shares insights about teaching by using photos from lessons and work samples from students to highlight effective mathematics teaching practices.

Your stories of practice, the retelling of your mathematics teaching experiences, have the potential to influence your colleagues’ practice and knowledge, leading to more impactful teaching.

What is a story of practice?

A story of practice is a narrative about a teaching episode that captures the intention and impact of a teaching strategy, approach, or action. The story is typically shared through a series of annotated photos that highlight key points in the teaching episode. The images could be ones focused on the teacher’s actions and responses from students, or student actions and responses from the teacher. The inclusion of photos of students in action and quotes from the students helps support the content of the annotations.

The annotations are done after the teaching episode. Those written descriptions should match carefully selected photos that highlight the intention and impact of teaching. Placing the images and annotations into a presentation application (e.g., PowerPoint) allows for greater ease of using the story of practice in a professional learning situation with teachers.

Posing reflective questions for teachers to engage in reasoning about pedagogy supports understanding of the intent and impact of the story of practice. The questions should prompt teachers to reflect on their pedagogy as a way of identifying what works well (WWW) and even better if (EBI). Questions that invite teachers to take action associated with the teaching episode can support how the story of practice facilitates practice development.

An example from my own practice

The following images are taken from a “story of practice” that I developed. In this teaching episode, I used Geo-Stix as a concrete material to support Grade 1 students’ learning about the properties of figures (2D shapes) and using properties to classify examples into groups.

In the example of my story of practice, the focus is on geometric reasoning and the use of non-examples to push precise mathematical descriptions. By combining annotated images with teacher-student dialogue, this story of practice becomes a tool for professional learning. It invites teachers to reflect on pedagogy, consider impact, and explore what they might try next in their own classrooms.

Stories of practice like this one make the work of mathematics teaching visible to teachers in powerful ways. The story highlights the pedagogical choices made in the moment, the ways the teacher responded to students’ thinking, and how the tasks and materials were used to deepen conceptual understanding.

Features of a story of practice

If you’re thinking about creating your own story of practice, here are a few features to keep in mind. These elements help ensure your story is meaningful, usable, and able to support professional learning with your colleagues:

  • Grounded in a real teaching episode
    Select a moment from your own teaching that highlights something important for your students’ mathematical learning. This could be a success, a challenge, a surprise, or a shift in student thinking.
  • Focused on the mathematics
    Keep the mathematics front and centre. What’s the big idea, concept, or knowledge to be learnt? What’s the mathematical purpose of the lesson? How are students enacting the proficiencies and ways of working mathematically that relate to the concept or knowledge being taught?
  • Includes annotated images
    Use a small set of photos that capture critical points in the lesson. Annotate them after the lesson with prompts, reflections, or notes that make your thinking visible.
  • Shows teacher and student thinking
    Include snippets of dialogue. This could be what you asked, what students said, and how you responded. This helps others see the pedagogical moves at play and the pedagogical reasoning happening in the moment.
  • Raises questions about practice
    Invite reflection. Pose questions that prompt colleagues to think about the pedagogy, the impact, and what they might try themselves.
  • Positions the story as an example, not an exemplar
    A story of practice isn’t about showcasing a perfect lesson. The story is about sharing something authentic that can support your teachers in thinking differently about teaching and learning.

Closing thoughts

Stories of practice are more than teaching anecdotes. They can become vital leadership tools for leading professional learning, building shared understandings, and developing mathematics teaching practice in ways that matter.

As a mathematics leader, your teaching experiences can become powerful catalysts for change when shared purposefully with colleagues. So next time you’re in the classroom, pay attention to those small moments that reveal something important and think about how you could create a story of practice.

They might just be the beginning of a story worth telling, and a step toward what’s not there yet…and a way of bringing about practice that could be in your school.

Leading professional learning about dialogue in mathematics lessons: A focus on Talk Moves

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In some of my recent work with primary school mathematics leaders in New South Wales, we have focused on ways of supporting teachers to engage their students in mathematical dialogue during mathematics lessons. We have focused on this considering that we know that effective teachers of mathematics know how to facilitate discussions in mathematics lessons. We also know that effective teachers can use that dialogue as a time for the students and the teacher to co-construct meanings about mathematics through that classroom talk.

My work with those mathematics leaders has been influenced by that of the NCTM, in particular, their work concerning the Principles to Action (2014). The NCTM endorsed eight practices that highlight ways of teaching mathematics effectively. Of those eight practices, one of them highlights a teacher practice concerned with the facilitation meaningful mathematical discourse. This is elaborated further with mention of the teachers’ use of knowledge and strategies that promote the building of shared understanding by and for the students. The focus of that discourse is on the analysis, comparison and contrast of student approaches, strategies, ways of thinking, and the arguments that take place through classroom dialogue.

As a way of enacting practices associated with mathematics dialogue in maths lessons, the mathematics leaders and I have explored the use of Talk Moves (e.g., Chapin, O’Connor, & Anderson, 2009). Since their introduction, the Talk Moves have been revised and interpreted in multiple ways. For example, the NSW Education Department has created some helpful resources on the Talk Moves based on the work of Chapin et al. (2009) which can be accessed via this URL (NSW Government numeracy resources: Talk moves). The Talk Moves, in whatever form they take, have provided the mathematics leaders with tools to support their leadership of school-based professional learning. This professional learning has focused on ways to develop teachers’ ability to highlight dialogue in their mathematics teaching by using the Talk Moves.

The Talk Moves on which we agreed to focus teachers’ professional learning on included: revoicing, restating, agree/disagree, reasoning, adding on, and wait time. After a coaching discussion with one mathematics leader, a dialogue move concerned with clarifying was also added. Working with some of the mathematics leaders in their schools, a general approach to leading teacher professional learning about the Talk Moves was developed.

Here are some of the leadership actions that were enacted by the mathematics leaders:

  • Auditing questioning practices and the types of dialogue already used by classroom teachers, using the Talk Move types to categorise the question and dialogue types
  • Reflecting on what aspects of dialogue practice is a strength of teachers’ activity in classrooms, and which Talk Moves had the potential to be focused on and developed
  • Engaging teachers in professional reading opportunities about the importance of dialogue and how questioning can support the facilitation of discussions in mathematics lessons. Some of the mathematics leaders shared the section on mathematical discourse by the NCTM (2014)
  • Posting Talk Move questions on anchor charts in the staffroom and teacher planning rooms where professional learning takes place and inviting teachers to add new questions that they find useful in developing each of the Talk Moves
  • Creating Mathematics Learning Walls (e.g., Stewart & Makin, 2017) where questions were posted as a means of prompting teachers to use the Talk Move questions and statement; further anchor charts were used on those learning walls as a way of making dialogue more visible to students
  • Setting goals with teachers to develop one or more of the Talk Moves, including planning evidence of goal achievement and strategies to make the goal a part of teacher practice
  • Co-teaching in classrooms so that teachers could see the mathematics leader model the use of Talk Moves and their associated questions, and so that the mathematics leader could provide feedback to classroom teachers
  • Creating Talk Move prompt cards (see below) which teachers used during mathematics lessons, and then using the cards with students as a way of encouraging more student-to-student talk

The ultimate goal of working with teachers to develop their practice with Talk Moves is so that students can engage in dialogue with each other using those dialogue moves. This is an ambitious goal to set for classroom teachers, but it is not unachievable. One of the mathematics leaders with whom I work has decided to create Talk Move cards (the final leadership action listed above).

These cards (downloaded from the link below) can be printed, cut out, and compiled, and then used by both teachers and students in mathematics classrooms. The mathematics leader intends to have their teachers use the cards as a prompt for their own teaching but also use the cards as a tool that encourages students to ask each other questions during mathematics lessons. There are plans for the mathematics leader to support teachers by encouraging them to focus on a specific Talk Move with their students for a set timeframe. Teachers will also be encouraged by the mathematics leader to set learning intentions and success criteria with students which are based on their students’ ability to engage in mathematics dialogue using the Talk Moves.

Talk move cards

Example of the Talk Move cards prepared for teacher and student use.

The mathematics leader also intends to ask teaching support staff members like the Classroom Support Assistants (CSAs) and Learning Support Officers (LSOs) to use the Talk Moves prompt cards. The reason for this is because the mathematics leader noticed that, in their work with supporting learning in classrooms, the CSAs and LSOs tend to support at-risk and vulnerable students by telling them what to say and what to think during mathematics lessons. That mathematics leader believes that the Talk Moves prompt cards would be helpful in moving the practices of the CSAs and LSOs from teaching focused on telling to teaching focused on questioning.

I would be keen to hear from primary school mathematics leaders who have focused their school-based professional learning leadership activity on the use of Talk Moves. The mathematics leaders with whom I work would appreciate hearing about your leadership actions so that they might enact similar activity that supports teacher professional learning in their schools.

The Talk Moves cards can be downloaded using the link below:

Talk moves prompt cards – Created by Matt Sexton

References

Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom discussions: Using math talk to help students learn, Grades K-6 (2nd ed.). Sausalito, CA: Math Solutions Publications.

National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM

Stewart, R., & Makin, L. (2017). Mathematics learning walls: The third teacher in the classroom. Prime Number, 32(4), 16-18.

Leading development of convincing practices in mathematics classrooms: Ideas for mathematics leaders’ professional learning activity

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It is really pleasing to see a greater focus on the role of the Proficiencies in the teaching and learning of mathematics in primary classrooms. At our recent MTLC conference for Primary Mathematics Teachers at Australian Catholic University (Melbourne Campus), we had eight workshops that ran during the day that specifically focused on the proficiencies of Understanding, Fluency, Problem Solving, and/or Reasoning. During that conference, I also presented a workshop on ways that teachers can highlight convincing in their mathematics lessons. In this post, I share some of my thoughts about this aspect of reasoning from a mathematics leadership perspective.

When it comes to teaching mathematics content, ideas, and knowledge, we need to ensure that the proficiencies, those important ways of thinking and working mathematically, are also taught in our classrooms. As mathematics leaders, it is important that we provide professional learning (PL) opportunities for our teachers that support their understanding of how to teach mathematical content along with the proficiencies for learning that content; both of these aspects of the curriculum are as important as each other. A way to start this focus on the proficiencies in your PL leadership activity could be through an exploration of convincing with your classroom teachers.

Defining and exploring convincing

Although not specifically mentioned in the Reasoning description within the Australian Curriculum documentation (ACARA, 2016), convincing is an important aspect of reasoning. It is a vital way of thinking and working mathematically. As mathematics leaders, our role is to influence the knowledge, practices, and dispositions of our classroom teachers through professional development. Exploring what convincing “looks like”, “sounds like”, and “feels like” in mathematics is one way of influencing the teaching of mathematics in classrooms. This exploring should lead to understanding, and a nice place to start is with a definition for convincing.

Through my reading, I have come to define convincing in mathematics as: a way to communicate mathematical reasoning (thinking) through discussions where evidence is used to persuade others’ reasoning and thinking, and thus supporting mathematics learning. Within this definition, it is assumed that our students take the lead in those discussions, which we can also call arguments. An argument in a learning situation is a form of discussion where all involved in that discussion seek to learn by engaging in practices such as explaining, elaborating, reasoning, and reflecting (Andriessen, 2006). I believe that these definitions for “convincing” and for “argument” are ones that work for teachers and students. I have actually shared these definitions with students in classrooms. They appeared to engage with the idea of a convincing argument really well (once, of course, we had clarified that a mathematical argument is a lot different to an argument outside of the classroom!)

Supporting convincing arguments by using the VLS framework

Within my definition of convincing, I mention the term “evidence”. Evidence is vital when we wish to persuade or convince others of our mathematical reasoning. It is important that as mathematics leaders we help our teachers to understand that evidence is important, and that they need to “keep a press” on students to create these evidence sources of reasoning.

One way that we can explore this need for evidence for convincing arguments is through visual, language, and symbolic representations of mathematical thinking. These three representations formed the basis of a framework which I developed from my work with leaders and teachers, and with my colleague, Leonie Anstey.

VLS framework

Figure 1. The Visual, Language, and Symbol (VLS) framework

This framework named the Visual, Language, and Symbol (VLS) framework (see to Figure 1) is influenced by the work of Fuson et al. (1997), Lesh, Post, and Behr (1987), and Mason, Burton, and Stacey (2010). The idea captured in this framework is that when engaging in substantive mathematical tasks, we create evidence of our mathematical reasoning using visual, language, and symbol representations. We then use that evidence to convince ourselves first (easiest convincing), then a friend, and then finally, a sceptic (most difficult convincing).  Through the convincing process, further understanding of mathematics is then co-created together with others through reasoning discussions or arguments. During those levels of convincing, our reasoning might be expanded, refined, and/or adjusted.

Exploring the VLS framework with teachers in professional learning opportunities

I believe that this framework has the potential to be used by mathematics leaders in their PL leadership activity with classroom teachers. For teachers to use an idea in their own classrooms, they need opportunities to explore that idea with their colleagues first. One of the characteristics of effective mathematics PL is that teachers undertake the role of the learner in the session, workshop, or meeting. This is a time when teachers can understand how mathematical knowledge is constructed, verified, and evaluated.

In terms of exploring the VLS framework, it is also a time when teachers can use the framework and see for themselves its use in supporting convincing practices. By using the framework in a PL opportunity, teachers can be better prepared for using it with their own students. Teachers are more likely to use the framework when they have had the appropriate amount of time to explore how it might be used with their colleagues first. For other teachers, they might need to observe you or their colleagues using the VLS framework during a mathematics lesson. For some of your other teachers, they might benefit from a co-teaching situation with you as their mathematics leader.

One of the important aspects of the VLS framework is that engagement in convincing arguments is best facilitated through the use of mathematical tasks where students can learn substantive mathematics. Richard Elmore, a well-known educationalist, has been known to share the phrase “task predicts performance”. This phrase, which is sound advice for any teacher of mathematics, means that if we only give students worksheets, “drill and practice” activities or lists of mathematics facts to memorise, then students are not encouraged to understand the need for mathematical arguments. They have diminished opportunities to learn how to reason in a mathematical way. The VLS framework works best when students have the opportunity to engage in challenging mathematical tasks where they can learn important mathematics content and important ways of thinking and working mathematically.

Ideas for professional learning about the VLS framework

In this next section, I offer some ideas to mathematics leaders for PL opportunities where teachers can explore the VLS framework. These are in no particular order and nor are they prescriptive. I offer them as a guide for those mathematics leaders who are keen to promote convincing practices in their schools. These ideas are best enacted in a PL workshop or meeting which is designed and facilitated by the mathematics leader.

It is also important that teachers have opportunities to take on the role of the mathematics learner and the mathematics teacher in your PL sessions. I offer too many ideas for the one PL opportunity, so there is the potential for you to explore these ideas over a number of sessions. I would be keen to hear from you if you take up, use, and expand upon any of these ideas in your own PL leadership activity.

  • Ask teachers to create convincing arguments for mathematical statements where they decide if the statement is “always true”, “sometimes true”, or “never true”. Ask the teachers to think about the statements, and then ask them to create evidence of their reasoning using visual, language, and symbol representations. Encourage your teachers to think and work by themselves first, and then work with a colleague to adjust or expand their convincing argument. Then they engage in a convincing argument where you and the other teachers play the sceptic, forcing your teachers to refine their mathematical arguments.
    • Examples of mathematical statements
      • The sum of two even numbers is an even number
      • Division leads to a smaller number
      • Multiplying a number by itself leads to a product which is larger
      • A square is a rectangle
  • Reflect on the responses to the mathematical statements and share the different representations that the teachers used to convince themselves and others. Engage teachers with questions where they decide which representations were more appropriate than others, how all three representations matched and aligned (or how they might become more aligned), which statements were the most challenging and why, and what are the accurate responses to those mathematical statements
  • Explore how the mathematical statements could be adjusted so that they are always true. Ask the teachers to create statements that they could use with their own students, where students need to engage in convincing practices using all three representations in attempts to convince themselves and others
  • Ask teachers to reflect on what convincing “looks like”, “sounds like”, and “feels like”. This is best done when teachers have engaged in mathematical tasks where they have used the VLS framework. Now that they know this about convincing, identify implications for their teaching and for student learning about convincing in mathematics lessons at your school
  • Share copies of the VLS framework, and ask teachers to discuss what ideas are conveyed in the framework, and how they enacted those ideas when creating their convincing arguments. Decide on ways of incorporating the VLS framework into teachers’ planning, teaching and assessment
  • Ask teachers to provide definitions for convincing in mathematics. Compare and contrast their definition with the one shared in this blog post. Discuss with teachers, “Why is it important that students seek to convince themselves, then a friend, and then finally a sceptic?” “Why is it important that we are now asking our students to respond to mathematical tasks using visual, language, and symbol representations?”
  • Explore with the teachers what they think students would do at each convincing level (convince yourself, convince a friend, and convince a sceptic). For example, what do we do when we attempt to convince ourselves? What specific actions do we undertake? What do we do when we convince our mathematical friend, and how is this different from convincing a sceptic? What specific actions do we undertake when convincing the sceptic? Discuss with teachers how they can support their own students to understand the levels of convincing and the actions required at each level (for most engagement with this task, it is best that teachers engage their students in the convincing levels and then debriefing afterwards with the students)
  • Focus on the Reasoning description in the Australian Curriculum, paying attention to the verbs. Ask teachers to reflect how students could enact the Reasoning proficiency through the use of the VLS framework. Identify which aspects of the Reasoning description are enacted when students engage in convincing practices. Encourage teachers to identify teacher actions that promote that proficiency enactment in their mathematics lessons
  • Brainstorm ideas of how to introduce and use the VLS framework in classrooms from Foundation to Year 6, with attention and focus on how students use the framework to support their own mathematics learning. Discuss ways of sharing and highlighting the VLS framework in classrooms (possibly through mathematics learning walls)
  • Use the VLS framework at each PL opportunity where you ask your teachers to engage in a mathematical task. Reflect on the task in relation to the evidence (the three representations of visual, language, and symbol) that they create. Discuss implications for mathematics teaching in the classroom, and ask teachers to set specific actions and enact them
  • Ask teachers to bring in evidence of their use of the VLS framework and share success stories of their own teaching and their students’ learning where the framework was used to support convincing practices in classrooms

These are only a few suggestions for mathematics leaders who are keen to establish a greater focus on convincing in mathematics lessons in their schools. If you are interested in accessing the article that I wrote for the May MTLC conference where I share further thoughts about the VLS framework, please feel free to email me. This article provides further background and advice on using the VLS framework and how it might be used in classrooms. Please email me at leadmemaths@gmail.com 

You can access the VLS poster which could be used in the classroom using the following link:

VLS framework poster

I will be presenting a workshop on the VLS framework at the MERGA Teachers’ Day on 29th June at Curtin University, Perth. If you are attending my workshop, I would love to hear from you.

If you are interested in attending any of our MTLC conferences at ACU (Melbourne Campus), please use the following link: www.acu.edu.au/mtlc

 

References

Andriessen, J. (2006). Arguing to learn. In K. Sawyer. (Ed). The Cambridge handbook of the learning sciences. (pp. 443-459). Cambridge, England: Cambridge University Press.

Australian Curriculum, Assessment and Reporting Authority. (2016). The Australian Curriculum: Mathematics. Canberra, Australia: ACARA.

Fuson, K., Wearne, D., Hiebert, J., Murray, H., Human, P., Olivier, A., Carpenter, T., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics, 28(2), 130-162.

Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem-solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Erlbaum.

Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed.). Essex, England: Person.