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This discussion forum is being moderated as an expression of servant leadership in teaching & learning. As a collaborative tool for brainstorming enriching experiences for students, teacher learning groups, and district learning teams, we can inspire and build experiences to help empower each of us to personal leadership in learning. Thank you, in advance, for your contributions and leadership to realizing outcomes for improving student achievement, equity and well-being.

FROM THE ARCHIVES

cognitive science...meet the thinking classroom

2/27/2019

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Welcome back to Flipping the Focus.

Introduction
In the last post (here), you read an interview, conducted by Flipping the Focus, with Ottawa Educator, Jaime DePippo (@MrsDePippo). The interview was Part 2 of a series that delved into the mindset and actions of an educator who is actively exploring the impact of the Thinking Classroom Framework (see Liljedahl, 2017) on student learning and teaching--in both Mathematics and English (Grades 9 and 10, Applied courses).
Picture
During a recent conversation that I had with a colleague, we reflected upon the following statement from that interview:

"Students are capable, critical thinkers and problems solvers.  If we deprive our students of these opportunities, they will not grow.  Students need to be actively engaged in their learning."

This quote, in turn, lead us to the following wonderings:

1-What does this mean for educators' planning? 
  • Given that adolescents are more likely to respond faster emotionally than rationally, how can we better support their learning? 
  • How can we engage adolescents' understanding that it's difficult for them to 'override automatic responses' because their pre-frontal cortex is still maturing?
2-How does novelty--in the context of the Thinking Classroom--fit into conversations (perhaps, larger ones) about student engagement?

The remainder of this post focuses on these questions--that is, not in an effort to answer them: As pointed out by the last wondering, the goal is our own learning...through conversations about exploring where novelty 'fits' into our teaching practices.

As you continue with the information presented from both cognitive science and the Thinking Classroom, invite a colleague to read, reflect on practice, and discuss next steps with you.
"Cognitive Science...Meet the Thinking Classroom"

Thinking Classrooms
In an earlier post, I listed and defined each of the elements (E) of the Thinking Classroom (here). These elements are clustered into, what Dr. Peter Liljedahl, refers to as gears. In the graphic (right), the 14 elements are colour-coded into their respective gear. Generally, as we evolve our pedagogical practice through the elements, we tend to begin with Gear 1 (G1) and progress by adding subsequent gears over time (G2 to G4).


Cognitive Science
Currently, I am helping support networked/regional professional learning around improving student outcomes in Mathematics through inquiry. The inquiries are being implemented by nine, district school boards in Eastern Ontario. These districts, for many years now, have been contributing to both networked learning and educational research through the Eastern Ontario Staff Development Network (EOSDN).

A complementary resource that educators are studying is David A. Sousa's How the Brain Learns Mathematics.
List of Peter Liljedahl's 14 Thinking Classroom Elements
Geared (4) Elements (14) of The Thinking Classroom
Recently, I've been reading and reflecting upon both brain development and cognitive processes that influence teaching and student learning. Below are some highlights from Chapter 5 (Teaching Mathematics to the Preadolescent Brain) and Chapter 6 (Teaching Mathematics to the Adolescent Brain). 
Highlights on Teaching Mathematics to Preadolescent and Adolescent Brains
Based on some the emerging themes, I couldn't help but to consider how these highlights might 'bump up against' the geared elements of the Thinking Classroom Framework. In relation to the wonderings presented (#1 and #2, above), how might the intersection of both cognitive science and Thinking Classrooms impact educators' planning and interaction with students in the evolution of their classroom practice?

In the section that follows, I will present a draft of results of some coding that I completed between the framework and Chapters 5 & 6. In terms of my own interpretation, I am currently seeing that aspects of both cognitive science (CS) and Thinking Classrooms (TC) play a reciprocal role in student learning. That is to say, a TC pedagogy is more successfully enacted when we attend to preadolescent and adolescent brain development, while student cognition is supported by engaging students with the geared elements of the framework.
 Results: "Cognitive Science (CS)...Meet the Thinking Classroom (TC)"
Thinking Classroom Gear Frequency Results - Preadolescence
Figure 1 - TC Gear Frequency Preadolescent
Thinking Classroom Gear Frequency - Adolescent
Figure 2 - TC Gear Frequency Adolescent
Figures 1 and 2 present the relative frequencies with which each of the gears (i.e., group of elements) related to the highlighted material in Chapters 5 and 6. Most notable are the differences between the relative frequencies of Gears 2 and 3 (see graphic for description of each gear's elements, above). 
Picture
Figure 3 - TC Elements Frequency (Combined Sets)
TC Elements-Adolescent
Figure 3 presents the frequencies for each of the 14 elements and for each stage in cognitive development--preadolescence and adolescence. The adolescent data portrays elements 9 through 11 as supporting adolescent cognition the most (i.e., "Gear 3")--specifically, managing hints and extensions; consolidating from the bottom; and student self-assessment.
Next to these, elements 4, 7 and 12 were also coded frequently--i.e., using verbal instructions, constructing meaningful notes, and communicating where students are in their learning/where they are going, respectively.

TC Elements-Preadolescent
Figure 3 also presents the frequencies for preadolescent learners. Notably, elements 4, 8 and 10 seem to be most influential in supporting preadolescent cognition (i.e., verbal instructions, building autonomy, and consolidating from the bottom, respectively), followed by elements 7, 9 and 12.
Discussion
​
Before wading further into the complexity of these results, I would like to re-emphasize that these are drafted--i.e., they, on the basis of my interpretation, suggest how CS and TC are related. They also provide you with a glimpse into my own reflective practice: not only is this how I inquire, in part, into examining the value of pedagogical practice on student engagement and learning; it is also representative of the collaborative inquiries we can be doing alongside and in support of future-ready learners.

​It's also important to consider those places where I have used the word, "most", in relation to support. For example, when thinking about the most frequently recorded skills for  preadolescents, it's not that learners already 'arrive' with these abilities; rather, we use the elements to actively engage learners and support the development of their executive functioning. Similarly with adolescents, learners may not enter in to your class already expressing the skills encompassed by the emerging themes (table, above). What it does imply is that we can attend to planning and interacting with learners in ways that will bring about growth in these areas. 

"Students are capable, critical thinkers and problems solvers.  If we deprive our
students of these opportunities, they will not grow.  Students need to be
actively engaged in their learning."

In addition to the thinking, above, each student develops and attains proficiency in different ways and at different times. To help students grow their mindset around becoming assessment-capable,  actively build, share and discuss students' learner profiles with them--especially as you plan tiered interventions with your "students of mystery". You can access a sample TC observation and student profiling template here to support your TC planning, documentation and profiling.

With respect to planning instruction for all (Chapter 2; Learning for All, 2013), we might interpret Figure 1 as Gear 2 elements being more widely used to setting the foundation for more complex interactions between preadolescent learners with elements of the "Assessment Loop" (Causarano & Coulombe, 2018)--for example, establishing assessment capability for deepened learning through self-assessment, goal setting and monitoring towards attaining goals. 

Moving towards adolescence, we might look upon the prevalence of Gear 3 elements as leveraging students' growing capacity to solving complex problems for deepening their conceptual understanding. We might also see how assigning fewer, meaningful "check your understanding" problems puts less strain on working memory, provides adequate time for elaborative rehearsal--lending itself also to developing procedural fluency.


Across all graphs (Figures 1 to 3), you'll note an under-representation of gear 1 and 4 elements. Gear 1 elements--good tasks, VNPSs and VRGs: these are the starting point for any educator's journey into enacting a TC pedagogy. Even for educators consistently practising Gear 4 elements, alongside their students, Gear 1 elements are a 'gateway' to engaging in deep, pedagogical practice. Notably, these elements can be implemented quickly and, perhaps, contribute indirectly--through classroom structures (physical space and groupings)--to student cognition. Gears 2 and 3 are comprised of a number of strategies that engage students to being active in their learning. Our engagement with these strategies means that we are consistently monitoring and communicating with learners over time. These actions, we might say, have direct impact on student learning and are more easily referenced against developmental markers from CS. Lastly, most of the Gear 4 elements relate to assessments of learning. These can be redressed by students and teachers in a formative manner but constitute a small amount of all assessments performed.
Conclusion
Beyond what has been presented in this post, there is so much that we can discuss and propose as inquiries that bridge the gap between the many opportunities we have to promote student learning and the actual learning that takes place for educators and their students.

In the opening, two wonderings were shared:

​1-What does this mean for educators' planning? 
  • Given that adolescents are more likely to respond faster emotionally than rationally, how can we better support their learning? 
  • How can we engage adolescents' understanding that it's difficult for them to 'override automatic responses' because their pre-frontal cortex is still maturing?
2-How does novelty--in the context of the Thinking Classroom--fit into conversations (perhaps, larger ones) about student engagement?

As you move forward with your own journey into enacting a TC pedagogy, I believe the following might serve as good guiding statements:
  • The Thinking Classroom can be enacted in such ways to support the development of student cognition.
  • The Thinking Classroom supports what is currently known about cognitive science and brain maturation.
  • When students understand how they learn and why a particular pedagogy is being used, they will be more likely to see themselves as confident and assessment-capable learners.
How Might I Share This Synthesis in My Context?
 
If the ideas presented in this post is something that you'd like to share within your own context (e.g., PLC, district team, learning network), I have posted a consolidation of the blog in the form of a slide-deck for your use. If you do happen to use it, please extend an invitation to me to discuss your plans.

You can view the slide-deck below, as well as downloading your own copy that includes notes I've used in speaking about the material in this blog, as well as future, potential directions for collaborative and action-based research.

Prefer to listen to a presentation of the synthesis? View the slides in the form of video? See the button below to listen/view.
MAKE A COPY
VIEW/LISTEN
Final Remarks
In closing, I can't help but to think of the conversations that can be inspired when we take collective action to improving student learning. As this blog is a means for readers to network and gradually change the context for how they teach and learn, we all benefit by drawing nearer to the perspectives shared here and shared beyond with our professional learning networks. Please feel free to share your comments and questions to this post using the "Leave a Reply" form provided, below.
​
I am more than happy to collaborate with you and make our learning visible, here, in this blog and across Flipping the Focus' social media platforms, as well as your own. I
f at any time, you have questions or comments, please feel free to reach out to me at Flipping the Focus. 

Sincerely,

Chris Stewart, OCT
Education Leader at Flipping the Focus
CONTACT CHRIS
BOOK CHRIS

References
Causarano, J., & Coulombe, H. (2018, September 14). The Assessment Loop: Merging Assessment and Instruction. Retrieved January 29, 2019, from https://harnessassessment.com/2018/09/04/the-assessment-loop-merging-assessment-and-instruction/
​
EduGAINS. (2013). Learning for All, A Guide to Effective Assessment and Instruction for All Students, Kindergarten to Grade 12. Retrieved from http://www.edugains.ca/newsite/Learning4All/whats_L4All.html

Liljedahl, P. (2017, October 17). Building a Thinking Classroom in Math. Retrieved from https://www.edutopia.org/article/building-thinking-classroom-math
​

Margaret Sinclair Memorial Award Lecture: Peter Liljedahl. (2019, February 14). Retrieved from http://www.fields.utoronto.ca/activities/18-19/Margaret-Sinclair-lecture 
​
Sousa, D. A. (2008). How The Brain Learns Mathematics. Thousand Oaks, CA: Corwin.

Thinking Classrooms: An Interview with Jaime DePippo - Part 2 [Online Interview]. (2019, February 18).

​
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the Equity through pedagogy series: thinking classrooms - Revisited

2/19/2019

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Welcome back to Flipping the Focus.
1-Introduction
Part 1 of the "Equity Through Pedagogy" series took readers through an examination of pedagogical practice through the lens of the Thinking Classroom--in secondary Mathematics and English.

According to Dr. Peter Liljedahl (2017), Thinking Classrooms are defined as spaces “...not [only] conducive to thinking but [that] also [occasion] thinking...a space inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion”.

It's important to note that these classrooms go well beyond engagement in vertical spaces--that is, well beyond the Stage 1 elements described in the Part 1 post and summarized by the graphic (see right).

As you reflect upon the elements from Stages 2 to 4, note that these elements serve the conversations we can have with students and one another about refining teaching and learning.

​Altogether, the last nine elements relate to the 'glue' that connects all aspects of our pedagogical practice: 
Formative Assessment. 
The 14 Elements of a Thinking Classroom
The 14 Elements of a Thinking Classroom

"...these classrooms go well beyond engagement in
​vertical spaces..."

As we collaboratively and continuously explore instructional practices that support students and promote respectful and caring learning communities, these elements can guide us in our efforts to better knowing both our learners and the learning.

Let's consider knowing our students and their learning through the context of the interview that follows.
2-The Search for Flow through Thinking Classrooms
Have you ever found yourself in a state where you were so engaged with a task that nothing else mattered? Think back to that time: Were you neither overwhelmed nor just coasting along? If you felt that you were in that space of 'neither', then you were in flow. Flow results from the dynamic relationship between challenge and skill development (Liljedahl, 2016). As teachers, we have a significant role to play in helping students find flow and what to do, alongside their peers, in this space.

In Part 1 of this series, I mentioned that there are a number of educators, in Eastern Ontario, journeying into building Thinking Classrooms. One of these educators is Jaime DePippo (St. Mother Teresa HS, Ottawa, ON). Jaime has been very gracious in speaking with me about her practice, and I am privileged and thankful to Jaime for the opportunity to feature her experiences, here, on Flipping the Focus.

Below, you'll see a continuation of our previous conversation, beginning with Question #4 (previously published). The remainder of the post focuses on Meaningful Notes, ​Spiralling, and Getting Started with a Thinking Classroom.

"Communicating where a student is and where they are going, alongside assessment AS learning, moves student learning forward."

3-A Conversation with Jaime DePippo
Question 4: You explained earlier that students are building and showing autonomy through their "meaningful notes". What does this look like in practice? What are you and your students doing?

Answer:
I use a gradual release model with my students in Applied English and Mathematics.

-A template, that provides some constraints--largely, the space provided for creating notes (i.e., not taking notes)--is shared with students towards the end of a period of learning.

-Students are encouraged to take notes when they need to.

-On a day-to-day basis, exit tickets are used to provide students with feedback.

-Using exit tickets allows me to provide suggestions for next steps. Students can map the next steps into the ‘what’ that they could be writing down in their notes.

-Students are not bound to making meaningful notes only at specific times in their learning.

-I accomplish this by spiralling the curriculum. Through interleaving the content, students have multiple opportunities to revisit and bring greater depth and meaning to their notes.

[The conversation continues, below, with a closer examination of meaningful notes.]

Question 5: In Stage 2 of Dr. Liljedahl’s elements is "Meaningful Notes". "Level to the Bottom" is in Stage 3.

-How fluid/fl
exible might we be in moving between these Stages? For example, should educators seek to fully level with student thinking, then build meaningful notes?
-Simultaneously level and build notes?

Altogether, how might a teacher best coach their students towards building autonomy?


Answer: 
​
At the beginning of a semester, I provide students with a template for creating their own meaningful notes. Essentially, the templates have only the key headings for the concepts students are learning.

[As a side-note, having the templates is important for educators who are teaching through spiralling their curriculum. For the math courses I teach, there are typically 4 spirals (or cycles). The first two cycles tend to be the longest, and by the time we reach cycle 4, there is less new material--i.e., the complexity of the tasks and the interconnectedness of concepts increases towards and throughout the final cycle.

As far as student learning is concerned, students feel more successful when spiralling: there isn’t a need or a feeling that you have to attain mastery right away. One of the primary goals of spiralling is to continuously go deeper with critical thinking and problem solving.]

[Back to the beginning of the semester...]

As we work towards consolidating student thinking in relation to a learning goal/goals, I’ll do some checks for understanding, which can be a small number of individual questions or related task so that students can see how they’re doing. I also use exit cards frequently to gain a better understanding of what students have been taking away from their time thinking alongside their peers. It’s through the exit cards, that I can create another opportunity to provide feedback to students. 

Above all of this, it’s important to remember that consolidation doesn’t always occur at the end of the task--student readiness is important: the exit card is key in my formative assessment and mentally-preparing students for self-assessment. 

With this feedback, students can go back to their notes (template) and write down those things that are important--i.e., meaningful--to them. For example, some of my early feedback would include notes to (and conversations with) students to write down the things that are not as familiar to them, as well as what their next step(s) would be.

By the time we get to Cycle 2, I find that students require less prompting to making their own, meaningful notes: student autonomy is always growing, cycle-by-cycle.

Earlier, I mentioned that conversations are a vehicle for providing students feedback. These principles are still in effect when we’re working on some test items. For example, if there are aspects of a student’s work that are not shining through on a test, I’ll ask them to show me their notes. It’s through their meaningful notes that a conversation about learning opens up (or re-opens). It’s during these moments that you can really tell if a student knows what they’re doing and what our next steps will be. In fact, there has been many times where a student’s assessment of learning has been enriched through conversation.
Question 6: There are many ways that educators can create safe and productively challenging learning environments with their students—some that do not include a Thinking Classroom.

If asked by a teacher why they should consider trying a Thinking Classroom, what advice might you provide to them? On getting started?


Answer:
Students are capable, critical thinkers and problems solvers.  If we deprive our students of these opportunities, they will not grow.  Students need to be actively engaged in their learning.

In any subject, nowadays, the goal of the student experience is not to remember everything. The goal is to create critical thinkers and people with strong collaboration skills, as well as their other global competencies. These are the skills and competencies that students will need moving forward. As educators, we have a great opportunity to show students that anything is possible when they develop and use these kinds of skills. And because we’re our own worst critics, sometimes students need us to point out their strengths: it’s through their strengths that they can develop/use these skills and grow as learners.

All of this said, teachers and administrators should start by attending a workshop or spending time in the practice of other teachers who are practising a Thinking Classroom model for teaching and learning. Asking questions of teachers and students who have immersed themselves in a Thinking Classroom everyday is important.  Hear their feedback about what has changed: engagement, curiosity, achievement, perseverance, etc.

Beyond this, I would say start small; take your time; and seek support from and collaboration with another person. There’s a lot to be said about planning and teaching together. To get a sense of the dynamic that’s possible between students and students and teachers--all of this in relation to the task that’s assigned--there’s a lot to be gained by spending time in and with these types of classrooms. Based on that experience, I would then encourage teachers to try the model by incorporating the Stage 1 elements--good tasks, visibly random groups and vertical non-permanent surfaces/whiteboards. For good tasks, there are plenty of resources available. 3-Act Math Tasks are a great base for teaching through problem solving--examples including Kyle Pearce & Jon Orr and Dan Meyer.

Realistically, it’s tough work to practise a Thinking Classroom model on a consistent basis, but it’s very gratifying and rewarding for both teachers and their students to see and hear themselves growing in autonomy and experiencing success. It has been and continues to be for me.

And I’m still learning! So if you’re starting out, don’t feel like you have to include all the elements of the Thinking Classroom right away. Dr. Liljedahl suggests you start with the top gear (visual included, above), and master that before you move on to the next. It happens in stages: not all right away. Start with something that you find manageable and continuously reflect and challenge you and your students to go deeper. Do some of your planning, teaching and reflecting with someone you trust. For feedback, also have them help you with how you’re going to challenge yourself in going for next steps.
4-Final Remarks
Successful Classroom StrategiesAchieving Excellence in Applied Courses (AEAC, 2017/18)
As this post draws to a close, I am certain of the following: my experience in teaching and learning has been enriched due to the collaboration that was experienced in seeing this post through from start to finish. 

As you reflect upon the final question, below, I would like to encourage you to think about inviting others to share in your experience--e.g., co-planning, co-teaching, debriefing and reflecting towards next steps with Thinking Classrooms and spiralling your curriculum.

Evidence reported from a large number of districts' administrators, across Ontario, is showing that successful classroom strategies for improving student performance in Applied-level Mathematics courses includes co-planning and co-teaching, as well as spiralling and classrooms that include the use of vertical spaces (AEAC, 2017/18; see graphic, right).

Might you consider incorporating a Thinking Classroom into your pedagogical practice?
In closing, I can't help but to think of the conversations that can be inspired when we take collective action to improving student learning. As this blog is a means for readers to network and gradually change the context for how they teach and learn, we all benefit by drawing nearer to the perspectives shared here and shared beyond with our professional learning networks.
​
I am more than happy to collaborate with you and make our learning visible, here, in this blog and across Flipping the Focus' social media platforms, as well as your own. I
f at any time, you have questions or comments, please feel free to reach out to me at Flipping the Focus. 

Sincerely,

Chris Stewart, OCT
Education Leader at Flipping the Focus
CONTACT
BOOK CHRIS

5-References
AEAC Project Team, Ontario Ministry of Education, Student Achievement Division. (2019). Achieving Excellence in Applied Courses: Dialogue with Principals, School Teams and Board Leads (Rep.).

Liljedahl, P. (2017, October 17). Building a Thinking Classroom in Math. Retrieved fromhttps://www.edutopia.org/article/building-thinking-classroom-math

Liljedahl, P. (2016). Flow: A Framework for Discussing Teaching. Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 203-210). Retrieved from http://www.peterliljedahl.com/wp-content/uploads/PME-2016-Flow-and-Teaching-1.pdf


Thinking Classrooms: An Interview with Jaime DePippo [Online Interview]. (2019, January 23).

Thinking Classrooms: An Interview with Jaime DePippo - Part 2 [Online Interview]. (2019, February 18).
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Connect 2 Learn: Episode 47 of "the Missing Link"

2/11/2019

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​Welcome back to Flipping the Focus.
1. Introduction
Recently, I had the awesome privilege to take part in a voicEdcanada Radiothon--a space that brought together many educators in the spirit of sharing their thoughts on the future of Education.

I’d like to express my deepest thanks to Rola Tibshirani (@rolat, Ottawa Catholic School Board) and Stephen Hurley and the team at voicEdcanada for the recent opportunity to share my learning journey with others across Canada. 


We engaged in a discussion of the confluence of thinking, assessment, connected learning (globalized competencies) and flipped learning practices upon student learning, as well as what learning we need to do, as educators and leaders, for continuously improving the conditions for teaching and learning.
tweet for episode 47 of the podcast, the missing link
Click on the image to access the podcast
These are timely, relevant and important considerations for Education.

​With respect to the last five posts, here on Flipping the Focus, listeners can experience the essence of each of them--i.e., a confluence of the themes--through the podcast.

The podcast can be accessed at: https://soundcloud.com/rola-tibshirani/rt-chris


Each of the contributing posts, from the "Equity Through Pedagogy" Series, are linked below for further, deep reflection.
2. Series: Equity Through Pedagogy
Post 1: Thinking Classrooms

Post 2: Global Competencies

Post 3: Flipped Learning

Post 4: Formative Assessment

Post 5: Pedagogical System for Teaching Mathematics
3. Provocation
​Once you've had an opportunity to listen to the podcast and/or read any one of the posts, consider sharing your perspectives to this blog and/or with your colleagues in response to the following provocation.

What affirmations, wonderings and/or challenges are you
and your teams experiencing?

​Final Remarks
In closing, I can't help but to think of the conversations that can be inspired when we take collective action to improving student learning. 

As this blog is a means for readers to network and gradually change the context for how they teach and learn, we all benefit by drawing nearer to the perspectives shared here and shared beyond with our professional learning networks.
​
I am more than happy to collaborate with you and make our learning visible, here, in this blog and across Flipping the Focus' social media platforms, as well as your own.

If at any time, you have questions or comments, please feel free to reach out through the "Contact" button (below).


Sincerely,

Chris Stewart
Education Leader, Flipping the Focus
CONTACT
BOOK CHRIS

Reference
The Missing Link: Episode 47 - Chris Stewart [Audio blog interview]. (2019, February 11). Retrieved from https://soundcloud.com/rola-tibshirani/rt-chris 
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equity through pedagogy - part 5: pedagogical considerations for Teaching mathematics

2/10/2019

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Welcome back to Flipping the Focus.
Leading up to the Mid-Atlantic Conference for Professional Learning, March 13-15, in Atlantic City, this marks the fifth (and final) post in this series devoted to pedagogical practices and frameworks that educators can leverage in their collaborative efforts to respectfully and equitably honour student voice.
1. Introduction
Envision learning environments where students and their teachers are engaged to interact in profound and meaningful ways--ways that...
Mid-Atlantic Conference for Professional Learning
MAC PL 2019, March 13-15, Atlantic City
  • help students to seeing “...themselves as powerful mathematics [learners]” (Anthony & Walshaw, 2009a); and
  • grow teachers’ and leaders’ confidence that a better prescription for student success in mathematics is one that’s grounded in occasioning students’ thinking.
In this post, the characteristics of The Pedagogical System for Teaching Mathematics (SIM K-12, 2017) are explained and questions are asked to engage you, your colleagues and students in thinking about how this system might better frame the teaching and learning that goes on in your class and school each and every day.

I’d also like to draw your attention to the following: You’ll not only find the framework described in this post as being effective in supporting mathematics teaching and learning, but also as having the potential to influence teaching and learning in other subject areas and aspects of school life.

As you continue with the post, consider framing your thinking against these, sample goals:
  • (Teacher Focus) To deepen your understanding of practices that engage students with differences in backgrounds, learning strengths, needs and interests.
  • (Leadership Focus) To inform your next best moves to supporting the growth of individual and collective teacher learning and practice.

"Having explored this framework--alongside many educators, students occasioning thinking, and complexed with other pedagogies--has
​been transformational for my own teaching and student
learning, as well as that of my colleagues."

2. Background
The Pedagogical System is a framework that provokes us to consider how teaching and learning needs to be...less about telling...less about prescriptive moves and responses. Really, it calls us to shift our mindset about teaching, leading, and learning to one that gives more value to student thinking. This philosophy, grown out of mathematical, educational research, is mirrored through several references and resources. Some suggested reading and resources are included in the references section to this post. 

This stance is one that depends on each and every one of us. Students, teachers and leaders all have a role to play in improving the conditions for student engagement, achievement and well-being.
Consolidated and adapted from the work of Anthony & Walshaw (2009a), System Implementation & Monitoring-K to 12 in Ontario (2017) has produced summary graphics and tools to help inform and guide educators as they inquire towards improving teaching and student learning.

The system (or framework) consists of four, interrelated components. This is so important to recognize and identify: none of the parts work in isolation of the others. These components include the following:

1-Worthwhile Mathematical Tasks
2-Tools & Representations
3-A Non-Threatening Classroom Environment
4-Classroom Discourse
The Pedagogical System for Teaching Mathematics
Let’s consider descriptions for each of these components in the form of reflective questions.

  • Worthwhile Mathematical Tasks
    • Do the tasks students engage in support how mathematics is viewed; how it can be understood through thinking and reasoning; and how it can be used?
  • Tools & Representations
    • How are students making their thinking visible? Are tools being used to organize thinking? Are multiple representations used, and are they connected for building a deepened understanding of mathematical concepts?
  • A Non-Threatening Classroom Environment
    • Does the classroom community encourage each of its members to think and reason, communicate ideas, and receive, provide, and take action on feedback?
  • Classroom Discourse
    • Do students value mistakes? Does the classroom community look forward to and honor the explanations of all students? What are the characteristics of productive, accountable discourse?​
Already you can get a sense of some criteria that you could use for monitoring inquiries into building communities of learners and a coherent school culture, where students see themselves and others as empowered learners of mathematics.

Let’s take a moment to look more closely at the first component.

What defines a task as being rich or worthwhile?

3. Worthwhile Mathematical Tasks
Let's start with considering problem-based teaching in mathematics.


The premise to problem-based teaching in mathematics is that students are able to meaningfully construct their knowledge by engaging in a variety of problem solving strategies. Coincident with developing their understanding, students also build procedural fluency since many of the concepts provide opportunities for students to work with numbers and expressions.

Ultimately, basing one’s approach to teaching and learning, in this way, moves us away from seeing mathematics as being about the right answer and/or getting to an answer TO seeing mathematics as valuable and connected to the real world.

According to Anthony & Walshaw (2009a), some of the characteristics of problems and tasks that are rich (or worthwhile) include the following:

They tend to...

  • Have a focus on original thinking;
  • Provide opportunities for productive struggle;
  • Be designed and presented at an appropriate level
    • E.g., Low-floor for entry and a high ceiling for extending (Boaler, 2016);
  • Be open-ended;
  • Include contextually-based problems that invite students to make sense of mathematics; and
  • Provide opportunities for students to develop procedural fluency in meaningful ways.​
4. Leading Through The Pedagogical System
Previously mentioned, you might have made some connections to criteria that you could use for monitoring inquiries into building communities of learners and a coherent school culture.

Linked to a tab called "Connections" (here), you’ll find a leadership-related tool with a variety of criteria for effective mathematics teaching. All of them are related to The Pedagogical System and are connected through formative assessment--both assessment FOR and AS learning (Anthony & Walshaw, 2009b).

Whether you’re conducting classroom walk-throughs, facilitating collaborative team learning, or observing and providing feedback to lessons, you might find these criteria essential to guiding discussions around effective practice and having discussions with students about their learning.

In your conversations and inquiries, consider how these criteria could allow you to better uncover and understand the hidden skills and talents of the educators in your school and your students.
5. Final Remarks
As you reflect, how are you seeking to co-create conditions that can give life to equity in the teaching and learning you do with students and your colleagues each and every day? 

In closing, I can't help but to think of the conversations that can be inspired when we take collective action to improving student learning. As this blog is a means for readers to network and gradually change the context for how they teach and learn, we all benefit by drawing nearer to the perspectives shared here and shared beyond with our professional learning networks.
​
I am more than happy to collaborate with you and make our learning visible, here, in this blog and across Flipping the Focus' social media platforms, as well as your own. If at any time, you have questions or comments, please feel free to reach out to me at Flipping the Focus. 


Sincerely,

Chris Stewart
Education Leader, Flipping the Focus
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6. References
Anthony, G., & Walshaw, M. (2009a). Characteristics of Effective Teaching of Mathematics: A View from the West. Journal of Mathematics Education, 2(2), 147-164.

Anthony, G. and Walshaw, M. (2009b). Effective Pedagogy in Mathematics. http://www.iaoed.org/downloads/EdPractices_19.pdf


Anthony, G. and Walshaw, M. (2009b). Characteristics of Effective Teaching of Mathematics: A View from the West. http://www.educationforatoz.org/images/_9734_12_Glenda_Anthony.pdf


Boaler, J. (2016). Mathematical Mindsets: Unleashing Students Potential Through Creative Math, Inspiring Messages, and Innovative Teaching. San Francisco, CA: Jossey-Bass & Pfeiffer Imprints.


System Implementation & Monitoring K - 12. (2017, February 23). The Pedagogical System with Reflective Questions. Retrieved from https://sim.thelearningexchange.ca/tag/the-pedagogical-system/

​
​
7. Suggested Reading
EduGAINS. (n.d.). Guides to Effective Instruction. Retrieved from http://www.edugains.ca/newsite/math/guides_effective_instruction.html

EduGAINS. (n.d.). Targeted Implementation and Planning Supports for Mathematics (TIPS4M). Retrieved from http://www.edugains.ca/newsite/math/tips.html


Ontario Ministry of Education. (2018). Focusing on the Fundamentals of Math: A Teacher's Guide. Retrieved from https://math.thelearningexchange.ca/


​Principles to Actions: Ensuring Mathematical Success for All
. (2014). Reston, VA: NCTM, National Council of Teachers of Mathematics.
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    I am passionate about leadership for learning and teaching and learning through inquiry. Through collaborative exploration of high-yield, pedagogical strategies, I have been able to further engage students to deepen their learning and fellow educators in continuously growing their practice--Flipped Learning, Thinking Classrooms, and culturing Student Voice as examples.  I hope that this site serves you well in your educational journey through teaching and learning by moving professional learning into your time ... your space. If you have questions or feedback, please feel free to contact me. Sincerely, Chris Stewart (OCT).

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