Mathematics class was never a place where I felt curious, a sense of anticipation or even reasonably challenged within a zone that kept me motivated and mentally engaged. Memories of math class across K-12 do not bring warm, fuzzy feelings or a smile to my face. My relationship with numbers, problem-solving and equations was strained; it lacked joy and always struck feelings of fear and anxiety about incompetence and a lack of intelligence. When a colleague talked about math, my pulse would pick up immediately. Panic would start to set in. Every memory of elementary math class consists of rote memorization, a quiet classroom watching a teacher perform operations and then mimicking equations without any conceptual understanding of why. I really had no idea why we were doing some of these problems, what was their connection to my world and the purpose. We did math in the abstract. There was no reference made to relevance, a real life context or the ways in which I could apply that for real world problem-solving.

Over time math concepts and equations became increasingly inaccessible. Such that I could not access prior knowledge to make meaning. As a visual person, I know I understand something well when I can use visualizing strategies to synthesize my thinking. I found I lacked the knowledge to visualize representations (model) and reason through problems to find a solution. It was such a relief to listen to Dr. Jo Boaler describe math trauma as I realized I was not alone. Many others globally suffered the same feelings of inadequacy. K-12 math evokes memories of tears, arguments and extreme frustration for me. Because of my experiences with math, I determined that students in my care would not carry those same feelings. I determined to ensure my approach to teaching mathematics brought a spirit of joy, evoked curiosity and created motivation. I would do my utmost to ensure my students could access the concepts and make sense of the problems they faced. In order to do so, I had to unlearn and relearn so much of what I had been taught down to how I saw, visualized and comprehended numbers, digits, place value, and sentences. It was a messy process, challenging my efficacy, mindset and resilience but over time, I experienced the results. How it warmed my heart to hear my learners shout with joy because it was time for math on the schedule!

A door opens to a new approach...
Every class in the GMU postgraduate VA certification program framed its content around international education, the IB and the PYP. When I arrived in July 2013, one of my first three classes was Mathematics in International Schools. To be completely honest, this class utterly terrified me. I walked in to the first day of math class feeling highly anxious. Dr. Baker was the first person to introduce me to the idea that there are strategies for problem-solving. We spent 8 days together wrestling with problems in a constructive, psychologically safe classroom. I discovered I was not the only one feeling a deep lack of confidence which was comforting. Over 7 days, I learned math can be constructed, visualized, accessible and even fun. However, 7 days did not help me overcome my math anxiety from 16 years of memorizing and mimicking. It did not help me unlearn and relearn math strategies for Number Operations. I knew I had a long road ahead of me so I challenged myself to develop myself at a pace that I could cognitively process.

Assessment for Process Errors
Assessment in mathematics involves the process of solving a problem. You can see that the solutions we analyzed were all traditional algorithms. At this stage in my math journey, I found it  comforting to work with familiar problems. Our class analyzed student process to identify the error within the process that led to an incorrect answer. The idea is to use this information to map out the teaching and learning response to support the student to move forward. Years down the road, I will learn that reasoning through a problem for efficient, clever solutions uses a process but it does not have to be through an algorithm only.

Problem of the Day
Every day, Dr. Baker introduced the problem of the day as a contextual meaningful problem that we could all access and solve in the way that best suited us. We enjoyed these so much as a class! I took these back to Mexico and used them in a new after school club - Math club for grade 3-6. My students had not been exposed to math and over time, I found them looking forward to math club which gave me great joy! This was my first baby step towards becoming a math teacher and I celebrated the small wins.​

Returning to the Classroom to learn the PYP, August 2014
Halfway through my M. ED. program, I decided to take a courageous step to leave my home of 11 years in Cancun, Mexico. I was an empty-nester, separated and ready for a new adventure so I accepted a position as a PYP Grade 2 Homeroom teacher in Erbil, Iraq (the Kurdistan Regional Government) at İhsan Doğramaci Bilkent Erbil College. I left Cancun for Iraq just a couple months after the war with ISIS began in June 2014. Besides learning to facilitate transdisciplinary inquiry, this would be my first experience to teach math. In Mexico, only the Spanish teachers taught mathematics. However, I was determined to do well, so I rolled up my sleeves, invested the time and pushed forward.

At IDBEC, we had incredibly experienced teachers on our faculty who I collaborated with and learned from. Two of those math champions were Perico Pineda (IBEN) and Nicole Panoho on my Grade 2 Team. One thing I quickly realized about Nicole was how much schema knowledge she brought to her classroom. She had no math anxiety. On the contrary, she loved math and had learned to solve problems with a wide variety of strategies as part of her elementary and secondary education.

As our Grade 2 team leader, Perico brought a passion to differentiate for our diverse learners to ensure all our learners developed as mathematicians; he urged us to use our Team Teaching time to challenge learners at their appropriate levels. These two colleagues helped me to explore and use strategies to teach our learners. Together, we developed visuals as anchor charts to unify our grade level language so that when we were team teaching we were using the same approaches. These were really for me more than anyone else!! I used them to teach myself so I could model strategies for my students in small groups and coach them. 

What I did not yet understand: 

• the difference between strategies and models
• how to make math contextual
• teaching reasoning over simply explaining a procedure to follow (reliance on memorization)
  • using number talks to build reasoning skills without encouraging mimicking
• running a problem string
• the best way to use the tools or manipulatives in my classroom
• how to spiral mathematical themes to ensure everyone was challenged

Vertical vs Horizontal
Our faculty meetings rotated between classrooms where the hosting teacher presented a lesson or shared a best practice prior to the start of the agenda. One afternoon, Frank Lewthwaite (New Zealand origin), Grade 4 teacher, put an addition problem on the board as a vertical algorithm. He shared that his students found it impossible to solve problems unless they moved the numbers from a horizontal number sentence to a vertical addition algorithm. His talk challenged us to move beyond the vertical algorithm and wrote memorization to teach the children to see numbers flexibly. This short number talk played a role in shifting my thinking about problem-solving.

​Strategy Posters (more for me than my learners)

IDBEC Years 2 into Year 3
After the first year with my grade two, I began to make conceptual connections between number (visualizing groups & partitioning), multiplication, division, fractions and time. I decided to leave Time (Measurement strand) to the end of the year after we had a solid amount of experiences with partitioning and working with fair shares, equal groups or fractions of a whole (one item or a group of items). Because of this, I decided to experiment with spiraling the concepts. I created more activities that could serve as warm up or independent small group work. Students would practice visualizing a number they chose (usually rolling a dice or two) and then to move towards thinking about equal groups. I found these to be very effective and engaging. I would make sure they understood what to do in a small group and then reinforce in an independent group when I felt they grasped it. Because it was group work, my students always had support and felt empowered to get tools when needed or to ask for support from a classmate. By leaving the Measurement Strand of Time to the end of the year, I noticed it was much easier for my learners to grasp the concept. They could understand the 60 minutes divided up into chunks based on groups of 5 much better and learning to read a clock was easier.

VIS shared the evolution of their mathematics curriculum as it connects to the IP/PYP for an inquiry-based approach. Chinyelu's impression of VIS's approach to curriculum mapping moved her to share this with our PYP team. As a team, we agreed with her proposal to move in a similar direction.

​VIS mapped beyond knowledge, understandings and skills by including strategies for operations. This approach both scaffolds and spirals the strategies across the primary for complexity. It builds common language between classrooms and grade levels. The accessibility of the mapping supports differentiation and more targeted assessment. The alignment of approaches had the aim of bring about increasing consistency across the division. Below you can see an example of Grade 2. We created a map for every grade level aligning it to Ontario Standards which were more inquiry- based and easier to follow than the PYP Mathematics Scope and Sequence available at the time. The hyperlinks led a teacher to a visual example of the strategy specific to that grade level. These pictures were captured from classroom examples and stored in a folder on our Google Drive.

Because we had the math curriculum mapped out like this for each grade level, it was easier to see how to respond to assessment data so we could personalize the learning. For those working below grade level, a teacher could use the previous years overview; and likewise, for those working ahead, the teacher could differentiate by offering extensions from the next grade level.

Materials for Differentiation
We had many resources spread across different closets that were difficult to access.  Our teachers were working so hard (without a teaching assistant), I felt the need to make not only curriculum accessible but to ensure that tools and manipulative materials could be located quickly. So after getting a space approved, I invested the time to organize a room with shelves for shared materials. They were organized by discipline and then by strand. It was amazing to identify, organize and catalog so many resources that were available for scaffolding the learning. And, we freed up space in the homeroom classrooms by putting these materials in that shared space. Working with all the material in this way across the grade levels enabled me to see what kinds of tools are available and how we can differentiate more for our learners.

October 2019
A year prior to my arrival at IICS,  mathematics learning and teaching in the PYP had begun a transformational process under Rob Grantham, Greta Hazlett and Monica Hoge's leadership. Through the Curriculum Review Cycle, the IICS PYP team had identified mathematics as an area of focus for complete revision and ongoing professional development. Teachers were unhappy with the written curriculum as it was viewed as a long laundry list of knowledge to be taught and ticked off. They were more unhappy with the taught curriculum and the pedagogy of math visible in the learning community. There was clear room for improvement through an intentional alignment of beliefs, practices and knowledge. Professional Development faculty-wide to shift practices towards to teaching for understanding and reasoning was needed.

This led to a two-week Faculty In Residence program with math specialist, Johnnie Wilson, from University of California Santa Cruz. The entire PYP team participated in the residency. And the initiative continued forward driven by a team of motivated teachers committed to the aims laid out with the guidance of Johnie Wilson. The Math Task Force met regularly to develop the IICS approach to mathematics actively involving all teachers. What was the objective, their goal? 

• to create a mathematics curriculum grounded in current research and best practice
• to bring alignment between the program and pedagogy with the mission and beliefs about learning 
  • this manifested as 6 tenets to united the PYP team
• to prepare students for success as mathematicians; to shift mindsets based on a common belief that 'we are all mathematicians'

These teachers met regularly. It was a pleasure to join the team as an interim PYP Coordinator and jump into these meetings. I listened and observed eager to learn from my new colleagues. As a team, NZ Mathematics (now hosted by Australia) was selected as the program to follow because it was accessible, conceptual and strategies based without an enormous list of knowledge to teach. There was the added bonus of two teachers on the team who were thoroughly trained in the program. NZ Mathematics provided a scope and sequence, student profiles to track knowledge and skill development, teacher materials with lessons and assessment tools to track stages (GLoSS, IKAN and JAM). A Program of Inquiry for mathematics was organized to make all the information and resources accessible through a single document on Google Drive.

The Math Task Force lead the ongoing work, making decisions about curriculum, resources and professional development. Many teachers and students participated in an online YouCubed course by Jo Boaler on mindsets. The Task Force drew on the research of Drs. Jo Boaler, John Hattie, Jeremy Bruner, and Van Der Walle et al., as well as the framework documents of the IB PYP. The IICS pedagogical approach to mathematics was elaborated and shared with the wider learning community.

​NZ Maths Pedagogy was enriched with other resources: 

• Van der Walle Books K - 2 and 3-6
• Number Talks by Sherry Parrish
• Mathematical Mindsets Big Ideas K-6 by Jo Boaler
• Cathy Fosnot's Contexts for Learning 
• Rich Learning with Dan Finkel and the games in Math for Love
• Youcubed Tasks​
• Which one doesn’t belong?
• Would you rather?
• Estimation 180 - Great warm ups
• Visual Patterns
• Number Strings
• Choral Counting
• Math Visuals
• GFletchy Progression Videos
• Same or Different?
• Always, Sometimes, Never
• Open Middle - Rich Math Tasks
• Dan Meyer’s Three Math Acts - Rich Math Tasks
• Yummy Math
• Robert Kaplinsky Lessons - Rich Math Tasks
• Inside Problem Solving
• A+ Click Maths
• NRich Maths
• K5 Math Resources and Math Journals
• Math Learning Center - Apps
• Build Math Minds
• ​Math Strat Chat Collections: Follow Pam Harris on Social Media - StratChat - to solve the problem of the week
• Love Maths Games

NZ Student Profiles
These tools support tracking student progress through ongoing assessment. Maintaining one for each child allows the teacher to personalize the learning by keeping track of student progress.

Volunteering for Grade 6 Learning Support Math Groups
That same year, we had some students in grade 6 who needed math support in proportional thinking and fractions. There were knowledge gaps that prevented them from moving forward. I wanted to close those gaps and build confidence for reasoning proportionally. So I decided to participate in providing personalized lessons to get to know the students, the curriculum and build my math capacity. I used the data from the GLoSS Assessment to target specific knowledge and skills through inquiry-based lessons. Knowing my former colleague, Perico Pineda, was teaching grade 5 in Luanda, I reached out to him for some tips. He sent me images of some fractions lessons he was doing with his learners. These activities inspired me to work with my learners creatively and also build my own understanding of the ways of visualizing fractions.

Perico's ideas inspired the following activities that turned into lessons over the remainder of the school year even as we transitioned online with the arrival of the pandemic in March 2020. I used an inquiry-based approach to problem-solving to motivate these learners and they grew. They took risks, developed confidence to participate more actively resulting in more joyful mathematical thinkers. It was wonderful to see my students make improvements in proportional thinking when assessed on GLoSS. Closing those gaps enabled them to move forward.

Jumping forward a few years to 2024 when I was in the middle of Pam Harris' Developing Powerful Fractions 1 course, I believe I can see ways to spiral proportional thinking between fractions, decimals, percentages and ratios. Though, I still need a lot of practice to feel a strong sense of self-efficacy.

Equal Groups Activities
Some of these were designed for language learners to reduce the amount of language to read and through our math groups, we talked about the context together.
      For example, for 3 groups 6, we talked about the unit  - What will we draw? 

• children
• animals
• tools
• dots
• candy
• sports equipment

And then decided together so there was context built in through our math group. As the year progressed, many times I observed my learners giving context themselves as they chose how to visualize numbers. I recognize that by doing these kinds of activities to visualize number and solidify strategies, I was still not really teaching reasoning skills. I believe that came later when I began to take classes with Pam Harris. The way she models problem strings, and dialogue about reasoning is something I aim to develop in myself.

Unpacking NZ Stages - a deep dive analysis to understand assessment indicators 
Following a year in the classroom, I stepped back into the role of PYP Coordinator. I decided to take a deep dive in NZ Stages. I needed to unpack it all for myself so I could make sense of the program, the approach, the stages and the concepts. I needed to make sense each stage; what it meant in terms of knowledge (understanding & facts), skills and strategies for reasoning.

​In order to launch this, I reached out a former colleague from IDBEC, Hannah Moorehouse of New Zealand and who was back in NZ teaching and leading. She thought it would be helpful to look at some of the previous documents and pointed me to the NUMPA. She also pointed me to Twinkle where the NZ Maths Stages were available in a variety of forms such as Posters which I downloaded and posted on my wall.​

Unpacking NZ Assessment Tools
​Following a year in the classroom, I stepped back into the role of PYP Coordinator. I decided to take a deep dive in NZ Stages. I needed to unpack it all for myself so I could make sense of the program, the approach, the stages and the concepts. I needed to make sense each stage; what it meant in terms of knowledge (understanding & facts), skills and strategies for reasoning. 

I began to create a visual using a table to show:

• Stage
• Grade Level (which grade is targeting that stage for year end achievement)
• Expected problem-solving strategies for that developmental stage
• Basic facts and knowledge needed to apply problem-solving strategies at that stage

Once I organized the information this way, I hyperlinked supplemental resources so that this document could enrich the Mathematics Program of Inquiry. In the middle of this process, I continued my journey to unlearn and relearn math by taking my first courses with Pam Harris and Math-is-Figure-Out-Able. I could not resist after listening to two of my colleagues at the time. Lauren Graham and Breda Hayes (see picture below) would share their insights and enthusiasm for building their toolbox of strategies every time I had a chance to sit with them. Both Lauren and Breda inspired me to find my new worry free math self, to continue building my capacity and to relentlessly pursue the joy in 'mathematizing' (invented by Pam Harris). Kendall Jackson pictured with Lauren below inspired me to learn how to use the Rekenrek which I eventually did at Brewster Madrid using Cathy Fosnot materials as a guide. I value these people who inspire me to dig in for ongoing growth. Thank you!

I began with Pam Harris' free course, Developing Mathematical Reasoning. I loved this course so much, that I decided to take Building Powerful Multiplication. This led me to ponder more deeply about spiraling strategies across grade levels, and developing student capacity to become increasingly efficient, accurate and clever with using what they know to solve a problem without rote memorization. 

Unpacking NZ Assessment Tools
​So I then began to analyze the strategies and map those across the grade levels for common metalanguage. I wanted to show how the same strategy can be used on more complex problems while some strategies are stepping stones to more complex strategies. This personal inquiry helped me so much to engage with teachers about math as I felt I could speak their language and make connections to challenges they faced. 

One of these challenges was the assessment tool we used for our lower grades. The NZ JAM was organized in a way that felt inaccessible. Having previously used the SENA (see above), I felt the urge to rearrange the information for flow and accessibility. One major adaptation we made attended to the breadth of numbers to be assessed. Assessing numbers to 10000 in the JAM felt quite daunting for us so we decided to break the content into two separate assessments for numbers to 100 and then up to 1000 moving to 10000. A second adaptation we made attended to the order of the assessment content. We wanted to make it flow more developmentally so Early Years teachers could hone in on the concepts and facts that were more relevant to their age group.​

Personal Math Journal
Throughout all this time, I kept a math journal where I tried out problems and wrestled with strategies. I recommend having a notebook to keep as a reference point as you learn anything new. It is important to try out the strategies so you can own them - flex your math muscles and try out problems on your own. It is great to see the changes in my thinking and my ability to apply strategies that occurred over time.

At the launch of the Brewster Madrid campus, I had the opportunity to make critical decisions about the program. One of these was the decision to join the Concept-Based Inquiry Mathematics Pilot Project. While attending the CBI Lesson Labs Conference 2023 in The Hague, I listened excitedly as Rachel French unveiled the CBIM project. Now I had the chance to participate! I was drawn to this project a couple of reasons:

• It was aligned to the CBI in Action inquiry cycle: I proposed this inquiry cycle to our team and they embraced the idea so it was a natural alignment.
• It would serve as a tool to support capacity building for the team to become CBI facilitators. Everyone also had a copy of Concept-Based Inquiry in Action.
• Understandings for each strand and phase or grade level are articulated and developmentally appropriate. This means teachers could structure the class to support their students to generalize about mathematics.
• The unit flow for each strand was designed for accessibility of the user and easily navigated.
• The responsiveness of PLI over time - I believed confidently in Rachel's team at PLC to build a quality math program and respond to feedback from participating schools to make improvements.
• I was eager to participate in a math program that aimed at conceptual understandings with meaningful, relevant problem-solving through CASE STUDIES!
• I believed this program would in the end, make the life of our teachers less stressful. My previous schools had so much curriculum, it overwhelmed! I wanted to reduce for a less is more approach. I wanted to work with a sustainable math program that would provide a common metalanguage and process to spiral across the Lower School.

At the introductory call, our team met with Kelvin Sparks, the math specialist shaping the program. We shared our challenge of opening a new school with zero assessment data on anyone. We needed to know as much about our students as quickly as possible so he suggested we continue with the NZ Assessments. So we began to track NZ Stages while building our understanding of the CBI Phases. 

The decision to join the CBIM project turned out to be precisely what I had hoped for. All the reasons for which I chose this direction came to fruition with the seeds of CBI popping up on the walls, the digital portfolios, conferences and in learning conversations.

​An inquiry into Visualizing Groups: Subitizing Leads to Groupitizing, and then Unitizing
This idea of Visualizing Groups fascinated me as once again, I was keenly aware of how much I had missed in my own children for a rich math education. Charaine Poutasi, our Early Years 3-4 teacher at IICS, used ten frames and student pictures to take attendance each day. She stressed to me the importance of asking, 'What do you see?' over "How many are there?"  I watched how she organized her playful lessons to spiral around subitizing. The impact on these learners by the time they reached grade 1 was impressive. They knew so much about number by then from a deep sense of understanding, not from a place of habit, memorization or mimicking. This captivated me! So, I finally made some time to dig into this idea. I needed to understand it more deeply especially after Jo Boaler released Mathish. There she highlighted new research on the idea of groupitizing!

Subitizing
Kaufman et al. first coined the term subitizing back in 1949 (Guillaume et al., 2023). Subitizing means learners develop the ability to instantly recognize and see patterns. Humans naturally have the ability to see and group numbers (Boaler, 2024). It is “considered as one of the core systems of numbers since it allows the precise representations of distinct objects” (Guillaume et al., 2023, p. 1). “Subitizing plays a critical role in the development of numerical skills, mostly in helping young children grasp the meaning of the first number words” (Guillaume et al., 2023 p. 1). We begin working with learners through subitizing and patterns over counting from 1 by posing these types of questions:

• How do you see that?
• What do you see?

This emphasis sets the children's focus on what they see and not to habitually begin counting from one. Subitizing is a skill that begins in the early years and continues throughout primary school. It develops the mental flexibility to easily make ten which can then be applied to 100, 1000 and greater when we scale up or down to decimals. Subitizing opportunities can be organized using dots, or patterns with any kind of object as well as pointing out nature or and the patterns around us in the world. It is a skill that can be practiced all the way through fifth grade with disappearing pictures of groups on a slides at which point, you have probably moved your learners to groupitize. Build Math Minds provides many nice ideas for building this skill.

Groupitizing
McCandliss et al. first coined the word ‘‘groupitizing’’ to capture the idea of “capitalizing on grouping cues during enumeration” (2010; Starkey and McCandliss, 2014). Starkey and McCandliss refer to “a phenomenon in which enumeration speed is enhanced by the presence of grouping cues, especially those that cue the presence of subsets in the subitizing range” (2014, p 122). Pam Harris defines groupitizing as the ability to use strategic grouping to parse arrays into subitizing chunks (2024). Jo Boaler explains these as “powerful mental models that are visual and physical and also foundational for a rich and deep understanding of mathematics” (2024, p. 142). Boaler goes on to highlight the importance of developing learner ability to use visual representations both physically and mentally as this has been found to be a powerful indicator for high achievement on high stakes math tests (Boaler, 2024, p. 144).
 
This is a strategy we want to encourage as it supports visualization strategies for sense-making with numbers. When we work with seeing groups (subsets) inside of 2-, 3-, and 4-digit numbers visually, we are groupitizing. We may also be working with groupitizing when we begin to think about fair shares and equi-partitioning. Here is a subitizing/ groupitizing resource or in these slides. Youcubed and Mahesh Sharma of Mathematics for All also have some resources.

Unitizing
“Fractions are relationships: they are defined in relation to an implicit or explicit whole or unit” (Neagoy, 2017). While unit is the root word, unitizing refers to the flexible thinking students begin to attain when thinking about the unit and it usually begins around third grade (Neagoy, 2017). According to Neagoy, “unitize means to separate, transform or classify something into discrete units” (2017, p. 85). Susan Lamon explains it further, “Unitizing refers to the process of constructing mental chunks in terms of which to think about a given quantity” (2020, p. 109). Lamon illustrates this with a 24 pack of drinks: How do you visualize that 24 pack? As 2 twelve-packs, 4 six-packs or 1 whole pack of 24? The way students chunk that visually affects the way they think about the quantity (Lamon, 2020).
 
The term unitizing applies to how we see fractions in relationship to the whole - the unit. A fraction is a quantity or a value. It is not a whole number. It is a small portion of a proportion of something - the unit. It is when we identify the unit of measure or the whole; we always identify a fraction that represents a unit of a whole. It is very important to always understand through unitizing what that whole thing is. Is the unit referring to one granola bar or a group of 5 granola bars? This supports making meaning for the fraction identified.
 
“The concept of a whole underlies the concept of a fraction.” (Behr and Post 1992, p. 213)
 
Continuous units: time, length, area, volume
Discrete units: sets or collections
Composite units: single entities that contain within them a set of items (case of cokes, dozen eggs)
Fractional units: the unit itself is a fractional quantity (i.e. ¼ km, ½ hour)

References

​Behr, M. J., & Post, T. R. (1992). Teaching rational numbers and decimal concepts. In T. R. Post (Ed.), Teaching mathematics in grades K-8: Research-based method. Boston: Allyn & Bacon.

Boaler, J. (2024). MATH-Ish: Finding creativity, diversity and meaning in mathematics. Harper One.

Guillaume, M., Roy, E., Van Rinsveld, A., Starkey, G.S., Uncapher, M.R. and McCandliss, B.D. (2023). Groupitizing reflects conceptual developments in math cognition and inequities in math achievement from childhood through adolescence. Child development [Online], 94(2), pp.335–347. Available from: https://doi.org/10.1111/cdev.13859.

Harris, P. (2024).  Building powerful fractions [Online]. Pam Harris Consulting, LLC. https://www.mathisfigureoutable.com/

Lamon, S. J. (2020). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers, 4th ed. Routledge.

McCandliss, B. D., Yun, C., Hannula, M., Hubbard, E. M., Vitale, J., & Schwartz, D. (2010). Quick, how many? fluency in subitizing and “groupitizing” link to arithmetic skills, Poster Presented at the Biennial meeting of the American Educational Research Association, Denver, CO, USA.

​Neagoy, M. (2017). Unpacking fractions: Classroom-tested strategies to build students’ mathematical understanding. ASCD.

Starkey, G.S. and McCandliss, B.D., 2014. The emergence of “groupitizing” in children’s numerical cognition. Journal of experimental child psychology [Online], 126, pp.120–137. Available from: https://doi.org/10.1016/j.jecp.2014.03.006.

Cognitive Load Theory and Time
And as I reflect on this 12 year Math Journey, which remains ongoing, I reflect on the time it has taken me to undo years of bad instruction. I still get nervous when someone mentions a math problem that I don't quickly understand, I have not arrived! But there is good news, I regularly now make ten mentally to solve problems quickly. I multiply and divide by ten to make the numbers more accessible. I think about larger chunks naturally. I have built a bank of strategies and models to use regularly and I enjoy working through problems, making mistakes, figuring out my error and moving forward. I have clear goals to continue building my own capacity as a mathematician.

I decided to share my Math Journey because I know I am not the only one and sharing our stories will help learn more about how teachers can overcome challenges. As I processed my journey, I began to realize something critical about resistance - there are hidden reasons for resistance and I am beginning to wonder if it is connected to cognitive load theory. The greatest lesson I am pulling from taking the time to articulate my personal learning journey is that this process took significant time. Content had to be made accessible to me over time in chunks. I could not process it all at the start. Going back to 2014 in Iraq, I remember the panic, the ringing in my ears moments where I mentally shut down. Panic would set in because of the extreme lack of self-efficacy as a mathematician. There were nights I would sit by myself and cry out of frustration, embarrassment and desperation.

Learning can be far from a neat, linear process. It can resemble a winding path with unexpected turns, requiring resilience and persistence to navigate. Each challenge and setback presents an opportunity for growth, pushing learners to adapt and develop a deeper understanding. As professionals, there are always ways we can push ourselves to evolve, and we have to recognize the supports we need to survive the messiness of the learning journey. If we as educators can embrace the process it can be so enriching and we experience the transformation. Embracing the complexities and uncertainties of the learning journey can lead to more meaningful insights and a stronger grasp of the subject at hand. By cultivating a mindset that welcomes mistakes as stepping stones, means we are authentically recognizing when we fall short and modeling this mindset for our community. This fosters a resilient approach that ultimately leads to personal and community success.

As I ponder working with teachers, I think on these:

• Funds of Knowledge and Identity
• Schema; Memories; Emotions connected to those memories
• Mathematics Understanding (knowledge, skills, processes, strategies, facts combined)
• New Curriculum - every school means learning a new curriculum 
• Mindset
• Cognitive Load Theory - Where is the correct starting point?

I am left with these questions which may surface as I work on my thesis: 

• How can I get to know the teachers I work with in a similar way I would get to know my students in a classroom?
• How can we as a learning community make decisions about professional development in response to learning needs and are accessible to the teachers? An approach for less panic, and more baby steps moving forward
• What ways can we leverage agency for addressing gaps (known/unknown)? No shame, just support!​​

GAP YEAR PLANS
Now that I am in a gap year, I am pondering what my next learning community will be curious about and what mathematics will look like. I know that I am interested in moving increasing towards ongoing assessment over spending hours and hours on summative assessment. The benefit of having growth markers against a baseline data point is that schools can track assessment growth over time and identify trends. Secondly, transparent assessment data supports shared ownership for learning. As a team, we can respond to the data and meet the needs of the learners. 

However, there is a downside to these. Number interviews take so much time; most times about 45-50 minutes. They generally lead to someone (like me, a learning support team member or a teaching assistant) pulling children for the teacher to conduct the interview. If it is a learning support team member then that person is not supporting students in their learning and their time is traded for assessment. I am unsure now if that is a valuable use of their time. It also means the homeroom teacher has not seen the student perform the assessment tasks. They are given a report with assigned stages thus internalizing what the assigned stages mean slows down drastically. I want to figure out an efficient system that allows us to track over time based on daily math observations and interactions. Our teachers work so hard so finding ways to improve our efficiency will ease the load. 

​I am so grateful to trusted colleagues like Christine Lewthwaite, Perico Pineda, Nicole Panoho of IDBEC; to Linda Allen and Chinyelu Ndubisi, of MEFIS; to Lauren Graham, Charaine Poutasi, Breda Hayes, Michelle Dirlik, Nichole Krissman, Rob Grantham, Monica Hoge and Greta Hazlett of IICS; and Olivia Popovitch, Celeste Hinshaw and Melanie McClean of Brewster Madrid all of whom supported me through moments when math anxiety unexpectedly surfaced, many times unbeknownst to them!​

Recently, I was invited to facilitate a workshop for a private school in Izmir on concept-based inquiry. The school had recently undergone the self-study process and a revision of its guiding statements. With a newly revised learning and teaching policy based on the new mission and vision, the process of shifting the approaches to teaching had to begin. I was asked to shift their beliefs about learning and teaching from a teacher led classroom to that of a thinking classroom through concept-based inquiry. Sometimes shifting mindsets can be quite a challenge because beliefs are challenged and affected by the need for self-efficacy. 

The school follows Cambridge International Curriculum (CIE) and the International Primary Curriculum (IPC). As the course was not solely for primary teachers but including the whole school staff, I had to consider ways to make the content contextual for all ages so every teacher could visualize concept-based inquiry in his or her context and age range. Thus, I chose the run the workshop through concept-based inquiry day and allow the teachers to learn about inquiry by experiencing it. Before I arrived that Saturday, I sent a short survey to pre-assess their understandings so I could be sure to address any misconceptions and try to find ways to answer their questions. 

Some of the questions and misconceptions I saw were:

1. How do I get my children to become inquiry based learners?
2. How do I transition student expectations from traditional to inquiry based and to build their confidence/willingness to fully engage? How does it work? How do I manage it when there is a wide ability range in the classroom? How it would look at different levels (ages)? How do I manage classroom time to meet demands of syllabus? How do I manage preparation time workload?
3. How can we transform the exam-based program (Cambridge exams) to a concept-based inquiry?
4. How do I match the success criteria of a concept-based inquiry unit in Maths to the CIE learning objectives? This type of learning is great for engaged students whom take responsibility for their own learning, but what about the students that aren't engaged, and don't care about their education? Wouldn't their concept-learning then need to become more teacher-led, in order to help them achieve the success criteria?
5. Isn't concept-based inquiry just another word for student centered?
6. Whereas concept-based learners develop skills related to collaboration and justifying their reasoning, many tests reward fact-based learning with multiple choice and short answer questions. This can cause issues when students take standardized tests, as they may not have the breadth of knowledge needed to achieve high scores. How can this be addressed?
7. I have tried to implement in past with Social Studies but felt it didn't work as it was a 'stand-alone' class and not worked on alongside other teachers/classes. How do we use inquiry-based practice and still meet very specific learning objectives in the Cambridge Curriculum? Is it possible to take elements, e.g. focus on skills and apply it to our curriculum?
8. Isn't it necessary for the students to have some kind of background knowledge and at least some basic skills before they can take up the challenge of research and inquiry? When are we sure that our students have acquired those skills? How do we build that previous background knowledge? Are the demands of that kind of knowledge too great for some students?

In order to be able to address these questions and misconceptions, I decided to begin the workshop with a jigsaw activity wherein different groups read articles about concept-based inquiry and created a headline with a summary of the content. During the sharing time, I observed discussion, critical thinking and reflection. Using this learning engagement, enabled the staff to begin to shed some light on their areas of concern. 

Then we used the Frayer Model to define inquiry-based learning before taking a look at a step deeper to concept-based inquiry.

As a time of application, I asked the teachers to plan one lesson that would be inquiry-based using the lesson planner acronym from Jane Pollack, GANAG. We did this rather than planning an entire unit because the workshop was for just one day. To guide the teachers, I gave them a template that explained this acronym more thoroughly with examples to show how it can be useful and relevant to the concept-based inquiry classroom. 

G - goal - with inquiry, we begin with a question or a series of questions
A - access prior knowledge
N - new information
A - application
G - goal review / reflection time

Finally, everyone filled out an exit ticket using the visible thinking tool, I used to think but now I think. This was even more rewarding than the conceptual understandings. It was an amazing day and I was so happy to see all teachers finding relevance to their context - from early years to grade 12.  Everyone left the workshop feeling challenged and empowered to try out concept-based inquiry in their classrooms.

Upon a visit to Reggio Emilia, Italy
When I first learned about Reggio Emilia’s approach to learning, I was forming my research questions for my final Action Research project to conclude an M.Ed. at George Mason University. I was interested in finding a variety of ways to capture the conceptual understandings of my students who had not developed sufficient English language ability to express themselves, as they would have liked. My professor recommended that I check out the work of Loris Malaguzzi and The Hundred Languages of the Child.  The deeper I delved into the resources I found about the municipality of Reggio Emilia and its approach to learning, the more I wanted to be a participant in an International Study Group. This April, I was able to attend along with one of our ECC teachers and what an inspiring experience it was for us both.  
 
Values and Beliefs
We saw the positive impacts of the goals set by the community of Reggio Emilia back in the late 1940’s at the close of World War II; goals to build a community that created new identities and new rights for women and children. This was based on a series of choices; cultural, ethical and political in nature. These goals were intended to foster community participation and innovation in education to make the child a priority in order to build a positive, respectful community for the present and future. A new life, a new way, new values and a new community by committing to the commons, values for the common good.

• Education is a right and responsibility of all and must be available to all
• Relationships – between the schools and the municipality, the families and the schools, the children and their teachers
• A New Image of the Child – a skilled child, building competence through relationships, who enjoys challenges, creating beauty and constructing knowledge through testing, observing and interacting with the world around them. A child full of ideas and thoughts to be honored.

 
Positive Impacts
As the school I’m working at is currently walking through a combined CIS/NEASC self-study, our staff is in in the process of re-evaluating our mission, vision, beliefs about learning and teaching and the roles we play as leaders and teachers. We are searching for the impacts of our endeavors, reflectively searching for evidence and pondering actions we might take. Have we achieved our mission? Are we on the road to achieving our vision? These are big questions and require a lot of thought.
 
This experience at Reggio Emilia enabled to me to see concretely what it looks like when a school system and the community work together to achieve their goals out of a common vision and commitment . The people of Reggio Emilia that I encountered were caring, respectful and helpful. They have a multitude of programs to reach children (0-12th), the elderly and for interaction between all age groups. Their annual municipality budget sets aside 13% for the early years programs. This community reaches out and welcomes visitors, immigrants (17%) and anyone who engages with their community.
 
Visible Learning
One strong value of the Reggio Emilia educational project was to make visible the educational contexts and children’s learning inside the schools and outside in the community. Learning is visible wherever one looks in the early childhood centers and schools. The materials and partially constructed projects left accessible in the ateliers, piazzas and classrooms show the thinking and understandings that are forming. The learning panels published and posted on the walls demonstrate the pedagogy (why), the process (how) and the conceptual understandings (what) the students have developed as a result of the progezzione (project). Publications are produced annually about the projects completed and gifted to parents. Projects are shared with the community in a variety of creative ways.
 
Participation
Parents are welcomed into the schools to demonstrate the value of participation. They are encouraged to come into the piazza, casually drop their children and chat with teachers about life to pass along important information or ask questions. The face-to-face interactions are highly valued and emails are avoided. Parents elect leaders to work alongside the teachers to promote learning and participation within the community. They can suggest field trips and assist with projects while in process and in the publication or presentation of projects to the community.
 
We had the opportunity to learn about two very important projects: the rights of the child and the hospital through the lens of the child. Both projects involved the community and resulted in action that impacted the community positively. Why? The community used the children’s ideas to publish and display these rights in all the schools as well as to make them available to the public through buttons and postcards. At the hospital, the children decided to gift their thoughts to make the hospital more welcoming. These thoughts were organized and artistically displayed by the Reggio Emilia atelieristas. Now patients in the hospital can get some respite from their fears by reading the thoughts of the children, thoughts that provoke smiles.
 
Learner and Teacher Agency
The PYP is releasing enhancements to update their framework and align it to the current research on learning and teaching. There is a new focus on agency and the learner’s voice, choice and ownership. I’ve been pondering this as I wondered what it looked like in action as well as what shifts our staff would have to make to say we both value and empower agency.
 
The teachers of Reggio Emilia value the child to the extent that is agency in action; it is embedded in their practice. They listen to the voice and ideas of the child, documenting the learning process and thoughts through pictures and anecdotal notes. The documentation guides the decisions about learning. It is visible in the centers on clipboards, binders and portfolios.
Teachers are knowledgeable about childhood development; stages of development and take research seriously but do not allow that knowledge to limit their ability to personalize the education for their students. They do not categorize their students into developmental boxes. It does not become a barrier. They honor and respect the child and the fact that each child is unique. All decisions appear to stem from their values and beliefs about the child. They allow their students to develop strategies for finding knowledge and support them throughout the process within their zone of proximal development. They allow their students to spiral back to what they know naturally as they attempt to spring to a new level of development.
 
Teacher agency is visible through the professional development time they use to collaboratively discuss their experiences each week about the learning that is happening. They work together to overcome difficulties, challenges and find the best way to respond to the ideas and thoughts of the children as they thoughtfully steer the projects through provocations, discussions, reflections and time for experimenting.  They work together to consider ways to re-launch a project that may have stalled slightly and continue extending the project to build new understandings. This belief statement resounded with me deeply, “I learn with you and you learn with me.” It is an atmosphere that does not value hierarchy but rather cooperation and collaborative reflective practice.
 
Creativity, Motivation and Curiosity
In this environment, creativity abounds. The walls and displays are student made. Beauty is everywhere. It is fostered and the children’s innate need to make things beautiful is honored. I never saw any child bored or acting out. The environment was relaxed, not tied to a rigid schedule, but allowing children the freedom and time to explore their curiosity. Children actively engage in projects for extended periods of time. The content is relevant and is founded in questions the children have so their interest is peeked. They investigate answers through concrete experiences with materials, field trips and experiments.  Technology is a tool that is used when they find themselves unable to solve problems without it. The environment is rich with materials and places to explore the answers to their questions.
 
They foster creativity by pursuing creativity themselves.  At Reggio Emilia, collegiality is the key to sustaining creativity. The teachers read about it, surround themselves with creative materials, attend museums and art exhibits. They discuss ideas together and allow ideas to flow uninhibited.  The children are their allies in creativity and are prompted to join in the discussion.
 
Tips from Reggio Emilia:

• Creativity comes from your hands; you have to get your hands dirty and use your hands so play with the concepts, materials and tools you’re offering your students
• When you fall in the love with the materials you’re offering the children, they will find it fun, also.
• Feed creativity by surrounding yourself with creative materials, visiting exhibitions and participating in cultural experiences
• Challenge yourself to change your perspective – look at the world around you differently

 
Dreams and Goals  
Now that I have had this experience, I am pondering ways to share my experience with my colleagues; an experience that left me profoundly impacted. I consider the ways we as a team can change the way we see our roles to minimize the hierarchy and increase the amount of collaboration for the benefit of all stakeholders. I’m excited about the shifts in the PYP and can now visualize learner agency (students and teachers). Together we can explore more deeply what that means for our students, parents and to each of us personally. We can reimagine learning and work to increase participation of all – our students, our parents, our teachers, our support staff and our leadership team.
 
I wish to experience that value and idea of professional development….”I grow with you and you grow with me.” and I look forward to exploring ways to make my beliefs about the child visible to our community alongside my colleagues.

Throughout the year, our students will be bringing home report cards and various projects. There is a wide variety of parental responses to the work that students show them. You may choose to reward your child with compliments on intelligence, or buy a treat as an indulgence. You may express disappointment or even get angry. Is there a better way? How can we speak to children about their learning constructively at home?

·      How can you, as parents, respond to these assessments? 
·      How can you help your child develop a growth mindset based on your feedback and that of his/her teacher?

Feedback can shape a child’s beliefs about him/herself. If the feedback is egocentric (feeding the ego with compliments), it can produce unrealistic ideas about self. Or it can decrease motivation or resilience to yield a child who looks for the easy way out more often than not (Mindset Works, 2017). For instance, praising your child for being such a  smart kid rather than for working hard and putting in a lot of effort. 

On the other hand, if teachers and parents focus on feedback that is non-egocentric, both motivation and resilience increase. This kind of feedback focuses on skills, effort, perseverance, goal setting and accepting challenges through which your child will develop a growth mindset. This kind of person accepts challenges, does not give up easily and is highly motivated. 

A fixed mindset person can be defined as…

“In a fixed mindset, people believe their basic qualities, like their intelligence or talent, are simply fixed traits. They spend their time documenting their intelligence or talent instead of developing them. They also believe that talent alone creates success—without effort. They're wrong (Dweck, 2010).”

While a growth mindset person is defined as… 

“In a growth mindset, people believe that their most basic abilities can be developed through dedication and hard work – brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment. Virtually all great people have had these qualities (Dweck, 2010).”

Over the last couple years, I’ve been consciously trying to develop my own skills for giving better feedback, even to my own young adult son and daughter. I look for ways to support them so I can challenge them to continue to grow as human beings rather than just feeding them with compliments. If you’re interested in learning more, you can continue to read some of these sites referenced below or download the book by Carol Dweck on your Kindle reader, Mindset. 
​
For examples of ways to give feedback, click here. 

Dweck, C. (2010). What is mindset? Retrieved from https://mindsetonline.com/whatisit/about/

​Mindset Works (2017). Dr. Dweck’s discovery of fixed and growth mindsets have shaped our understanding of learning. Retrieved from https://www.mindsetworks.com/science/

Following that session, we continued to make connections between our professional learning communities (PLC's) and CBCI to develop further our understandings about the value of CBCI. I provided each PLC with a T-Chart to compare CBCI with their inquiry topic. While some were investigating Understanding by Design, others were looking into methods of inquiry, multilingualism, strategies for ELL to use in inquiry or visible thinking tools. As a staff, we were able to make many connections between CBCI and our PLC's.

Finally, I shared some of my own learning journey as I had moved from project-based learning to concept-based inquiry. I shared some of my personal questions, frustrations and ideas I had found to both deepen and capture conceptual understandings. My goal was to show them the complexity and validity of the journey I had pursued by making it tangible. I shared my struggles, my fears and victories, and my passion for growing and deepening my own understanding about CBCI in the classroom.

We resumed the CBCI course content with a different vibe in the room. There was more interest, more questioning and buy in. Teachers actively engaged in discussions to plan their units of inquiry using the framework of CBCI. At the conclusion of the training, I saw some true shifts in understandings. The most important lesson I learned from my  reflection is that learning must be relevant for all - both students and teachers. They must know why.

During our staff orientation, we decided to look at building a common understanding about teaching and learning through the question of why.  This approach to analyzing our beliefs, our school's belief and the International Baccalaureate's (IB) beliefs enabled us to write common belief statements and became a part of our staff essential agreement. Simon Sinek's idea of changing the order of thew way we sell things from the what we do to the why we do what we do is far more sellable (2011). This approach to building a common understanding strongly appealed to me for several reasons. When we, as educators, agree to work at a school, in essence we are agreeing to promote the mission of that school. In addition, when that school is IB PYP, there is an additional agreement to consider and that is the mission and vision of the IB. Finally, every educator has his/her own philosophy of education which is a strong influence on pedagogical decisions about teaching and learning. I saw this approach as the means of bringing the three ideas together.

We began this journey by using a Venn Diagram with three circles to collaboratively analyze the following: 

Personal beliefs
MEF IS beliefs
IB PYP beliefs

The step was for the staff to discuss their personal beliefs about teaching and learning together in collaborative groups. These beliefs were recorded on the circle for personal beliefs. Then I asked them to read the MEF International School mission statement and vision to identify the beliefs of our organization. Afterwards, everyone read the mission statement of the IB. When all the beliefs were identified, I asked them to begin to find the common themes and write those in the center. From those ideas in the center, I asked them to write one big idea statement.  These belief statements were recorded and posted in the center of our concentric circle under 'why.' 

Throughout the remainder of our orientation we reviewed the school's expectations about teaching and learning, policies and procedures as well as the nitty gritty of teaching. Details about 'how' and 'what' were slowly added by the staff as an exit ticket each day. Below you can see how our draft appeared at the orientation as well as the Venn Diagrams. It was a worthwhile process from which we were able to create a printed version for everyone to post in their classrooms as a reminder of what we believe about teaching and learning as a staff. It strengthened our commitment to implement both the school and IB mission statements.

Sinek, S. [Tedx Talks]. (2011, April 6). First why and then trust [Video File]. Retrieved from https://www.youtube.com/watch?v=4VdO7LuoBzM

Tunguz, T. (2015, July 13) When selling, start with the why [Blog post].  Retrieved from: https://www.linkedin.com/pulse/when-selling-start-why-tomasz-tunguz/