 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! Stage 1: George Mason University M. Ed. ProgramA 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. Setting the stage Dr. Baker modeled math routines through the way she organized our learning day at GMU that summer. We solved a problem of the day, experienced a math read aloud and played with strategies that she showed us. Below are some of the topics we talked about which all made sense to me and gave me ideas for how I would approach math instruction for my young learners at IDBEC. 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. Math Picture Books Every day, Dr. Baker included a read aloud which I absolutely loved! This inspired me to begin my own collection of math related pictures books. And when I arrived to IDBEC in Iraq, we had Marilyn Burns collections available which I readily put to use with my learners. Later I discovered Book Source which has collections of math read alouds. Stage 2: My initial steps as mathematics teacher at IDBEC... 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:
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)  Introduction to Count Me In and Count Me In, Too! Our school was following the Ontario Mathematics Standards supported by Australia's First Steps Math, Count Me In and Count Me In, Too as well as some printed Abacus workbooks. It was overwhelming for me to get my head around; huge amounts of information that I could not easily digest. Stacks of materials that were in essence inaccessible because of the time it would take me to read them and then make sense of them. Over time, I began to create my own games for the children to play using dice to build number sense. I also spent a lot of money on Teachers Pay Teachers to download games. My Australian colleagues introduced me to Partitioning as a tool to build number knowledge beginning with numbers to 10, 20 and then 100. Then we had in-house training sessions on using games to build number fluency. It was a monumental work on the part of our colleagues who organized this program for our team. Karen Zuvich and Gail Houghton brought deep knowledge about mathematics to our team. I remember sitting in meetings listening to them elaborate on math strategies which were so far ahead of me cognitively, I could not follow as I would have liked. This manifested as resistance; however, I was not actually resistant. Simply, I just did not understand yet. I could not make sense of it in my head and get a vision for math. I needed someone to plan math with me every week, to show me how to use those games, to help me understand what knowledge and skills I was assessing through those games and how I was helping children build fluency. There remained so many unanswered questions:
I continued in survival mode, buying materials from Teachers Pay Teachers and creating my own materials. Schools frequently provide Literacy Specialists to support teachers with the complexity of teaching students to be communicators but a Mathematics Specialist is rare. I would have loved to have ongoing access to a math mentor but they had full time jobs in other grade levels. Years down the road, now that I have spent a couple years taking math courses to relearn math, I want to go back to the resources of Count Me In, Too. I want to analyze the progression and how they build fluency. I want to play with those math games with students to notice where it supports their conceptualization or comprehension of math facts and concepts. 
Designing Math Games Over time, I began to create my own games for the children to play using dice to build number sense. I also spent a lot of my own money on Teachers Pay Teachers to download games that made sense to me and connected to the Ontario math standards. My colleague, Karen Zuvich, introduced me to Partitioning as a tool to build number knowledge beginning with numbers to 10, 20 and then 100. This is a strategy I have held onto and continue to use and build with learners working on fluency. Fun Partitioning Game:
 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. Assessment at IDBEC Our school decided to follow Ontario, Canada's mathematics standards for number knowledge and strategies. However, as a school we used the Australian SENA to assess and track number knowledge and strategies. We did not tackle the work to see where the SENA aligned with the Ontario standards and where there were gaps. Using the SENA 1 and SENA 2 helped me to build understandings about the content I should be spiraling in the classroom. This led me to implement Number of the Day for number talk warm up with the whole group. I then began to include Number of the Day in our weekly centers work (examples below the sena assessments).
Stage 3: Mathematics at MEF International School...Curriculum Mapping Opens Doors to Differentiation (2017-2019) When I joined MEFIS as PYP Coordinator, we had a significant number of new staff in the primary. The timing was relevant for a team review of our Scope and Sequence documents. We looked for ways to bring alignment of pedagogical practices and curriculum across disciplines including mathematics. Our grade 4 teacher and STEAM specialist, Chinyelu Ndubisi, was keen on joining me in this endeavor so MEFIS sent her to a Mathematics in the PYP Conference in Vienna, Austria. While attending Vienna International School's Mathematics conference, Chinyelu had the opportunity to not only learn more about problem-solving strategies, but also obtained access to a tremendous amount of resources. She was also "able to get a 'whole picture' of how the PYP works" through the lens of mathematics. Chinyelu came back inspired.
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.  Linda Allen, our principal, was from New Zealand so she joined our committee to conduct the review and support the mapping process. It was a deep dive into strategies for me serving as an incredible year of math professional development. Then, to dive deeper into assessing for differentiation, we applied the SENA Assessments across the entire school. Linda and I pulled students to assess their knowledge and use of strategies for solving problems. This experience with students through grade 5 enabled me to build deeper understandings about the assessment of fractions. Investing time in conducting these number interviews provided me, the observer, with multiple ways of learning how our students visualized and approached problems. This informed our teaching, our professional development and how we use resources. At the time, I was able to download these continuums (see below) from Australia's New South Wales Education website. They are no longer available but I have added those PDFs here.
 Mathematics & Connections to CBCI When I arrived at MEFIS, I had recently attended and graduated the CBCI Institute in Haarlem, The Netherlands (2017). Dr. Jennifer Chang Wathall presented on mathematics demonstrating ways to teach the concepts of math through experiences and rich real world problems. Mathematics involves both processes and knowledge which are important for us to be aware of so we make space for those in the classroom. As we went through our curriculum, I noticed the lack of understandings. Most curriculum focuses on the skills to develop and facts/algorithms/rules to drill for memorization.  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. Stage 4: IICS Math Task Force nurtures curiosity for ongoing inquiry... 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?
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:
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.
Stepping back into the Classroom: Grade 2 Distance Learning (2020-2021) For the school year 2020-2021, I made the decision to return to the classroom as I really loved the team at IICS and wished to remain. I was hoping that a leadership position would open up in the near future. This provided me the opportunity to further develop my capacity as a mathematics instructor building on what I had learned at MEFIS and in the past year at IICS. We made the decision for that school year to use Eureka Math from New York State as it provided slide decks and printable sheets that we could use for Distance Listening. Every day, we provided small group instruction online for math and literacy. For each week, I edited the slides for a week of math warm ups, number talks and application time. Learners had printed packets at home to practice with their parents - something I would not do in the classroom on a regular basis but given the circumstances, I had to choose the best approach to simplify learning for parent involvement that minimized frustration. The nature of teaching young grade 2 learners online was tough however, using my tools creatively while posing questions, I was able to keep my learners engaged. I used my iPad as a doc camera so I could demonstrate when necessary or document visually what a learner was explaining when solving a problem. We honed in on learning strategies for efficient, clever solutions such as making ten, compensation, and using place value. We learned to model on a number line, to use arrays for moving towards multiplicative thinking. We did a lot of work with equal groups and for those working at a more advanced level, we began to explore multiplying and dividing by 10. As a visual person, I began to analyze what I was doing and make visuals. We began with a simplified version of the process and practices of a mathematician for our learners. From that I created visual for myself to extend on it.
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?
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:
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.
As a team we trialed these and made changes over time to improve usability. Here you can see Number Interview I (stages 0-4 and limited to numbers to 100). The front cover checklist was organized with the intent to track a learner's progress over time. We were pondering what it would look like to use this to track development without necessarily conducting the interview. Interviews take a considerable amount of time so learning to track development as an ongoing process is a paradigm shift. Over time, after working with these for several years, I realized that I had internalized the indicators for Stages 0-5. Because of the deep dive work I did to build my understanding and then following it up with administering these assessments, I was able to work with a children and quickly see what stage they were working at and what would be the best instructional response to move them forward. This was useful when I took on the role of Founding Head of Lower School at Brewster Madrid.
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.  Stage 5: Brewster Madrid joins the CBIM Pilot Project... 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:
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.  Math Learning Support Over the next two years, I supported identified learners with significant gaps in basic knowledge to be able to move forward. These learners were dependent upon non-efficient strategies and facing issues with accuracy. This left some with low levels of confidence and self-efficacy. So I needed to work on mindset as well as to shore up the basic knowledge that would open the door to efficient, clever problem-solving. Using problem strings, I was able to document their thinking and talk through the problems together. Pam Harris urges problem strings over number talks because it is accessible to all learners in the room. According to Pam Harris, "everyone grows and everyone participates" as we engage with the string, make observations together and strategically asking share our reasoning as a community. It is not a 'how to' talk where we explain how to solve the problems. I found using these with my students built confidence to communicate as the mathematized - learning from mistakes and identifying efficient, clever strategies together. At the same time, I continued to unlearn and relearn math with Pam Harris and Journey. I took Building Powerful Division and then her Building Powerful Fractions 1. Then I began to practice this with my students using her Problem Strings. I wanted to learn to facilitate the thinking in the group without teaching them a procedure to be memorized and mimicked. I wanted my students to internalize ways to see problems that helped them move away the inefficient, clunky strategies they were currently struggling with. Pam Harris provides many free resources which you can try out to get a taste of her approach to math - instructional routines, reasoning and thinking together about problems. Check out her #MathStratChat on social media for the problem of the week. And listen to her Podcast with Kim Montague when you have a chance as it always makes me smile! Now that Pam Harris has released Numeracy Problem String Books for Kindergarten to Grade 5, I have a copy of each. I love them! My students engage in problem-solving enthusiastically, they developing reasoning skills and the joy of working with numbers surfaces. We ordered these resources for all our classrooms as a supplement to the CBIM project so that Number Operations becomes an ongoing endeavor across the school year. 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:
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. Stage 6: My Next Steps... 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:
I am left with these questions which may surface as I work on my thesis:
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!
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AuthorAs an international educator, I work with colleagues in my local and global network regularly to implement inquiry through concept-based approaches to learning and teaching. It is a journey of discovery, learning and growing our own understandings about the ways children learn. Categories
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