Numicon is a research-based collection of resources to support the teaching and learning of mathematics, published by Oxford University Press in the United Kingdom. Several research case studies can be found on their website HERE.
Numicon is based on the Concrete, Pictorial, Abstract (CPA) model of learning mathematics. This is also sometimes known as the CRA model in the USA (Concrete, Representational, Abstract) and are BC curriculum uses the language of Concrete, Pictorial, Symbolic (CPS).
There are teaching resources, both physical and online, student books and the materials that are called “apparatus”. I have been particularly intrigued by what are called the Numicon Shapes after seeing many twitter posts by Simon Gregg that include them.
An individual set of Numicon Shapes includes the numbers 1-10 as in the picture below.
A box of 80 Numicon Shapes is also available:
I have created some investigation cards to use with the Numicon Shapes.
There area also number lines and other tools available to use with the Numicon Shapes. One tool I have invested in is the baseboards which are great for creating symmetrical or “double” designs on or for visualizing and making 100 in different ways..
baseboard overlays to explore different math concepts
The Numicon Shapes apparatus can be ordered directly through the Oxford University Press website HERE but I have also found them available in Canada online through Chapters Indigo and Amazon.
A collection of free online resources are available through Oxford University Press HERE and I have collated some of them here:
I have always been curious to learn more about Froebel’s work and this summer I decided to do a bit of a deep dive and read more and play with the materials.
Friedrich Froebel lived in the 19th century in Germany and developed the first kindergartens. The Froebelian approach has been adapted all over the world and is based on the following principles:
respect
connectedness
play, talk and first-hand experiences
creativity
freedom and guidance
play outdoors
community
positive relationships
well-informed and qualified educators
Although these principles may seem what is typical is most early childhood education settings now, they were radical ideas at the time.
The blocks that Froebel developed and how children engage with them are grounded in three ideas connected to the above principles:
forms of life – using the blocks to create and represent things and events from the world around them
forms of beauty – focus on the aesthetic, symmetry, pattern, order, design, etc
forms of knowledge – exploring mathematical forms and scientific concepts such as size, shape, area, stability and balance
forms of life: connecting to architecture
forms of knowledge: thinking about fractions
forms of beauty: symmetry and design
Froebel’s Gifts are early play materials intended for use by individual students and designed by Froebel and made by artisans in the communities where the kindergartens were. The German Froebel designed the first six “play gifts” (spielgabe or shortened to gabe) as educational materials that children would progress through and make connections as they played and built with the geometric solids . Further gifts followed and Froebel then developed what he called “occupations” to extend the learning experiences of the gifts. The occupations included weaving, sewing and working with clay. The occupations often transformed the materials unlike the gifts which were intended to go back in their boxes to be used again and again.
The following photographs are of Gifts 1-6 which focuses on the “solids”or what may also be called 3D geometric shapes.
The progression of the gifts and occupations begin with 3D solids, move to 2D shapes then to lines, then to points and then circle back to using lines, points and 2D shapes to create 3D shapes. This progression is connected to the contemporary research of van Hiele’s hierarchy of geometric thinking.
There are clear connections to our mathematics curriculum as students learn about describing and comparing both 2D and 3D shapes. There is also a focus on spatial reasoning in how the gifts are used as students move the shapes, observe them from different perspectives, compose and decompose and transform their orientation.
One of the many interesting stories I came across on the impact of these gifts was about architect Frank Lloyd Wright. His mother was a Froebelian educator and he has distinct memories of playing and building with the the three distinct maple wood blocks. He commented that those shapes were “in” him through those early experiences and inspired many of his designs.
Two books that I read this summer:
Some resources I found online that were helpful include:
This site explains each gift in detail including forms of life, knowledge and beauty. Videos are included. A brief introduction to kindergarten and the Froebel philosophy are included.
This website has a lot of information about the history of Froebel’s philosophy and the use of the gifts. It also includes current research and resources to support early learning. There are a set of downloadable pdf “pamphlets” including one on Froebel gifts and block play that makes connections to how blocks are often used in current early childhood settings.
3. Garden of Children Project
The Garden of Children Project looks at the Froebelian approach in the USA and has collected video, transcripts, images and interviews in order to compile a documentary. Short video clips can be viewed on the website here:
They have a Froebel tab in their shop which has the gifts and some of the occupations.
Current math education research is highlighting the importance of spatial reasoning to overall mathematics development and achievement. See here if you are interested in reading more. The more I learn about the Froebelian approach, I am reminded how these ideas have been highlighted since the 19th Century. We just sometimes let things drop or prioritize other areas of learning when we need to remember to think more holistically and that learning in ultimately about relationships and connections. I am thinking about ways to share these “gifts” with students and thinking of other ways to use these ideas and will share these investigations on twitter and Instagram.
Usually in June, I am reflecting on the professional learning projects from the school year and writing my annual summary report. But this year was a little different…and for purposes of reflection and documentation, I have written about my year of supporting Transitional Learning.
screenshot on Zoom just before a math studio session
We began the 2020-2021 school year with uncertainty, not sure how things were going to unfold. On September 14 I was re-assigned from the district teacher consultant role to being what our HR department called “general teacher” and being responsible for supporting 4000 K-7 students in mathematics and science who were learning from home due to the pandemic while several thousand students attended school face to face in our school buildings. We had just a few days to imagine what this could be and within a week we were “live” with weekly learning plans posted to our portal and live webinars and Zoom meetings scheduled. I was part of a new team of re-assigned teacher consultants who were also supporting students at home in the areas of French Immersion, Language Arts/Social Studies/Career Education and Arts Education. The first few weeks tested me in new ways and I do not think I have ever felt that degree of stress and responsibility in my entire teaching career.
I learned how to use a whiteboard within a confined space that was visible during webinars and how to teach to hundreds of students at a time that I didn’t know and layer language, gestures, visuals and other supports for their learning.
I learned all the new tools involved with this type of remote learning – webinars. Zoom meetings, using an iPevo mirror-cam and a HUE document camera. I used sites such as desmos and a variety of virtual manipulatives and online resources from Mathigon, the Math Learning Center and Toy Theater.
where I stood for the four “live webinars I did every week
Each week I developed and wrote four math and science plans (K&1, grades 2&3, 4&5, 6&7) prepped and delivered four live webinars, filmed and edited a weekly overview video, held a weekly webinar for parents and caregivers and hosted a Zoom session for TL teachers each week. It was a lot. I kept telling myself that this was my way of serving the district this year until February 1…
One of the highlights of the fall and winter was inviting our K-3 TL students to submit videos of themselves counting from 1-10 or 20 in their home languages. I then shared these videos during the primary math webinars and we practiced counting in different languages including sign language. I now have a collection of counting videos in over thirty languages!
I continued to develop a SD38 YouTube elementary math channel that I started in the spring of 2020 when we all were staying safe at home. I never thought that I would have a YouTube channel. My sons think it is quite hilarious! It has many many videos now – weekly videos for the TL learners and their families teachers are unlisted but all the math games and projects are public. You can find it here: https://bit.ly/SD38mathyoutube
In schools, TL teachers supported their students with regular Zoom meetings and assessed the mathematics assignments that students posted to their eportfolios. I found it increasingly difficult to see that there were hundreds of students in the webinars but not be able to see them or even see their names. It is hard to connect and develop relationships through the camera on your laptop! During the week before the winter holidays, I offered some optional Zoom math studio sessions which were well-attended and I finally felt some joy in the work I was doing.
The Board then voted to extend Transitional Learning to spring break. Some students returned to their schools on February 1 but most continued to stay at home. I held weekly Zoom math studio sessions for primary and intermediate students and added a primary math storytime each week. These sessions were the highlight of my week and I began to get to know many of the TL students through this format. During this time, one of our teacher consultant colleagues was asked to take on the science component of the planning and I was able to streamline a few things in my weekly plans in order to make time for a 20% return to my teacher consultant work. For this part of my work I focused on our district’s numeracy visioning and framework development and the Ministry’s numeracy proficiency standards project.
One of the highlights of doing Transitional Learning was seeing the math and numeracy tasks and projects that the students did. Some teachers shared their students eportfolios with me and other teachers or parents would send me emails with student work attached for me to see. For some tasks and math studio projects, I created padlets for students to share their projects with each other.
tangram pictures from our math studio sessions
And then just before spring break, the Board voted to extend Transitional Learning until the end of the school year. This was a hard and difficult year, not just for me and my colleagues supporting Transitional Learning but for everyone. I kept reminding myself of the students, many who had been at home since March 2020, and continued to do what I could to serve and support the students, families and teachers I had been tasked with supporting this year.
June has brought a lot of “lasts” – last video filmed and edited, last project posted, last math game shared, last primary math storytime, last set of studio sessions and I took a photo of the portal screen when I posted the final week of TL math plans – Week 36. And on Wednesday, June 23, I hosted my last live math webinars – what a year.
I never expected in September 2020 that this would have been my year. I did a completely different job than what I had “signed up” for. I learned a lot about myself, my district, our families and students. And I learned that regardless of the context, in person or remote, the heart of teaching and learning is relationships. So yes, there has been a LOT of math for me this year, but learning to nurture connection and care through a web cam has been at the essence of my work this year.
For the final week, I asked students to reflect on and celebrate their math learning from this year. We had done some different weaving projects and I invited students to use those techniques to weave together their reflections, goals and celebrations of learning. I decided to do the week 36 project as well and invited students to share their final projects on a padlet. It was so lovely to read over what the students shared.
It was both an honour and a pleasure teaching this group of students and I hope as many of them return to our schools in September that I might actually get to meet some of them in person!
In February 2021 I was able to carve out one day a week from my Transitional Learning duties to work on some district numeracy projects.
K-12 Numeracy Vision and Framework
Our school district’s board has a new strategic plan for 2020-2025 and the first priority is Inspired Learners. One of the goals for this priority is: the district builds literacy, numeracy and digital literacy through innovation and a commonly held vision. And the objective I have been focusing on is 2. develop and implement a K-12 numeracy vision and framework. The team I have been working with includes secondary curriculum teacher consultant Shaheen Musani and Kate Campbell of the District Support Team. We have had regular meetings this last term with the district executive team and our parallel literacy team to develop a vision statement and framework, including a visual image that we hope will be “clickable” online and lead to resources and supports for educators and family and community members. We have been able to get feedback from teachers in the district who have been involved with numeracy working groups and projects over the last two years and look forward to continued development of the framework and stakeholder feedback. The overall goal of this vision and framework is to consider what is a numerate learner and how do learners experience numeracy across all areas of learning and contexts.
Ministry Proficiency Project
The BC Ministry of Education is developing new literacy and numeracy performance standards. Last summer myself and some other teacher consultant colleagues were in the first development group and shared literacy and numeracy tasks from Richmond classrooms. A provincial K-5 working group has further refined what are currently being called proficiency benchmarks and this spring several teachers in our district trialled numeracy tasks and provided evidence/illustraitons of learning for the different numeracy aspects and sub-aspects. The provincial working group will continue to refine the proficiency standards this summer and the grades 6-12 proficiency standards should be available next year with final numeracy performance standards available by 2023.
The following table shares the current numeracy aspects and sub-aspects that are included in the assessment tool:
This spring some updates, revisions and additions have been made to the resources and they will added to the above link as well as to the Mathematics and Numeracy tile on the portal.
Numeracy Presentations from May 21
On our district professional learning day on May 21, I shared a short informational session for K-5 on the different numeracy initiatives and projects in our district. Shaheen Musani joined me for a second session with grades 6-9 teachers. The slides from both sessions can be found on the district portal in the Mathematics & Numeracy tile.
SD38 Numeracy Tasks and Resources
On the portal, under Learn 38. under Resources for Teaching and Learning 2020/21, under Curriculum Specific Links and then in the Mathematics & Literacy tile, you will find several links to numeracy resources. A direct link is HERE for SD38 employees.
Also on the portal in Learn 38 are all of this years’ weekly math plans for Transitional Learning. For each grade, in each week, there is a numeracy task. Some of these tasks are intended to help students build their understanding of what numeracy is and others are tasks that students need to use the numeracy processes of interpret, apply, solve, analyze and communicate.
What are some actions you can take to support the development of numeracy within our district?
Who could have predicted when we left for spring break on Friday, March 13, that we would not return to school as we have known it, for the rest of the school year. This spring has brought announcements, updates, change of plans, curveballs and new information and expectations on a regular basis. A global pandemic has brought changes to our lives, and to teaching and learning.
I never expected to have a YouTube channel, to be Zooming multiple times a day or to be doing live online math studio sessions but that’s what happened!
A highlight of each week was Thursday mornings when I took our math studio experiences online via Zoom. I hosted a primary session that usually had 30-40 students join in, and then an intermediate session that grew to about 15 students. I bought a special arm to hold my phone to use as a document camera so that I could do demonstrations for the students.
Zoom studio set up in my dining room
Some of their favourite projects were mazes, origami, cardboard shape structures and using the digital microscope.
using the digital microscope on Zoom on Global Outdoor Classroom Day
The children in the Zoom sessions found ways to connect with each other – waving, calling out and commenting in the chat box. I loved hearing their little voices calling out, “Ms Novakowski, I need help…” as they worked on a project, holding it up to the screen for me to see, with a look that suggested that maybe I could reach through the screen to help them. These Zoom sessions highlighted the importance of the need for interaction during this time.
I also was fortunate to continue my learning with the grades 5&6&7 class an their teachers at Quilchena through Zoom math chats and numeracy tasks. More information can be found HERE.
I had regular weekly or bi-weekly Zoom chats with several small groups of educators throughout the term. Sometimes we were just checking in with each other but other times were time for planning together, sharing ideas and collaborating.
I also spent more time focused on content creation this spring than I have in a long time – from K-7 weekly math plans, interdisciplinary projects, math game resources and at-home learning ideas for families. It was an opportunity for me to think about connecting the dots across the grades as well as across curricular areas. Most these new resources are curated on our district portal, as well as on a page on this blog here:
Every Wednesday at 2pm, about twenty Lower Mainland educators in similar roles as mine met via Zoom to share ideas and resources around math education during this time. As events and protests here in North America amplified the important discussions around anti-racist practices in education, Shaheen and I were asked to sharing some of the work we have been doing in the district around re-humanizing mathematics through culturally responsive pedagogies and investigating social issues. This focus re-framed much of the discussion within this group and our planning for professional learning opportunities for next year. Some reading and resources can be found HERE.
At this point, as we come to the end of the school year, it is still unknown what September will bring. We have learned to much and developed so many new skills over the last three months and we will adapt again in September. For now, I am going to let go of some of my wonderings – wondering how The Studio and its focus on materials might look in the fall and wondering what format my district role and responsibilities will take on.
I’m looking forward to a summer of rest and rejuvenation through spending time with my family, my backyard, reading, creating, learning new things and enjoying the outdoors (and spending less time on Zoom).
As the province moved to remote learning from home due the COVID-19 pandemic, the Quilchena teachers and I met via Zoom to plan numeracy experiences for the students that were connected to our current context. I joined in on the class’ Zoom meetings for Math Chats where we did routines like number talks and WODBs and also built content knowledge to support projects and numeracy tasks.
sharing mathematical modelling connected the COVID-19 virus during a Zoom Math Chatbuilding understanding of fractions during a Math Chat on Zoom
Two the at-home numeracy tasks the students engaged in this spring were Acts of Kindness and Garden Design projects.
In response to the pandemic and in-school instruction being suspended, we thought about ways for the students to connect to each other and the community. They were asked to participation Acts of Kindness. Some students painted rocks to leave around the neighbourhood while others created signs to put in the front windows of their homes. We also asked the students reach out to their classmates through phone calls. We asked the students to consider the “ripple effect” of their acts and to create a visual model to show the impact. The following are some of the students’ visuals which some also connected to ways to prevent the spread of the COVID-19 virus.
Another numeracy task the students did was a plan and design task, creating garden beds, boxes or containers. During our math chats, we learned about different kinds of polygons, how to calculate area and perimeter and then volume. Students were also asked to research and budget for how much soil they would need.
The following is the final task criteria:
And some examples of student submissions:
I am looking forward to continuing this numeracy project with this class next year. With thanks to teachers Jen Yager, Sam Davis and Christine Gerencser.
On February 25, on what would be our last face to face time together, I spent the morning with the grades 5&6&7 class at Quilchena. At that time, I was still thinking I was going to Reggio Emilia, Italy for a study group on spring break and I shared a project from that community with the students to connect to their class’ Agents of Change and Human Rights project. In Reggio Emilia, the community has organized a Flags 4 Rights project where local children, as well as children from around the world, have submitted artwork sharing their thinking about children’s rights. The artwork was then transferred to large flags that were hung throughout the community.
A primary class from Grauer created artwork that was sent to Reggio Emilia. The following is just one example:
We invited the intermediate students at Quilchena to engage in a plan and design numeracy task, considering creating flags for their school, what size they would, where they would be hung and what their budget would be.
The students measured, estimated, visualized and drew their plans to think about about where they would hang the flags and what size they could be.
The students shared and compared their plans. It would have been great to have been able to follow through on this task and think about what students could design and create for their school after spring break but alas, that was not to be!
On January 29, I spent the morning with the grades 5&6&7 class at Quilchena thinking about numbers in the news. We connecting this idea to our study of infographics and how numbers can be used to inform, persuade or evoke emotion in us. We wondered how headlines might use numbers to pull us into a news article and how we can use numbers to make sense of the world around us.
We began by asking the students what their sources of news were. Many replied hearing it on the radio when they were driving with their parents. We shared sources of both local and national/international news.
We then shared some different current headlines with the students and asked them to analyze them with the following questions:
We then invited students to choose a current event that they were interested in, either local or global, and to find a news article about it that included the use of numbers, whether it be quantities, measurements or percentages.
The students investigated their news articles individually or in pairs or triads. The events of most interest to the students included the wildfires in Australia and the recent plane crash involving Kobe Bryant.
The students shared their findings at the end of their time together and compared data from different sources.
The COVID-19 virus outbreak was just beginning to be a focus in the news and some students were interested in learning more and chose this as their current news. Who could have ever predicted at the end of January what this would have meant for us here in Richmond and all the mathematical modelling that has been involved with our public health guidelines in the province?
The Richmond School District’s Math Play Space popped up at several school and community events this school year.
In the fall we were at Blundell Elementary for their Meet the Teacher Night/Open House:
Blundell Elementary, September 19 2019
We visited Kingswood Elementary during one of their afternoons of parent-teacher conferences:
Kingswood Elementary, October 16, 2019
Gilmore Elementary invited us to their school during an afternoon of conferences with families:
Gilmore Elementary, February 18, 2020
We set up in the school gym at Diefenbaker Elementary during an afternoon of conferences with families. We set up earlier in the afternoon so some of the classes could come by for a Math Play Space experience as well.
Diefenbaker Elementary, March 3, 2020
Our last school visit of the year was at Walter Lee Elementary where the Math Play Space was set up in one of their wings during an afternoon of student-led conferences:
Walter Lee Elementary, March 5, 2020
We also set up on the main floor of the Brighouse branch of the Richmond Public Library for two afternoons:
October 17
February 13 – What different ways can you make a heart with the materials? (and other playful creations)
A public blog with an events calendar and information for parents related to the Math Play Space can be found HERE.
Unfortunately due to the COVID-19 pandemic, our events needed to be cancelled in the spring. We look forward to re-imagining how the Math Play Space might be experienced as we move forward with new health and safety protocols.
Just before spring break, I was fortunate to collaborate with secondary colleagues from our department, Shaheen Musani and Baren Tsui on a professional learning series for secondary school teams. The series is called Math for Action and looks at the intersection of mathematics and social issues with a focus on student agency. A goal of the series is to elevate the importance of numeracy across curricular areas.
A variety of resources were shared. A book list that supports this series can be downloaded here:
We looked at this Social Justice Lens, as created by the BCTF’s Social Justice committee.
It felt like we just got started with this thinking together and then in-school instruction was suspended. We look forward to continuing this series in some format when we return to school in the fall.
