Archive for the ‘primary’ Category

September thinking together: What is math?

Posted on: September 19th, 2017 by jnovakowski No Comments

This year I am going to share a monthly focus as a way for educators in our district (and beyond, of course!) to think together, collaborate and share ideas around K-12 mathematics education. On the list are number sense, estimation, reasoning, spatial awareness…it is a list in progress so suggestions are welcome.

My intentions are to begin each month with a blog post highlighting the focus area in our BC mathematics curriculum and connecting it to the broader field of mathematics education. I plan to share links to websites and resources, share books that I have found helpful and provide examples of mathematical tasks from Richmond classrooms. During each month, I will also tweet out related links, ideas, blog posts and photographs from classrooms.

For September, I thought we’d start with What is math?

I am fortunate to have opportunities to sit around tables with educators from many contexts – elementary, secondary, post-secondary as well as working with parents and students. What I have found over many years of having the conversation around What is math? is that there is much diversity in definitions and responses. Some views are quite narrow and focus on number, computation and operations while other views are much broader in topic but also in what it means to be a mathematician.

Mathematicians such as Fields medallists Maryam Mirzakhani and Cédric Villani have said that mathematics is a creative, collaborative endeavour. Other mathematicians emphasize that mathematics is more about justification and proof than getting the “right answer”. One thing that pretty consistently comes up from those who engage in mathematics is that it takes time – sometimes a problem or proof takes days, weeks, years.

How do these ideas about mathematics resonate with you? with your mathematical story?

As a classroom teacher and when working with pre-service teachers at UBC, I began the year with an individual brainstorm or web around “What is math?” – and these responses were added to a collective chart or web. For the pre-service teachers I worked with, I also asked them to tell my a little a bit about their background and experience with mathematics. These short narratives and webs usually gave me quite a bit of insight and where we needed to begin as a class.

What assumptions, conceptions and understandings about mathematics do students carry with them into our classroom communities? What feelings and beliefs do they hold?

This week a grades 2 and 3 class visited The Studio at Grauer and we began by talking about what is math? I then invited them to explore the materials, images and books in the studio space and to investigate something that piqued their interest.

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As we gathered back together, the students added to our original list. Interestingly, there additions were much more focused on mathematical experiences, different from naming mathematical content.

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What do our classroom environments say about what mathematics is? Do the images, books and materials we offer inspire our students and nurture connections? Do all students see openings to engage in mathematics? 

 

Ideas to nudge students’ thinking about what mathematics is:

What is math?

Create a class chart or math graffiti wall that can be added to as the school year progresses. Students can add images, diagrams, words, phrases, etc. Students can also use materials to create representations of what math is. The following is “math sun” created by a kindergarten student at Grauer last year – when I asked her what made it a math sun, her reply was that it was “full of math”.

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Where do we see math? What math lives here?

Encourage students to think about math beyond the classroom and school. Where do they experience and see math outdoors? in the community? at home? Create an area in the classroom to add photographs and materials found in the local environment that might inspire mathematical thinking and connections.

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A grade 4&5 class I worked with in The Studio at Grauer shared some of the math they experienced over the summer and then the grade 2&3 class added to their list this week:

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Many elementary classrooms have “wonder windows” to encourage students to observe and wonder the local environment. This year, we have added a math window to The Studio.

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I facilitated a K-7 place-based mathematics project at Byng a few years ago and one of the tasks classes engaged in was math walks around the school and in the community. Sometimes a specific focus was selected such as What shapes can we see? but we mostly looked for math to world connections. One class created a photo book while others created math problem posters (sharing problems the students posed inspired by their photographs) or concept panels.

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For the last three summers I have participated in a Twitter challenge with mathy types from around the world. Each week a math concept is posted and the challenge it to take photographs of the world around us that connect to that concept. Concepts such as estimation, tessellation and scale were explored this year. You can find this year’s posts on Twitter by searching #mathphoto17 – and here is a photo book I created of my photos and tweets from this year’s challenge:

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I will be launching a district-based K-12 math photo challenge on Twitter soon – hashtag #mathphoto38 (the 38 for school district 38) with plans to document and share the photos over the school year. We will begin with photos that respond to the question: What is math?

Make mathematics visible in your classroom and school

I also try and make different ideas about what math is visible to students, to parents and to colleagues. The following panel was created with images of the representations created by Kindergarten students as they responded to the question, What is math?

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A middle school teacher created an interactive bulletin board based on the instructional routine Which One Doesn’t Belong? to engage the whole school population in mathematical reasoning and communication – important mathematical work and this idea builds mathematical community in a school.

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source: Twitter

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source: Twitter

And for high schools – I think this is inspiring and helps to expand students’ notions of what math is. High school math teacher, Sara VanDerWerf, from Minnesota, has created a play table space in her classroom to engage students in thinking and playing with mathematics in different ways. She shares photos on twitter HERE and shared a blog post about play tables in high school classrooms HERE. Is there a secondary math classroom in Richmond that would like to set up a play table? I’d be happy to help.

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Think about how mathematics is experienced across cultures, across the world and across time.

Mathematics is a human construct and is often narrowly define through a Western or European lens. There is much evidence that mathematics as it is typically defined, existed in Asia long before it was “discovered” by Europeans.  There is a long history of cultural practices across cultures from all over the world that we would now label as mathematics. Alan Bishop has done considerable research in this area and describes six mathematical practices or activities that exist in all known cultures – counting, locating, measuring, designing, playing and explaining. I have found students find it interesting to learn about different number systems or how measurement is often contextual to a culture and environment. Some examples of these cultural practices are included in the elaborations for the learning standards in our BC math curriculum.

Seeing and experiencing mathematics as a creative endeavour

For the past three years, Dr. Jo Boaler and her “youcubians” have launched a week of inspirational math to begin the school year. There are a variety of videos and open mathematical tasks available for grade bands from K-12. The focus is developing a mathematic mindset  with messages such as: we can all learn math and we learn from mistakes. Resources can be found HERE.

There are many videos available that show mathematics as creative and inspiring but a particularly interesting youtube channel is created by self-defined mathemusician Vi Hart, daughter of acclaimed mathematical sculptor George Hart. I think her videos are particularly great for students in grades 5-12. Her channel is HERE.

There are lots of ways to nurture the creative thinking core competency (BC curriculum) while engaged in mathematics.

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The three facets of the competency are: novelty and value, generating ideas and developing ideas. I see these facets enacted when children engage in number talks and consider different strategies for solving mental math questions, when children engage in a rich open task or problem or when they apply mathematics to create or design something.

 What is math?

How will you investigate this idea yourself and how will you investigate and extend your students’ thinking about this over this school year?

~Janice

 

References

A Mathematician’s Lament by Paul Lockhart (with new books Measurement and Arithmetic)

Mathematician Keith Devlin’s blog: Devlin’s Angle

Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms by Tracy Zager

The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph

Bishop’s six universal cultural activities

Mathematical Mindsets by Jo Boaler

summer professional reading: Spatial Reasoning in the Early Years

Posted on: August 2nd, 2017 by jnovakowski No Comments

IMG_6498Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations

by Brent Davis and the Spatial Reasoning Study Group

published by Taylor & Francis, Routledge, 2015

 

 

 

 

This book isn’t really a typical professional resource for teachers – it is much more of an academic read. The Canadian authors are all affiliated with universities and their areas of research intersect around spatial reasoning, particularly in the early years. Their work draws about mathematics, education, neuroscience, psychology, art and more.

I have had this book for awhile as this is a particular area of interest for me and is inspiring our work for our BCAMT Reggio-Inspired Mathematics project. I attended some presentations affiliated with this group’s work at the PME conference in Vancouver in 2014 and I think it is important work for classroom teachers to consider as we try to broaden our thinking about what mathematics is, what different entry points there are for children seeing themselves as mathematicians and for using instructional tasks in classrooms that are research supported.

The authors begin with a working definition of spatial reasoning and a list of dynamic process in verb form such as locating, balancing and visualizing. The book concludes with a refined definition/framework for spatial reasoning. The chapters in-between zoom in on aspects of spatial reasoning written by small groups of contributing authors. For example, there is a chapter dedicated to the research that supports that spatial reasoning is not a fixed trait and is something that can be developed. Much research on embodied knowing and embodiment is shared as well as interactions between two and three dimensions. Throughout the book, as an imperative for re-imaging what we think of in terms of school mathematics and “geometry,” the authors refer to the research studies (such as Duncan et al 2007) that claim that spatial reasoning in four and five year olds is a strong indicator in overall school success as well as more specifically, literacy and numeracy.

IMG_6499There are a few helpful tables/graphics to help understand key concepts and that I will share in my work with teachers. This one to the left is a typology of different types of spatial reasoning which I think is important to consider in instructional planning and assessment.

 

 

 

IMG_6065One of the reasons I chose to read this book this summer is that some of the contributing authors in this book worked together on the Math for Young Children (M4YC) research project in Ontario which is the foundation for the wonderful book Taking Shape: Activities to Develop Geometric and Spatial Thinking K-2 – a book that is well worth the investment to have in elementary school professional libraries. This book is an excellent example of how a professional resource for teachers can show the theory to practice flow and intersection.

~Janice

summer professional reading: Teaching Mathematical Thinking

Posted on: July 25th, 2017 by jnovakowski No Comments

IMG_6362Teaching Mathematical Thinking: Tasks & Questions to Strengthen Practices and Processes

by Marian Small

foreward by Linda Dacey

published by Teachers College Press, 2017

 

 

In this recently published book, well known math educator and author Marian Small highlights an important aspect of the discipline of mathematics – the thinking practices and processes that are “the doing of mathematics” when engaging in mathematical problems and learning content.

For those wanting clear examples of practices such as mathematical modeling, structure and argument are – the author clearly defines these with examples from across grade bands (K-2, 3-5 and 6-8).

For each practice/process, the author includes:

1) a definition with examples

2) where that practice/process is seen in K-8 mathematics

3) examples of problems, across grade bands, that might bring out that practice/process, often with examples of student responses

4) assessment questions for the educator to use to help notice and reflect on the students’ use of the practice/process

5) a short summary

I can’t think of another book that makes such careful nods to the Canadian mathematics education landscape. Although the focus is on the eight American Common Core standards for mathematical practice, the author connects these to our mathematical processes/competencies in Canada (with slight differences in different provinces/regions). Because our Canadian emphasis on visualization and mental math and estimation is not explicit in the American practices, the author has added a final chapter dedicated to these processes.

The problems are chosen to connect to each practice/process but should not be considered practice-specific. There are different types of problems – if you are familiar with Marian Small’s other books, you will understand the type of open-ness, differentiation and complexity built into the problems provided. For each practice/process she provides at least one problem for each grade band and then discusses how students take up the problems, with student examples.

I highly recommend this book. So so many wonderful problems for K-8 students and great information for teachers to help us think about the discipline of mathematics.

~Janice

playful storytelling celebration

Posted on: June 13th, 2017 by jnovakowski

We held our year end celebration for our Playful Storytelling through the First Peoples Principles of Learning project on June 1 at Grauer. Teachers and school teams came together to share what they had been working on during this school year. It is always interesting to hear how each school makes the project its own.

An overview of our session at the beginning of the year is HERE

After an acknowledgement of territory and a welcome to some visiting educators from Manitoba, in circle each teacher introduced themselves and shared the principle that they have most connected to this school year.

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A group of us involved in this project visited the Vancouver Art Gallery in early May to see the Susan Point exhibit. We made so many connections between her work and story and came up with several questions to guide our practice when having students engage with art. We shared this with the group and also shared how some teachers had already taken up these ideas with their classes.

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School teams then shared how they have investigated playful storytelling in their schools.

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We noticed a clear move to many classes taking storytelling outdoors and this will be something we continue to investigate next year.

After a lovely dinner together prepared by Marie Thom, we moved to The Studio at Grauer to engage in some sensory experiences with materials that can be used to enhance the storytelling experience. Experiences with clay, watercolour and wet felting were provided, for teachers to consider – what stories live in these materials?

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We all left inspired by each other and full of new ideas to enact with our students. Looking forward to continue this work in our district next year!

~Janice

The Studio at Grauer

Posted on: June 11th, 2017 by jnovakowski 2 Comments

The Studio at Grauer came to be based on a need we felt existed in our district – a space to engage in professional learning experiences for teachers as well as for a learning environment for students that could be left  “set up”. My office partner, Marie Thom, and I have been talking about this for the last couple of years. The notion of a “pop-up” classroom emerged and Andrew Ferguson, the principal at Grauer, was approached to see if we could use one of the school’s unused classrooms.

December 16 2018 – Room 102

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Room 102 was being used as a storage room until we began our transformation of it in January 2017. Blending Marie’s background in learning environments and my understanding of mathematics teaching and learning, we developed a space focused on mathematics, filled with inspiring materials in a learning environment designed for learners K-Adult. Our goal was to create a flexible, responsive and inclusive learning environment.

Room 102 – January 10 2017

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January 13 2017

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The learning environment is set up to create opportunities for choice, collaboration, personalized learning and inquiry. More specifically for mathematics, our hope was to broaden understanding of what mathematics is and what the teaching and learning of mathematics can be. Often school mathematics is perceived as “arithmetic” and mathematics is a much broader discipline that this. We wanted students and teachers to see math all around them and be inspired to think about mathematics in different ways – to see mathematical ideas in the materials, in pinecones, in buildings and structures, in images of our community, in art, in stories.

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As we discussed how we were going to use the space, we decided to call it The Studio, inspired the notion of an atelier, a studio space used in the early childhood schools of Reggio Emilia.

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Classes from Grauer, as well as visiting classes on “math field trips” visit The Studio to engage in mathematical inquiry. Marie and I take on the role of atelierista, working with the classroom teacher to facilitate learning experiences through different materials in the space. We have intentionally curated both mathematically structured materials like pattern blocks and Cuisenaire rods alongside materials often known as loose parts such as ribbons, gems, rocks, pinecones, etc. We also have art materials available to the students such as paint, clay, charcoal, yarn and wool so that students can express themselves and think using different languages. Students also have access to various tools to support their investigations such as measuring tapes, protractors, grids and ten frames.

The first class to visit The Studio – the grades 3&4 students from Grauer on January 18 2017

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The learning environment is intentionally flexible with choices in seating and tables available for both students and adults. Interestingly, although we have some chairs available, none of the students using The Studio have used them, preferring instead to stand or sit and lie on the carpet or use pillows. We have observed the flow of movement in the space and intentionally have large open spaces for students to move through. Shelves filled with baskets of materials are open and accessible to students. Students can choose the materials they want to use and take them to where they would like to engage.  We took doors off of some cupboards to create more open shelving. All of the furniture, except for four small Ikea open shelves, was found in school storage rooms and thrift shops.

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Provocations are set up on tables for students (and educators) to inspire mathematical thinking and inquiry. Inspired by one of the students’ interests in optical illusions, the grades 3&4 students from Grauer investigated the mathematics embedded in optical illusions. I gathered materials and tools that I hoped would provoke their thinking about optical illusions and the students also accessed and were inspired by other materials in The Studio.

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As we have more classes through The Studio, we have developed documentation to share in the space. Panels, photographs and photobooks are available for students and educators to engage with, to reflect upon and to inspire new experiences.

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One area of pedagogical intention in The Studio has been on noticing, naming and nurturing the Core Competencies and the Mathematics Curricular Competencies from our BC curriculum framework. A focus has been on both communication and creative thinking in mathematics. We intentionally create opportunities for students to engage in different types of communication and to reflect on how they are doing.

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We have had many groups of educators also visit The Studio. Our District Support Team, educators attending our Playful Inquiry professional learning series and teams from schools in our district. Many BC educators involved with our BCAMT Reggio-Inspired Mathematics project have visited the space as well. We have also had visitors from Manitoba and Sweden! We often focus the visits with the questions – what do you notice? what do you wonder?

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We also have a dedicated professional learning library with the teacher resources we recommend around teaching and learning mathematics, the use of loose parts and the importance of the learning environment.

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We had hoped The Studio would inspire similar learning environments in our district but recognize that many schools do not have access to a dedicated room for a studio space – we hope that teachers will be inspired by elements of The Studio for their own classroom learning environments. What has been exciting for Marie and I is that this little project has had a huge ripple effect at Grauer, in our district, and beyond!

~Janice

 

 

creating spaces for playful inquiry: April 2017

Posted on: May 28th, 2017 by jnovakowski

For our third session of our Creating Spaces for Playful Inquiry dinner series, Richmond educators came together at Grauer Elementary to share and learn together. This year at our sessions we have focused on broad themes or big ideas that cut across curriculum areas and grade levels, beginning with community, then identity and for our third time together this year, we chose to focus on place. Those that attended our Lower Mainland study tour to  the Opal School in Portland created panels reflecting on their experience. Many of our playful inquiry mentors set up either pedagogical provocations or shared provocations they developed to engage their students.

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Four teachers shared their experiences visiting Opal for the first or second time – what had an impact on them and how it is affecting their practice. Thank you to April, Louesa, Laurie and Karen for your thoughtful and passionate presentations!

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Based on feedback from the mentor groups during our January session, Michelle Hikida and I did a short professional learning presentation on playful mathematical inquiry and how we plan around a big idea, use provocations and projects based on students’ interests and curiosities and how we extend and sustain a math inquiry.

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After dinner together, we spent time in our mentor groups, zooming in on an area of interest and sharing and learning from each other.

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We are looking forward to continuing working together next year to support professional learning and building a playful inquiry community across our district.

~Janice

primary teachers study group: sixth session

Posted on: May 28th, 2017 by jnovakowski

For our sixth and final session of the year, the primary teachers study group met at Blair Elementary. Thank you to Karen, Robyn and Tanyia for hosting us!

We broke into small groups and worked on our environmental inquiry visual – still a work in progress! Lots of great feedback from the group.

We then toured the outdoor learning spaces at Blair. I was the teacher-librarian at Blair for three years when I introduced Spuds in Tubs to the school – since then the school has embraced school gardens and many of the teachers have made outdoor learning and integral part of their programs. April and Karen shared how they use some of the typical suburban school spaces around the school for outdoor learning and also shared how they involved the district Works Yard employees in creating an outdoor classroom space and storage.

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There was much talk about school gardening as a way to get students outdoors and feeling connected. We left thinking about ways to continue our work around environmental inquiry and creating opportunities for students to take action and take care of the outdoor spaces at their schools.

Next year, our primary teachers study group will be entering its fifteenth year as a professional learning structure in our school district!

~Janice

primary teachers study group: fifth session

Posted on: May 27th, 2017 by jnovakowski

For our after spring break session, we met at Steves Elementary, hosted by Kathleen Paiger. After sharing some of the outdoor learning experiences we were engaging in, we began to co-construct a visual of sorts to show the process of environmental inquiry that we have been working towards together this school year. For many teachers in the group – the first stages have been the focus: getting outside, noticing and naming local plants and animals, being curious and connecting. Drawing upon the work of David Suzuki, Ann Pelo and many others, we know these initial stages are essential if children are to care about the environment.

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What we discussed, and connecting to last year’s Natural Curiosity seminar that some of us attended, was that getting outdoors and gardening, although important, is not the overall goal of environmental inquiry. We want to find opportunities for students to take action, to take up a concern or issue that emerges or that they are noticing in their environment.

Kathleen regularly takes her kindergarten students for walks in their neighbourhood which includes the west dyke. We ventured out together and learned about some of the story of this place – the farmland, the coyotes and the birds that frequent the wetland areas.

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And as the weather eventually warms up, we know there will be lots of opportunities for our classes to get outside, to notice and wonder. We are using these three picture books this spring to inspire us.

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~Janice

Big Mathematical Ideas for Grades 3-5: year four

Posted on: May 25th, 2017 by jnovakowski

This after school series is in its fourth year, focused specifically on the foundational mathematical concepts in grades 3-5 such as operations, place value and fractions. This year, we met once a term after school in Jennifer Plummer’s French Immersion grades 3&4 classroom at Homma Elementary (thank you for hosting Jennifer!).

For our first session in the fall, we focused on number sense and operations and the curricular competencies of reasoning and communication. I had recently been to the NCTM Regional Conference in Phoenix and shared a great game that I was reminded of in one of the sessions there.

The Product Gameboard

The Product Gameboard instructions

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At our second session, in January, we focused on financial literacy and mathematical inquiry. I shared the new Pizza Co. game for the iPad from Osmo as well as some other resources and children’s books to support the financial literacy content in the BC Mathematics Curriculum.

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BC K-5 Mathematics Big Ideas

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For our third session in April, teachers in the series requested a focus on assessment, place-based learning and connecting the core and curricular competencies.

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BC K-5 Math Communication

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We explored curricular content and competencies by investigating with power polygons.

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Looking forward to continuing the conversation around big mathematical ideas next year!

~Janice

Provincial Numeracy Project in Richmond – year two

Posted on: April 24th, 2017 by jnovakowski

For the second year, Richmond is one of several districts in the province that are taking part of an adapted version of Changing Results for Young Readers in the form of Changing Results for Young Mathematicians. Provincially, this project is not just focused on young students but for K-12 students and their teachers and has been called the Provincial Numeracy Project.

This year, three school teams of primary teachers and a Learning Resource Teachers are participating – teams from Whiteside, Grauer and McNeely.

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Several routines such as Counting Collections (see tedd.org) and High Yield Routines (book published by NCTM) were shared with teachers as elements that could be introduced as both whole class and small group instruction.The  importance of assessment to inform instruction was discussed and the importance of spending time being present with our students – listening, noticing and talking with them. This is a critical practice in this project – really zooming in on students’ mathematical thinking.

One of the handouts provided to teachers was a collection of Number Sense tasks and background information connected to the Kathy Richardson book we would be using. The handout package can be downloaded here: Didax Number Sense guide

number concepts kathy richardsonUnfortunately, our second session together was cancelled due to a lack of TTOCs. Between sessions, copies of Kathy Richardson’s book How Children Learn Number Concepts was delivered to each teacher in the project. Teachers were asked to zoom in on one section of the book and make connections to what they were noticing as their students engaged in counting or other number routines.

Each Orange Had 8 Slices Teachers were also provided with a copy of the classic math picture book Each Orange Had 8 Slices and were encouraged to share some of the pages with their students and engage in playful and joyful mathematics – counting, problem posing and problem solving.

The one-pager that was created to accompany the book to send home with students to engage in joyful math with their parents can be downloaded here: Each Orange Had 8 Slices.

For our third and final session in March, school teams shared what they had been trying and what they had been noticing in their classrooms.

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A specific focus on the reasoning and analyzing section of the curricular competencies was explored through estimation. Estimation is a good indicator of students’ number sense and is an important competency to develop and use in many contexts – both computational and with measurement. The estimation information and connections to literature that was developed as part of the BCAMT Reggio-Inspired Mathematics project can be downloaded here: Estimating final

We also looked at the curricular competency: “use reasoning to explore and make connections” through routines such as Number Talk Images and Clothesline.

How many ducks are there? How do you know? What different ways can you see the quantity?

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A blog post of how clothesline math has been used in kindergarten classes in Richmond can be found HERE.

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Teachers in the project each received a copy of Which One Doesn’t Belong? by Christopher Danielson and connections to our BC curricular competencies in math were made – communication, reasoning, justifying and explaining.

As part of the project, teachers were asked to pay attention to a child who they were wondering about – a child whose math learning was causing questions for the teacher. Based on the findings from Changing Results for Young Readers, we felt that focusing on the learning of one child, and how that child  responds to new ways of approaching math learning, would influence the class’ learning as a whole. Teachers were asked to reflect on this as well as write a short professional narrative reflecting on their own professional inquiry during the project. A collection of excerpts from the teachers’ reflections are included below:

“We are having to unlearn the ways we learned math in order to think about ways we can help the students build a solid number sense foundation.”

“I noticed increased student ownership over their own learning. They are choosing collection at their just right level and are trying to figure out, on their own, ways to count their collection in multiple different ways. Students are beginning to be able to communicate their thinking about the strategies they use to count their collections.”

“I have a better understanding of number concepts and where to go next to help/challenge a student.”

“High Yield Routines had high engagement levels and encouraged a lot of math talk in our classroom.”

“I noticed that I was taking chances in my teaching during this project, allowing myself to learn alongside my students.”

“I learned that math is about exploring and sense-making.”

“Students were engaged with math talk images of actual objects rather than dots or ten frames.”

“My students are more engaged and more hands on with the math now as a result of the changes I made.”

“Students are now expected to voice and articulate and justify their thinking about why they feel their answer could be right.”

“I learned about recognizing the phases of mathematical development and how foundational skills contribute to deeper meaning and understanding for students in subsequent years.”

“I noticed that I need to step back and invest time to delve into student learning and understanding beyond the correct answer.”

*****

~Janice