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: THINQ Kindergarten (and Grades 4-6)

Posted on: July 28th, 2017 by jnovakowski 2 Comments

IMG_6380THINQ Kindergarten: Inquiry-based learning in the kindergarten classroom

by Joan Reimer and Deb Watters

THINQ series authors: Jennifer Watt and Jill Colyer

published by WAVE Learning Solutions, Canada, 2017

accompanying website: www.wavelearningsolutions.com

This is a relatively new series of books, written by Canadian (Ontario) authors. The Kindergarten book just came out this spring. One of the many things I like about this book is that it acknowledges that there are many interpretations of inquiry and not “one way” to engage in inquiry. There is a focus on remembering that being inquiry-minded is part of being human and that we are born with curiosity. I also like the recognition of the importance of the learning environment and the emphasis on developing inquiry dispositions. The “Inquiry in Action” sections share learning stories or case studies from classrooms. As I read the book, I added lots of post-it notes to pages to go back to, particularly connections I was making to our BC competencies – both core and curricular.

There are seven chapters:

1) Inquiry-based learning in kindergarten

2) Wondering and questioning

3) Creating an inquiry environment

4) Negotiating the curriculum

5) Documentation

6) Inquiry assessment in kindergarten

7) Final thoughts

IMG_6381The layout for each chapter is very similar. There is lots of “white space” and use of text boxes and colourful visuals to support the content of the chapter. Each section has a big idea and often quotes from well-known educators and authors.

 

 

 

 

IMG_6382At the end of each chapter there is a chapter summary with some questions to provoke reflective thinking. There are also “thumbnails” of the blackline masters/printables that accompany each chapter and can be found at the back of the book.

 

 

 

 

 

 

IMG_6383The printable resources at the back of the book correspond to each chapter and focus on the big ideas from the chapter as well as templates for educators to use for planning and assessment.

 

 

 

 

The Grades 4-6 book was release last year and is very similar in format to the Kindergarten book. Interestingly, in this book, the assessment chapter is up front and then assessment ideas are woven throughout the rest of the book. I wonder if that is because this is a pressing issue for intermediate teachers – how to assess student learning during the process of inquiry? I know this question comes up a lot in my work with teachers. There is also much more of an emphasis on questions of personal significance, inquiry approaches across disciplines and the importance of providing feedback (often through questioning_ during the inquiry process in the Grades 4-6 or Junior book. The printable resources focus more on student self-assessment templates than the Kindergarten book.

The Grades 1-3 book and the Grades 7-9 book are supposed to be released this summer or fall.

~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

summer professional reading: The Beach Book

Posted on: July 23rd, 2017 by jnovakowski 1 Comment

IMG_6293The Beach Book: loads to do at lakes, rivers and the seaside

by Fiona Danks and Jo Schofield

published by Frances Lincoln Limited, 2015

related website – www.goingwild.net

 

This little book packed with inspiration is part of a broader Going Wild series of books of which I have many (yes, I have a book problem). I also have The Stick Book, The Wild Weather Book, The Wild City Book, Nature’s Playground, Go Wild! and Make It Wild! All of these books are full of inspiring photographs of children experiencing and learning outdoors.

The Beach Book is divided into seven sections – bookended with an overview of things to think about before heading to the beach and a final section on important safety reminders. The seven sections are: beach adventures, beach wildlife, beach games, beach art, beach imagination, beach at night and beach rubbish. Some of the explorations in each section are very open while others are more structured. They all make use of the natural environment, the materials the beach offers, and what is available to see and do at a beach. There are lots of curricular connections – big ideas, core competencies, place, story, mathematics, science, visual arts and more.

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The authors define beach as “where water meets land” – and make a case for visiting beaches all year round so that we understand that these are special places worth taking care of.

spiral with driftwood at Garry Point, Steveston, BC

spiral with driftwood at Garry Point, Steveston, BC

I know this is a book that I will pull out again and again for ideas and inspiration. Am looking forward to finally digging into the other books in this series as well.

~Janice

summer professional reading: Teaching Math with Google Apps

Posted on: July 20th, 2017 by jnovakowski No Comments

IMG_6290Teaching Math with Google Apps: 50 G Suite Activities by Alice Keeler and Diana Herrington

 

Foreward by Jo Boaler

 

Published by Dave Burgess Consulting, Inc. 2017

 

This book opens with a foreward by Jo Boaler, with a call for educators to transform math classes. She references the Forbes list of skills needed for employment such as teamwork, problem solving, communication – all of which she argues can be enhanced through collaboration with technology. She also addresses the issue of “speed” and mathematics and how some students believe they are not “math people” because they are not fast. Boaler explains how the simple submissions of thinking and solutions on a Google form can take away the focus on speed in mathematics.

Authors Alice and Diana have both been math teachers at the high school and college levels. They emphasize the importance of digital tools in reimagining the math class with a focus on collaboration. They outline seven ways to use Google Apps to teach math:

1) Post Directions

2) Watch Students Work

3) Collaboration

4) Shift Students to Higher DOK Levels

5) Students Research

6) Shift to Facilitator

7) Conversations for Depper Understanding

The majority of the book is dedicated to overviewing 50 activities to teach math with Google Apps such as “Small Group Investigation,” “Discuss Strategies,” “Analyze Data Sets” and “Create Geometry Constructions”. The authors suggest asking yourself, “how does this activity make learning better?” Most of the activities use Google Classroom, Google Docs, Google Sheets or Google Slides and provides the advantages of using each format. Also used are Google Search, Google Forms, Google Drive, Google Chrome, Google Drawing, and Google Flights,

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Links to examples and tutorials are provided.  Some key reminders are interspersed throughout this section:

Teach like YouTube and Google exist.

The person doing the work is the person doing the learning.

We are a community of learners and we help each other get better.

The back matter shares examples from classrooms and highlights DOK levels (Depth of Knowledge), the 4 Cs (creativity, critical thinking, communicate and collaborate), mathematical practices for the CCSS and the 5E instructional cycle (engage, explore, explain, elaborate and evaluate).

There are lots of great ideas for tech integration and student collaboration throughout this book. Be mindful that some districts have policies or concerns regarding students having gmail accounts and as Alice has clearly said on Twitter – Outlook and Google apps aren’t really compatible. If having gmail accounts for students is frowned upon, like in my district (Richmond), take some of the ideas from this book and figure out how to make them work with the platforms that you are able to use! That will be one of my goals for the coming year as I see so much opportunity in technology enabling  our secondary students to engage in in-class, cross-class and cross-school collaboration around mathematics.

~Janice

summer professional reading: Redesigning Learning Spaces

Posted on: July 17th, 2017 by jnovakowski No Comments

IMG_6202Redesigning Learning Spaces by Robert Dillon, Ben Gilpin, A.J. Juliani and Erin Klein, published in 2016 by Corwin.

 

This volume is part of the Corwin Connected Educators Series.

 

 

This short book has five chapters:

Leading Change Through Classroom Learning-Space Design

This chapter focused on listening to students about what works for them. It also cites the much referenced research study looking at how environmental print and stimulus in the classroom affects student learning. I appreciated the emphasis on creating “truly beautiful places to learn” and how we shouldn’t underestimate the importance of the aesthetic element. The authors discuss the intentional design of learning spaces to focus on exploration and student centred experiences and how this type of design gives the message to students that “I respect you as learners.”

Learning-Space Change as a Lever to Shift School Culture

The overall message of this book is emphasized in this chapter: Learning spaces matter and they impact the entire teaching and learning community. The authors cite a 2012 study that indicates that classroom environment can affect a child’s academic achievement by up to 25%.

Shaping Learning-Space Change for the Community

“Redesigning spaces to maximize learning is primarily a shift in culture and mindset.” The authors share examples of how small changes in schools can have impacts on systems and whole school design can shift the school culture and that of the greater community.

Learning Space as a Lever for Systemic Change

This chapter looks more broadly at systemic change but begins with the metaphor for the classroom of  ”habitat” and how a supportive habitat helps students’ learning power to be magnified. The importance of technology and “connected classrooms” as part of a learning ecosystem is emphasized, but acknowledges that the learning environment or habitat of the classroom, seems to be supportive of this.

Systemic change can begin with: 1) new options for lighting, seating, work spaces, ideation spaces, 2) looking for innovative partnerships outside of the school with industry and in community, 3) a “laser” focus on meaningful learning as opposed to what the authors call Pinterest-based learning and 4) with seeing all spaces as potential places for learning such as hallways as ideation spaces. These changes in classrooms spread in schools and then in districts.

Models of Excellence and a Place to Start

The final chapter shares some specific examples and challenges educators to be agents of change.

I appreciated this book as a short and succinct read that I will draw upon when advocating for changes to learning environments in our classrooms and schools. The authors have curated educational and design research that supports shifts in classroom and school learning spaces.

~Janice

summer professional reading: Engaging Minds in Science and Math Classrooms

Posted on: July 11th, 2017 by jnovakowski No Comments

Sharing some of my summer reading here on the blog.

First professional read of the summer -

IMG_6149Engaging Minds in Science and Math Classrooms: The Surprising Power of Joy by Eric Brunsell and Michelle A. Fleming. Published in 2014 by the ASCD.

This book is a follow-up to Engaging Minds in the Classroom: The Surprising Power of Joy by Michael F. Opitz and Michael P. Ford. These two original authors edited this volume. They define joyful learning as “acquiring knowledge or skills in ways that cause pleasure or happiness.” They surmise that when students are engaged learners, joy emanates from the learning process. Their joyful learning framework is the foundation for this follow-up book.

This book has four short chapters -

1) Understanding Joyful Learning in Science and Math

Drawing upon current research, the authors outline the joyful learning framework and answer the question Why joyful learning? with:

  • it capitalizes on what we know and how to best motivate students.
  • it enables us to build upon what we currently know about engagement
  • it enables us to focus on the whole child
  • it acknowledges that the learner is influenced by the contexts in which learning takes place

2) Evaluating and Assessing Joyful Learning

This chapter outlines frameworks to evaluate learners, ourselves as teachers, texts and materials, assessments and school-wide configurations. The frameworks for evaluating learners parallels the one for evaluating teachers and both provide some thoughtful questions for consideration.

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3) Implementing Joyful Learning in Science and Math

Strategies, structures and examples of ways to implement joyful learning are provide for several contexts: school community, classroom environment, whole-group instruction, small-group instruction and individual instruction.

4) Using Joyful Learning to Support Education Initiatives

The final chapters makes connections to standards, accountability and assessment, RTI, achievement gaps and professional development, drawing upon research studies to support the importance of engagement and interest in learning to standardized test results.

The book ends with a reminder to teachers to assess their own joyful learning and to look for joy in unexpected places and a quote from author Henri Nouwen:

“Joy does not simply happen to us. We have to choose joy and keep choosing it every day.”

The ideas of identity, student self-efficacy, challenge, choice, creativity and goal orientation resonate throughout the book. This speaks to me about students’ understanding of what it means to be a learner and what their role in that is – not as a passive, compliant recipient, but as a fully engaged, curious learner.

One issue that the authors return through out the book is that for students to be engaged in joyful learning, they need to focus themselves on “mastery” goals (learning that focuses on learning content) versus performance goals (learning for the purpose of getting a grade or being compared to others). After hearing Megan Franke’s keynote presentation at the CGI Conference in Seattle this year, I bristle at the term “mastery” and would rather consider these goals as just learning goals.

Another area of interest that reading this book re-ignited for me was the concept of engagement. I have thought about this a lot over the years and read quite a bit in this area during my doctoral studies. The authors look at the relationship between motivation and engagement but don’t tease apart what they mean by engagement very thoroughly even though they come back to and use this term throughout the book. They describe engagement as “being attentive, committed, persistent, and seeking meaning.” There are many types of engagement – physical, emotional, cognitive etc and sometimes I think compliance can actually be perceived as engagement which is a concern.

As I zipped through this quick read, I made many connections to both of the books Mathematical Mindsets by Jo Boaler and Embracing a Culture of Joy: How Educators Can Bring Joy to Their Classrooms Each Day by Dean Shareski. I highly recommend both of these books!

~Janice

reflections and highlights from 2016-2017

Posted on: June 29th, 2017 by jnovakowski No Comments

To say this has been an interesting year in our district is an understatement. We began the year with a new email system, a new “portal”, a new eportfolio platform, a huge changeover in our district curriculum department all while under the umbrella of the full launch of BC’s K-9 curriculum framework across all subject areas. New layers were added as information about communicating student learning and core competencies self-assessment were added to the mix. We began the year with uncertainty about school closures and ended it with concerns about whether we would have our jobs and where money would come from to fund teaching jobs in our district.

In all of these twists and turns, there were some downs but also many ups.

There were many opportunities for educators in our district to come together and figure everything out (an ongoing process for sure). Professional development days, our Curriculum Implementation Day, professional learning events and series and the power of twitter to share ideas and collaborate.

In thinking back on this year, I will remember all the times I was surrounded by inspiring colleagues and in classrooms thinking alongside students about big ideas – this is what fills me up and sustains me.

Some professional highlights for me include:

  • the Creating Spaces for Playful Inquiry dinner series – this large group of K-7 teachers continues to come together to engage in provocations and think about playful inquiry across the curriculum, this year looking at broad themes of community, identity and place
  • visiting the Susan Point exhibit at the Vancouver Art Gallery twice with colleagues and how that inspired connections in our classrooms – place, ecosystems, environmental sustainability, stories and math
  • having the opportunities to share our thinking from #sd38learn in other places –  Coquitlam, Kamloops, Sooke, Niagara Falls, Seattle, Burnaby, Vernon, Vancouver, Victoria and Cranbrook
  • being part of the BC Numeracy Network – this team of enthusiastic educators from across BC have collaborated to create a website that will support K-12 educators as they think about balanced numeracy through the lens of BC’s curriculum
  • being on the program committee for the National Council of Teachers of Mathematics (an international organization) for their annual conference in San Antonio in April
  • hosting teams of educators as they visit our district (from across BC, Manitoba and Sweden) – it is always interesting to hear about what they are noticing and what is resonating for them
  • working in collaboration with the Musqueam Language and Culture department to develop our first project together
  • being a part of the BCAMT Reggio-Inspired Mathematics collaborative professional inquiry project – this project continues to grow and we are working on publishing a book this summer with contributions from ten districts

But when I think about the biggest highlight of the year for me and what I believe has affected both students and educators on many levels, I think about the creation of The Studio at Grauer. My heart is so full when I think how Marie Thom and I transformed an old classroom being used for storage into a studio space to investigate mathematics through a variety of materials.

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And my summer reading “stack” – yes, there will be novels, travel books and magazines as well!

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I hope to share some of my thoughts on these books on the blog over the summer.

Wishing you a summer full of adventure and time to refresh!

~Janice

thinking about fractions

Posted on: June 21st, 2017 by jnovakowski

One of the foundational concepts in grades 3-5 is an understanding of fractions – some of the questions we investigate with students are:

What is a fraction? What makes a fraction a fraction?

How can we order and compare fractions?

When do we use fractions in our daily life?

What different ways can we represent fractions?

What are equivalent fractions?

In our BC mathematics curriculum, the big idea for grade 3 that we guide students to understand by the end of the school year is that “fractions are a type of number that can represent quantities” – this is a significant concept. I often have discussions with older students who have the conception that fractions have something to do with shapes/geometry, possibly because of the models used in school to represent fractional numbers (think circles/pies/pizzas or rectangles/chocolate bars). An intentional focus is to provide opportunities for students to see and represent fractional amounts using different models – area/region, set and linear. In grade 4, students learn about the relationship between fractional and decimal numbers and in grade 5, consolidate their understanding of fractions by working with equivalent fractions. During these investigations, students may begin to see relationships and connections to whole number operations when working with fractions and decimals. We want students to be able to think flexibly with fractions and decimals, just as they do with whole numbers and think about composing and decomposing, benchmark numbers, etc as they consider addition, subtraction, multiplication and division.

This term I have been thinking a lot about fractions with students and teachers. I visited two classes at Homma and a class at Wowk and hosted classes at The Studio at Grauer to investigate fractions including a grade 4 class from Woodward, the grades 3&4 class from Grauer and the Richmond School Program students from Blundell.

The following are some images sharing our investigations. As you scroll through the images, consider:

What do you notice? What do you wonder?

What instructional routines and structures have we used to support students in their understanding of fractions?

What different materials have been provided to create opportunities to think about fractions?

What conceptions do students reveal in their representations of fractions? What might you ask students? How does this information guide where we go next?

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IMG_2690 IMG_2691 IMG_2692 IMG_2693 IMG_2694 IMG_2695 IMG_2696 IMG_2697

IMG_2699

IMG_2701 IMG_2702 IMG_2703 IMG_2704 IMG_3804 IMG_3805 IMG_3806 IMG_3807 IMG_3808 IMG_3809 IMG_3810

IMG_3814 IMG_3815

IMG_3817 IMG_3818

IMG_3821 IMG_3822

IMG_3828

IMG_4767 IMG_4766 IMG_4772 IMG_4774 IMG_4775 IMG_4785 IMG_4817 IMG_4823 IMG_4825 IMG_4816 IMG_4836 (1) IMG_5062 IMG_5061 IMG_5055 IMG_5054

~Janice