What is number sense?
How are numeracy and number sense the same and how are they different?
What routines nurture the development of number sense?
What is number sense?
How are numeracy and number sense the same and how are they different?
What routines nurture the development of number sense?
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.
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.
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”.
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.
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:
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.
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.
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:
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?
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.
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.
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.
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?
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
Mathematical Mindsets by Jo Boaler
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.
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
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,
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.
Building on interest from an ILC (Inclusive Learning Community) project Shelley Moore and I facilitated with grade 8 teachers at Boyd Secondary, we held an after school session in October looking at inclusive practices in mathematics for grades 6-9 teachers. These practices are particularly mindful of the personal, social, intellectual and physical needs of students in the middle school age range.
Shelley began the session by sharing Richmond’s history with inclusive education and sharing some frameworks she has developed for thinking about inclusion (bowling pins, Fisher-Price stacker toy, planning pyramid, etc). She refers to inclusions lenses – personal, social and intellectual as well as places – different classrooms and places in the school as well as out of the school.
In using the planning pyramid, Shelley considers goals, tasks and questions for all students, some students and a few students, starting where ALL students can access the unit or lesson. And here’s Shelley doing the tree pose – using the analogy that everyone/all could start this yoga pose by using the wall for support!
Shelley shared the two year project with the grade 8 teachers and students at Boyd, with the first year addressing the Shape and Space curriculum and the second year examining the linear equations part of the curriculum. One example of a planning framework for an initial lesson on geometry looks like this:
I shared some of the practices and structures that we considered during the ILC project at Boyd and that can be used as a guide for planning mathematics lessons and units with inclusion in mind.
Some of the choices that students were provided were what types of materials they might use. For example, during our lesson together about the volume of prisms, some students built prisms with cubes, some students used centimetre graph paper to create nets for their prisms and other drew 3D drawings that represented the measurements they were working with. Another choice was the range within the concept being addressed – for example, in the geometry lessons, identification of basic 2D shapes (faces) was an access point for all while some students investigated a range of 3D prisms. In the study of linear equations, choices of equations to investigate and represent with balances and other materials were provided, increasing in complexity or number of operations. Students were also provided with choices in how they processed or representing their thinking, for example, iPad technology was available and students could use the camera to take video or photos and then use a choice of screencasting apps to provide evidence of their understanding of the concept. Non-permanent vertical surfaces (NPVS) aka whiteboards or windows provide another choice for students who may not want to sit and work at a desk or table or use paper and pencil. The research-based practice of using NPVS has been shown to increase engagement and mathematical discourse, particularly at the middle-school age range.
I shared the idea of mathematical routines such as number talks as inclusive practices with starting points for all and a way to build an inclusive mathematical community in the classroom. These routines also focus on the nurturing and development of the curricular competencies which are the same for grades 6-9. One of the routines shared was WODB (Which One Doesn’t Belong?). This routine has become very popular in Richmond classrooms as it provides an opportunity for the clear connection between curricular competencies and content. Four items are presented and they all belong to a set/group of some sort – integers, polygons, etc but each item is unique is some way. The goal of the routine is for the students to analyze and use reasoning to justify or defend which one they think doesn’t belong in the set and why. WODBs for geometry, number, graphs, etc are available at WODB.CA - a site curated by an Ontario secondary math teacher.
Shelley has posted a pdf version of our slides from the session on her blog. They can be found HERE.
Because of interest, we will be facilitating a repeat of this session on December 6 from 3:30-5:00pm at IDC – register on our district’s event page with further follow-up sessions planned in the new year.
The Richmond School District hosted its first IGNITE event on Monday, May 9th at the Big River Brew Pub. The first IGNITE took place in Seattle in 2006 and is now a movement that is international in scope. An IGNITE talk is a five minute presentation consisting of 20 slides, auto-advancing after 15 seconds whether the speaker is ready or not. The IGNITE tagline is….”Enlighten us, but make it quick!” More information about the IGNITE movement can be found here.
Having presented a few ignite talks and experiencing the inspiration and fun that goes along with these social events, I really wanted to be able to bring this professional learning format to our district and my colleagues Rosalind Poon and Lorraine Minosky were on board and we ran with it. Chris Loat created our logo for us…
And both Chris and Lisa Schwartz agreed to be our technical support for the event. We found a site and had a meeting at the Big River Brew Pub to see where attendees would sit and how the technology and food service would play out.
We invited Richmond educators representative of primary, intermediate and secondary as well as a balance of teachers and administrators to share a story about something they are passionate about. We also invited two out of district colleagues to add to our Richmond stories.
Two weeks before the event, we hosted a rehearsal especially for educators who were new to the ignite format. It gave them a chance to meet other igniters and to practice their presentation in front of an audience. By seeing and discussing what we appreciate about others’ presentations, I think it also gave presenters some ideas for their own ignites. And its always great to have sushi…and pens.
As we were setting up on May 9th, we were all so excited to see everything fall into place. The venue was great and it was a beautiful day so the patio was open, the technology was cooperating and the tables were set with programs and sweet treats from Sinfully the Best for our guests.The burger bar was a hit and the company was great. Unfortunately two of our igniters (Neil Stephenson and Sarah Garr) had to pull out due to personal reasons.
Some of the guests…
And then the talks began! I was live streaming the talks using the Periscope app and people that weren’t able to attend the event could still watch the talks live. Between each “set” there was a 15-minute break for guests to chat about the talks, etc.
Twitter was alive with #sd38ignite…we were trending!
It was such a positive, passionate event and such a great way to build community amongst colleagues. All our igniters shared their own personal narratives within their professional narratives and these stories are what connect us and make us better together.
We will be releasing the IGNITE talks on youtube soon…watch twitter for announcements!
A HUGE thank you to our igniters…you are what made the event the success it was!
For the third year, our district has invited grade 6-9 teachers to participate in a Core Competencies project. This year, Rosalind Poon and myself hosted the project, focusing on the Creative Thinking core competency in the BC curriculum. A blog post about our first session together can be found here.
Our second session was held in January and teachers shared what they had been trying in their classrooms, nurturing students’ creative thinking focusing on the three facets of novelty & value, generating ideas and developing ideas. Leah, a grades 5&6 teacher shared a force and motion invention fair she co-coordinated at her school with students applying creative thinking to their invention design and purpose. Irene shared the creatures her grade 8 Home Ec students created, considering the facets of creative thinking.
During the session we looked at the book, Weaving Creativity into Every Strand of Your Curriculum by Dr. Cyndi Burnett and Julia Figliotti, that we have been using in this series and in our school teams, used one of the ideas from the book to think about a concept we were studying with our students. Teachers were asked to use materials to create a soundscape to represent an idea or concept. Challenging! One teacher commented on how a student really has to understand the idea/concept and synthesize thinking in order to do this task.
Our third and final session of the year was held at the end of April. Teachers shared ho they were weaving opportunities for creative thinking throughout their classroom programs. We asked teachers to write-up a lesson idea that was successful for them and their students and we will be compiling these ideas to share with teachers in our district.
During the session, we introduced teachers to the electronic components littleBits which were piloted in the district two years ago for the purposes of nurturing creative thinking through STEAM experiences. Different ways teachers and students have been using littleBits in the district were shared and then the teachers in the project were asked to create something that does something – being mindful of the three facets of the creative thinking core competency: novelty & value, generating ideas and developing ideas.
As teachers engaged with the materials, they could see the possibilities for using the littleBits beyond the electricity component of the science curriculum in grades 7 and 9. More information about littleBits can be found on the website here.
We also asked teachers to record a short professional narrative about their involvement in this project. The following is one teacher’s response:
“Creative Thinking is the process of having original ideas of value. The Creative Thinking Challenges we do each Wednesday encourages us to think creatively when faced with a problem. Itís very important to have unique and innovative ideas, and to have different ways of overcoming challenges, especially with our rapidly changing society. We also do something called Monday Morning Provocations. Every Monday, we use a variety of materials to reflect on a question. We have more ways to show what we know, and explain our thinking visually. Not just with pencil and paper.We do a lot of activities helping us learn in different ways, not just reading a textbook, copying facts, and memorizing it. We do discussion circles where everyone shares an opinion and we discuss it. We use iPads and take pictures of our work to communicate and show our learning to parents and teachers. We use an app called FreshGrade. After every activity, we have a self-assessment/feedback frame to let us reflect on our work. Instead of the teacher giving us marks, we actually think about what we did well on, what we need to improve on etc. Overall, we think the new BC curriculum develops a creative and positive mind, which will help us in the near future.” ~ L&L
How valuable to have students reflect and comment on their learning experiences!
Roz and I look forward to continuing our work with the core competencies project next year!
Several schools in our district are participating in Inclusive Learning Communities projects, with a school team working alongside Learning Services staff to consider inclusive principles and practices. Hugh Boyd Secondary is continuing with a its second year in the project, investigating the question: How can we meet the diverse needs of our students in our grade 8 math classes?
A summary of last year’s project can be found HERE.
This year, the school team met with Shelley Moore and myself to look at class profiles, develop performance tasks for the beginning of a unit of study, consider and plan for inclusive practices within lessons and discuss ways of assessing students. This year, the teachers chose to focus on algebra (solving equations) and linear relations (graphing). We looked at the prescribed learning outcomes, considering what students needed to know to be successful at the grade 8 level – language, concepts, processes, skills. I introduced the team to the number balance as a way to think about balancing equations and emphasizing the concept of equivalence.
New this year was a collaboration with SetBC – the school was provided with a half-class set of iPad tablets loaded with apps suitable for communicating learning in mathematics. SetBC facilitators provided support to the Boyd teachers in learning how to use screencasting apps such as doceri.
One of the lessons we developed together was having the students use the number balances to represent algebraic equations and then use the app ShadowPuppet to capture their process and understanding.
A short video of this lesson can be viewed HERE. The teachers noticed the high engagement of the students and how some students were able to demonstrate their understanding of the mathematics in ways that played to their strengths.
The teachers followed up with lessons, continuing with the students creating screencasts using iPad technology. The students were provided with choices of algebraic equations to create a table of values for and then graph using the doceri app (it has various graph paper backgrounds to choose from).
During our debrief session before spring break, the teachers identified the new practices that helped to meet the needs of their students and that they felt would endure for them: using a performance task before the unit of study to assess where students are in their understanding, using manipulatives/models, using open-ended learning experiences, presenting three or more different entry points for students (different complexity of questions or problems) and using iPad technology.
For the third year in our district, Rosalind Poon and I are facilitating a Core Competencies Project. This series takes place over the school year, with release time provided to grades 6-9 teachers. This year, we have narrowed our focus to closely examine the Creative Thinking Competency and ways we can provide opportunities for students to develop this competency across disciplines.
For our first session together, we asked teachers to engage in a “chalk talk” about what creativity is- a strategy to record your thoughts about something and make connections between others’ ideas and yours.
We watched a short video with Sir Ken Robinson discussing what creativity is and how we can assess it. Watch the video HERE.
We unpacked what Core Competencies are and then specifically looked at the Creative Thinking Competency.
The Creative Thinking Competency has three facets – novelty & value, generating ideas and developing ideas. Each teacher or school team of teachers was asked to choose one facet to play around with in their classrooms between our first and second sessions. Roz Poon documented our first session using Pages:
We are using the Spirals of Inquiry cycle to engage in professional learning together. Teachers each received a copy of a teacher resource book that is full of ideas for the “taking action” part of the cycle.
In order for teachers to connect to the Creative Thinking profiles and illustrations, we facilitated a series of mini-challenges from Destination Imagination and had teachers consider the facets of the Creative Thinking competency – novelty & value, generating ideas and developing ideas – and consider how they might describe their own competencies in these areas.
Information about BC’s Core Competencies can be found in this short video HERE.
The information about the Creative Thinking Competency can be found HERE.
We are meeting together in January and I look forward to seeing how the creative thinking competency is coming alive in Richmond classrooms!
As part of the Northwest Math Conference, I was invited to present an Ignite session- defined in the program as “math educators present on topics that ignite their individual passions. Each talk is 5 minutes long and consists of 20 slides that auto-advance every 15 seconds, whether the speaker is ready or not!”
Five minutes is fast – as is 15 seconds per slide! I had to really think about one big idea and how I would tell the story I wanted to tell.
I decided to title my presentation “Where’s the wonder in math?” and discuss the importance of students posing their own problems and asking authentic questions that they are interested in finding answers for. I referred to Alfie Kohn’s recent publication in this area, which can be found here. I also shared references from a Google Ad that can be found on youtube here – it’s called “A Question Waiting to be Answered”. I wanted to provoke educators’ thinking with the question: What openings are we creating for wonder in our math classes? And then I shared my story of what I have seen when the classroom is turned over to students and they are asked to notice and wonder and investigate questions and problems that are important to them. I shared several examples from Richmond classrooms and discussed ways to nurture a stance of inquiry in our classrooms. Lots packed into five minutes!
I have done a lot of presenting over the years to different sizes of crowds but for the first time, I really got the jitters. The room was set for 400 people or so and people kept streaming in, till there were educators standing all around the edges of the room. The fast-paced timing of the whole thing was out of my control and I know this is what made me anxious – usually not my style of presenting at all! The other presenters around me (three others also first-timers) were also displaying different stress behaviours – rustling notes, bouncing knees up and down, talking about how nervous they were. Thankfully, Marian Small was sitting next to me and reminded me that the slides will just keep moving and it will be what it is – not to fret.
We did it! All of the other talks were so inspiring and I look forward to watching them when the BCAMT posts them on their website (although I’m not sure I want to see mine – will likely skip over it).
After my heart stopped racing and I had some time to reflect…someone asked me if I would do it again. I am not sure about how I feel about doing it in this large of an arena again but I have been thinking what a great structure Ignites might be for staff meetings, professional learning etc. I am envisioning beginning a professional learning day off with a panel of Ignite sessions and then breaking out into EdCamp sessions based on on topics raised by the Ignites. And of course, what a great way for students to share what they know about a big idea. Lots of possibilities!